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RADICAL POLYMERIZATION

Introduction

Free radical polymerization is one of the most studied chemical processes. Thisis not surprising, because free radical polymerization is carried out on a large in-dustrial scale; the world production of polymers by this method is in the range of100 million tons per year. This enormous production corresponds to nearly 50% ofall synthetic polymers. The importance of free radical polymerization is likely toincrease in the coming years as new controlled/living radical polymerization tech-niques find industrial applications. Free radical polymerization has been knownfor more than 60 years. As far back as the 1950s, the basic theory and compre-hension of radical polymerization was established (1–5). It included the thoroughunderstanding of the mechanism of the process, encompassing the chemistry andkinetics of the elementary reactions involved, with the determination of the corre-sponding absolute rate constants, the structure, and concentrations of the growingspecies, as well as a correlation of the structure of the involved reagents and theirreactivities. Similar to other chain reactions, the radical polymerization processmay be subdivided into initiation, propagation, transfer, and termination steps asdepicted in Scheme 1. The radicals necessary to initiate the chain process have tobe generated in situ in most cases.

A multitude of monomers can be used in radical polymerization and it isimpossible to list them all. The fundamental feature of the vast majority ofmonomers in question is the vinylic double bond. Thus the simplest monomeris ethylene, which, however, can only be polymerized under high pressure andhigh temperature to the commercially very important low density polyethy-lene. Common monomers are monosubstituted or unsymmetrically (1,1-) disubsti-tuted ethylenes, CH2 CHR or CH2 CRR′. The substituents R and R′ determinethe properties of resulting polymers and also the kinetic and thermodynamic

359Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

360 RADICAL POLYMERIZATION Vol. 11

Scheme 1.

polymerizability of monomers. Polymerizable monomers under radical condi-tions include styrene and its substitution products, dienes, mono- and disubsti-tuted ethylene derivatives, such as vinyl acetate, acrylonitrile, (meth)acrylates,(meth)acrylamides, and vinyl chloride, and a variety of halogenated alkenes.Carbon–heteroatom (N,O) double bonds only rarely polymerize via radical poly-merization (eg, CF3 CHO (6)).

The essential polymerization step is a repetitive free radical addition tothe monomer double bonds, forming chains of carbon atoms constructed of units

( CH2 CHR ) or ( CH2 CRR′ ) linked together predominantly head-to-tail (the substituted carbon atom is denoted as the head). Since the free radicalis essentially sp2 hybridized, very limited control of tacticity is observed, withmany monosubstituted monomers providing atactic polymers and disubstitutedlike methacrylates showing (thermodynamic) preference for syndiotacticity.

Monomers with exocyclic double bond can polymerize via ring-opening poly-merization and incorporate heteroatoms to the backbone. The systematic study ofring-opening polymerization started in the 1950s (7). Ring-opening polymerizationhas several characteristic features, and it can produce a wide variety of polymers,many of which have found important industrial applications (8,9). A characteristicfeature of ring-opening polymerization is the smaller volume shrinkage of cyclicmonomers during polymerization in comparison with vinyl monomers (10). Con-sequently, cyclic monomers are better as adhesives, curing resins, and moldingand filling materials. Vinyl monomers show volume shrinkage about two timeslarger than the shrinkage of cyclic monomers of identical molecular weight. Thisis due to the compact structures of cyclic monomers and the compensation of thecloseness of monomer molecules by ring opening during polymerization.

Most cyclic monomers polymerize ionically, but there are several monomersundergoing radical ring-opening polymerization (RROP). The copolymerizationof radically polymerizable cyclic monomers with common vinyl monomers canadd functions to common vinyl polymers. Scheme 2 illustrates the typical pat-terns of RROP, depicting vinyl-substituted cyclic and exo-methylene-substitutedmonomers. The radical species formed by the radical addition to the double bondundergoes ring opening to form a propagating radical species. RROP is commonly

Vol. 11 RADICAL POLYMERIZATION 361

Scheme 2.

accompanied by so-called vinyl polymerization without ring opening of the cyclicstructure.

Although fewer monomers undergo RROP than cationic, anionic, and coor-dination ring-opening polymerizations, novel monomers undergoing RROP aresteadily being developed, some of which were recently examined as monomers forliving radical polymerization (11). RROP can provide a wide variety of functionalpolymers with industrially promising radical polymerization.

1,6-Dienes having vinylic double bonds that are not polymerizable can formpolymers consisting of repeat units made up of five- or six-membered rings.Such processes are called cyclopolymerization (qv), and typically the intramolec-ular cyclization followed by intermolecular addition of the cyclized radicals (seeScheme 3) is orders of magnitudes faster than individual propagation via thecarbon–carbon double bonds (12–15).

Typical cyclopolymerizable monomers include N-substituted dimethylacry-lamides where the nonpolymerizability of the N,N-disubstituted methylacry-lamido group results in the generation of five-membered cyclic structuresinvolving head–head linkages. In cases where one of the carbon–carbondouble bonds is severely sterically hindered, cyclopolymerization (qv) maynot occur and a linear polymer with pendant double bonds is formed(16). Other examples of monomers that undergo cyclopolymerization are 4-(N,N-diallylamino)pyridine, 5,6-di-O-isopropylidene-D-mannitol, 1,5-hexadiene,N-methyl-N-methallyl-2-(methoxycarbonyl)-allyl amine, and dipropargyl com-pounds (13). One of the most efficient ways to achieve structural control in radicalpolymerization is via cyclopolymerization. For example, the free radical cyclopoly-merization of a diacryloyl monomer involving a chiral template results in thegeneration of stereogenic centers and four different stereoisomers (17).

Another essential component of free radical polymerization systems are ini-tiators which should provide initiating free radicals via homolytic cleavage of co-valent bonds, redox reaction, photochemical or other stimuli. In addition, various

Scheme 3.


compounds which can regulate molecular weights (transfer agents) or rates(retarders/inhibitors) can be used. Free radical polymerizations tolerate manyprotic impurities such as water or alcohols but require absence of oxygen, whichact as a powerful inhibitor. Various solvents can be used which should not (unlessdesired) participate in transfer.

Initiation

The initiation process constitutes the first reaction step in free radical polymer-ization, leading to the generation of (primary) radicals. The kinetics of the initia-tion process, ie its rate and effectiveness, are of fundamental importance in boththeoretical studies and commercial applications. Commercial procedures mainlyrely on the formation of primary radicals via thermal decomposition processes us-ing azo- and peroxy-type compounds. Investigative kinetic studies are—to a largeextent—carried out using photoinitiators, which decompose upon irradiation withUV or visible light. The main reason for this choice is the possibility to define exactstart and end times of the initiation and subsequently the polymerization process.

The decomposition scheme (eq. 1) describing the generation of radicals iscommon to both thermal and photoinitiators.

(1)

The measurable decrease of the initiator concentration [I] in a polymerizingsystems with time is given by

− d[I]dt

= kd[I] (2)

Integration of equation 2 leads to equation 3, an expression which describesthe decreasing initiator concentration as a function of time.

[I] = [I]0 e− kdt (3)

However, the rate of the formation of initiating primary radicals is of greaterinterest in kinetic studies. The rate of generation of radicals that are capable ofinitiating the polymerization process, Rd, is described via the following generalfirst-order rate law:

Rd = d[I•]dt

= − 2fd[I]dt

= 2f kd[I] (4)

where kd corresponds to the rate coefficient of initiator decomposition and f de-notes the initiator efficiency (see below). It should be noted that in the case ofphotoinitiation, kd is a composite of various variables.

In order to initiate the polymerization process via reaction with a monomerunit, the generated primary radicals, I1

• and I2•, have to leave the solvent cage

that surrounds them. The ability of the primary radicals to leave the solvent cage


unreacted and to start the polymerization process is quantified by the initiatorefficiency f , with theoretical values between zero and unity. Not all generated pri-mary free radicals initiate polymer growth. Shortly after decomposition, the freeradicals are very close to each other and recombination can occur. In addition,they can also react in alternative ways before they can react with a monomerunit. An efficiency of zero corresponds to no initiation taking place, whereas f = 1indicates that every generated primary radical escapes the solvent cage and sub-sequently initiates polymerization. Typical values of f are between 0.5 and 0.8,depending on the viscosity of the reaction medium, indicating that the escapingprocess is diffusion controlled. It should be noted that in the case of an unsymmet-rical initiator molecule, I1

• and I2• do not necessarily display the same reactivity

toward the monomer unit (18–20). Hence, the initiation process may be describedby equations 5 and 6:

(5)

(6)

where I1• and I2

• represent initiator fragment 1 and 2, respectively, M indicates amonomer unit, R1

• corresponds to a macroradical of chain length 1, and ki(1) and

ki(2) refer to the individual initiation rate coefficient of the respective fragments.

The rate of initiation, Ri, is given by equation 7:

Ri = d[R1•]

dt= − d[I1

•]dt

− d[I2•]

dt= k(1)

i [M][I1•] + k(2)

i [M][I2•] (7)

Because [I1•] = [I2

•] = [I•]/2, the rate coefficient of initiation, ki, is a compositeof the individual rate coefficients of initiation for the initiator fragments I1

• andI2

•.

Ri = ki[M][I•] with ki =k(1)

i + k(2)i

2(8)

Thermal Initiation. Thermally decomposing initiators (mainly) fall intotwo classes: azo- and peroxy-type molecules. The general structures of azo- andperoxy-initiators are represented by 1 and 2 respectively.

An important quantity of a thermal initiator is its half-life t1/2 (at a certaintemperature), given by equation 9. The half-life is the time period during whichhalf of the initiator molecules initially present is decomposed.

t1/2 = ln2kd

(9)


There are initiators available for virtually any desired decomposition ratein a given solvent (or monomer) at a given temperature. Initiators are often char-acterized by the temperature at which their half-life is 10 h. These temperaturesrange from 20 to 120◦C, depending on the structure of the initiator. Extensive dataon initiator decomposition rates and their activation parameters can be found inthe Polymer Handbook (21). In the following, the basic features of thermal initia-tion systems are discussed.

Azoinitiators. This widely used class of thermally decomposing initiatorsgenerates both carbon- and oxygen-centered radicals by either C N (dialkyl di-azenes) or O N (dialkyl hyponitrites) bond scission driven by the expulsion of astable nitrogen molecule (22–25). Dialkyl diazenes are generally used at highertemperatures than dialkyl hyponitrites. Most of the azoinitiators generate twoidentical radical species upon fragmentation. The most common azoinitiatorsare 2,2′-azobisisobutyronitrile (AIBN, 3), dimethyl 2,2′-azobisisobutyrate (4), 1,1′-azobis(1-cyclohexanecarbonitrile) (5), and 2,2′-azobis-2,4-dimethylvaleronitrile(6). Both the delocalization of the free electron in the generated radical (22,23)and the steric demand of the leaving radical (24) significantly affect the decom-position rate coefficient kd: The more the free electron in the resulting radicalis delocalized and the bulkier the leaving radical, the larger is kd. The choice ofthe solvent, however, has comparatively little effect on the rate of azoinitiatordecomposition (26).

The concept of azoinitiators has been extended toward the generation ofmacroazo-initiators for the synthesis of block copolymers. The development oftechniques for block copolymer preparation was especially important before theadvent of living free radical polymerization processes such as reversible addition–fragmentation chain transfer (RAFT) polymerization or atom transfer radicalpolymerization (ATRP). Macroazoinitiators are low molecular weight polymerscontaining one or more N N units (27–32). Usually, the macroazo-initiator isprepared by a technique other than free radical polymerization, ie polyconden-sation of low molecular weight azoinitiators with suitable end groups. A typicalexample of such a strategy is the polycondensation of 2,2-azobis(2-cyanopropanol)with isocyanate functionalized prepolymer. The generated polymer can subse-quently be used to initiate the free-radical polymerization of styrene. The rate ofmacroazo-initiator decomposition may be different from the parent initiator, butstill follows first-order kinetics (32). Macroazo-initiators have also been used inconjunction with (NMP) (nitroxide-mediated polymerization) and ATRP for the


synthesis of block copolymers (29,33). It is interesting to note that polymers con-taining azo-moieties along the backbone have been reported to be biodegradable(34).

Peroxide-Based Initiators. The use of peroxides as thermally decomposinginitiators is—especially in an industrial context—more widely spread than the useof azoinitiators (35,36). Peroxy compounds range from alkyl, acyl, and perestersto inorganic peroxides (26,37). The most employed peroxides are di-tert-butyl per-oxide (7), benzoyl peroxide (8), di-tert-butyl peroxalate (9), cumene hydroperoxide(10), and di-n-propyl peroxydicarbonate (11). The primary—oxygen-centered—radicals that are generated by organic peroxide decay may undergo further re-actions such as β-scission, which leads to the formation of secondary radicals. Acommon reaction pathway is the expulsion of carbon dioxide from the generatedprimary radicals originating from acyl peroxides. For example, the decomposi-tion of benzoyl peroxide leads to the formation of benzoyloxy and phenyl radicals.The mechanism of peroxide decomposition is still under investigation, especiallyin the case of alkyl peroxyesters. Peresters may decompose—depending on theirstructure—by either a concerted two-bond scission or a one-bond homolysis ofthe oxygen–oxygen bond (38). The one-bond homolysis results in the formationof acyloxy and alkoxy radicals, whereas the concerted two-bond fragmentationgenerates alkyl and alkoxy radicals plus carbon dioxide.

The rate of peroxide decomposition is strongly dependent on the structureof the molecule in question. The rate of decomposition for diacyl peroxides andperesters increases when going from aryl to primary alkyl to secondary alkylto tertiary alkyl substituents (39). In the case of peresters, the structure of themolecule governs both the rate of fragmentation and the above mentioned mode


of fragmentation (ie one-bond homolysis vs concerted two-bond cleavage). Exper-imental data indicate that the rate of decomposition increases when going fromprimary to secondary to tertiary carbons connected to the (CO)O2 moiety in theperester. The concerted mechanism is favored in the case of secondary and ter-tiary carbons (38,40,41). Because of the strong dependence of the rate of decom-position on the peroxide structure, a wide variety of decomposition rates existsfor a given temperature. The general guideline is that dialkyl peroxides and hy-droperoxides are mostly used at elevated temperatures, whereas dialkyl perox-ydicarbonates are best suited for lower temperatures. The solvent in which thedecomposition is carried out affects the rate to much larger extent than for azoini-tiators. For example, the decomposition rate coefficient for the diacyl peroxidebis(3,5,5-trimethylhexanoyl)peroxide increases by a factor of 7 when going frompentadecane to acetonitrile (at 150 MPa and 80◦C) (42). The rate of diacyl peroxideand peroxy ester decomposition has also been studied in supercritical CO2 and—contrary to the above—no significant change in decomposition rate compared ton-heptane as solvent has been observed (43).

Photoinitiation. An attractive alternative to thermally decomposing ini-tiators are photoinitiators that decay upon irradiation with UV or visible light (44–48). The use of photoinitiators is (in most cases) restricted to applications involv-ing accurate kinetic measurements, whereas their usage in industrial processesis very limited because of the technical problems associated with the uniform ir-radiation of large reaction volumes. Some exceptions are applications involvingcoatings and surface polymerizations. The main advantage of photoinitiators usedin polymerizing systems is the possibility to define exact start and end points ofthe polymerization process via the duration of the irradiation period. In addition,the rate of (most) photoinitiator decomposition is almost independent of the re-action temperature, but depends strongly on the (UV) light intensity. However,weak temperature dependencies of the primary quantum yield (see below) havelong been known for the photoinduced decomposition of azoalkanes (23). The weakdependence of the rate of decomposition on the reaction medium temperature isdue to the large amount of energy which is deposited into the initiator moleculesvia the light source. This energy exceeds the thermal energy of the surroundingmedium by orders of magnitude. An ideal photoinitiator for a specific polymeriza-tion may be defined via the following criteria:

(1) The photoinitiator should decompose upon irradiation with the (UV) lightsource; eg, an absorption should coincide with the radiation wavelength. Themonomer(s) used in the specific polymerization process should not absorblight at the selected wavelength.

(2) The efficiency of the initiator should be high, preferably close to one; ie, allradicals generated should start growing chains.

(3) At best, there should be only one type of free radical species that is formedupon laser irradiation.

On the basis of the mechanism by which initiating radicals are formed, pho-toinitiators are generally divided into two classes: Type I photoinitiators undergoa unimolecular bond cleavage upon irradiation to yield free radicals. Type II


Fig. 1. Simplified Jablonski diagram of the photochemically initiated acetophenone typeinitiator decomposition.

photoinitiators undergo a bimolecular reaction where the excited state of thephotoinitiator interacts with a second molecule (a co-initiator) to generate freeradicals. However, visible light photoinitiators belong almost exclusively to theType II class of photoinitiators. Within the Type I initiator class, there are severalstructural variations. The most widely used class are photoinitiators containing abenzoyl group as the effective initiating moiety (acetophenone type). Their generalformula and primary decomposition products are given in equation 10.

(10)

It is a standard practice to show the processes that can take place upon irra-diation (ie assuming that each photoinitiator molecule absorbs a certain dose ofenergy) in a so-called Jablonski diagram. Such a diagram is a simplified illustra-tion of the relative positions of the electronic energy levels of a molecule.

Upon (UV) irradiation, the molecule is converted from its (singlet) groundstate to its first excited state. The Jablonski diagram given in Fig. 1 shows thatthere is more than one possibility to deactivate an excited initiator molecule. Thismultitude of deactivation modes is quantified in the quantum yield for the primaryfree radical production, �. � is composed of three parts, which can be assigned tothe reactions shown in Scheme 4.

The quantum yield for intersystem crossing (Ib) is rather high for ketones(49), so that to a good approximation �ISC can be set to one. Thus, the overall

Scheme 4.


quantum yield of the initiation process depends only on �R and �RM. From theT1 triplet state, parallel reactions can lead to a decrease in the quantum yield�R. In free radical polymerizations these reactions are deactivation by molecu-lar oxygen and deactivation by the monomer (50). This is one of the reasons whythe monomer mixture should be thoroughly degassed prior to the polymerizationprocess. The longer the lifetime of the triplet state, the higher the chances thatdeactivation processes can reduce the quantum yield. For example, 2,2-dimethyl-2-phenylacetophenone (DMPA) has a rather short-lived T1 state [τ < 0.1 ns (51),whereas the respective value for 1,1-dimethoxy-1-hydroxyacetophenone (Darocur1173) is close to 20 ns (52)], which results in a high �R value. The quantumyield of the third reaction, which leads to the formation of macroradicals, hasbeen termed initiator efficiency, f ≡ �RM. The initiator efficiency is influencedby the ability of the radicals formed by the laser pulse to diffuse from the sol-vent cage to the reaction site and its definition is analogous to the definition ofthe efficiency factor for thermally decomposing initiators. This process has beentermed cage effect. The initiator efficiency of the typical acetophenone initiatorDMPA (at 30◦C and ambient pressure) is close to 0.4 (53). However, the situationfor most acetophenone-type initiators is more complex than for a photoinitiatorwhich yields only a single type of free radical, R•, upon irradiation. It is generallyaccepted that most acetophenone-type initiators (eg, DMPA, Darocur 1173, or ben-zoin) decompose into two radical species, R1

• and R2•, as detailed in equation 10.

Generally, both species are distinctly different in their character. The carbonylradical is very efficient in terms of starting macromolecular growth, whereas thesecond radical is often not [for the example of DMPA, the methoxy-type radical isonly involved in termination steps (54)]. Initial radicals that come from a commoninitiator molecule, but have clearly different efficiencies, may be termed “effective”and “ineffective” (primary) initial radicals, respectively. The sum of both radicalconcentrations is termed “overall” radical concentration and is two times ρ, withρ being the concentration of primary radicals that are capable of starting macro-molecular growth. The concentration of effective and ineffective initial radicals isthe same and equal to ρ. The occurrence of primary radicals with markedly differ-ent reactivities may have serious consequences for the polymerization kinetics.

Another widely used structural variation of class I photoinitiators are azoini-tiators, which are also employed as thermal initiators. However, their mechanismof photodecomposition is markedly different from acetophenone-type initiators.Equation 11 shows the UV-induced decomposition of the widely used azoinitiator2,2′-azobisisobutyronitrile (AIBN).

(11)


Upon irradiation by laser light, the initiator molecule changes its stereo-chemical configuration from the trans to the cis isomer. This cis/trans isomerismbecomes of importance if azoinitiators are employed in pulsed laser experiments:The time at which the laser pulse hits the reaction mixture is no longer identicalwith the generation of the primary radicals. The time delay is usually in the orderof microseconds and may be observed in time-resolved pulsed laser experiments.

It is possible to calculate the effective primary free radical concentration ρ

generated by monochromatic irradiation of a reaction mixture containing a pho-toinitiator. The radical concentration ρ, which is generated by a specific numberof absorbed photons, is given by equation 12:

ρ = 2�nabs

V(12)

where � is the primary quantum yield (see above), nabs is the number of absorbedphotons, and V is the irradiated volume. According to Beer–Lambert’s law, thenumber of absorbed photons may be calculated by equation 13:

nabs = Ep

Eλ

(1 − 10− εcd) (13)

where Ep is the energy deposited, Eλ is the energy of 1 mole of photons at theirradiation wavelength λ, ε is the molar absorption coefficient of the initiatormolecule at the laser wavelength λ, c is the photoinitiator concentration, and d isthe optical path length.

The rate of photochemical initiation is given by the product of the intensityof absorbed light (in moles of light quanta per liter and second) and the primaryquantum yield. A good survey on different photoinitiators and their photodecom-position by UV light was given by Gruber (55).

Self-Initiated Polymerization. Free radical polymerizations can also beinitiated by the monomer itself or peroxy compounds that are formed via exposureof the reaction mixture to molecular oxygen. The processes (mainly) take place athigh temperatures. However, under very pure conditions in exhaustively purifiedreaction vessels, most monomers do not tend to polymerize spontaneously by in-creasing the temperature. Styrene is one of the rare monomers that—also in itspurest state—does exhibit initiation processes without additionally added initia-tor. The self-initiation of styrene has been studied in great detail with respect toits kinetics and mechanism (56–58). The underlying reaction is a self-Diels–Aldercycloaddition of two styrene moieties as shown in equation 14:

(14)

The self-initiated bulk polymerization of styrene has a substantial activa-tion energy: a 50% monomer conversion needs 400 days at 29◦C, but only 4 h at127◦C. However, the produced polystyrene is very pure because of the absence of


initiators and other additives. Methyl methacrylate was also thought to undergoself-initiated polymerization, but increasing evidence indicates that this may notbe the case (59–64). It should be noted that certain pairs of monomers undergospontaneous copolymerization without the addition of a thermal initiator. Exam-ples include maleic anhydride/styrene (65) and acrylonitrile/styrene (66,67). Themechanism of initiation of such copolymerization is still unclear, but pathwaysanalogous to that operative in styrene homopolymerizations but with acryloni-trile acting as the dienophile in the formation of the Diels–Alder adduct is de-bated. Other postulated mechanisms for spontaneous initiation include hydrogenatom transfer between the solvent and charge-transfer complex (68) and electrontransfer followed by proton transfer to yield macroradicals (69).

Redox-Initiated Polymerization. Redox-initiating systems are very of-ten used, especially in aqueous media. However, the redox initiation can also becarried out in organic media. These systems consist of a reducing and oxidizingagent (70,71). Redox initiators very often form only one radical (eg, classic Fen-ton system of ferrous salts and hydrogen peroxide or other peroxides) which, byeliminating cage termination, increases initiation efficiency:

Fe2 + + RO OR → Fe3 + + RO− + RO• (15)

Commercially important is also the use of ceric salts, which can reduce alco-hols to alkoxy radicals and enable grafting from cellulose and other surfaces withOH groups.

Some redox systems provide two radicals such as persulfates with thiosul-fates and metabisulfates:

(16)

(17)

The electrostatic repulsion between radical anions also reduces the proba-bility of cage termination.

Transition metals in redox initiation usually act as reducing agents but mayalso act as oxidants:

Co3 + + −OH → Co2 + + •OH (18)

There are also several nonaqueous redox systems which proceed by eitherouter- or inner-sphere electron transfer (OSET or ISET). As already stated before,aromatic amines accelerate decomposition of peroxides by electron transfer andformation of peroxide radical anions as depicted in equation 19:

(19)


Trialkylboron derivatives are also powerful reducing agents and their inter-action with benzoyl peroxide allows to initiate radical polymerization at very lowtemperatures, approximately −50◦C. A similar low temperature redox-initiatingsystem consists of trialkylboron derivatives and oxygen.

Another class of redox-initiating systems involves transition metal com-plexes which either alone provide radicals by homolytic cleavage or react withorganic halides. The latter system applied under reversible conditions forms thebasis of ATRP, one of the most powerful controlled/living radical polymerizationsystems (vide infra).

Some transition metal chelates, such as Mn(II), Co(II), Ce(IV), Fe(III), orCu(II) acetylacetonates, initiate polymerization of alkenes at wide temperaturerange 25–100◦C, depending on monomer and solvent, which both can affect thestability of the complexes.

The reaction of transition metal complexes with alkyl halides may occur viaOSET or ISET, depending on the redox potential of the components and also on thecoordination sphere of the transition metal compound. The saturated coordinationsphere favors OSET or requires ligand dissociation. The open coordination sphereenables ISET process, as in ATRP. The most common alkyl halides used in thisprocess are polyhalogenated methanes (CBr4, CCl4, HCCl3) and also halogenatedesters (CCl3CO2R, ClCH2CO2R), sulfonyl halides, N-haloamines, or N-haloureas.

Mt0 + Cl CCl3 → MtICl− + •CCl3 (20)

Low oxidation state transition metal complexes include Mo(CO)6, andNi(P(OR)3)4, and Ni(CO)4; more recently many Cu(I) complexes with bi-, tri-, andtetradentate N-ligands and Fe(II), Ni(II), and Ru(II) complexes with phosphineligand as well as CO, Cp, and other ligands have been reported.

Other Methods of Initiation. Apart from the above-mentioned methods ofinitiating free radical polymerizations, there are several more alternatives whichcan be selected from, if required for the specific application. These techniquesinclude (1) ionizing irradiation, (2) plasma initiation, (3) electroinitiation, and (4)ultrasonic initiation.

(1) Radioactive sources and ionizing particles (such as γ -irradiation (72), elec-trons (73), neutrons, and α-particles) are used to initiate free radical poly-merization processes. The interaction of these radiations with matter are—compared to the interactions of less energetic irradiation (such as UV orvisible light)—much more complex. High energy radiation applied to vinylicmonomers usually results in the formation of anions, cations, and primaryfree radicals. These species may then initiate chain growth. Whether thisgrowth is of the free radical type depends largely on the reaction conditions.Ionic chain initiation is predominant at low reaction temperatures, becausethe formed ions tend to dissociate into free radicals at higher reaction tem-peratures. High-energy-radiation-initiated free radical polymerizations areless applied in scientific and commercial applications, which is mainly dueto the complex initiation mechanism and the safety issues associated withthe usage of this type of irradiation.


(2) When a gaseous monomer under low pressure is exposed to a high voltageelectric discharge, plasma polymerization may occur (74). The plasma con-sists of ionized molecules. A large number of monomers undergo plasmapolymerization and form high molecular weight products. The polymeriza-tion process appears to be rather complicated but seems to involve both ionicand free radical species. Plasma polymerization is especially attractive forthe generation of thin polymer films for a variety of applications (75).

(3) Electroinitiation—which is not to be confused with initiation by an electronbeam—is done by direct electrolysis of the reaction mixture. The reactionmixture usually contains an organic solvent, the monomer, and an inorganiccompound that allows to conduct the current or participates in the ionizationprocess itself (76). The polymerization proceeds via either free radicals orions, depending on the reaction conditions and the substances present inthe reaction mixture.

(4) Sonication—the application of high intensity ultrasound—produces highconcentrations of H• and OH• radicals in water, which can lead to polymer-ization of aqueous monomer solutions. Free radicals are also produced inorganic solvents and vinyl monomers, however, to a lesser extent than inwater. Most of the physical and chemical effects caused by ultrasound aredue to the process of cavitation, or the formation and collapse of microscopicbubbles as the sound wave passes through the liquid. Although the originof chemical effects has been attributed to electrical discharge processes, themost widely accepted theory assumes the generation of very large temper-atures and pressures during an adiabatic bubble collapse (77–79).

Initiator-Derived Radical/Monomer Reactions. An initiator-derivedradical (also called a primary radical) can undergo a multitude of reactions in-cluding hydrogen abstraction (from the solvent, the monomer, or already presentpolymer), tail/head addition to the monomer, fragmentation, and/or primary rad-ical termination. Which of these reactions primarily occurs depends on the struc-ture of the generated radicals, the monomer, and to some extent the solvent andthe general reaction conditions (ie pressure and temperature). The reaction path-ways of the primary radicals may be studied via the nitroxide trapping techniqueand for some initiator/monomer systems these have been successfully mapped out(80–82). For example, the specificity of reaction with styrene monomer is foundto decrease in the series cyanoisopropyl ∼ methyl ∼ tert-butoxy > phenyl > ben-zoyloxy. While cyanoisopropyl (83), tert-butoxy, and methyl radicals almost exclu-sively afford tail addition, (84), phenyl radicals give tail addition and 1% aromaticsubstitution (84). Benzoyloxy radicals have been reported to undergo head ad-dition, tail addition, and aromatic substitution (84,85). Interestingly, in methylmethacrylate free radical polymerizations, the same radicals display a very dif-ferent order of reactivity. Here, the regiospecificity decreases in the order cyanoiso-propyl ∼ methyl > phenyl > benzoyloxy > tert-butoxy. Methyl and cyanoisopropylradicals almost exclusively yield tail addition. Phenyl and benzoyloxy radicals alsotend to react with the double bond, whereas hydrogen abstraction is very minor(86,87). tert-Butoxy radicals react with methyl methacrylate via 63% tail addition


Scheme 5.

(88) and hydrogen abstraction mainly from the α-methyl group or the ester methylgroup (see Scheme 5) (88,89).

Inspection of Scheme 5 indicates that a significant proportion of thepoly(methyl methacrylate) chains will have an olefinic end group, when tert-butylperoxide is employed as free radical initiator. Such chains may themselves un-dergo reaction in the polymerization, leading to a variation in the desired productproperties. It is thus mandatory to carefully match the monomer with the initiatorto achieve the required end group functionalities.

Some initiators generate two types of radicals upon fragmentation, whichgreatly differ in their reactivity and/or reactions toward the monomer units. Suchinitiators include the above-discussed photoinitiators, such as DMPA or benzoin,but also encompass acyl phosphine oxides and the thermal iniferter phenyla-zotriphenylmethane. The reactivity differences between the individual radicalscan be quite dramatic. It has been proposed that the acetal-type radical gener-ated by DMPA decomposition is not involved in initiation reactions at all, whileits counter benzoyl radical is very effective in initiating macromolecular growth(18). Similarly, the addition rates to vinyl monomers of (2,4,6-trimethyl ben-zoyl)diphenylphosphine oxide derived radicals have been found to differ greatly(90–92). The carbon-centered radicals add to vinyl monomers about 2 orders ofmagnitude slower than the phosphorus-centered ones.

Initiator Efficiency. A prerequisite for primary radical/monomer reac-tions is the successful escape of the generated radicals from their “solvent cage”.If the formed radicals remain inside the cage, they may undergo reactions withthemselves. Potential ‘in-cage’ reactions are the main reason for initiator efficien-cies, f , being below unity (0.3 < f < 0.8, typically) (93). In-cage reactions includeradical recombination or disproportionation. If the initiator is regenerated by anin-cage reaction, the initiator efficiency is not affected, but the rate of initiatordecomposition is reduced. The importance of reactions within the solvent cagestrongly depends on the rate of diffusion of the generated radicals away fromeach other. Therefore, both the size and the reactivity of the primary radicals aswell as the medium’s microviscosity determine the initiator efficiency. As increas-ing monomer-to-polymer conversion goes hand in hand with a large increase insystem viscosity, the likelihood of in-cage reaction will increase with monomer con-version (94,95). The species generated by reactions within the solvent cage canbe reactive under the polymerization conditions or they may have detrimentaleffects on the polymer properties. For example, cyanoisopropyl radicals formed by


Scheme 6.

the fragmentation of AIBN can react to yield methacrylonitrile, which undergoescopolymerization and will be incorporated in the final polymer (83,96). In addi-tion, the toxic tetramethylsuccinonitrile is generated (97). The in-cage reactionpathways of primary cyanoisopropyl radicals generated from AIBN are depictedin Scheme 6.

These reactions are almost exclusively taking place within the solvent cagewhen styrene monomer is present. In the absence of monomer, however, with onlya solvent present (eg, benzene), close to half of the products are formed outsidethe solvent cage, indicating that the addition of the cyanoisopropyl radicals tothe monomer units is effectively competing with self-reactions of the primaryradicals (94).

The efficiency of peroxide initiators is affected by the mode of their fragmen-tation. Concerted decomposition results in an alkoxy and in an acyloxy radical,whose recombination will not affect the initiator efficiency (38). Such processeshave also been termed ‘cage-return’ reactions. However, concerted two-bond scis-sion yields alkyl and alkoxy radicals, which—when coupled inside the cage—leadto a lower f value. Experimental investigations into the initiator efficiency canbe carried out via radical trapping techniques (98), infrared spectroscopy (99),and product or polymer-end-group analysis via NMR (94). Measurements of thetemperature dependency of the initiator efficiency are extremely scarce; however,it has been found that in the AIBN-initiated bulk free radical polymerization ofstyrene at 100 MPa, f increases by approximately a factor of 1.5 when going from40 to 80◦C (99). The same study indicated that the initiator efficiency decreaseswhen the reaction pressure is increased (f decreases by close to one third whengoing from ambient pressure to 200 MPa bar at 70◦C).

As mentioned earlier, initiation efficiency is much higher in the redox processsince in many systems only one radical is generated. However, with transitionmetals, the radicals may be formed reversibly either by abstracting back halogenor by formation of organometallic species.

Propagation

The propagation step in free radical polymerization has been in the center of scien-tific interest within the last decade, because of the advent of novel methods for the


accurate determination of propagation rate coefficients. The addition of a macro-radical to a monomer unit can be described via the following rate law expression:

− d[M]dt

=∑

i

kip[Ri

•][M] (21)

with kpi being the propagation rate coefficient of a macroradical with chain length

i, Ri, [M] the monomer concentration, and [Ri] the free radical concentration. Itis generally accepted that the propagation reaction is chemically controlled up tohigh monomer conversions, ie high viscosities of the reaction medium. This impliesthat the propagation rate coefficient is independent of monomer conversions up to80%. The chemical control of the propagation step is impressively demonstratedwhen comparing the average collision frequency of 1012 s− 1 in the liquid phaseat room temperature (100) to the frequency of successful propagation reactions(typically close to 103 s− 1). These numbers indicate that (approximately) onlyevery 109th collision leads to a successful addition of a monomer unit to a macro-radical. The propagation rate coefficient given in equation 21 carries the index i(representing the chain length of the growing chain), because it is beyond doubtthat kp is a chain-length-dependent rate coefficient. This is especially true for thefirst few addition steps, which proceed at a markedly increased rate compared tolong-chain propagation. In the case of methyl methacrylate, for example, the firstpropagation step at 60◦C is approximately 16 times faster than the long-chainpropagation (101–106). The difference in the polymerization of monosubstitutedmonomers may be smaller (107). Nevertheless, recent evidence points toward achain length dependence of the product of kp and the monomer concentration [M],up to chain lengths of hundreds of monomer units (108–111). Unfortunately, to thisdate, there is no method to determine kp independently of the monomer concen-tration. It is thus quite possible that the apparent chain length dependence of kpreflects a structuring of the monomer concentration at the propagating chain end.

For most free radical polymerizations, the propagation step is fast (see below)and exothermic. The thermodynamics of free radical polymerization propagationhave been reviewed by a few authors including Busfield (112), Sawada (113), andAllen and Patrick (114). Heats of free radical polymerization have been commonlydetermined from calorimetric data using standard thermochemical techniques.Entropies of polymerization are more scarce. It should be noted that the additionof radicals to vinyl monomers is—at least in principle—a reversible process. De-propagation is favored entropically and its importance increases with increasingtemperature. Details about depropagation reactions can be found in the relevantsubsection.

The propagating radical has been studied in great detail via electron spinresonance (ESR) spectroscopy. The monomers under ESR investigation includestyrene and substituted styrenes (115–117), acrylic esters (118,119), methacrylicesters (120–124), vinyl chloride (125), α-substituted acrylic esters, and othermonomers (115,121). In the overwhelming majority of cases, free radical propaga-tion proceeds in a regioselective way to afford a polymer chain almost exclusivelyconsisting of head–tail linkages. For mono- and 1,1-substituted monomers, it isusual to call the less substituted side the ‘tail’ and the more substituted side the


‘head.’ For more complex monomer substitution patterns (such as 1,2-di-, tri-, andtetrasubstituted monomers), the above definition is less stringent and the terms‘head’ and ‘tail’ are used more arbitrarily. A terminology has been suggested forpolymerization regioselectivity, where head–tail linkages are called isoregic, head–head linkages syndioregic, and tail–tail linkages aregic (126). Despite the high re-giospecificity of the propagation step, it is in general not stereospecific, because ofan addition of the very shallow pyramidal propagating radical center to an almostplanar vinyl double bond.

For the formation of high molecular weight polymer, the propagation stepmust be sufficiently fast to compete with other elemental reactions such as trans-fer and termination. In fact, range of factors may prevent fast propagation, asdiscussed below.

Free radical propagation is poorly stereocontrolled, with nearly equal pro-portion of meso and racemic dyads in polymerization of monosubstituted alkenesand a preference for syndiotactic placement for disubstituted monomers such asmethacrylates (rr = 0.62, mm = 0.04). The sequence distribution follows a first-order Markov model with a slight deviation from Bernoulian statistics. However,for very bulky substituents, as in polymerization of triphenylmethyl methacry-late, the preference for isotacticity was observed (mm = 0.64, rr = 0.12). Recently,complexation with Lewis acids and acidic solvents enabled to enhance stereocon-trol in polymerization of vinyl esters and acrylamides, and to a smaller degree inpolymerization of methacrylates (127–129).

Complexation with Lewis acids has also been used to alter sequences incopolymerization and prepare alternating copolymers in systems which usuallyprovide very small tendency for alternation like styrene/methyl methacrylate(rS ∼ rMMA ∼ 0.5) (130,131).

Monomer Structure and Reactivity: Electronic and Entropic Effects.The absolute value of the propagation rate coefficient is governed by the nature ofthe monomer unit and the reactivity of the propagating radical. Both entropic andelectronic factors influence the absolute value of the propagation rate coefficientand its activation parameters, EA and A. It is important to note that the reactivityof the propagating radical and the reactivity of the monomer units correlate recip-rocally. For example, methacrylates are several times more reactive then acrylatesbut polyacrylate radicals are significantly more reactive than poly(methacrylate)radicals. For polymerization and chain growth to take place, it is mandatory thatthe free macroradical lives long enough to survive the above-mentioned 109 inef-fective collisions. The addition of the monomer in a propagation reaction has totake place before any possible decomposition reaction or other side reaction. Forexample, acetone cannot be polymerized at ambient reaction conditions—besidesthermodynamic reasons, which make the polymerization unfavorable, because theassociated free radicals quickly decompose into methyl free radicals and a ketone.

The substituent that is introduced into the ethene molecule to facilitate poly-merization via activation of the double bond and stabilization of the propagatingradical, respectively, has also the effect of adding steric hindrance to the propa-gation step. A mechanistic model has been proposed for the propagation of ster-ically hindered monomers (132). In such a model the propagating radical andthe monomer attain the maximum overlap of the π -electrons and the unpairedelectron at a certain angle between the Cα Cβ bonds of the monomer and the


Scheme 7.

propagating radical. If the propagation reaction is sterically hindered, the bondformation is energetically not suppressed, but the probability that the radicaland the monomer approach each other such that bond formation can occur is de-creased. It is thus evident that it is close to impossible to completely separateenthalpic from entropic effects.

An attempt to separate individual contributions using different monomersis illustrated in Scheme 7. While electronic effects should be reflected in the ac-tivation energy EA, steric effects are associated with the preexponential factor A.When going from methyl methacrylate to dimethyl itaconate (133), a substantialdecrease in the preexponential factor of about one order of magnitude is observed(r.h.s. of Scheme 7). This decrease is due to the sterically more demanding natureof the propagating dimethyl itaconate radical. When going from methyl methacry-late to ethyl α-hydroxy methacrylate (EHMA), the preexponential factor remainsunchanged, but the activation energy is reduced due to the electronic effect of theadditional β-oxygen atom (l.h.s. of Scheme 7). Both effects may be combined inone monomer (ethyl α-acetoxymethacrylate, EAMA, bottom of Scheme 7), with aresulting decrease in the activation energy and the preexponential factor.

This illustrates the effect of different contributions (ie steric and electronic)to the activation parameters using different monomers.


Table 1. Activation Parameters for the Propagation Step for Various Monomers,Obtained via PLP–SECa

EA, A, kp at 60◦C,Monomer kJ · mol− 1 L · mol− 1 · s− 1 L · mol− 1 · s− 1

Methyl methacrylate 22.3 2.65 × 106 833Ethyl methacrylate 23.4 4.07 × 106 873Butyl methacrylate 22.9 3.80 × 106 976Isodecyl methacrylate 20.8 2.19 × 106 1,590Dodecyl methacrylate 21.0 2.51 × 106 1,2802-Ethylhexyl methacrylate 20.4 1.87 × 106 1,190Cyclohexyl methacrylate 22.3 4.88 × 106 1,560Glycidyl methacrylate 21.9 4.41 × 106 1,620Hydroxyethyl methacrylate 21.9 8.89 × 106 3,270Hydroxypropyl methacrylate 20.8 3.51 × 106 1,900Isobornyl methacrylate 22.5 4.28 × 106 1,2903-[Tris(trimethylsilyloxy)silyl]propyl 19.9 1.44 × 106 1,092

methacrylateb

Dimethyl itaconatec 24.9 2.20 × 105 27Dicyclohexyl itaconated 22.0 1.74 × 104 6Methyl acrylatee 13.9 3.61 × 106 24,000Butyl acrylate 17.4 1.8 × 107 33,700Dodecyl acrylatee 15.8 1.09 × 107 36,400Styrene 32.5 4.27 × 107 341p-CH3-Styrene 32.4 2.84 × 107 236p-Cl-Styrene 32.1 4.48 × 107 415p-F-Styrene 32.0 3.50 × 107 336Vinyl acetate 20.4 1.49 × 107 9,460aFrom Ref. 134, unless otherwise indicated.bRef. 136.cRef. 133.dRef. 135.eExperiments carried out at 10 MPa (100 bar).

The propagation rate coefficients of similar monomers are also similar, whichallows them to be arranged in groups or families. This is the case for structurallysimilar monomers such as acrylates or methacrylates. The homologous series ofacrylates with increasing ester groups, such as methyl, ethyl, butyl, and dodecylacrylate, have approximately the same kp value as do the corresponding methacry-late systems. The propagation rate coefficients and Arrhenius parameters [deter-mined via the pulsed laser polymerization–size exclusion chromatography (PLP–SEC) method] for these and additional monomers are given in Table 1 (data takenfrom References 134 and 137).

The propagation rate coefficient for most monomers is pressure dependentwith negative activation volumes; ie, an increase in pressure leads to an increasedkp value. Typical activation volumes V# (cm3 · mol− 1) for the propagation step forsome common monomers are as follows: −11.7 (methyl acrylate) (138), −11.7 (do-decyl acrylate) (138), −16.7 (methyl methacrylate) (139), −16.0 (butyl methacry-late) (140), and −16.5 (dodecyl methacrylate) (140).


It should be noted that the termination rate coefficient is unexpectedly de-creasing with increasing pressure, owing to viscosity effects. The opposite behaviorof kp and kt with respect to an increase in pressure results in markedly increasedoverall polymerization rates at higher reaction pressures.

Effect of the Reaction Medium. Since the transition state in propaga-tion is relatively nonpolar and the propagation reaction is chemically controlled(up to high monomer conversions of ∼80%), there is weak solvent influence onthe propagation rate coefficients. Extensive studies have been carried out, whichmainly confirm this small influence of the solvent on the propagation rate coeffi-cient (141–143). Larger effects in solvents have only been observed for specificmonomers, eg, EHMA (144), vinyl acetate (145), vinyl benzoate (146), or spe-cific solvents like supercritical CO2 (147–151). Furthermore, the kp values forthe polymerizations of methacrylic acid and acrylic acid in water are significantlyaffected by the monomer concentrations (152–154). In the case of the solvent ef-fects on EHMA, where kp falls between 580 L · mol− 1 · s− 1 (tetrahydrofuran) and1860 L · mol− 1 · s− 1 (xylene), it was proposed that the solvent is playing a specificrole in the transition state of the reaction by possibly affecting the geometry ofthe transition state via specific interactions. This suspicion is underpinned by theobservation that the solvent also affects both the activation energy and the pre-exponential factor. However, it should always be kept in mind that any existingtechnique to assess the propagation rate coefficient is only capable of measur-ing the product of kp and the monomer concentration. Results obtained from freeradical polymerizations carried out in supercritical CO2 suggest that the actualmonomer concentration at the reaction site is somewhat different from the solu-tion concentration; ie, a structuring of the solution occurs. Since at present thereis no methodology to separate the propagation rate coefficient from the monomerconcentration, these interpretations are somewhat speculative.

Effects of solvents on the tacticity of the generated polymeric material havebeen reported (155). When fluoroalcohols were used as solvents, the stereochem-istry of the polymerizations of some vinyl esters was affected to varying extents,depending on the structure of the acyl group of the monomer (156–159).

It has been shown that for conversion exceeding 80%, the propagation reac-tion becomes diffusion controlled, which leads to a marked decrease of kp. For ex-ample, in the case of the bulk free radical polymerization of methyl methacrylate,both the propagation rate coefficient and the initiator efficiency decrease signifi-cantly at high monomer conversion in a highly viscous regime (160–162). However,exact measurements in this highly viscous reaction regime are extremely difficultto carry out and thus reliable data are scarce.

Transfer

The measured average molecular weights, eg, obtained from molecular weight dis-tributions, of macromolecules generated by free radical polymerization processesare often lower than those predicted by accounting for initiation, propagation, andtermination processes. This experimental observation can be attributed to chainstopping events via chain transfer reactions (163,164). The transfer reaction canbe described via equations 22 and 23.


(22)

(23)

The propagating macroradical of chain length i, Ri, abstracts a weaklybonded atom X (eg, a hydrogen or halogen atom) from the transfer agent TX.A dead polymer chain with X attached to the end group is generated, as well asa new small free radical (T•), which can in turn react with a monomer unit. Thetransfer agent may be the monomer itself, the initiator, the solvent, or any otherdeliberately added transfer agent. The transfer reaction does not change the freeradical concentration—at least in its ideal form—but transfers the radical centerfrom one end of the growing polymer chain to another molecule. The overall rateof polymerization hence remains unchanged. The kinetic chain length, which de-scribes the amount of monomer units that are polymerized by one radical duringits lifetime between initiation and termination, remains unaltered. However, thetransfer process relocates the radical center from one molecule to another and en-ables the radical site to polymerize more than one molecule to completion. Hence,the actual physical chain length, ie the degree of polymerization, of the producedpolymer is reduced and concomitantly the number of polymer molecules increases(see Figure 2).

The decrease in the degree of polymerization should not be confused withthe effect of a decrease of monomer concentrations and increase of initiator con-centration, which also leads to an effective reduction of the kinetic chain length.It should be noted that an increase in the initiator concentration is always as-sociated with a considerable increase in the rate of polymerization according toequation 56, which may lead to a loss over the control of the polymerization.

The fragments of the chain transfer agent (T and X) are incorporated intothe polymer chain as end groups instead of the end groups otherwise resultingfrom the initiation and termination process. A transfer reaction can therefore beused to introduce specific end groups into the polymeric material. This technique ishowever limited by the fact that the degree of polymerization is strongly decreased

Fig. 2. Schematic description of chain transfer: �denotes the initiator-derived end group,�denotes the end group generated by the termination process. Additional transfer-agent-derived end groups (T and X) are incorporated into the polymeric material by the transfermechanism and the chain length of the polymer is reduced.


when using a transfer agent and that an efficiency of 100% regarding the endgroup introduction can never be achieved—even with very effectively transferringcompounds—because the bimolecular termination reaction is always present infree radical polymerization.

The chain transfer reaction is mainly used in industrial processes to ef-fectively reduce the molecular weight of the generated polymer. This procedureprevents the significant increase of the viscosity, which is related to the weight-average degree of polymerization. This facilitates the heat transfer inside thereactor and the processing of the final polymeric material. It should be kept inmind that conventional chain transfer agents are consumed according to equa-tions 22 and 23 during the reaction. However, there are catalytic chain transferagents based on transition metal complexes [eg, Co(II)] that are capable of effec-tively transferring the radical center without being consumed. Such agents werediscovered in the late 1970s and are now commonly used in industrial applications(see below).

The rate of the transfer reaction is described by the consumption of thetransfer agent TX, expressed by the following equation:

− d[TX]dt

= ktr[R•][TX] (24)

where ktr is the transfer rate coefficient, [TX] is the concentration of the transferagent, and [R•] is the free radical concentration. The transfer rate coefficient isoften reported as a ratio with the propagation rate coefficient kp. The resultingquantity is called the transfer coefficient, C:

C = ktr

kp(25)

The transfer coefficient C is, in principle, not constant, since it is dependenton the same external parameters as the propagation rate coefficient (ie chainlength, temperature, pressure, etc). However, C is also often denoted the chaintransfer constant. The chain transfer coefficient C describes the effectiveness ofthe transferring compound to reduce the degree of polymerization. Its value de-pends on structural features of the attacking radical and the transfer agent. Achain transfer coefficient of one has been called optimal, because the transferagent/monomer ratio will remain constant throughout the polymerization (165).The degree of polymerization therefore stays constant up to high conversions andthe polydispersity of the product remains at the theoretical value of close to 2. IfC � 1, the transfer agent will be consumed rapidly during the polymerization andthe molecular weight will increase with conversion. A small value of the transfercoefficient (C � 1), on the other hand, will lead to a decrease of the molecularweight, accordingly. In both cases, a broadening of the molecular weight distribu-tion with conversion occurs.

The ideal chain transfer reaction does not alter the overall rate of polymer-ization, Rp. However, if the chain transfer reaction has a measurable effect onRp, this rate depends on the size of the propagation, reinitiation, and transferrate coefficients. That is, the free radical which is newly formed by the transfer


step is not always of the equal activity to that which is lost. Four cases can bedistinguished:

(1) kp � ktr and kri ≈ kp leads to ‘ideal’ chain transfer, decreasing molecularweights, and no measurable effect on Rp.

(2) kp � ktr and kri ≈ kp leads to telomerization (166), a large decreasing effecton molecular weights, and no measurable effect on Rp.

(3) kp � ktr and kri < kp leads to retardation (167), a decrease in molecularweights, and a decrease in Rp.

(4) Finally, kp � ktr and kri < kp results in degradative chain transfer (168), alarge decrease in molecular weights, and a large decrease in Rp.

It can thus be easily inferred that chain transfer may alter the propertiesof the polymeric product in an undesirable way, or—in contrast—may be usedadvantageously to specifically reduce the molecular weights obtained in a specificpolymerization process or to introduce specific end groups. The first case discussedabove (which refers to the kinetic concept of chain transfer) is assumed for thederivation of the Mayo equation, which is frequently used to derive chain transferrate coefficients. This specific case does not alter the overall rate of the polymer-ization system, because it does not change the overall free radical concentration,but rather influences the molecular weight distribution.

There are at least two basic mechanisms for chain transfer: (1) atom or grouptransfer and (2) addition–fragmentation (169). The two mechanisms differ mainlyin the nature of the intermediate that is formed by the two participating molecules.The atom transfer mechanism—the most common mechanism—proceeds via anS 2 mechanism (substitution, homolytic, bimolecular) as depicted in equation 26.

(26)

The intermediate of this reaction is identical with the activated complex atthe transition state (170), and hence is destabilized and immediately decomposesinto the final products (see Figure 3a).

Tedder (171) suggested guidelines to predict qualitatively the rate and out-come of radical transfer processes:

(1) When there is little polarity in the transition state, the relative rates of theatom transfer by a specific radical will correlate with the strengths of thebonds being broken.

(2) The strength of the bond being formed is important for the absolute rate ofthe transfer event.

(3) The steric strain released or incurred when going from the reactants tothe products is especially important for endothermic or thermoneutral re-actions.

(4) Nucleophilic radicals will preferably attack electron-rich sites, and elec-trophilic radicals will attack electron-poor sites. For example, transfercoefficient for nucleophilic trialkylamine with partially positively charged


Fig. 3. Schematic representations of the energy profile along the reaction coordinate for(a), atom or group transfer and (b) addition–fragmentation transfer.

β-hydrogens is over 1000 times higher with electrophilic polyacrylonitrileand polyacrylate radicals rather than with polystyrene radicals. On theother hand, in dimethyl formamide, the electrophilic FeCl3 with partiallynegatively charged Cl atoms transfers 15 times faster with nucleophilicpolystyrene radicals than with polyacrylonitrile radicals (172).

The addition–fragmentation mechanism, on the other hand, proceeds viathe formation of a relatively stable intermediate, as depicted in equations 27 and28, with kβ being the rate coefficient of the addition reaction and k−β the ratecoefficient of the fragmentation. The corresponding energy profile along the re-action coordinate is outlined in Figure 3b. The absolute depth of the sink mayvary according to the stability of the intermediate. The reactivity of the trans-fer agent, toward the propagating species and the properties of the intermediateradical—especially its stability—are both important to characterize the overalltransfer reaction. If the fragmentation rate coefficient is small, ie the intermedi-ate exhibits a significant lifetime, it is possible that it reacts via different pathwayother than fragmentation. Depending on the structure of the intermediate radical,reversible and irreversible bimolecular termination reactions may occur, leadingto inhibition and/or retardation. If the intermediate radical adds on to monomer,ie initiates polymerization, the compound is not a transfer agent, but rather acomonomer and a branching agent.

(27)

(28)

Transfer to Monomer. The only transfer reaction that cannot be avoidedis transfer to monomer. Thus, the maximum molecular weight that can be reachedunder a given set of reaction conditions is limited by the transfer-to-monomerreaction, assuming the absence of all other transfer events. Fortunately, thetransfer-to-monomer rate coefficients in free radical polymerization are usually


rather low, approximately between 3 × 10− 5 and 20 × 10− 5 L · mol− 1 · s− 1. It is im-portant to be aware that the transfer-to-monomer reaction cannot be suppressedby a decrease in monomer concentration, because the number-average degree ofpolymerization is independent of the monomer concentration for the transfer-to-monomer step. This is, because the transfer-to-monomer effect is governed bythe ratio of the transfer rate to propagation rate and both the denominator andnumerator of this ratio contain the monomer concentration, which consequentlycancels out. (This is illustrated in eq. 91). However, the activation energies forpropagation and transfer to monomer differ for most of the common monomers.Thus, a decrease in the temperature normally lowers the monomer transfer coef-ficient CM. Table 2 gives the transfer-to-monomer constants CM for some commonmonomers.

Table 2. Transfer Coefficients to-Monomers, CM, at 60◦Ca

Monomer T,◦C CM (10− 5)

Acrylamide 60 6.0b

Acrylonitrile 60 3.3–10.2c

1-Butene 60 73Butyl acrylate 60 3.33–12.5o-Chlorostyrene 50 2.5–2.8Ethyl acrylate 60 5.79Ethylene 60 4–42d

110 11–90130 16–112

Methacrylonitrile 60 58.1Methyl acrylate 60 0.36–3.25e

Methyl methacrylate 0 1.28–1.4830 1.17–2.650 5.15f

100 3.8α-Methylstyrene 50 412f

Styrene 25 3.5g

60 7.8–8.790 15–16.5

Vinyl acetate 0 5.0–9.625 9.0–10740 12.9–13.260 18h

Vinyl chloride 60 108–160Vinylidene chloride 60 380i

aData From Ref. 173, unless other reference is noted.bRef. 174.cRef. 175.dRef. 176.eRef. 177.f Ref. 178.gRef. 179.hRef. 180.iRef. 181.


The transfer-to-monomer reaction mostly proceeds via an abstraction of ahydrogen atom; however, it is often not clear which hydrogen is involved. Thisabstraction mechanism is reasonable for monomers with aliphatic hydrogens(eg, methyl methacrylate, vinyl acetate), but different mechanisms seem to be op-erational when only vinylic and/or aromatic hydrogens are present. The transferto monomer in the free radical ethylene polymerization, for instance, also involveshydrogen atom transfer from the radical to the monomer (182). Although this hy-drogen transfer (see eq. 29), is thermoneutral—reactants and products are thesame—it is disfavored—due to the high strength of the vinylic hydrogen–carbonbond that has to be broken—over the endothermic hydrogen abstraction reactiondepicted in equation 30.

CH3 CH2• + CH2 CH2 → CH2 CH2 + CH3 CH2

• (29)

CH3 CH2• + CH2 CH2 → CH3 CH3 + CH2 CH• (30)

The transfer to styrene monomer is unlikely to proceed via a direct hydrogenabstraction, because the chain transfer coefficient is higher than that of ethylben-zene with its weaker bound hydrogen atoms. The transfer step rather involves theabstraction of a weakly bound hydrogen of the Diels–Alder adduct (see eq. 14),formed by two styrene molecules (183).

The chain transfer to vinyl chloride monomer involves a chlorine atom trans-fer reaction, following a head-to-head addition of a monomer molecule to the grow-ing macroradical (184–186). The head-to-head addition leads to a highly unsta-ble radical that stabilizes itself via a chlorine abstraction. This reaction pathway(eq. 31) accounts for CM being much greater than for other commercially importantmonomers.

(31)

Allylic compounds (CH2 CHCH2X) are usually reluctant to undergo ho-mopolymerization, because of the activation of the allylic hydrogen atom towardabstraction. The generated allylic radical is highly stabilized both by the sub-stituent X and by delocalization of the free electrons into the double bond. Theseradicals add to monomer very slowly and perform side reactions that in turn leadto retardation.

Transfer to Initiator. In most polymerization systems the transfer reac-tion to the initiator is also unavoidable. However, this form of chain transfer isto some extent controllable via the choice of the employed initiator. In addition,the effects on the overall polymerization kinetics and on the molecular weightdistribution are small if low concentrations of initiator are employed. Typicaltransfer-to-initiator constants CI are presented in Table 3.

A special case of chain transfer to initiator is the reaction involving thi-urams. The photosensitizers tetraalkylthiuram disulfides can either be cleaved


Table 3. Transfer Coefficients to Initiators, C I, at 60◦Ca

CI

Initiator Styrene Methyl acrylate Methyl methacrylate

2,2′-Azobis(isobutyronitrile) 0.09–0.14b — 0.02c

tert-Butyl peroxide 0.00023–0.0006d 0.00047e (65◦C) 0.0001f (20◦C)2-Butanone peroxide 0.46 (50◦C) 0.05 (65◦C) 0.0025–0.00698 (65◦C)tert-Butyl hydroperoxide 0.035 0.01 —Ethyl peroxide 0.00066 — —2,2′-Azobis(2,4,4-trimethyl 0.59 (25◦C) — —

valeronitrile)Benzoyl peroxide 0.101g 0.0246e 0.02h

aData from Ref. 173, unless other reference is noted.bRef. 187.cRef. 188.dRef. 189.eRef. 177.f Ref. 190.gRef. 191.hRef. 192.

photochemically to initiate polymerization or exhibit a transfer reaction via groupabstraction with another macroradical, as depicted in Scheme 8. The resultingthiyl radical can either initiate macromolecular growth or terminate a growingchain to give a polymer chain with R2NC(S)S groups at both ends (193,194). Un-der the photoirradiating conditions this termination reaction is reversible, leadingto an equilibrium of propagating and dormant radicals. This equilibrium can ef-fectively reduce the free radical concentration and hence bimolecular terminationreactions. The tetraalkylthiuram disulfides are known as iniferters (195,196) andthey function as initiator, transfer agent, and terminator simultaneously.

Transfer to Transfer Agents and Solvents. In contrast to transfer tomonomer and initiator, the transfer to a solvent molecule is of considerable im-portance, because solvents are used in high concentrations in most industrial

Scheme 8.


polymerization processes. Many organic solvents exhibit transfer coefficients sim-ilar to those found for common monomers and their significant transfer effect canbe largely attributed to their usage in high concentrations. There are a few solventsthat show significantly higher transfer coefficients, such as carbon tetrachloride.In the case when the transfer coefficient of a solvent is too high, it may not beused as a solvent but rather as a modifier, ie a ‘conventional’ chain transfer agent.Chain transfer agents with chain transfer coefficients greater than one are veryuseful, as they can be employed in low concentrations. Every solvent that containsabstractable hydrogen atoms is capable of acting as a chain transfer agent. How-ever, most of the common solvents show only minor transfer activity as can be seenin Table 4. The transfer reaction is strongly dependent on the nature of the prop-agating radical. The very reactive vinyl acetate radical, for example, has a muchhigher tendency to transfer than the relatively low reactive methyl methacrylateradical. However, many transfer agents exhibit significant retardation when usedin vinyl acetate polymerization (203). The factors influencing the reactivity inhydrogen atom abstraction reactions are discussed above in general terms. Thetransfer of aromatic compounds may involve the addition of the growing radical tothe aromatic system, generating a cyclohexadienyl radical adduct, which in turnmay restore its aromaticity via the transfer of a hydrogen to another macroradi-cal. A solvent can also be deliberately used to introduce specific end groups intothe polymeric chain if its transfer coefficient for the specific system is moderate.The compound then serves as both solvent and transfer agent, and side reactionsmay hence be suppressed (204).

Thiols. There are various thiols (R SH) that are used as chain transferagents and they are employed in many different free radical polymerization pro-cesses (205). A considerable disadvantage of these transfer agents is their oftenvery unpleasant odor. The general transfer process of these compounds is depictedin equations 32 and 33.

(32)

(33)

The vast majority of the generated polymeric material possesses a thioetherend group. The chain transfer coefficients of some thiols in styrene and methylmethacrylate polymerizations are given in Table 4. It should be noted that thevalue of the chain transfer coefficient is dependent on the polymerization sys-tem, ie monomer, temperature, pressure, etc. However, the tabulated values areguidelines for the transfer ability of the transfer agents in other monomer sys-tems. A number of functional thiols have been used to prepare end-functionalpolymeric material. These were used as starting materials for block (206,207)and graft (208) copolymers for the application as emulsifier, thermoplastic elas-tomers, compatibilizer, and adhesives. Thiols react more rapidly with nucleophilicthan electrophilic radicals. Hence, thiols have a greater transfer coefficient withstyrene and vinyl esters than with acrylates and methacrylates. Aromatic thiolshave a higher transfer ability; however, they exhibit more retardation. The thiyl


Table 4. Transfer Coefficients to Solvents and Additives, CT, at 60◦Ca

CT (10− 4)

Transfer agent Styrene Methyl methacrylate

2-Aminoethanthiol hydochloride — 1,100b

2-Butanone 4.98 0.45Acetaldehyde 8.5 6.5Acetic acid 2.22 (40◦C) 0.24 (80◦C)Acetone 0.32c 0.195Acetonitrile 0.44 —Aniline 2.0 4.2Benzaldehyde 4.5–5.5 0.86–2.5Benzene 0.018–0.04 0.04–0.83Benzenesulfonyl chloride 4,330 5Benzenethiol — 27,000Carbon tetrabromide 2,500,000d 1,500–2,700Carbon tetrachloride 69–148 0.5–20.11Chloroform 0.41c 0.45–1.77Copper(II) chloride 108 2 × 107

Cumene 0.8–3.88 1.9–2.56Cyclohexane 0.024–0.063 0.1–0.2 (80◦C)Diethyldisulfide 45 (99◦C) 1.3Dibenzyldisulfide 100 63Diphenyldisulfide 1,500 110Ethyl acetate 15.5 0.1–0.46Ethyl ether 5.64 —Ethyl iodoacetate 8,000 —Ethyl bromoacetate 430 —Ethyl tribromoacetate 100,000 —Ethyl trichloroacetate 100 —Heptane 0.42 1.8 (50◦C)Iron(III) chloride 5,360,000 4,0002-Propanol 3.05 0.583Methanol 0.296–0.74 0.2Mercaptoacetic acid methyl ester 14,000e 3,000e

N,N-Dibenzylhydroxylamine 5,000 —N,N-Dimethyl acetamide 4.6 —N,N-Dimethyl formamide 4.0 —n-Butyl alchol 1.6 0.394n-Butanethiol 220,000 6,600n-Dodecanethiol 150,000 9700–12300f

Pentaphenylethane 20,000 —Phenyl ether 7.86 9.13Pyridine 0.6 0.176 (70◦C)Tetraethylthiuram 3,200g —Tetrahydrofurane 0.5 (50◦C) —Toluene 0.105–2.05 0.17–0.45Trichlorotoluene 57.5Triethylamine 1.4–7.5 8.3Water 0.006–0.31aData from Ref. 173, unless other reference is noted.bRef. 197.cRef. 198.dRef. 199.eRef. 200.f Ref. 201.gRef. 202.


radical produced by the transfer reaction of a thiol is electrophilic and, when usedin copolymerizations, will more likely react with electron-rich monomers.

A wide range of functional thiols, eg, mercaptoethanol (OHCH2CH2SH) andthioglycolic acid (OH(CO)CH2SH), have been used to produce monofunctionalpolymers (165,209,210).

Sulfides. A wide range of various disulfides have been employed as chaintransfer agents: dialkyl disulfides (211), diaryl disulfides (212,213), diaroyl disul-fides (214), and xanthogens (215). The reaction of these compounds leads to the in-corporation of sulfur-containing end groups at both ends of the polymer, as shownin equations 34 and 35. They therefore have found some applications in the prepa-ration of telechelic polymers.

(34)

(35)

The transfer coefficients of disulfides are extremely low for styrene andmethyl methacrylate (see Table 4), but are close to one for the vinyl acetate poly-merization. In general the xanthogens and thiurams have higher chain transferability, which has been attributed to the iniferter mechanism described above.Monosulfides have lower transfer coefficients in comparison to disulfides. Thismay reflect steric factors and the relative strength of the C S bond, which issignificantly stronger than the S S bond.

Catalytic Chain Transfer. A highly useful variant of chain transfer wasdiscovered in the 1970–1980s in the Soviet Union (216). A number of reviewshave been published in recent years (217–221) on this synthetic method whichhas acquired the nomenclature of either catalytic chain transfer (CCT) or specialchain transfer (SCT). The most commonly adopted catalysts are based on low spincobalt macrocycles, although other metal-containing complexes have also beensuggested in the patent and scientific literature. Some typical catalyst structuresare shown as 12 and 13.

Some of the highest chain transfer constants ever measured have been re-ported for COBF (14), as shown below:


Monomer Cs (10− 3) at 60◦C

Methyl methacrylate 28.1 (222)tert-Butyl methacrylate 15.1 (222)Dimethyl itaconate 9.5 (222)Styrene 9.0 (223)

These high transfer values would only be of curiosity value for synthetic pur-poses if a conventional chain transfer mechanism was operative. However, thesecobalt complexes appear to catalyze transfer to monomer and they do not getincorporated into the polymer chains, nor do they get consumed in the transfer re-action. A catalytic cycle involving methyl methacrylate as the monomer is thoughtto be a good representation of the mechanism. The cobalt complex abstracts a β-hydrogen from a growing radical (creating terminal unsaturation) and a cobalthydride, which then reinitiates a monomer unit via hydrogen-atom addition orinsertion into the double bond (eqs. 36 and 37).

(36)

(37)

An interesting variation of this reaction (224,225), termed catalytic chaintransfer isomerization, has been described whereby transfer occurs from a radicalwith a β-CH2OH group, as shown in Scheme 9. A keto–enol isomerization occursand an aldehyde end group results.

Catalytic chain transfer is most efficient for tertiary propagating radicalssuch as methyl methacrylate. When secondary radicals are involved, such as


Scheme 9.

acrylates or styrene, the efficiency of the reaction is diminished. It is now thoughtthat this is primarily caused by cobalt–carbon bonding acting as a significantside-reaction, effectively removing some active catalyst from the reacting system(226). This cobalt–carbon bonding has been observed directly in methyl acrylatepolymerization using MALDI (227) (see MASS SPECTROMETRY).

There remains some minor unsolved mechanistic mysteries in the CCT re-action. The rate constant for transfer in methacrylates is similar to those forradical–radical termination reactions; these are known to be diffusion controlled.This has lead to speculation that the CCT reaction is also diffusion controlled (229)and there is some evidence to support this. However, this evidence is not absoluteand the most prudent analysis is that the CCT reaction of methacrylates falls ina regime where both features of diffusion and chemical control can be observeddependent on the reaction conditions. Hopefully further experimental work willallow clarification of this situation.

The CCT reaction has attained commercial utility as it is ideal for synthe-sizing low molecular weight polymers with terminal unsaturation. The resultantmacromonomers can be used as either chain transfer agents (useful for makingblock structures) or polymerizable monomers, useful in the synthesis of comb- andstar-type molecules. The reaction is tolerant of many functionalities and of water,making it amenable to emulsion reactions.

Alkyl Halides. Transfer to alkyl halides depends on structures of both alkylgroup and halogen, but also on the polymeric radicals. Thus, nucleophilic radicals(styrene) react faster with alkyl halides than electrophilic radicals (acrylates).Hence, the C values with styrene and vinyl acetate are higher than those withacrylic monomers.

The halogen abstraction ability decreases according to I > Br > Cl. For ex-ample, in styrene polymerization, transfer coefficient for iodoacetates is C = 0.8,for bromoacetates C = 0.043, but only C = 0.0001 for chloroacetates. The C valueincreases with number of halogens, for example for dibromoacetates C = 0.27and C > 10 for tribromoacetates. Among the most often use alkyl halides arehalomethanes and aforementioned halogenated esters. Both of them—including


chloroform, carbon tetrachloride, carbon tetrabromide, bromotrichloromethane—have widely been used in the preparation of telomers. The application of effectivetransfer agents in very high concentrations leads to so-called telomerization, ie theexclusive formation of oligomeric (dimeric, trimeric, etc) species. The kinetics ofsuch telomerization is far from simple, because of the disparate radical reactivitiesdisplayed by the very short—halogen-substituted—radicals involved: The shorterradicals show a smaller reactivity toward propagation and transfer reactions thando the longer ones. In the case of carbon tetrachloride mediated polymerizations,this has been attributed to the stabilizing effect of the CCl3 end group (229–231).The chain length dependence of the chain transfer coefficient most likely reflectsa variation of both kp and ktr with the length of the macroradical. However, com-mon techniques for the determination of C do not allow for the separation of sucheffects.

The perhalomethanes react in the chain transfer process (as shown ineq. 38) to give a perhaloalkyl radical, which in turn is capable of initiating thepolymerization.

(38)

With hydrohalomethanes, two pathways for the transfer step may be envis-aged: halogen- or hydrogen-atom abstraction. For chloroform the hydrogen trans-fer is favored because of the relative bond strengths, especially for electrophilicradicals.

Very fast transfer which results in the inhibition occurs with transition metalhalides such as FeCl3 and CuCl2. In dimethyl formamide at 60◦C transfer coeffi-cients for CuCl2 decrease in the order, styrene (C = 10,000) > MMA (C = 2000) >

acrylonitrile (C = 100). FeCl3 is less efficient transfer agent and C values de-crease in the following order: vinyl acetate (C = 626) > styrene (C = 536) > vinylchloride (C = 85) > MMA (C = 4) and acrylonitrile (C = 2) (232). Ligands which sol-ubilize transition metal halides provide additional possibility for tuning transfercoefficients. Moreover, the resulting lower oxidation state transition metal com-pounds can activate the alkyl halides, resulting from the transfer process. Such areversible activation/deactivation is used for ATRP.

Atom Transfer Radical Polymerization. ATRP is one of the most successfulcontrolled/living radical polymerization (CRP) systems, in addition to NMP anddegenerative transfer processes, such as RAFT (5,233,234). The key feature of allof them is the dynamic equilibration between the active radicals and various typesof dormant species (see LIVING RADICAL POLYMERIZATION).

Mechanistically, these systems are quite similar to conventional radical poly-merizations. Polymeric radicals grow and terminate with similar rate constants.The main difference is how the radicals are generated. They are generated slowlyand irreversibly in the conventional processes through dissociation of peroxidesor diazo compounds which typically have a half-life in the range of 10 h. Propaga-tion is rapid with an approximate frequency of monomer addition of ∼1 ms. Thismeans that within 1 s, a polymer chain with the chain length P ∼ 1000 is typ-ically formed. Within approximately the same time, chains terminate by either


coupling or disproportionation. During such a short time, it is not possible toperform any kind of macromolecular engineering by adding another monomer,functionalizing reagent, etc. In controlled/living reactions, however, the radicalformation is reversible. Similar values for the equilibrium constants during initi-ation and propagation ensure that the initiator is consumed at the early stages ofthe polymerization, generating chains which slowly and continuously grow, as in aliving process. Perhaps the most important difference between the two approachesis the lifetime of the propagating chains, which is extended from ∼1 s to more than1 h. The second major difference is the dramatic increase in the initiation rate,which enables simultaneous growth of all the polymer, chains. Both parametersallow various macromolecular engineering techniques to be applied, such as mak-ing well-defined star polymers, block and grafts, end functional polymers, andmany other well-defined, materials.

Termination processes cannot be avoided in CRPs. Since termination is a bi-molecular process, increasing the polymerization rate increases the concentrationof radicals and enhances the termination process. Termination also becomes moresignificant for longer chains and at higher conversion. However, this process issomehow self-tuned, since termination is chain length dependent. Polymers pre-pared by CRPs will never have 100% functionality and 100% blocking efficiency.For monomers with low values of rate constants of radical propagation (kp), suchas dienes, styrenes, and methacrylates, it is necessary to slow down the polymer-ization rate to avoid excessive termination; for those with high kp value, suchas acrylates, this is less important. Since propagating species are free radicals,chemo-, regio-, and stereoselectivities are similar to those found in conventionalradical polymerizations. This means that similar reactivity ratios, similar sensi-tivity to transfer reagents, and similar tacticities are observed.

Each of the controlled/living radical polymerization systems has some limi-tations and some special advantages, and it is expected that each technique mayfind special areas where it would be best suited synthetically. For example, NMPcarried out in the presence of bulky nitroxides cannot be applied to the polymer-ization of methacrylates owing to fast β-hydrogen abstraction. ATRP cannot yet beused for the polymerization of acidic monomers, which can protonate the ligandsand complex with copper. RAFT is very slow for the synthesis of low molecularweight polymers because of retardation effects, and may provide branching owingto trapping of growing radicals by the intermediate radicals. At the same time,each technique has some special advantages. Terminal alkoxyamines may act asadditional stabilizers for some polymers. ATRP enables the synthesis of specialblock copolymers by utilizing a halogen exchange and has an inexpensive halogenat the chain end. RAFT can be applied to the polymerization of some less reactivemonomers, such as vinyl acetate.

A successful ATRP (and any controlled/living radical polymerization process)should meet three requirements: initiator should be consumed at the early stagesof polymerization to form polymers with degrees of polymerization predeterminedby the ratio of the concentrations of converted monomer to the introduced initiator(DP = [M]/[I]0); the number of monomer molecules added during one activationstep should be small, resulting in polymers with low polydispersities; the contri-bution of chain breaking reactions (transfer and termination) should be negligibleto yield polymers with high degrees of end functionalities, and allow the synthesis


Scheme 10.

of block copolymers. In order to reach these three goals, it is necessary to selectappropriate reagents and appropriate reaction conditions.

ATRP is based on the reversible transfer of an atom or group from a dor-mant polymer chain (R-X) to a transition metal (Mt

n/Ligand) to form a radical(R•), which can initiate the polymerization, and a metal halide whose oxidationstate has increased by one (X Mt

n+1/Ligand); the transferred atom or group iscovalently bound to the transition metal. A catalytic system employing copper(I)halides (Mt

n/Ligand) complexed with substituted 2,2′-bipyridines (bpy) was suc-cessful for polymerization of styrenes, various (meth)acrylates, acrylamides, acry-lonitrile, and other monomers (235–238). Other metal centers have been used,such as ruthenium, nickel, iron, and other transition-metal-based systems. Cop-per salts with various anions and polydentate complexing ligands were used, suchas substituted bpys, pyridines, and linear polyamines. The rate constants of theexchange process, propagation and termination shown in Scheme 10 relate tostyrene polymerization catalyzed by CuBr/2bpy and initiated by 1-phenylethylbromide at 110◦C.

According to this general ATRP scheme (Scheme 10), the rate of polymeriza-tion is given by the following equation:

Rp = kp[M][RX]0ka[CuI]/(

kda[X CuII])

(39)

Thus, the rate of polymerization is internally first order in monomer, exter-nally first order with respect to initiator and activator, Cu(I), and negative firstorder with respect to deactivator, XCuII. However, the kinetics may be more com-plex because of the formation of XCuII species via the persistent radical effect. Theactual kinetics depend on many factors, including the solubility of activator anddeactivator, their possible interactions, and variations of their structures and re-activities with concentrations and composition of the reaction medium. It shouldalso be noted that the contribution of persistent radical effect at the initial stagesmight be affected by the mixing method, solubility of the metal compound andligand, etc (239–242).

One of the most important parameters in ATRP is the dynamics of exchange,and especially the relative rate of deactivation. If the deactivation process is slowin comparison with propagation, then a classic redox initiation process operates,leading to conventional, and not controlled, radical polymerization. Polydispersi-ties in ATRP are defined by the following equation, when the contribution of chain


breaking reactions is small and initiation is complete:

Mw/Mn = 1 + {(kp[RX]0)

/(kda[X − CuII]

)}(2/p − 1) (40)

Thus, polydispersities decrease with conversion, p, the rate constant of de-activation, kda, and also the concentration of deactivator, [XCuII]. They, however,increase with the propagation rate constant kp and the concentration of initiator,[RX]0. This means that more uniform polymers are obtained at higher conversions,when the concentration of deactivator in solution is high and the concentrationof initiator is low. Also, more uniform polymers are formed when the deactivatoris very reactive and monomer propagates slowly (styrene rather than acrylate).Polydispersities can also be expressed as a function of the reaction time t (243):

Mw/Mn = 1 + 1/(

2ka[CuI]t)

(41)

The most important benefit of living polymerizations is that they allow prepa-ration of new macromolecules with precisely designed and controlled compositions(homopolymers, random, periodic, block, graft, and gradient copolymers), topolo-gies (linear, star, comb, (hyper)branched, networks, etc), and functionalities placedat different parts of macromolecules or various combinations of these. Some of thepossibilities are outlined below (244,245).

The advent of controlled/living radical polymerization enables preparationof many new materials such as well-defined components of coatings (with narrowmolecular weight distribution, precisely controlled functionalities, and reducedvolatile organic compounds), nonionic surfactants, polar thermoplastic elas-tomers, entirely water-soluble block copolymers (potentially for crystal engineer-ing), gels and hydrogels, lubricants and additives, surface modifiers, hybrids withnatural and inorganic polymers, various biomaterials and electronic materials.Extensive discussion of more detailed aspects, such as the monomers, initiators,catalysts, and media, can be found in Reference 5.

Addition–Fragmentation Chain Transfer. The addition–fragmentationchain transfer process is induced by the addition of a growing radical to a sim-ple organic compound with a reactive double bond. The general structure of theaddition–fragmentation chain transfer agents and the transfer process is depictedin equation 42: A is part of the activated double bond, Z is both the activating groupfor the double bond and the stabilizing group for the intermediate radical, and Band the leaving group R are connected to each other by a weak single bond. Thepropagating radical adds to the activated double bond with the rate coefficientkβ , and an intermediate radical is formed. This intermediate radical in turn frag-ments with the rate coefficient k− β via generation of a new double bond and asmall radical. The success of the transfer process is dependent on three factors:(1) the reactivity of the double bond toward radical addition must be similar orhigher to that of the polymerizing monomer double bond, (2) the leaving abilityof the released moiety R must be substantial, and (3) the reactivity of the ex-pelled radical R• toward reinitiation of the polymerization process has to be high(246,247).


(42)

Several different types of addition–fragmentation chain transfer agents maybe envisaged, when considering different types of atoms and/or groups for A, B,Z, and R in the general structure given in equation 42. The first class (A = B= CH2) represents the allylic compounds, which include the allylic sulfides (R =SR) (248–250), allylic peroxides (R = OOR) (251,252), allylic bromides (R = Br)(248,253,254), allylic sulfones (R = S(O)R) (248,253), allylic phosphonates (R =P(O)(OR)2) (248,253), and allylic stannanes (R = SnR) (248,253). The other twoclasses are the vinyl ethers (A = CH2 and B = O) (248,255) and the thiocarbonylcompounds (A = S and B = O) (256,257). R is any good homolytic leaving group andZ can be varied to give optimum transfer coefficients. The addition–fragmentationchain transfer coefficients of the various compounds are very high compared tocommon transfer agents with C values in the order of one and above (0.2–20) (246)depending on the structure of the transferring compound and the reacting radical.

The addition–fragmentation chain transfer is a versatile process being ap-plicable to a wide range of monomers (styrene, acrylates, methacrylates) to intro-duce functionalized end groups (246,258). End functional polymers—the type ofend group may be governed by the chosen transfer agent and its different A, B,Z, and R groups—are very useful for building block, graft, and network polymerstructures (259–261).

Methyl methacrylate macromonomers act as a chain transfer agent by anaddition–fragmentation mechanism as depicted in equation 43. By that, an olefinicgroup is installed at the end of the polymer chain. These products are themselvesmacromonomers that can react in the same way during the polymerization. In theacrylate polymerizations these macromonomers are readily copolymerized, henceleading to branching (262). The polymerization of methacrylates in the presenceof methacrylate macromonomers will produce new macromonomers, which are inturn participating in the described addition–fragmentation mechanism. This re-versible addition–fragmentation process gives rise to living behavior, as describedin the next section.

(43)

Reversible Addition–Fragmentation Chain Transfer. The RAFT repre-sents a process for performing living free radical polymerization. Many simpleorganic compounds possessing a thiocarbonylthio (S C S) moiety are employed


as mediating agents which exhibit a great effectiveness and versatility towardcontrolling the free radical polymerization. The general scheme of these dithio-compounds is depicted as structure 15. The ability of a RAFT agent to controlthe polymerization activity is governed by two characteristic groups, the Z and Rgroups, also denoted as the ‘stabilizing’ and the ‘leaving’ group.

Different substituents as Z and R groups lead to a wide array of differentRAFT agents: (1) dithioesters (Z = phenyl, methyl, etc), (2) trithiocarbonates (Z =S R), (3) xanthates (Z = O R), and (4) dithiocarbamates (Z = NR2). A controlledpolymerization involving xanthates is referred to as MADIX (263,264). The RAFTpolymerization is performed by adding a chosen quantity of an appropriate RAFTagent to a conventional free radical polymerization mixture. The process yieldspolymers of predetermined chain length and narrow polydispersity. Polydisper-sities indices of less than 1.1 can be usually achieved under optimal conditions.The RAFT process generates polymeric material with active thiocarbonylthio endgroups, which can be used for chain extensions and/or further chemical reactions,to yield complex polymeric architectures. Polymeric structures ranging from starand comb polymers to block copolymers can be prepared with relative ease for avariety of monomers (265–272).

The mechanism of the living free radical RAFT polymerization with dithiocompounds involves two important equilibrium reactions that are superimposedon a conventional free radical polymerization (see Scheme 11). As in conventionalfree radical polymerization, initiation leads to the formation of propagating chains.In the early stages of the polymerization the macroradicals react with the initialRAFT agent (species 1 in Scheme 11), generating the intermediate radical (species2 in Scheme 11), followed by the decomposition of these species by formation ofpolymeric RAFT agent (species 3 in Scheme 11) and the leaving group radical.The first equilibrium reaction may be simplified by an overall transfer step in-volving the initial RAFT agent as transfer agent. The leaving group radical R•

initiates in turn a new polymeric chain with the rate coefficient kp,1 by reactionwith monomer and leads to macromolecular chain growth of the free radicals withthe propagation rate coefficient kp. The reaction step (IV) constitutes the core ofthe RAFT process and has been termed the main or core equilibrium. The additionreaction proceeds with a rate coefficient kβ , whereas the formed intermediate—the macro-RAFT radical (species 4 in Scheme 11)—fragments proceed with therate coefficient k− β . Finally, the reaction step (V) considers the termination re-actions of the linear macroradicals, which proceed with a chain-length-averagedtermination rate coefficient 〈kt〉.

The equilibrium between the active propagating species (Pn• and Pm

•) andthe dormant polymeric RAFT agent (species 3 in Scheme 11) provides an equalprobability for all chains to grow. Because of the reversible nature of the transferreaction, the transfer agent (ie the polymeric RAFT agent) is not consumed and theactive dithio groups are incorporated into the final polymeric material (273). Each


Scheme 11.

RAFT agent molecule is transformed into the end group of one polymer chain.Hence, the final chain lengths of the polymeric material can easily be controlledvia varying the ratio of RAFT agent concentration to conversion. It should be notedthat the free radical concentration—at least in principle—is not reduced in theRAFT process. Hence, the overall rate of polymerization is unchanged in an idealRAFT polymerization compared to a conventional free radical polymerization.

A successful RAFT process is, on the one hand, dependent on the effective ad-dition of the growing radical to the RAFT agent compound. On the other hand, theleaving and reinitiating ability of the leaving group also governs the effectivenessof the transfer step. This very effective transfer reaction—which has to be compet-itive with the propagation step—is illustrated by the large transfer coefficients forthe simplified preequilibrium: transfer coefficients of more than 5000 in styrenepolymerization can be observed for some dithiobenzoates (274). The transfer stepis strongly enhanced when the RAFT agent possesses a well-stabilizing Z group(eg, phenyl). The stabilizing ability of the Z group, however, leads to relativelystable RAFT intermediate radicals, causing rate retardation, that is, a decreasein the overall rate of polymerization up to high conversions with increasing ini-tial RAFT agent concentration. This kinetic phenomenon may be explained eitherby intermediate radicals with very long lifetimes (275) or by reversible (276–278)and/or irreversible (279) termination of this species. The different leaving abilitiesof various R groups sometimes affect the preequilibrium and leads to inhibitionphenomenon, that is, a time period without any polymerization activity at early


Table 5. Transfer-to-Polymer Constants, CPa

Polymer T,◦C CP (10− 4)

Poly-1,3-butadiene 50 11Polyacrylonitrile 50 4.7Polyethylene 175 108.4Poly(methyl methacrylate) 50 0.22–1.5Poly(N,N-dimethyl acrylamide) 50 0.61Polystyrene 50 1.9–16.6Poly(vinyl acetate) 50 0.06–10.2aData from Ref. 173.

reactions times, dependent in extent on the initial RAFT agent concentration(280).

Transfer to Polymer. At higher monomer conversion, transfer processesto the formed polymer are becoming significant. Interestingly, the transfer-to-polymer rate constants are considerably higher (approximately by a factor of 10)than those observed for the corresponding monomer. Transfer to polymer is result-ing in so-called long-chain branching if the reaction takes place intermolecularly(281). In the case of intramolecular transfer to polymer, the reaction is described as‘backbiting,’ leading to short-chain branching, as observed in ethylene polymeriza-tions. The length of the branched side chains generated via backbiting is usuallysmaller than five carbons. Typical values for the transfer-to-polymer constants CPof some common monomers are collected in Table 5.

Transfer coefficients to polymers are not as readily determined as othertransfer coefficients because the process does not necessarily lead to a reduc-tion of the molecular weight. Polymerization in the presence of polymer yields amixture of the polymer initially present and the new polymer formed, and thusthe decrease in the molecular weight cannot be accurately evaluated. However,chain transfer coefficients to polymers are accessible via the structural investiga-tions of the generated branched macromolecules, eg, via NMR (282,283). As NMRmeasurements of branching rely on the determination of the branching points onthe polymeric backbone, a quantitative distinction between long- and short-chainbranching is a major analytical problem. The determination of the number of endgroups per polymeric chain is another viable technique to quantify the branchingof macromolecules (284). For some polymers, the value of CP depends on the poly-mer molecular weight. This may account for the wide range of values for CP in theliterature.

Detailed structural analysis of poly(vinyl acetate) has revealed that hydro-gen abstraction from the acetyl group is the predominant chain transfer mecha-nism (285). Abstraction from the methine hydrogen of the main chain leads to asmall amount of branching (286,287).

Mid-chain radicals may be formed by intermolecular abstraction of a ter-tiary hydrogen atom from the polyacrylate main chain. This pathway is favoredunder the conditions of low monomer and high polymer concentration such asduring emulsion polymerization, where the particles consist of large amount ofpolymer and small amount of polymer (288–290) and at high conversion duringsolution/bulk polymerization (281). It has been proposed that this process and


the consequent formation of branches may contribute to the early onset of the geleffect in the polymerization of acrylates.

The presence of long-chain branches in low density polyethylene accountsfor the difference in properties (eg, better melt strength, greater toughness) whencompared to completely linear polyethylene chains. The long-chain branchingcan be routinely detected by determining the end group—ie methyl—content ofthe polyethylene via infrared spectroscopy (291,292). Other studies involve NMR(293–295) and SEC (size exclusion chromatography) (296,297). The extent of long-chain branching is known to be strongly dependent on the reactor design and thereaction conditions employed.

Termination

The radical–radical termination reaction in free radical polymerization is the mostcomplex reaction in the polymerization process (298,299). Its termination ratecoefficient kt is influenced by a multitude of different factors, which are not easilyseparated. Only within the last 15 years new methods have become availablethat allow for an accurate measurement of this rate coefficient. The scatter of thetermination rate coefficients given in the Polymer Handbook (173) reported for thesame monomer at the same reaction temperature is a direct manifestation ofthe influence of these various parameters on kt. This appreciable disagreement ispartly explainable by the frequent use of incorrect values for the propagation ratecoefficient kp, which is always needed to determine kt from the coupled form of thetwo coefficients. However, the situation has improved greatly with the inventionof the PLP–SEC method.

Combination vs Disproportionation. There are two modes of termina-tion: one is the direct coupling (combination) of two free macroradicals to give adead polymer chain of chain length i + j, with the rate coefficient kt,c. The othermode is the so-called disproportionation, where a hydrogen atom is transferredfrom one of the radical chain ends to another radical, yielding two stabilized poly-mer chains, of which one carries a double bond. This reaction is associated withthe rate coefficient kt,d. The process is illustrated in equation 44 using polyethy-lene macroradicals as the example. It is important to notice that—in the case ofmacroradicals derived from other monomers—in principle any β-hydrogen maybe abstracted.

(44)

The reactions between carbon-centered radicals generally give a mixtureof disproportionation and combination. Which termination mode dominates de-pends largely on the structure of the monomer unit, and to a lesser extent on thereaction temperature, pressure, and solvent (300). Disproportionation is (slightly)favored at higher reaction temperatures. The reasons for this behavior have yet


Fig. 4. Temperature dependence of the contribution of disproportionation to the over-all termination process, δ, for a methyl methacrylate polymerization at ambient pressure(selected references: Refs. 305–308).

to be clarified, but there is some evidence pointing toward a different temperaturedependence of the corresponding preexponential factors in the Arrhenius expres-sion for kt,c and kt,d. Combination and disproportionation should be consideredas two different reactions with two distinct transition states; this is supported bytheoretical studies (301–304). However, the observed effects are small and asso-ciated with a large experimental scatter, as indicated in Figure 4, showing thetemperature dependence of the contribution of disproportionation to the overalltermination process, δ, defined according to equation 45, for a methyl methacrylatepolymerization at ambient pressure.

δ = kt,d

kt,d + kt,c(45)

For a given series of radicals, δ increases with the number of β-hydrogenatoms. However, there is no direct relationship and it is clear that a number ofother factors are involved (300,309). In addition, it is generally observed that theimportance of disproportionation increases with increasing substitution at theradical center. Steric effects play a very important role as demonstrated bythe self-reaction of cumyl radicals 16 and of the tert-butyl-substituted equiva-lent 17. The termination reaction of radical 16 shows predominantly combina-tion, whereas radical 17 gives predominantly disproportionation, although thereare less β-hydrogen atoms. This finding may imply that the combination reac-tion is more suppressed by the steric hindrance than the disproportionation reac-tion. The statistical effect that favors disproportionation when more β-hydrogenatoms are available is hence to be considered less pronounced than the steric ef-fect (310). The steric crowding can lead in extreme cases to persistent radicals


(eg, di-tert-butyl methyl radical (311) and triisopropylmethyl radical (312)) thatare relatively reluctant to undergo radical–radical reactions.

The increase of δ with polar solvents is attributed to the fact that the tran-sition state of disproportionation has polar character while that for combinationis neutral (304).

It should be noted that the mode of termination has no influence on the kinet-ics of the free radical polymerization process. However, the generated molecularweight distributions are strongly influenced by the termination mode and it ishence vital to the understanding of structure–property relationships. Since somemethods for the determination of the termination rate coefficient rely on the anal-ysis of the full molecular weight distributions, it is mandatory to have reliabledata on the termination mode of a specific monomer. In addition, the terminationmode governs the end groups of the generated polymer (one initiator derived endgroup with disproportionation and two with combination). Table 6 gives data oncombination–disproportionation modes for various monomers at ambient pressureand various temperatures.

The dead polymer chains generated via disproportionation carry an unsatu-rated end group which may be reactive during polymerization. Copolymerizationof these macromonomers is a possible mechanism for the formation of long-chain

Table 6. Contribution of Disproportionation to the OverallTermination, δ, for Different Monomers

Monomer T,◦C δ Reference

Acrylonitirile 10–90 0 313–315Butyl methacrylate 80 0.54 316Dicyclohexyl itaconate 25 0.8–1.0 294,317Ethyl methacrylate 80 0.42 316Methacrylonitrile 25 0.65 318Methyl acrylate −34 ∼0.25 317Methyl methacrylate 0 0.61 308

25 0.67 30560 0.73 30590 0.81 319

α-Methylstyrene 55 0.091 320Styrene 20–50 0.17 321

30 0.14 322,32350 0.2 32260 0.1–0.2 8390 0.054 319


branches (316,324). A comprehensive review of the literature known data on com-bination vs termination modes for various monomers and model systems can befound in Reference 325. However, unequivocal numbers for α are not yet availablefor most of the polymerization systems and in most cases there is only a qualita-tive agreement between different literature values. Despite these discrepanciesfour different generalizations can be made:

(1) Termination of polymerizations involving vinyl monomers involves predom-inantly combination.

(2) Termination of polymerizations involving α-methylvinyl monomers alwaysexhibit a significant contribution of disproportionation.

(3) The hydrogen atoms of the α-methyl group are much more prone to abstrac-tion during the disproportionation reaction than the methylene hydrogens.

(4) Within a series of vinyl or α-methylvinyl monomers, δ apparently decreasesaccording to the radical-stabilizing ability of the substituent.

Kinetic Aspects of the Termination Reaction. It is generally acceptedthat the termination rate coefficient depends on the following factors and experi-mental parameters: (1) the system viscosity, (2) the chain length of the terminatingfree macroradicals, (3) the temperature, (4) the pressure, and (5) the monomer-to-polymer conversion.

The rate law expression for the termination step that describes the variationof the free radical concentration with time reads (326)

− d[R•]dt

=∑

i

∑j

2ki, jt [Ri

•][Rj•] (46)

The indices i and j indicate the individual chain lengths of the terminatingmacroradicals. There has been considerable confusion in the past on whether toincorporate the factor 2 from the rate law expression into the termination ratecoefficient. The factor 2 is necessary if the rate law describes the rate of the loss ofmacroradicals; however, it is unnecessary if only termination events are consid-ered. Nevertheless, the IUPAC ruling on this is clear; termination rate coefficientsare to be reported without the incorporated factor 2. All termination rate coeffi-cients given in this article are in accordance with the IUPAC guideline.

Because of the above-mentioned chain length dependence of the terminationrate coefficient (327) and its strong dependence on the monomer conversion, it israther difficult to report tabulated values for kt. However, it is possible to give achain-length-averaged kt value for a specific monomer conversion, 〈kt〉. It is im-portant to note that for many of the reported chain-length-averaged kt values, it isunclear to which chain length region they correspond. Since most measurementsof the termination rate coefficient have been carried out at low monomer conver-sion, it is sensible to report kt values for the low conversion regime (see below fora discussion of high conversion kt data).

Average termination rate coefficients have been reported over several ordersof magnitude, ranging from approximately 50 to 109 L · mol− 1 · s− 1 (173). Thisimpressive range may be explained by a shift in the rate-determining step when


Scheme 12.

going from one monomer to another. The termination reaction can be broken downinto a three-stage process (328–330), as can be seen in Scheme 12:

(1) The translational (ie center of mass) diffusion of the individual macroradicalcoils toward each other through the reaction medium.

(2) The so-called segmental diffusion of the radical chain ends toward eachother. This segmental diffusion process brings the chain ends into a positionthat enables them to react.

(3) The chemical reaction of the two radicals which yields the polymeric prod-uct(s).

The slowest reaction step—in a sequence of reaction steps—will always de-termine the rate of the overall reaction according to equation 47, which describesthe inverse overall termination rate coefficient as the sum of the inverse ratecoefficients of the individual reaction steps:

1kt

= 1kTD

+ 1kSD

+ 1kR

(47)

with kTD being the rate coefficient for the translational diffusion, kSD the rate co-efficient for the segmental diffusion, and kR the rate coefficient for the chemicalreaction of radical–radical recombination. Because of the extremely fast reactionrate of the radical combination/disproportionation reaction (ie strictly speakingthe termination itself) with kR being in the order of 1010 L · mol− 1 · s− 1, thecenter of mass and segmental diffusion steps are (in almost all the cases) therate-determining steps. At low monomer conversions, the rate-determining stepfor reasonably high molecular weight macroradicals (of length 50 or above) is be-lieved to be almost exclusively segmental diffusion. Typical examples for high ktmonomers are methyl methacrylate and styrene, ie, 〈kt〉 ≈ 108 L · mol− 1 · s− 1. Dif-ferences in the (average) termination rate coefficients that can be observed whencomparing different monomers are most likely due to differences in segmental dif-fusion coefficients. However, lower values of kt, as observed for example for dodecylmethacrylate, ie 〈kt〉 ≈ 106 L · mol− 1 · s− 1, have been attributed to the stericallydemanding substituents, leading to a (partly) chemically controlled terminationprocess (331). For reported ultralow kt monomers, such as the fumarates (332,333)or itaconates (135,334) with 〈kt〉 < 500 L · mol− 1 · s− 1, a chemically controlledtermination process may also be envisaged, but evidence for this hypothesis isnonexistent and other explanations can by no means be ruled out.

As indicated above, the termination reaction for most common monomers isbelieved to be segmental diffusion controlled at low and intermediate monomer


conversions and center of mass diffusion controlled at high conversion. If the dif-fusion controlled nature of the termination reaction is accepted, this has the im-mediate consequence that the termination rate coefficient should be chain lengthdependent. However, the chain length dependence of kt is markedly different forthe two diffusion mechanisms.

The chain length dependence of kt is normally quantified by the followingequations: A macroscopic—ie experimental—kt value may be obtained by the fol-lowing averaging procedure (335):

〈kt〉 =∑

i∑

jki, jt [Ri

•][Rj•](∑

i[Ri•]

)2(48)

The microscopic kt value kti,j corresponds to the individual termination rate

coefficient involving two free macroradicals with the chain length i and j. Theaveraging procedure is advantageous because the chain length distribution ofthe macroradicals in free radical polymerization is normally highly disperse. Theindividual termination rate coefficient kt

i,j may be described via the followingequation:

ki, jt = k0

t ( ¯i, j)− α (49)

with ¯i, j denoting some average [eg, the harmonic (336) or the geometric (337)mean] of the two chain lengths i and j involved and α being a positive constant. Asa consequence, the problem of determining kt is not solved by the evaluation of onespecific value but comprises the evaluation of an entire functional dependence. Thechain length dependence of the macroscopic 〈kt〉 is consequently often expressedvia the following power law:

〈kt〉 = k0t P− α

n (50)

This power law correlates the average value of the termination rate coeffi-cient 〈kt〉 with the number-average degree of polymerization, Pn. The chain lengthdependence of kt is now easily described and quantified by the exponent α in equa-tion 50.

If center of mass diffusion is the rate-determining step in the terminationreaction, an exponent of 0.5–0.6 results, depending on the thermodynamic qual-ity of the solvent. This result can be explained by considering the translationaldiffusion properties of the polymer coil. The diffusion coefficient D of the polymercoil is inversely proportional to its hydrodynamic radius. If the hydrodynamic ra-dius is equated with the radius of gyration 〈s2〉 0.5, it can be inferred that for atheta solution 〈s2〉 ∝ i and for an athermic solution 〈s2〉 ∝ i1.176 (338). This leadsdirectly to the chain length dependence of the diffusion coefficient of D ∝ i− 0.5 fortheta solutions and D ∝ i− 0.59 for athermic solutions. According to various theoriesfor diffusion controlled reactions, the diffusion coefficient is directly proportionalto the termination rate coefficient. The majority of these models is based on thework by Smoluchowski (339), which describes the rate coefficient for a diffusion


controlled reaction (see eq. 51):

ki, jt = 2pπ (Di + Dj)σNA (51)

in which p is the spin multiplicity, σ is the capture radius of the reaction, and NAis the Avogadro’s number.

The segmental diffusion coefficient is generally chain length independent.However, a slight chain length dependence of the termination rate coefficient ofclose to α = 0.16 is predicted for athermic solvents, owing to thermodynamicshielding effects of the chain. The chain shields its own reactive center owingto its size from the second chain, which displays a similar shielding effect. Theexponent of −0.167 has been confirmed in various theoretical studies applying thescaling theory (336,340), as well as in chain statistical simulations (341). The chainlength dependence of kt is almost negligible in theta solvents for the segmentaldiffusion controlled regime, with α being close to 0.05.

Experimentally, evidence has been gathered for both exponents, ie 0.1–0.2for segmental diffusion and 0.5–0.6 for center of mass diffusion, depending on thesize of the macroradicals undergoing termination. Consequently, the chain lengthdependence of the termination rate coefficient is best described by a step functionfeaturing two (main) domains of different chain length dependencies, as depictedin Figure 5.

Domain A is associated with center of mass diffusion controlled termination,owing to the small size of the macroradicals (ie, α ≈ 0.5–0.6), whereas domain Bis controlled by segmental diffusion processes (ie, α ≈ 0.05–0.16). Domain C rep-resents an intermediate region associated with the change in mechanism. Thesenumbers have been confirmed experimentally by various research groups in thepast (18,342–347) and the general shape of the curve could be confirmed in recentwork (348,349). Styrene and methyl methacrylate have been studied particularlyextensively with respect to the chain length dependence of the termination ratecoefficient in the segmental diffusion region. These chain length dependencies

Fig. 5. General step function for the termination rate coefficient as a function of the chainlength of terminating macroradicals. For details see text.


Table 7. Activation Parameters of the Termination Step for Various Acrylates andMethacrylates

EA, V#, log(kt),Monomer kJ · mol− 1 cm3 · mol− 1 L · mol− 1 · s− 1 Reference

Methyl acrylate 8.0 (100 MPa) 16.0 (30◦C) 8.13 350Butyl acrylate 6.0 (100 MPa) 16.0 (40◦C) 7.55 351Dodecyl acrylate 3.0 (100 MPa) 20.0 (40◦C) 6.38 350Methyl methacrylate 6.0 (100 MPa) 15.0a (40◦C) 7.39 352,353Butyl methacrylate — — 6.70 352,353Dodecyl methacrylate — 10.8 (30◦C) 6.22aThis activation volume is pressure dependent; the value is valid for the pressure range from 100 to150 MPa. To convert MPa to bar, multiply by 10.

may be described by the following (345):

MMA(25◦C

): 〈kt〉i = 7.56×107×i− 0.18 Styrene

(25◦C

): 〈kt〉i = 1.21×108×i− 0.16

where 〈kt〉i denotes the average experimental termination rate coefficient for amean chain length i.

Although the termination rate coefficient of a particular monomer is in-fluenced by the system viscosity and the chain lengths of the terminating freemacroradicals, it is important to notice that the pressure and, to a lesser extent,the reaction temperature have a significant effect on its value. Especially thereaction pressure influences the absolute value of the termination rate to a fargreater extent than the change in kt associated with its chain length dependence.Table 7 gives activation energies and activation volumes for selected acrylate andmethacrylate termination rate coefficients along with its value at 100 MPa and40◦C at low monomer conversions. The numbers given in Table 7 were mainlyderived from single pluse–pulsed laser polymerization (SP–PLP) experiments.

Inspection of Table 7 shows that the activation energies are rather low andbetween 3 and 8 kJ · mol− 1. This observation is consistent with the diffusion(either segmental or translational) controlled nature of the termination reaction.Strictly speaking, these activation energies correspond to the temperature depen-dence of the inverse system viscosity. Up to this date, there is no data on the ac-tivation energy of the actual termination reaction itself. However, it is very likelythat the termination reaction itself has a close to zero activation energy. This con-clusion may be deduced from the activation parameters observed for small radicaltermination (354,355). It is interesting to note that the activation volume of thetermination reaction is positive, ie the value of kt is reduced at higher reactionpressures. The activation volumes of a given reaction is defined by

d lnkdp

= − V#

RT(52)

where V# is the activation volume and p is the pressure.


Fig. 6. Termination rate coefficients in bulk methyl acrylate (MA) and dodecyl acrylate(DA) homopolymerizations as a function of monomer conversion. The reaction conditionswere 40◦C and 100 MPa.

This experimental observation can also be correlated with the pressure de-pendency of the viscosity. The reaction medium tends to be more viscous at higherreaction pressures, thus slowing the rate of termination; ie, the activation volumeof the termination rate coefficient is very close to the corresponding activationvolume that characterizes the pressure dependence of the inverse of the monomerviscosity (356). It is important to note that the pressure dependencies of the ter-mination and propagation rate coefficients display opposite behavior, ie allowingfor increased rates of polymerization at elevated pressures.

As has been indicated above, the termination rate coefficient is—dependingon the monomer in question—strongly dependent on the overall monomer con-version. The conversion dependencies of the (average) termination rate coefficienthave been reported for several monomers, with most measurements being donevia the SP–PLP technique, which allows to point-wise probe the kinetics of thepolymerization reaction up to high overall monomer conversions. A typical kt vsmonomer conversion dependence is given in Figure 6 for the example of methylacrylate and dodecyl acrylate. The data are taken from Reference 331.

Inspection of Figure 6 shows that both monomers behave very differently.While in the case of methyl acrylate different regions for the change of the termi-nation rate coefficient with conversion can be identified, dodecyl acrylate exhibitsa constant (average) kt value up to high conversions. The different regions in themethyl acrylate kt vs conversion dependence may be attributed to segmental diffu-sion (0 to ∼15%), followed by translational diffusion and the onset of the so-calledgel effect, which is characterized by a strong autoacceleration of the reaction (seebelow). At very high monomer conversions, the reaction medium is highly viscousand the macroradicals are no longer capable of movement via translational dif-fusion. Instead, the addition of new monomer units to the chain end results in


Fig. 7. Schematic dependence of the (average) termination rate coefficient on the overallmonomer conversion, indicating different regimes of reaction control.

a change in the position of the radical chain end. This process has been termedreaction diffusion or propagation diffusion. Buback has introduced a model for theabove-described dependence of the termination rate on the monomer conversion,which considers segmental, translational, and reaction diffusion processes (357).This model has been extremely successful in describing a large set of data up tohigh monomer conversions (358–360). In the case of dodecyl acrylate, the segmen-tal diffusion region seems to extend to high monomer conversions, although thesystem viscosity changes by orders of magnitude when going from 0 to 80% con-version. However, in a monomer behaving like dodecyl acrylate, it has not beenconclusively clarified up to this point (1) whether the entire conversion range is inthe segmental or translational diffusion controlled regime and (2) what the exactcause for a nonchanging (average) termination rate coefficient is.

Figure 7 shows the three different regimes of the termination rate coefficientin general, which can be described as follows:

(1) Low conversion: this regime—prior to the onset of the gel effect—is char-acterized by highly mobile propagating species. Segmental diffusion is therate-determining step for the termination reaction.

(2) Medium-to-high conversion: The diffusion mechanism of the propagatingradicals becomes complex after the onset of the gel effect. Large chains be-come immobile; however, the chain ends may move by reptation or reactiondiffusion. Monomer and short species may still be highly mobile in the poly-merizing system. Translational center-of-mass diffusion may become therate-determining step for radical–radical termination.

(3) Very high conversion: The polymerizing system becomes a glassy matrix.Reaction diffusion is the only remaining pathway for radical movement.


The precise conversion ranges are determined by the nature of the monomer,the molecular weight of the polymer that is produced, and the solvent. For bulkpolymerizations, these regimes are (a) typically <10–20%, (b) between 15 and90%, and (c) >90%. When using a solvent and/or chain transfer agents to reducethe molecular weight of the polymer, the initial regime (a) can be extended and(c) may not even occur.

It has been observed that the average kt value slightly increases up to 5–10%of conversion and then starts to decrease (361–364). An explanation for such be-havior was first offered by North and Reed who assumed faster segmental diffusionto be responsible (328). The initial increase of kt with increasing monomer conver-sion has also been associated with a change in solvent quality: kt increases whengoing from a good to a poor solvent, because of either diminishing coil sizes (whichincreases the segmental diffusion coefficient) or decreasing repulsions betweenmacroradicals. This behavior of kt has not only been observed experimentally, butis also predicted theoretically (365,366).

The conversion dependence of the termination rate coefficient is linkedto the above-mentioned autoacceleration effect of the polymerization (367,368),which can be observed with some monomers at increased conversion, eg, methylmethacrylate (369), styrene (370), vinyl acetate (371), and methyl acrylate (372).Also known as the ‘Trommsdorff,’ ‘Norrish–Smith,’ or ‘Norrish-Trommsdorff’ ef-fect, this effect can cause problems within both an industrial and scientific con-text ranging from a product mixture to reactor explosion, owing to its exothermicnature (373,374). It is important not to confuse the gel effect with the autoacceler-ation that is observed when a polymerization is carried out under nonisothermalconditions, so that the reaction temperature increases with increasing monomerconversion, owing to the exothermic nature of the polymerization reaction. Thegel effect is observed under isothermal reaction conditions. The cause of the geleffect has been debated over 50 years and various theories have emerged whichcan explain all or part of the experimental data (excellent reviews on the topiccan be found in References 375 and 376). In theory, any given model that linksthe termination rate coefficient to the increasingly difficult diffusion of macro-radicals through the reaction mixture as the monomer conversion increases andthus accounts for a decrease in the termination rate, is capable of explaining thegel effect, without introducing a drastic change in the physical chemistry of thepolymerizing system. The debate does not so much focus on the fact that the ter-mination reaction becomes increasingly hindered as the mobility of the polymerchains decreases, than on the details of exactly what type of mobility is important.The main explanation for the gel effect that is offered concentrates on the forma-tion of chain entanglements that hinders the diffusion of the macroradicals, thuscausing a decrease in the rate of termination.

Primary Radical Termination. Primary radical termination refers to thebimolecular reaction between a propagating radical and a primary, initiator-derived radical. This process leads to lower rates of initiation and propagation,thus causing a deviation from the predictions of classical kinetics (377). In anal-ogy with the termination reactions between propagating radicals, primary rad-ical termination can also occur by either combination or disproportionation. Ithas been shown that primary radical termination between cyanoisopropyl radi-cals and polystyryl radicals occurs mainly via combination at 98◦C with extremely


high AIBN concentrations, generating polymeric material with degree of polymer-izations of 10 and below (378). Primary radical termination in conventional AIBN-or benzoyl peroxide-initiated polymerizations occurs only to a significant extentwhen high initiator or low monomer concentrations are used. Hence, the effect ofprimary radical termination becomes more pronounced at high conversions andhigh dilutions (319,379–381). Even under experimental conditions that would fa-vor the primary radical termination reaction, more than 85% of the terminationevents occur between two macroradicals in AIBN-initiated styrene polymerizationat 100◦C (83). In conventional bulk polymerizations of the same system under con-ventional conditions (ie initiator concentration in the order of 10− 2 mol · L− 1) lessthan 2% of the end groups in the generated polymeric material are due to primaryradical termination events. When using BPO as initiator, approximately 8% ofthe end groups have been formed through transfer or termination with primaryradicals (85).

Many industrial photopolymerization processes exploit unusually high ini-tiation rates in order to achieve a rapid cure and a high final degree of conversion.These conditions favor primary radical termination reactions (380,382). Certainprimary radical fragments derived from photoinitiators exhibit a relatively highstability and show reluctance to initiate polymerization. This is often observedwith benzoin and benzoin ethers that fragment upon UV irradiation into an ini-tiating species (the benzoyl fragment) and an inhibiting species (the benzyl al-cohol fragment in the case of benzoin, and the acetal fragment in the case ofbenzoin ethers) (19,20,54,383). A manifestation of such a photoinitiator nonide-ality has been reported for 2,2-dimethoxy-2-phenylacetophenone- (DMPA-) andbenzoin-initiated SP–PLPs of various monomers (18). A decrease in monomerconversion per single laser pulse with increasing photoinitiator concentrationswas observed. MALDI-TOF investigations of benzoin and DMPA photoinitiatedpolymer show that the two radical fragments generated upon pulsed laser irra-diation show markedly different reactivity toward methyl methacrylate: whereasthe benzoyl fragment—common to both DMPA and benzoin—clearly participatesin the initiation process, the acetal and benzyl alcohol fragments cannot be iden-tified as end groups in the polymer (384). Another example of how comparativelylow rates of addition of primary radicals to monomer may lead to higher ratesof primary radical termination is the polymerization of hindered monomers, eg,fumarates and itaconates, initiated by AIBN (379,385).

Rate of Polymerization

Stationary Polymerization. A stationary polymerization system is char-acterized by a constant free radical concentration as given by equation 53:

d[R•]dt

= 0 (53)

In the past it has been a standard practice to derive a simple but generalexpression for the rate of polymerization, Rp. This expression correlates the rate ofpolymerization with the initiator and monomer concentrations on one side and the


kinetic rate coefficients kd, kp, and kt on the other side. Although this expressioncontains many approximations, it is surprisingly successful in describing the ex-perimental reality. Detailed kinetic studies have revealed the shortcomings of thisequation. However, it remains the basis of standard free radical polymerizationkinetics. The approximations that are made in its derivation are

(1) chain length and conversion independent rate coefficients kt and kp;(2) instantaneous establishment of a steady-state free radical concentration;(3) assumption that monomer is only consumed by chain propagation and not

via the initiation process or chain transfer. This assumption allows to equatethe rate of the loss of monomer with the rate of polymerization;

(4) all reactions are irreversible;(5) the effective concentration of initiator-derived free radicals is constant

throughout the polymerization.

The core of the derivation for the rate of polymerization expression is theassumption that the rate of initiation equals the rate of termination, equation 46in its simplified form. This assumption is mandatory for the establishment of aconstant free radical concentration (eq. 54).

2f kd[I] = 2kt[R•][R•] (54)

Rearrangement of equation 54 and insertion into the simplified form of equa-tion 21

Rp = − d[M]dt

= kp[M][R•] (55)

yields the final expression for the rate of polymerization, Rp:

Rp = − d[M]dt

= kp

(f

kd

kt

)0.5

[M][I]0.5 (56)

Equation 56 indicates a first-order dependence of the rate of polymerizationon the monomer concentration and a square-root dependence on the concentrationof the initiator. These dependencies have been confirmed for the example of manypolymerizing systems. It should be pointed out that deviations from equation 56(such as chain-length-dependent rate coefficients or primary radical termina-tion) are manifest in a change in the exponents associated with the initiator andmonomer concentrations (386,387). The rate of polymerization will scale with aweaker than square-root dependence on [I] and a stronger than linear dependenceon [M]. Extreme dilution of monomer can also change the exponents of monomerand initiator concentration. Equation 56 is easily integrated to yield an expressionwhich directly correlates the monomer conversion with the observed kinetic ratecoefficient kobs.


ln(

11 − p

)= kobst where kobs = kp

(f

kd

kt[I]

)0.5(57)

where p is the conversion of monomer to polymer. The temperature dependenceof the polymerization rate is given by the temperature dependencies of the indi-vidual rate coefficients. Each rate coefficient follows its own Arrhenius law, k =A exp(−EA/RT), where A is the preexponential factor and EA denotes the acti-vation energy. The overall activation energy of the rate of polymerization, ERp

A ,equals the sum of the weighted activation energies of the elementary reactions,propagation (E p

A), initiation (E iA), and termination (E t

A).

ERp

A = EpA + 1

2Ei

A − 12

EtA (58)

Activation energies for commonly used thermal decomposing initiators, E iA,

are in the order of 120–150 kJ · mol− 1. The E pA values for most common monomers

lie within the range of 20–40 kJ · mol− 1, and E tA is generally in the range of

4–10 kJ · mol− 1. Hence, typical values for overall activation energies for therate of polymerization initiated by a thermally decomposing initiator are close to80 kJ · mol− 1. This corresponds to a two- or threefold increase in rate for a 10◦Ctemperature increase. Photochemical polymerization rates have a much lower ac-tivation energy of about 20 kJ · mol− 1, according to close to zero activation energyof the photoinitiation process.

Dead-End Polymerization. To reach a steady state in free radical poly-merization it is important that the initiator concentration is constant over a sig-nificant time span to ensure a constant rate of initiation. However, the initiatordecomposes according to equation 3 and its concentration unavoidably decreases.This can often be neglected when the decomposition rate of the initiator is verysmall in comparison with the rate of polymerization. To perform a steady-stateexperiment an appropriate initiator should be chosen with a rate coefficient ofdecomposition, kd (at the given temperature), that ensures a maximum decreasein initiator concentration of no more than 10% over the entire reaction time. Es-pecially for fast decomposing initiators or very long reaction times the decreasinginitiator concentration has to be accounted for. If all initiator molecules are de-composed before the end of polymerization, the reaction ceases. This experimen-tal conditions are referred to as dead-end polymerization (388–391). However, thepolymerization can be reinitiated by adding new initiator. Insertion of equation 3in equation 56, separation of the variables, and integration from [M]0 to [M] andfrom t = 0 to t = t yields

− ln[M][M]0

= − ln(1 − p) = 2kp

(f [I]0

kdkt

)0.5

(1 − e− kdt/2) (59)

where p is the monomer-to-polymer conversion and is defined by p =([M]0−[M])/[M]0. At long reaction times (t → ∞), the monomer concentration andconversion reach the limiting values of [M]∞ and p∞.


− ln[M]∞[M]0

= − ln(1 − p∞) = 2kp

(f [I]0

kdkt

)0.5

(60)

The example for polymerizations of styrene using AIBN at 60◦C shows howthe maximum conversion depends on the concentration of the initiator (kp =341 L · mol− 1 · s− 1, kt = 5 × 107 L · mol− 1 · s− 1, kd = 1.35 · 10− 5 s− 1, f = 0.5):

[AIBN] = 0.001 mol · L− 1 p∞ = 44.4%[AIBN] = 0.01 mol · L− 1 p∞ = 84.4%[AIBN] = 0.1 mol · L− 1 p∞ = 99.7%

A higher initiator concentration results in a higher polymer yield. However,the molecular weight of the polymer produced decreases at the same time.

Nonstationary (Instationary) Polymerization. Especially at the begin-ning and at the end of the polymerization process the steady-state principle doesnot hold. The generation of new free radicals by the initiator decay exceeds theirconsumption via termination events at early reaction times (preeffect) (392). Afterthe initiation process ceases, the free radical concentration decreases according tothe termination rate law expression (aftereffect) (393). Consequently, the rate ofpolymerization is not constant over the entire time period of the polymerization(394–396). Measurement of the polymerization rate in the pre- and aftereffectregions allows for the determination of the coupled form of termination and prop-agation rate coefficients, kp/kt, whereas in the steady-state region, they can onlybe accessed as the ratio kp

2/kt. The individual propagation and termination ratecoefficients can be calculated by combining the above two ratios. The determina-tion of kp/kt and kp

2/kt via the pre- and aftereffects and stationary polymerizationexperiments, respectively, has long been the only possibility to determine propaga-tion and termination rate coefficients. However, recent studies reveal that seriousproblems were associated with this approach, owing to the chain length depen-dence of kt, (397), which was not accounted for. This may be the reason for thelarge discrepancies observed when comparing values for kp and kt from differentsources. The situation has improved drastically by the invention of the pulsedlaser polymerization (PLP) method, which has been introduced in the late 1980s.This improvement is impressively demonstrated when comparing recent data forthe propagation rate coefficient in styrene homopolymerizations with earlier data(see Figure 8).

Although the steady state is reached within a couple of seconds after startingthe initiation process, the generation of free radicals exceeds their loss by termi-nation (ie the preeffect) in this period. Therefore, the concentration of free radicals[R•] is not constant but a function of time. The rate of free radical production isthe rate of initiation minus the rate of termination, as given by equation 61:

d[R•]dt

= Ri − 2kt[R•]2 (61)

Assuming a constant rate of initiation, integration leads to

[R•] = [R•]Stanh[(2ktRi)

0.5t]

(62)


Fig. 8. Comparison of propagation rate coefficients for bulk styrene homopolymerizationsfrom two generations: PLP–SEC data from an IUPAC benchmark publication (Ref. 398 ( �),and data compiled from Ref. 399 ( �).

where [R•]S represents the steady-state free radical concentration. The examplefor polymerizations of styrene using 10− 3 mol · L− 1 AIBN at 60◦C shows how fastthe steady state is reached: kp = 341 L · mol− 1 · s− 1, kt = 5 · 107 L · mol− 1 · s− 1,kd = 1.35 · 10− 5 s− 1, f = 0.5 (see Figure 9). The polymerization can be consideredto proceed in a steady state after approximately 3 s.

After stopping the initiation process no new free radicals are available andthe concentration decreases according to a second-order rate law. Integration of

Fig. 9. Increase of the radical concentration in a conventional AIBN-initiated bulk poly-merization of styrene. For kinetic parameters see text.


equation 61 with Ri = 0 yields the concentration of free radicals as a function oftime in the so-called aftereffect region:

[R•]− 1 − [R•]− 1S = 2ktt (63)

Insertion into the rate law of propagation (eq. 54)—assuming that the rateof loss of monomer equals the rate of polymerization, Rp—gives

Rp = kp[M]

2ktt + [R•]− 1S

(64)

which contains the individual rate coefficients kp and kt in a combination differentfrom that obtained via steady-state experiments. Thus, the ratio kp/kt is accessibleby equation 64 via the measurement of the polymerization rate in the aftereffectregion.

Pseudostationary Polymerization. Conversions in the pre- and after–effect regions are rather low, making it experimentally challenging to determinethem accurately. To overcome this problem a structured continuous initiation pro-file is chosen by which the system is facing pre- and aftereffects in sequences.This can be achieved by using a photo-polymerizable system, which is exposedto a succession of light and dark periods, leading to a pseudostationary state,which provides a continuous polymerization of a system being in a non-steadystate (400,401). It is characterized by a constant mean free radical concentration,averaged over one cycle period, but a permanently changing actual concentra-tion of the reactive intermediates. While the technique was introduced by usinga rotating sector (402–404)—where light periods are considerably long, ie about1/4 of the whole cycle time t0—the technique improved by using a pulsed laseras light source (51,405–407). The extremely short duration of the laser flash—in the order of nanoseconds—allows to assume an instantaneous formation offree radicals. Strictly speaking, instantaneous radical generation is equivalentto neglecting the preeffect and termination during the laser pulse, respectively.Figure 10 shows a typical time profile of the free radical concentration in a pulsedlaser experiment.

The maximum free radical concentration [R+•] is reached immediately after

the laser pulse has been applied. It is the sum of the radical concentration formedat each laser flash, ρ, and the amount of free radicals still in the system, whichare produced by former pulses, [R−

•]. This value is identical to the minimum freeradical concentration in the polymerizing system. Radical formation during thedark time is neglected. Assuming that termination is not dependent on the chainlength, the rate law for termination can be written as

− d[R•]dt

= 2kt[R•]2 (65)

Integration of equation 65 over the whole pulse period t0 yields

[R−•]− 1 − [R+

•]− 1 = 2ktt0 (66)


Fig. 10. Typical radical concentration profile generated by the pulsing action of laser on areaction mixture of photoinitiator and monomer. The maximum free radical concentration[R+

•] is reached immediately after the laser pulse has been applied. It is the sum of theradical concentration formed at each laser flash, ρ, and the amount of free radicals still inthe system produced by former pulses, [R−

•].

The pseudostationary state is defined by the exact compensation of the lossof free radicals during the dark period via termination by the radical formation ofthe laser pulse.

ρ = [R+•] − [R−

•] (67)

Combination of equations 66 and 67 leads to

[R+•] = ρ

[12

±(

14

+ 12ρktt0

)0.5 ](68)

The free radical concentration at any moment during the pulse period is

[R•] =(

1[R+

•]+ 2ktt

)− 1

= [R+•]

1 + 2[R+•]ktt

(69)

Averaging over the whole pulse period leads to a mean value for the freeradical concentration

[R•] = 1t0

∫ t0

0

[R+•]

1 + 2[R+•]ktt

dt = 12ktt0

ln(1 + 2[R+•]ktt0) (70)

By insertion of equations 68 and 70 in equation 55, the expression for thepolymerization rate in pseudostationary PLPs is obtained.


Rp = kp[M]2ktt0

ln

{1 + ρktt0

[1 +

(1 + 2

ρktt0

)0.5]}

(71)

This expression contains the ratio of kp/kt, which is typically derived fromnonstationary polymerizations. The measurement of the rate of polymerizationunder pseudostationary conditions may hence allow for the determination of theratio of kp/kt (408). However, the major advantage of pseudostationary polymer-ization does not lie within the possibility to separate the ratio of k 2

p/kt into its indi-vidual components, but to generate structured chain length distributions. Thesestructured distributions allow for the accurate determination of the propagationand termination rate coefficients.

A special case of the PLP is the SP–PLP technique (409,410). The equationsneeded to analyze the monomer conversion vs time traces obtained from SP–PLPexperiments are easily derived. Integration of the rate law expression for termina-tion, assuming a chain length independent, average termination rate coefficientkt (eq. 65), yields

[R•] =(

2ktt + 1[R+

•]

)− 1

(72)

Substitution of equation 72 into the rate law expression for the propagationstep (eq. 55) and subsequent integration yields the change in relative monomerconcentration after a single laser pulse:

[M][M]0

= (2kt[R+•]t + 1)− kp/2kt (73)

Expression 73 can be fitted to time-resolved monomer conversion vs timetraces obtained from SP–PLP experiments (see below), with the two fit variablesbeing kt[R+

•] and kp/kt. With knowledge of the primary free radical concentra-tion [R+

•], it is possible to determine kt and kp from a single conversion vs timetrace. Unfortunately, in reality, [R+

•] is often not known (because of insufficientknowledge about initiator efficiencies) and kt can only be assessed via knowledgeof the propagation rate coefficient from independent experiments (eg, PLP–SEC).However, in recent studies the accurate determination of primary radical concen-trations produced by a laser flash have been reported in nonpolymerizing systems(411,412), giving promising results for future work.

The Chain Length Distribution

The kinetic rate coefficients of the various steps involved in the polymerizationreaction are controlling the rate of polymerization, Rp, and the overall free radicalconcentration. Keeping in mind that the polymerization is a chain mechanismleading to macromolecules, it is self-evident that the same kinetic parametersmay be employed to calculate the sizes of polymeric intermediates and thepolymer generated. For this purpose it is necessary to solve the complete set of


coupled differential equations, one equation for each chain length plus one forthe initiation step.

The rate law for the concentration change of a macroradical with chain lengthi can be written as

d[Ri•]

dt= ki − 1

p [M][Ri − 1•] −

(ki

p[M] + kMtr [M] +

∑k

kTktr [Tk] + 2

∞∑j = 1

ki, jt [Rj

•]

)[Ri

•]

(74)

This rate law reflects a polymerization process during which a free radicalwith chain length i is solely formed by a propagation step from the free radicalwith chain length i − 1. The free macroradical has then the possibility to groweither by reaction with a monomer molecule M, giving a chain length of i + 1, orit can undergo a transfer reaction with the monomer or any other transferringmolecule Tk (eg, transfer agent, solvent, initiator, polymer). Alternatively, it maybe terminated by another free radical. The analytical solution of this problemis obviously very difficult. However, numerical solutions of this set of equationsbecame possible because the rapid increase in computer power and simulations ofchain length distributions of polymers are of increasing importance for academicand industrial applications (413).

The chain length distribution of a polymer is defined as the fraction ofmolecules xP that contains P basic monomer units. It should be noted that thedegree of polymerization P is equivalent to the chain length i. The living macro-radicals by which the dead polymer is generated through any chain-stopping eventshow a chain length distribution, too. Both distributions are closely related to eachother and the chain length distribution of the dead polymer can be calculated viathe derivative of the distribution of the living macroradicals.

Like any other distribution function, the chain length distribution is de-scribed by its statistical moments, which are defined as

m(k) =∞∑

P = 1

PkxP (75)

By combination of such moments one can easily calculate mean values for thedegree of polymerization, P, which characterize the chain length distribution. Thedistribution is only fully described if all moments are known. However, in practicethere are two mean values calculated by the first three statistical moments, whichare extensively used: the number-average degree of polymerization, Pn, and theweight-average degree of polymerization, Pw.

Pn = m(1)

m(0)=

∑∞P = 1PxP∑∞P = 1xP

Pw = m(2)

m(1)=

∑∞P = 1P2xP∑∞P = 1PxP

(76)

Stationary Polymerization.Average Degree of Polymerization. To calculate the number-average de-

gree of polymerization, Pn, of a polymer produced by a steady-state polymerization,


it is mandatory to know how many propagation steps occur before the chain reac-tion is stopped. A distinction has to be made between the term ‘chain’ used in amolecular sense and ‘chain’ used as a kinetic concept. The kinetic chain length ν,assuming that every radical I• initiates polymerization, is defined as

ν = Total number of polymerized monomer unitTotal number of initiation step

=∫ t

0

d[M]dt

dt/∫ t

0

d[M]dt

dt

(77)

In a system that has reached the steady state, the integrands of equation 77may be constant over a significant time span. By substituting equations 4and 21—with the assumption of chain-length-independent rate coefficients—equation 77 can therefore be rewritten as

ν = Rp

Rd= kp[R•][M]

2f kd[I](78)

Elimination of [R•] by means of equation 54 leads to an expression for thekinetic chain length ν that shows the dependence of the different kinetic parame-ters. One important characteristic of the free radical polymerization is hereby wellillustrated: The sizes of the macromolecules produced are inversely proportionalto the square root of initiator concentration. Increasing the initiator concentrationleads to smaller size polymer molecules.

ν = kp[M]

2(f kdkt[I]

)0.5(79)

Disregarding any transfer effect as a first approximation correlates the ki-netic chain length with the number-average degree of polymerization, Pn. In thecase of termination by disproportionation one polymer molecule is produced perevery kinetic chain

Pn = ν (80)

Termination by combination leads to one polymer molecule per two kineticchains, reflecting the combination mechanism.

Pn = 2ν (81)

Any mixture of both mechanisms can be described by using the value δ (seeeq. 45), the contribution of disproportionation to the overall termination process.

Pn = 21 + δ

ν (82)

The mean kinetic chain length can be experimentally determined by usingmarked initiator molecules, eg, 14C radiolabeled (414). By these means the number


of initiator fragments per weight of polymer can be measured, and therefore thenumber of monomer units polymerized by each initiator step may be calculated.This represents one method to determine the mode of termination. With the valuesfor the AIBN-initiated styrene polymerization (10− 3 mol · L− 1 AIBN at 60◦C, kp =341 L · mol− 1 · s− 1, kt = 5 × 107 L · mol− 1 · s− 1, kd = 1.35 × 10− 5 s− 1, f = 0.5)the average kinetic chain length is about 3500 and it will take approximately 1.2 sto be finished.

Without changing the free radical concentration, ‘normal’ chain transfer pro-cesses remain hidden in any experiment measuring the rate of polymerizationalone. The kinetic chain length is also unaffected by the transfer process, becausethe growing free radical center generated by the initiation step stays alive afterany chain transfer event, although more than one polymer chain is produced. Forthis reason, equation 82 does not hold true any longer if chain transfer reactionsare taken into account. In many kinetic measurements based on the analysis ofthe molecular weight distributions, transfer processes are neglected and are seenas disturbing factors. However, determination of the chain length distribution of apolymer remains the only possibility to measure the rate coefficients for transferprocesses.

Taking chain transfer into account, the number-average degree of polymer-ization, Pn, can be described as

Pn = Total number of polymerized monomer unitsHalf the number of formed end groups

(83)

The various reactions within the polymerization process generate differentamounts of end groups per initiation step:

Initiation 1 end groupPropagation 0 end groupsTransfer 2 end groupsTermination by disproportionation 1 end groupTermination by combination 0 end groups

Again the steady state with its general approximations is assumed in whichthe concentrations of the reactants, such as the monomer, free radicals, and trans-fer agent, do not vary with time. Hence, in equation 83 the number of polymerizedmonomer units can be substituted with the rate of polymerization and the num-bers of end groups by the rate of their formation.

Pn = Rp

(1/2)(Ri + Rt, d + 2Rtr)(84)

Insertion of the simplified rate laws of the different processes

Initiation Ri = 2f kd[I] = 2(kt, d + kt, c)[R•]2 (85)


Propagation Rp = kp[M][R•] (86)

Termination(disproportionation

)Rt,d = 2kt, d[R•]2 (87)

Chain transf er Rtr = ∑k

kTktr [Tk][R•] + kM

tr [M][R•] (88)

and subsequent inversion leads to

1

Pn= 2kt,d + kt,c

k2p[M]2

Rp + kMtr

kp+

∑k

kTktr

kp

[Tk][M]

(89)

[Tk] is the concentration of any molecule that is capable of taking part ina chain transfer reaction, including solvent S, initiator I, polymer P, and addedchain transfer agent T. It is usual to define chain transfer coefficients C for thedifferent molecules:

CM = kMtr

kpCS = kS

tr

kpCI = kI

tr

kpCP = kP

tr

kpCT = kT

tr

kp(90)

Thus, equation 89 becomes

1

Pn= 2kt,d + kt,c

k2p[M]2

Rp + CM + CS[S][M]

+ CI[I][M]

+ CP[P][M]

+ CT[T][M]

(91)

This equation gives the fundamental correlation of the number-average de-gree of polymerization with the rate of polymerization and the various chain trans-fer coefficients. Equation 91 constitutes the basis for determining the various chaintransfer coefficients.

Performing a polymerization experiment with only low conversion ofmonomer to polymer, the concentration of polymer is often too low to show signifi-cant chain transfer. The same holds true for the initiator, which is mainly used inthe range of low concentrations. Without addition of solvent and additional chaintransfer agent, equation 91 reads after introduction of equation 45

1

Pn= (1 + δ)kt

k2p[M]2

Rp + CM (92)

Hence, a plot of the inverse number-average degree of polymerization, Pnagainst the rate of polymerization, Rp, (the rate of polymerization can easily bevaried by the concentration of the initiator) yields the monomer chain transfercoefficient CM as intercept and the ratio (1 + δ)kt/(k 2

p[M]2) as slope of a linear plot.The value of CM entails an inevitable limit for the maximum number-averagedegree of polymerization, Pmax

n . The value for Pn is increased by lowering therate of polymerization Rp according to equation 92. The limit is reached when Rpbecomes zero.


limRp→0

(1

Pn

)= CM (93)

Hence, the maximum number-average degree of polymerization, Pmaxn , which

is feasible is given by

Pmaxn = C− 1

M (94)

Methyl methacrylate, for instance, has a monomer chain transfer coefficientof about CM = 5 × 10− 5 at 60◦C, leading to a maximal mean chain length of about20,000, whereas in a free radical polymerization of vinyl acetate with a monomerchain transfer coefficient of CM = 2 × 10− 4 at 60◦C, the limit is already reachedat a number-average degree of polymerization of 5000.

The Full Chain Length Distribution. So far, only the average degree of poly-merization has been considered. To calculate the distribution function itself for asteady-state polymerization it is convenient to choose a statistical approach basedon kinetic parameters. A probability factor α of propagation is defined as the prob-ability that a radical will propagate rather than terminate. The factor α is the ratioof the rate of propagation over the sum of the rates of all possible reactions themacroradical can undergo.

α = Rp

Rp + Rtr + Rt(95)

Firstly, we assume that termination is solely by disproportionation and thatthe propagation probability factor is equal for each chain length. The probabilityfor the occurrence of a polymer chain—hence its distribution function—with thelength P is given by the probability of P − 1 propagation steps and the probabilityof one chain-stopping event (termination or transfer).

xP, d = αP − 1(1 −α) (96)

The molecular weight averages can be evaluated by calculating the momentsof this distribution function by insertion of equation 96 into equation 75,

m(0) =∞∑

P = 1

xP, d = 1 (97)

m(1) =∞∑

P = 1

PxP,d = (1 −α)− 1 (98)

m(2) =∞∑

P = 1

P2xP,d = (1 + α)(1 − α)− 2 (99)

and subsequent insertion into equation 76.


Pn,d = m(1)/m(0) = (1 − α)− 1 Pw,d = m(2)/m(1) = (1 + α)(1 − α)− 1 (100)

The ratio of the weight-average and the number-average degree of poly-merization, Pw/Pn, describes the polydispersity of a chain length distribution.It becomes unity if all chains have the same length—called a monodispersedistribution—and values greater than one, if the distribution exhibits a broadershape.

Pw,d

Pn,d= m(2)m(0)

m(1)m(1)= 1 + α (101)

It should be noted that the propagation step must be highly favored overchain-stopping events to produce polymer with a significant chain length andthe value of α must be near to one. Hence, equation 101 shows that for a chainlength distribution of a polymer produced in a stationary experiment, where chain-stopping events are termination by disproportionation or transfer, the polydisper-sity becomes nearly 2. This holds true for chain length distributions which arecontrolled by termination via disproportionation and also for distributions wherechain transfer is the dominant chain-stopping event. It should be kept in mindthat this theoretical value for the polydispersity is valid for an instantaneouschain length distribution. The polydispersities in real polymerizing systems arealways higher than the theoretical value, because of changes in the concentrationof the participating compounds (ie initiator, monomer, transfer agent), which leadto an overlay of a multitude of different chain length distributions.

Expressions may also be derived for the chain length distribution produced,when the termination process is by combination. The expression for the proba-bility of the occurrence of a chain with the chain length P is now given by thecontributions of two chains with the chain length n and m, which form the desiredmolecule by combination. Hence, the auxiliary condition n + m = P must be valid:

xP, c =P − 1∑n = 1

αn − 1(1 − α)αm − 1(1 − α) = (P − 1)αP − 2(1 − α)2 (102)

Evaluating the moments of this distribution function by insertion of equation102 into equation 75 as above

m(0) =∞∑

P = 1

xP, c = 1 (103)

m(1) =∞∑

P = 1

PxP, c = 2(1 − α)− 1 (104)

m(2) =∞∑

P = 1

P2xP, c = (4 + 2α)(1 − α)− 2 (105)


leads to

Pn,c = m(1)/m(0) = 2(1 − α)− 1 Pw,c = m(2)/m(1) = (2 + α)(1 − α)− 1 (106)

The breadth of the distribution is therefore given by

Pw,c

Pn,c= m(2)m(0)

m(1)m(1)= 1 + α

2(107)

Keeping in mind that α has a value close to one, equation 107 leads to apolydispersity of 1.5 for a polymer produced in a polymerization process wheretermination is by combination. The corresponding chain length distribution issomewhat narrower than that generated by disproportionation, because of thestatistical coupling of two chains with different sizes.

Almost every polymerization system shows both disproportionation and com-bination modes. In order to combine the two modes the general expression forthe polydispersity of any given termination-controlled chain length distributionsreads

Pw

Pn= 1

2(3 − δ)(1 + δ) (108)

Because the value of α is close two one, the expression ln(α) ≈ α − 1 does holdtrue, leading to α ≈ exp[−(1 − α)]. With this correlation in mind the combinationof equation 96 with the l.h.s of equation 100 gives

xP = 1

PnαP − 1 ≈ 1

Pnexp

[− (P − 1)

Pn

]≈ 1

Pnexp

(− P

Pn

)(109)

with the factor (P − 1) substituted by P, because the chain length P is assumedto be much larger than 1, ie P � 1. Equation 109 demonstrates that the chainlength distribution of the polymer formed by disproportionation or chain transferfollows an exponential function in the limit of infinite chain length.

The same calculation procedure, starting with equation 102, also leads toan exponential expression for the chain length distribution for termination bycombination.

xP = 4P

P2n

exp(

− 2P

Pn

)(110)

However, equation 110 has the independent variable, the chain length P,in the preexponential factor, giving the chain length distribution of the polymerformed by combination a different shape.

Evaluation of equation 95 immediately leads to

α = kp[M]

kp[M] + 2(f kdkt[I])0.5 + ktr[T]

(111)


All derived distribution functions and average degrees of polymerization maynow be expressed via the kinetic coefficients.

Nonstationary (Instationary) Polymerization. Calculation of chainlength distributions of a polymer formed via a nonstationary polymerization (alsosometimes denoted as instationary polymerization) is reasonable only if the non-stationary process is not linked to another polymerization process, eg, the pre- andaftereffect of a steady-state polymerization. One possibility to perform such an un-coupled nonstationary polymerization is the so-called single pulse experiment. Asingle laser pulse produces a free radical concentration ρ; the polymerization pro-cess is started, and no new free radicals are generated. The kinetic equations forsuch an experiment can be solved analytically, even if a chain-length-dependenttermination rate coefficient is assumed. This is the case, because in such an ex-periment all macroradicals have the same chain length—within a narrow Poissondistribution and neglecting chain transfer—at any given time.

The free radical concentration formed by the laser pulse, ρ, decays accordingto the termination rate law expression.

− d[R•]dt

= 2kt[R•]2 (112)

Assuming a perfect correlation of time and the degree of polymerization,P = kp[M]t, the average frequency of the propagation steps, and the postulatedlaw for the chain length dependence of kt, kt = k 0

t P− α, equation 112 can be writtenas

− d[R•]dt

= 2k0t (kp[M])− αt− α[R•]2 (113)

Solving this differential equation yields the concentration of free radicals asfunction of time, where ρ is the free radical concentration at t = 0:

[R•]t ={

1ρ

+ 2k0t (kp[M])− α

1 − αt1 − α

}− 1

(114)

The termination process by disproportionation transforms the living macro-radicals into dead polymer chains with exactly the same chain length. This trans-formation process can be written as

− d[R•]t

dt= [PP]kp[M] (115)

because the loss of free radicals must equal the generation of dead polymer. Thefactor kp[M] allows for the transformation of time to chain length. The concentra-tion of dead polymer with chain length P, [PP], is the chain length distribution interms of concentrations. Division of [PP] by the free radical concentration at thebeginning, ρ, according to xP,d = [PP]/ρ with

∑[PP] = ρ), insertion of equations

113, 114, and 115, and rearrangement yields the number chain length distributionfor a single pulse experiment with termination by disproportionation.


xP,d = 2k0t ρ

kp[M]P− α

{1 + 2k0

t ρ

(1 − α)kp[M]P1 − α

}− 2

(116)

The combination process gives a dead polymer chain with exactly doublethe chain length of the living macroradical, because all macroradicals have thesame size at any given time. The transformation process from time to chain lengththerefore reads

− d[R•]t

dt= 4[P2P]kp[M] (117)

Because two free radicals are leading to one dead polymer,∑

[P2P] = ρ/2must hold true. With this in mind and insertion of equations 113, 114, and 117and subsequent rearrangement gives the number chain length distribution for asingle pulse experiment with termination by combination.

xP, c = 2k0t ρ

4kp[M]

(P2

)− α{

1 + 2k0t ρ

(1 − α)kp[M]

(P2

)1 − α}− 2

(118)

According to the different termination modes the overall number chainlength distribution can be calculated via

xP = 21 + δ

[δxP,d + (1 − δ)xP,c] (119)

with δ being the contribution of disproportionation to the overall terminationprocess.

Pseudostationary Polymerization. Throughout many decades thepseudostationary polymerization was carried out—mainly with a rotating sector—to measure a different ratio of the individual rate coefficients kp and kt as thatobtained from steady-state experiments. The rate of polymerization was the onlymeasured value, and the chain length distribution of the polymer produced dur-ing such an experiment was not evaluated, mainly because of the lack of suitableanalytical techniques. The invention and improvement of SEC paved the way fordetailed investigations of the chain length distribution formed throughout a pseu-dostationary polymerization experiment (415). This improvement eventually leadto the invention of the PLP–SEC method (416), which turned out to be the bestimprovement in polymerization kinetic measurements long since. This techniqueallows for the direct measurement of the individual propagation rate coefficientkp (133,417).

The principle of the pulsed laser technique is simple but ingenious. Amonomer/photoinitiator mixture is irradiated by a pulsed laser beam. Each laserflash generates free radicals which initiate a polymerization process. No newfree radicals are formed during the dark time periods. All growing macroradicalsformed by one specific laser pulse have the same chain length within a narrowPoisson distribution. As the free radical concentration decreases owing to termi-nation processes, the rate of termination decreases according to equation 46, too.


Fig. 11. Simulated and normalized number distribution for a pulsed laser experiment,with termination exclusively via disproportionation (a) without and (b) with Poissonbroadening.

After the dark period, t0, the next laser flash irradiates the system and instanta-neously increases the free radical concentration. Hence, the termination rate issuddenly highly increased, too, leading to a significant amount of dead polymerwith the chain length L0 being the chain length of a macroradical that grew forone dark time period, t0. Taking into account that the radicals have a certain prob-ability to survive the laser flash and to terminate at any later pulse, the relativeconcentration of polymer with the chain length 2L0, 3L0, and so on, is increased,too. The described polymerization conditions therefore produce a well-structuredchain length distribution with additional peaks at the chain length of L0 and itsmultiples (see Figure 11). The radicals which are formed at the laser pulse arevery small. The mode of termination is therefore not overly important for theformation of the additional peaks of the chain length distribution.

Assuming a low conversion of monomer to polymer, the monomer concen-tration can be expected to be constant. The propagation rate coefficient and themonomer concentration can therefore be combined into a new rate coefficient,kp = kp[M], which is associated with the following first-order rate law:

Rp = kp[R•] with kp = kp[M] (120)

The average time span between two first-order reaction steps, τ , is given by

τ = k− 1 (121)

with k being the first-order rate coefficient. Insertion of equation 120 intoequation 121 yields the time of an average propagation step τp, assuming themonomer concentration is constant.

τp = 1

kp= 1

kp[M](122)

The chain length L0,n of a macroradical that grows n laser periods, nt0, isnow easily correlated with the propagation rate coefficient kp.


L0, n = nkp[M]t0 (123)

The evaluation of the additional peaks occurring in the chain length distri-bution of a PLP experiment and the corresponding values of L0,n by means of SECenables the calculation of kp, because the monomer concentration and the laserfrequency are known.

Because of the statistical process of the chain growth, the macroradicalsproduced by the same laser pulse do not exhibit the same chain length at anygiven time, but rather show a narrow Poisson distribution with L being the meanvalue.

xP = e− L LP

P!(124)

The theoretical chain length distribution is therefore subject to a broadening,losing its discontinuities (see Figure 11b). It turned out that the inflection pointon the low molecular weight side of the additional peak is in most cases the bestmeasure for the real value of L0 (418). Only at the so-called high termination limit,where the free radical concentration produced by each laser pulse is extremelyhigh, the maximum of the additional peak may be a better choice (419). However,second or higher points of inflection can often be evaluated, even when there is nopeak maximum visible. In addition, because of being a point of second order, theinflection point is less affected by baseline errors occurring during the SEC.

Analytical solutions for the chain length distribution generated by PLP (415,416,420,421) are very complex, even in their simplest form, assuming no Poissondistribution, no transfer, and no initiation process during the dark time period.The chain length distribution for assuming a strict coupling of time and degree ofpolymerization for termination by disproportionation can be written as

x(n)P, d = ρ

{CL0

[1 + C

L0(P − nL0)

]− 2

(1 + C)− n

}(125)

where

0 ≤ P ≤ L0 for n = 0

nL0 < P ≤ (n + 1)L0 for n = 1, 2, 3. . .

with the definitions

C = C[R+

•]ρ

(126)

C = 2ρktt0 (127)

L0 = kp[M]t0 (128)


Especially the value C is of importance, because it governs the overall shapeof the chain length distribution. A typical value of C is 0.5–10, whereas highervalues of C yield more pronounced first additional peaks (high termination limit).The shape of a PLP distribution can be (in principle) held constant by balancingthe product of the laser period t0 and the free radical concentration produced byeach laser pulse, ρ, to give constant values of C. It should be noted that the extrapeaks are sharper greater the value for L0.

The expression for the chain length distribution for termination by combi-nation reads

x(n)P,c = ρ

[n(1 + C)− n + 1 xP − (n − 1)L0, c

ρ+ (n + 1)(1 + C)− n xP − nL0, c

ρ

](129)

with

xP,c = ρ

2C

2L0

(1 + C

P2L0

)− 2

(130)

The statistical moments of the chain length distribution in the long-chainlimit can be calculated by insertion of equations 125 and 129 into equation 75.

For termination by disproportionation,

m(0)d = ρ (131)

m(1)d = ρ

[L0

ln(1 + C)C

](132)

m(2)d = ρ

(2L0

L0

C

)(133)

For termination by combination,

m(0)c = ρ

2(134)

m(1)c = ρ

[L0

ln(1 + C)C

](135)

m(2)c = ρ

(3L0

L0

C

)(136)

The number- and weight-average degrees of polymerization can be calculatedvia combination of these moments according to equation 76. For termination bydisproportionation,

Pn,d = L0ln(1 + C)

C(137)

Pw,d = 2L01

ln(1 + C)(138)


For termination by combination,

Pn,c = 2L0ln(1 + C)

C(139)

Pw,d = 3L01

ln(1 + C)(140)

The polydispersity of the chain length distribution of a polymer produced viaa pulsed laser experiment for disproportionation reads

Pw,d

Pn,d= 2

C

[ln(1 + C)]2(141)

and for combination

Pw,c

Pn,c= 3

2C

[ln(1 + C)]2(142)

As can be easily seen, the breadth of the PLP distribution is not only depen-dent on the termination mode, as this is the case for steady-state experiments. Thepolydispersity is additionally controlled by the polymerization conditions like ini-tiator concentration and laser intensity, which both influence the value of the freeradical concentration which is produced at each laser flash, ρ. The polydispersityis also different for different pulse periods, t0.

Thermodynamics

Especially at elevated temperatures the propagation step can no longer be con-sidered irreversible. The propagation step is in fact reversible, leading to a ther-modynamic equilibrium. This equilibrium can be described by the magnitude ofthe free energy difference Gp between the polymer and the monomer. The poly-merization process is thermodynamically favored if Gp is negative. The value ofthe free energy difference is given by the fundamental equation

Gp = Hp − TSp (143)

For long polymer chains the enthalpy and entropy changes in the propagationreaction are effectively those of the overall polymerization reaction (113,422). Thepolymerization enthalpies Hp of most free radical polymerizations are negative,with typical values of −30 to −100 kJ · mol− 1 as can be seen in Table 8.

The values for the polymerization entropies are negative, too, reflecting theloss of degrees of freedom of the monomer becoming a part of the polymer chain.Typical values for the polymerization entropies are −100 to −120 J · K− 1 · mol− 1

(see Table 8). Hence, the two terms on the r.h.s. of equation 143 are antagonistic.Under normal temperature conditions, the exothermicity of the reaction exceedsthe entropic term and Gp becomes negative. However, at elevated temperatures


Table 8. Standard Polymerization Enthalpies ∆H0p and Polymerization

Entropies ∆S0p of Various Monomers for the Reaction of Liquid

Monomers to Condensed Polymersa

Monomer −H 0p, kJ · mol− 1 −S 0

p, J · K− 1 · mol− 1

α-Methylstyrene 35 110α-Vinylnaphthalene 36 —Acetone 0 —Acrylamide 79 —Acrylonitrile 76 109Methyl acrylate 78 —Methyl methacrylate 54 112Styrene 70 105Sulfur, S8 −19 −31Tetrafluoroethylene 138.1 112Vinyl acetate 89 —Vinyl chloride 108.8 —Vinylidene chloride 60 106aData from Ref. 173.

the entropic term becomes significantly larger and finally equals the enthalpicterm at the so-called ceiling temperature Tc.

Tc = Hp

Sp(144)

At this temperature the free energy difference is zero and no polymerizationprocess occurs—the system is in equilibrium (423). A few systems are known inwhich both the enthalpy and the entropy changes of the polymerization reactionare positive. The polymerization of sulfur of the eight-membered ring conforma-tion is one example (424–426): the entropy increases because the S8 ring is rigidand the ring-opening reaction makes additional conformations available. It fol-lows from equation 143 that in such cases there exists a floor temperature withpolymerization possible only above a certain temperature.

The kinetic interpretation of the thermodynamic effects describes the propa-gation reaction as reversible with a propagation and depropagation step (113,427):

(145)

(146)

The rate coefficient of depropagation is written as kdp. The activation energyof the depropagation reaction, EA,dp, is generally much higher than that of thepropagation reaction, EA,p, the difference being equal to the enthalpy change of


the polymerization reaction:

EA,p − EA,dp = Hp (147)

For many typical free radical polymerization systems and conditions, de-propagation does not occur to any appreciable extent. However, for some 1,1-disubstituted ethylene monomers, it is possible to polymerize at conditions wherethe effects of the reverse reaction cannot be neglected. The classical example ofsuch a monomer is α-methylstyrene, with a low ceiling temperature of around60◦C for bulk polymerization, arising from its relatively low heat of polymeriza-tion (Table 8), and the equilibrium monomer concentrations are correspondinglyhigh (0.76 mol · L− 1 at 0◦C) (427). Methacrylate and styrene monomers also ex-hibit depropagation, although at much higher temperatures (220 and 310◦C, re-spectively, for bulk polymerizations). The depropagation process lowers the rateof polymerization according to

Rp = kp[M][R•] − kdp[R•] =(

kp − kdp

[M]

)[M][R•] = keff

p [M][R•] (148)

The effective rate coefficient of propagation is therefore defined as

keffp = kp − kdp

[M](149)

The depropagation effect is inversely proportional to the monomer concen-tration, because it is part of the thermodynamic equilibrium. The effective prop-agation rate coefficient can be determined via the PLP–SEC method. Figure 12

Fig. 12. Temperature dependence of the effective propagation rate coefficient k effp ob-

served in the bulk polymerization of dodecyl methacrylate. The data are taken fromRef. 429.


shows the deviation of keffp from the linear slope of an Arrhenius plot at higher

temperatures, where depropagation becomes important (429).At the ceiling temperature, the effective rate coefficient becomes zero, that

is, kdp equals kp[M]. Using this and equation 147 leads to

Tc = Hp

R ln(Ap/Adp) + R ln[M](150)

when introducing Arrhenius relations for kp and kdp with Ap and Adp being thefrequency factors for the propagation and depropagation reactions. Comparisonof equations 144 and 150 shows that

Sp = R ln(Ap/Adp) + R ln[M] = S0p + R ln[M] (151)

where S p0 is the standard entropy change for [M] = 1 mol · L− 1, so that

Tc = Hp

S0p + R ln[M]

(152)

The considerations leading to equation 152 enable the ceiling temperatureto be defined as the temperature above which no polymerization to high polymerscan occur, with monomer at the specified concentration [M]. It should therefore bekept in mind that the ceiling temperature is a function of the monomer concen-tration. It is determined by thermodynamic parameters and is independent of themechanism, eg, whether ionic or free-radical. The applicability of the above re-lations depends on depropagation being the exact reverse of propagation. Hence,the observation of a ceiling temperature requires the presence of active centers. Intheir absence polymers can exist without change at temperatures well above theirceiling temperatures. When active centers are present there exists an equilibriumconcentration of monomer given by equation 152. The introduction of active cen-ters, eg, via UV radiation and/or high temperatures, and subsequent depropaga-tion is one pathway for the degradation of polymers. By adding stabilizers theseradicals can be trapped and the depropagation may hence be suppressed.

With many monomers the equilibrium monomer concentrations at ambienttemperatures are too small to be measured directly since the ceiling temperaturesare relatively high. However, they may be obtained via extrapolation (430). Someof the deviations from simple kinetic behavior at high temperatures recorded inthe older literature can be explained by the occurrence of depropagation. Use ofthe effective monomer concentration, defined as the total concentration minus theequilibrium concentration, removes the discrepancies.

The thermodynamic definition of the equilibrium constant K reads

Gp = − RT ln K (153)

whereas the kinetic expression for K is the ratio of the rate coefficients of the for-ward reaction to the rate coefficient of the backward reaction. This can be set equalto the thermodynamically definition of K (eq. 153). If the degree of polymerization


is very large, the concentrations of growing chains i and i + 1 can be considerednearly identical, which leads to

K =→k←k

= kp

kdp= [Ri + 1

•][Ri

•][M]e≈ 1

[M]e(154)

where [M]e is the equilibrium monomer concentration (eg, 10− 6 mol− 1 · L− 1 forstyrene at 25◦C). By combining equations 149 and 154 it can be seen that whenthe monomer concentration equals the equilibrium monomer concentration, theeffective propagation rate coefficient keff

p becomes zero, which is the definition forthe ceiling temperature. This implies that there exists a specific ceiling tempera-ture for every given monomer concentration. The maximum ceiling temperatureis reached for the bulk polymerization system. It has been recently demonstratedthat effective molecular weight control in copolymerizations may be achieved bythe judicious selection of monomers which display a low ceiling temperature (431).

Equilibrium concentrations and ceiling temperatures are additionally influ-enced by the pressure. According to the Clausius–Clapeyron equation, a changein the molar volume change Vc = Vp − Vm at the ceiling temperature generallyleads to

dTc

dp= Tc

Vc

Hp(155)

The polymer usually has the smaller molar volume with respect to monomerunits, V being therefore negative, eg, −14.7 cm3 · mol− 1 for α-methylstyrene(432). This leads, together with a negative reaction enthalpy, to a positive pres-sure dependence of the ceiling temperature. The Tc value, for instance, of α-methylstyrene rises from 60◦C at 0.1 MPa (1 bar) over 131◦C at 421 MPa to171◦C at 648 MPa. With increasing pressure, however, the melting points of themonomers are also raised, and polymerizations of crystalline monomers are farfrom being easy.

Inhibition and Retardation

Since the polymerization reaction should proceed under controlled reactionconditions it is undesirable that polymerization activity is induced duringthe monomer preparation, storage, or purification. However, premature radi-cal formation from the pure monomer and from impurities or contaminantscan easily lead to unwanted polymerization. If a substance can quantitativelyscavenge these initiating radicals the propagation reaction is effectively sup-pressed (167,433). When this scavenger compound is consumed by reactionwith the generated radicals, the polymerization proceeds with the same rateas without any additives. These additives are consequently denoted as in-hibitors. The inhibition period, ie the time span that is necessary for thecomplete consumption of the scavenger molecules, is often measured to cal-culate the initiation rate (see Figure 13b). A retarder is a substance that


Fig. 13. Typical conversion versus time profiles for conventional free radical polymeriza-tion (A) with addition of inhibitor (B) and retarder (C).

decreases the overall polymerization rate without any induction period, andis not completely consumed throughout the polymerization (see Figure 13c).The distinction between inhibition and retardation is not always clear. However,both effects are based on the concept of effectively preventing the generated rad-icals from initiating polymerization. Two general cases can be identified:

(1) The scavenger molecule reacts with the initiator-derived—or very smallpolymeric—radical to form a relatively stable radical that by itself is notcapable of initiating polymerization. However, this radical may be activetoward other radicals in the system and may hence terminate irreversiblyor reversibly via radical–radical reactions.

(2) The scavenger molecule is by itself a radical and reacts with any otherradicals in the system to generate nonreactive products. Because of thestability of the radical compounds employed for such inhibition/retardationreactions, the generated bond is very weak and may homolytically cleaveat elevated temperatures to give back the radical reactants. This reactionbehavior is exploited in the living free radical polymerization techniqueusing nitroxides as mediators (434,435).

Stable free radicals are frequently employed as inhibitors (436,437). Themost commonly used species are nitroxides, eg, 2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPO) 18. They are far too stable to be able to initiate polymerization,but they are reactive enough to undergo reaction with other free radicals (438).Nitroxides are very efficient inhibitors, being capable of producing induction peri-ods when present in concentrations of less than 10− 4 mol · L− 1. Nitroxides, suchas TEMPO, react with carbon-centered radicals at close to diffusion controlledreaction rates (439–441). The stoichiometry between the number of the chains


terminated and the number of the nitroxide molecules consumed is 1:1, makingthese compounds very useful for quantitative measurements of free radical con-centrations (81,82,442). The coupling process of the nitroxide with the propagatingradical is reversible, especially at elevated temperatures. This equilibrium enablesa living free radical polymerization by capping the reactive chain ends. The freeradical concentration is therefore extremely decreased, suppressing the termina-tion process. Temporarily uncapped free radicals are adding monomers, leadingto a very slow, but controlled polymerization process (435,443,444). Recent reviewarticles on nitroxide-mediated living free radical polymerization are References434 and 445.

Another commonly used stable radical is 1,3,5-triphenylverdazyl 19 (446,447). It is less thermally stable than TEMPO. Both TEMPO and the verdazyl rad-ical do not react with oxygen-centered radicals or oxygen. If an initiator generatesan oxygen-centered radical the nitroxide will capture the carbon-centered radicalthat is generated via the first addition step involving a monomer. Galvinoxyl 20and 1,3-bisdiphenylene-2-phenylallyl (or Koelsch’s) radical 21 can also be used asinhibitors. Diphenylpicrylhydrazyl 22 is used much less frequently because of itscomplicated reaction mode of inhibition (448).

The overall polymerization rate decreases with increasing concentration ofretarder. An induction period as with inhibitors is not observed. The rate retarding


reactions can be considered very similar to the inhibition reactions; however, thereactivity of retarding agents toward radicals present in the polymerization mix-ture is smaller than that of inhibitors, and the compounds are therefore con-sumed slower, not depleting during the polymerization. Some retarder-derivedradicals may even reinitiate the polymerization slowly—therefore constituting acomonomer—again leading to decreased polymerization rates. This slow reinitia-tion rate of the retarder-derived radicals effectively increases their concentrationin the polymerization mixture, hence leading to an increased probability of ter-mination reactions and other side reactions. In some cases it is very difficult tocompletely identify the real reason for rate retardation, because of the couplednature of all the different reactions involved.

The most common retarders—denoted as stabilizers, because of their appli-cation in stabilizing monomers during their storage—are substituted phenols (eg,p-methoxyphenol, 2,6-di-tert-butyl-4-methyl phenol, used as stabilizer for styrene)(449). They readily undergo subtraction of the weakly bonded phenolic hydrogento yield low reactive phenoxy radicals. The retardation ability of these compoundsis strongly dependent on the type of monomers, since the phenoxy radicals mayadd on to monomer molecules with very different rates, hence acting as a transferagent with a slow reinitiation rate. Phenols react slowly with electrophilic radi-cals and essentially do not inhibit polymerization of acrylates and methacrylatesin the absence of oxygen (450,451).

Benzoquinone, which also may be employed as retarder (used as stabilizerfor methyl methacrylate), gives a phenoxy radical as a result of a rapid addition ofthe radicals present in the system to the C O double bond. The addition is facil-itated by aromatic stabilization (452–454). The relative reactivities of alkyl- andmethoxy-substituted benzoquinones have been correlated to their redox potentialsand steric factors of the substituents (455,456). Industrially, hydroquinone is oftenadded to monomers, although hydroquinone itself acts neither as an inhibitor noras a retarding agent. However, if oxygen or peroxides are present in the system,the hydroquinone will be oxidized to quinone. Hydoquinone therefore acts in twoways: it reduces the peroxides that may initiate the polymerization and acts asan inhibitor in its oxidized quinone form.

A further important class of retarders, generally less effective than quinones,are the aromatic nitro compounds. They show very different effectiveness towardthe retardation of different monomer types: the polymerization of vinyl acetateis inhibited, the styrene polymerization is retarded, but there is nearly no influ-ence on the polymerization rate of acrylates and methacrylates. The effectiveness,however, increases with the number of nitro groups per molecule.

Transition metal [eg, Fe(III), Cu(II), Ce(IV), Hg(II), and Ag(I)] salts suchas halides or pseudohalides are also employed as inhibitors (457). The reactionmechanism involves the effective electron transfer (redox process) or a ligandtransfer mechanism. The carbon carrying the unpaired electron is converted to aσ -bonded carbon or ionic species. In nonaqueous systems the most used retarderis iron(III) chloride which shows no reinitiation abilities, hence being consideredas an ideal inhibitor or retarder (205). The Fe(III) is reduced to Fe(II) duringthe reaction. This Fe(II) formation can be monitored by UV/vis spectroscopy toinvestigate the reaction rate. Equation 156 shows the reaction scheme for theretarding reaction involving ferric chloride.


(156)

CuCl2 is an even more powerful inhibitor/transfer agent than FeCl3. It reactswith growing polystyrene radicals 20 times faster (C = 10,000 vs 500) and withpoly(methyl methacrylate) radicals 50 times faster (C = 2000 vs 4) (458). In thepresence of a suitable ligand the resulting alkyl halides can react reversibly withCu(I) species and establish an atom transfer equilibrium, which is at the essenceof ATRP.

One of the most well-known retarders of free radical polymerization is molec-ular oxygen (459,460). Because of the diradical character of oxygen it reacts readilywith carbon-centered radicals present in the polymerization system via an addi-tion process to yield peroxy radicals. These peroxy radicals may reinitiate thepolymerization, effectively incorporating the molecular oxygen into the polymericchain, so that oxygen is a potential comonomer. In many cases the reinitiationreaction is slower than the propagation and oxygen acts as a retarder. The gen-erated polyperoxides may act subsequently as initiator (461–463). Consequently,the kinetics of polymerizations performed in presence of oxygen may be complex.

Interestingly, the polymerization rates of vinyl esters are remarkably re-tarded by small amounts of styrene. The highly reactive vinyl ester radicals read-ily react with the activated styrene monomer, which results in a relatively stablebenzyl-type styrene radical. The vinyl ester monomer molecule is not activatedenough for the addition of the styrene radicals and the reaction ceases (464).

Buckminsterfullerene (C60) has attracted much interest, not only becauseof its structure and physical properties, but also because of its reaction behaviorin radical polymerization systems. C60 reacts rapidly with radicals to yield ESR-detectable radical species C60

• (465). C60 was found to act as an effective inhibitorin vinyl acetate polymerization and all consumed C60 molecules were incorporatedinto the polymeric chains during the polymerization. The relation between theinduction period and the initiation rate revealed that one C60 molecule can trap15 radicals (466). The rate of polymerization of styrene and methyl methacrylate isalso significantly retarded in the presence of C60 (467–471). The average numberof C60 molecules incorporated in the polymer chains increases when increasing theC60 concentration, maybe because of coupling of polymer–C60

• to give polymer–C60–C60–polymer (472).

The kinetics of the retardation effect for a stationary polymerization can beanalyzed by adding an additional reaction to the basic scheme of polymerization,including initiation, propagation, and termination (473,474).

(157)

where Q is the retarder or inhibitor and kQ is the rate coefficient of the retardationreaction.


The kinetics is simplified by assuming that the generated radical Q• doesneither reinitiate nor show any transfer behavior. The steady-state assumption—which is only a very rough approximation until all inhibitor is consumed (475)—can now be written as

− d[R•]dt

= Ri − 2kt[R•]2 − kQ[R•][Q] = 0 (158)

which yields in combination with equation 55 (again assuming that the rate of theloss of monomer, −d[M]/dt, equals the rate of polymerization, Rp)

Ri −2ktR2

p

k2p[M]2

− kQRp[Q]kp[M]

= 0 (159)

The ratio of the rate coefficients for retardation, kQ, and propagation, kp, isoften referred to as the inhibition constant, z = kQ/kp, which reflects the abilityof a molecule to cause inhibition. Table 9 gives selected values for the inhibitorconstants of some inhibitors in conjunction with a specific monomer.

If the inhibition constant is large (z � 1), the second term of the l.h.s. ofequation 159 will become much smaller than the third one. In this case, the rateof inhibition is much larger than the rate of termination. Equation 159 then reads

Rp = kp[M]Ri

kQ[Q](160)

Table 9. Inhibition Constants z of Various Inhibitors to SelectedMonomers at 50◦Ca

Inhibitor Monomer z

Nitrobenzene Methyl acrylate 0.00464Styrene 0.326Vinyl acetate 11.2

1,3,5-Trinitrobenzene Methyl acrylate 0.204Styrene 64.2Vinyl acetate 404

Chloranil Methyl methacrylate (44◦C) 0.26Styrene 2,040

Oxygen Methyl methacrylate 33,000Styrene 14,600

Phenol Methyl acrylate 0.0002Vinyl acetate 0.012

TEMPO Styrene 335,712CuCl2 Styrene 10,000

Methyl methacrylate 2,000FeCl3 Styrene 536

Methyl methacrylate 4aData from Ref. 173.


Equation 160 shows that the polymerization rate is inversely proportionalto the inhibitor concentration. It should be kept in mind that the inhibitor con-centration will decrease during the reaction. Each free radical generated by theinitiation process will consume one inhibitor molecule. If the inhibitor concentra-tion finally becomes sufficiently low, propagation can become competitive with theinhibition reaction.

Dividing the rate law for the loss of inhibitor, −d[Q]/dt, = kQ[R•][Q], by therate law of propagation, −d[M]/dt, = kp[R•][M], leads to

d[Q]d[M]

= z[Q][M]

(161)

and subsequent integration with [Q]0 and [M]0 being the concentration of inhibitorand monomer at the beginning of the reaction.

ln(

[Q][Q]0

)= z ln

([M][M]0

)(162)

It is apparent from equation 162 that if z is large, the monomer conversionremains nearly zero until the inhibitor is consumed.

Experimental Methods

The aim of most kinetic experimental methods has always been to determineaccurate values for the individual rate coefficients that govern the free radicalpolymerization process, especially kt, kp, and ktr. Up to the late 1980s termina-tion and propagation rate coefficients were accessible only in their coupled form,kp/kt

0.5, or individually via combination with pseudo flickering techniques like therotating sector or spatially intermittent polymerization methods in combinationwith stationary polymerization measurements. The only exception has been thedirect determination of the propagation rate coefficient via the measurement ofthe steady-state free radical concentration by ESR experiments in combinationwith rate measurements. However, the detection of such low free radical con-centrations (typically close to 10− 8 mol · L− 1) has always been subject to largeuncertainties, leading to a large scatter in the values for the rate coefficients. Thesituation has dramatically improved with the invention of the PLP technique inthe late 1980s. Since then, this technique (and its spin-offs like SP–PLP) has beenextensively used to collate propagation and termination rate coefficients for var-ious homo- and copolymerizations. Today, the PLP method is almost exclusivelyused to determine propagation and termination rate coefficients and has been rec-ommended by the IUPAC for the measurement of propagation rate coefficients.In contrast, the methods available for the determination of the chain transferrate coefficient have not changed significantly over the last decades. It should,however, be mentioned that the interpretation of the transfer controlled molecu-lar weight distributions generated in stationary free radical polymerizations hasbeen somewhat refined with the introduction of the Clay–Gilbert method.


Methods for the Measurement of kd.Direct Measurement of the Initiator Concentration as a Function of Time.

The rate coefficients of initiator decomposition, kd, can be assessed by variousmethods. A straightforward approach is to directly assess the thermal decay of theinitiator via the measurement of its concentration as a function of the reactiontime by equation 3. The concentration can be measured via any quantity it isdirectly proportional to, eg, a spectroscopic infrared absorption that can be easilyfollowed with reaction time. The connectivity of the initiator concentration and theintensity of the absorption (at a specific wavelength) is given by Beer–Lambert’slaw. Infrared spectroscopy has been used extensively in the past to study thedecay of organic peroxides in various reaction media (including supercritical CO2).Organic peroxides are widely used to initiate polymerizations, and knowledgeabout their rate of decomposition at various temperatures and their mechanismof decomposition are vital for optimizing polymerization processes (476,477).

Dead-End Polymerization. Dead-end polymerization refers to a polymer-ization process where the initiator concentration significantly decreases to a verylow value during the polymerization. The measurement of the conversion ofmonomer to polymer, p, according to such an experiment, allows to determine therate coefficient of initiator decomposition, kd, and the calculation of the efficiencyfactor f .

Dividing equation 59 by equation 60, rearranging, and subsequently takingthe logarithms of both sides yields

− ln[1 − ln(1 − p)

ln(1 − p∞)

]= kdt

2(163)

The value of kd can now easily be evaluated from a slope of a plot of the l.h.s.of equation 163 against the polymerization time t. If the ratio of kp

2/kt is knownby other studies, a value for the frequency factor f can be estimated by insertioninto equation 56 or equation 60.

Methods for the Measurement of kp.Pulsed Laser Polymerization–Size Exclusion Chromatography. The

careful determination of the chain length distribution of the polymer producedvia a pseudostationary pulsed laser experiment allows to obtain accurate val-ues for the propagation rate coefficient kp. The polymerizable system—containingmonomer and photoinitiator, occasionally solvent and transfer agents—is irradi-ated by a pulsed laser beam and the chain length distribution formed is subse-quently analyzed by SEC. The determination of the additional peaks and its pointsof inflection on the low molecular side, respectively, gives a value for L0,n whichcan easily be inserted into equation 123.

L0, n = nkp[M]t0 (123)

The propagation rate coefficient kp is now available because the monomerconcentration [M] and the time interval between laser pulses, t0, are known. Thismethod has developed into the IUPAC recommended method for kp determination(398,478,479).


Prior to the PLP experiment the monomer should be purified to remove thestabilizer, which is added to most of the commercially available monomers. Thiscan be achieved by either distillation or column chromatography. Both methodshave different advantages: distillation removes small amounts of polymer dis-solved in the monomer, but many inhibitors are very volatile and distillation doesnot completely remove them. However, PLP experiments without cleaning themonomer were reported and have shown the robustness of this method. Photoini-tiators are added in typical concentrations of millimoles per liter. Degassing thesamples by a number of freeze–pump–thaw cycles on a high vacuum line or purg-ing with inert gases like nitrogen or argon makes sure that the dissolved oxygen,which disturbs the kinetic measurements by an inhibition effect, is removed.

For a successful PLP experiment, care must be taken to ensure a homoge-neous intensity profile over the whole optical cross section of the reaction cell soas to produce homogenous reaction conditions. In addition, absorption of the laserlight by initiator and monomer molecules should be accounted for. Accurate tem-perature control is necessary to dissipate the heat of reaction and any possibletemperature increase induced by absorbed laser energy. Typical laser sources arepulsed Nd/YAG solid-state lasers (355 or 532 nm) or XeF excimer lasers (351 nm)with a pulse energy up to 80 mJ per pulse and a pulse width of 5–20 ns. Laser rep-etition rates between 100 and 0.1 Hz have been used in the past. The value of ρ (iethe free radical concentration generated by each laser pulse) can be varied by theinitiator concentration and the laser pulse energy. Samples are exposed to pulsedlaser irradiation to allow for a maximum conversion of monomer to polymer ofabout 1–3%, with typical pulsing times between 1 min and 5 h.

It should be noted that a pulsed lamp or even a rotating sector in combina-tion with a continuous lamp as a pulsed radiation source leads to well-structuredmolecular weight distributions, allowing for the determination of kp. However,optimum results are obtained by the use of a pulsed laser.

After the monomer solution has been irradiated, the produced polymer isanalyzed by a SEC system, which is calibrated by narrow polymer standards orby absolute molecular weight detection methods. The values of L0,n can easilybe determined by differentiating the chain length distribution. The use of differ-ent types of distributions (size exclusion, mass, or number distribution) leads toslightly different values for kp. Figure 14 shows a typically data sheet for styrenebulk polymerization. It should be noted that the obtained kp value is slightly de-pendent on the type of molecular weight distribution (ie size exclusions, weight,or number distribution) that is used for the data analysis (480).

Electron Spin Resonance Spectroscopy—Stationary Polymerization.The experimental determination of kp data usually proceeds via the IUPAC recom-mended PLP–SEC procedure (see above). However, under certain circumstances,kp data are also available by direct determination of the concentration of prop-agating free radicals via ESR spectroscopy, accompanied by the measurement ofthe overall polymerization rate (481) (see ELECTRON SPIN RESONANCE). The calcu-lation of kp then proceeds via either the differential (eq. 55) or the integrated formof the propagation rate law expression:

ln[M]1

[M]2= kp[R•](t1 − t2) (164)


Fig. 14. Typical data sheet detailing the outcome of a successful PLP–SEC experimentfor a styrene bulk polymerization at 25◦C.

where [M]1 and [M]2 are the monomer concentrations at the reaction times t1 andt2, respectively. The determination of the monomer conversion at various reac-tion times can proceed via independent experiments (under the same experimen-tal conditions as the ESR measurements) using, eg, NMR spectroscopy, infraredspectroscopy, gravimetry, or chromatographic methods.

Inspection of the existing literature indicates that kp values derived fromESR are in poor agreement with kp data from PLP–SEC measurement (482–485).Only a few recent studies indicate a somewhat better agreement (117,120), whichis partly due to a significant increase in the quality of the ESR signals by usingoptimized ESR cavities and spectrometers (482). The studies indicating betteragreement seem to have been carried out under specific experimental conditions,allowing for a reliable measurement of the long chain limit of the propagation ratecoefficient. These studies indicate that particular attention needs to be paid to thesize of the steady-state free radical concentration, which should be in an interme-diate range. The type of initiator also seems to be important as is the fact that theonly free radical in the system is the propagating species. This point is of specialimportance in photochemically induced polymerizations where, in addition to thephotoinitiator, the monomer may also be excited. Additionally, the experimentalconditions must be such that the majority of propagating radicals are of sufficientlylarge size to avoid determination of untypically high propagation rate coefficients,


which are known to occur with free radicals of short chain length. Because of thislimitation, studies into higher kp monomers may be advantageous (unless poly-merization rates become too fast to precisely determine monomer conversion vstime profiles). Moreover, it should be checked that high molecular weight materialis indeed produced. A specific problem of ESR spectroscopy is the calibration ofthe spectrometer. Usually, calibration proceeds via the measurement of the signalof a known concentration of a stable radical (eg TEMPO) in the same medium asthe polymerization is carried out. The calibration may be the most serious draw-back of the ESR method for kp determination, since (1) it is very likely that thetwo different radicals, ie the radical used for calibration purposes and the prop-agating radical, are not comparable in terms of their detector signal, and (2) thecalibration is—strictly speaking—only valid for the hypothetical case of 0% con-version and is immediately invalidated as the polymerization process changes thebulk viscosity of the reaction medium. It should also be noted that—similar to thePLP–SEC technique—only the product of the propagation rate coefficient andthe monomer concentration is the measured quantity. Up to this date, no tech-nique exists that allows the direct measurement of kp.

Because these problems, the ESR technique remains problematic for the re-liable determination of propagation rate coefficients and may only be applicablein very specific cases. However, the ESR technique also has the indisputable ad-vantage of actually probing the propagating radicals directly and—in contrastto the PLP–SEC method—does not rely on the interpretation of secondary datamaterial such as a molecular weight distribution. Valuable insight into the poly-merization process, particularly at high conversions, has been deduced from ESRspectra recorded at different monomer conversions.

It should be briefly mentioned that the ESR technique may (via equation 65)also be applied toward the measurement of termination rate coefficients. Thereare several publications reporting on the kt measurements via ESR. However,these measurements are associated with same problems as the kp determinations(see for example References 161,486,487).

Quenched Instationary Polymerization Systems. Quenched instationarypolymerization systems (QUIPS) (488,489) are characterized by the complete de-activation of all radicals by reaction with an inhibitor after a certain time. Aphotopolymerizable mixture passes through a capillary system, is irradiated ata specific location, and polymerizes in the capillary during a well-defined darkperiod until it drops into a quenching bath. The parameters determining the typeof QUIPS are the duration of the initiation period (tL) and an eventual followingdark period (tD). The method is based on the intentional limitation of the max-imum active chain length Lact,max the radicals can achieve. For this purpose allradicals present at a certain time in the system are deactivated by reaction withan inhibitor. Therefore, the kinetic scheme for the free radical polymerization isextended by the quench reaction with the rate constant kq. A number of stable freeradicals are known which fulfill the conditions of an extremely fast and efficientquench reaction (for instance nitroxyl radicals below 100◦C). It is essential thatthe products of the quench reaction are stable and are not mediators for consecu-tive reactions. Experimental conditions can be chosen to ensure that the quenchreaction is complete within the time span necessary for an ordinary propagationstep, namely 1/(kp[M]). Under these conditions the radical spectrum present at


the beginning of (the short) quench period is immediately converted into inactiveproducts.

Measuring the chain length distribution by SEC reveals a structured shapewhich allows to determine kp in accordance to the same equation 123 used inPLP–SEC, which relates the active lifetime of a growing macroradical with itsmaximum active chain length. Recently, Vana and co-workers (135) modified theQUIPS technique in an attempt to make it applicable to hindered monomer sys-tems. The original QUIPS method was changed into a batch operation; ie, a multi-pulse sequence of laser pulses was applied to the reaction mixture to effect a highprimary radical concentration. Subsequently, the reaction mixture was quenchedafter a quenching time, tq, by rapid mixing with an inhibitor containing solution.The propagation rate coefficient (at varying temperatures) can be assessed via themaximum of the formed polymer distribution, Lmax. In addition, the multipulsequenched instationary polymerization also allows for easy access to terminationrate coefficients kt. The novel strategy has so far been successfully applied to thehindered monomer dicyclohexyl itaconate.

Methods for the Measurement of k tr. The evaluation procedures andtheoretical basis for the determination of transfer coefficients C have been ex-tended and perfected over the past decade and a great number of transfer coeffi-cients have been determined by various research groups [see for example Refer-ences 490 and 491 or the multiple entries in the Polymer Handbook (492)]. Thetransfer coefficient is defined by equation 90 as the ratio of the transfer rate coef-ficient ktr to the propagation rate coefficient kp.

C = ktr

kp(165)

The classical procedure for transfer coefficient measurement (to themonomer itself, any other added substance, or deliberately added transfer agent,such as a thiol or a catalytic chain transfer agent) has always been the Mayomethod (493). However, Gilbert and co-workers recently introduced the chainlength distribution (CLD) method (494) as an alternative way to determine trans-fer coefficients. The Mayo and CLD methods in themselves also provide differentchoices of data analysis. Both methods have been carefully compared and thor-oughly discussed with the conclusion that both are theoretically equivalent (201).

In terms of applied experimental procedures, there are only two conceptu-ally different methods available. The first one is the conventional or classicalpolymerization via thermal polymerization experiments, with specific amountsof thermally decaying initiator being present in the reaction mixture. Variousgroups have used this technique to determine transfer coefficients, as the impres-sive collection of data in Reference 492 indicates. Kukulj and co-workers (178), forexample, investigated the transfer-to-monomer constant of methyl methacrylate,styrene, and α-methylstyrene at 50◦C using thermal polymerization. To generatea sufficiently low radical flux to achieve transfer-dominated reaction conditions,the stock solution of the initiator in monomer is successively diluted. This givesa series of solutions with decreasing initiator concentrations, yielding increasingmolecular weights upon polymerization. In many cases and when the initiator


concentrations are chosen correctly, a limiting molecular weight is reached whichmay then be used to determine the transfer to monomer constant via either theMayo or the CLD method.

A second and more recent experimental technique became possible with theadvent of pulsed laser systems: PLP in combination with subsequent SEC of theresulting polymer is the standard and IUPAC recommended technique forthe determination of propagation rate coefficients kp (398). However, several re-search groups have used PLP to determine transfer rate constants to a range ofchain-transfer agents. For example, Hutchinson and co-workers (495) determinedthe transfer coefficients for transfer to n-dodecyl mercaptan in methyl methacry-late, styrene, ethyl methacrylate, and butyl methacrylate homopolymerizations inthe temperature range between 20 and 80◦C. By adding sufficient transfer agent,these authors were able to generate transfer-dominated conditions, as seen by theloss of the PLP characteristics in the molecular weight distributions obtained bySEC analysis of the polymers. In addition, Buback and co-workers (496) designeda method to determine the value of the propagation rate coefficient kp and thetransfer-to-monomer rate coefficient ktr

M from a single PLP experiment. Pack-ages of high frequency pulses separated with long dark time intervals give rise totwo polymer distributions. This procedure has been termed the ‘rail-road’ experi-ment, because of the characteristic pulse profile of the light source that resemblesthe sound patterns generated by a moving train. The polymer produced during thehigh frequency pulse packages may be used to determine kp, while the polymerproduced in the longer dark time may be used to determine CM (413,496).

PLP, however, is normally not the method of choice for measuring transfer-to-monomer rate coefficients via the CLD method, because the CLD method requirestermination to be an unimportant or even absent route of radical chain stop-page as compared to transfer to monomer. In addition, the CLD method is, likethe Mayo method, derived for steady-state polymerizations. Moreover, the radicalflux must be low enough so that transfer, rather than termination, is the mainchain-stopping event, so that the polymerization is truly transfer dominated. PLPis essentially used as a flickering termination rate technique, ie as a techniquethat makes essential use of effective termination. PLP, can however, be madeapplicable for studying even transfer to monomer, if combined with a rotating re-actor/cuvette assembly (179). This assembly permits time efficient experimentswith very long pulse periods and thus enables high molecular weight material tobe produced at very low radical concentrations or termination rates, respectively.The rotating reactor/cuvette allows for acceptable polymerization rates even forslowly propagating monomers such as styrene.

The mathematical methods used to analyze molecular weight distributionsgenerated in transfer-controlled free radical polymerization experiments, ie theMayo and CLD methods, are discussed below.

The Mayo Method. The overall chain transfer coefficient C is defined as theratio of the chain transfer and propagation rate coefficients, ktr/kp. For example,CT is the ratio of the rate coefficient for chain transfer to the chain transfer agentT, and the rate coefficient for propagation. It is a measure for the reactivity ofa chain transfer agent. The higher the value of CT, the lower the concentrationof the chain transfer agent required for a particular molecular weight reduction.This effect on the molecular weight of the polymer is quantitatively described by


the Mayo equation 91, which expresses the reciprocal of Pn, the number-averagedegree of polymerization, as a function of the rates of chain growths and chain-stopping events. Neglecting chain transfer to solvent, initiator, and polymer—thisassumption is nearly fulfilled in bulk polymerizations at low overall monomerconversions—equation 91 reads after insertion of equation 45.

1

Pn= (1 + δ)kt

k2p[M]2

Rp + CM + CT[T][M]

(166)

The usual procedure for measuring the chain transfer coefficient CT—henceforth referred to as the Mayo procedure—involves the determination of theaverage degree of polymerization for a range of [T]/[M] values and plotting the dataas (Pn)− 1 vs [T]/[M], ie, a so-called Mayo plot. The value of CT is then determinedas the straight-line slope of this plot. This procedure assumes that the interceptis independent of the variation of [T]/[M]. However, the chain-length-dependenttermination rate coefficient implies that the first term on the r.h.s. of equation 166is in general not a constant, and therefore a Mayo plot is not necessarily linear.This is a principal weakness of the Mayo method. However, in practice, this ef-fect does not seem to be significant, which suggests that in systems with addedchain transfer agent, the first term on the r.h.s. of equation 166 generally makesa negligible contribution to (Pn)− 1.

CLD Method. The transfer process shifts the chain length distributiongenerated by the polymerization process to lower molecular weights. The Mayomethod, described above, relies on the accurate determination of the number-average degree of polymerization, Pn. Especially if the chain length distributionshows considerable amounts of low molecular weight material, the application ofthe Mayo method is problematic. In the cases when SEC is chosen to measure thenumber-average degree of polymerization, Pn, baseline errors in the low molecularweight region show a significant influence on its value.

To overcome this problem, Gilbert and co-workers introduced a method whichputs more emphasis on the high molecular weight region of the polymer chainlength distribution (497).

The general rate law for the polymerization process reads

d[Ri•]

dt= kp[M][Ri − 1

•] −(

kp[M] + kMtr [M] + kT

tr[T] + 2∞∑

j = 1

ki, jt [Rj

•]

)[Ri

•] (167)

Introducing the steady-state principle, d[R•]/dt = 0, yields an recursive ex-pression for the concentration of macroradicals with chain length i.

[Ri•] =

(1 + kM

tr [M] + kTtr[T] + 2

∑∞j = 1ki, j

t [Rj•]

kp[M]

)− 1

[Ri − 1•] (168)


A repeated substitution leads to an expression of this free radical concentra-tion as a function of the concentration of radicals being composed of one monomericunit, [R1

•].

[Ri•] =

(1 + kM


∑∞j = 1ki, j

t [Rj•]

kp[M]

)1 − i

[R1•] (169)

The rate law for radicals of chain length one, [R1•], combines the rate of

formation of these radicals via the initiation process, Ri, the rate of their formationby transfer, and the rate of their loss by propagation, transfer, and terminationevents.

d[R1•]

dt= Ri +

(kM

tr [M] + kTtr[T]

)µ0 −

(kp[M] + kT

tr[T] + 2∞∑

j = 1

ki, jt [Rj

•]

)[R1

•]

(170)where µ0≡

∑∞i = 1 [Ri

•] is the overall free radical concentration.Introducing the steady-state assumption and inserting equation 170 into

equation 169 leads to

[Ri•] =

(Ri + kM

tr [M]µ0 + kTtr[T]µ0

kp[M]

) (1 + kM


∑∞j = 1ki, j

t [Rj•]

kp[M]

)− i

(171)

If transfer and termination events are much less probable than propagation,ie ktr � kp, and 2

∑∞j = 1 ki,j

t [Rj•] � kp [M], the r.h.s. of equation 171 can be rewritten

as a progression with only two terms, using ex ∼= 1 + x,

[Ri•] ∝ exp

(− kM


∑∞j = 1ki, j

t [Ri•]

kp[M]i

)(172)

For transfer-dominated systems the shape of the polymer distribution is thesame as that of the free radical distribution, [Ri

•] ∼ [PP], keeping in mind thatthe chain length i equals the degree of polymerization, P. It turns out that thissituation is in fact more general: Clay and Gilbert (497) have shown that for partlytermination controlled distributions also, the following equation holds true, inwhich the slope of the logarithmic number distribution (xp ≡ [Pp]/

∑∞P = 0[PP]) is

correlated with the kinetic parameters:

d lnxP

dP= − kM


∑∞j = 1ki, j

t [Rj•]

kp[M](173)

Equation 173 indicates that a semilogarithmic number distribution shouldbe linear, as can be seen in Figure 15.

A plot of the slopes of semilogarithmic number distributions, d ln xP/dP,against the ratio of the transfer agent-to-monomer concentration, [T]/[M], allows


Fig. 15. Transformation of a gel permeation chromatography (GPC) trace of a stationarystyrene bulk polymerization at 25◦C with carbon tetrachloride as transfer agent, into asemilogarithmic number distribution according to equation 173.

to calculate the transfer coefficient C = ktr/kp. It should be kept in mind thatbecause of the chain length dependent termination process, equation 173 is onlyvalid in the long chain limit. However, reasonable results are achieved at lowermolecular weights as well. This method has been studied extensively in the lastfew years (201) and is now frequently used as an alternative to the classical Mayomethod. Although it was recently shown that both methods are in essence thesame (498), there may be situations which makes the CLD method preferableover the Mayo method.

As mentioned earlier, the number-average degree of polymerization, Pn, isassociated with a certain degree of error. This is a difficulty of the Mayo method,which therefore encounters problems with low molecular weight polymers. An-other situation when the CLD procedure is more advantageous than the Mayomethod, is when one must analyze a contaminated polymer sample. A contamina-tion (of an arbitrary nature) may alter the molecular weight distribution and willtherefore significantly change the molecular weight averages, rendering the Mayoprocedure useless. However, if a region in the molecular weight distribution can beidentified in which the distribution is less affected by the contaminant, this regioncan still be used in the CLD procedure. The CLD procedure is also expected to bemore robust when one has systematic errors in the SEC calibration, because thenthe obtained molecular weight averages will not be accurate, but the systematicerror in values of xP can be expected to cancel out to some extent when the slopeof a ln xP plot is taken.

Methods for the Measurement of k t.Single Pulse–Pulsed Laser Polymerization. Applying PLP in conjunction

with infrared or near-infrared spectroscopic measurement of monomer conver-sion induced by a single laser pulse (SP–PLP) allows for the determination of theratio of termination to propagation rate coefficients, kt/kp, in wide ranges of tem-perature, T, pressure, p, and monomer conversion. The SP–PLP technique waspioneered in the 1980s by Buback and co-workers (409). The monomer conversioninduced by a single laser pulse, typically of 20 ns width, is measured by on-line


IR/NIR spectroscopy, with a time resolution of microseconds. The distribution, offree radical chain lengths after a single laser pulse is close with a Poisson distribu-tion, with chain length being linearly correlated with time, unless chain transferinterferes. As a consequence, SP–PLP experiments may provide access to inves-tigations into the chain length dependence of kt. SP–PLP has first been used forstudies into ethene kinetics and subsequently to measure free radical terminationof methyl acrylate, butyl acrylate, and dodecyl acrylate (350,351). Because of thehigh academic and technical interest in detailed kt studies on slowly propagat-ing monomers, the technique has also been applied to methyl methacrylate (353),butyl methacrylate, and dodecyl methacrylate and various acrylate/methacrylatebinary and ternary comonomer systems (499). Figure 16 shows the spectroscopi-cally measured relative monomer concentration vs time profile of a methyl acry-late/dodecyl acrylate copolymerization at equimolar amounts of both monomersin the reaction mixture. The relative change in overall monomer concentrationinduced by the single laser pulse in this particular experiment is 1% after about600 ms. Equation 73 has been used to evaluate the kinetic trace. The differencebetween measured and fitted data is illustrated by plotting the residuals (res) inthe lower part of the figure, indicating the excellent applicability of equation 73to represent the kinetic data.

The majority of SP–PLP experiments reported in the literature have beencarried out under high pressure to increase the conversion per single pulse. ‘True’single pulse experiments can readily be carried out for the acrylates (eg, methylacrylate, butyl acrylate, or dodecyl acrylate). However, for slowly propagatingmonomers like methyl methacrylate and styrene, the signal-to-noise ratio of a‘true’ single pulse experiment is too poor to allow a meaningful kinetic analy-sis of the monomer conversion vs time trace. Nevertheless, slowly propagatingmonomers may be studied by enhancing the signal-to-noise ratio by co-adding alarger number of individual SP–PLP signals recorded under virtually the sameconditions, with only a minor decrease in both monomer and photoinitiator con-centration between ‘true’ single pulse experiments. As the concentration of freeradicals originating from the previous pulse decays to a very low level and assmall primary radicals are generated by each pulse, the range of chain lengthcovered during successively recorded individual concentration vs time traces isidentical. The number of co-added signals is limited by the requirement of deriv-ing kt/kp for a small range of monomer conversion extending over no more thanabout 2% to stay close to the intention of ‘pointwise’ probing the kinetics. To ef-fectively probe the kinetics of a given monomer, a minimum of 0.001% monomerconversion per single laser pulse is required for a sufficient signal-to-noise ratioafter approximately 100 co-additions of single pulse signals. A particular strengthof the SP–PLP technique is that it offers the possibility to probe the conversiondependence of the termination rate coefficient. At present, the technique is lim-ited to studying monomers with a ln(kt/kp) of less or close to approximately 10,otherwise the resulting conversion per single pulse is insufficient to be detectedeven with co-addition of true single pulses. In the past and again recently, theSP–PLP technique has been used for investigations into the chain length depen-dence of the termination rate coefficient. Since all chains grow linearly with timeupon initiation with a laser pulse, the time-resolved monomer conversion vs timesignal contains information about the chain length dependence of kt. The linear


Fig. 16. Monomer concentration vs time trace of a methyl methacrylate/dodecyl acrylatecopolymerization at equimolar amounts of both monomers measured in a reaction mixtureat 40◦C, 100 MPa, and 5 wt% polymer concentration. The difference between measuredand fitted data (eq. 73) is illustrated by plotting the residuals (res) in the lower part of thefigure. Reprinted with friendly permission by the American Chemical Society.

chain growth–time relation is, however, valid only if chain transfer, does not inter-fere. Early investigations into kt

i,i using the SP–PLP technique were carried outon ethylene (500,501), whereas more recent data are available for styrene, methylacrylate, butyl acrylate, and dodecyl acrylate (18). It should be mentioned that theSP–PLP equipment requires a relatively high level of technical expertise to be suc-cessfully operated. Furthermore, the related equipment costs are high. However,to this date it probably remains the most powerful tool for detailed investigationsinto the termination rate coefficient.

The CLD Method. The shape of the chain length distribution of a polymer isdirectly governed by the kinetic parameters of the polymerization process by whichthe distribution is generated. It is therefore self-evident to try to extract theseparameters from such a distribution, which is convincingly realized by measuringkp by PLP or determining the chain transfer coefficient C via the Mayo or CLDmethod. Determining the rate coefficient of termination, kt, via the evaluation ofthe chain length distributions is possible only in combination with polymerizationrate measurements. However, these methods also allow for the determination ofthe chain length dependence of kt. The chain length dependence of the termination


rate coefficient should always be taken into account if reasonable results are to beachieved.

The termination rate coefficient is strictly speaking not a constant, but isproven to vary with the chain length of the two radicals involved in the terminationprocess (342).

ki, jt ∝ F(i, j) (174)

where kti,j is the rate constant of termination between radical chains of length i

and j. F(i, j) is usually represented by a power law of the type

F(i, j) = ( ¯i, j)− α (175)

with ¯i, j denoting some average [eg, the harmonic (336) or the geometric (337)mean] of the two chain lengths i and j involved and α a positive constant. Asa consequence, the problem of determining kt is not solved by the evaluation ofone specific value but rather comprises the evaluation of an entire functionaldependence.

Most of the methods employed to deduce kt and kp are based on the assump-tion of a chain-length-independent termination rate coefficient and yield kp and ktin some combination (kp/kt or kp

2/kt, respectively). In the past, before kp could bedetermined directly, the usual procedure was to use two different ratios of kp andkt and to split them into their individual components via combination. In view ofthe chain length dependence of kt it is clear that any (single) experiment aimingat the determination of kt will render only an average value kt, which is definedby the method used and the experimental conditions chosen (eg, monomer andinitiator concentration, type of initiation). Combination of kp/kt and kp

2/kt ratios,which are invariably taken from different experiments, therefore always involvesaverages of kt, which are not consistent with each other.

Any serious approach aimed at the determination of kt should therefore avoidthe shortcomings outlined above. This implies that one should (1) preferentiallyrefer to single point measurements, (2) avoid combinations of two different ratiosof kp and kt, (3) take advantage of the possibility of determining kp directly (eg byPLP–SEC), and (4) make sure that the average of kt is a fair replicate of the trueaverage of kt, 〈kt〉 (see eq. 48), which is operative in the respective experiment.

With these premises fulfilled it is possible to treat a kt value originating froman experiment dealing with reactions between radicals of different chain lengths,as an average kt specific for the experimental conditions chosen. Assuming thisaverage kt to be physically correct, a power law of the form

kt = k0t ν

′ −α (176)

with the same exponent α as in equation 175 will be obtained. In order to establishthe correct chain length dependence of kt, the averages of kt have to be correlatedwith a quantity characteristic of the population of terminating radicals in eachexperiment. The best solution to this problem appeared to chose a quantity ν ′

which marks the number-average chain length of the radicals at the moment


of their termination. ν ′, which is proportional to the number-average degree ofpolymerization, Pn, of the polymer formed

ν′ = Pn(1 + δ)/2 (177)

thus in itself is independent of the mode of termination (combination or dispro-portionation). The preexponential factor in equation 176, k 0

t , is not to be confusedwith the termination rate coefficient of two radicals of chain length one, but israther a proportionality factor with no physical meaning associated with it.

Two successful approaches, designed for systems with negligible chain-transfer, are (1) the analysis of the second moment of the chain length distri-bution, represented by the product of the rate of polymerization, Rp, and theweight-average degree of polymerization, Pw, and (2) the formal solution for ktof the rate equation for polymerization with periodic laser pulses. Both methodswere originally derived for chain-length-independent termination (502,503). Inboth cases additional quantities have to be known: Apart from kp, the quantityδ (contribution of disproportionation to overall termination) must be available inthe first case and the quantity ρ (the concentration of new radicals formed in eachlaser pulse) in the second (implicitly the knowledge of δ is necessary, too).

The two averages of kt, kmt and kt∗, are calculated as follows:

(1) kmt from the product of rate of polymerization, Rp, and weight-average de-

gree of polymerization, Pw (which represents the second moment of thechain length distribution per time)

PwRp = k2p

2kmt

[M]2(3 − δ) (178)

(2) ¯kt∗ from the rate of polymerization under the same conditions given by

Rp = kp[M]

2k∗t t0

ln

1 + ρk∗

t t0

[1 +

(1 + 2

ρk∗t t0

)0.5 ] (179)

As indicated above, both equations were originally developed for chain-length-independent termination; their use in the context of chain length depen-dence, however, is permissible only if the kt data obtained are explicitly treatedas averages.

The quantity ρ can be determined as follows. In systems with negligible chaintransfer the following relation holds:

ρ = 2Rpt0

Pn(1 + δ)(180)

Once Pn is known, ρ can be calculated for a given δ. A reasonable estimateof δ is sufficient because δ never appears as an isolated factor in the calculation[either as (3 − δ) or as (1 + δ)]. However, errors in δ do not significantly influence


absolute values of kt. The values for α, representing the chain length dependenceof kt, remain nearly unchanged.

The polymerization is performed via a PLP–SEC experiment, which allows tocontrol the number-average degree of polymerization easily by choosing differentlaser frequencies and gives the exact chain length distribution of the producedpolymer. The value of the propagation rate coefficient kp can be obtained by thesame experiment. Measuring the rate of polymerization, Rp, by determining theconversion of the polymer produced per time, and the number- and weight-averagedegrees of polymerization, Pn and Pw, allow to calculate the absolute values ofaverage kt according to equations 178 and 179. Insertion into equation 176 givesvalues of α, which is a measure for the chain length dependence of the terminationrate coefficient for macroradicals of sufficiently large size [according to region Bin Figure 5].

Another method to measure the chain length dependence of kt via evaluationof the chain length distribution of the polymer formed in a single pulse experimentuses equation 181 to measure model independent values of the termination ratecoefficient kt at any given chain length (504,505).

ki,it = Vkp[M]x2P(∫ ∞

0 x2P dP − ∫ P0 x2P dP

)2= Vkp[M]x2P(∫ ∞

P x2P dP)2

(181)

where x2P is the differential number distribution for termination by combinationand V is an arbitrary volume.

There are two uncertainties in this development which prevent a direct ap-plication of equation 181. The first one is the quantity V, which plays the role of ascaling factor, and the second refers to the fact that the integration of x2P cannotbe carried out to infinity in practice, because of the effect of chain transfer in thehigh molecular weight region. This implies that some maximal chain length, Pmax,has to be introduced which corresponds to a specific residual radical concentra-tion [R•]res present at the moment at which integration is stopped. The calibrationproblem as a whole might be solved by estimating values for [R•]0 as well as for[R•]res. They may be calculated via k1,1

t from the Smoluchowski equation or fromliterature data and by combination with average kt data linking [R•]0 and [R•]0or alternatively from SP–PLP traces.

The chain length dependence of kt can also be assessed by fitting a chainlength distribution obtained from a single pulse experiment to the theoreticaldistribution given by equation 119 (506). However, with this method it is also notpossible to measure absolute termination rate coefficients, because of unknownabsolute scaling factors.

Stationary Polymerization Methods. The determination of the kinetic ratecoefficients kt and kp in their coupled form k2

p/kt has long proceeded via mea-surement of the rate of polymerization and the calculation of the k2

p/kt via equa-tion 56. With the advent of pulsed laser techniques that allow to obtain muchmore accurate and detailed information about the kinetic rate coefficients (suchas chain length dependencies), these techniques became less important. Neverthe-less, measurements of the rate of polymerization are still widely performed and are


especially used for systems where the novel techniques are not applicable. This isespecially true for measurements of the termination rate coefficient kt, where rateof polymerization measurements, in combination with independent experimentsfor the kp determination, are sometimes the only possibility to get an estimate forkt. However, it should always be kept in mind that the determination of termina-tion rate coefficients from stationary polymerization experiments yields only anaverage and approximate value for kt, as has been outlined above. The polymer-ization rate can be measured via any quantity that it is directly proportional to.Possible quantities are the density (dilatometry), the refractive index (refractrom-etry), the heat of polymerization (calorimetry), the polymer mass (gravimetry), ora spectral absorption (IR/NIR and NMR spectroscopy).

The two most common methods are dilatometry and spectroscopic methods.Dilatometry has the advantages of being easy to perform and having low equip-ment costs. This technique utilizes the volume change that occurs upon polymer-ization to follow monomer conversion vs time. It is well applicable to free radicalpolymerization, because of the large difference in the densities of polymer andmonomer. For example, the density of methyl methacrylate changes by approx-imately 22% when going from its monomeric to its polymeric form. The densitychanges in other polymerizing systems are of the same order of magnitude. Thedensity change is followed in a volume calibrated dilatometer. In modern dilatome-ters the volume change is followed via the computer controlled observation of themeniscus of a solvent in a capillary on top of the reaction mixture. It is importantthat the solvent does not mix with the reaction mixture.

The spectroscopic measurement of the rate of polymerization is inherentlymore elegant than any other of the above-mentioned methods, since it directlyprobes the reaction mixture on a molecular level and does not rely on the inter-pretation of a secondary quantity. In addition to providing the rate of polymer-ization, spectroscopic methods provide real-time insight into the reaction process.The major disadvantage of spectroscopic methods is the relatively large price of,eg, NMR or FTIR spectrometers. Kinetic spectroscopic measurements normallyproceed via the recording of a part of the spectrum where a spectroscopic absorp-tion is directly associated with the monomer in the reaction mixture. Care hasto be taken that during the course of the reaction no formed product displays anabsorption in the same frequency region as the monomeric species. This require-ment is met for most monomers in kinetic 1H NMR spectroscopic investigations,where the vinylic absorption(s) can be easily used to probe the progress of thereaction (507). However, NMR spectroscopic investigations are limited in theirtime resolution. For faster proceeding reactions, NIR/FTIR spectroscopy has beensuccessfully performed to follow free radical acrylate and methacrylate polymer-izations, using the first overtone of the C H stretching vibration on the doublebond at roughly 6200 cm− 1 (351).

Using Living Free Radical Polymerization for kt Determination. Living freeradical polymerization presents—at least in theory—an ideal tool for investiga-tions into chain-length-dependent termination rate coefficients, since in livingsystems the monomer conversion and the molecular weight, ie the chain lengthof the growing polymer, are closely interlinked. With increasing reaction time, iemonomer conversion, the terminating polymer chains increase linearly in length.If termination events are occurring, it should be possible to extract information


about the chain length dependence of the termination rate coefficients from rateof polymerization measurements. Thus, the use of a living free radical polymer-ization system is similar to the time-resolved SP–PLPs described above. In someliving free radical polymerization systems, however, the concentration of propa-gating radicals is greatly reduced, thus almost completely suppressing termina-tion events from occurring. A typical example of such systems are NMP (434) andATRP (508,509). The RAFT process (268,269), however, makes use of repeatedreversible transfer events during the polymerization without, changing, at leastin principle the free radical concentration so as to induce an equilibrium of dor-mant and living chains. In such an ‘ideal’ living RAFT system—in which no rateretardation effects and other kinetic phenomena occur that may alter the radicalconcentration, the overall rate of polymerization, Rp, and the rate of the individualreaction steps remain unaltered. The specific characteristics of the RAFT processthus allow for undisturbed investigations into the termination reaction of freeradical polymerizations. Because the termination reaction is present throughoutthe polymerization, RAFT is a suitable tool to study radical–radical terminationevents. A study of the chain length dependence of an average termination ratecoefficient for the termination reaction of two radicals with (approximately) thesame chain length i, ki,i

t , is therefore possible.The basis of the RAFT kt method (as mentioned above) is the direct determi-

nation of Rp as a function of time throughout a free radical polymerization.

kt(t) = k2p

[Rp(t)]2(2f kd)[I]0e− kdt

[[M]0 −

∫ t

0Rp(t) dt

]2

(182)

Equation 182 gives kt as a function of time during a free radical polymeriza-tion when considering the concentration changes of the participating reagents. Itshould be noted that this expression assumes steady-state conditions.

When using a RAFT polymerization, the terminating radicals are consideredto be approximately of the same chain length—ideally Poisson-distributed—atevery single instant in time. The time axis may hence easily be converted to achain length axis following the simple procedure of calculating the conversion vstime data from the measured polymerization rate and assuming that the chainlength of the polymer chains increases linearly with conversion. The chain lengthP may therefore be calculated as a function of time by

P(t) =∫ t

0Rp(t) dt[RAFT]0

(183)

with the initial RAFT agent concentration [RAFT]0. The termination rate coeffi-cient is now accessible by measuring the polymerization rate as a function of timein a RAFT-agent-mediated polymerization to ensure that all terminating chainshave the same chain length. This task may be carried out by isothermal differentialscanning calorimetry, which directly determines the polymerization rate by mon-itoring the reaction heat. The chain length dependence and the absolute valueof kt are therefore accessed via one experiment. This very novel procedure has to


this date been used to successfully determine chain-length-dependent terminationrate coefficient of styrene (510). The main advantage of this novel RAFT-basedtechnique for kt determination is its simplicity in combination with the enormousamount of information that is obtained: A single polymerization rate measure-ment yields the conversion dependence, the chain length dependence, and theabsolute value of kt. It should be noted that it is very important to choose theRAFT agent carefully and the selected RAFT agent/monomer system must notshow any retardation and inhibition phenomena. In addition, the technique issuited only for monomers that polymerize fairly rapidly in order to allow for aprecise measurement of the polymerization rate.

Determining the Mode of Termination: Disproportionation vs Combina-tion. Several experimental approaches have been taken to determine the modeof chain termination in free radical polymerization. These have been extensivelyreviewed by Moad and Solomon (511,512). In the case of termination by dispropor-tionation, a chain is generated with one initiator fragment, whereas in the case ofcombination, a chain with two initiator fragments is formed. The determinationof the ratio of number of end groups to the number of monomer units consumedby the polymerization process allows for the calculation of δ, via the additionalknowledge of the number-average degree of polymerization, Pn. Unfortunately,identification and quantification of chain ends are not simple as they give onlysmall signals (relative to the rest of the polymer chain) in a spectroscopic analy-sis. This can be overcome to some extent by isotopic labeling of the initiator endgroups by 14C or by using initiator fragments containing fluorine or phosphorus asNMR-sensitive molecules. Other complications in the analysis include isolatingthe long-chain termination process from other chain-stopping mechanisms suchas chain transfer and primary radical termination. Because of these experimentaldifficulties, there remains considerable uncertainty in existing termination modemeasurements and there is a large scatter in the obtained results. The applicationof soft ionizing mass spectrometry techniques, such as the matrix-assisted laserdesorption ionization time-of-flight mass spectroscopy and electrospray ionizaton,to the problem of end group analysis of polymers brought some promising resultsto this field of polymerization kinetics (317,319).

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PHILIPP VANA

University of GottingenCHRISTOPHER BARNER-KOWOLLIK

THOMAS P. DAVIS

The University of New South WalesKRZYSZTOF MATYJASZEWSKI

Carnegie Mellon University

Vol. 11 RHEOLOGICAL MEASUREMENTS 473

RAMAN SPECTROSCOPY. See VIBRATIONAL SPECTROSCOPY.

RAYON. See CELLULOSIC FIBERS, REGENERATED.

REACTIVE POLYMERS. See POLYMER-SUPPORTED REAGENTS.

RECYCLING, PLASTICS. See Volume 7.

REINFORCEMENT. See Volume 4.

RELEASE AGENTS. See Volume 4.

radical polymerization'. in: encyclopedia of polymer science and...

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