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ARTICULO INVITADO /INVITED PAPER

Statistics of the Formation of Polymer Networks by a Jilragmentation Approach

Roberto J.J. Williams

Instituto de Investigaciones en Ciencia y Tecnología de Materiales (INTEMA), Universidad Nacional de Mardel Plata - Consejo Nacional de Investigaciones Científicas y Técnicas, Juan B. Justo 4302, (7600) Mar delPlata, Argentina

Synopsis

A fragrnentation approach for the generation of average statistical parameters in pre- and postgel stages, is presented and applied for

ideal and non-ideal polymerizations. In the former case the analysis is illustrated with an A homopolymerization an A +B addition. 3 ' 4 2

reaction and an A4 + A2 free radical reaction. Non-ideal polymerizatíons, characterized by complex reaction schemes, including the

possibility oí unequal reactivity offunctionalities, substitution effects and formation oí cyclic species in the pregel stage, are illustrated

with the synthesis oftwo-stage phenolic resins (novolacs), and an epoxy - amine reaction takinginto account the possibility oí asimulta-

neous etherification. It is shown that the desired information may be obtained by using very simple mathematical arguments, provided

that the rates of the different reactions involved in the polymerization are known.

1. INTRODUCTIONThermosetting polymers constitute a signifi-

cant segment, i.e. 15%- 20%, of the plastics mate-rials. The most representative members of this fa-mily are phenolics, urea and melamine-formalde-hyde resins, unsaturated polyesters, alkyds, epoxyresins and some polyurethanes. A common charac-teristic of all these polymers is that they form tridi-mensional networks through the chemical reactionof small molecules (monomers or oligomers). In ge-neral, this reaction may be represented by the ho-mopolymerization of an A species (f indicates thefunctionality, Le. the number of reactive sites permolecule), or the polymerization of an A + B sys-temo In both cases, the necessary condidon fdr theproduction of a tridimensional network is that fand/or g be higher than two. For exarnple, phenolicresins are formed by condensign phenol (A ) withformaldehyde (B ), epoxy - amine networks'resultfrom the additioñ of an epoxy resin (B or B ) to adiamine (A), unsaturated polyesters{Uf')" arisefrom a free tadical mechanism involving a linear UP(A, f. 2) and styrene (B ).

f 2Several factors alter this simple reaction sche-

me. For example, in cases where there are severalpossible reactions involving the same functionali-ties, the structure of the resulting network will behighly dependent on the prevailing mechanism atthe selected reaction (cure) conditions. It is alsopossible that one of the species shows a distribution

of functionalities, i.e. A .f = 1, 2, 3... , a fact whichhas also abearing onfith~ structure of the resul-ting network.

In any case, in order to predict and corre lateproperties of the polymeric material, it ·ísextremelyimportant to be able to characterize the structure ofthe network as a function of the reaction extent.Being this network an amorphous material, itsstructural description relies on the use of a statisti-cal analysis ofthe polymerization reaction. This hasbeen a central problem in the field of thermosettingpolymers since the 40's.

The type of description that a statistical met-hod has to provide, is illustrated in Figure 1 for theaddition reaction of a trifunctional monomer (A )and a bifunctional one (B ). At time = O only monó-mers are presento The prégel stage of the reaction ischaracterized by the presence of finite moleculeswith a broad distribution of molecular weights. At aparticular reaction extent an "infinite" molecule(i.e., limited by the walls of the reactor or mold), ea-lled the gel, appears in the reaction medium. At thistime the material changes irreversibly from a vis-cous liquid to a rubber-like material. The knowled-ge of the particular conversion at which this trans-formation occurs is absolutely necessary for the de-sign of correct processing conditions.

The structure of the material in the postgelstage determines its macroscopic behavior. Thus,the sol is the mass fraction that may be leached from

Revista Latinoamericana de Metalurgia y Materiales, Vol. 9, Nos. 1-2, 1989 5

B IIIIIIIIIIIIIIII

1. Different stages during the formation of a network by the nridition reaction of a trifunctional monomer (Ag), with a bifunc-tional monomer (1:l2). (a) = imtial stage, (hl = tinite molecules present m the pregel stage. (e) = mf'irnt.e molecule (gel frac-tion) and firiite rnolecules (sol fraction) present in the postgel stuge. C. P. = crosslinking points; P. C. = pendant chain;arrows indicate that the network follows to infinite.

BB

Fig,

the material by using an adequate solvento Moreo-ver, as at hign conversions the sol fraction is mainlycomposed by unreacted monomers, the sol may bealso associated with the emission of monomers tothe sorroundíng atmosphere, thus leading to envi-ron mental contamination (the formaldehyde emis-sion from agglomerated wood panels containing anurea-formaldehyde resin as an adhesive, is a wellknown example), Crosslinking points (C.P.) arefragments of the gel structure with three or morearms going to infinite. The concentration of C.P. isrelated to the glass transition temperature, T , ofthe material. Materials with high values of Tg

, asthose required for the use in composites with gcar-bon fibers (i.e. epoxy amine networks), are charac-terized by a very high density of crosslinkingpoints, and behave as glasses at room temperature.On the other hand, materials with low concentra-tion of C.P. behave as rubbers at room temperature.Polymeric chains extending between two crosslin-king points are elasticalIy active network chains(EANC). The elastic modulus of the rubbery mate-rial is proportional to the EANC's concentration.Figure 1 also shows the presence of pendant chainsin the gel structure. These chains are associatedwith the viscous dissipation experienced by the ma-terial upon deformation.

STATISTICS OF NETWORK FORMATION

The scientific basis of the statistics of networkformation may be found in the pioneering work of

GEL

Flory [1] and Stockmayer [2]. The main assump-tions of the Flory - Stockmayer treatment are:

(i) equal reactivities of functionalities ofeach species;

(ii) absence of substitution effects (i.e. thereaction of a particular site does not alterthe reacti-vity oí the remaining unreacted sites);

(iii) absence ofintramolecular cycles in fini-te species

These conditions define an ideal polymeriza-tion.

Using combinatorial arguments, the distribu-tion function of the i-mers as a function of the reac-tion extent may be obtained. This information ena-bles one to calculate the various statistical parame-ters associated with the polymerization reaction.The tremendous mathematical complexities makeit very difficult to extend this approach to non-ideal polymerizations.

In 1962, Gordon [3] showed that average sta-tistical parameters could be obtained directly, wit-hout going through the distribution function, fromthe theory of stochastic branching processes. Themodel has been extensively used to deal with non-idealities in the polymerization scheme [4-10]. Al"ternative approaches leading directly to statisticalaverages have been presented in the literature [11-17].

The aim of this paper is to present a fragmenta-tion approach forthe generation of average statisti-

6 LatinAmerican Journal of Metallurgy and Materials, Vol. 9, Nos. 1-2,.1989

cal parameters characterizing polymer networks.For polymerizations following simple reactionschemes, this approach may be regarded as equiva-lent as those reported in the literature [3-17]. Howe-ver, for complex reaction schemes, i.e. polymeriza-tions with consecutive or parallel competing reac-tions, the fragmentation approach enables thestatistical analysis usingvery simple mathematicalconcepts. We have already used this approach forthe case of phenolic resins [18-19] and epoxy -amine networks, taking into account the possibilityof a simultaneous polyetherification [20]. Here, ageneral analysis of the fragmentation method willbe presented and its application to different parti-cular systems will be illustrated. Ideal polymeriza-tions will be analyzed first in order to discuss theapproach, and then, more complex systems will bedealt with.

IDEAL POLYMERIZATIONS

A homopolymerization3

Let us consider the homopolymerization of anA molecule. At any time during thereaction thenétwork may be regarded as composed by the frag-ments depicted in Figure 2. If all the sites have thesame reactivity and there are no substitution :ef-fects, Le. the reaction of one of the sites does notalter the reactivity of the remaining sites, the con-centration of the different fragments as any reac-tion extent, p (fraction ofreacted functionalities), isgiven byA = (1 - p)3 A

2 oB = 3(1 - p) p A (1)C = 3(1 - p)p2 A 1\

3 oD=pA

owhere A is the initical concentration of A molecu-les. For ~xample, the probability of havink one un-reacted site together with two reacted sites is givenby the product (1- p)p2. However, as the unreactedsite may be any one of the three sites, there arethree possibilities of obtaining the fragrnent C.

A

Ae D

Fig. 2. Fragments present during the homopolymerization of an A3molecule. A: unreacted monomer, B: one reacted functiona=lity, C: two reacted functionalities, D: three reacted func-tionalities.

!hus, its concentration is given by 3(1 - p)p2 A , asmdicated by eqn. o (1)

Let us call y the average weight hangingfrom areacted functionality, A particular reacted functio-nality may bejoined to fragment B, probability: B/(B + 2C + 3D), orC, probability: 2C/ (B + 2C + 3D),or D, probabiblity: 3D/ (B+2C+3D). Thus,y = [B/(B+2C+3D)]M + [2C/(B+2C+3d)] (m+ Y)

+ [3D/(B+2C+3D)] (m+2Y) (2)

where M is the mas s of the A molecule. Note thatwhen entering fragment C, ar1 event with probabi-lit y 2C/(B+2C+3D), the mas s attached is the one ofthe triangle (M) plus the mass attached to the otherreacted functionality (Y). The same is valid forfrag-ment D.

Solving for Y and using eqn (1) leads toY = M/(l - 2p) (3)

The weight average molecular weight is defi-ned asMv = ~ fragments (mass fraction)x(attachedweight) (4)M.v = (A/~i¿)l\•.+ (B/Ao) (M+Y) + (C/A) (M+ 2Y)+ (D/Ao) (lVl+3Y, o (5)By replacing eqns (1) and (2) into eqn (5), wegetMv = M (1 + p)/(1 - 2p) (6)

Figure 3 shows the increase in the weight ave-rage molecular weight with reaction extent. For~ ---:0.5, Mv - 00, i.e. a giant macromolecule, onlylimited by the walls of the reactor vessel, appears inthe system. This particular reaction extent is calledthe gel point, an irreversible transition from a liquidto a rubber. During processing, gelation has to take

20r--------------,-,-----------------,

Mw : w.1

M 1

! x31

II,

W.

~1111 r111111

10 el1 0.5

01I.-:

~!'"1

-11." 1

mI!1

!11

i1

O 0.25 0.5 0.75 OP

Fig. 3. Statistical parameters for the homopolymerization of an A3monomer leading to a network.


place when the part has the final shape. Obviously,gelation must be avoided in the polymerization reac-tor, i.e. used to advance the reaction like in one-stage phenolic resins, or in the conducts that feedthe mold in a processing machine.

Properties of the system in the postgel stagemay be obtained in a similar way, making simplestatistical derivations. Let us call Z the probabilitythat, when looking out from a reacted functionality,a finite set of fragments be obtained, i.e. this direc-tion does not lead to the reactor walls. By a similarargument to that used in the derivation of eqn(2),we get

Z = B/(B+2C+3D) + [2C/(B+2C+3D)] Z + [3DI(B+2C+3D)] i (7)Using eqn(l) and solving for Z, leads to two roots:Z= 1 (8)Z = (1 _ p)2/p2 (9)

Equation (8) is the root with physical sense in thepregel stage (all chains are finite); eqn(9) is the de-sired root in the postgel stage (for p = gel = .O.?,Z = 1; for p = 1, Z = 0, i.e, there are no finitechains).

The mass faction of soluble material, w (solfraction), is obtained as s

Ws = (AI1\) + (BI Ao)Z+ (CI1\)'C ~ (DI1\)Z3 (10)With the aid of eqns(l) and (9), it resultsw = (1 _ p)3/p3 (11)

sThe rapid decrease of sol fraction with reaction

extent in the postgel stage, is shown in Figure 3.Calculations show that at high reaction extents, theterm (Al A ) constitutes the major contribution tothe sol fra~tion, Le. the remaining unreacted mono-mer is the last fragment to join the network. Thishas an important bearing in practice because freemonomer, even in very small amounts, may showtoxic effects.

Another statistical parameter of interest is thefraction of crosslinking points, i.e. triangles withthree arms going to infinite. This is given by~ = (DI Ao) (1 - Z)3 (12)

Replacing eqns (1) and 99) into (12), we get~ = (2 - 1/pt (13)

Figure 3 shows the increase in the fraction ofcrosslinking points (C.P.) after gelation. As three-halves chains arise from each C.P., the concentra-tion of elastically active network chains is givenbyEANC = (3/2) ~ Ao (14)

The elastic modulus of the material in the rub-bery state, i.e. the state at which the reaction is ta-king place after gelation [21,22], increases with theEANC concentration. In order to eject a part from

the mold where the cure is taking place, it is neces-sary to attain a reaction extent such that Xg is highenough to avoid deformation after ejection (usuallya cured part shows a profile of reaction extents; inthis case it is convenient to state the cure conditionsat or near the surfaces because this material makesthe largest contribution to the elastic modulus).

As depicted in Figure 1, after gelation the sys-tem may be regarded as consisting of a sol fractionand a gel fraction. This last one is composed by elas-tically active network chains and pendant chains.For the A homopolymerization, pendant chains areconstituted by fragments with only one arm joinedto the gel. Its mass fraction is given by

W = (BI Aa) (1 - Z) + (CI Aa2Z(1 - Z) + (DI Aa)3Z(1P-Z) (15)

Replacing eqns (1) and 99), we get

(16)

The fraction of pendant chains is zero at p = Pl= 0.5 and at p = 1, going through a maximum int~eintermediate conversion range.

The mass fraction of EANC's is obtained as

w =l-w-we s p(17)

A + B addition reaction-4 -2

A good example of this reaction is the forma-tion of an epoxy - amine network, startingfrom a di-functional epoxy molecule (B) as bisphenol Adiglycidylether (DGEBA) and a tetrafunctionaldiamine (A ). In general, the reactivity of a secon-

4dary amine is different than the one of a primaryamine. That is to say, substitution effects are pre-sent. However, for the particular case of ethylenediamine (EDA), these effects may be neglected[23,24].Thus, thesystemDGEBA-EDA may be regar-ded as ideal from the network formation point ofview. As both epoxy gropus (B functionalities in ge-neral), and both amine groups (A functionalities ingeneral) are independent, the analysis may be per-formed by using the fragments depicted in Fi-gure 4.

The stoichiometric ratio of amine/epoxy equi-valents is given by

r = 2A/Bo (18)

The conversion ofA and B functionalities, p and p ,is related through A B

p = rpB A (19)

8 LatinAmerican Journal of Metallurgy and Materials, Vol. 9, Nos. 1-2, i989

A B

e DFig. 4. Fragments present during the addition polymerization of an

A4 with aB2' A: unreactedaminegroup (halfof an A4),B: un-reacted epoxy group (half of a B2), C: partially reacted amine .group, D: totally reacted amine group. Arrows are joinedwith arrows, and segments among themselves.

The concentration of every fragment along thereaction is given byA = (1 _pA)2~

B = (1 - PB)Boe = 2PA (1 - PA)~ (20)

D = p2AA o

where A+C+ D = A and B+C+2D = B .o o

Let us call Y and W the average weights han-ging from an arrow and a segment, respectively.Then,

y = (B/Bo) (~/2) + (CIBo) [(~ + M)/2 + W] +(2D/Bo)[M

B+(M

A/2)+W+Y] 21)

W = (A/Ao) (M)2) + (C/~) [(MB+M)/2 + Y] +(DI Ao) [~+(M)2)+2Y] (22)

where (BIBo)' (C/Bo) and (2D/Bo} are the probabili-ties of joining, by an arrow, fragments B,C and D,respectively; and (Al Ao)' (CIAo) and (DI Ao) are theprobabilities ofjoining, by a segment, fragments A,C and D, respectively. MAand ~ are the molecularmasses of both monomers.

Replacing eqn(22) in (21), and using eqns (18)to (20), we get

Y= [MAPB+ ~(0.5+0.5PB +PAPB)]/(l- 3PAPB) (23)

W = 0.5~ + PA~ + 2PAY (24)

Both Y and W become infinite whenPAPB~ 1/3 (25)

This is, precisely, the gel condition for an A + Bpolymerization. In general, for an ~ + B polymeri:zation, it may be shown that gelation oc~urs when

PAPB= 1/[(f - 1) (g - 1)] (26)

The weight average molecular weight is calcu-lated with the aid of eqn(4), asMv = wA (M)2 + W) + wB(~/2 + Y) + Wc[(MA+~)/2 + Y + W] + WD[~ + (M)2) +W+2Y](27) (27)

where w. is the mass fraction of fragment i (i =A,B,C or'D), which may be readily calculated fromthe concentrations and particular masses of eachfragmento

Properties of the system in the postgel stagemay be obtained in a similar way as in the ~ horno- .polymerization. Let us callR and S the probabilityof finding a finite chain leaving a particular frag-ment from an arrow and a segment, respectívely.

R = ~fragments (fraction of total arrows associa-ted with a particular fragment)x (probabilitythat all branches leaving from the fragment, al-ready linked by an arrow, are finite) (28)

Then,

R = (B/Bo) + (C/Bo)S + (2D/Bo)RS (29)

Similarly,

S = (Al Ao) + (CIAo)R + (DI Ao)R2 (30)

Solving the system of two equations in twounknowns and using eqns(18) to (20), the values ofR and S may be obtained as a function of the reac-tion extent.

The sol fraction may be calculated as

Ws = =, S + wB R + WC

RS + wD R2S (31)

Any fragment is part of a pendant chain if onlyone ofits arms isjoinedto the gel. Thus, mas s frac-tion of pendant chains is given by

w =w (l-S)+WB(l-R)+wc[R(l-S)+S(l-R)]+p A

wD[2RS(1 - R)+R\l- S)] (32)

The mass fraction of material pertaining toelastically active network chains results from

w = 1- w - w (33)e s p

The fraction of amine groups acting as cross-linking poínts with three branches going to infinite,is given by

~ = (D/Ao) (1 - R)2(1 - S) (34)I

However, it is necessary to realize that if the diamí-ne is very much shorter than the epoxy resin, cross-linking points (C.P.) are better regarded taking the


whole diamine unit as a single fragmento In thiscase, one may generate C.P. offunctionality 3 and 4,from the fragments shown in Figure 4, or, what isbetter, by changing the initial description of frag-ments including complete diamine molecules.A + A free radical reaction-2 -4 .

This case may represent the copolymerizationof styrene (A

2) with divinylbenzene (A ), by a free

radical mechanism. A characteristic offuis polyme-rization is that once a monomer has been activated,i.e. by an initiator, there is rapid propagation reac-tion until a termination step occurs. Then, at everyreaction extent, the system consists of unreactedmonomer, polymer and a extremely low concentra-tion of free radicals. (activated chains).

Two main termination steps are possible: a)combination, b) disproportionation. In the first me-chanism there is a reaction of two polymeric chains;in the second one there is an hydrogen abstractionleaving an unsaturated end in one chain and a satu-rated one in the other.

Then, the different events that a free radicalmay undergo are propagation (addition of a mono-mer), with rate r , termination by combination, withrate r , and termination by disproportionation, withrate;d' We may assume that the ratioq = r/(rp + re + rd), (35)is known and remains constant along the polym~ri-zation (often q lies in the order of 10-2 to 10- ).

At any reaction extent p, the system is compo-sed by the fragments shown in Figure 5. In such afragmentation it has been assumed that the chainends originated in the disproportionation step areinert, i.e. the unsaturated end will not be activatedagain. There is no distinction between propagationand termination by combination. In both cases thechain goes on to another unit.

I-:=: --¡ t----¡

T111\

~~~

1) I~ l~Fig. 5. Fragments present during the free radical polymerization of

an A2 with an A4' A: unreacted A2 monomer; B: unreactedhalf of an A4 molecule (arrows of different fragments arejoi-ned among themselves); C: a2 monomer which has been acti-vated and has then undergone propagation or termination bycombination; D: half of an A4 monomer which has undergonethe same process as fragment C; E: fragment originated bythe termination by disproportionation of an A2 monomerwhich has been previously activated; F: fragment originatedby the terminatian by dispropartionation of half of an A4 mo-nomer which has been previously activated.

The concentration of the different fragmentsalong the polymerization is given byA = (1 - p)AoB = (1 - p)BoC = p(l - q)AoD = p(l - q)Bo

(36)

E = pqAoF = pqBo

The initial fraction ofunsaturations belongingto A4 molecules is defined as

a4

= (2A)/(2A4 + A2) = B/(Bo + Ao) (37)In order to calculate the gel point we define: Y

= average weight hanging from a m and W = ave-rage weight hangingfrom an arrow. These averageweights are calculated asy = [2C(M

A+Y) + 2D(~/2 + Y + W) + EM

A+

F(~/2 + W)]/(2C+2D+E+F) (38)

W = [B(~/2) + D(~/2 + 2Y) + F(~/2 + 2Y) +F(~/2 + Y)]/(B+D+F) (39)

where M and M are the molecular masses of A2A "'I! . 1

and A4 monomers, respective y.Solving for Y with the aid of eqns(36), (37) and

(39), we get

Y= (2-q) [MA(1- a) + ~a4]/[q - a4P(2-ql] (40)

Gelation takes place when Y (and W) --(l!). Thisoccurs whenPgel = q/[a4 (2 - qn (41)

Equation (41) differs from previous results obtai-ned by approximated approaches [11]. This repre-sents aclear advantage ofthe use ofthefragmenta-tion method to deal with the network formation byfree radical mechanisms[25].

Other statistical parameters in the pre- andpostgel stages, may be obtained by similar proce-dures a those used in the previous cases.

NON-IDEAL POLYMERIZATIONSSo far, we have dealt with ideal systems in

which the concentration of the different fragmentsalong the polymerizatíon could be obtained directlyfrom the conversion level (including the knowledgeof the probability oftermination by disproportiona-tion in free radical mechanisms).

Non-ideal polymerizations are characterizedby complex reaction schemes, including the possibi-lity of unequal reactivity of functionalities, substi-tution effects and formation of cyclie species in thepregel stage. The derivation of statistical parame-

10 LatinAmencan Journai 01MetaUurgy and Maierials, Vol. 9, Nos. 1-2, 1989

ters in this case requires the knowledge of all thereactions taking place, as well as their relativerates. That is to say, a kinetic scheme is a necessaryprerequisite for the statistical analysis.

Non-ideal polymerizations will be illustratedby two particular systems: the formation of novo-lacs, Le. two-stage phenolic resíns, and the forma-tion of epoxy - amine networks including the possi-bility of a simultaneous polyetherification, i.e. an~poxy - hydroxy eornpeting reaction.

Formation of Novolacs

Figure 6 illustrates the two steps taking placein the formation oí novolacs, the acid-catalyzedpolymerization of phenol and formaldehyde with adeffect of the latter. The first step is the addition oíformaldehyde, represented as methyleneglycol inwater solution, to a free re active po sitio n (o, o' or p)of the phenolic ring, to give a methylolphenol. Thesecond one is the condensation of the methylol witha free re active position of another phenolic ring, togive a methylene bridge. The novolac chemistry ischaracterized by a high condensation rate with res-pect to the addition rate[26]. This implies that con-centration of ofmethylol phenols will be low throug-hout the polymerization. Moreover, the concentra-tion of fragments having two or three methylols inthe same phenolic ring may be considered comple-tely negligible.

addition ••HOCH2 OH

OH©-CHz OH + H2 o

Fig. 6. Addition and condensation steps taking place in the forma-tion oí novolacs.

The different fragments present along thepolymerization are depicted in Figure 7. These mo-lecular species include the monomers and all thepossible forms in which the phenolic rings may bepresent in the reaction mixture (as single species oras part of molecular chains). The para position ofthe phenolic ring is denoted by 2. The ortho positionis denoted by 1, when both o and o' positions arefree, and by L' when one o position is reacted. Whentwo re active positions of the phenolic ring are usedup for methylene brídges, the remaining position isrepresented by the subscript i (internal site).

P 2

PHENOLlC RING :06.0

1 = ,6.,F.ORMALOEHYOE 0--0

METHYLOL Me

1/2 METHYLENE BRIDGE rvvvSPECIES ANO FRAGMENTS

~1i

2 ,¿, 2

,D, 0-0 .D"A B D E F

2i A Me¡¡ 2 Me2.c..1~' Me,D1, .c:

G H J K.s. 2 A .s..rs:L M N O

Fig. 7. Different fragments present during the formation of novo-lacs.

All addition and condensation steps are taken assecond order reactions. The following reactivity ra-tíos, expressed per reaction site, characterize novo-lac chemistry under strong acid conditions [18,26]:

r(2/1) = 2.4 (with B)r(2/1) = 12 (with any Me)r(Me/Me1) = 2.4 (with any free reactive position)r(Me/B) = 8 (with any free re active position)r(i/e) = 1/3The para position (2) is more re active than the orthoposition (1), both inthe addition (factor 2.4) and thecondensation step (factor 12). Also, a methylolloca-ted in a para position undergoes condensation at arate 2.4 times higher than a methylol in an ortho po-sition. A factor 8 characterizes the ratio betweencondensation and addition steps. Internal positionsare 1/3 as re active as external positions due to ste-ric hindrance.

Rate expressions for every species and frag-ment may be written by summing up all possiblesteps leading to or consuming each particular frag-ment[18]. The numerical solution of the resultingset of differential equations give us the concentra-tion of every fragment as a function of reaction ex-tentlt S]. Once this information is known, statisticalparameters may be calculated by the same proce-dures as those used for ideal polymerizations. Inorder to illustrate the method, the analysis of thegelation condition of a novolac will be shown.

Revista Latinoamericana de Metaluroia. '1/ Materiales, Vol. 9, Nos. 1-2, 1989 11

As in previous examples, we will call Y the ave-rage weight hanging from half a methylene bridge(m). The concentration of methylene bridges isgiven byZ = (1/2) [D+EK+L+M+2(F+G+N+O)+3H]

(42)y = (l/2Z) [(D+ E)100 + 2(F+G) (106+ Y) +3H(112+2Y) + (K+L+M)130 + 2(N+O) (136+Y)

(43)Rearranging,Y= [(D+E)50+ (F+G)106+ H168+ (K+L+M)65+ (N+O)136]/[Z - (F+G+N+O+3H)] (44)

The novolac gels when Y -> =. From eqns (42)and (44), we obtain

3H = D + E + K + L + M (45)Then, the polymer gels when the number of methy-lene bridges joined to triplyreacted phenolic rings(3H) equals the number of methylene bridges joinedto terminal phenolic rings (D+ E+ K«L+ M). Whenthe rate equations are solved for different B lA ra-tios, it is found that in order to avoid gelation affullformaldehyde conversion, B lA ,0.90 [18]. Howe-ver, well before gelation thefe i~ a practicallimita-tion associated with the high viscosity attained bythe reaction mixture at the end ofpolymerization. Apractical operating limit is B/ Ao = 0.85.Epoxy - amine networks with simultaneousetherification

Figure 8 shows different possible reactions inan epoxy - amine system. Steps (1) and (2) indicatethe reaction ofprimary and secondary amine hidro-gens with epoxy groups. Step (3) shows the forma-tion of an ether bridge by reaction of an epoxygroup with asecondary hydroxyl arisingfrom steps(1)and (2) (The OH may be also present in the initialepoxy .molecule). Step (3) acts as a polyadditionreaction without termination because the OH groupos regenerated in the reaction. The relevance ofstep (3) with respect to the two other reactions issignificant in the case oí aromatic diamines and, ob-viously, when an excess oí epoxy over the stoichio-metric amount is used.

®9H

-CH- + -CI-I-CH2 --+ -CI-I-

\0/ 60-12 CH-IOH

Fig. 8. Possible reactions in an epoxy - amine system.

Figure 9 shows the different fragments pre-sent during the cure of the epoxy resin. Implicit inthis notation is the fact that both epoxy groups ofthe same diepoxide molecule, and both amine groupsof the same diamine molecule, have independentreactivities. The fragments that may be distinguis-hed as the cure proceeds are shown in Figure 10.Fragment E is part of a diepr xide molecule.Arrows are Joined to arrows, segments to seg-ments, adn (r+) asterisks to (-) asterisks. The reac-tion scheme may be written as:

E2 + El -> EgE2 + Eg -+ E5 + Elo

E2 + E3 -- E4E + E -- E + 10Ef+E4-+E6 E

2 5 6

Ez + E6 -+ E7 + E10

Ez + Es -+ E9 + E10

Ez + E10 -- ElO+ En

0->1

4 5 6

--*<+)7

--* <-)

8

Fig. 9. The various structuraTelements present during the cure ofthe epoxy resin: 1, unreacted epoxy; 2, reacted epoxy; 3, un-reacted amine; 4, partially reacted amine; 5, completely reac-ted amine; 6, hydroxy group; 7 and 8, half-linkages that arejoined between themselves, i.e. + with ».

All reaction steps may take place both by anoncatalytic mechanism and by a path catalyzed byOH groups [27]. The rate of disappearence of epoxygroups is given by

-clE¿dt = E~..Jkl'2~} + kl (OH)2Ea+~' (Es +E5) +~ (OH)(E3 + ,l!;5) + ~ (OH) + ~ ( H)2] (46)where 1\, Is' and ~ are noncatalytic specific rateconstants, while ~ , ~ and k, are the specific rateconstants for the reactions catalyzed by OH gropus

12 LatinAmerican Journal of Metallurgy and Materials, Vol. 9, Nos. 1-2, 1989

~E4

: ;:~?- *(+)( +)

~( )I

*E8

( +)E9

~*(-) ~*(-)

(+)

E"

Fig. 10. Structural fragments present during the cure of-thl:!epuxyresin.

(E3 + 2E + E6 + ES+ ElO)' Subscripts 1,2 and 3 co-rrespond to the three reaction paths described byequations (1) to (3).

We define:

(47)

(reactivity ratio of secondary to primary aminehydrogens).

(48)

(ratio of the etherification reaction with respect tothe addition to a primary amine).

M=lY~ (E2)O=l{j~(E2)O=Rj~(E2)O (49)

(ratio of the noncatalytic to the catalytic reaction).

The rate of variation of the different fragmentsalong the polymerization may be stated using thereaction scheme previously described. The numeri-cal solution ofthe differential equations leads to theconcentration of every fragment as a function of theepoxy conversion, provided the values of N, L andM are known for the particular system under analy-siso For example, for the cure of a commercial digly-cidylether of bisphenol A (DGEBA, Araldit GY 250,Ciba-Geigy) with diaminodiphenylsulfone (DDS,HT 976, Ciba-Geigy), at 200°C, the kinetic ratiostake the followingvalues [20,27]: N = 0.4, L = 0.14,M = 5.76 X 10-2(for a stoichiometric mixture) or M =5.01x10-2 (for a mixture with 100% epoxy excess).AIsoJEg)o)(E

2)o = 0.061.

Once the concentration of fragments is known,any statistical parameter in pre- and postgel stagesmay be estimated. In order to illustrate the method,the concentration of crosslinking points with func-tionality 4 and 5, Le. associated with the etherifica-tion reaction, will be calculated.

Let us call R, S, Tp and TNthe probability offin-ding a finite chain leauing a particular fragmentfrom an arrow, a segment, a (+) asterisk and a (-)asterisk, respectively. As in previous examples,these probabilities are calculated as

R = (E2 + E~S + 2E4RS + E5STp + 2E6RSTp +2E

7RSTp + 2EgR + 2E9RTp + ElOTN +

En TpTN)/[(E2)o + 2(Eg)o] (50)2 2

S = (El + E3R + E4R + E5RTp + + E6R Tp +E7R

2rr;)/(El)o (51)

Tp = (ElOR + En RTp)/(ElO + En) (52)2 2

TN = (E5RS + E6R S + 2E7R STpEuRTN)/(ElO + En)

+ E R2 +9

(53)

Solving the system of four equations in fourunknowns, thevalues of R, S, Tp and TN may be ob-tained as a functions of the reaction extent.

The concentration of crosslinking points perinitial epoxy equivalent, with four and five bran-ches going to infinite, is calculated as

X4

= [1/(E2)o]{E6(1 - R)2(1_ S)(1 - Tp) + E7 [(1 - R)2(1 - Tp)2S + 2(1 - R)2(1 - S)(l - Tp)Tp +2(1 _Tp)2(1 - S)(1 - R)R]} (54)

~ = [1I(E2)o]E

7(1 - R)2 (1 - T/(1 - S) (55)

Figure 11 shows the evolution of X4 and ~ forthe cure of DGEBA with DDS, at 200°C and diffe-rent epoxy/amine initial ratios. Working with an

x

0.1

1.0

.# Ix. /# /

I//

/1.

o 0.6 0.80.4

Fig. 11. Concentration off-functional crosslinking points (f + 4 and 5),per initial epoxy equivalent. as a function ofthe epoxy conver-sion: dashed fines represent a toichiometric epoxy - aminemixture. full fines indicate a lOO:;¿ epoxy excess.

Revista Latinoamericana de Metalurgia y Materiales, Vol. 9, Nos, 1-2, 1989 13

epoxy excess leads to a higher concentration of 4-and 5-functional crosslinking units. This structuraldifference will be reflected in distinct properties ofthe material. For example, both the glass transitiontemperature, Tf' and the elastic modulus in the rub-bery state wil increase with X4 and X¡, concen-trations.

CONCLUSIONSA fragmentation approach for the generation

of average statistical parameters characterizingpolymer networks has been presented and appliedfor ideal and non-ideal polymerizations. In the for-mer case the analysis is equivalent to other approa-ches reported in the literature. However, when com-plex reaction schemes are to be dealt with, as inmost thermosetting polymers of industrial practice,the fragmentation approach gives the desired infor-mation with the use oí very simple mathematical ar-guments, provided that the rates oí the differentreactions involved in the polymerization are known.This was illustrated for the synthesis oí two-stagephenolic resins (novolacs) and the build up oí epoxy-amine networkds with simultaneous etherification.

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