dspace cover pageAuthor(s): Butté, Alessandro
Publication Date: 2000
Permanent Link: https://doi.org/10.3929/ethz-a-004101826
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ETH Library
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH (ETHZ)
for the degree of
Doctor of Technical Sciences
born July 14, 1972
Dr. G. Storti, co-examiner
1.2 Living Free Radical Polymerization in Miniemulsion 6
1.3 Structure of this work 8
2 Fractionation Techniques for the Solution of Molecular Weight Distri¬
bution Equations 11
2.1 Introduction 11
2.3 Model Solution 15
2.3.1 Detailed Solution 16
2.3.3 Partitioning According to the Number of Branches (PANB) .... 17
2.3.4 Numerical Fractionation 18
2.5.1 Base Cases 22
2.6 Conclusions 40
2.7 Nomenclature 42
Evaluation in Bulk FRP. 44
3.1 Introduction 44
3.3 Development of the KR method 48
3.4 Results and Discussion 55
3.4.1 Reconstruction of the CLD 57
3.4.2 Chain transfer to polymer 61
3.4.3 Cross-linking 67
3.4.5 Branching distribution 73
Evaluation in Emulsion FRP. 79
4.1 Introduction 79
4.4 Development of the Numerical Solution 85
4.5 Illustrative calculations 87
4.5.3 Application to a living system in emulsion 99
4.6 Conclusion 105
4.7 Nomenclature 107
5 Comparative Analysis of the "Living" Free Radical Polymerization in
Homogeneous and Heterogeneous Systems. 109
5.1 Introduction 109
5.3 Living process in bulk 112
5.3.1 LFRP by NMLP 112
5.3.2 LFRP by ATRP 122
5.3.3 LFRP by degenerative transfer 122
CONTENTS iii
5.4.1 The role of particle segregation 126
5.4.2 LFRP by NMLP and ATRP 130
5.4.3 Degenerative Transfer 139
5.5 Conclusions 144
5.6 Nomenclature 147
6.1 Introduction 148
6.4 Application to other monomers 159
6.5 Influence of different reaction conditions 164
6.6 Formation of block copolymers 169
6.7 Conclusions 170
7.1 Introduction 172
7.4 Experimental Part 182
7.5 Experimental results 184
A Overall Moment Equations for a FRP in Bulk 196
B Fractionated CLD Equations for a FRP in Bulk 198
C KR Equations for a Miniemulsion Polymerization in the Limiting Values
of the Indexes 200
D Application of KR Method to a Miniemulsion Polymerization with De¬
generative Transfer 202
E Overall Moment Equations for a Bulk LFRP 205
F Analytical Polydispersity for Ideal Systems in LFRP 206
G S-E Equations for NMLP without Segregation of the Trapping Agent 208
H Application of the KR Method to a bulk LFRP with RAFT 210
I Formulation of a model for the Trommsdorff effect 212
J Diffusion Model for RAFT Agent in ab-initio Emulsion polymerization 215
Abstract
In this work, the living free radical polymerization in emulsion has been developed.
The concept of living free radical polymerization (FRP) has been originally introduced by
Moad at the beginning of the '80, but it is only in the second half of the '90 that the first
applications and developments started to appear in the literature. Since then, many appli¬
cations of this technique in homogeneous system have been introduced. On the contrary
of a classical living system, where all the chains grow simultaneously by starting all at
the beginning, this is impossible with free radical polymerization because of the presence
of radical recombination, or bimolecular termination. The idea behind each living FRP
is to reversibly trap a large fraction of the chains by means of a chemical species. This
decreases the radical concentration and allows carrying out the process with a negligi¬
ble amount of terminated chains. It is worth recalling, in fact, that, while bimolecular
termination rate exhibits a second order kinetics with respect to radical concentration,
propagation exhibits only a first order kinetics. Since terminations are only minimized
with respect to the growing chains and not avoided as in a classical living system, this
process is sometimes also referred to as pseudo-living FRP. The drawback of this approach
is clear. The reduction of the radical concentration leads to a corresponding reduction of
the polymerization rate, thus making this process not attractive for industrial application.
A further way to obtain a reduction of the number of termination is through the
segregation of the radical chains. This is best exploited in emulsion polymerization, where
the polymer particles provide a natural segregation for the growing radicals. Accordingly,
using living FRP in emulsion polymerization it is no more necessary to reduce the radical
concentration is order to minimize the final number of terminated chains, so to better
approach true living conditions.
In this work, the main aspects of living FRP are first theoretically analyzed. In par¬
ticular, the attention is first focused on the living FRP in homogeneous systems, in order
to understand which are the main peculiarities and requirements to be fulfilled in order to
better approach living conditions. This is done with respect to all the living mechanisms
developed in the literature for FRP. The same analysis is then repeated for heterogeneous
system, i.e. to emulsion polymerization. At the end of this analysis, it will become appar¬
ent that not all the living mechanisms are suitable to be applied to emulsion in order to
obtain a reduction of the termination without decreasing at the same time the polymer¬
ization rate. This analysis represents the most important contribution and improvement
to the common knowledge of this work. This part of the work is preceded by a part where
suitable mathematical models to correctly account for the molecular weight distribution in
emulsion are developed. This is done in order to select the proper mathematical approach
to the solution of these problems. Finally, the correctness of all the theoretical suppo¬
sitions argued in the first part of the work is experimentally demonstrated both when
dealing with homogeneous and heterogeneous systems.
Sommario
In questo lavoro é stata sviluppata la polimerizzazione radicalica "living" in emulsione.
II concetto di polimerizzazione radicalica "living" (PRL) é stato introdotto originariamente
da Moad all'inizio degli anni 80, ma é stato solo nella seconda meta degli anni 90 che le
prime applicazioni e sviluppi sono iniziati. Da allora, molte applicazioni di questa tecnica
sono state riportate in letteratura per sistemi omogenei. Al contrario dei sistemi living clas-
sici, dove tutte le catene possono crescere insieme partendo all'inizio del processo, questo
é impossibile in una polimerizzazione radicalica per la presenza della terminazione per
combinazione tra radicah. L'idea che sta dietro la PRL é invece di "intrappolare" in modo
reversibile la gran parte dei radicah per mezzo di una specie chimica. Questo provoca una
diminuzione della concentrazione di radicali e permette di portare a termine il processo
con un numéro di terminazioni trascurabile. Vale la pena infatti di ricordare che mentre la
terminazione bimolecolare é del secondo ordine rispetto alia concentrazione di radicali, la
propagazione é solo del primo. Tuttavia, poiché le terminazioni sono solamente minimiz-
zate rispetto al numéro di catene dormienti e non sono eliminate come in un sistema living
classico, questa polimerizzazione é spesso chiamata polimerizzazione radicalica pseudo-
living. L'inconveniente di questa soluzione é chiaro. La riduzione della concentrazione
di radicali porta ad una corrispondente diminuzione della velocitâ di polimerizzazione,
rendendo il processo poco attraente per sviluppi industriali.
Un modo ulteriore per ottenere una riduzione del numéro di terminazioni puö essere
ottenuto attraverso la segregazione dei radicali. Questo meccanismo viene utilizzato nelle
polimerizzazioni in emulsione, dove le particelle di polimero contribuiscono a separare i
radicali. Per questo motivo, utilizzando la polimerizzazione living in emulsione non é piü
necessario diminuire la concentrazione di radicali per ridurre il numéro di terminazioni
cosi da meglio avvicinare condizioni living.
In questo lavoro, i principali aspetti delle polimerizzazioni living sono stati prima
analizzati da un punto di vista teorico. In particolare, si é prima focalizzata l'attenzione
sulla polimerizzazione in bulk in modo da capirne le peculiaritâ e quali sono i prerequisiti
da soddisfare in modo da ottenere una buona polimerizzazione living. Questo studio é
stato ampliato a tutti i meccanismi living riportati in letteratura per sistemi omogenei.
Quindi, la stessa analisi é stata ripetuta per i sistemi segregati, ovvero per l'emulsione.
Alla fine di questa analisi, é diventato evidente che non tutti questi meccanismi sono adatti
per essere applicati ad una polimerizzazione in emulsione allô scopo di ridurre il numéro
di terminazioni preservando le velocitâ di reazioni tipiche di questi sistemi. Questa analisi
rappresenta il piû importante contributo, nonché miglioramento, apportato durante questo
lavoro aile conoscenze generali su questi processi. Questa parte é preceduta da una studio
dove si sono sviluppati modelli numerici per prédire la distribuzioni di peso molecolare
in emulsione. Questo é stato fatto in modo da poter selezionare il modello numerico piû
opportuno per simulare questi sistemi. Infine, la correttezza délie supposizioni teoriche
discusse in questo lavoro sono state confermate mediante prove sperimentali sia per i
sistemi omogenei che per quelli eterogenei.
Ringraziamenti
Innanzitutto, un grande ringraziamento va ai Prof. Massimo Morbidelli e Giuseppe
Storti, per avermi offerte l'opportunità di svolgere questo lavoro a Zurigo e per avermi
sostenuto dandomi consigli utili per il mio futuro professionale.
I thank Prof. U. Suter for having reviewed this thesis.
Un ringraziamento particolare a Gianmarco Zenoni e Giovanni Biressi, che in numero-
sissime occasioni mi sono stati vicini per aiutarmi a superare le difficoltà del lavoro. In
particolare ringrazio Gianmarco per I'enorme aiuto pratico che mi ha dato durante il mio
lavoro sperimentale.
Ancora grazie a Alessandro Ghielmi, Francesca Maienza, Marinella Sanna, Massim-
iliano Valentini, Francesca Quattrini, Cristiano Migliorini, MariaPia Pedeferri, Stefano
Melis, Bernardo Neri, Andrea Vaccaro, Marco Lattuada, Sergio Panza, Davide Bonifazi,
Barbara Bonavoglia, Gianmario Francesconi, Stefano Lodi, Andrea Portaluri, Orazio Di
Giovanni e tutti gli altri compagni "italiani" incontrati nelle varie fasi della mia vita
"zurighese".
I thank all the "non-italian" part of the group (Annie Rosell, Florian Lode, Sebastian
Böcker, Thomas Hug, Diane Carini, Alex Hipp, Marcel Verduyn) for the help and the
good moments we have spend together.
Ringrazio i tesisti (o affini), ovvero Paolo Mellini, Marzio Ghezzi e in particolare Gion-
ata Frasca, per il grande aiuto che mi hanno dato durante il mio lavoro.
Per ultima, ma non per importanza, ringrazio la mia ragazza, Alessandra Magistrato,
per l'aiuto, il conforto e la pazienza dimostrata durante il lavoro di dottorato.
Chapter 1
Free-radical polymerization (FRP) is a very important commercial process for preparing
high molecular weight polymers such as polystyrene, polymethyl methacrylate, polyvinyl
chloride and random or alternating copolymers, such as SBR (styrene-butadiene rubber),
ABS (acrylonitrile-butadiene-styrene), EVA (ethylene-vinylacetate) and many different
acrylic resins. About 30% of the overall polymer production of the western countries
(about 108 tons per year) is made by FRP [33]. To a great extent, this success is due
to the mild reaction conditions required [61], being FRP tolerant to water and to many
impurities, such residuals of inhibitor and traces of oxygen, and able to operate on a large
temperature range (-20 to 200°C).
In spite of these advantages, FRP suffers of three important drawbacks. The first one
is intrinsically related to the nature of the reaction scheme, involving radical chains which,
due to the presence of radical combination, have a limited lifetime. Since the final chain
length is related to this lifetime through the frequency of propagation, when dealing with
homogeneous systems the only way to get high molecular weights is to operate with low
radical concentrations and, thus, small polymerization rates. This problem may be over¬
come using an heterogeneous process, such as emulsion polymerization. In this process,
the growing radicals are confined into extremely small particles. In the limit of very large
"segregation", the radicals can propagate separately one from the other and the poly¬
merization kinetics is controlled by the rate of radical entry in the polymer particle. As
shown in Figure 1.1, when a radical enters a polymer particle, it can propagate alone for
1
Figure 1.1: Classical evolution of the number of radicals per particle for an emulsion
polymerization.
an average time equal to 1/p, where p represents the frequency of radical entry. As soon
as a second radical enters the particle, practically instantaneous bimolecular combination
takes place, resulting in a particle with no active chains. As a result, the average number
of active chains per particle becomes independent of p and approaches the value of 0.5.
Note that the lifetime of a growing radical is much shorter in a bulk polymerization, where
in the absence of segregation time for propagation is controlled by the fast bimolecular ter¬
mination. In addition, since in emulsion polymerization the number of particles is always
very large, the overall radical concentration is much larger than that in the equivalent
homogeneous polymerization, thus leading to very large polymerization rates. This leads
to high molecular weights and large polymerization rates which characterize emulsion over
bulk polymerization. Finally, it is worth noting that the use of emulsion also eliminates
the large increase of medium viscosity thus facilitating the polymerization heat removal,
a major drawback of the homogeneous processes.
The second limitation of FRP is related to the production of copolymers. As men¬
tioned above, the success of FRP is partly due to the possibility of producing large va¬
rieties of alternating or random copolymers. These copolymers are very attractive since
their end-use properties are a weighted combination of the properties of each single ho-
3
mopolymer. Therefore, a wide range of applicative properties can be covered by properly
tuning the copolymer composition. However, for specific applications it is desirable that
each monomer in the chain somehow maintains its identity, and this is the case of block
copolymers. A block copolymer can be a single sequence of two homopolymers in the same
chain or a series of homopolymer branches grafted to a homopolymer backbone. They are
generally produced by living processes, such as ionic or polycondensation polymerization,
since an essential requirement is the capability of the homopolymer chain to restart, i.e. to
be still alive when the second monomer is charged. This cannot be realized in FRP, since
the chains are rapidly initiated and terminated. However, this situation changed after
Moad in 1982 [59] found a way to establish living conditions also in FRP, the so-called
living free radical polymerization (LFRP), thus opening the possibility of producing block
copolymers by a free radical process.
The basic requirement to approach living conditions by FRP is the minimization of
terminations. This has been carried out in the literature by different reaction mechanisms.
It is worth mentioning the nitroxide-mediated living polymerization (NMLP) [37], the
atom transfer radical polymerization (ATRP) [54], the degenerative transfer (DT) [36]
and the reversible addition-fragmentation transfer (RAFT) polymerization [18]. In spite
of the differences involved in these mechanisms, the idea behind them is the same. Since
it is impossible to let all the chains growing throughout the entire process because of the
presence of fast radical recombination, most of them are reversibly trapped by a so-called
"trapping agent", a molecular species the nature of which identifies the specific living
mechanism. Therefore, a reduction of radical concentration is achieved and, in turn, the
probability of termination is largely depressed with respect to that of chain growth. While
bimolecular termination rate exhibits in fact a second order kinetic with respect to radical
concentration, propagation exhibits only a first order kinetic. Accordingly, the impact of
termination events can be strongly reduced, resulting in a negligible amount of terminated
chains at the end of the process. However, since radical combination is always present,
this process is also referred to as pseudo-living free radical polymerization, to highlight
the difference with respect to an intrinsically living system, such as ionic polymerization,
where no termination mechanism is possible. It is clear that, under these conditions, LFRP
can be used effectively to produce block-copolymers, since most of the chains grown during
4
the formation of the first block are still alive when the second monomer is charged. Since
the monomers polymerized by LFRP are in principle all the monomers polymerized by
FRP, the living process appears to have a major potential to produce large varieties of
new block copolymers.
Finally, the third drawback of FRP is the poor control of the polymer microstructure,
which is again an intrinsic limitation of the reaction mechanism. Due to the presence
of radical combination, the lifetime of chains in FRP is extremely short with respect to
the characteristic time of the process and new chains have to be produced all along the
polymerization process to sustain the reaction. Accordingly, they experience very different
process conditions (e.g. monomer and initiator concentrations, temperature, etc.) during
their growth, thus resulting in a broad distribution of the chain structure. On the other
hand, in living systems all the chains grow together throughout the entire process, thus
experiencing very similar conditions. As a consequence, the final polymer microstructure
is much more homogeneous. For example, in the case of homopolymers, this results in
very narrow distributions of the chain length.
In LFRP the homogeneity of the resulting polymer is a direct consequence of the
reversible trapping reaction. This reaction forces every growing chain to undergo a series
of regularly alternating periods where the chain is either active or dormant. If the exchange
between the active and the dormant state is fast enough so that during each active period
the chain has the possibility of adding only a limited number of monomer units before
being trapped again, a very homogeneous growth of all polymer chains is established. To
better explain this fundamental point, in Figure 1.2(a) the case of two chains undergoing
only two active periods during the polymerization is shown. Their final chain length is
given by summing up the number of monomer units added during each active period,
calculated as the ratio between the frequency of propagation and that of trapping, or
deactivation:
Jd,i
where ATadd represents the total number of monomer units added by a polymer chain,
while fp>l and f^t are the frequencies of propagation and deactivation during the \th
active period, respectively. Assuming f^t constant (as it is the case for most of the
living mechanisms), and fpt = kpMt, being kp the propagation rate constant and Mt the
fi ©
(a)
Time
Figure 1.2: (a) Case of two dormant chains undergoing two active periods only; (b) Case
of a dormant chain undergoing many active period. Activation time and corresponding
monomer concentration are indicated by the arrows.
6
monomer concentration during the ith active period, eq (1.1) reduces to:
AUi = ^£M, (1.2) Id
t
This relation states that the final chain length of the growing chains is proportional to
the sum of the monomer concentration values at the time at which they become active.
Going back to Figure 1.2(a), it is clear that the chain length of the first growing chain is
larger than that of the second one, since the first chain became active at larger monomer
concentration values. On the contrary, when the chain undergoes many active periods (as
sketched in Figure 1.2(b)), differences in the final chain length are much smaller. In other
words, by increasing the rate of the exchange reaction, the number of active periods expe¬
rienced by all chains increases and the continuous growth typical of an ideal living process
(without terminations) is approached, thus resulting in a remarkable homogenization of
the chain growth. This simple example reveals the importance of a fast exchange between
the active and the dormant state in order to get an homogeneous growth of the polymer
chains, thus obtaining a good control of the polymer microstructure.
1.2 Living Free Radical Polymerization in Miniemulsion
Summarizing, all drawbacks reported for FRP are related to the chemistry of this process
which involves the presence of highly-reactive radical chains. Homogeneous LFRP is ef¬
fective in solving both the problems of poor control of the chain growth and of inability
to produce block copolymers. However, the resulting process has a very low productivity
and typical operating problems related to high viscosity and difficult heat removal re¬
main. On the other hand, an heterogeneous process like emulsion polymerization could be
very effective in reducing the impact of chain termination by taking advantage of radical
segregation. This effect could be very helpful to increase the process productivity while
removing also all limitations related to viscosity increase during the reaction. Therefore,
it comes natural to explore the potential of LFRP in emulsion and this is the aim of this
work.
The first experiment of LFRP in emulsion was reported by Bon et al. [6]. Since
then, further attempts have been carried out using different emulsion processes, such as
ab-initio emulsion polymerization [15, 60], seeded [6] and miniemulsion polymerization
[47, 12]. Among them, we have selected the latter as the most favorable way to carry out
7
a living segregated polymerization. The motivation of this choice is briefly explained in
the following.
The typical recipe of a miniemulsion polymerization is completely equivalent to that
of a standard emulsion process except for an additional (small) amount of a hydrophobe
in the oil phase, such as hexadecane [5], cetyl alcohol [86] or a polymer [58]. The emulsion
droplets can in fact degrade (i.e., evolve to a less segregated situation) by two processes:
coalescence and monomer diffusion. The first process is prevented by the presence of the
surfactant on the droplet surface, while the hydrophobe prevents the second one. More
specifically, its presence decreases the total Gibbs energy of the droplet thus reducing
the driving force for monomer diffusion out of the particle. As a result, small miniemul¬
sion droplets can be obtained by imposing to the initial aqueous suspension of monomer
droplets large shear rates. Very small droplets (few tens of nanometers in size) are formed
in this way and, in the presence of the hydrophobe, they are kinetically stable, which
means that the diffusional degradation is slow enough with respect to the polymerization
characteristic time. Different devices can be used to impart the large shear rate needed
to form small miniemulsion droplets, such as the sonicator and the microfluidizer [5]. The
particular device used affects the process in terms of size of the produced miniemulsion
droplets, a parameter which has a major impact on the process performance. The quality
of the initial monomer miniemulsion is in fact a key feature from several viewpoints. First,
small droplets have a large specific surface area, thus reducing the surfactant concentration
in water phase well below the critical micellar concentration. Accordingly, direct droplet
nucleation by radical entry is statistically predominant, while micellar and homogeneous
nucleation are largely reduced or even completely prevented. Second, the droplet size
affects the global kinetics of the polymerization process. Stages I and II, as described by
the classical Smith-Ewart kinetics, are not present and the polymerization directly starts
from stage III, where the monomer concentration inside the particle decreases similarly
to bulk polymerization. Moreover, because of the small size of the formed particles, rad¬
ical segregation is very strong and plays a significant role in reducing terminations, thus
increasing the polymerization rate. A final remark about miniemulsion polymerization is
that the fact of having monomer droplets which are transformed directly into polymer
particles without forming a new polymer phase, provides the opportunity of introduc-
8
ing water-insoluble components directly in the polymerization loci. This feature is very
important for LFRP, since the species used to trap the active radicals are necessarily
water-insoluble. This becomes a problem when dealing with ab-initio emulsion polymer¬
ization, where the trapping species are initially stored in the monomer droplets which,
differently from mini-emulsion, are not the polymerization loci. Therefore, after the stage
of particle formation, they have to diffuse to the polymer particles in order to participate
to the reaction. The low compatibility of the trapping compounds with water would result
in a largely hindered diffusion and, therefore, in a significant delay of the trapping activity.
This justifies the failure of some attempts in the literature to operate LFRP in ab-initio
emulsion polymerization [87].
In conclusion, miniemulsion is representing an optimal mode to carry out a LFRP in a
heterogeneous process since it offers all the typical advantages of emulsion without posing
any limitation to the choice of the trapping agent, which can be selected as the best one
with respect to the living mechanism for each specific application.
1.3 Structure of this work
The structure of this thesis reflects the actual chronological evolution of the work carried
out to join together LFRP and miniemulsion polymerization. This problem was first
attached from a theoretical point of view, while the experimental activity started only at
a later stage. The reason for this approach is twofold: first, it was not understood, and
no suitable model was available in the literature to understand the complex interaction
between radical segregation and living mechanism. Second, but partly as a consequence
of the first point, it was not clear whether all or none of the currently available living
mechanisms could be successfully transported in emulsion. All the experimental reports
in the literature about this point were in fact negative.
Accordingly, the first part of the work was the development of a mathematical model
to simulate the kinetic behavior of the system, in order to provide a reliable tool to
compare the characteristics of the different possible living mechanisms. A key feature
of such a model is the ability to evaluate the entire MWD of the final product, which
is the essential quantity to evaluate the quality of the process. This poses a significant
modeling and numerical problem, since the evaluation of the MWD of a polymer produced
9
in emulsion still represents a field of active scientific research. A popular approach is
based on the Numerical Fractionation [77]. The main idea is to partition the infinite set
of chains into smaller classes characterized by similar properties. Accordingly, the leading
moments of the distribution of the polymer inside each class are calculated and the MWD
is estimated by perturbing suitable model distributions. In Chapter 2, a check of the
reliability of this approach is performed and it is found that its performance is strongly
affected by the choice of the model distribution. A wrong choice here would results in a
MWD exhibiting shoulders which could not be justified by the kinetic mechanism. Since a
suitable choice of the model distribution is very difficult a priori, this aspect reduces to a
significant extent the potential of such techniques. Since for our study we actually need an
accurate model which computes the entire MWD without introducing any artifact, a direct
method based on a discretization technique recently introduced by Kumar and Ramkrishna
[44, 45, 46] has been developed in Chapter 3. This method, originally proposed to solve
PBEs describing particle size distribution, has been modified and applied to compute
the MWD in bulk polymerization. The satisfactory results obtained also in the case of
complex kinetic schemes, including branching, cross-linking and gel formation, motivated
the extension of the method to emulsion polymerization. This is described in Chapter 4,
after having developed a new concept for modeling simultaneously segregated and living
systems, which could not be handled by any of the models currently available in the
literature.
The developed model has been extensively used in Chapter 5, where a more funda¬
mental analysis of LFRP is performed, starting from homogeneous systems. The aim of
this analysis was to identify the main requirements to be fulfilled in order to obtain a high
quality process as well as to quantify the sensitivity of the process to the different reaction
parameters. This allowed us to conclude that, no matter which is the living mechanism,
living conditions can be approached in bulk systems only at the expenses of a drastic re¬
duction of the polymerization rate, especially when a high MW polymer is the final target.
This same study has been extended to emulsion polymerization, by properly accounting
for the effects of radical segregation through the use of the mathematical model developed
in Chapter 4. Here probably the most important conceptual result has been obtained. In
order to preserve reaction rates typical of emulsion polymerization along with the prod-
10
uct quality (i.e., the "livingness" of the process), not all living mechanisms used in bulk
polymerization are equivalent when applied to emulsion. In particular, it is shown that
only RAFT and DT assure large reaction rates and, at the same time, the MWD control
typical of living systems.
The above modeling results have been validated experimentally in Chapter 6 and
7. The experimental results on RAFT bulk polymerization are discussed in Chapter
6. Even though the general conclusions drawn in Chapter 5 are confirmed, additional
features specific of this mechanism and not included in the theoretical analysis, have been
identified. In particular, it is shown that the knowledge of the effect of viscosity on the
trapping efficiency of the RAFT reaction is fundamental to improve the quality of the
process. In Chapter 7, RAFT is extended to miniemulsion. Several experimental runs are
discussed and it is shown that a good control on the polymer growth and the production
of block copolymers becomes feasible by combining miniemulsion polymerization with the
RAFT living mechanism. In all cases, the reaction rates were quite high, thus confirming
that the major disadvantages encountered in bulk polymerization have been now overcome.
This novel polymerization process opens the possibility of using LFRP at the production
scale and therefore it has been covered by a proper patent application.
Chapter 2
2.1 Introduction
When dealing with LFRP, the evaluation of the MWD represents a fundamental aspect
to account for. Its solution is usually approached in emulsion polymerization by using
methods based on the partitioning of the polymer into classes, such as the numerical frac¬
tionation method (NF), developed by Teymour and Campbell [77]. Accordingly, each class
is characterized by similar dimensions and/or properties and the corresponding MWD is
calculated by computing the moments of the distribution and applying a model distribu¬
tion for the unknown MWD as an approximation to the correct solution. The aim of the
present chapter is to verify the accuracy and the robustness of this approach. It is worth
noting that the problem has been faced from a general point of view, i. e. considering a
"classical" kinetic scheme for a FRP. In particular, broad distributions originated by the
presence of a branching mechanism, namely chain transfer to polymer, have been ana¬
lyzed. This reaction is responsible for the appearance of peculiar shoulders in the MWD
profile obtained by NF [77, 30] and, therefore, represents a good test for the reliability of
these approaches.
The whole analysis is carried out referring to a bulk system. This is done because the
equations for such a system are simpler, and accurate reference solutions are obtained more
easily. Nevertheless, the similar peculiar aspects exhibited by the approximate solution
methods based on polymer fractionation in homogeneous [77] and segregated [30] systems
11
12
assure that the conclusions reached can be extended to emulsion polymerization.
Considering free-radical polymerization in homogeneous systems, the modeling of the
active and inactive species of all possible lengths in the reaction system leads to PBEs
which consist of a very large set of ordinary differential equations with the reaction time
as the independent variable. The dimension of this system of equations varies according
to the maximum chain length achieved in the reactor, and typically ranges from 103
to 105 or more. A great variety of approaches has appeared through the years in the
literature to solve this system efficiently. Significant examples are lumping [3], passage
to continuous variable [40], the method of moments [2], and other methods of weighted
residuals (including discrete weighted Galerkin [88] and discrete collocation methods [14]).
Approaches of statistical nature, based on the mathematics of the Markov chains and on
Monte-Carlo simulation [52], have also been used.
Of the methods above, some involve simplifying assumptions or reduction of the equa¬
tions before their numerical solution. An example is given by the method of moments:
in the presence of certain reaction mechanisms, such as chain transfer to polymer, this
method requires the quasi-steady-state assumption (QSSA) for the active species to obtain
a set of equations in closed form. If this assumption is not used, a closure equation has to
be added to the system.
When considering accuracy and flexibility versus computational effort, excellent per¬
formances have been obtained using the discrete Galerkin h-p-method [89]. This provides
in fact accurate solutions in the presence of comprehensive kinetic schemes, allows the
treatment of complex reaction steps, e.g. chain-length dependent kinetics, and requires no
model reduction. However, at least two situations may be illustrated where the application
of this approach is not convenient.
The first situation is provided by those heterogeneous polymerization systems, such
as emulsion polymerization, where the active chains grow segregated in the reaction loci
(polymer particles). In such systems, more internal coordinates than simply the chain
length are required to describe completely the distribution of the active chains. First
of all, one has to account for the number of radicals which are present in the polymer
particle where the chain grows ('particle state'), since the relative probabilities of the
various reactions depend upon this number. For instance, bimolecular terminations cannot
13
occur in a particle containing just one active chain, while they are allowed in particles
containing two or more active chains. Moreover, when considering bimolecular termination
by combination, since the length of the resulting chain is given by the sum of the lengths of
the two coupling active chains, and since these must belong to the same particle in order to
come in contact, it is necessary to model the simultaneous growth of both chains [48, 32].
Accordingly, the internal coordinates of the distribution increase from one (chain length)
to at least three (particle state plus two chain lengths). If one considers additionally that
the polymer particles are not all the same in size, a further coordinate may be added to
the distribution (particle size). In problems of such a high dimensionality, using discrete
methods on every variable becomes indeed rather heavy. In this case less accurate methods
may result more efficient and provide anyhow a satisfactory solution.
The second situation arises when evaluating the MWD in polymerization systems where
the formation of a gel phase occurs, corresponding to the formation of chains of untreatably
high lengths. In this case all solution methods fail unless the kinetic equations are modified
in some way or an upper limit to the calculated chain length is imposed, most likely with
a closure problem arising.
To deal with these situations, approximate methods have been adopted consisting of
the subdivision of the overall polymer into classes, namely the mentioned NF [77] and
a method identifying the polymer chains according to the number of branches [74]. As
briefly recalled earlier, the basic idea of these methods is to subdivide a broad and complex
distribution into a number of narrower distributions, the description of which can be made
by simpler means. In such a way a single problem of difficult solution is decomposed into
several sub-problems easier to solve. This simplification undoubtedly represents a great
advantage when dealing with segregated systems [30, 1], where the starting problem is
very complex, as discussed above.
It is worth noting that the NF technique has been specifically developed to deal with
the problem of gel formation. By properly subdividing the polymer into classes one can
isolate a class which is identified with the gel phase and which is not described explicitly,
thus avoiding the related numerical difficulties. In principle, this feature is shared by all
methods which partition the polymer into classes (therefore also by the method subdividing
the polymer according to the number of branches) as long as a finite number of classes is
14
able to describe the whole sol phase, from the linear chains to the very large and complex
macromolecules which are the precursors of the gel phase. It must however be noticed that
the NF technique, besides constituting a device to solve the mathematical problem, also
offers a remarkable physical insight into the process of gel formation. The passage from
one class to the following is given in the frame of this method by chain coupling (through
any possible reaction mechanism), and this is also identified as the step which can lead to
the formation of the huge molecules of the gel.
Finally, in order to assess the potentiality of the methods based on the partitioning
of the polymer into classes, reference solutions have been obtained through a numerical
technique which, though computationally intensive, permits to calculate the correct MWD.
The computational details of all these techniques are reported in the following.
2.2 Kinetic Scheme and Molecular Weight Equations
A kinetic scheme constituted by classical free-radical polymerization steps has been con¬
sidered, involving chain transfer to polymer as a source of chain branching:
• initiator decomposition rate
• chain transfer to monomer
• chain transfer to modifier
• chain transfer to polymer
• bimolecular termination by combination
15
iin + iim —> rn + _rTO Ttd = hdHm±in
where all symbols are defined in the Notation. On the right-hand side the expression for
the rate of each reaction has been reported. All kinetic rate constants have been considered
chain-length independent. Note that a R' molecule has been assumed to be formed by
chain transfer to monomer instead of i?* as usually considered. This has been done to
make the following equations more compact and does not modify the final results due to
the high chain lengths of the macromolecular species under examination. Finally, in the
kinetic scheme above it is assumed that the polymer chains can have at most one active
center (monoradical assumption).
For the calculation of the concentrations of the active and terminated chains in a batch
reactor, the following PBEs apply:
au*
+kfpX0nPn + Sn,o[(kfmM + kftT)X0 + Hi] (2.1)
dP 1 ra
+kfppiR'n - kfpX0nPn (2.2)
where Ao = X^m=o ^m 1S the zeroth-order moment of the distribution of the active chains
(i.e., the overall radical concentration), p\ = X^m=o m^m is the first-order moment of
the distribution of the terminated chains (i.e., the overall concentration of polymerized
monomer), önß is the Kronecker index and IZi is the initiation rate (IZi = 2r/kil, with r]
an efficiency parameter).
2.3 Model Solution
In this section four numerical methods are described for solving the model eqs (2.1) and
(2.2). The first method described is rather time consuming but provides the correct solu¬
tion (at least in the frame of the QSSA, which is assumed valid by all methods considered).
This solution is used as a term of comparison for testing the reliability of the approximate
solutions provided by the other numerical techniques.
16
2.3.1 Detailed Solution
The rigorous solution of system (2.1)-(2.2) to obtain the CLD would require integration
for all values of the chain length n up to values corresponding to negligible polymer chain
concentrations. A simpler but equally accurate method can be developed by applying the
QSSA to eq (2.1) for the radical species, so that the differential system (2.1)-(2.2) reduces
to the following algebraic-differential system:
R. =
dP 1 ra
m=0
+kfppiR'n - kfpX0nPn (2.4)
From system (2.3)-(2.4) it is seen that, as long as the moments Ao and p\ are known
independently, the concentration of chains of length n depends only on that of chains of
the same length or shorter. Accordingly, the CLD calculation can be truncated at any
value of n, no matter how low, without introducing any error. The equations for the
moments Ao and p\ can be derived by applying the method of moments to eqs (2.3) and
(2.4), as described in the next section. The equations obtained are rigorous (in the frame
of the QSSA) and allow an independent evaluation of the moments Ao and p\, which can
then be used to solve system (2.3)-(2.4).
It is worth noting that to obtain an accurate solution it is not necessary to solve
eqs (2.3)-(2.4) for all n values up to the selected maximum value. In fact, eqs (2.4) for
the terminated polymer can be integrated only for some selected values of n, while for
the intermediate n values the concentrations Pn are obtained by low order interpolation.
Here, parabolic interpolation was used. A convergence check with an increasing number
of nodes was performed for each simulated reaction to verify that the adopted number of
nodes was sufficient.
2.3.2 Overall Method of Moments
The method of moments applied to the calculation of the CLD of a polymer typically
consists of calculating integral properties of the distribution (the moments) and recon¬
structing it by selecting a model distribution and imposing that its moments are the same
as those calculated. The A^-order moments of the active and terminated chain distribu-
17
tions, Xk = S^Lo n -R-n an(i f'k = S^Lo n Pni respectively, can be calculated by applying
the moment operator X^o n^ to ecls (2.1)-(2.2). When this is done, however, the resulting
moment equations are not in closed form since, due to the presence of the chain transfer to
polymer terms, the moments of order k of both the active and terminated chains depend
on the moment of order k + 1 of the terminated polymer. If the QSSA assumption is
applied, the following system of equations is obtained (see Appendix A):
MÄf (2'5)
and for k > 1:
fc-i / , \ , fc-i
(2.7)
^ = kPM Y, ( ) ) A, + i*fc E ( • ) X^-i (2-8) j=0
^ ^ 3=1
Note that, though the moment of order k of the active chains depends on the moment of
order k + 1 of the terminated chains, the latter depends only on lower order moments of
the active chains. This implies that, given a maximum moment order k.max-, the system
which calculates the moments up to Pkmax for the terminated chains and up to Xkmax-i for
the active chains, is in closed form. Accordingly, moments of any order can be calculated
from eqs (2.5)-(2.8) in a rigorous manner (of course, in the limit of the QSSA).
Here, the distribution selected for the reconstruction of the CLD from its moments is a
re-scaled T-distribution perturbed to the unknown distribution by a polynomial expansion
in terms of associated Laguerre polynomials orthogonal to the T-distribution weighting
function [39].
2.3.3 Partitioning According to the Number of Branches (PANB)
According to this method [74], each polymer chain is characterized not only by its length
but also by the number of branches it bears. The following PBEs result:
dR* u
+kfpX0nPn,b-1 + ôny0ôb,0[(kfmM + kftT)X0 + ft/] (2.9)
18
(kfmM + kftT)R'nb + -ktc /2 /2 Kn,kRn-m,b-k + hd^Rnfi k=0 m=0
+kfp/j,iR'lb - kfpX0nPnjb (2.10)
where R'n b and Pn^ represent the concentration of active and terminated chains, respec¬
tively, of length n and with b branches.
To obtain the overall polymer CLD, eqs (2.9)-(2.10) are solved to obtain the CLDs of
the polymer chains for each number of branches b. Then, these are summed up over all
possible branch numbers to obtain the desired quantity: Pn = X^o Rn,b-
The distributions Pn^ are obtained by reducing eq (2.9) for the radical species by means
of the QSSA and by applying the moment operator X)nLo n^ to the resulting system, in
order to obtain the first three CLD moments for each number of branches (zeroth-, first-
and second-order moments). The formulae and passages required to derive these moment
equations are the same as those reported in Appendix A in the frame of the overall method
of moments. From the first three moments, each distribution is reconstructed using a re-
scaled T-distribution as model distribution [39]. With respect to this, the previous finding
[78, 79] that the polymer fractions constituted by chains with a given number of branches
follow a T-distribution in a randomly branched polymer suggests this to be a reasonable
choice. Note that, unlike the overall moment eqs (2.7) and (2.8), the moment equations
resulting in this case are not in closed form, i.e., the dependence of the terminated polymer
moments on the next higher order moments cannot be eliminated. Accordingly, a closure
equation is required. The following equation for the calculation of the third-order moment
of the CLD of the terminated polymer with a given number of branches has been adopted:
t^3,b = ß2,b 2 M2,fe
_
Plfi (2.11)
ßl,b ß0,b.
This closure formula is in agreement with the model distribution (re-scaled T-distribution)
which has been chosen for the reconstruction of each CLD from its first three moments.
2.3.4 Numerical Fractionation
In the original version of NF proposed by Teymour and Campbell10-1, the polymer chain
population is subdivided into ng + 1 classes (or 'generations'): one linear and ng branched
classes, each constituted by chains similar in size. The subdivision of the polymer into
classes is not operated according to a structural feature of the chains (such as the number
19
of branches in the PANB method) but is defined by the rules describing the transition of
chains from one class to the other: these rules should assure that each class is made up
by chains similar in size.
The transition rules are the following: the passage from the linear generation to the
first branched generation occurs when a linear chain adds a branch, i.e., when it undergoes
chain transfer to polymer. The passage from the first generation to the second occurs when
two first-generation chains couple (through bimolecular combination in the kinetic scheme
considered) to form a single chain. The same rule applies for the successive generations,
i.e., coupling of two second-generation chains yields a third-generation chain, and so on.
In such a way a geometric growth is obtained passing from one generation to the following
one. This provides a unique and enlightening description of the physical process which
leads to the formation of a gel. However, when regarding this fractionation strategy in
the frame of a numerical method for accurate MWD evaluation, it probably results to
be too coarse. This can be understood, for instance, by considering that, according to
the scheme above, a chain can keep adding branches within a certain generation without
being transferred to the next higher. This results in broad generations which are difficult
to describe numerically (even though much less than the original overall distribution).
To improve the description of the MWD, a refinement of the fractionation scheme has
been proposed in the context of emulsion polymerization by Arzamendi and Asua [1]. The
polymer is still subdivided into linear and branched chains, but the latter are subdivided as
follows: each of the first nb generations is constituted by chains having the same number
of branches (coinciding with the generation index g), while for the generations nb + 1
and higher the original rules allowing a geometrical growth in the chain dimensions are
adopted. In other words, not only the linear chains but also chains with up to nb branches
are described separately, before introducing the fractionation developed by Teymour and
Campbell.
The following transfer rules result in this case: the linear chains (g = 0) pass on to the
first branched generation when they undergo chain transfer to polymer. Within the next
nb branched generations, transfer occurs either by chain transfer to polymer (to the next
higher generation) or by combination (to the generation corresponding to the number of
branches of the resulting chain or to generation nb + 1 if the number of branches exceeds
20
0<:9i,92 < «ft
iin + Rm ktc
Rn + Rm ktc
Table 2.1: Reactions giving transfer between generations.
nb). From generation nb + 1 onwards, transfer to the next higher generation occurs when
two chains of the same generation couple together through termination by combination.
This generation transfer scheme is summarized in Table 2.1. In this table, Rn and Png
represent an active and a terminated chain, respectively, of length n and belonging to
generation g. The parameter ng indicates the number of branched generations in the sol
phase. Accordingly, generation ng + 1 corresponds to the gel phase (or better 'pseudo-gel'
phase, where this term indicates everything which is lumped together with the gel phase
into a single class).
This scheme allows a more detailed description of the moderately branched polymer
which is instead lumped into the first very few branched generations (only one in the limit
of no termination by combination) in the original version of NF.
According to the transfer rules summarized in Table 2.1, the PBEs (2.1)-(2.2) must
be modified in order to account for the passage of chains between generations. The re¬
sulting 'fractionated equations' are reported in Appendix B. Note that when nb = 0
these equations reduce to those corresponding to Teymour and Campbell's original gen¬
eration transfer scheme [77], which can therefore be considered as a special case of the
fractionation scheme here adopted. In the following, we will refer to this special case as
21
'classical NF'. Instead, when ng = nb and the equation for g = ng + 1 is neglected, the
fractionation method considered reduces to the partitioning of the polymer based on the
number of branches described in the previous Section 2.3.3, which does not account for
the geometrical growth in size of the chains approaching gelation typical of the classical
NF.
To solve the fractionated PBEs (cf. Appendix B), as for the method presented in
Section 2.3.3, after reduction of the equations for the radical species by means of the QSSA,
the method of moments has been applied in order to obtain the first three moments of each
generation. Again, the resulting moment equations are not in closed form. The closure
formula (2.11) is adopted within each generation to calculate the third-order moment as
a function of the lower-order moments. This closure formula results from the re-scaled
T-distribution which has been chosen for the reconstruction of the CLD of each generation
from its first three moments [39]. The overall CLD of the sol phase is obtained by summing
up the contributions of each generation.
2.4 Gel Weight Calculation
From a mathematical point of view, a polymer molecule is identified as belonging to the
gel phase when its dimensions are so large that the molecule can be considered infinite in
size. Therefore, the formation of a gel phase in the polymerization system is revealed by
the divergence of the second- and higher-order moments of the CLD. This implies that
the overall method of moments fails at the gel point and provides no means of calculating
the amount of gel or any other quantity beyond the gel point. On the other hand, for
the other methods discussed above, integration can be carried out across the gel point
without introducing any discontinuity. To this aim, it is sufficient to establish a maximum
chain length, or number of branches, or any other feature related to molecular dimension,
above which the polymer molecules are assumed to belong to the gel and are no longer
simulated. This also permits, if this maximum size is selected so as to actually include all
the molecules of the sol, to calculate correctly the amount of sol polymer (and thus of gel)
and to have a complete description of the sol MWD. What ensures that this is possible,
is that in the process of gel formation a gap appears between the gel, infinite in size, and
the sol phase, where very large molecules may be present but still finite in size, so that
22
they can be simulated (cf. e.g. ref. [77]).
In the case of the detailed solution method, it has been shown that the differential
system can be truncated at any chain length without introducing any error in the CLD
calculation up to that chain length. The sol CLD is thus calculated correctly up to any
selected um value. Instead, for a correct evaluation of the amount of gel, the value of
um must be taken so large as to have a negligible amount (in terms of weight and not
only of number) of soluble chains that exhibit this chain length. This can be verified by
examining the calculated weight CLD of the sol phase. The amount of gel pfe is then
obtained by subtracting from the overall polymer p\ the amount of polymer in the sol
phase, given by plot = YTn=onRn- Thus, the gel is considered to be made up of chains
with a length greater that n«.
In the case of the PANB method, a number of branches bu can be selected, above
which the polymer chain is considered to belong to the gel phase. The parameter bu must
be large enough so that the amount of soluble chains having this number of branches is
negligible. The amount of gel can be calculated as pfe = p\ — X]&=o A^i,^ wnere ßi,b 1S the
amount of polymer constituted by chains with b branches.
In the case of NF, the index reflecting the dimension of the molecule is the generation
index. Accordingly, a sufficiently large number of generations ng has to be used to describe
the sol phase. Teymour and Campbell suggest to check convergence of all quantities
predicted by the model, and in particular the gel point, to verify that the number of
generations selected for the description of the sol is large enough (note that this criterion
can be applied also with the other solution techniques to check if the 'dimension parameter'
has been taken large enough). The amount of gel can be calculated as pf6 = pi —
zCo=o An ! where pf is the amount of polymer belonging to generation g.
2.5 Results and Discussion
2.5.1 Base Cases
Two main case studies have been selected for comparing the results obtained through the
various solution methods. In the first one, bimolecular termination is assumed to occur
by disproportionation while in the second one it occurs by combination. In all cases chain
transfer to polymer is present as a branching mechanism. Accordingly, in the second case
23
kfp 5 • 10"1 1 mol"1 s"1
kT 1.18 • 10"6 s"1
ktc 0 ^ 5.97 • 106 1 mol"1 s"1
fcw 0 ^ 5.97 • 106 1 mol"1 s"1
rj 1
Initial concentrations
M0 8.43 mol 1_1
T0 0 mol r1
Table 2.2: Kinetic parameters and initial concentrations for the model simulations
the formation of a gel phase is possible, while in the first one the absence of a mechanism
connecting chains together prevents the formation of the huge-sized molecules constituting
such a phase. Within each case study, two different reaction rates, given by two different
initiator concentrations, have been examined. All the values of the kinetic parameters and
the initial concentrations selected for these simulations are reported in Table 2.2.
2.5.2 Bimolecular Termination by Disproportionation
For these simulations, the value of ktc has been set to zero, i.e., bimolecular termination
is assumed to occur only by disproportionation. Two reaction rates have been examined,
corresponding to initial initiator concentrations I0 = 1 10-3 mol 1_1 and I0 = 1 10_1
mol 1_1.
The case of a lower reaction rate (lower initiator concentration) is first analyzed. The
distribution is calculated at 65% conversion through the four solution methods described
above. In the case of NF, it is first calculated with nb = 0, i.e., using the fractionation rules
by Teymour and Campbell (classical NF). This is done because the classical method often
predicts, in the presence of the mechanisms under consideration (chain transfer to polymer
and disproportionation) a pronounced shoulder in the high molecular weight portion of
the MWD or even a bimodal behavior, the existence of which in the real solution has
to be checked. Fig. 2.1(a) shows a comparison between the results of the classical NF
(solid curve) and the detailed solution method (dashed curve). It is apparent that the
high molecular weight maximum is an artifact of the NF technique. By examining the
24
Figure 2.1: Chain length distribution at 65% conversion in the case of termination by
disproportionation and low initiator concentration, (a) — : classical NF; — : reference
solution calculated by the detailed method; (b) classical NF with the contribution of the
individual generations.
25
contributions of the single generations to the overall CLD curve, as shown in Fig. 2.1(b),
it can be observed that this maximum is due to the first branched generation, which
is in fact a single generation containing all chains which are nonlinear. Teymour and
Campbell's scheme does not in fact admit the transition to a second branched generation
in this case, and the bimodality can therefore be related to the accumulation in the first
branched generation of chains adding a growing number of branches. This observation has
suggested the more detailed fractionation of the polymer chain population [1] discussed in
Section 2.3.4.
Note that when chain branching occurs through different mechanisms, namely, terminal
double bond propagation or crosslinking, this problem does not arise. This is because the
branching mechanism joins two polymer chains together and is therefore directly responsi¬
ble for the geometrical growth in chain size which rules the transfer between generations in
the classical fractionation scheme. Thus, this scheme is enough to provide a good descrip¬
tion of the branching process, which is not true in the case of chain transfer to polymer,
where the branching mechanism is not a chain coupling mechanism. This is also suggested
by the fact that the distributions calculated through the classical NF in the presence of
terminal double bond propagation or crosslinking (and in the absence of chain transfer to
polymer) do not exhibit the peculiar shoulders discussed here above [31, 57].
In Fig. 2.2 the correct solution is used as a term of comparison for the solution obtained
by the overall method of moments at an increasing number of moments (from kmax = 2 to
kmax = 35, where kmax is the maximum order of the moments used for the reconstruction
of the CLD). It can be seen that a very large increase of the maximum moment order
corresponds to a limited improvement of the obtained solution, which is in any case very
far from the 'true' solution. Increasing further the number of moments, severe oscillations
begin to appear, in agreement with previous results reported in the literature using La-
guerre polynomials [71, 84]. Therefore, when the full MWD is required, the overall method
of moments fails to provide a satisfactory solution to the problem, at least with the model
distribution chosen (perturbed T-distribution).
With the aim of testing to what extent a more detailed fractionation scheme can
improve the accuracy of the solution, the CLD has been reconstructed using NF with
increasing nb values. The results are shown in Fig. 2.3(a), where values of nb from 5 to 40
26
6
/ \
' /s i /j^"
log(n) 6
Figure 2.2: Chain length distribution at 65% conversion calculated by the overall method of
moments for an increasing number of moments (kmax = 2 to 35) in the case of termination
by disproportionation and low initiator concentration; — : reference solution calculated
by the detailed method.
log(n) 6
Figure 2.3: Chain length distribution at 65% conversion in the case of termination by
disproportionation and low initiator concentration, (a) Refined NF for increasing nb values;
(b) refined NF (with nb = 40) with the contribution of some of the generations.
28
have been used (note that the value of ng — nb is irrelevant to the result since with the
considered reaction mechanisms the maximum generation which can be reached is nb + 1).
It can be observed that for increasing nb values the high molecular weight maximum is
shifted to higher chain lengths while its intensity is reduced. This maximum is due to
the polymer chains accumulated in the last branched generation, constituted by polymer
chains with more than nb branches. This can be clearly seen in Fig. 2.3(b), where the
contribution of some of the generations has been reported under the overall CLD for the
case of nb = 40. It is evident that, for increasing nb values, the quantity of chains with
more than nb branches has to become negligible and the maximum has to disappear,
while the polymer tends to be entirely subdivided into generations each constituted by
chains with a given number of branches b. This suggests the more appropriate description
to be given by the PANB method, as long as a large enough number of branches bu is
chosen. In fact, subdividing the chains according to the number of branches and selecting
increasing bu values, the calculated CLD approaches the 'true' solution. This is shown
in Fig. 2.4, where the solution given by the PANB method is plotted for increasing bu
values (Fig. 2.4(a)) and at convergence to the correct solution (Fig. 2.4(b)). Therefore, in
this case the PANB method results optimal when a detailed MWD profile is required.
If the number bu is chosen large enough, so as to cover the entire molecular weight
range, the PANB solution method also provides in a straightforward way a detailed char¬
acterization of the polymer in terms of branching. In fact, the concentration of polymer
chains with a given number of branches b is provided by the zeroth-order moment po)b.
This quantity is reported in Fig. 2.5(a) for the case under examination.
Another quantity of interest for a complete description of the branching properties of
a polymer is the average number bn of branches per chain in chains of a given length n.
This is given by: V°M bP u
l«=W^ (2-12) 2^b=0 ^n,b
For the case under examination, this function is plotted in Fig. 2.5(b). It is interesting
to observe that the average number of branches increases linearly with chain length. This
feature gives a hint on a possible way to quickly estimate the number of branches needed
to cover the whole polymer CLD without seeking convergence on the CLD profile. In
fact, from Fig. 2.1, it can be seen that, though yielding an incorrectly shaped CLD, the
29
Figure 2.4: Chain length distribution at 65% conversion in the case of termination by
disproportionation and low initiator concentration, (a) PANB method for increasing bu
values; (b) — : PANB method with bu = 145; — : reference solution calculated by the
detailed method.
Chain Length (x10=3)
Figure 2.5: Branching properties at 65% conversion in the case of termination by dispro¬
portionation and low initiator concentration, (a) Concentration of chains as a function of
the number of branches; (b) average number of branches per chain as a function of chain
length.
31
classical NF is able to predict correctly the maximum chain length reached in the system.
Accordingly, the CLD can be first computed through the classical NF, which usually
requires short computational times. Once the maximum chain length is known, and once
the slope of the bn vs. n line is obtained by plotting eq. (2.12) with a low 6m value,
extrapolation of this line to the maximum chain length gives an estimate of the 6m value
required for the correct calculation of the CLD up to high chain length values. As a check
of the method, note that the value of about 140 branches estimated from Fig. 2.5(b) to
reach the maximum chain length 2 • 105 (see Fig. 2.1(a)), coincides with the value at which
convergence on the CLD is achieved (see Fig. 2.4), i.e., it coincides with the minimum
number of branches required for the correct solution.
A second case has been examined in which the reaction rate is enhanced by the use
of a higher initial initiator concentration (I0 = 1 • 10_1 mol 1_1). The CLD obtained in
this case by the classical NF is reported in Fig. 2.6 at 50% conversion (solid curve) and
compared to the CLD given by the detailed solution method (dashed curve). In these
conditions no bimodality is present and the dimensions of the shoulder are limited, so that
the solution provided by the classical NF is not far from being satisfactory as it is, without
the need for a more refined fractionation scheme. However, if a more accurate solution is
required, this is provided by the PANB method (with 6m = 80), as in the case of a slow
reaction rate previously examined.
The solution provided by the overall method of moments has also been analyzed for
comparison. In Fig. 2.7 it can be seen that this solution is nearer to the correct solution
than in the case of low initiator concentration. This can be ascribed to the fact that the
overall distribution is narrower. However, the solution given by the overall method of
moments is still far from the correct solution even at very high moment orders. Therefore,
also in this case the overall method of moments fails in providing a satisfactory CLD.
Summarizing, two cases have been examined, corresponding to two different reaction
rates. In both cases the classical NF has been shown to give an improved MWD solution
compared to the overall method of moments. However, it has also been shown that there
exist situations where the classical NF gives birth to bimodalities or pronounced shoulders
which do not appear in the true MWD. In this case, the refined NF technique proposed by
Arzamendi and Asua [1] can be used as a quick way to verify whether the shoulders are
32
1.8
1.6
*«
log(n)
Figure 2.6: Chain length distribution at 50% conversion in the case of termination by
disproportionation and high initiator concentration; — : classical NF; — : reference
solution calculated by the detailed method.
1.8
1.6
Ä 1.2
*•
log(n)
Figure 2.7: Chain length distribution at 50% conversion calculated by the overall method of
moments for an increasing number of moments (kmax = 2 to 35) in the case of termination
by disproportionation and high initiator concentration; — : reference solution calculated
by the detailed method.
33
an artifact of the NF solution method. It is enough to repeat the MWD calculation with
a small nb value, but greater than zero, and observe if the result changes. However, this
technique does not remove the basic problem of the classical NF which, with the kinetic
mechanisms considered, consists of the accumulation in the last generation (nb + 1) of the
chains with more than nb branches. For a correct calculation of the MWD, the optimal
solution method has been found to be the PANB method in both cases examined.
2.5.3 Bimolecular Termination by Combination
For these calculations, the value of ktd has been set to zero, i.e., bimolecular termination
is assumed to occur only by combination. Two reaction rates have again been examined,
corresponding to I0 = 1 • 10_1 mol 1_1 and I0 = 1 • 10-3 mol 1_1. Unlike the case of
bimolecular termination by disproportionation, the presence of a mechanism connecting
together chains which form branches through chain transfer to polymer can lead to the
formation of a gel phase. In both cases examined gelation is in fact predicted by all four
solution methods. In the case of the overall method of moments, the gel point is predicted
as the instant where the second- and higher-order moments diverge and integration of
eq (2.8) cannot be carried further. In the other cases, the amount of gel is calculated as
discussed in Section 2.4.
The case of a lower reaction rate (lower initiator concentration) is first examined.
In Fig. 2.8, the number- and weight-average chain lengths are reported as a function of
conversion. The solid curve refers to the overall method of moments, while the dashed
curve is the solution given by the NF (with nb = 0, i.e., classical NF). Note that up to
the gel point (vertical dotted line), which is predicted to be the same by the two methods,
these provide exactly the same answer. This means that, though the fractionated equations
require a closure formula, the average chain length values which they provide coincide with
the rigorous values given by the overall method of moments. Additionally, while the overall
method of moments fails at the gel point, NF permits the calculation of the average chain
length values of the sol phase up to complete conversion.
Regarding the calculation of the complete CLD, the solution obtained from the overall
method of moments at 65% conversion (just prior to gelation) is compared to the correct
one in Fig. 2.9. Since the distribution is very wide, the overall method of moments provides
a solution which is completely unsatisfactory. Moreover, increasing the maximum order
34
Gonwersion
Figure 2.8: Number- and weight-average chain length as a function of conversion in the
case of termination by combination and low initiator concentration; — : overall method
of moments; — : classical NF.
Figure 2.9: Chain length distribution at 65% conversion calculated by the overall method
of moments for an increasing number of moments (kmax = 2 to 3) in the case of termination
by combination and low initiator concentration; — : reference solution calculated by the
detailed method.
of moments, oscillations already start to appear at kmax = 3.
In Fig. 2.10(a) the solution given by the classical NF (solid curve) is compared to
the correct solution (dashed curve) at 65% conversion. The improvement in comparison
to the overall method of moments is apparent. However, as in the case of bimolecular
termination by disproportionation, it appears that NF is responsible for the arising of a
marked shoulder at high molecular weights which does not exist in the true CLD. From
Fig. 2.10(b), where the contributions of the single generations are reported under the
overall CLD, it can be seen that the presence of the shoulder is due to the first branched
generation. Although in this case the passage to the second generation is guaranteed by
the presence of combination, the first branched generation still contains an accumulated
amount of chains which have repeatedly added branches without being able to pass over to
the second generation, as they didn't undergo a combination event with another branched
chain. This problem can therefore be solved, as for the case of bimolecular termination by
disproportionation, by operating a finer subdivision of the less branched chains according
to the number of branches.
When this kind of partitioning is adopted, the marked shoulder actually tends to be
reduced. This is shown in Fig. 2.11, where the CLD calculated for increasing nb values is
reported (with ng — nb, i.e., the Teymour and Campbell generations, fixed and equal to 6).
Since for increasing nb values, the polymer present in generation nb+l (which is responsible
for the shoulder) tends to disappear, and the same happens to the polymer belonging to the
higher Teymour and Campbell generations, the description of these generations becomes
redundant and one can more appropriately pass to the PANB solution method.
The solution given by the PANB method with bu = 160 is reported in Fig. 2.12 (solid
curve) and can be seen to be practically coincident with the correct solution (dashed
curve). Therefore, also in this case the PANB method proves to be the most efficient
when a detailed MWD is required. Moreover, as shown in the case of termination by
disproportionation, it permits a complete description of the chain branching properties of
the polymer. The number of branches bu = 160 required to cover the whole MWD has
been obtained through the extrapolation method discussed earlier in Section 2.5.2. It can
be seen in fact from Fig. 2.10(a) that the classical NF provides a correct indication of the
maximum chain length in the system and it can be verified that the average number of
36
Figure 2.10: Chain length distribution at 65% conversion in the case of termination by combination and low initiator concentration, (a) — : classical NF; — : reference solu¬
tion calculated by the detailed method; (b) classical NF with the contribution of some
individual generations.
37
Figure 2.11: Chain length distribution at 65% conversion calculated by the refined NF
technique for increasing nb values (nb = 5 to 40) in the case of termination by combination
and low initiator concentration.
Figure 2.12: Chain length distribution at 65% conversion calculated by the PANB method
with 6m = 160 in the case of termination by combination and low initiator concentration; — : reference solution calculated by the detailed method.
38
Figure 2.13: Chain length distribution at 80% conversion (after gel formation) in the
case of termination by combination and low initiator concentration; — : PANB method
(6m = 160); - •- : classical NF; — : reference solution calculated by the detailed method.
branches vs. chain length relation is linear, as in the case of bimolecular termination by
disproportionation.
The calculation of the CLD has been repeated at 80% conversion, after gelation has
occurred, to examine the quality of the solution provided by the various methods after the
gel point. Of course, no solution can be obtained by the overall method of moments, since
the second- and higher-order moments diverge at the gel point. The detailed method has
been shown in Section 2.3.1 to provide a correct solution to the PBEs (2.3)-(2.4) up to
any selected chain length n«. This is true for all conversions, no matter if a gel phase
has formed or not. Therefore, the detailed solution method is able to provide a reliable
reference solution also after the gel point. The requirement of a sufficiently large number
of nodes at which the integration is performed has of course to be fulfilled.
In Fig. 2.13 the solution provided by the detailed method is used as a reference for
the solutions given by the classical NF and PANB methods. The picture is unchanged
compared to the pre-gel situation. The NF solution presents a shoulder that is an artifact
of the method and the PANB technique provides a solution practically coinciding with the
39
Figure 2.14: Chain length distribution at 50% conversion in the case of termination by combination and high initiator concentration; — : classical NF; — : reference solution
calculated by the detailed method.
correct one. Accordingly, the PANB method proves to be an efficient and reliable method
for the characterization of the soluble polymer fraction also in the post-gel period of the
reaction.
Finally, the case with a larger initiator concentration has been investigated. In Fig. 2.14
the solution obtained by the classical NF at 50% conversion (just prior to gelation) is
shown together with the solution given by the detailed method. The shoulder appears to
be reduced in dimension and the solution is not far from being satisfactory. Once more,
it can be shown that by adopting a finer subdivision of the polymer chain population, the
solution can be further improved. A solution indistinguishable from the rigorous one can
be obtained through the PANB method with 6m = 180.
As for the case of termination by disproportionation, two different situations have been
examined, corresponding to two different reaction rates. In the case of higher reaction
rates, the classical NF provides an acceptable solution. However, in the other case, where
both the pre- and post-gel reaction periods have been investigated, a marked shoulder
appears which is an artifact of the solution method, rather than a result of the reaction
40
mechanisms. A reliable solution has instead been shown to be obtained in all cases when
the polymer chains are subdivided according to the number of branches (provided that a
sufficiently large number of branches is considered). This approach also has the advantage
of providing a straightforward description of the polymer in terms of branching properties.
2.6 Conclusions
Several numerical methods have been developed in the literature to calculate the MWD in
free-radical polymerizations where the occurrence of chain branching leads to rather large
polydispersity values and possibly the formation of a gel phase, which implies infinitely
large molecules. In this work, a detailed numerical method has been developed which,
though computationally intensive, allows to compute the correct MWD. Using this method
it was possible to check the reliability of the previously proposed methods which, based
on some subdivision of the unknown polymer chain population, provide an estimate of the
MWD in reasonable computational times. These approximate methods result particularly
helpful when dealing with systems where several internal coordinates appear in the relevant
PBEs, e.g. segregated systems.
Of these techniques, the NF method proposed by Teymour and Campbell was found
to constitute an enormous improvement with respect to the classical method of moments,
requiring an increased but still very limited computational effort (10 generations are usu¬
ally more than enough to have convergence of the method). However, in some cases it is
not able to provide a correct description of the MWD, giving rise to artificial shoulders
at high molecular weights. This is due to the nature of the partitioning into generations
proposed by Teymour and Campbell, which can lead to the accumulation of chains with
a very different number of branches in the first branched generation. This problem is
only partially removed by the refined fractionation technique proposed by Arzamendi and
Asua. The best solution appears to be to simulate the chains subdividing them according
to the number of branches, as proposed by Soares and Hamielec. This usually requires
integration of a rather large number of ODEs, let's say, ten times as many as classical NF,
which implies correspondingly larger computational times, but a very accurate solution is
obtained.
It has to be stressed that the geometric growth mechanism proposed by Teymour and
41
Campbell as a route to gel formation is in any case extremely valuable not only from a
physical point of view, but also as a remarkably simple first approximation to a rather
difficult problem. Moreover, if average quantities of the MWD and not its detailed shape
are of interest, which is often the case, NF always provides the correct solution (without
being limited to the pre-gel phase as the overall method of moments).
If an accurate complete MWD is required, a quick-solution strategy has been indi¬
cated: through the classical NF the maximum chain length achieved in the system is first
estimated. Then, the number of branches corresponding to this maximum chain length
is estimated from the slope of the number of branches vs. chain length relation obtained
by the PANB method using a low number of branches. This step obviously requires the
number of branches to be a linear function of molecular weight (at least to a fairly good ap¬
proximation). Note that this requirement was completely fulfilled in all cases examined.
Finally, the MWD is computed through the PANB method using the calculated maxi¬
mum number of branches. This method provides, along with the MWD, the branching
distribution function.
bu total number of branches used in the PANB method
bn average number of branches in chains of length n
I (concentration of) initiator molecules
kfm rate constant for chain transfer to monomer
kfp rate constant for chain transfer to polymer
kft rate constant for chain transfer to chain transfer agent
ki rate constant for initiator decomposition
kp rate constant for propagation
he rate constant for termination by combination
hd rate constant for termination by disproportionation
M (concentration of) monomer molecules
nb number of branched generations whose index coincides with the
number of chain branches in the refined NF technique
ng total number of branched generations used in NF
um maximum chain length in the detailed solution method
Pn (concentration of) terminated chains of length n
Pn>b (concentration of) terminated chains of length n with 6 branches
Pn (concentration of) terminated chains of length n belonging to
generation g in NF
R'n (concentration of) active chains of length n
R'n b (concentration of) active chains of length n with 6 branches
R'n (concentration of) active chains of length n belonging to
generation g in NF
Greek letters
Afc kth-order moment of the CLD of the active polymer
Pk k -order moment of the CLD of the terminated polymer
Pk b kth-order moment of the CLD of the terminated polymer 6 branches
p^' kth-order moment of the CLD of the terminated polymer
generation g in NF
Bulk FRP.
3.1 Introduction
Kinetic models for simulating the MWD of polymers produced by FRP are based on a
set of PBEs which have to be solved numerically. A popular approach is based on the
method of moments, which is a general technique based on the evaluation of a finite
number of integral (or average properties), expressed in terms of the leading moments
of the unknown distribution. By perturbing a suitable "model distribution", e.g. a re¬
distribution [39], the corresponding complete MWD can be estimated from a very limited
set of moments. Accordingly, the method of moments represents an "indirect" approach