Ind. Eng. Chem. Res. 2009, 48, 5193–5210
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Particle Formation in Vinyl Chloride Emulsion Polymerization: Reaction Modeling Hugo M. Vale†,‡ and Timothy F. McKenna*,†,§ LCPP-CNRS/ESCPE-Lyon, BP 2077, 43 Bd du 11 NoVembre 1918, 69616 Villeurbanne Cedex, France
A mathematical model is developed to help quantify and improve our understanding of the formation of particles and the evolution of the particle size distribution (PSD) in the emulsion polymerization of vinyl chloride monomer (VCM). A special feature of the model is the computation of the coupled radical number and particle size distributions by means of the zero-one-two population balance equations. This approach is shown to be necessary when modeling processes that involve particle nucleation. Preliminary results are presented to illustrate the ability of the model in interpreting experimental data. 1. Introduction
2. Previous Work
Modeling the entire course of an emulsion polymerizations including the formation of the latex particles and the evolution of their particle size distributionsis an aspiration that is perhaps as old as the process itself. However, this problem remains largely unsolved, despite the significant progress made over the past six decades in understanding emulsion polymerization and quantifying the microstructure of emulsion polymers. A detailed discussion of this subject can be found in earlier publications from our group,1,2 as well as in numerous references cited therein.
2.1. Polymerization Kinetics. One of the most significant results in this area is attributed to Ugelstad et al.,7 who derived an approximate expression for the rate of polymerization by neglecting particles with more than two radicals and termination in the aqueous phase:
State-of-the-art models of emulsion polymerization are based on population balances1,3-5 (this is the reason for which they are also designated by population balance models). These models have, in theory, the ability to account for all of the phenomena that affect particle size and number: nucleation, coagulation, growth by polymerization, and radical kinetics. Nevertheless, despite such potentialities, their predictive power is still limited: even in the case of well-studied systems (e.g., styrene), model results are, more often than not, qualitative.1,2 The objective of the present work is to contribute to the development of this type of mechanistic model of emulsion polymerization and, more particularly, to the modeling of particle formation and particle size distribution (PSD) in the emulsion polymerization of vinyl chloride (VCM). Experimental results on the kinetics of polymerization and particle formation have been reported in a previous publication (see the first part of this series of papers6). Here, we will focus on the modeling of this system. The paper is organized as follows. In section 2, we review previous work concerning the modeling of VCM emulsion polymerization. Then, in section 3, the model that we developed for the free-radical batch emulsion polymerization of VCM is described in detail. Model simulations and comparisons against experimental data are presented in section 4. Finally, concluding remarks are given in section 5. * To whom correspondence should be addressed. E-mail:
[email protected]. † LCPP-CNRS/ESCPE-Lyon. ‡ Current address: Synthetic Rubber Group, Michelin Technology Center, Clermont-Ferrand, France. § Current address: Department of Chemical Engineering, Queen’s University, Kingston, Ontario, Canada.
Rp )
(
kp[M]p(2f kd[I ])1/2 Vp N + NA 2kt 2kdes
)
1/2
(1)
To complete this expression, the same authors further assumed that the radical desorption frequency (kdes) was dependent on particle size, according to the following relation: kdes )
3DE
(2)
rs2
where DE is an effective diffusion coefficient for radical desorption. The reader is referred to the Nomenclature section for the meaning of the various symbols. Ugelstad and co-workers,7,8 as well as Friis and Hamielec9 (using data from ref 7), found that eqs 1 and 2 could correctly describe Interval II kinetics over a wide range of particle number, concentration of initiator, and monomer conversion. Examples of kt and DE values determined by fitting kinetic data are given in Table 1. We note that some of the discrepancies observed are only apparent and result from differences in the values of kp, [M]p, and fkd assumed by the various authors, as attested by the two right-most columns in Table 1. Indeed, what one independently determines are the two groups of constants (or combinations of these) shown at the right of Table 1. Unfortunately, it is not rare to find works in the literature where this principle is ignored and inconsistent sets of constants are used. Ugelstad and co-workers also claimed that the good agreement between eq 1 and experimental results might indicate that Table 1. Kinetic Parameters of VCM Emulsion Polymerization (at 50 °C) kt reference (m3 mol-1 s-1) 7 8 9 10
8.4 × 104 1.34 × 105 3.04 × 104 2.04 × 105
DE (m2/s) 3.6 × 10-14 7.0 × 10-14 3.7 × 10-14 1.9 × 10-13
10.1021/ie801406n CCC: $40.75 2009 American Chemical Society Published on Web 05/05/2009
kp[M]p(fkd/kt)1/2 (mol1/2 m-3/2 s-1)
DE/kt (mol/m)
0.23
4.3 × 10-19 5.4 × 10-19 1.2 × 10-18 9.7 × 10-19
0.23 0.22
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aqueous-phase termination could be neglected under the chosen experimental conditions.10,11 Ugelstad et al.10 performed some calculations to quantify the importance of aqueous-phase termination under typical polymerization conditions and found that, unless the efficiency factor for radical entry is orders of magnitude below unity, termination in the aqueous phase can be safely neglected. Lee and Poehlein,12 in their simulations of a seed-fed CSTR, verified that the model deviations were higher for nonzero values of the term for aqueous-phase termination (used as an adjustable parameter). A rather different point of view was expressed by Nilsson et al.,13 who performed a calorimetric investigation of the emulsion polymerization of VCM. They concluded that aqueous-phase termination was of measurable importance in Interval II and was the dominating termination reaction in Interval III. Nevertheless, these authors did not show numeric values of nj, nor did they compare the conversions that were determined by calorimetry with gravimetric measurements. Tauer and Petruscke14 performed seeded experiments to determine the influence of the number and size of the seed particles on the polymerization rate. They found that the rate of polymerization could be expressed as Rp ) AN1/2 + B (where A and B are parameters that are dependent on the particle size), in agreement with the dependence on N given by eq 1. 2.2. Particle Formation. According to Ugelstad et al.,10 the high rate of radical desorption is the factor that is responsible for the special features of particle formation in VCM emulsion polymerization: N is independent of [I], there is no abrupt change in N at the CMC (note, however, that the results presented in the first part of this series of papers6 show that this is not necessarily the case), and the values of N are high in comparison to other monomers under similar conditions. Two arguments are presented to justify this idea: (i) desorbed radicals participate in particle nucleation, and (ii) the desorption/re-entry mechanism decreases the particle growth rate, which, in turn, decreases the rate of micelle consumption. The authors base their hypothesis on some theoretical results, as shown below. Hansen and Ugelstad15 analyzed the effect of radical desorption on the rate of micellar nucleation by means of a simplified treatment. They observed that radical desorption increased the particle number and changed the order of the particle number, with respect to the concentrations of initiator and surfactant (as predicted by the Smith-Ewart theory). For VCM emulsion polymerization, the calculated orders with respect to [I] and [S] were determined to be zero and one, respectively, in agreement with experimental results previously obtained by the authors,7 although no comparison between data and theory was actually presented. To bring the experimental and calculated particle numbers into agreement, a very low value of the radical capture efficiency of a micelle, relative to a particle, had to be assumed, as in similar studies with vinyl acetate.16 Ugelstad et al.11 investigated the effect of radical desorption on the rate of homogeneous nucleation by extending the model of Hansen and Ugelstad.17 In this new treatment, the balance equations for the aqueous-phase species account separately for the presence and length of transfer-derived radicals. As a result, the expression for the particle nucleation rate contains an additional term that corresponds to nucleation by transferderived radicals. Although the equation shows how the exit of radicals might increase the number of particles nucleated, the actual contribution of this phenomenon is not clear. Finally, we note that this expression was not tested against experimental data.
Melis et al.18 developed a model to analyze the influence of aggregation phenomena in the emulsion polymerization of VCM. The model was specifically conceived to describe seeded polymerizations and does not account for particle nucleation. Particle coalescence is assumed to occur only by perikinetic aggregation. In the Fuchs’ stability ratio, the hydrodynamic interaction is neglected and the total particle interaction energy is computed from the DVLO theory. It is assumed that every particle has the same diffuse potential (ψd). The evolution of the PSD is described by the pseudobulk model. A series of seeded batch reactions were simulated to assess the effect of ψd and ionic strength on the conversion, the number of particles, nj, and the final PSD. The authors concluded that the stability of the system is very sensitive to ψd (a variation as small as 5% can completey change the stability of the system) and less sensitive to the ionic strength. 2.3. Models. Min and Gostin19 were perhaps the first to establish a mathematical model to predict the time evolution of the PSD in the semibatch emulsion polymerization of VCM. The model is a simplification of the general modeling framework presented by Min and Ray.3 Homogeneous nucleation is neglected and, thus, new particles can only be formed by micellar nucleation. Particle coagulation is taken into account, with the coagulation rate coefficients β given by
( )
β ) β0 exp
E* kBT
but the authors do not mention how they calculate the parameter E* (this is the energy barrier for aggregation). Judging from Min and Ray,3 E* is probably related to the square of the surface charge density of the particles. The evolution of the PSD is described by the pseudobulk model. The population balance equation (PBE) is solved by the method of moments,20,21 and a Laguerre polynomial expansion is used to reconstruct the differential distribution. The authors presented a series of simulations to illustrate the capabilities of the model in predicting the effect of several variables on the final latex PSD: seed particle size, quantity of seed, solid content of the seed, and initial amount of initiator. A single comparison between experimental and predicted PSD (bimodal) was made, with the authors claiming excellent agreement. Penlidis et al.22 proposed a model for the batch and continuous emulsion polymerization of VCM that is essentially a modification of the work previously developed by Kiparissides et al.23 for vinyl acetate. The model is based on an age distribution analysis in which classes of particles born in a given time interval are followed through the reactor. By integrating over all classes of particles in a CSTR, a set of ODEs (termed total property balance equations) is obtained for the total particle size properties, number of particles, etc. Micellar as well as homogeneous nucleation is considered, but particle coagulation is ignored. The model was first tested against experimental batch polymerization data obtained by Ugelstad et al.7 By adjusting two parameters (the diffusivity of the monomer in the aqueous phase and the ratio of the radical capture rate constant to the rate coefficient of micellar nucleation), the authors were able to correctly predict the conversion profiles for different concentrations of initiator (potassium persulfate) and surfactant (SDS). The particle numbers computed from the model were also in good agreement with the experimental results, although it is not clear how this was possible. Actually, the authors show a plot of N versus [SDS] in the range 0.3 g/L < [SDS] < 3.0 g/L, where we can see that the value of N predicted by the model continually increases with [SDS]. Because particle coagulation is not taken into account, [SDS] can only exert an influence
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over N by controlling the extent of micellar nucleation, and, thus, only for [SDS] > CMC. Therefore, it seems that the authors assumed a value for the CMC of SDS below 0.3 g/L, which would require some justification (the CMC of SDS in water is ∼2.6 g/L at 50 °C).6 In a subsequent paper,24 this same model was also compared with data from Berens.25 Nevertheless, the authors seem to have made a strange conclusion: they interpreted Berens’ batch data as ab initio polymerizations, when they were actually seeded experiments. Of course, under such conditions, comparing the number of particles estimated by the model is completely meaningless. The authors also simulated experimental CSTR data from Berens25 and showed that their model predicted the same trends and led to the same confusion that Berens had discussed in his article. Lee and Poehlein12 employed the model developed by Poehlein et al.26 to simulate the emulsion polymerization of VCM in a seed-fed CSTR. The model basically accounts for the effect of the residence time distribution of a CSTR on the growth of the seed particles, neglecting both particle nucleation and coagulation. The simulation results could be reasonably well-fitted to the PSD data published by Berens.25 Nevertheless, the value of the ratio DE/kt estimated by fitting (DE/kt ) 5.2 × 10-20) was 1 order of magnitude lower than expected (cf. Table 1). This discrepancy was attributed to the eventual formation of new particles in the reactor, which is an aspect that was not taken into account by the model. Tauer and Mu¨ller27 proposed a simple model to describe the sustained oscillations in the continuous emulsion polymerization of VCM. The model is based on the monodispersed approximation and takes particle coagulation into account. Regarding particle formation, empirical expressions are used both for homogeneous nucleation (the concentration of free surfactant in the aqueous phase, [S]w, is kept below the CMC); for coagulation, the rate of homogeneous nucleation is assumed to be proportional to [S]w, while the coagulation rate is unaffected by this variable. Such dependencies are rather uncommon (the opposite would be expected). Simulations were then performed to analyze the impact of some model parameters upon the occurrence of sustained oscillations. The authors concluded that particle coagulation was essential to model said oscillations. Forcolin et al.28 improved the model of Melis et al.,18 which has been described previously. The new model considers the possibility of both micellar and homogeneous nucleation. With respect to particle coagulation, the coagulation rate coefficients are evaluated using the approach proposed by Melis et al.,29 the essential feature of which is the ability to account for orthokinetic and perikinetic aggregation simultaneously. The model, without coagulation, was used to interpret experimental conversion and molar mass data. The agreement, with respect to the conversion profile and molar mass distribution, was reasonable in some cases, but no information was given on particle number or size. Regarding latex stability and PSD, the authors only performed numerical simulations to assess the influence of the concentrations of emulsifier and buffer and the agitation rate. The results obtained showed a limited contribution of the flow to the coagulation mechanism, which, according to the authors, corresponds to an underestimation of this effect (no data were presented). Kiparissides et al.30 recently proposed a model to evaluate the effect of oxygen concentration on the kinetics and PSD of VCM emulsion polymerization. Material balances, including monomer partitioning, are solved according to the methodology presented by Richards et al.31 Micellar and homogeneous
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nucleation are considered. The evolution of the PSD is described by the pseudobulk model. Note that the authors seem to account for re-entry of desorbed radicals twice in the total entry frequency (cf. eq 43 of ref 30). Particle coalescence is assumed to occur only by perikinetic aggregation, and the Fuchs’ stability ratio is calculated by a semiempirical expression that was attributed to Feeney et al.32 Note, however, that this expression, which was adjusted for polystyrene particles in a particular range of sizes and diffuse potentials, is used here, regardless of such restrictions. In the computation of the surface charge density of the particles, the authors just take into account the contribution, relative to the ionic end groups, because only emulsifierfree systems are analyzed. The surface charge density is estimated as a function of the particle radius by means of an expression previously proposed by Kiparissides.33 Nevertheless, this expression, which has a primordial role in determining the colloidal stability of the particles, raises several questions: (1) The said expression, σ ) (C/4π)r p-2, where C and p are adjustable parameters, has no theoretical or empirical basis. (2) The parameter C was estimated from a single value of the zeta potential of primary suspension particles, which is highly questionable. First, because of the obvious problems of obtaining a correlation out of a single point. Second, because of the difficulties related to the conversion of zeta potentials (or, more exactly, electrophoretic mobilities) to surface charge densities.34 Third, and more important, because the nature and mechanism of formation of the surface charges in primary suspension particles have nothing to do with those of emulsion particles. In suspension, the superficial charge is due to Cl- ions formed during the polymerization process itself,35 whereas, in emulsion, it results from sulfate and eventually carboxylate groups (see, e.g., refs 6 and 36). (3) The parameter p was set to 2.5, meaning that σ continuously increases with particle size. This evolution does not agree with experimental observations.36 (4) The surface charge density values obtained by such correlations are orders of magnitude smaller than those found experimentally. For instance, for a 150-nm particle, the correlation predicts σ ) 0.06 mC/m2, whereas experimental values are ∼5 mC/m2 (see, e.g., refs 6 and 36). The predictive capabilities of the model were tested against experimental measurements of conversion and average particle diameter over a range of oxygen and initiator (APS) concentrations, in the absence of surfactant. The model could successfully explain the effect of the oxygen concentration on the evolution of conversion and particle size. With increasing oxygen concentration, the polymerization rate increases and the particle diameter exhibits a U-shaped curve (attributed to the competition between the increase in radical production and the decrease in particle stability caused by the formation of HCl). We note that no inhibition periods are observed and no mention is made of the use of buffers or other substances, apart from VCM, oxygen, and APS, which seems surprising, in view of the results reported by Mørk et al.37,38 Indeed, according to the latter authors, we would only expect to observe an influence of [O2] upon Rp in the presence of bases. Besides, an inhibition period would be expected. Kiparissides et al.30 also performed some simulations to analyze the influence of conversion and initiator concentration on the shape of the latex PSD, but no comparison between experimental and predicted PSDs was given. 2.4. Current Status and Objectives. The literature review shows that the situation regarding the modeling of vinyl chloride emulsion polymerization is no different from the general panorama presented in recent publications from this group.2,39
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Actually, the opposite would hardly be expected, given the particular challenges posed by this system: significant homogeneous nucleation, experimental difficulties associated to the manipulation of a liquefied vapor, flammability, toxicity, and so forth. Among all subjects, the kinetics of polymerization is the more investigated one and, apparently, there is a consensus about the major questions. As for the quantitative description of particle formation, the situation is clearly less favorable. In fact, most models are either too simplisticssome neglect homogeneous nucleation, others neglect particle coagulation, and yet others use inconsistent stability models, just to cite the most evident shortcomingssor do not pay sufficient attention to the experimental validation. Consequently, it becomes very hard to extract useful information from these studies (e.g., what works and what does not, and why?). The most significant result, although not fully explored, is the idea defended by Ugelstad and collaborators,11,15 that desorbed radicals participate in micellar and homogeneous particle nucleation, because this mechanism would offer a plausible explanation for the absence of any effect of [I] upon N below and above the CMC (see the first of this series of papers6). Given the aforementioned difficulties, the presentation of a robust model here that is capable of simulating in detail the emulsion polymerization of VCM would be unrealistic. Instead, our goal is to develop a model that can help us to clearly identify what we do (and do not) know about the process (rather than simply fitting curves to experimental data). To do so, we will build a first-principles model, taking into account, as much as possible, all the major phenomena that are believed to occur in the emulsion polymerization of VCM and confront its predictions to the experimental data collected in the first of this series of papers.6 In this manner, we hope to answer certain questions concerning the formation of particles and the evolution of the PSD in this system, as well as to identify axes for further progress.
Table 2. Comparison between the Rate of Aqueous-Phase Propagation and the Rate of Re-entry of Exited Radicals, According to the Methodology of Casey et al.40 (T ) 50 °C and rs ) 10 nm)
N (part/L)
rate of propagation, kpw ′ [M]w (s-1)
rate of re-entry, kepE(0)N/NA (s-1)
propagation/ entry
Styrenea 1.0 × 10
1.1
1.0 × 10 1.0 × 1017 1.0 × 1018
1.8 × 10 1.8 × 103 1.8 × 103
16
1.9 × 103
6 × 10-4
2.5 × 103 2.5 × 104 2.5 × 105
0.70 0.07 0.01
VCMb 16
3
a Parameter values taken from Casey et al.40 summarized and discussed in section 3.7.
b
Parameter values
Figure 1. Scheme of the aqueous-phase and phase-transfer events considered in the present work.
3. Model This section describes the model that we have developed for the free-radical batch emulsion homopolymerization of vinyl chloride. We begin by discussing some aspects related to the chemistry that is assumed and then concentrate on the complete set of equations that constitute the model. The meaning of all symbols and variables is summarized in the Nomenclature section. 3.1. Chemistry. 3.1.1. Reaction Scheme. As discussed in section 2, the literature suggests that, in the case of VCM emulsion polymerization, transfer-derived radicals might be directly involved in particle nucleation (micellar and homogeneous). In addition, a comparison of the estimated rates of aqueous-phase propagation and re-entry of exited radicals (Table 2) shows that, in vinyl chloride emulsion polymerization, the probability of propagation might not be negligible, with respect to that of re-entry. This is in contrast to styrene emulsion polymerization, where the aqueous-phase propagation of exited radicals is frequently ignored a priorisand has essentially ′ [M]w in the case of involved with a possibly high value of kpw VCM. These two facts demonstrate the need to consider a scheme where (i) initiator-derived and transfer-derived radicals are properly taken into account and distinguished, and (ii) transfer-derived radicals, just like initiator-derived radicals, are allowed to propagate and terminate in the aqueous phase, and to nucleate new particles. The contribution of these
Figure 2. Possible mechanisms of chain transfer to monomer in the freeradical polymerization of vinyl chloride.
phenomena is determined by the magnitude of the corresponding rate coefficients. The reaction scheme assumed in this work (Table 3) satisfies these requisites and can be seen as an extension of the model of Gilbert and co-workers5 to incorporate the ideas of Ugelstad et al.11 (cf. section 2.2). The reader is referred to Figure 1 for an illustration of the reaction scheme. 3.1.2. Chain Transfer to Monomer. The details of the chain-transfer mechanism are important, because they determine the nature of the exited species E• and the associated properties and rate coefficients. Unfortunately, as discussed next, some doubts remain, with regard to the actual mechanism of chain transfer to monomer in the free-radical polymerization of vinyl chloride. Nowadays, it is widely accepted that radical activity is not transferred directly from the polymer chain to the monomer by abstraction of an H or Cl atom, but through a more complex multistep mechanism.10,42-44 The first two steps are believed
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to be a head-to-head addition of monomer to the growing chain, followed by a 1,2-Cl migration (see Figure 2). The steps leading to the formation of monomeric radicals from the last species are not consensual and two alternatives dominate the discussion. A first and older possibility (mechanism A) involves the loss of a Cl · radical, which may then start a new chain.43 More recently, Starnes et al.45 gathered strong evidence for a new mechanism (mechanism B) involving either a direct interaction with VCM or a reaction via the scavenging of Cl- ions from a molecular cage. In this case, the amount of kinetically free chlorine radicals is very limited. Starnes and co-workers also defend the existence of an auxiliary transfer mechanism44-46 that is concurrent with the first mechanism. Again, only monomeric radicals of type ClCH2-C˙HCl are involved, not chlorine radicals. Classically, the high desorption frequencies found in VCM emulsion polymerization have been explained in terms of the formation of Cl · radicals (readily desorbable). Nevertheless, the findings of Starnes and collaborators show that this question is not yet closed. To address this problem, we will have to resort to a sensitivity analysis of the rate coefficients of E · . 3.2. Aqueous-Phase and Phase-Transfer Events. 3.2.1. Balances to Initiator-Derived Radicals. The material balances to the initiator and the initiator-derived oligoradicals present in the aqueous phase can be written as follows:
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d[I] ) -kd[I] dt
(3)
d[IM·1] ) 2fkd[I] - kpw[IM·1][M]w - ktw[IM·1][T·] dt
(4)
d[IM·i] · ) kpw([IMi-1 ] - [IM·i])[M]w - ktw[IM·i][T·] dt (2 e i e z - 1) (5) d[IM·i] · ) kpw([IMi-1 ] - [IM·i])[M]w - ktw[IM·i][T·] dt N 〈kepI(i)〉[IM·i] - kemI(i)[IM·i][MIC] (z e i e jcr - 1) NA (6)
( )
where [I] is the initiator concentration, [IMi· ] the concentration of oligoradicals of length i, [T · ] the total concentration of radicals that can undergo termination, [M]w the monomer concentration in the aqueous phase, [MIC] the concentration of micelles, kd the initiator decomposition rate coefficient, f the initiator efficiency, kpw the aqueous-phase propagation rate coefficient, ktw the aqueous-phase termination rate coefficient, 〈kepI(i)〉 the average rate coefficient for entry of initiator-derived radicals of length i into particles, kemI(i) the rate coefficient for
Table 3. Reaction Scheme Assumed for the Emulsion Polymerization of VCM reaction step
expression
reaction step
Aqueous-Phase Reactions
Phase-Transfer Processes
initiator decomposition
I-I 98 2fI· + (1 - f ) others
propagation
I· + M 98 IM·1
kd
kpw
IM·i + particle 98 R·i
re-entry into particles
EM·i + particle 98 R·i
kepE(i)
(1 e i e jcr - 1) exit from particles
k′pw
kpw
ktw
termination
IM·i + T· 98 inerts ktw
EM·i + T· 98 inerts
(z e i e jcr - 1) (0 e i e jEcr - 1)
kdM
E·(particle) 98 E· + particle kpw
E· + M 98 EM·1 · EM·i + M 98 EMi+1
kepI(i)
entry into particles
kpI
· IM·i + M 98 IMi+1
expression
homogeneous nucleation
IMj· cr-1 + M 98 new particle kpw
(1 e i e jEcr - 1)
EMj· Ecr-1 + M 98 new particle
(1 e i e jcr - 1) micellar nucleation
IM·i + micelle 98 new particle (z e i e jcr - 1)
(0 e i e jEcr - 1)
kemI(i)
kemE(i)
Polymer-Phase Reactions kp
propagation
· R·n + M 98 Rn+1
terminationa
R·n + Rm· 98 dead polymer
transfer to monomer
R·n + M 98 Pn + E·
reinitiation
E· + M 98 R·1
a
EM·i + micelle 98 new particle (0 e i e jEcr - 1)
kt
kfM
kp′
Bimolecular termination is believed to occur mainly by combination,41,42 but this question is actually of no consequence for our purposes.
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entry of initiator-derived radicals of length i into micelles, and N the number of particles per unit volume of aqueous phase. In deriving eq 4, it was assumed that the homolysis of the initiator is the rate-determining step in the initiation sequence; that is, kpI . 2fkd.47 The rate coefficients kpw and ktw were considered to be independent of radical length and/or type, because of the lack of more-precise information (see section 3.7). 3.2.2. Balances to Transfer-Derived Radicals. The material balances to the transfer-derived radicals present in the aqueous phase can be written as follows:11 d[E·] ) Φdes - kpw ′ [E·][M]w - ktw[E·][T·] dt N 〈kepE(0)〉[E·] - kemE(0)[E·][MIC] (7) NA
( )
where [EMi· ] is the concentration of transfer-derived radicals with i monomer units added, Φdes the rate of radical desorption, ′ the rate coefficient for propagation of species E · in the kpw aqueous phase, 〈kepE(i)〉 the average rate coefficient for entry of species EMi· into particles, and kemE(i) the rate coefficient for entry of said species into micelles. The total concentration of radicals that can undergo termination in the aqueous phase is given by jcr-1
Rnuc ) Rhom + Rmic
(17)
( ) ( )
kepI(i) ) 4πrsNA
( )
( )
(16)
Independent of their origin, all nucleated particles are assumed to have the same unswollen radius, rnuc. It can be shown by physical as well as numerical arguments that this simplification is plausible.1 3.2.4. Entry Rates. The rate of radical entry is assumed to be diffusion-controlled, for which there is experimental evidence obtained, in particular, from competitive growth experiments performed with VCM.8 As a first approximation, the rate coefficients for entry of initiator-derived radicals into particles and micelles are computed by the following formulas,48 which are valid for highly diluted polymer dispersions:
d[EM·] ) kpw ′ [E·][M]w - kpw[EM·][M]w - ktw[EM·][T·] dt N 〈kepE(1)〉[EM·] - kemE(1)[EM·][MIC] (8) NA d[EM·i] · ) kpw([EMi-1 ] - [EM·i])[M]w - ktw[EM·i][T·] dt N 〈kepE(i)〉[EM·i] - kemE(i)[EM·i][MIC] (2 e i e jEcr - 1) NA (9)
Rmic ) Rmic,I + Rmic,E
w DM
i1/2
kemI(i) ) 4πfmicrmicNA
w DM
i1/2
(18)
(19)
where rs is the swollen particle radius, rmic the micelle radius, w the fmic the efficiency of radical entry into micelles, and DM diffusion coefficient of the monomer in the aqueous phase. The entry rate coefficients for the transfer-derived radicals, EMi· , are given by similar expressions, but with i f i + 1. This means that, for diffusion purposes, an E unit is assumed to count as an M unit. Note that recent works49,50 have suggested that the aforementioned formulas might significantly underestimate the entry rate coefficients in nondiluted systems. In addition, there is no consensus regarding the exponent of i; however, Coen et al.48 reported that model predictions are insensitive to the value of the exponent in the range of 0.5-1. The average entry rate coefficients are related to the coefficients just defined by
jEcr-1
(10)
〈kepI(i)〉 )
1 N
∫
kepI(i) f (r, t) dr
(20)
3.2.3. Nucleation Rates. The rates of homogeneous nucleation due to initiator-derived and transfer-derived radicals are given, respectively, by
〈kepE(i)〉 )
1 N
∫
kepE(i) f (r, t) dr
(21)
[T·] )
∑
∑ [EM ]
[IM·i] +
· i
i)1
i)0
Rhom,I ) kpw[IMj· cr-1][M]w
(11)
Rhom,E ) kpw[EMj· Ecr-1][M]w
(12)
∞
0 ∞
0
where f (r,t) is the number density function. Finally, the total radical entry frequency into a particle of swollen radius rs is given by jcr-1
F)
∑
jEcr-1
kepI(i)[IM·i] +
i)z
Analogously, for the rates of micellar nucleation we have jcr-1
Rmic,I ) [MIC]
∑k
· emI(i)[IMi]
(13)
i)z
∑k
· epE(i)[EMi]
(22)
i)0
where the first term accounts for the entry of initiator-derived radicals and the second term for the re-entry of transfer-derived radicals. 3.2.5. Exit Rate. The rate of desorption of E · radicals from the particles to the aqueous phase is given by
jEcr-1
Rmic,E ) [MIC]
∑k
· emE(i)[EMi]
(14)
i)0
Finally, the expressions for the total homogeneous nucleation rate, total micellar nucleation rate, and the total rate of particle nucleation read, respectively, as follows: Rhom ) Rhom,I + Rhom,E
(15)
Φdes )
1 NA
∫
∞
0
kdesnj f (r, t) dr
(23)
where kdes is the desorption frequency and nj is the average number of radicals per particle. Several expressions for kdes have been proposed, namely, by Ugelstad and Hansen,51 Nomura, Harada, and co-workers,52,53 and Asua and co-workers,54,55 just to cite the most well-known.
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Figure 3. Comparison of the rates of propagation (k′[M] p), desorption (kdM), p and termination (cML) of a monomeric radical, as a function of the swollen radius for VCM emulsion polymerization at 50 °C.
These relationships differ on, among other things, the assumptions made about the fate of the exited radicals. In the present work, we decided to utilize the semiempirical expression recommended by Ugelstad et al.11 for the emulsion polymerization of VCM (cf. eq 2): kdes )
3DE rs2
where DE is the effective diffusion coefficient for radical desorption. We do not claim that this equation is superior to others; we adopt it because of its simplicity (a single adjustable parameter, DE) and because it has been used with success to describe the polymerization kinetics of this system. Obviously, for a system where radical exit is thought to be so important, finding a theoretically sound, coherent expression for kdes should be a future priority. Alternatively, the use of a relationship for kdes may be avoided by distinguishing between monomeric and polymeric radicals in the particle population balances (at the expense of an increase of the complexity of the PBEs). 3.3. Particle Population Balance. 3.3.1. Available Models. Two types of models are most commonly used to describe the particulate phase of emulsion polymerization systems:1 the zero-one (0-1) model and the pseudobulk (PB) model. By definition, the 0-1 model is restricted to systems where intraparticle termination is pseudo-instantaneous (i.e., termination is so fast that it is not rate-determining). Because of this constraint, the use of the 0-1 model is inherently limited to small particles. An estimate of the maximum particle size up to which the assumptions of the 0-1 approach are valid can be obtained by comparing the relative rates of propagation, desorption, and termination of a monomeric free radical.40 Following this method, we conclude that, in the case of vinyl chloride, the hypothesis of pseudo-instantaneous termination is not valid for particles of swollen radius greater than ∼5-10 nm (see Figure 3), because exit of the monomeric radical is more likely than propagation and termination. This is analogous to that found with methyl methacrylate.40 Clearly, such a condition upon rs is too severe for any practical application. The PB model is, by far, the most employed in the polymer reaction engineering field and has been utilized in all population balance models of VCM emulsion polymerization developed
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so far (see section 2.3). Nevertheless, the PB approach is not without its limitations; namely, using this approach, the average number of radicals per particle is computed by an approximate expression that neglects the effects of nucleation, coagulation, growth, and dynamics.1 Although this simplification works well under certain conditions, it generally is not valid. In particular, we may expect it to lead to severe errors when simulating particle formation processes, because the contribution of the effects neglected can then be significant. Examples of cases where the PB model works well and fails will be presented below. To surmount the difficulties encountered with the 0-1 and PB models, we must find an alternative that allows for the fact that intraparticle termination is not instantaneous and that correctly accounts for particle size as well as number of radicals per particle. The simplest option is the zero-one-two (0-1-2) model, which is described next. 3.3.2. Zero-One-Two Model. In a zero-one-two (0-1-2) system,56-58 the entry of a radical into a particle that contains two radicals is assumed to result in pseudo-instantaneous termination. This simplification is valid when nj is low, but termination is rate-determining. Therefore, the 0-1-2 model is well-suited for VCM emulsion polymerization, seeing that nj is usually in the range of 0.001-0.1 and that the 0-1 model is not applicable. In fact, the 0-1-2 approach has been used with success by Ugelstad et al.7 to describe the kinetics of this system. Based on the 0-1-2 model, the full set of PBEs for a perfectly mixed batch reactor reads as follows: ∂f0 ) -Ff0 + kdesf1 + 2cf2 + ∂t 1 ν-νnuc β(ν - ν′, ν′)[f0(ν - ν′)f0(ν′) + 2 νnuc ∞ f2(ν - ν′)f2(ν′)] dν′ - f0(ν) ν β(ν, ν′)f(ν′) dν′ (24)
∫
∫
nuc
∂f1 ∂ ) - (Kf1) + Ff0 - (F + kdes)f1 + (F + 2kdes)f2 + ∂t ∂ν
∫
ν-νnuc
νnuc
β(ν - ν′, ν′)[f0(ν - ν′)f1(ν′) + f1(ν - ν′)f2(ν′)] dν′ f1(ν)
∫
∞
νnuc
β(ν, ν′)f(ν′) dν′ + δ(ν - νnuc)Rnuc (25)
∂f2 ∂ ) -2 (Kf2) + Ff1 - (F + 2kdes + 2c)f2 + ∂t ∂ν ν-νnuc β(ν - ν′, ν′)f0(ν - ν′)f2(ν′) dν′ + ν
∫ ∫
nuc
1 2
ν-νnuc
νnuc
β(ν - ν′, ν′)f1(ν - ν′)f1(ν′) dν′ f2(ν)
∫
∞
νnuc
β(ν, ν′)f(ν′) dν′ (26)
where fn(V,t) is the number density function for particles having n radicals, V the unswollen volume of a particle, K the rate coefficient of volume growth (K ) kpMM[M]p/(NAFp)), c the pseudo-first-order rate coefficient for termination in the particles (c ) 3kt/(4πNArs3)), and β the binary coagulation rate coefficient. The overall number distribution is given by f(V,t) ) ∑nfn(V,t). Note that (i) in the limit c f ∞, the 0-1 equations are recovered; (ii) by summing the three subequations, the model reduces to the PB model with nj(V,t) ) (f1 + 2f2)/f, for f * 0. To the best of our knowledge, eqs 24-26 have not previously been used to describe the evolution of PSD in emulsion polymerization (neither for VCM, nor for any other system). 3.4. Monomer Balance. The total monomer consumption rate (Rp) is given by the sum of the polymerization rates in the
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particles (Rpp) and in the aqueous phase (Rpw), which may be written as Rpp )
kp NA
∫
∞
[M]pnjf(r, t) dr
0
(
Rpw ) 2fkd[I] + kpw[M]w
jcr-1
(27) jEcr-1
)
∑ [IM ] + ∑ [EM ] · i
· i
i)1
i)0
(28)
Typically, Rpp . Rpw, except during the very first instants of the polymerization. For a batch reactor, the mass balance for the monomer then reads dmM ) -(Rpw + Rpp)MMVw dt
(29)
where MM is the molar mass of the monomer. The vinyl chloride monomer is distributed between four phases: the vapor phase, the aqueous phase, the particles, and the monomer droplets. Assuming that the agitation of the reactor contents is good enough such that the mass-transfer rate between phases is sufficiently high, in comparison to Rp, the monomer concentrations in each phase can be computed by supposing thermodynamic equilibrium. Therefore, up to the end of Interval II (i.e., while a separate “pure” monomer phase exists), we have [M]w ) [M]wsat and [M]p ) [M]psat. The modeling of Interval III will not be addressed here, because our experiments do not include this period. Nevertheless, the interested reader can find useful information regarding this issue in refs 43, 59, 60, and 61. 3.5. Surfactant Balance. Neglecting the amount of surfactant adsorbed on the monomer droplets, the material balance to the surfactant reads [S] ) [S]w + [MIC]nagg + Γ
( ) Ap Vw
(30)
where [S] is the total amount of surfactant per unit volume of aqueous phase, [S]w the concentration of free surfactant in the aqueous phase, nagg the aggregation number of the micelles, Γ the surface concentration of surfactant adsorbed on the particles, and Ap/Vw the total area of the particle phase per unit volume of aqueous phase, which is given as Ap ) 4π Vw
∫
∞
0
rs2f(r, t) dr
(31)
To compute the concentrations of emulsifier in the various phases, thermodynamic equilibrium is assumed. As noted by certain authors,1 this hypothesis is perhaps invalid (especially for the small precursor particles), but, to date, there is no viable alternative. Accordingly, the adsorption isotherms determined in a previous study39 are supposed to apply:
(
Γ ) Γ∞
Ks[S]wb 1 + Ks[S]wb
)
(32)
To compute [S]w, Γ, and [MIC], eqs 30 and 32 must be solved together with the following set of conditions:
{
[S]w < CMC ∧ [MIC] ) 0, or [S]w ) CMC ∧ [MIC] g 0
(33)
3.6. Particle Coagulation Rates. The computation of the particle coagulation rates in emulsion polymerization systems
is a very difficult subject, because of the complexity of the phenomena involved and the lack of suitable physical models. In this regard, it is worth recalling that there is strong experimental evidence that the well-known and widely used DLVO theory generally is not able to predict the aggregation rate of colloids quantitatively. Additional information about this issue can be found in the work of Vale and McKenna1 and references cited therein. In practice, these difficulties mean that only approximate treatments of particle coagulation are actually possible. In the present work, collisions involving three or more particles are neglected (as implicitly considered in eqs 24-26) and it is assumed that the rate-determining step of the binary aggregation process is the encounter between the two latex particles. Additional mechanisms, such as particle coalescence induced by radical desorption,62,63 are not taken into consideration. Moreover, given the experimental findings reported in the first part of this series of papers,6 the contribution of shear aggregation is neglected, with respect to that of perikinetic (Brownian) aggregation. Under these assumptions, the coagulation rate coefficient between particles of swollen radii rs and rs′ is estimated from the Fuchs’ modification of the Smoluchowski equation:64,65 β)
(
rs rs′ 2kBT 2+ + 3µW(rs, rs′) rs′ rs
)
(34)
where µ is the viscosity of the aqueous phase and W is the Fuchs’ stability ratio. Because the DLVO theory of colloidal stability is not a suitable method to compute W in emulsion polymerization systems, we decided to use a simple empirical approach, similar to that reported by Arau´jo et al.66 The approach consists of defining a critical particle size r* and dividing the latex particles in two distinct populations: unstable (rs < r*) and stable (rs > r*). The stability ratio is then discretized accordingly:
{
Wuu (rs, rs′ < r*) W ) Wus (rs < r* ∧ rs′ > r*) ∞ (otherwise)
(35)
where Wuu is the stability ratio between two unstable particles and Wus is the stability ratio between one unstable particle and one stable particle. Stable particles do not coagulate among themselves. The values of r*, Wuu, and Wus are determined by fitting the model to experimental data. 3.7. Model Parameters and Numerical Solution. The values of all parameters used in the model are listed in Table 4. Regarding these values, the following should be noted: (1) Given the range of values reported in the literature for the ratio DE/kt (cf. Table 1), we decided to determine this parameter ourselves, by fitting the experimental data of Ugelstad et al.8 Using literature data in the fitting makes it possible to utilize our data as an independent validation set. The value obtained was DE/kt ) 7.1 × 10-19 mol/m. (2) Experimental data are not available for the diffusivity of vinyl chloride in water. Values in the range (1.0s2.5) × 10-9 m2/s have been used in previous works;18,28,30 the Wilke-Chang correlation67 gives a value of 2.4 × 10-9 m2/s at 50 °C. Therefore, the value assumed in Table 4 should be a reasonable estimate. (3) The value of fkd has been measured by Neelsen et al.68 under reaction conditions and has been determined to increase slightly with the concentration of SDS. On the other hand, Ugelstad et al.69 determined that the polymerization rate
Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009 Table 4. Summary of the Parameters Used in the Model (50 °C) parameter
value
reliabilitya
reference
DE w DM jcr jEcr fmic fkd kd kp kpw kpw ′ kt ktw [M]wsat [M]sat p naggb rmicb z FM FP FW
7.1 × 10-14 m2/s 2 × 10-9 m2/s 13 6 1 2 × 10-6 s-1 5 × 10-6 s-1 11 m3 mol-1 s-1 11 m3 mol-1 s-1 11 m3 mol-1 s-1 1.0 × 105 m3 mol-1 s-1 1 × 106 m3 mol-1 s-1 1.6 × 102 mol/m3 4.5 × 103 mol/m3 40 1.5 nm 6 850 kg/m3 1380 kg/m3 988 kg/m3
2 2 2
this work this work 75 this work this work 68 70 71 76 40 10 76 this work 86 81 85 75 43 43 87
2 2 2 1 1 2 1 3 3 2 2 2 3 3 3
a The reliability criterion used in this table is purely subjective. When the data are considered to be reliable, they are either directly measured or found to be similar in independent studies. On the other hand, parameters with limited sources or that represent unsubstantiated or rough estimates are considered to be less reliable. b For SDS.
measured in seeded experiments was not influenced by the amount of SDS used (up to 100% surface coverage). For simplicity, we use a constant value for the product fkd, taken as the average value determined by Neelsen et al.70 (4) The rate of persulfate decomposition is known to be affected by many factors (monomer, surfactant, etc.), and so kd values that have been determined under polymerization conditions should preferably be employed. Neelsen et al.70 measured kd under VCM polymerization conditions and determined that it increased with SDS concentration, despite some scatter in the data. For simplicity, this dependence was ignored and an average kd value was assumed. This is a safe procedure because, in any case, kdt , 1 and, thus, [I]
) [I]0 exp(-kdt) ≈ [I]0
As a result, the initiation rate, 2fkd[I], is essentially controlled by the value of the product fkd (as discussed previously). (5) For vinyl chloride, there are no kp values determined by the IUPAC-recommended method (PLP). The only value at 50 °C has been determined by Burnett and Wright,71 using the rotating sector method. Nevertheless, measurements by electron spin resonance at 25 °C72 and ab initio molecular orbital calculations at 25 °C73 and 58 °C74 are in reasonable accord with data from Burnett and Wright at the corresponding temperatures. Therefore, the value reported in Table 4 should be a reasonable estimate. (6) In the absence of information about kpw, the chain-length dependency of this parameter (which possibly is important for i j 4)75 was neglected and its value is set equal to kp, as suggested by Morrison et al.76 The results of the model are not significantly dependent on the value of kpw for i < z, because, in the present system, the rate of aqueous-phase termination is small, with respect to the rate of aqueous-phase propagation (see below). ′ , we assumed the (7) In the absence of information about kpw ′ ) kp, as recommended by Casey et al.40 relationship kpw (8) Accurate values for kt are difficult to obtain, and there is not yet a recommended method to determine this kinetic parameter.77,78 For VCM, one can find values varying over
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9,42,60
several orders of magnitude, depending on the polymerization method and/or the assumptions done to extract kt. The value given in Table 4 was recomputed from the value of kp[M]p(fkd/kt)1/2 (the experimentally accessible quantity) that was determined by Ugelstad et al.10 (cf. Table 1). (9) In previous works, aqueous-phase termination has either been neglected or it has been assumed that ktw ) kt. Nevertheless, aqueous-phase termination is thought to be so fast that it is diffusion-controlled.5,75 With experimental data being nonexistent for VCM, the order-of-magnitude estimate that was recommended by Morrison et al.76 was used. The results of the model are not significantly affected by the value of ktw in the range of (0s5) × 106 m3 mol-1 s-1, confirming the idea of Ugelstad and co-workers10,11 (see section 2.1). (10) The literature shows that the aggregation number of SDS micelles is dependent on temperature, surfactant concentration, electrolyte concentration, and measurement technique. The same can be said about the micelle radius, which is intimately related to nagg. At 25 °C, typical values are nagg ≈ 6579-81 and rmic ≈ 2 nm.82-84 As the temperature increases, nagg decreases, and, at 40 °C (the closest value we could get to the reaction temperature), one has nagg ) 40;81 this is the value that was adopted. The micellar radius was computed from nagg by a formula due to Paul et al.;85 if necessary, corrections to this value can be taken into account for fmic. (11) For certain systems, z may be determined (by fitting) from experimental measurements of the frequency of radical entry into particles.75 For vinyl chloride, however, such an approach cannot be applied, because aqueous-phase propagation is very rapid, in comparison to termination (the estimated value of the ratio kpw[M]p/ktw1/2 ≈ 50 is very high). As a result, the capture efficiency is very close to 100% and almost insensitive to z. Although, at first, this may seem a problem, it is actually an advantage, because it means that the model outcome is insensitive to z. Given this fact, the semiempirical expression of Maxwell et al.75 can be safely used to obtain an estimate of z. (12) The value of jcr has a small effect on the outcome of the simulations, because the role of the exited radicals on nucleation is dominant. Given this fact, the expression from Maxwell et al.75 can be safely used to obtain an estimate of this parameter. The resolution of the 0-1-2 PBEs is accomplished by the numerical method described in Vale and McKenna.88 The technique is based on the finite-volume method and uses highresolution schemes and a generalized fixed pivot technique to discretize the growth and aggregation terms, respectively. Implicit-explicit methods are used to integrate the semidiscrete equations in time. The material balances to the initiator-derived and transferderived radicals are solved by making use of the quasi-steadystate approximation. 4. Results and Discussion In this section, the model is solved for a few cases and the results obtained are analyzed and compared against the experimental data of the first part of this series of papers.6 We begin by addressing the modeling of particle growth by polymerization and then focus on the more-intricate subjects of particle formation and particle size distribution. 4.1. Particle Growth. 4.1.1. Comparison of Particle Population Balance Models. To illustrate what was discussed about the PB, 0-1, and 0-1-2 models in section 3.3, we simulated the growth of an arbitrarily defined PVC seed latex, according to these three approaches (see Figure 4). Figure 4 shows that the simulations indicate that the 0-1-2 model and
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Figure 4. Growth of a PVC seed: comparison of the final PSDs and nj histories computed with the PB, 0-1, and 0-1-2 models.
the PB model agree very well for the seed latex example, which implies that they are probably both equally capable of describing particle growth for this system. The simulations also show that the 0-1 model does not give satisfactory results, despite the low nj values of the system (a dangerous temptation!), because intraparticle termination is rate-determining. 4.1.2. Comparison against Experimental Data. To evaluate the part of the model that specifically involves the kinetics of polymerization (particle growth), its predictions were compared with monomer conversion data obtained from batch unseeded experiments (after the initial period of particle formation) and from seeded experiments without the occurrence of secondary particle formation. Figure 5 summarizes the comparisons relative to the ab initio polymerizations performed with SDS. For runs A15, A12, A6, A4, A13, and A8, the fit is very good; there was no need to tune any parameter. In contrast, to obtain good agreement in runs A21, A20, A17, and A19, the value of fkd given in Table 4 must be multiplied by a factor of 1.5. This change in the kinetics seems to be correlated with the transition observed in the N-[SDS] curve that was identified in the experimental part:6 the first and the second set of experiments correspond, respectively, to the parts of the curve before and after the sudden increase in particle number. To explain this observation, one
Figure 5. Comparison between experimental and simulated (s) conversion profiles of ab initio polymerizations stabilized by SDS: (a) [APS] ) 1.0 mM and (b) [APS] ) 2.0 mM.
could invoke an effect of [SDS] on the rate of initiation and, indeed, the factor of 1.5 is consistent with the dependence of fkd on [SDS] that was measured by Neelsen et al.68 However, note that the latter authors found a continuous increase of fkd with [SDS], whereas our results suggest an abrupt variation of fkd. Figure 6 presents the comparisons for the ab initio polymerizations performed with SDBS. As already discussed in the first part of this series of papers,6 the runs that were performed at [APS] ) 1.0 mM show an atypical behavior, more specifically, the rate of polymerization does not increase with particle number, as expected from the kinetic equations. On the other hand, for [APS] ) 2.0 mM, the agreement between measured and calculated conversion histories is comparable to that found in the SDS runs. Note that, in this case, no parameter was adjusted. Finally, Figure 7 shows what happens in the case of the seeded polymerizations. As we can see, the agreement is globally good; however, to obtain such results, an inhibition time of 10 min had to be introduced. In fact, at the beginning of the runs, there seems to be a retardation period, over which the polymerization rate is lower than expected. At this time,
Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009
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Figure 6. Comparison between experimental and simulated (s) conversion profiles of ab initio polymerizations stabilized by SDBS: (a) [APS] ) 1.0 mM and (b) [APS] ) 2.0 mM.
Figure 7. Comparison between experimental and simulated (s) conversion profiles of seeded polymerizations without occurrence of secondary particle formation.
we cannot state if this is due to an imperfect removal of oxygen from the reactor or to some impurity released from the ionexchange resins during the washing of the seed latexes. In summary, despite some unresolved issues, the quality of the kinetic model can be considered satisfactory, especially when SDS is used as the emulsifier; we just need to be aware that an adjustement of fkd is needed after we reach certain surfactant concentrations. The results obtained with SDBS suggest that some sort of interaction between this surfactant and the initiator (APS) is occurring. Further experiments are necessary to determine if this effect is only relevant at low initiator concentrations. 4.2. Mechanistic Issues in Particle Formation. We start the analysis of particle formation by addressing some mechanistic issues. To this purpose, we will consider the hypothetical case where particles are formed uniquely by micellar nucleation. This is, of course, not truesexcept perhaps in the limit of high surfactant concentrations, [S] . CMCsbut it will allow us to draw some interesting conclusions about the role of desorbed radicals in particle nucleation, the significance of particle
coagulation, and the insufficiencies of the PB model. To “switch off” homogeneous nucleation, we set jcr f ∞ and kpw ′ ) 0 (or jEcr f ∞). 4.2.1. Nucleation by Exited Radicals. Let us simulate a series of ab initio polymerizations, at constant [SDS] (e.g., 3 g/L) and increasing [I] values, discarding the possibility of particle nucleation by exited radicals (i.e., by setting kemE ) 0) and assuming negligible particle coagulation (β ) 0). If we do so, we observe that the particle number predicted by the model increases with initiator concentration, as one might expect from the analysis of the equations. Although the dependency is weak (N ∝[I]0.2), it does not agree with the experimental results of Ugelstad et al.,7 which show a constant N value in an range of [I] extending over more than an order of magnitude. Repeating the same exercise (β ) 0), but now allowing for the entry of desorbed radicals into micelles, it can be seen that the particle number becomes independent of [I], in agreement with experiment and with the idea defended by Hansen and Ugelstad.15 This observation corroborates that nucleation by exited radicals should be taken into consideration a priori, not
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Figure 8. Simulation results for run A21, neglecting homogeneous nucleation and particle coagulation, using the parameter fmic ) 1 × 10-5, adjusted to give the correct particle number.
only in micellar nucleation, but also in homogeneous nucleation (where the same line of reasoning applies). However, the problem of including nucleation by desorbed radicals is that the predicted N values (>1019) are an order of magnitude higher than the experimental values. This brings us to the next topic. 4.2.2. Significance of Particle Coagulation. If we keep on insisting that β ) 0, the only way (recall that homogeneous nucleation is being neglected) to bring the computed particle numbers closer to the experimental values is to lower the efficiency of radical entry into micelles. However, to obtain N values of the same order, we are forced to decrease fmic to 10-5. Such a low efficiency is in agreement with the values found by Hansen and Ugelstad15 for VCM, and by Nomura et al.16 for vinyl acetate, based on their simplified models of particle formation (their models neglect, among other things, homogeneous nucleation and particle coagulation). Nevertheless, no explanation has yet been found for these low values,89 which cannot be attributed to inaccurate estimates of rmic, nagg, CMC, or any other ill-defined parameters. Analyzing the PSDs predicted by the model (Figure 8) can help us determine the origin of the problem. As we can see, even long after the particle nucleation process is finished, there
Figure 9. Simulation results for run A21, neglecting homogeneous nucleation, but considering particle coagulation. Parameters: fmic ) 1, r* ) 10 nm, Wuu ) Wus ) 6 × 105, all adjusted to give the correct particle number.
is still a significant amount of rather small particles (
