Ind. Eng. Chem. Res. 2001, 40, 3883-3894
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Modeling of Seeded Semibatch Emulsion Polymerization of n-BA Christophe Plessis,† Gurutze Arzamendi,‡ Jose´ R. Leiza,† Harold A. S. Schoonbrood,§,| Dominique Charmot,§,⊥ and Jose´ M. Asua*,† Institute for Polymer Materials POLYMAT and Grupo de Ingenierı´a Quı´mica, Departamento de Quı´mica Aplicada, Facultad de Ciencias Quı´micas, The University of the Basque Country, Apartado 1072, E-20080 Donostia-San Sebastia´ n, Spain, Departamento de Quı´mica Aplicada, Universidad Pu´ blica de Navarra, 31006 Pamplona, Spain, and Rhodia, Centre de Recherches d’Aubervilliers, 52 rue de la Haie Coq, F-93308 Aubervilliers Cedex, France
A mathematical model for the computation of kinetics, branching frequency, sol molecular weight distribution, and gel fraction for the seeded semicontinuous emulsion polymerization of n-BA is presented. The model incorporates mechanistic features that have been found to play an important role in the polymerization of n-BA, such as the intramolecular transfer to polymer, so-called backbiting, and the low reactivity of the tertiary radicals resulting from such a reaction. Model parameters for which values are not available in the literature were obtained by fitting the model predictions to the kinetic data and structural properties of the polymer (fraction of gel, sol molecular weight distribution, and level of branches) gathered in seeded semicontinuous emulsion polymerizations of n-BA carried out at 75 °C with potassium persulfate as the initiator. The model fits all of these experimental data quite well. Introduction Mathematical modeling of emulsion polymerization of gel-forming monomers is a challenging task because it is necessary to account for a complex kinetic scheme (multiple reactions with chain-length-dependent reaction rates) occurring in a compartmentalized system. This means that growing radicals are segregated in the polymer particles and termination must involve radicals present in the same particle. This yields a complex product whose complete characterization involves many different aspects (gel fraction, sol molecular weight distribution (MWD), number of long and short branches, etc.). Several attempts to model branching and gel formation have been reported.1-6 Charmot and Guillot1 developed a model in which the instantaneous distributions of macroradicals and dead chains containing an increasing number of cross-links were computed. They calculated the amount of gel assuming that polymer chains containing a number of cross-links higher than a critical value can be considered to be gel. Even though the model did not account for the compartmentalization of radicals or for chain-length-dependent termination, it was able to fit styrene-butadiene emulsion polymerization experimental data well. Arzamendi et al.2 developed a model for the kinetics of long-chain branching in emulsion polymerization. This model computes the average molecular weights and was used to analyze the effect of the monomer addition policy on the average molecular weights of the copolymers obtained in the * To whom correspondence should be addressed. E-mail:
[email protected]. † The University of the Basque Country. ‡ Universidad Pu ´ blica de Navarra. § Rhodia. | Current address: Wacker Chemicals Australia, Pty. Ltd., Suite 3, 11 Leicester Avenue, Glen Wakerley, 3150 Victoria, Australia. ⊥ Current address: SYMYX Technologies, Inc., 3100 Central Expressway, Santa Clara, CA 95051.
emulsion polymerization of ethyl acrylate and methyl methacrylate. Teymour and Campbell3,4 proposed an interesting method for the calculation of the MWD of gel-forming systems in bulk polymerization: the numerical fractionation approach. The polymer chain population is subdivided into n + 1 classes (generations). One class includes all of the linear chains, and n classes contain the branched chains; the chains included in each generation are of similar length. The molecular weights of the branched generations are chosen in such a way that as one moves from one generation to the next the average molecular weight grows geometrically. A polymer chain passes from the linear generation to the first branched generation when it undergoes chain transfer from a growing radical to its chain (chain transfer to polymer). The transition from one branched generation to the subsequent generation requires the coupling of two polymer chains (e.g., by bimolecular combination or propagation to terminal double bonds). The description of the sol MWD obtained by using numerical fractionation is often too coarse, yielding artifacts as shoulders or even bimodal MWD.5 To overcome these limitations, Arzamendi and Asua5 used a refinement of the fractionation technique to model gelation and sol MWD in emulsion polymerization. Numerical simulations showed that, in the absence of chain-transfer agent, the higher the gel fraction, the lower the molecular weight of the sol fraction. Arzamendi et al.2,5 accounted for different rates of reaction in particles containing a different number of radicals but failed to describe correctly the bimolecular termination by combination because the effect of compartmentalization was only partially accounted for. Ghielmi et al.6 combined the concept of distinguished particles (particles in which the model keeps track of the length of the growing free radicals) with the numerical fractionation to calculate the gel fraction and the sol MWD. This model correctly accounts for the compartmentalization of the system,
10.1021/ie000427e CCC: $20.00 © 2001 American Chemical Society Published on Web 08/08/2001
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although it does not provide a correct description of the sol MWD, giving rise to artificial shoulders. None of these models accounted for the chain-lengthdependent termination rate. The rigorous modeling of this phenomenon in emulsion polymerization systems requires the balance of the radicals in the particles and aqueous phase and the balance for active and inactive polymer chains of each length to be solved simultaneously. This is a formidable task, and only partial solutions involving simplifications which allowed an independent calculation of both the number of radicals per particle and the MWD have been reported.7,8 Tobita et al.9 proposed an approach based on the Monte Carlo technique, which allows accounting for branching and gelation, compartmentalization, and, in principle, chain-length-dependent termination. However, it is rather intensive from a computational point of view, which is a typical disadvantage of the Monte Carlo methods. From the previous paragraphs, it is clear that, in order to model branching and gelation in emulsion polymerization systems more accurately, increasingly complex models should be used. In practice, one would like to use a model that is as simple as possible yet satisfactory. In this context, it is worth mentioning that so far rather simple models have been able to fit experimental data1,2 well and that the more complex models have not been validated by fitting experimental data. In addition, one has to bear in mind that there is no practical point in using model refinements that yield prediction improvements that are within the experimental error of the experimental procedure. Lichti et al.10 and Ghielmi et al.11 used simulations to study the effect of accounting properly for compartmentalization on the calculated average molecular weights. It was reported that errors can be made in the calculation of the instantaneous polydispersity ratio of a polymer produced in emulsion polymerization in the presence of termination by combination if compartmentalization is not taken into account properly. However, the error in the cumulative properties (which are the values measured experimentally) is less critical. Thus, for linear polymers the number-average molecular weight, Mn, is not affected by compartmentalization, whereas the weight-average molecular weight, Mw, is overestimated by only 10%. Arzamendi et al.12 showed for linear polymers that even rather simplified models (in terms of the description of compartmentalization) provided MWDs which are within the experimental error. Ghielmi et al.11 also included branched polymers in their simulation study and reported on the effect of accounting properly for compartmentalization on Mn, Mw, and the gel fraction of a polymer produced in a batch reactor. They found that compartmentalization has almost no effect on these properties when chain coupling is only by propagation to terminal double bonds. On the other hand, when termination by combination is the main mechanism for chain coupling, compartmentalization strongly affects Mw and the gel fraction at low conversion but has no effect at high conversions. These findings seem to indicate that compartmentalization may not be critical in simulations of semicontinuous emulsion polymerization processes carried out under starved conditions. Tobita13,14 analyzed by simulation the effect of a chain-length-dependent termination rate constant on
the MWD. Negligible effects were found when termination occurred by combination and when polymerization was carried out at high conversions. In the polymerization of n-BA, termination occurs by combination. The present model will be used to analyze semicontinuous emulsion polymerizations carried out under starved conditions. Because of these findings and the uncertainties associated with the parameters included in the chain-length-dependent termination model, a chainlength-independent termination rate constant was used in this work. In the mathematical modeling presented in this paper, compartmentalization was accounted for by means of the partial distinction approach12 and sol MWD was described in detail by means of a refined numerical fractionation approach.5 The model was able to predict kinetics, gel fraction, sol MWD, and branching frequency of the seeded semibatch emulsion polymerization of n-BA. The kinetic parameters were estimated by comparing model predictions and experimental data from n-BA emulsion polymerizations. Distinctive Features of the n-BA Emulsion Polymerization Industrially, emulsion polymerizations of n-BA are carried out in a semicontinuous reactor under starved conditions in order to control the temperature of the reactor. Under these conditions a highly branched polymer (% branches ) 0.9-3.4) which contains about 50-60% gel is formed.16 Interestingly, there is no direct correlation between the branching frequency and gel content. This strongly suggests that a significant part of the branches do not contribute to gel formation; namely, they are short branches resulting from intramolecular chain transfer. Both intermolecular and intramolecular chain transfer to polymer yield tertiary carbon radicals with much lower reactivity than the secondary carbon radicals involved in the linear growth of the poly(n-BA) chain.17 The exchange between these types of radicals can be illustrated with the following reaction:
where R1 and R2 are the secondary and tertiary radicals, kfp1 is the intermolecular chain transfer to polymer rate constant, [Q1]p is the concentration of polymer in the polymerization loci (polymer particles); kfp2 is the intramolecular chain transfer to polymer rate constant; kp1 and kp2 are the propagation rate constants of radicals R1 and R2, respectively, and [M]p is the concentration of monomer in the polymer particles. According to eq 1, the effective propagation rate constant can be expressed as follows:
kp ) kp1P1 + kp2P2
(2)
where P1 and P2 are the probabilities of having a radical of type R1 and R2, respectively. Assuming that a pseudosteady-state applies to these radicals, one obtains
(kfp2 + kfp1[Q1]p)R1 ) (kp2[M]p)R2
(3)
Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3885
R ) R1 + R2
(4)
By combination of eqs 3 and 4, the probabilities of having a radical of type R1 or R2 can be obtained:
P1 )
kp2[M]p R1 ) ; P2 ) 1 - P1 R kp2[M]p + kfp2 + kfp1[Q1]p (5)
Equation 5 is more general than that proposed by Plessis et al.18 for P1 and P2, because the contribution of the intermolecular chain transfer is included. As will be shown below, for n-BA kfp1[Q1]p , kp2[M]p + kfp2, and eq 5 reduces to
P1 )
kp2[M]p kp2[M]p + kfp2
; P2 ) 1 - P1
(6)
Equations 2 and 5 (or 6) show that the effective propagation rate constant depends on the monomer and polymer concentrations in the polymer particles. Plessis et al.18 showed that the value of the effective propagation rate constant in emulsion polymerization is substantially lower than that measured by pulsed laser polymerization (PLP) in bulk.19 Mathematical Model The description of the model has been structured in different sections for easier understanding. The following assumptions were made: (i) The amount of polymer formed in the aqueous phase is negligible compared with that formed in polymer particles. (ii) From a kinetic point of view, the whole population of polymer particles can be represented by a monodisperse population of particles. (iii) The free radicals are uniformly distributed (in a statistical sense) in the monomer-swollen polymer particles. This means that the effect of the anchoring of the hydrophilic end group of the growing polymer chain on the surface of the particle was neglected. This is a reasonable assumption for systems in which small mobile radicals are produced by a chain-transfer reaction and for small polymer particles.20 (iv) The pseudo steady state for the concentration of free radicals was applied. (v) The growing time of a polymeric chain was less than that required to increase the volume of the polymer particle to an extent that affects the values of the kinetic parameters that control the number of radicals in the polymer particles. Therefore, during the growth of the polymer chain, the distribution of particles containing j radicals remains unchanged. However, the number of radicals in the particle in which the polymer chain is growing can change during the growth of the polymer chain. (vi) Kinetic parameters are independent of the length of the growing polymer chains. Only monomeric radicals are able to exit to the aqueous phase. (vii) Radicals entering the polymer particles are considered to be of length 1. (viii) The polymer particles can contain a maximum number of free radicals, m. Therefore, instantaneous termination occurs if a radical enters a polymer particle already containing m radicals.
The model accounts for the following kinetic events: (i) initiation of polymer chains from radicals entering into the particles or produced by a chain-transfer reaction, (ii) linear propagation; (iii) chain transfer to monomer and chain-transfer agent, CTA; (iv) intermolecular and intramolecular (backbiting) chain transfer to polymer; and (v) termination by combination.21 Reactant Balances: Evolution of the Kinetics The material balances of reactants for a semicontinuous process are
di/dt ) Fi ( Ri
(7)
where i is the total number of moles of compound i in the reactor, Fi the molar feed rate of compound i, and Ri its rate of appearance or disappearance by reaction. The rate of decomposition of a water-soluble initiator like persulfate can be expressed as
RI ) 2fkII2
(8)
where f and kI are the efficiency factor and the rate constant for initiator decomposition, respectively, and I2 is the number of moles of initiator present in the reactor. According to the polymerization mechanism proposed by Plessis et al.18 and with the assumption that the extent of aqueous-phase polymerization is negligible, the rate of monomer consumption can be expressed as
nNp Rp ) kp[M]p NA
(9)
where kp is the effective propagation rate constant given by eq 2, [M]p is the monomer concentration in the polymer particles (calculated by means of an iterative algorithm using the partition equilibrium equations and the monomer material balance22,23), Np is the total number of polymer particles in the reactor, NA is Avogadro’s number, and n is the average number of radicals per particle. As mentioned above, the model was developed for the seeded semicontinuous emulsion polymerization of n-BA. In the experiments used to validate the model,16 Np was almost constant throughout the polymerizations and it was not affected by the operation variables employed (initiator concentration and monomer feed flow rate). Therefore and for the sake of model simplicity, a polynomial was fitted to the evolution of the experimental particle diameter, and this was used to account for the total number of particles. To integrate the monomer balance, the evolution of n during the reaction is required, and for this, the approach proposed by Ugelstad and Hansen24 (eq 10) and the radical balance in the aqueous phase (eq 11) were used:
n) m+
a2/8 a2/4 a2/4 m + 2 + ... kdvpNA and a2 ) 8ka[R]w (10) where m ) kt
m+1+
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d[R]w/dt ) 0 ) 2fkII2NA + kdnNp - ka[R]wNp 2ktw[R]w2NA (11) where ka is the entry rate constant of radicals into polymer particles, kd is the exit rate constant of radicals from polymer particles to the aqueous phase, kt and ktw are the termination rate constants in the polymer particle and in the aqueous phase, respectively, vp is the polymer particle volume, and [R]w is the concentration of radicals in the aqueous phase. The instantaneous and overall conversions are given by
Xinst )
Mt - M + Mseed ; Mt ) M0 + Mt + Mseed Xglob )
∫0t FM dt
Mt - M + Mseed MTot + Mseed
(12)
(13)
where M0 represents the amount of monomer in the initial charge, MTot the total amount of monomer in the formulation, Mseed the polymer in the seed, Mt the total amount of monomer fed until time t, and M the amount of unreacted monomer in the reactor. Branching Frequency In the polymerization of n-BA, branching points can be formed by intermolecular and intramolecular (backbiting) transfer to polymer. The former leads to the formation of long branches (LCB) and the latter to short branches (SCB). However, when the branching frequency is measured by 13C NMR, SCB and LCB cannot be distinguished. The balance of the total number of branching points is given by
dB/dt ) kfp1[Q1]pnNp + kfp2nNp
(14)
The frequency of branching is calculated as the ratio between branching points and the monomeric units polymerized, Q1, as
νB ) B/Q1
(15)
where Q1 is given by
Q1 ) (Mo +
∫0tFM dt - M)NA
(16)
Molecular Weights of the Soluble and Gel Fractions (Mnsol and Mwsol) In the n-BA emulsion polymerization, gel is formed through intermolecular chain transfer to polymer and termination by combination. The accurate calculation of the polymer structure requires a good description of the sol MWD and accounting for the compartmentalization of radicals. To describe the sol MWD in detail, a refined numerical fractionation approach was used.5 The polymer is divided into generations based on size, and the polymer belonging to higher generations is considered to be gel. The first generation (e.g., n ) 0) is formed by linear chains. The next ne generations are composed of polymer chains having the same number of long branching points
(e.g., all chains in generation 4 have four long branching points), and for generations higher than ne + 1, the geometrical growth of Teymour and Campbell3,4 was adopted. A more detailed description of the transfer between the different generations can be found elsewhere.5 In the original method, gel was considered to be the polymer whose molecular weight approaches infinity (numerically, belonging to generations higher than nc). However, experimentally, gel is the polymer which cannot be dissolved in a good solvent, i.e., THF in this work. An indication of the maximum molecular weight that can be dissolved by THF can be obtained from gel permeation chromatography (GPC) traces of the soluble polymer. For the series of experiments presented in this paper,18 no polymer was observed over a molecular weight of 7 × 106. It has to be pointed out that this number may be higher as some polymer might be retained in the columns [it was checked that the filtration devices did not affect the size exclusion chromatography (SEC) trace]. However, the use of an arbitrary higher limit (e.g., 107 molecular weight) has even fewer grounds than the use of the experimental value. Therefore, a polymer with a molecular weight exceeding this value was considered to be gel in the model. Compartmentalization was accounted for by means of the partial distinction approach.12 This method uses an intermediate level of description of compartmentalization, and its predictions agree very well with those of the more detailed models.10 The idea of the partial distinction approach can be explained by considering a 0-1-2 emulsion polymerization system. This is a system in which the maximum number of radicals per particle is 2, namely, that instantaneous termination occurs if a radical enters into a particle already containing two radicals. In such a system, the probability of having two long radicals in the same particle is almost negligible because they would have to grow together for a long time, and this is unlikely because of the large probability of the bimolecular termination in a 0-1-2 system. Therefore, only one long and one short, or two short radicals, can be in the particles at the same time. This rationale was followed in the model, and the freeradical population is divided into “short” (q) and “long” (p) radicals. Because there is not a clear-cut difference between short and long radicals, these terms must be defined. For the current system, it has been considered that linear radicals are short whereas branched polymers are long. It is worth pointing out that the reason for including the length distinction is that when termination occurs by combination, the length of the resulting polymer molecule depends on the length of the reacting radicals. Therefore, it is important to determine the types of radicals that coexist in the same particle. On the other hand, following Tobita’s findings,13,14 a chainlength-independent termination was used. Using this description of short and long radicals, the number- (Mn) and weight-average (Mw) molecular weights of the sol fraction can be expressed as follows: nc
Mn,sol )
∑ n)0 nc
nc m
Q1(n) +
p1i(n) + q1i(n) ∑ ∑ n)0i)1 nc m
p0i(n) + q0i(n) ∑ Q0(n) + n)0 ∑∑ i)1
Pm (17)
n)0
Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3887 nc
Mw,sol )
∑ n)0
nc m
p2i(n) + q2i(n) ∑ ∑ n)0i)1
Q2(n) +
nc
Pm (18)
nc m
p1i(n) + q1i(n) ∑ Q1(n) + n)0 ∑∑ i)1
Parameter Estimation
n)0
where Qk(n) is the kth-order moment of the chain-length distribution of the dead polymer of generation n, qki(n) is the kth-order moment of short radicals of generation n in particles containing i radicals (generation zero is the linear generation), pki(n) is the kth-order moment of long radicals of generation n in particles containing i radicals, and Pm is the molecular weight of the monomeric unit. The fraction of gel was calculated by the difference between the total amount of polymerized monomer units and the monomer units in the polymer chain generations up to nc:
Gel ) nc
m
Q1 +
∑ i)1
(p1i + q1i) - (
∑ n)0
nc m
Q1(n) +
p1i(n) + q1i(n)) ∑ ∑ n)0i)1
The mathematical model described above contains numerous parameters. Some of them are available from literature (see Table 1), but others are not available or the range of values given in the literature is very broad. Therefore, several parameters were estimated by fitting model predictions to the experimental data gathered in seeded semicontinuous emulsion polymerizations of n-BA,16 namely, (i) the evolution of instantaneous conversion, Xinst, (ii) the branching frequency, νB, (iii) the gel fraction, Gel, and (iv) the weight-average molecular weight, Mw. The estimation was carried out using the Nelder and Mead algorithm26 of direct search furnished by the DBCPOL subroutine in the IMSL library. This algorithm minimizes the following objective function:
J)
m
Q1 +
where Xn(n,s,i) and Xw(n,s,i) are the number- and weight-average chain lengths of the polymer of generation n formed in polymer particles with i radicals through the process s.
(p1 ∑ i)1
i
i
+ q1 ) (19)
nc is the generation with an average molecular weight lower than or equal to 7 × 106. The calculation of the moments of chain-length distributions of both dead and active polymers is given in Appendix I.
[
∑ ∑ N
n1(N)
n1(N)
∑ ∑ N n3(N)
n2(N) w3
n3(N)
(
dW(x) dt
nc
)
s
dQ1,si(n) w(x,n,s,i) dt i)1 m
∑ ∑∑
n)0
(20)
where W(x) is the cumulative weight fraction of the sol polymer of length x and w(x,n,s,i) is the instantaneous weight fraction of chains of length x of generation n, formed by process s, in a particle with i radicals. The instantaneous MWD of polymer chains produced by the different mechanisms was calculated under the assumption that the Schultz distribution25 is applicable.
w(x,n,s,i) )
y(n,s,i) {xy(n,s,i)z(n,s,i) exp[-xy(n,s,i)]} Γ(z(n,s,i)) (21)
where
z(n,s,i) )
[
1
Xw(n,s,i)
-1
Xn(n,s,i)
y(n,s,i) )
z(n,s,i) + 1 Xw(n,s,i)
]
(22)
(23)
2
+
)
νB exp - νB mod
(
νB exp
∑ ∑ N
2
+
)
Gelexp - Gelmod Gelexp
MWD of the Sol Fraction The overall MWD of the sol polymer was calculated by adding the instantaneous MWDs of the generations up to nc and integrating over time.
)
Xinst exp w2
n2(N)
∑ ∑ N
(
Xinst exp - Xinst mod
w1
n4(N)
w4 n4(N)
(
2
+
Mwexp - Mwmod Mwexp
)] 2
(38)
where N is the number of experiments, wi the weighting factor, and ni(N) the number of experimental points in each experiment; Zexp and Zmod are the experimental and simulated points of the variable Z. The values of wi used were w1 ) 103, w2 ) 5 × 105, w3 ) 1, and w4 ) 1. These values were chosen such that each of the terms in eq 38 have similar values. The adjustable parameters were as follows: (i) ka*: Radical entry was considered to occur through the diffusional mechanism.27 Therefore, the rate constant for radical entry into the polymer particles was considered to vary with the radius of the swollen polymer particle: ka ) ka*rp, where ka* was taken as the adjustable parameter. (ii) kd*: For sparingly water-soluble monomers, the rate constant for radical exit was expected to be28 kd ) kd*/rp2, where kd* was taken as the adjustable parameter. (iii) kp1 and kp2: propagation rate constants of the n-BA radical and the tertiary radical produced by chain transfer to polymer, respectively. Propagation rate constants for n-BA have been recently determined by means of PLP. Thus, Beuermann et al.19 measured the effective propagation rate constant between 5 and 30 °C using bulk polymerization. Extrapolation of Beuermann data to 75 °C yields a value for the effective propagation rate constant of 26 500 L mol-1 s-1. This
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Table 1. Values of Parameters Taken from the Literature Used for the Polymerization of n-BA at 75 °C kfm/kp ) 8.8 × 10-5 Kw,p ) 2.083 × 10-3 Kw,g ) 1.035 × 10-3 f ) 0.6 kI ) 4.354 × 10-5 s-1
ref 29 ref 30 ref 30 ref 31
Table 2. Estimated Values of the Adjustable Parameters ka* ) 8.1 × 10-11 dm2 mol-1 s-1 kd* ) 4.5 × 10-13 dm2 s-1 kp1 ) 53.46 × 103 L mol-1 s-1 kp2 ) 100 L mol-1 s-1 kfp2 ) 9.24 × 102 s-1 kto ) 6.55 × 106 L mol-1 s-1 a1 ) 2.23 kfp1 ) 0.178 L mol-1 s-1 Table 3. Recipe Used in the Seeded Semicontinuous Emulsion Polymerizations (T in °C) initial charge polymera (g) water (g) surfactant A (g) n-BA (g) acrylic acid (g) K2S2O8 (%)c
stream 1
20 97.5b 1.25
20
0.0188-0.30d
0.0188-0.30d
stream 2 255 2.5 225.4 4.6
a Mass of polymer in the amount of seed added. b Mass of water including the water mass present in the seed. c Weight percentage based on the total monomer weight. d This amount was the sum of the initial charge plus that added in stream 1. The ratio of the initiator between the initial charge and stream was 1/1.
Table 4. Summary of the Seeded Semicontinuous Emulsion Polymerizations experiment
initiator concn (wt %)
feeding time (h)
1 2 3 4 5 6 7 8
0.0188 0.0375 0.075 0.15 0.30 0.30 0.30 0.30
3 3 3 3 3 1 2 4
value would be equal to kp1 if P1 ) 1. However, there was no proof that P1 ) 1, and hence both kp1 and kp2 were taken as adjustable parameters. (iv) kfp1: chain transfer to polymer rate constant. (v) kfp2: backbiting rate coefficient. (vi) kt0: termination rate coefficient at zero polymer content. (vii) a1: The decrease of the termination rate constant due to the gel effect was accounted for by using the following empirical expression: kt ) kt0 exp(-a1φpp), where φpp is the volume fraction of polymer in the particle and a1 was taken as an adjustable parameter. (viii) nc: Beyond this generation, all of the polymer is considered as gel. In practice, gel is the polymer that cannot be dissolved by THF. An indication of the minimum molecular weight that cannot be dissolved by THF can be obtained from the maximum molecular weight obtained in the GPC trace of the soluble polymer. In the samples considered here (Table 4) no polymer was observed over a molecular weight of 7 × 106 (measured using polystyrene standards). Therefore, a polymer with a molecular weight over this value is considered as gel. Under these circumstances, the model is rather insensitive to the values of ne and nc. In the simulation, a large enough value of ne was used (ne ) 50) and nc was not limited.
Figure 1. Comparison of model predictions (lines) and experimental values of instantaneous conversion for the experiments in which the initiator concentration was varied. Legend: (b and s) 0.0188 wt %; ([ and - -) 0.0375 wt %; (1 and ‚‚‚) 0.075 wt %; (2 and ‚ - ‚) 0.15 wt %; (9 and - - -) 0.30 wt %. Points and lines are experimental and model prediction values, respectively.
Table 1 presents the kinetic parameters of the model taken from literature. Table 2 presents the estimated values of the adjustable parameters obtained by fitting the model to the experimental results obtained in the seeded semicontinuous emulsion polymerization of n-BA. These values agree with those presented in ref 18, although it should be noted that the units for kt0 and kfp2 in that work should be cm3/mol‚s instead of L/mol‚ s. Table 3 presents the typical recipe of a seeded semicontinuous emulsion polymerization of n-BA. The seed [dp ) 97 nm, solids content ) 20.5 wt %, gel content ) 10 wt %, Mw ) 2.1 × 106, and Mn ) 210 000 and prepared using sodium lauryl sulfate (Merck)], the surfactant (provided by Rhodia; its name or composition cannot be disclosed for proprietary reasons), the initiator potassium persulfate, and water were initially charged into the reactor. The feed was divided in two streams: one was a preemulsion of the monomer and the other an initiator solution. The different experiments considered in this work are summarized in Table 4. The final solids content was about 40 wt %. Samples were withdrawn from the reactor and conversion was measured gravimetrically, the particle size by dynamic light scattering and by capillary hydrodynamic fractionation, CHDF. The sol MWDs were determined by SEC using polystyrene standards and a dual detector system consisting of a differential refractometer and a viscometer. The amount of gel was determined by means of an extraction process under reflux conditions in THF. The level of branching was measured by solid-state 13C NMR by analyzing the percentage of quaternary carbons in the spectrum as reported in the literature.15,16 In the following section the model predictions using the parameters of Tables 1 and 2 will be compared with the experimental results. Results and Discussion Figure 1 compares the simulated and experimental instantaneous conversions for runs 1-5 where the initiator concentration was varied in the range 0.01880.30 wt %. It can be observed that the higher the initiator concentration, the higher the instantaneous conversion, and all polymerizations evolved under rather starved conditions. A reasonably good agreement was found between the experimental and model predictions. Because of the distinctive mechanism of the n-BA
Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3889
Figure 2. Evolution of the predicted effective propagation rate constant, kp, for the experiments in which the initiator concentration was varied. Legend: (s) 0.0188 wt %; (- -) 0.0375 wt %; (‚‚‚) 0.075 wt %; (‚ - ‚) 0.15 wt %; (- - -) 0.30 wt %.
Figure 3. Evolution of the experimental and predicted fraction of gel for the experiments in which the initiator concentration was varied (runs 1-5). Legend: (b and s) 0.0188 wt %; ([ and - -) 0.0375 wt %; (1 and ‚‚‚) 0.075 wt %; (2 and ‚ - ‚) 0.15 wt %; (9 and - - -) 0.30 wt %. Points and lines are experimental and model prediction values, respectively.
polymerization, the effective propagation rate constant, kp, depends on the concentrations of monomer and polymer in the polymerization loci (eqs 2 and 5). Figure 2 shows the effective kp calculated by the model for the five experiments carried out with different initiator concentrations. It can be seen that the higher the initiator concentrations (which results in lower monomer concentrations and higher polymer concentrations in the polymer particles), the lower the effective kp. Simulations show that the effect of intermolecular chain transfer to polymer on kp is negligible (kfp2/kfp1[Q1]p ) 800); in other words, eq 6 can be applied. It is worth mentioning that the values of kp in Figure 2 are significantly lower than the value (26 500 L‚mol-1‚s-1) obtained in bulk polymerizations by Beuermann et al.,19 who used PLP. However, when a monomer concentration typical of bulk conditions ([M] ) 6.6 mol/L) is used in eqs 2 and 6, the effective propagation rate constant obtained is ca. 24 000 L mol-1 s-1, which compares well with the value obtained by Beuermann et al.19 It is worth pointing out that this value is still significantly lower than kp1 (53 460 L mol-1 s-1) because even under bulk conditions P1 < 1. Figure 3 presents the comparison between the experimental and model predictions for the fraction of gel produced in experiments 1-5. It can be seen that the fraction of gel increases from 10%, corresponding to the amount present in the seed, to 55-60 wt %, in a fashion mostly independent of the initiator concentration. The
Figure 4. Evolution of the experimental and predicted weightaverage molecular weight for the experiments in which the initiator concentration was varied. Legend: (b and s) 0.0188 wt %; ([ and - -) 0.0375 wt %; (1 and ‚‚‚) 0.075 wt %; (2 and ‚ - ‚) 0.15 wt %; (9 and - - -) 0.30 wt %. Point and lines are experimental and model prediction values, respectively.
model predicts the evolution of the gel content fairly well: a sharp initial increase followed by a plateau, although this predicted evolution is slightly more sensitive to the initiator concentration than the experimental data suggest. According to the model, gel can only be produced if both intermolecular chain transfer to polymer and termination by combination are operative. No gel was predicted by the model when kfp1 ) 0 or when termination was considered to occur by disproportionation. An additional condition is that the termination by combination is significant and therefore the population of polymer particles with two radicals must not be negligible. In other words, the average number of radicals, n, should be greater than 0.5. According to the model, n ranges from 0.47 in run 1 to 1.46 in run 5. Figure 4 presents the evolution of the weight-average molecular weight in runs 1-5. It can be seen that, in agreement with the experimental results, the model predicts a decrease of Mw with the initiator concentration and a continuous decrease of Mw during each reaction. The decrease of Mw with the initiator concentration is a result of two effects. First, as the initiator concentration is increased, the entry rate increases, inducing an increase of the termination rate and a decrease of the average lifetime of the growing chains. Second, the increase of the initiator concentration causes the process to proceed under more starved conditions, which increases intermolecular chain transfer to polymer. Contrary to what one would expect at first sight, the model shows that this effect yields lower sol molecular weights because the probability of suffering transfer is higher for long chains, and hence they are preferentially incorporated into the gel. Figure 5 compares the simulated and measured sol MWDs of the polymer obtained at the end of the process. It can be seen that although the prediction is not perfect, the trend is well captured by the model. It is worth pointing out that, to our best knowledge, this is the first time that experimental results and model predictions of the sol MWD of a gel-forming monomer system are presented at this level of comparison. Figure 6 presents the comparison between experimental data and model predictions for the branching frequency (both short and long) in the backbone. An excellent agreement between simulated and experimental results is obtained. It can be seen that the higher
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Figure 5. Evolution of the experimental and predicted final MWD for the experiments in which the initiator concentration was varied. Legend: (b and s) 0.0188 wt %; ([ and - -) 0.075 wt %; (2 and - - -) 0.30 wt %. Point and lines are experimental and model prediction values, respectively.
Figure 7. Evolution of the experimental and predicted instantaneous conversions for the experiments in which the feeding time was varied (runs 5-8). Legend: (b and s) 1 h; ([ and - -) 2 h; (1 and ‚‚‚) 3 h; (2 and ‚ - ‚) 4 h. Point and lines are experimental and model prediction values, respectively.
Figure 6. Evolution of the experimental and predicted level of branches of the final latexes for the experiments in which the initiator concentration was varied (runs 1-5). Legend: (b) 13C NMR and (s) model predictions.
Figure 8. Effect of the feeding time on the experimental and predicted fraction of gel (runs 5-8). Legend: (b and s) 1 h; ([ and - -) 2 h; (1 and ‚‚‚) 3 h; (2 and ‚ - ‚) 4 h. Point and lines are experimental and model prediction values, respectively.
the initiator concentration, the higher the level of branches. On the other hand, Figure 3 shows that the amount of gel was only slightly affected by the initiator concentration. This is evidence of the importance of backbiting because if only long chain branching were involved, the gel fraction would increase with the branching frequency. Actually, the number of long-chain branches, LCB, predicted by the model is 2 orders of magnitude lower than the number of short-chain branches, SCB; the main contribution to the total of branches comes from SCB formed by the backbiting mechanism. It has to be pointed out that this does not mean that LCB has a negligible effect on the polymer microstructure. On the contrary, LCB is necessary for gel formation. The number of SCB increased with the initiator concentration because the monomer concentration decreased, and the lower the monomer concentration, the higher the probability for backbiting, and hence the higher the level of branches. The formation of LCB is proportional to the concentration of polymer in particles, which is much less sensitive to variations in the initiator concentration than the monomer concentration. In Figures 7-11 the effect of the feeding time in experiments 5-8 is presented. Figure 7 presents the simulated and experimental instantaneous conversions. It can be observed that the longer the monomer feeding time, the higher the instantaneous conversion. A fairly good agreement between model predictions and experimental data was obtained.
Figure 9. Effect of the feeding time on the experimental and predicted weight-average molecular weights (runs 5-8). Legend: (b and s) 1 h; ([ and - -) 2 h; (1 and ‚‚‚) 3 h; (2 and ‚ - ‚) 4 h. Point and lines are experimental and model prediction values, respectively.
Figure 8 presents the effect of the feeding time on the experimental- and model-predicted gel fractions. It can be seen that for most of the feeding times the model fitted the experimental data quite well, but for the shortest feeding time (run 6), the model predicted a gel fraction lower than that observed experimentally. Figures 9 and 10 present the effect of feeding time on sol Mw and MWD, respectively. It can be seen that the model predicts the observed evolution adequately and that the faster the feeding rate, the higher the molecular
Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3891
Figure 10. Comparison between the experimental- and modelpredicted MWDs for experiments with different feeding times (runs 5-8). Legend: (b and s) 1 h; ([ and - -) 2 h; (1 and ‚‚‚) 3 h; (2 and ‚ - ‚) 4 h. Point and lines are experimental and model prediction values, respectively.
features that have been found to play an important role in the polymerization of n-BA, such as backbiting and the low reactivity of the tertiary radicals obtained in such a reaction. The compartmentalization of the system was accounted for by means of the partial distinction approach, and the sol MWD was described by using a refined numerical fractionation approach. The mathematical model was validated by fitting experimental data of kinetics, branching frequency, sol MWD, and gel fraction. A good fit of the experimental data was achieved. The model indicated that most of the branches were SCB produced by backbiting. This mechanism affects also the propagation rate constant because the tertiary radicals are less reactive than the secondary ones. The probability of backbiting increases as the monomer concentration decreases. Therefore, the effective kp for starved emulsion polymerization is substantially lower than the value determined by PLP in bulk. It was also found that the sol MWD and the gel formed during the process are intimately related and that under starved conditions, commonly used in industry, it is difficult to modify the amount of gel produced by manipulating the initiator concentration and monomer flow rate. Appendix I. Moments of the Chain-Length Distributions of Dead and Active Polymers
Figure 11. Evolution of the experimental and predicted final level of branches for the experiments in which the feeding time was varied. Legend: (b) 13C NMR and (s) model predictions.
The balances for the kth-order moments of the overall chain-length distribution of the dead polymer in generation n, Qk(n), can be expressed as the sum of the contributions of each of the processes, s, leading to the formation of a dead polymer in the particles containing different numbers of radicals, i.
dQk(n) weights. This result is due to the fact that faster feeding rates yielded higher monomer concentrations, which resulted in longer chain lengths. The higher monomer concentrations also resulted in a reduction of the number of chains that are transferred to the gel fraction through chain transfer to polymer and bimolecular termination by combination. Figure 11 presents additional evidence that the backbiting reaction is the predominant mechanism for branching in the seeded semicontinuous emulsion polymerization of n-BA. When the feeding time is increased, the total level of branches increases but the fraction of gel is not significantly affected. The proposed mechanism predicts this behavior well because an increasing feeding time leads to a significant decrease of the monomer concentration (see Figure 7), and this promotes the backbiting reaction, in turn leading to a more branched polymer. On the other hand, the feeding time only has a limited effect on the polymer concentration. Therefore, intermolecular chain transfer to polymer (and hence gel content) is only slightly affected. Conclusions A mathematical model for the computation of kinetics, branching frequency, sol MWD, and gel fraction for the seeded semicontinuous emulsion polymerization of n-BA was presented. The model incorporates mechanistic
dt
m
)
∑s ∑ i)1
dQk,si(n) dt
(I-1)
To initialize the integration algorithm, the values of Qk(n) at time zero are required. These were calculated by discretizing the experimentally measured MWD of the sol polymer of the seed in 10 equally weighted distributions of polydispersity equal to two and given Mw, in such a way that the sum of these distributions resulted in the experimentally measured distribution for the sol polymer of the seed. These 10 distributions (different numbers of distributions were attempted but no significant effects observed) were considered the initial generations and the values of Qk(n) calculated from the molecular weights of these distributions. The kth-order moments of the chain-length distribution of a dead polymer produced by the different mechanisms are
Chain transfer to monomer and CTA dQk,fi(n) ) (kfm[M]p + kfCTA[CTA]p)[pki(n) + qki(n)] dt (I-2) Chain transfer to polymer dQk,fpi(n) ) kfp1[Q1][pki(n) + qki(n)] dt
(I-3)
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Termination by combination dQk,ctci(n) dt
i
) ctc
{[
1
∑
i-1
j)0,k)i-jNi,j,k 2p0i(n) q0i(n)
j-1 i 2 i p0 (n) µk (pn,pn) + j
µki(pn,qn) +
]
k-1 i 2 i q0 (n) µk (qn,qn) δn)0 + k n j-1 i p0 (h) p0i(l) µki(ph,pl) + j h)0,l)n-h
[
∑
ne+1
∑ ∑
h)0 l)ne+1-h
]
[
i
] ]
i
dq1i(0) ) 0 ) ka[R]w[Ni-1 - q1i(0)] dt kd[iNi + (i - 1)q1i(0)] +
j-1 i 2 i p0 (n) µk (pn,pn) δn)ne+1 + j j-1 i p0 (n-1)2µki(pn-1,pn-1) δne+1enenc + j n j-1 i p0 (h) µki(ph,pn) + 2p0i(n) j h)0
[
∑
[
(kfm[M]p + kfCTA[CTA]p)[iNi - q1i(0)] -
]
}
]
2q0i(h) µki(ph,qn) δne+1enenc
kfp1[Q1]q1i(0) - ci(i - 1)q1i(0) + kp[M]pq0i(0) (I-12) dq2i(0) ) 0 ) ka[R]w[Ni-1 - q2i(0)] dt kd[iNi + (i - 1)q2i(0)] + (I-4)
Termination by disproportionation (not operative in the polymerization of n-BA) dQk,tdi(n) ) 2ctd(i - 1)[pki(n) + qki(n)] dt
(I-5)
where kfm and kfCTA are the chain-transfer rate constants to monomer and CTA, respectively, ctd is the pseudo-first-order rate coefficient for termination by disproportionation, ctc is the pseudo-first-order rate coefficient for termination by combination, δ is the Dirac delta function, and µki(ph,q1) is the instantaneous kthorder moment of inactive polymer produced through the combination of a radical pi(h) and a radical qi(l) expressed as32
µ0i(ph,ql) ) 1 µ1i(ph,ql) ) i
µ2 (ph,ql) )
p2i(h) p0i(h)
+
q2i(l) q0i(l)
p1i(h) p0i(h) +
+
(I-6) q1i(l)
(I-7)
q0i(l)
[ ] [ ] p1i(h)
p0i(h)
2
+
q1i(l)
q0i(l)
The zeroth-, first-, and second-order moments of length distribution of radicals q (belonging to the linear generation) are expressed as
ci(i - 1)q0i(0) (I-11)
2p0 (h) q0 (l) µk (ph,ql) δn)ne+1 -
[
(I-10)
kfCTA[CTA]p)[iNi - q0i(0)] - kfp1[Q1]q0i(0) -
j-1 i p0 (h) p0i(l) µki(ph,pl) + j
i
∑ Ni,j,k k)1
dq0i(0) ) 0 ) ka[R]w[Ni-1 - q0i(0)] dt kd[iNi + (i - 1)q0i(0)] + (kfm[M]p +
2p0i(h) q0i(l) µki(ph,ql) δ0
