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Modeling semicontinuous emulsion terpolymerization
Chemical Engineering Science, Vol. 47, No. 9-11, Printed in Great Britain.
MODELING
pp. 2579-2584,
SEMICONTINUOUS
Agustfn Urretabizkaia,
1992.
EMULSION
Gurutze Alzamendi
Grupo de Ingenierfa Qufmica, Departamento Universidad de1 Pafs Vasco, Apdo.lM2.20080
Mm-2509/92 $5.00+0.00 Pcrgamon Pleas Ltd
TERPOLYMERIZATION
and Jose M. Asua*
de Qufmica Aplicada. Facultad San Sebastian. Spain.
de Ciencias
Qufmicas.
Abstract A mathematical model for the emulsion terpolymerization process was developed. The outputs of the model are the time evolution of the number of polymer particles, conversions of the monomers, latex solids content and terpolymer composition. The model was checked during the seeded semicontinuous emulsion tetpolymerization of methyl methacrylate, butyl acrylate and vinyl acetate carried out using high solids content latexes. The parameters of the model were estimated using all the experimental results by means of a direct search algorithm. An excellent fitting of the experimental data was achieved. INTRODUCTION Despite the industrial importance of emulsion terpolymers only scarce studies on emulsion terpolymerization have been reported in the literature (Rfos et al., 1980; Rfos and Guillot. 1982; Jerman and Janovic, 1984; Huo et al., 1988). In addition, most of these works focussed on experimental aspects of emulsion terpolymerization and they included almost no modeling. Recently, Storti et al. (1989 a, b) presented a so-called “pseudo-homopolymeriization approach” to model the kinetics of the multimonomer emulsion polymerization. The basic idea is to use the classical homopolymerization equations with averaged kinetic coefficients. These averaged kinetic coefficients involved the time averaged probabilities of finding a growing polymer chain with an ultimate unit of a given type (Nomura et al, 1983; Fomada and Asua, 1985). Storti et al. (1989 a. b) using values of the parameters taken from literature achieved a good agreement between experimental results and model predictions for the time evolution of the conversion and terpolymer composition. However, Storti et al. (1989 b) reported that the agreement between experimental and calculated conversion was restricted to a limited range of experimental conditions. In addition, no comparison of the experimental and calculated time evolution of the number of polymer particles was reported. In the present work, a mathematical model for the emulsion terpolymerization process was developed. This model was checked during the seeded emulsion terpolymerization of methyl methacrylate, butyl acrylate and vinyl acetate carried out in a semicontinuous reactor. MATHEMATICAL
MODEL
The mathematical model was developed taking into account the following assumptions: i) From a kinetic point of view, the whole population of polymer particles could be represented by a monodisperse population of particles; ii) the concentration of the three monomers in the different phases were at the thermodynamic equilibrium; and iii) pseudo-steady state assumption for radicals in the polymer particles and in the aqueous phase applied. In addition. all balances are referred to the whole reactor. Material
Balances
The material
balances
* To whom correspondence
for the initiator and water are: should be addressed
2579
2580
AGIJST~N URRETABIZKAIA
CII = dt
-klI
dV L=F dt
+ FI
W
El
etol.
(mol s-l)
(1)
(2)
The material balance of monomer i is as follows: di
flA + kEEi P;
-‘kpAi
dt=
- ( kpAiPT
+ k,,Pz
+ kXi
P;
+ kpciP2)
)
“p;
NP
[i],:,
+ Fi
(mol s-i)
(3)
This equation includes the time averaged probability of finding a free radical with ultimate unit of type i in the phase], P,! that can be easily calculated assuming the pseudo-steady state conditions for the radicals (Arzamendi et al. 1992). In order to calculate the concentrations of the monomers in the different phases it was assumed that they are at the thermodynamic equilibrium. The approach used by Delgado et al. (1988) was expanded to the case of terpolymerization and the equilibrium equations were written in terms of the partition coefficients. Polymer Particles Population
Balances
Following Hansen and Ugelstad ( 1978 ). it has been considered thatnucleation occurs in two steps in series. In the first, the particle precursors are produced by either entry of free radicals into micelles or self-precipitation of growing oligoradicals when a critical degree of polymerization is reached. In the second, the precursors grow until they become mature polymer particles. The water solubility of the monomers used in the present work is significant, therefore it was considered that the formation of particle precursors occurred mainly through oligoradical self-precipitation. The rate of formation of polymer particle precursors is the rate of formation of oligomers of a length greater than a critical value. Although this critical value may depend on the composition of the oligoradical. in this work a single critical value was used for all types of radicals. dN; dt
=
(kp 1 i
I, Iav Rj crir NA
(4)
whenz N,,’is the number of particle precursors in the reactor, R,;the critical length in the reactor, and (kp [ i I,), is given by:
‘kpl i I,,), =
‘r.
i=A.B.C
PI + CkpAi
kpBi
pB”+
number of moles of oligomers of
kpci
Pzl[ i lw
(5)
The p~cursors are colloidal unstable species thatshould reach some size in order to become stable. The growth can occur through polymerization and by aggregation with other precursors. Because of the small size of the precursory. both the monomer concentration and the average number of radicals per precursor should be rather low and hence, most likely, the growth will occur through aggregation. Detailed models for the aggregation have been proposed by Hansen and Ugelstad (1978) and Richards et al. (1989). However, these models involved a large number of parameters which are very difficult to estimate. Therefore, a pragmatic approach was used in the present work. Thus, the rate of production of polymer particles was defined as follows: dNP dt
=
(dN;/dt).P,
(6)
where P, is the probability that the precursor becomes a polymer particle given by the following empirical equation:
Modeling
El
semicontinuous
2581
emulsion terpolymerization
1
PN =
k, Np l+%Er
l+
where E, is the amount of emulsifier in the reactor. Equation (7) accounts for the main experimental findings. namely, that nucleation ceases when the number of polymer particle8 reached a given value and that this number inceases when the amount of emulsifier increases. It should be pointed out that this is a kinetic criterion for the end of nucleation in contrast to the classical theories of particle nucleation (Smith
and Ewart, 1948) which assumed that nucleation stopped when the total area of the polymer particles was equal to the area that could be covered by the emulsifier, i.e., these theories assumed complete coverage of the surface of the polymer particles. However, partial coverage is enough to stabilize the polymer particles during the polymerization (Powers, 1950. Urquiola et al., 1991). Radical Balances in the Polymer Particles The model was applied to a seeded emulsion terpolymerization system where, although secondary nucleation occurred, the relative increment of the total number of polymer particles wa8 reduced by the presence of the seed. Therefore. it was assumed that pseudo-steady state assumption applied for the
radicals in the polymer particles. Under these circumstances. the average number of radicals per particle can be calculated using the approach proposed by Ugelstad and Hansen (1976): n =(a2/8)/[m+
1 + [(a2/4)/(m+2+
[(a2/4)/(m+
. ..)I)11
(8) ’
where Kv
a=8kaN
vp
ATI;
;
m=kdN
(9)
A vp ’ ‘TV
tP
k_ is the entry rate coefficient, vp the volume of the monomer swollen polymer particle, and zW and k, the average termination rate constant and the overall desorption rate coefficient, respectively. itp
= k,,
(P,pj2 + ktBB a;>’
+ 2k,,,
P;
+ ktcc P;>”
+ 2 k,,
P’A p;
+ 2k,,
p: p:
+
(10)
P;
kd = kdA + kdB + kdc
(11)
where k, is the rate coefficient for desorption of radicals of type i that are calculated using the approach proposed by Asua at al. (1989) and Forcada and Asua (1990). In addition, an average gel effect factor was used: ktP = klpo expl a, c$
+ b, (d::23
(12)
where a,,’is the volume fraction of polymer in the polymer particles. Radical Balances in the Aqueous Phase In order to calculate I$_ and RI, the radical balances in the aqueous phase should be developed. Taking into consideration the pseudo-steady state assumption. these balances are as follows: Radicals with one monomeric
d RI
-=“=kd?iN dt
unit
Np + A
2fkII
- (kp[ilWjar
R,
- i?##R,F
- kaVRl Np N W W A
(13)
2582
AGUST~N
f-Oligomer d Rj -=
dt
URR~TABIZKAIA
et al.
El
Radicais
0 E (k,[i],),
(Rj_l
- Ri)
- k
Rj %
W
R*Np - ke V’N W A
(14)
The total amount of radicals in the aqueous phase is:
j=l
CHECK OF THE MODEL The model was checked during the seeded semicontinuous emulsion terpolymerization of methyl methacrylate (MMA), butyl acrylate (BuA) and vinyl acetate (VAc) carried out feeding to the reactor pteemulsified cleaned monomers and a solution of initiator in separate streams. The seed was prepared thmught a semicontinuous process. Polymerizations were carried out at 8CFCunder industrial conditions. i.e., producing high solids content latexes (55wt%). Alipal CO436 (ammonium salt of sulfated nonylphenoxy poly(ethyleneoxy) ethanol (4 ethylene oxide)) was used as emulsifier and in order to enhance the latex stability a 1 wt % (based on monomers) of acrylic acid (AA) was included in the recipeTaking the recipe given in Table I as a reference, the following effects were investigated: i) Total feeding time (4,6 and 9 hours); ii) concentration of initiator (0.18 wt%, and 0.3796, based on monomers); iii) total amount of emulsifier (0.7 wt% and 1.4 wt%. based on monomers); iv) distribution of the emulsifier between the initial charge and the feed (lODO and 50/50); v) amount of seed (5 wt% and 15 wt%. based on the whole recipe): and vi) solids content (33 wt%. 55 wt% and 60 wt%). Samples were withdrawn from the reactor at appropriate intervals and the polymerization was short-stopped with hydroquinone. The overall conversion was determined gravimetrically. The terpolymer composition was determined measuring the residual monomer by gas chromatography and the particle size of the latexes was measured by dynamic light scattering. The model developed contained a large number of parameters. Sxd 0.12 The parameters given in Table II have been obtained from haMA 0.1227 literature. In addition, the radical entry rate coefficient, k,; BirA 0.2241 the desorption rate coefficient, k,,; the critical length of the VAC 0.0452 growing oligomer&; the parameters in eq.(7) (k, and kJ; AA 3.9 x l&X and the parameters associated with the gel effect factor (a, Wls 0.100 0.1281 0.060 and b,, eq(12)) were estimated by means of the Nelder and Emulrifii 0.25 x lo-3 2.5 x lo-3 %T?Z% 0.6 S. lo-’ 0.9 x IO-3 Mead method of direct search. The objective function was NfiC% 0.6 x 10-a 0.9 x lo-3 the best fit of all experimental results. namely, time evolution of overall conversion, tcrpolymer composition, and total number of polymer particles of all experiments carried out. It was found that polymerization proceeded under Smith-Ewart case 3 conditions ( ?i z=> 0.5). Under these conditions desorption is not kinetically significant and themfore the values of k,, are irrelevant. The estimated values of the rest of adjustable parameters were: k.= 2.21 x 105 j,= 6
al= -5.08 k,= 0.12 x lo-”
b,= - 0.76 k= 49.9
This value of k. means that termination in the aqueous phase is negligible. Figures l-3 present some examples of the comparisons between experimental results and model predictions. Figure 1 shows the effect of the feed flow rateon the evolution of the fractional conversion. This is defined as the fraction of the monomer already fed to the reactor that has been transformed into terpolymer. It can be seen that the longer the feeding time the less the monomer accumulated in the mactor. In addition, Figure 1 shows that an excellent
TableII.-VdnuofIhepamuerof1110modsl:Tr8~’C 4.48
1.16
rAtI.%C* qa***-
0.07
0.06
22.21
Qc, ‘CA, qn”*=
2.64
3.07
0.315
*J&y;;
kf=”
0.234
be. b.‘. kc” (m3 moi-’ S-1) k,,P. kaP. k&‘-•
4.06 101
2.03 lo*
036 101
0.936 101
0.526 l(rl
0.855 1tH
kirr*.ks*, ke$-
0.261101
0.11 10-l
a22 102
2 1DS
0.5
k1b-% f
El
Modeling
semicontinuous emulsion terpolymerization
2583
agreement between experimental results and model predictions was achieved. Figure 2 presents the effect of the feed flow rate on the evolution of the cumulative terpolymer composition. It can be seen that at the beginning of the process, the MMA content of the terpolymer was much higher than the feed composition (given by the horizontal solid lines). In addition, VAc was difficult to incorporate into the terpolymer. This behavior was due to the widely different Eactivity ratios of these monomers mable II). Figure 2 also shows that the cumulative tepolymer composition approached the feed composition when the feeding time was increased from 4h to 9h. Longer feeding times reduced the I .o amount of monomers accumulated in the reactor bringing the system nea=r the true starved conditions where the ter- l polymer formed has the same composition than the feed. Figure 2 shows that a good agreement between experimental s ,,= n~ults and model predictions was achieved.Figme 3 presents 1 the effect of the total amount of emulsifier on the evolution of j 0.4 the number of polymer particles. It can be seen that no & secondary nucleation was observed in the case of the low emulsifier concentration whereas a significant number of 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 polymer particles were newly formed when the amount of Rclalirc The emulsifier was 0.37 wt%. In terms of the model used in the present work, this means that the precursors became more Fkurc 1.. Effect of IIX feed mic on Ihe evolution of the W collvaial: stable when the amount of emulsifier increased and hence PN MOdd Fediagtimr Fsxp increased. Figure 3 shows that a good agreement between ex. 9h perimental results and model predictions was achieved. 6h a .. . . I 4h CONCLUSIONS 4 0.6 A mathematical model for the emulsion terpolymerization process was developed. The model takes into account the following processes: i) nucleation of polymer particles; ii) entry and exit of free radicals into and from the polymer particles; iii) termination of free radicals in both aqueous phase and polymer particles and iv) polymerization in both polymer particles and aqueous phase. In addition, the concentrations of the monomers in the different phases are assumed to be under thermodynamic equilibrium. The outputs of the model are the time evolution of the number of polymer particles, conversions of the monomers, latex solids content and terpolymer composition. The model was checked during the semicontinuous emulsion terpolymerization of MMA, BuA and VAc. The effect of the feed flow rate; initiator concentration; total amount of emulsifier and its distribution between the initial charge of the reactor and the feed; amount of seed; and solids content on the time evolution of the conversions of the monomers, terpolymer composition and number of polymer particles was investigated. Polymerizations were carried out under industrial conditions using high solids content latexes. The parameters of the model were estimated using all the experimental results by means of a direct search algorithm. An excellent fitting of the experimental data was achieved. ACKNOWLEDGEMENTS financial support by the Diputaci6n Foral de Gipuzkoa, the scholarships for A. Urretabizkaia from the Basque Government and for G. Alzamendi from the Ministerio de Educaci6n y Ciencia are gratefully appreciated. NOMENCLATURE Parameter given by eq.
o_s Oe4 0.3 0_2 “’ 0.0 0.0
0.4
0.2
(9).
0.8
0.8
1.0
1.2
1.4
PcbUv*The FI~W~ Z- EEL octi fd - QPrbeWOI,.~~O~ of ti wmuluivccapdymscomposilim: F&diagI.imsEXL? Model 9h 6h 4h
f ; -
001
---
-10 .I+
-.-a-
1.0
o8 ’
0 -. t 08 0 _. .a < . .p.. ...‘. 0 2 o 4 - -o- - = c A *- - * - - ;- - -.-_” . l 6 . : I
I
100
200
0.2
The
a
? 1 z $ 3 z t ;
0
300
400
nlar (tin)
Figure 3-e EftcctoftbebetMlllamr evolutioa
W Fm&ifia0.18% 0.37% .bvadO_
dcmulsiIklonabe
of tbc toUl number
Exp . -
of polymer
Model ---
2584
A. B. C Ci I, F
V”
4
AGUST~N URRETABIZKAIA et al.
El
Total number of moles of VAc. MMA and BuA. respectively, in the reactor (mol).
Concentrations of monomer i in phase j (mol m”). Total amount of emulsifier (kg). Efficiency factor for initiator decomposition. Molar feed rate of monomer i (mol s-l). Molar feed rate of initistor (mol 6-l). Volumetric feed rate of water (m’ s-l) Total number of moles of initiator in the reactor (mol) Critical length of the growing oligomers Entry rate coefficient (m’ mol-’ s-t) Overall desorption rate coefficient (cl) Rate coefficient for desorption of radicals of type i (s-l) Partition coefBcient of monomer i between the phasej and the aqueous phase. Rate constant for initiator decomposition (s-r) Propagation rate constant (m’ mol-’ s-l) Average termination rate constant in phase j (m’ mol-’ s-l) Average termination rate constant in the polymer particles at polymer content (my mol-’ s-l) Parameter given by eq.(9) Average number of radicals per particle Avogadro’s number. Total number of polymer particles in the reactor Total number of particle precursors in the reactor Tie averaged probability of finding a free radical with ultimate unit of type i in the phase j. Probability that a precursor becomes a polymer particle. Moles of radicals of lengthj in the reactor (mol). Moles of oligomers of critical length in the reactor (mol). Moles of free radicals of any kind in the aqueous phase (mol). Volume of one monomer swollen polymer particle (m’). Volume of aqueous phase, monomer droplets and monomer swollen polymer particle, respectively. in the reactor (m3) Volume of water in the reactor (m’). Volume fraction of component j in phase i.
REFERENCES Arzamendi. G.. de la Cal, J. C. and Asua. I. M., 1992, Die Angewandte Macromol. Chem., 194.4764. Asua, J. M., Sudol. E. D. and El-Aasser, M. S.. 1989. J. Polym. Sci., Polym. Chem. Ed., 22.3903-3913. Brandntp. 3. and Immergut. E.H.. Eds.. 1989. “Polymer Handbook” 2nd. Ed. Wiley-Interscience.
New York.
Del ado. J..El-Aasser- M-S.. Silebi, C.. Vanderhoff. J.W. and Guillof J.. , 1988. I. Polym. Sci.; Part B: Polym. Phys.. a. 1513. Dimitratos Y.N..
1495
1989. Ph. D. Thesis. Lehigh University.
Forcada, J. and Asua, I. M.. 1985. I. Polym. Sci., Polym. Chem. Ed., z,
19551962.
Forcada, J. and Asua, I. M., 1990. J. Polym. Sci.: Part A: Polym. Chem.. a, Hskoila E.. 1963,“Annales
Universitatis Turkuensis” Ser. A&,
Hansen. F.K. and Ugelstad, J.. 1978.
51.
3. Polym. Sci.. Polym. Chem. Ed.,fi.
Huo, B. P.. Hamielec. A. E. andMacGregor.
987-1009.
1953-1979.
J. F., 1988. J. Appl. Polym. Sci.. 21. 1409-1420.
Jerman. M. B. and Janovic. Z., 1984, J. Macromol. Sci. Chem. 821.887-899. Matejicelc, A., Pivonkov& A., Kasha, J.. Dilt, P. and FormBnek. L., 1988, J. Appl. Polym. Sci.. s., Nomura, M., Yamamoto. K., Horie. I. and Fuji& K., 1982. J. Appl. Polym. Sci., z
583-591.
2483-2501.
Powers, J.R.. 1950. U.S. Patent 2.520.959. Rios. L.. Pichot. C. and Guillot, J., 1980. Makromol. Chem., 181. 677-700. Rios. L. and Guillot, J.. 1982. Makromol. Chem.. 183.531548. Richards, J. R.. Congalidis. J. P. and Gilbert. R. G.. 1989. J. Appl. Polym. Sci.. z Smith, W. V. and Ewart. R. H.. 1948, J. Chem. Phys.. x.
2727-2756,
592-599.
Storti. G.. Carra, S.. Morbidelli. M. and Vita, G.. 1989 a, J. Appl. Polim. Sci.. 22. 2443-2467. Storti, G.. Carra. S. and Morbidelli, M.. 1989 b, in “Corn uters Applications in Applied Polymer Science II: Automation, Modelling and Simulation”, T. Provder. Ed., ACS Symp. Sp er. 404. Ugelstad. J. and Hansen. F. K.. 1976. Rubber Chem. Tecbnol.. e.
536-609.
Ur uiola. B., Araamendi. G.. Leiza, J. R.. Zamora. A.. Asua, J. M.. Delgado. J.. El-Aasser. M. S. and Vanderhoff, J. W., 1991, J. ?olym. Sci.: Part. A: Polym. Chem.. 2!& 169-186. Walling, C.. 1957. “Free Radicals in Solution”, Wiley-Interscience,
New York.
MODELING
pp. 2579-2584,
SEMICONTINUOUS
Agustfn Urretabizkaia,
1992.
EMULSION
Gurutze Alzamendi
Grupo de Ingenierfa Qufmica, Departamento Universidad de1 Pafs Vasco, Apdo.lM2.20080
Mm-2509/92 $5.00+0.00 Pcrgamon Pleas Ltd
TERPOLYMERIZATION
and Jose M. Asua*
de Qufmica Aplicada. Facultad San Sebastian. Spain.
de Ciencias
Qufmicas.
Abstract A mathematical model for the emulsion terpolymerization process was developed. The outputs of the model are the time evolution of the number of polymer particles, conversions of the monomers, latex solids content and terpolymer composition. The model was checked during the seeded semicontinuous emulsion tetpolymerization of methyl methacrylate, butyl acrylate and vinyl acetate carried out using high solids content latexes. The parameters of the model were estimated using all the experimental results by means of a direct search algorithm. An excellent fitting of the experimental data was achieved. INTRODUCTION Despite the industrial importance of emulsion terpolymers only scarce studies on emulsion terpolymerization have been reported in the literature (Rfos et al., 1980; Rfos and Guillot. 1982; Jerman and Janovic, 1984; Huo et al., 1988). In addition, most of these works focussed on experimental aspects of emulsion terpolymerization and they included almost no modeling. Recently, Storti et al. (1989 a, b) presented a so-called “pseudo-homopolymeriization approach” to model the kinetics of the multimonomer emulsion polymerization. The basic idea is to use the classical homopolymerization equations with averaged kinetic coefficients. These averaged kinetic coefficients involved the time averaged probabilities of finding a growing polymer chain with an ultimate unit of a given type (Nomura et al, 1983; Fomada and Asua, 1985). Storti et al. (1989 a. b) using values of the parameters taken from literature achieved a good agreement between experimental results and model predictions for the time evolution of the conversion and terpolymer composition. However, Storti et al. (1989 b) reported that the agreement between experimental and calculated conversion was restricted to a limited range of experimental conditions. In addition, no comparison of the experimental and calculated time evolution of the number of polymer particles was reported. In the present work, a mathematical model for the emulsion terpolymerization process was developed. This model was checked during the seeded emulsion terpolymerization of methyl methacrylate, butyl acrylate and vinyl acetate carried out in a semicontinuous reactor. MATHEMATICAL
MODEL
The mathematical model was developed taking into account the following assumptions: i) From a kinetic point of view, the whole population of polymer particles could be represented by a monodisperse population of particles; ii) the concentration of the three monomers in the different phases were at the thermodynamic equilibrium; and iii) pseudo-steady state assumption for radicals in the polymer particles and in the aqueous phase applied. In addition. all balances are referred to the whole reactor. Material
Balances
The material
balances
* To whom correspondence
for the initiator and water are: should be addressed
2579
2580
AGIJST~N URRETABIZKAIA
CII = dt
-klI
dV L=F dt
+ FI
W
El
etol.
(mol s-l)
(1)
(2)
The material balance of monomer i is as follows: di
flA + kEEi P;
-‘kpAi
dt=
- ( kpAiPT
+ k,,Pz
+ kXi
P;
+ kpciP2)
)
“p;
NP
[i],:,
+ Fi
(mol s-i)
(3)
This equation includes the time averaged probability of finding a free radical with ultimate unit of type i in the phase], P,! that can be easily calculated assuming the pseudo-steady state conditions for the radicals (Arzamendi et al. 1992). In order to calculate the concentrations of the monomers in the different phases it was assumed that they are at the thermodynamic equilibrium. The approach used by Delgado et al. (1988) was expanded to the case of terpolymerization and the equilibrium equations were written in terms of the partition coefficients. Polymer Particles Population
Balances
Following Hansen and Ugelstad ( 1978 ). it has been considered thatnucleation occurs in two steps in series. In the first, the particle precursors are produced by either entry of free radicals into micelles or self-precipitation of growing oligoradicals when a critical degree of polymerization is reached. In the second, the precursors grow until they become mature polymer particles. The water solubility of the monomers used in the present work is significant, therefore it was considered that the formation of particle precursors occurred mainly through oligoradical self-precipitation. The rate of formation of polymer particle precursors is the rate of formation of oligomers of a length greater than a critical value. Although this critical value may depend on the composition of the oligoradical. in this work a single critical value was used for all types of radicals. dN; dt
=
(kp 1 i
I, Iav Rj crir NA
(4)
whenz N,,’is the number of particle precursors in the reactor, R,;the critical length in the reactor, and (kp [ i I,), is given by:
‘kpl i I,,), =
‘r.
i=A.B.C
PI + CkpAi
kpBi
pB”+
number of moles of oligomers of
kpci
Pzl[ i lw
(5)
The p~cursors are colloidal unstable species thatshould reach some size in order to become stable. The growth can occur through polymerization and by aggregation with other precursors. Because of the small size of the precursory. both the monomer concentration and the average number of radicals per precursor should be rather low and hence, most likely, the growth will occur through aggregation. Detailed models for the aggregation have been proposed by Hansen and Ugelstad (1978) and Richards et al. (1989). However, these models involved a large number of parameters which are very difficult to estimate. Therefore, a pragmatic approach was used in the present work. Thus, the rate of production of polymer particles was defined as follows: dNP dt
=
(dN;/dt).P,
(6)
where P, is the probability that the precursor becomes a polymer particle given by the following empirical equation:
Modeling
El
semicontinuous
2581
emulsion terpolymerization
1
PN =
k, Np l+%Er
l+
where E, is the amount of emulsifier in the reactor. Equation (7) accounts for the main experimental findings. namely, that nucleation ceases when the number of polymer particle8 reached a given value and that this number inceases when the amount of emulsifier increases. It should be pointed out that this is a kinetic criterion for the end of nucleation in contrast to the classical theories of particle nucleation (Smith
and Ewart, 1948) which assumed that nucleation stopped when the total area of the polymer particles was equal to the area that could be covered by the emulsifier, i.e., these theories assumed complete coverage of the surface of the polymer particles. However, partial coverage is enough to stabilize the polymer particles during the polymerization (Powers, 1950. Urquiola et al., 1991). Radical Balances in the Polymer Particles The model was applied to a seeded emulsion terpolymerization system where, although secondary nucleation occurred, the relative increment of the total number of polymer particles wa8 reduced by the presence of the seed. Therefore. it was assumed that pseudo-steady state assumption applied for the
radicals in the polymer particles. Under these circumstances. the average number of radicals per particle can be calculated using the approach proposed by Ugelstad and Hansen (1976): n =(a2/8)/[m+
1 + [(a2/4)/(m+2+
[(a2/4)/(m+
. ..)I)11
(8) ’
where Kv
a=8kaN
vp
ATI;
;
m=kdN
(9)
A vp ’ ‘TV
tP
k_ is the entry rate coefficient, vp the volume of the monomer swollen polymer particle, and zW and k, the average termination rate constant and the overall desorption rate coefficient, respectively. itp
= k,,
(P,pj2 + ktBB a;>’
+ 2k,,,
P;
+ ktcc P;>”
+ 2 k,,
P’A p;
+ 2k,,
p: p:
+
(10)
P;
kd = kdA + kdB + kdc
(11)
where k, is the rate coefficient for desorption of radicals of type i that are calculated using the approach proposed by Asua at al. (1989) and Forcada and Asua (1990). In addition, an average gel effect factor was used: ktP = klpo expl a, c$
+ b, (d::23
(12)
where a,,’is the volume fraction of polymer in the polymer particles. Radical Balances in the Aqueous Phase In order to calculate I$_ and RI, the radical balances in the aqueous phase should be developed. Taking into consideration the pseudo-steady state assumption. these balances are as follows: Radicals with one monomeric
d RI
-=“=kd?iN dt
unit
Np + A
2fkII
- (kp[ilWjar
R,
- i?##R,F
- kaVRl Np N W W A
(13)
2582
AGUST~N
f-Oligomer d Rj -=
dt
URR~TABIZKAIA
et al.
El
Radicais
0 E (k,[i],),
(Rj_l
- Ri)
- k
Rj %
W
R*Np - ke V’N W A
(14)
The total amount of radicals in the aqueous phase is:
j=l
CHECK OF THE MODEL The model was checked during the seeded semicontinuous emulsion terpolymerization of methyl methacrylate (MMA), butyl acrylate (BuA) and vinyl acetate (VAc) carried out feeding to the reactor pteemulsified cleaned monomers and a solution of initiator in separate streams. The seed was prepared thmught a semicontinuous process. Polymerizations were carried out at 8CFCunder industrial conditions. i.e., producing high solids content latexes (55wt%). Alipal CO436 (ammonium salt of sulfated nonylphenoxy poly(ethyleneoxy) ethanol (4 ethylene oxide)) was used as emulsifier and in order to enhance the latex stability a 1 wt % (based on monomers) of acrylic acid (AA) was included in the recipeTaking the recipe given in Table I as a reference, the following effects were investigated: i) Total feeding time (4,6 and 9 hours); ii) concentration of initiator (0.18 wt%, and 0.3796, based on monomers); iii) total amount of emulsifier (0.7 wt% and 1.4 wt%. based on monomers); iv) distribution of the emulsifier between the initial charge and the feed (lODO and 50/50); v) amount of seed (5 wt% and 15 wt%. based on the whole recipe): and vi) solids content (33 wt%. 55 wt% and 60 wt%). Samples were withdrawn from the reactor at appropriate intervals and the polymerization was short-stopped with hydroquinone. The overall conversion was determined gravimetrically. The terpolymer composition was determined measuring the residual monomer by gas chromatography and the particle size of the latexes was measured by dynamic light scattering. The model developed contained a large number of parameters. Sxd 0.12 The parameters given in Table II have been obtained from haMA 0.1227 literature. In addition, the radical entry rate coefficient, k,; BirA 0.2241 the desorption rate coefficient, k,,; the critical length of the VAC 0.0452 growing oligomer&; the parameters in eq.(7) (k, and kJ; AA 3.9 x l&X and the parameters associated with the gel effect factor (a, Wls 0.100 0.1281 0.060 and b,, eq(12)) were estimated by means of the Nelder and Emulrifii 0.25 x lo-3 2.5 x lo-3 %T?Z% 0.6 S. lo-’ 0.9 x IO-3 Mead method of direct search. The objective function was NfiC% 0.6 x 10-a 0.9 x lo-3 the best fit of all experimental results. namely, time evolution of overall conversion, tcrpolymer composition, and total number of polymer particles of all experiments carried out. It was found that polymerization proceeded under Smith-Ewart case 3 conditions ( ?i z=> 0.5). Under these conditions desorption is not kinetically significant and themfore the values of k,, are irrelevant. The estimated values of the rest of adjustable parameters were: k.= 2.21 x 105 j,= 6
al= -5.08 k,= 0.12 x lo-”
b,= - 0.76 k= 49.9
This value of k. means that termination in the aqueous phase is negligible. Figures l-3 present some examples of the comparisons between experimental results and model predictions. Figure 1 shows the effect of the feed flow rateon the evolution of the fractional conversion. This is defined as the fraction of the monomer already fed to the reactor that has been transformed into terpolymer. It can be seen that the longer the feeding time the less the monomer accumulated in the mactor. In addition, Figure 1 shows that an excellent
TableII.-VdnuofIhepamuerof1110modsl:Tr8~’C 4.48
1.16
rAtI.%C* qa***-
0.07
0.06
22.21
Qc, ‘CA, qn”*=
2.64
3.07
0.315
*J&y;;
kf=”
0.234
be. b.‘. kc” (m3 moi-’ S-1) k,,P. kaP. k&‘-•
4.06 101
2.03 lo*
036 101
0.936 101
0.526 l(rl
0.855 1tH
kirr*.ks*, ke$-
0.261101
0.11 10-l
a22 102
2 1DS
0.5
k1b-% f
El
Modeling
semicontinuous emulsion terpolymerization
2583
agreement between experimental results and model predictions was achieved. Figure 2 presents the effect of the feed flow rate on the evolution of the cumulative terpolymer composition. It can be seen that at the beginning of the process, the MMA content of the terpolymer was much higher than the feed composition (given by the horizontal solid lines). In addition, VAc was difficult to incorporate into the terpolymer. This behavior was due to the widely different Eactivity ratios of these monomers mable II). Figure 2 also shows that the cumulative tepolymer composition approached the feed composition when the feeding time was increased from 4h to 9h. Longer feeding times reduced the I .o amount of monomers accumulated in the reactor bringing the system nea=r the true starved conditions where the ter- l polymer formed has the same composition than the feed. Figure 2 shows that a good agreement between experimental s ,,= n~ults and model predictions was achieved.Figme 3 presents 1 the effect of the total amount of emulsifier on the evolution of j 0.4 the number of polymer particles. It can be seen that no & secondary nucleation was observed in the case of the low emulsifier concentration whereas a significant number of 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 polymer particles were newly formed when the amount of Rclalirc The emulsifier was 0.37 wt%. In terms of the model used in the present work, this means that the precursors became more Fkurc 1.. Effect of IIX feed mic on Ihe evolution of the W collvaial: stable when the amount of emulsifier increased and hence PN MOdd Fediagtimr Fsxp increased. Figure 3 shows that a good agreement between ex. 9h perimental results and model predictions was achieved. 6h a .. . . I 4h CONCLUSIONS 4 0.6 A mathematical model for the emulsion terpolymerization process was developed. The model takes into account the following processes: i) nucleation of polymer particles; ii) entry and exit of free radicals into and from the polymer particles; iii) termination of free radicals in both aqueous phase and polymer particles and iv) polymerization in both polymer particles and aqueous phase. In addition, the concentrations of the monomers in the different phases are assumed to be under thermodynamic equilibrium. The outputs of the model are the time evolution of the number of polymer particles, conversions of the monomers, latex solids content and terpolymer composition. The model was checked during the semicontinuous emulsion terpolymerization of MMA, BuA and VAc. The effect of the feed flow rate; initiator concentration; total amount of emulsifier and its distribution between the initial charge of the reactor and the feed; amount of seed; and solids content on the time evolution of the conversions of the monomers, terpolymer composition and number of polymer particles was investigated. Polymerizations were carried out under industrial conditions using high solids content latexes. The parameters of the model were estimated using all the experimental results by means of a direct search algorithm. An excellent fitting of the experimental data was achieved. ACKNOWLEDGEMENTS financial support by the Diputaci6n Foral de Gipuzkoa, the scholarships for A. Urretabizkaia from the Basque Government and for G. Alzamendi from the Ministerio de Educaci6n y Ciencia are gratefully appreciated. NOMENCLATURE Parameter given by eq.
o_s Oe4 0.3 0_2 “’ 0.0 0.0
0.4
0.2
(9).
0.8
0.8
1.0
1.2
1.4
PcbUv*The FI~W~ Z- EEL octi fd - QPrbeWOI,.~~O~ of ti wmuluivccapdymscomposilim: F&diagI.imsEXL? Model 9h 6h 4h
f ; -
001
---
-10 .I+
-.-a-
1.0
o8 ’
0 -. t 08 0 _. .a < . .p.. ...‘. 0 2 o 4 - -o- - = c A *- - * - - ;- - -.-_” . l 6 . : I
I
100
200
0.2
The
a
? 1 z $ 3 z t ;
0
300
400
nlar (tin)
Figure 3-e EftcctoftbebetMlllamr evolutioa
W Fm&ifia0.18% 0.37% .bvadO_
dcmulsiIklonabe
of tbc toUl number
Exp . -
of polymer
Model ---
2584
A. B. C Ci I, F
V”
4
AGUST~N URRETABIZKAIA et al.
El
Total number of moles of VAc. MMA and BuA. respectively, in the reactor (mol).
Concentrations of monomer i in phase j (mol m”). Total amount of emulsifier (kg). Efficiency factor for initiator decomposition. Molar feed rate of monomer i (mol s-l). Molar feed rate of initistor (mol 6-l). Volumetric feed rate of water (m’ s-l) Total number of moles of initiator in the reactor (mol) Critical length of the growing oligomers Entry rate coefficient (m’ mol-’ s-t) Overall desorption rate coefficient (cl) Rate coefficient for desorption of radicals of type i (s-l) Partition coefBcient of monomer i between the phasej and the aqueous phase. Rate constant for initiator decomposition (s-r) Propagation rate constant (m’ mol-’ s-l) Average termination rate constant in phase j (m’ mol-’ s-l) Average termination rate constant in the polymer particles at polymer content (my mol-’ s-l) Parameter given by eq.(9) Average number of radicals per particle Avogadro’s number. Total number of polymer particles in the reactor Total number of particle precursors in the reactor Tie averaged probability of finding a free radical with ultimate unit of type i in the phase j. Probability that a precursor becomes a polymer particle. Moles of radicals of lengthj in the reactor (mol). Moles of oligomers of critical length in the reactor (mol). Moles of free radicals of any kind in the aqueous phase (mol). Volume of one monomer swollen polymer particle (m’). Volume of aqueous phase, monomer droplets and monomer swollen polymer particle, respectively. in the reactor (m3) Volume of water in the reactor (m’). Volume fraction of component j in phase i.
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Ur uiola. B., Araamendi. G.. Leiza, J. R.. Zamora. A.. Asua, J. M.. Delgado. J.. El-Aasser. M. S. and Vanderhoff, J. W., 1991, J. ?olym. Sci.: Part. A: Polym. Chem.. 2!& 169-186. Walling, C.. 1957. “Free Radicals in Solution”, Wiley-Interscience,
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