Kinetics of Emulsion Polymerisation in the Unsteady State BY BRIAN W. BROOKS* Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire LE11 3TU Received 28th February, 1978 Equations are developed which show how the number of free radicals in polymer particles changes with time in emulsion polymerisation. The treatment is confined to cases where the total number of particles is constant. Unlike the procedures which have been published previously, the present treatment, which is relatively simple, allows for both radical desorption from the particles and radical combination within the particles. Inclusion of intraparticle combination leads to predictions of more realistic values of the numbers of radicals within the particles. The new procedure is shown to apply over a wider range of operating variables than the previous treatments.If the rate at which radicals enter the particles is time-independent, the efficiency of radical capture by the particles is expected to be low. In a recent paper Birtwistle and Blackley discussed the kinetics of emulsion polymerisation. Their treatment dealt with a fixed number of reaction loci which would be found in the post-nucleation stages of the polymerisation or in a seed latex. Attention was focused on reaction systems which had not attained a steady state, with a view to obtaining expressions which showed how the distribution of free radicals in the particles changed with time. They considered cases in which radical desorption from the polymer particles is important.In common with previous workers,2* Birtwistle and Blackley used a modified version of the Smith-Ewart recursion eq~ations.~ However, they assumed that the rate constant for chain termination within the polymer particles is zero. This assumption was not justified. Gilbert and Napper made a similar assumption, saying that the low values of i, (the average number of radicals per particle) for which their equations apply, are consistent with negligible bimolecular termination within the particles. Although this consistency is apparent, it cannot be assumed that a low value of i implies negligible bimolecular termination. Rapid combination of radicals in the particles would help to maintain a low value of i. The purpose of the present paper is to present a treatment which allows for chain termination in the particles.This treatment is much simpler than those of previous worker^,^'^ and is more useful because it applied to a wider range of i. Implications for the efficiency of radical capture by the particles will also be discussed. It will be assumed that no new particles are formed and that existing particles do not agglome- rate. In addition to bimolecular termination, radicals may disappear from the particles by first-order-processes. Any changes in particle volumes are assumed to be small. KINETIC EQUATIONS Using the notation of Birtwistle and Blackley the recursion equations can be (1) written as dnJdt = (ni-l -n,)o+[(i+ l)nf+l -int]k+[(i+2)(i+1)ni+,-i(i- l)nl]k,/v 3022B . W. BROOKS 3023 where nf is the number of particles in unit volume of emulsion which contain i radicals, 0 is the average rate at which radicals enter the particles, v is the volume of a single particle and k, is the rate coefficient for radical-radical termination reactions in the particles.k is the rate coefficient for first-order processes in which radicals are lost from the particles. In many cases, this loss will result from the migration of radicals into the aqueous phase. When this happens, k = koa/v, where a is the surface area of a particle and ko is a mass transfer coefficient. The time t is taken to be zero when radicals from the aqueous phase start to enter the radical-free particles. In cases where i is 41 we may take i < 2. This approximation that i =P 2 in cases where kt # 0 is reasonable and, in the subsequent treatment, is shown to be valid even when i is as high as 0.4.The recursion expression (1) is now converted into a set of three simultaneous differential equations. When i = 0 This set of differential equations and their solutions are similar to those encountered in isotopic exchange processes.5* For convenience the quantities nl/N will be found as functions of time. The boundary values to be used are now t = 0 dnoldt = -noo+n,k+2n2k,/v (2) no/N = 1, n i = n 2 = 0 , t = O where N is the total number of particles. The solutions to the equations now become where Ti = ~ ( l - L i ) , L1 = (k-2kt/~)/(3k+ 6- P i ) , L z = (k-2kt/~)/(3k + o - P ~ ) , Pi = (20 + 3k + 2kt/~)2 + J, P2 = (20 + 3k + 2 k t / ~ ) / 2 - J T2 = ~(l-Lz), and J = [f2k'iv-3k)"+(~-2k)(k-Ik,jv) 4 I .DISCUSSION OF SOLUTIONS In order to compare the above solutions with those of Birtwistle and Blackley we will first use the same values of the key variables as they did, i.e. Q = 1 x s-l, k = 5 x 10-4s-1 and k, = 0. In this case the predictions of eqn (3)-(5) are identical to those obtained by the previous workers.l, This is demonstrated in fig. 1. If k,/v is now given a value, say 10 s-l, the effect on no and nl is small but the change in n2 is pronounced (see fig. 1). In practice kt ranges between 1 x lo6 and 2 x lo9 dm3 mol-i s-l depending on the monomer and ternperat~re.~ Since particle3024 KINETICS OF EMULSION POLYMERISATION I N THE UNSTEADY STATE diameters are often between 2 x lo-* and 2 x lo-' m [see ref.(S)], 10 s-l is not an unreasonable value of k,/u although higher values will often be found. (The Avogadro number is used in the evaluation of k,/u). If higher values of Q are used the effect of taking k, # 0 is more important. For example, in a mol dm-3 solu- tion of potassium persulphate at 50°C the production rate of free radicals * will be about l~lO~~drn--~s-~. If the emulsion contained 1015 to 10'' particles dm-3 (not unrealistic values) then would exceed 1 X ~ O - ~ even if the efficiency of radical 1.0 0 0.8 0.4 0.2 0 0 I 5 t x 10-31s FIG. 1.-Fractional populations as a function of time. s-' and k = 0. Curve 0, no/N; curve 1, n l / N x 4 0 ; curve 2, nz/Nx2000; curve 3, n2/Nx107 when u = 1 x s-l, k = 5 x kt/v = 10. 0.8 '**P===. 0.4 O*.i 8 10 15 t x 10-31s FIG.2.-Fractional populations as a function of time. Q = 1 x s-l and k = 5 x s-l. Curve 0, no/N with kf = 0 ; curve 4, no/N with kt/u = 10 ; curve 1, nl/Nx 4 with kt = 0 ; curve 5, n l / N x 4 with kr/u = 10; curve 2, nJNx 20 with kt = 0; curve 6, n2/Nx lo5 with kf /v = 10. capture was very s-l, radical-termination in the particles has a noticeable effect on nl(t). As 0 is raised, not only does k,/u become more important, but also the time required to reach the steady state becomes much shorter, as shown in fig. 3. It can be seen (for example in fig. 3) that the value of n2 can be very small when values of 1' are relatively high. Thus, the assumption of i < 2 is still valid. The present treatment is applicable to a much wider range of conditions than that of Birtwistle and Blackley or of Weiss and Di~hon.~ For example, when CJ lo Fig.2 shows that, even when 0 is still only 1 xB . W. BROOKS 3025 and k both equal 5 x s-l, the equations of the previous workers (which always set k, = 0) indicate that as t + co the values of n2/N and 5 approach 0.184 and 1.0, respectively. These values are far too high (in fact Weiss and Dishon limit the application of their equations to cases where t' c 1); the present treatment with k,/u = 10 s-l shows that n2/N should approach 8.3 x and i should become 0.333. If t~ and k have the values given in fig. 3 the equations of the previous workers would not apply since they would predict n2/N = 0.27 and i = 2 when t + co. 0.01, the value of i is not greatly influenced by small changes in k,/u.Examination of the equations given in this paper shows that, when k,/u exceeds 2 0.4' 0.3- 3 0.2- 4 0.1 - 0.0- I I 5' 10 t x 10-31s FIG. 3.-n2/N and ias functions of time. s-'. Curve 1 , i when kt/v = 0.01 ; curve 2,: when kt/v = 10; curve 3, nJNx 10 when kl/v = 0.01 ; curve 4, n2/Nx 5000 when kt/v = 10. u = 1 x s-I and k = 5 x EFFICIENCY OF RADICAL CAPTURE An assumption which has been made in this and previous treatments is that Q is time-independent. This has important implications. If the rate at which radicals desorb from the particles (which is time-dependent) does not affect Q significantly, then the desorption rate has little effect on the concentration of radicals in the aqueous phase, CR. This must be so because Q depends on CR. Thus the course of the radical reactions in the aqueous phase is unperturbed by the presence of the particles.This must mean that the majority of initiator radicals combine mutually in the aqueous phase (as they would do in the absence of the polymer particles) and do not enter the particles. Therefore, in any real emulsion polymerisation which conforms to the description given in this and previous papers,I* Q will be time independent and the particles will be expected to exhibit a low efficiency of radical capture. Thus, in the example of vinyl acetate polymerisation quoted by Gilbert and Napper and Birtwistle and Blackley the low efficiency of radical capture should not be regarded as an unexpected discovery but as an inherent feature of any system for which Q is virtually time-independent . D. T. Birtwistle and D. C. Blackley, J.C.S. Faraday I, 1977, 73, 1998. R. G. Gilbert and D. H. Napper, J.C.S. Faraday I, 1974,70, 391. G. H. Weiss and M. Dishon, J.C.S. Faraday I, 1976,72,1342. W. V. Smith and R. H. Ewart, J. Chem. Phys., 1948,16,592. Z. G. Szabo, Comprehensive Chemical Kinetics, ed. C . H. Bamford and C. F. H. Tipper (Elsevier, Amsterdam, 1969), vol. 2, p. 31.3026 KINETICS OF EMULSION POLYMERISATION I N THE UNSTEADY STATE 8 6 7 R. A. Alberty and W. G. Miller, J. Chem. Phys., 1957, 26, 1231. G. C. Eastmond, Comprehensive Chemical Kinetics, ed. C. H. Bamford and C. F. H. Tipper (Elsevier, Amsterdam, 1977), vol. 14A. D. C. Blackley, Emulsion PoZymerisation (Applied Science, Barking, 1975). J . Ugelstad and P. C. Mork, Brit. PoZymer J., 1970, 2, 31. lo B. M. E. Van der Hoff, J. PoZymer Sci., 1960, 44,241. (PAPER 8/366)