Radical Desorption Kinetics in Emulsion Polymerization. 1. Theory

Ind. Eng. Chem. Res. 2008, 47, 9795–9811

9795

Radical Desorption Kinetics in Emulsion Polymerization. 1. Theory and Simulation Hugo F. Hernandez* and Klaus Tauer Max Planck Institute of Colloids and Interfaces, Colloid Chemistry Department, Research Campus Golm, Am Mu¨hlenberg 1, 14476 Golm, Germany

Radical desorption is one of the most important physical processes influencing the kinetics of free-radical emulsion polymerization. Desorption of any molecule from a polymer particle can only take place if the molecule reaches the particle surface during its random diffusion through the polymer phase. For doing this the molecule must survive all possible competitive reactions taking place simultaneously inside the particle. Once at the surface it must also overcome a certain energy barrier in order to leave the particle. This energy barrier is determined mainly by the difference in the chemical potential of the molecule between the particle and the continuous phase (including interfacial tension effects) and by additional barriers such as the presence of a stabilizer layer. In this molecular picture of the desorption process all molecules inside the polymer particle can desorb from the particle at a rate determined by its energy barrier and diffusion coefficient. It is for this reason that desorption of the smaller and more hydrophilic molecules is predominant. In this paper we propose the use of Brownian dynamics (BD) algorithms for simulating the desorption of radicals from polymer particles and estimate the corresponding desorption rate coefficients. The results obtained in simple systems are found to be in very good agreement with the rate coefficients of desorption determined theoretically. For more complex systems, such as core-shell particles, nonspherical particles, hollow particles, or particles with a gradient in monomer concentration, BD simulations open the possibility to obtain easy, reliable estimations of the desorption rate coefficients, which are difficult to obtain using the experimental or theoretical methods currently available. This article is also intended to be a comprehensive and critical review of the different available theories of radical desorption in emulsion polymerization. 1. Introduction For almost one century emulsion polymerization has been a very important industrial method for the synthesis of very high molecular weight polymers for a wide variety of applications including paints, adhesives, paper making, high-performance materials, functional materials, and many others.1 One very important feature of emulsion polymerization that makes it very competitive is its ability to produce high molecular weight polymers at very high reaction rates. Since the early work of Smith and Ewart,2 a great effort has been placed on investigation of the kinetics of emulsion polymerization. Although the kinetic model of Smith and Ewart was very successful for hydrophobic monomers such as styrene, it was found that more hydrophilic monomers, such as vinyl chloride3 and vinyl acetate,4 did not follow Smith and Ewart’s famous Case 2 of emulsion polymerization kinetics. After careful experimental investigations, different researchers arrived at the conclusion that desorption of radicals from polymer particles plays a major role in the kinetics of emulsion polymerization (even for hydrophobic monomers) and cannot be neglected.3-9 Ugelstad et al. concluded that precise quantitative estimations of the kinetics of emulsion polymerization of vinyl chloride requires detailed knowledge of the rate of radical desorption.3 They were the first to propose that the rate of radical desorption was inversely proportional to the surface of the particle, but * To whom correspondence should be addressed. E-mail: [email protected].

quantitative results were not possible at that time. Nomura et al.,4 Harriot,5 and Litt et al.6 almost simultaneously suggested that the chain transfer to monomer inside the particles and subsequent desorption of the radicals into the aqueous phase was responsible for the different kinetic behavior in the emulsion polymerization of vinyl acetate and not the high water solubility of the monomer. Nomura and co-workers, aware of the necessity of estimating quantitatively the rate coefficient for radical desorption from the particles for prediction of rates of emulsion polymerization, developed the first quantitative model of radical desorption.4,10 They continued improving this model during more than one decade.11-13 Asua et al. complemented the model of desorption by considering the fate of the radicals in the aqueous phase.14 Further improvements of the model have been proposed by considering the layer of stabilizer around electrosterically stabilized polymer particles as an additional resistance to the desorption process.15,16 According to the present theory of emulsion polymerization only the monomeric radicals generated by transfer to monomer can desorb from polymer particles because of their higher solubility in water.17 However, from a molecular point of view not only any radical but also any molecule or macromolecule present inside the polymer particles, independently of its size or chemical nature, can diffuse throughout the particle and eventually desorb (Figure 1). The rate at which any given molecular species abandons a particle depends on the concentration and diffusion coefficient of the molecule inside the particle, the size of the particle, the

10.1021/ie800304t CCC: $40.75  2008 American Chemical Society Published on Web 11/12/2008

9796 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008

there is a preferential desorption for certain molecules (small, hydrophilic) which is caused by the difference in chemical potentials, interaction forces, and diffusion coefficients, but it must be clear that the desorption process is not selective per se. The molecular picture of desorption presented here is valid also for the reverse processes (absorption and adsorption), and it might be used as the basis for a better understanding of chemical equilibrium and dynamics in polymer dispersions (e.g., radical dynamics, swelling equilibrium and dynamics, surfactant adsorption, polymer aggregation, etc.). Figure 1. Molecular picture of the processes of desorption and absorption of molecules by polymer particles. The transfer of any molecule through the interface is possible only if the energy (E) of the molecule is higher than the barrier for transfer (Edes for desorption, Eabs for absorption): M, monomer; I, initiator; R*, primary radical; T, chain-transfer agent; T*, transfer-derived radical; Pn*, polymer radical; Dm, dead polymer; X, any other molecule. Some of the reactions that take place inside the particle are included.

magnitude of the energy barrier for desorption, and the rate of competitive processes that consume the molecules before they leave the particle. The diffusion coefficient inside the particle is closely related to the size and shape of the diffusing molecule and the magnitude of intermolecular forces acting on the molecule. The energy barrier for desorption depends basically on the chemical potential of the molecule between the polymer particle and the continuous phase, which includes intermolecular interactions and the interfacial free energy involved in the process. The idea of an energy barrier opposing the transfer of molecules between phases, which is very reasonable and physically correct, can be safely used instead of the usual assumption of a stationary film around the particle surface (see section 5.2). For the particular case of radicals there are three main types of competitive reactions taking place simultaneously inside the particles: termination, propagation, and chain transfer. While termination completely suppresses desorption of a couple of radicals, propagation and chain transfer only affect the rate of desorption. Propagation will slow down desorption of the radicals, while chain transfer to low molecular weight species will speed it up. The competition between desorption and the chemical reactions where radicals are involved is determined by the characteristic rate of each event. The higher the rate of a given process, the higher its probability of occurrence. This is the basic principle of the kinetic Monte Carlo (kMC) simulation method, also known as the stochastic simulation algorithm.18 In the case of propagation, the characteristic rate of reaction is given by kp[M]p, where kp is the rate coefficient of propagation and [M]p is the concentration of monomer inside the polymer particles. Clearly, one of the most important factors determining the competition between propagation and desorption is the concentration of monomer in the polymer phase. Despite the fact that experimental methods are available for determination of bulk monomer concentration at equilibrium swelling, at present it is not possible to obtain fully reliable experimental data of the concentration of monomer inside individual particles during polymerization. The Morton-Kaizerman-Altier thermodynamic equation is usually employed to estimate the equilibrium concentration of monomer; however, it has already been demonstrated that this method lacks the required accuracy.19 From the analysis of all these factors it can be concluded that desorption of a macromolecular hydrophobic radical is a rather unlikely but not an impossible event. It is evident that

Although many different models have been developed for describing in detail the process of radical desorption, it had not been possible to determine which of these models is the most accurate. Basically, the rates of radical desorption cannot be measured directly but must be inferred from experimental kinetic data, and for this, many assumptions and uncertain parameters have to be used which goes in detriment of the accuracy of the results. On the other hand, the sensitivity of the kinetics of effective radical desorption to some of the parameters of the model is very low, and therefore, reliable experimental validation of radical desorption models is very difficult. Recently, a numerical method of Brownian dynamics (BD) simulation known as the Monte Carlo random flight (MCRF) algorithm, based on the solution of Langevin’s equation for Brownian motion, was found to be a very valuable alternative for investigation of diffusion and chemical reactions in complex systems.20-22 This MCRF method has been used in emulsion polymerization for determination of radical capture kinetics in emulsion polymerization.23,24 Rawlings and Ray stated that “possibly no other facet of emulsion polymerization has generated more discussion and controversy than the process of radical entry and desorption”.25 In this paper we propose the use of BD simulation as an accurate and a reliable computational method for validation of radical desorption models, and we expect that it may offer new insights for the understanding of this important event in heterogeneous polymerization. The paper is structured as follows: In section 2, some fundamentals of radical desorption are presented. Taking into account that the definition of radical desorption has usually been ambiguous,16 four different types of radical desorption are presented with their corresponding rate coefficients. At the end of section 2 a short review of the methods for estimation of diffusion coefficients in polymers is presented. In section 3 the emphasis will be placed on the simple radical desorption process. A theoretical derivation of the simple desorption rate coefficient is presented, leading to an expression structurally similar but quantitatively different with respect to some of the previous models. The validity of this new expression is confirmed by BD simulation results. In section 4 apparent desorption rate coefficients considering the competition of additional simultaneous events are investigated. A critical review of the definition of equilibrium radical desorption is presented. At the end of this section a hybrid kinetic Monte Carlo-Brownian dynamics simulation is used to determine apparent desorption rate coefficients, and the results are compared with the theory. In section 5 the most relevant radical desorption models available in the literature are compared to the results obtained by BD simulation. Finally, desorption in the presence of a surfactant layer is simulated using BD and compared with the most recent models of radical desorption.

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2. Fundamentals of Radical Desorption Kinetics 2.1. Definition of Radical Desorption. Before introducing any equation describing the kinetics of radical desorption it is extremely important to clearly define what is radical desorption as different interpretations of the desorption process have led to ambiguous results in the past. On the basis of the most common scenarios for radical desorption considered in the literature, the following definitions will be used throughout this paper. (1) Simple Radical Desorption. A radical is considered to be simply desorbed as soon as it abandons the polymer particle after diffusing through the particle in the absence of any competitive reaction. Therefore, simple radical desorption is the result of the diffusive motion of the radicals when no reactions are considered inside the polymer particles. In this definition the fate of the radicals after desorption is not considered. (2) Equilibrium Radical Desorption. The equilibrium desorption of a radical takes into account the different solubility of the radicals between the polymer particles and the aqueous phase. In this case, radicals are assumed to reach an equilibrium distribution between the bulk aqueous phase and the polymer particles, represented by the partition coefficient. In this definition competitive reactions involving the radicals are neglected, and the fate of the radicals in the bulk aqueous phase is not considered. In section 5.2 it will be shown that the effect of the solubility of the radical can also be described considering desorption of radicals in the presence of an energy barrier at the interface, originated basically by the difference in chemical potential of the radical between both phases. (3) Net Radical Desorption. The net desorption of a radical occurs when the radical escapes the polymer particle after surviving the competitive reactions taking place inside. The possibility of radical reabsorption and redesorption is not considered. (4) Effective Radical Desorption. A radical is considered to be effectively desorbed from the particle only after it reacts in the aqueous phase. This definition accounts for the fact that desorbed radicals which diffuse through the aqueous phase may be reabsorbed by a polymer particle and continue reacting therein without significantly affecting the kinetics of emulsion polymerization. 2.2. Rate Coefficients of Radical Desorption. From the previous definitions it follows that the kinetic expressions for each type of radical desorption must be different. The radical desorption rate coefficients presented next are all expressed in units of frequency (s-1). The general expressions for the different rate coefficients of radical desorption areas follows. (1) Rate Coefficient of Simple Radical Desorption (k0). The rate coefficient of simple radical desorption (k0), also known as the maximal desorption rate coefficient, is determined by the velocity at which the radical diffuses out of the particle and is a function of the particle size (dp) and the diffusion coefficient of the radical inside the polymer particle (Dp). k0 ) λ

Dp d2p

(1)

In eq 1 λ is a constant. In section 3 it will be shown that λ ) 60. However, there are many different reported values for λ in the literature ranging from 2 to 12 (see section 5.1). This expression is obtained by describing diffusion of the radicals inside the particles using Einstein’s equation of Brownian motion in 3 dimensions26 and averaging with respect to the spatial density probability distribution of the radicals (a detailed

derivation is presented in section 3.1). This rate coefficient can also be determined from random-walk considerations.27 (2) Rate Coefficient of Equilibrium Radical Desorption (k0*). The rate coefficient of equilibrium radical desorption (k0*) can be related to the simple radical desorption rate coefficient by k*0 )

(

k0 Dp δw 1+m Dw δp

)

(2)

where Dw is the diffusion coefficient of the radical in the aqueous phase, m is the partition coefficient of the radical between the polymer particle and the aqueous phase, δw is the thickness of the stagnant layer in the aqueous phase, and δp is the thickness of the diffusion layer in the polymer phase. This equation is obtained after balancing the flux of radicals in both phases (see section 4.1). The partition coefficient for radicals derived from chain transfer to monomer is usually approximated by the ratio of monomer concentration between both phases: m ) [M]p/[M]w. In the limit where m f 0 or Dw.Dp, the equilibrium desorption rate coefficients becomes equivalent to the simple desorption rate coefficient, k0* ) k0. In section 5.2 it will be shown that an alternative expression for the equilibrium desorption rate coefficient is given by the effect of a net energy barrier for desorption (Edes)

( )

k*0 ) k0 exp -

Edes kBT

(3)

where kB is the Boltzmann constant and T is the temperature of the system. The magnitude of the energy barrier for desorption is determined by the difference in chemical potential of the radical between the particle and the aqueous phase and by friction forces. The energy barrier for desorption takes into account all intermolecular forces acting on the radical at the interface during the phase change. It is important to notice that k0* is an Arrhenius-type kinetic rate coefficient which does not necessarily imply equilibrium, although equilibrium is strongly influenced by the magnitude of this energy barrier. (3) Rate Coefficient of Net Radical Desorption (K0). The rate coefficients of net radical desorption are obtained by considering that a fraction of the radicals that can desorb undergoes a chemical reaction before they reach the aqueous phase. This fraction is expressed by the probability of reaction Pp of a radical and defined as follows

∑k

i,p

Pp )

i

k*0 +

(4)

∑k

i,p

i

-1

where ki,p represents the rate (s ) for the ith competitive reaction inside the particle. The rate coefficient of net radical desorption (K0) is then determined by K0 ) krin(1 - Pp) ) (krgen + kabs) ·

k*0 k*0 +

∑k

(5)

i,p

i

where krin is the rate of appearance of radicals inside the particles either by generation krgen or by absorption kabs. Possible sources of primary radical generation are decomposition of oil-soluble initiators or chain-transfer reactions inside the particles. (4) Rate Coefficient of Effective Radical Desorption (kdes). Since only the radicals undergoing reactions in the aqueous phase are considered as effectively desorbed, the rate coefficient

9798 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008

of effective radical desorption depends on the fraction of desorbed radicals that reacts in the aqueous phase (Pw)

∑k

i,w

Pw )

i

kabs +

∑k

(6)

i,w

i

where ki,w represents the rate coefficient (s-1) for the ith competitive reaction in the aqueous phase. After a balance of radicals in steady state, the following expression for the rate coefficient of effective radical desorption is obtained kdes ) krgen ·

Pw(1 - Pp) 1 - (1 - Pw)(1 - Pp)

(7)

In the absence of radical absorption events, the net and effective desorption rate coefficients become identical since kabs ) 0 and Pw ) 1, and eq 7 becomes kdes ) krgen · (1 - Pp) ) K0

(8)

2.3. Rate of Radical Desorption. From the point of view of emulsion polymerization kinetics, the most important type of radical desorption is the effective radical desorption. In the previous section a general expression for the effective radical desorption rate coefficient was presented (eq 7). However, this expression is only illustrative of its mathematical structure because it did not consider the effect of the size and type (attached end groups) of the radicals. The effective rate of desorption (in radicals per second per particle) for radicals of degree of polymerization j, derived after chain transfer to monomer, from a polymer particle h of diameter dp,h is proportional to the number of such radicals inside the polymer particle (Rj,h•) rdes,j,h ) kdes,j,h · R•j,h

(9)

kdes,j,h is the effective desorption rate coefficient of the radicals of degree of polymerization j in the particle h. In the case of radicals with an initiator fragment (I) or chain-transfer agent fragment (T) as attached end groups, similar expressions can be obtained I I rdes,j,h ) kdes,j,h · IR•j,h

(10)

T T ) kdes,j,h · TR•j,h rdes,j,h

(11)

The total rate of effective radical desorption from a single polymer particle can be determined as ∞

rdes,h )

∑

∞

rdes,j,h +

j)1

∑ j)0

∞

)

∑ (k

∞

I rdes,j,h +

∑r

T des,j,h

j)0

• I • T • des,j,h · Rj,h + kdes,j,h · IRj,h + kdes,j,h · TRj,h)

(12)

j)0

and the total rate of effective desorption of radicals in the polymer dispersion (rˆdes) is (in radicals per second per unit volume) Np

∑

rdes,h rdes h)1 ) rˆdes ) V V Np

)

∞

∑∑

1 I T (k · R• + kdes,j,h · IR•j,h + kdes,j,h · TR•j,h) (13) V h)1 j)0 des,j,h j,h

where Np is the total number of particles in a volume V of dispersion. After observing eq 13 it can be concluded that

determination of the total rate of effective radical desorption requires precise knowledge of the individual effective desorption rate coefficients (which depend on the simple desorption rate coefficients and reaction rate coefficients in both phases), the particle size distribution (PSD) in the dispersion, and the radicals concentration and distribution for every particle. Although the PSD of the dispersion can be measured, the distribution of radicals in the particles cannot yet be determined experimentally and its estimation requires solution (numerical or analytical) of a large system of differential equations. Due to the complexity of the system a simplified approach to the mathematical description of the system considering the averagenumberofradicalsperparticlenj isoftenemployed.14,15,28-30 Some important assumptions are usually made during this approach: (i) only nonionic primary radicals derived from chain transfer to monomer are able to desorb from the particle or reenter into the particle, (ii) the system is in pseudo-steady state, i.e., the concentration of radicals inside the particles and in the water phase is constant for the time scale considered, and (iii) the system is monodisperse. The final expression obtained is simply rdes ) kdesn¯Np

(14)

where kdes obtained from eq 7 is kdes ) kfm[M]p ·

Pw(1 - Pp) 1 - (1 - Pw)(1 - Pp)

(15)

This approach can be better understood using an analogy with electrical circuits. If the rate of change in the number of radicals is considered analogous to the electrical current (IE), the average time for each event (the inverse of the rate coefficient) to the electrical resistance (RE), and the number of molecules of a key chemical species to the electrical potential or voltage difference (∆VE) then from Ohm’s law31 the following general expression is obtained ∆VE dR• ) kS (16) w dt RE where k is the rate coefficient in s-1 and S is the number of molecules of the key reacting species. From this representation we can also deduce that the rate coefficient k corresponds to the electrical conductivity, that the analogous electrical charge is the number of radicals, and that a capacitor represents accumulation of radicals in a given phase. Generation of radicals is represented by a power source with a certain potential, while their consumption is represented by the ground. The corresponding schematic representation of the balance of primary radicals derived from chain transfer to monomer in quasi-steady state as an electrical circuit is presented in Figure 2. The nodes N1 and N2 indicate the concentration of primary radicals in the particles and in the aqueous phase, respectively. The net radical desorption rate (K0) can be obtained by determining the current passing through B (IEB) as a function of the potential in A (∆VEA) making use of the rules of Kirchhoff31 for electrical circuits IE )

K0 )

IEB 1 ) E RAB ∆VAE

(17)

Neglecting reabsorption of radicals, the net radical desorption rate becomes (see dotted path in Figure 2) K0 ) kfm[M]p

k*0 k*0 +

∑k

i,p

i

(18)

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9799

Figure 2. Simplified representation of the balance of primary radicals generated by chain transfer to monomer. Dotted circuit: Net radical desorption rate (from A to B, eq 18). Bold circuit: Effective radical desorption rate (from A to C, eq 19).

If termination and chain-transfer reactions inside the particles are neglected, then eq 18 corresponds to the Friis and Nyhagen expression for the rate coefficient of radical desorption.8 The effective radical desorption rate coefficient which is the rate coefficient for the reaction of desorbed radicals in the aqueous phase (considering absorption, redesorption, and reabsorption) can be determined as the reciprocal of the apparent resistance between A and C (bold circuit in Figure 2). If generation of monomer radicals in the aqueous phase by chain transfer to monomer is neglected, the rate coefficient of effective radical desorption is (using Kirchhoff rules)

( ) ( ) ∑k

i,w

i

IEC 1 ) kdes ) E ) RAC ∆VAE

kabs +

∑k

k*0

i,w

i

∑k

i,w

i

kabs +

∑

ki,w

k*0 +

kfm[M]p

∑k

i,p

i

i

(19)

which is the corrected expression for effective radical desorption obtained by Asua.15 2.4. Diffusion of Radicals Inside Polymer Particles. It is clear from eq 1 that the kinetics of radical desorption depends strongly on the diffusion coefficient of the radicals inside the polymer particles. Experimentally, it has been observed that diffusion of small molecules in a polymer phase depends on the state of the polymer.32,33 If the polymer is in its hard glassy state, i.e., the temperature of the system is below the glasstransition temperature Tg of the polymer, the diffusion coefficients are orders of magnitude lower than in the soft rubbery state of the polymer and much lower than in the polymer melt.

Therefore, the difference between the temperature of the system and the glass-transition temperature of the polymer determines the mobility of small molecules in polymer media. On the other hand, the glass-transition temperature of the polymer depends on the concentration of plasticizing agents in the mixture. The higher the concentration of plasticizing agents, the lower the Tg of the polymer. In emulsion polymerization systems the monomer usually acts as plasticizing agent for the polymer particles and thus the diffusion coefficient inside the polymer particles becomes a function of the concentration of monomer in the particles. Additionally, for hydrophilic polymers (e.g., poly(vinyl acetate), poly(vinyl alcohol), poly(N-isopropylacrylamide), etc.), the continuous phase (water or aqueous surfactant solution) may swell the particles and plasticize the polymer.34 A mathematical expression relating the diffusion coefficient of a small molecule in a polymer was obtained by Vrentas and Duda35 using the free volume theory developed by Fujita.32 In this case, the diffusion coefficient is given by the equation

( )

Dp ) Dp01 exp -

(

)

(1 - wp)V1 + wpξvV2 E exp -γ (20) kBT VFH *

*

where the subscripts 1 and 2 represent the small molecule and the polymer respectively, E is the attractive energy between the small molecule and its polymer neighbors, Dp01 is a pre-exponential factor, γ is a correction factor for the free-volume overlap, wp is the weight fraction of polymer, Vi* is the specific critical hole free volume of the component i, ξv is the ratio of the critical molar volume of the small molecule jumping unit to that of the polymer jumping unit, and VFH ) (1 - wp)K11(K21 + T - Tg1) + wpK12(K22 + T - Tg2) (21)

9800 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008

where Kij are parameters of the system and Tgi is the glasstransition temperature for each component. A different expression was obtained by Yasuda et al.,36 who found that the diffusion coefficient of the small molecule in the polymer can be expressed as

(

Dp ) D0p exp -

βxv(1 - R) 1 + Rxv

)

(22)

where Dp0 is the self-diffusion coefficient of the small molecule in its own medium, β ) V*/Vfm, R ) Vfp/Vfm, xv ) H/(1 - H), V* is the critical free volume fraction necessary for diffusion to take place, Vfm and Vfp are the free volume fractions of small molecules and polymer, and H is the volume fraction of small molecules. A simplified approach was presented by Chang et al.37 They expressed the diffusion coefficient of a small molecule inside the polymer particle as Dp ) K(H - H0)2

Dp )

η2⁄3p1⁄3

(

∆Gq exp 3kBT

)

(24)

where p qis Planck’s constant, η is the viscosity of the mixture, and ∆G is the activation free energy for the diffusion of the molecule. An additional problem arises when propagation of radicals is considered because the diffusion coefficient is reduced as the molecule grows by addition of monomer units. In this case, some empirical equations have been developed for estimation of diffusion coefficients as a function of the radical size and monomer fraction in the particles.39,40 For example, Griffiths et al.39 obtained an empirical equation for estimating the diffusion coefficients of oligomers inside monomer-swollen polymer particles as a function of the degree of polymerization of the oligomer and the weight fraction of polymer in the particle Dpi(wp) )

Dp1(wp) 0.664+2.02wp

i

3. Simple Desorption Rate Coefficient 3.1. Theoretical Derivation. In this section the rate coefficient of simple desorption is derived theoretically following the description of radical desorption presented by Grady.27 First, let us consider a small spherical nonionic molecule located exactly at the center of a homogeneous spherical polymer particle of diameter dp which is dispersed in a continuous medium. This molecule will diffuse by Brownian motion throughout the particle (with a constant diffusion coefficient Dp) and will, on average, reach the particle surface in a time given by

〈 ( )〉 τdiff

(23)

where K is a constant and H0 is a parameter defined as a function of Tg. H0 is zero at about 20-30 °C above Tg, positive at lower temperatures, and negative at higher temperatures. An additional alternative for estimation of diffusion coefficients, valid only for low polymer concentrations, is the expression for high-viscosity liquids obtained by Hiss and Cussler based on Eyring’s theory of absolute reaction rates.38 In this case kBT

estimation of diffusion coefficients in polymers because they consider explicitly the intermolecular forces between the polymer and the diffusing molecule.

(25)

This expression was found to be valid for a variety of monomers when the concentration of polymer in the particles is above a critical concentration at which polymer chains start to overlap. A different approach for estimating the diffusion coefficients of radicals in polymer particles is use of molecular modeling simulation methods, such as molecular dynamics, coarsegrained, or Monte Carlo methods.41-43 In this approach the molecular motion of a radical of a given size in a polymer phase of a given composition is simulated taking into account the different intermolecular forces exerted by the neighbor molecules (polymer chains, monomer). A suitable autocorrelation function is then used to calculate the diffusion coefficient of the radical from the molecular trajectories obtained. Additional information about the modeling of diffusion in polymers can be found in the review paper of Masaro and Zhu.44 Molecular modeling simulation methods as well as the model of Vrentas and Duda are perhaps the most relevant approaches for

dp |0 2

d2p d2p 4 ) ) 6Dp 24Dp

(26)

Equation 26 is derived directly from Einstein’s equation of Brownian motion in three dimensions. If the molecule is not located exactly at the center of the particle but at a certain distance x to the center, then the average time required by the molecule to reach the particle surface will be

〈 ( )〉 〈 ( )〉 τdiff

dp |x 2

) τdiff

dp |0 2

- 〈τdiff(x|0)〉 )

(

2 1 dp - x2 6Dp 4

)

(27) Since for simple desorption it is assumed that there is no energy barrier for desorption, then the average desorption time will correspond to the average diffusion time required by the molecules to reach the particle surface. The average desorption time is calculated considering the radial probability density distribution inside the particle Fd

∫

dp⁄2

〈τdes 〉 )

0

〈 ( )〉 τdiff

dp |x Fd(x) dx 2

∫

Fd(x) dx

dp⁄2

0

(28)

If the probability of finding the molecule inside the spherical particle is uniform (homogeneous particle), then 4πx2 dx 24x2 dx ) d3p (πd3p ⁄ 6)

Fd(x) dx )

(29)

Substituting eqs 27 and 29 in eq 28 and integrating result in

∫

dp⁄2

0

〈τdes 〉 )

〈 ( )〉

dp 24 2 |x x dx 2 d3p ) dp⁄2 24 2 x dx 0 d3p

τdiff

∫

[

2 1 dp 6Dp 4

∫

dp⁄2 2

0

x dx -

∫

∫

dp⁄2 4

0

dp⁄2 2

0

x dx

]

x dx

)

d2p (30) 60Dp

Therefore, the simple desorption rate coefficient (s-1) is k0 )

60Dp 1 ) 2 〈τdes 〉 d

(31)

p

In general, for a macromolecule of chain length j in a particle of size dp,h, the simple desorption rate coefficient will be given by

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9801

k0,j,h )

60Dp,j

(32)

d2p,h

Although a uniform probability of finding the radicals inside the particles has been assumed, nonuniform radical distributions are also possible. Almost all authors have considered either a uniform spatial distribution inside the particles or that all radicals are initially at the center of the particle. Chang, Litt, and Nomura37 also considered the case where radicals are absorbed from the continuous phase. In general, any reaction or physical event that change the number of radicals inside the particles can lead to a nonuniform distribution provided that it takes place faster than diffusion of the radicals inside the particles. This is the case, for example, when the desorbing radicals are generated exclusively by chain transfer to monomer and there is a radial monomer gradient inside the particle.45 In a more realistic situation there is also an energy barrier for desorption (and also for absorption) caused by the difference in chemical potential of the molecule being exchanged between the two phases: by the intermolecular forces acting on the molecule and/or by nonconservative (path-dependent) forces present during the exchange process. In this case, the rate coefficient of equilibrium desorption will be given by

( )

k*0 ) k0 exp -

(

Edes 60Dp Enc,des + ∆µ ) 2 exp kBT kBT dp

)

(33)

where Edes is the total energy barrier for desorption, Enc,des is the contribution of nonconservative forces (like friction forces, for example) to the energy barrier for the desorption process, and ∆µ is the difference in the chemical potential of the molecule between the two phases. 3.2. Brownian Dynamics Simulation of Simple Radical Desorption. As an alternative to the theoretical derivation presented above, the rate coefficient of simple desorption can be estimated from the simulation of molecular diffusion inside the polymer particle by means of Brownian dynamics (BD) simulation. Brownian dynamics simulation consists on the numerical solution of Einstein’s equation of Brownian motion (eq 34) which is equivalent to the solution of Langevin’s equation of Brownian motion (eq 35)46 if the time scale considered allows for motion relaxation, i.e., if zero acceleration of the particles is assumed 〈x2 〉 ) 2ndDt

(34)

kT dx d2x dx +X (35) ) -6πηa + X ) dt D dt dt2 where x is the position of a particle, 〈x2〉 is the average squared distance that an ensemble of identical particles will travel along nd spatial dimensions during a time interval t, m is the mass of the particle, a is its hydrodynamic radius, D is the diffusion coefficient of the particles, η is the viscosity of the system, and X is a random force acting on the particles. In this section simple radical desorption rate coefficients are obtained from simulation of the Brownian motion of radicals inside the particles using an optimized (variable time step) Monte Carlo random flight (MCRF) method20-22 similar to the method used for estimating radical capture by polymer particles.23,24 The diffusive displacement dxj of a polymer radical of chain length j on each direction during a time step dt is obtained from m

dxj ) ξ√2Dp,jdt

(36)

where ξ is a random number obtained from a Gaussian distribution with mean zero and variance 1 and Dp,j is the diffusion coefficient

Figure 3. Average desorption time of radicals (dr ) 0.5 nm, mr ) 1.6 × 10-22 g) from the center of a particle. Data points represent the BD simulation results for different particle diameters and diffusion coefficients. Dotted lines represent the average desorption time calculated using eq 26.

of the radical of chain length j inside the particle. This Gaussian distribution arises as a result of the large number of collisions between the diffusing radical and the neighboring molecules present in the medium, according to the central limit theorem. The variable time step dt is constantly updated to reduce computation time according to the following expression

(

dt ) max R

d2min mr,jDp,j , Dp,j kBT

)

(37)

where dmin is the shortest distance between the radical surface and the particle interface, mr,j is the mass of the radical which is function of the chain length j of the radical, mr,jDp,j/kBT represents the relaxation time for the Brownian motion of the radical of size j inside the particle, and R is a damping factor with a chosen value of 0.01852 such that the probability of a radical displacement larger than dmin in a time step dt is lower than 10-7.23 Equation 37 is used to minimize the error involved in estimation of the desorption time by increasing the time resolution of the simulation as the radical approaches the particle surface. The time resolution limit is given by the characteristic relaxation time of the diffusing radical. The BD algorithm was tested with the simulation of Brownian motion for a large sample of radicals (>103) placed at the center of a polymer particle. All simulations were performed for radicals of 0.5 nm in diameter and 1.6 × 10-22 g in mass. The average desorption time obtained from the simulation reproduced the behavior predicted by eq 26 for a wide range of particle diameters and diffusion coefficients (Figure 3). The rate coefficients for simple desorption of radicals initially placed at a distance x from the center of the particle determined by BD simulation for different particle sizes and different initial positions are presented in Figure 4. These results are in very good agreement with the values predicted by eq 27. Assuming a uniform probability distribution of the radicals inside the polymer particle (homogeneous distribution), the theoretical rate coefficient of simple radical desorption was found to be 60Dp/dp2 (eq 31). Figures 5 and 6 present the rate coefficients of simple radical desorption obtained from BD simulations performed under the homogeneous distribution assumption for different particle diameters and radical diffusion coefficients. Notice that the desorption rate coefficients are estimated using average values of the desorption time obtained from many single-radical simulations (usually between 2000 and 20000

9802 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008

4.1. Equilibrium Radical Desorption. The definition of equilibrium radical desorption involves reaching a steady state in the concentrations of radicals in the particle and aqueous phase and establishment of concentration profiles on both sides of the interface (boundary layers) as depicted in Figure 7. The model of equilibrium radical desorption was developed by Nomura and co-workers,10-13 and it has been widely used to describe radical desorption in heterophase polymerization.29,30,47 However, some underlying assumptions involved in the derivation of this model may not be fulfilled by a radical desorption process in a real emulsion polymerization system. This will be explained in detail later. When the equilibrium condition is reached, the flux of radicals from the particles to the interface (Jrp) is given by Figure 4. Average desorption time of radicals placed initially at a distance x from the center of the particle. Data points represent the BD simulation for different particle diameters and initial positions. Solid lines are obtained using eq 27. Dp ) 1 × 10-9 m2/s.

Jrp ) Kmp([R•]p - [R•]*p) )

Dp • ([R ]p - [R•]*p) δp

(38)

where Kmp is the mass-transfer coefficient in the boundary layer of the polymer phase (of thickness δp), [R•]p is the equilibrium concentration of radicals in the bulk of the polymer particle, and [R•]p* is the equilibrium concentration of radicals in the polymer at the interface. Similarly, the flux of radicals from the interface to the bulk of the aqueous phase (Jrw) is given by Jrw ) Kmw([R•]w* - [R•]w) )

Dw • * ([R ]w - [R•]w) δw

(39)

where Kmw is the mass-transfer coefficient in the boundary layer of the aqueous phase (of thickness δw), [R•]w is the equilibrium concentration of radicals in the bulk of the aqueous phase, and [R•]w* is the equilibrium concentration of radicals in the aqueous phase at the interface. At equilibrium [R•]p* ) m[R•]w* and Jrp ) Jrw, and assuming that [R•]w ) 0, then Figure 5. Simple radical desorption rate coefficients for homogeneous spherical particles. Data points represent BD simulation results for different particle diameters and radical diffusion coefficients. Solid lines are obtained from eq 31.

simulations). In Figure 5 it is possible to compare the results obtained using eq 31 (solid lines) with the simulation results (data points). A very good agreement between theory and simulation is again obtained. In Figure 6 the linear regression of the simple desorption rate coefficient as a function of Dp/ dp2 for the complete set of simulation data is presented. The fitted value of the slope for the simulation data was 57.14, in very good agreement with the value of 60 predicted by the theoretical model. In section 5 these results will also be compared to previous models of radical desorption presented in the literature. 4. Apparent Desorption Rate Coefficients In section 2.1 different types of radical desorption were defined, but in section 3 only the rate coefficient of simple radical desorption was considered. In this section the rate coefficients for apparent radical desorption will be presented. The different types of apparent radical desorption are equilibrium, net, and effective radical desorption. They are apparent because in all of them the rates of radical desorption are affected by the presence of additional competitive events (absorption, reactions in the polymer phase, and reactions in the aqueous phase, respectively).

[R•]*p )

(

[R•]p Dw δp 1+ mDp δw

)

(40)

Therefore, the equilibrium radical desorption rate per unit volume is rˆ*0 ) πd2pJrp ) πd2p

Dp • ([R ]p - [R•]*p) ) δp

πd2pDp [R•]p δw mDp δp 1 + δp Dw (41)

)

(

and the equilibrium radical desorption rate coefficient is k*0 )

6Dp λDp 6dp ) 2 ) δw mDp δw mDp dp δpdp 1 + λδp 1 + δp Dw δp Dw 6dp k0 (42) δw mDp λδp 1+ δp Dw

(

)

(

( )

)

(

)

It is also possible to estimate the thickness of the boundary layer in the aqueous phase for the specific geometry and flow regime of the system from semiempirical relations for mass transfer in spheres48 δw 1 1 ) ) dp Sh 2 + 0.6Re1⁄2Sc1⁄3

(43)

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9803

Figure 6. BD simulation results for simple radical desorption rate coefficients as a function of Dp/dp2. Data points represent BD simulation results. The solid lines correspond to the best linear fit (left) and the theoretical prediction (right).

Figure 7. Equilibrium distribution of radicals at the polymer-water interface.

where Sh, Re, and Sc are the dimensionless numbers of Sherwood, Reynolds, and Schmidt, respectively. Assuming a stagnant boundary layer, Re ) 0 and dp (44) 2 On the other hand, comparing the rate of simple radical desorption (m ) 0) with the rate of mass transfer inside the particle it is found that δw )

δp 6 ) dp λ

(45)

Using the value of λ ) 60 previously obtained in section 3 dp (46) 10 The theory of mass transfer by Fickian diffusion through stationary films may not be suitable for describing radical desorption because this theory was developed for macroscopic mass-transfer processes. In the case of transfer of molecules at very low concentration, as is the case of radicals in a polymer particle where only one radical may be present, it may not be possible to define a radical concentration profile around a single particle, so the macroscopic theory of mass transfer cannot be safely used. Nomura and Harada were aware of this situation when they presented the model.11 Although for a large number of identical particles a probability distribution profile inside and δp )

outside the particles is observed which may be interpreted as the average concentration profile for a representative particle, this probability distribution does not lead to formation of a stationary film around the particles as depicted in Figure 7 because it would require the presence of a concentration gradient around each individual particle. Furthermore, if the concentration of radicals inside the particles is compared to the concentration of radicals in the bulk of the aqueous phase, it is clear that the latter cannot be neglected as is also assumed in this model. On the other hand, the equilibrium radical desorption model considers that a radical is desorbed after it reaches the bulk of the aqueous phase. However, immediately after the radical abandons the particle surface it is able to react with other molecules present in the aqueous phase (i.e., monomer, chaintransfer agents, other radicals) or re-enter the particle. All events are possible even in the boundary layer surrounding the particle, as demonstrated by Thickett and Gilbert, who reported chaintransfer and termination reactions taking place in the stabilizer layer outside the particle.17 Therefore, the radical is desorbed just after abandoning the particle surface (as is assumed by the simple desorption model) and not after abandoning the boundary layer. A very clear evidence of the failure of the boundary layer model was presented by Grady.27 He calculated the thickness of the stationary layer around the polymer particles and found that increasing the volume fraction of particles the boundary layers of neighbor particles begin to overlap, and around a volume fraction of 15% the mass-transfer boundary layer of the particles occupy the whole aqueous phase volume. Considering the traditional equilibrium model this means that in polymer dispersions with volume fractions above 15% it is impossible to observe radical desorption since the radicals will never abandon the boundary layers of the particles, and this is evidently not the case. One last inconsistency of this model is that according to Nomura and Harada λ is equal to 2,11 and therefore, from eq 45 the thickness of the mass-transfer film in the polymer phase should be δp ) 3dp, which is physically impossible. The equilibrium radical desorption model has been successfully used because it is able to explain the effect of the hydrophobicity of the radicals on the desorption rates. However, this effect can also be explained using energy barriers for phase transfer in the simple radical desorption model (eq 33) instead of assuming stagnant boundary layers around the particles and using radical partition coefficients. A more detailed explanation of the success of Nomura’s model for radical desorption in spite of the deficiencies exposed here is presented in section 5.2.

9804 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008

Figure 8. Probability of reaction inside the polymer particle (Pp) for different values of the propagation rate coefficient. Dp ) 10-11 m2/s, dp ) 100 nm, [M]p ) 5.5 mol/L, kfm ) 0.02 L/mol · s.

Figure 9. Net desorption rate coefficient (K0) for different values of the propagation rate coefficient. Dp ) 10-11 m2/s, dp ) 100 nm, [M]p ) 5.5 mol/L, kfm ) 0.02 L/mol · s.

4.2. Hybrid BD-kMC Simulation of Apparent Radical Desorption. In this section we present an example of the integration of two different simulation techniques, Brownian dynamics (BD) and kinetic Monte Carlo (kMC) simulation,49,50 for determination of apparent radical desorption rate coefficients. BD simulation is used to follow the trajectory of the radicals inside the particle and determine the precise moment of desorption, while at the same time a kMC algorithm simulates the reactions taking place simultaneously inside the particles or in the aqueous phase. In the example considered, monomerderived radicals are generated by chain transfer to monomer inside the particles and are assumed to be the only species capable of desorption with zero energy barrier for desorption; higher molecular weight radicals are assumed to have an infinite energy barrier for desorption. Inside the particles the only competitive reaction considered is propagation of the radicals. There will be no radical reabsorption, that is, only the net rate coefficient of radical desorption will be determined. All these assumptions are considered only for comparison with the available analytical results. The hybrid BD-kMC simulation method can be used also without any of these restrictions. Figures 8-11 show the results obtained using the hybrid BD-kMC simulation, varying the propagation rate coefficient (Figures 8 and 9) or the chain transfer to monomer rate

Figure 10. Probability of reaction inside the polymer particle (Pp) for different values of the chain transfer to monomer rate coefficient. Dp ) 10-11 m2/s, dp ) 100 nm, [M]p ) 5.5 mol/L, kp ) 240 L/mol · s.

Figure 11. Net desorption rate coefficient (K0) for different values of the chain transfer to monomer rate coefficient. Dp ) 10-11 m2/s, dp ) 100 nm, [M]p ) 5.5 mol/L, kp ) 240 L/mol · s.

coefficient (Figure 10 and 11). In both cases, the probability of reaction of the radical inside the polymer particle Pp and the net radical desorption rate coefficient K0 obtained by simulation are presented and compared to the corresponding theoretical model (eqs 4 and 5). It is observed that the values for the probability of reaction inside the particles and the net radical desorption rate coefficient obtained by the hybrid BD-kMC method are in very good agreement with those predicted by the theory (eqs 4 and 5, respectively). When the rate coefficient for propagation is high, the rate coefficients for net radical desorption obtained by simulation are slightly higher than those calculated from eq 5 (Figure 9). This difference can be explained by the fact that the simulations are performed under non-steady-state conditions. That means that at the beginning of the simulation only the radicals which desorb faster (i.e., close to the particle surface) are able to leave the particle, leading to a relatively higher initial desorption rate. In Figure 10 the inherent variability of the stochastic simulation method is evident when compared to the theoretical model (eq 4). The variability of the simulation results can be reduced by increasing the number of radicals simulated, but this will demand more computer time. A tradeoff between accuracy and speed is always unavoidable when using computational methods.

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9805

Use of the hybrid BD-kMC method can be extended to determination of effective radical desorption rate coefficients simply by incorporating into the kMC model the reactions in the aqueous phase and including in the BD model the reabsorption of radicals into the particles. In addition, a higher degree of accuracy can be obtained by considering the energy barrier for absorption or desorption as a function of the radical size (chain length). In this case, the molecular weight distribution of the radicals inside the particles and in the aqueous phase can be obtained also. 5. Validation of Radical Desorption Models Using Simulation Methods 5.1. Simple Radical Desorption. After the initial idea of desorption presented by Smith and Ewart2 more than a dozen different models of radical desorption in emulsion polymerization have been proposed. In this section the most representative models of radical desorption are compared to the model of simple radical desorption presented in section 3. The mathematical formulation for radical desorption given by the most relevant radical desorption models is presented in Table 1. All rate coefficients are expressed in units of s-1 and always as a function of the particle diameter (not the particle radius). The nomenclature in these equations may change with respect to the original formulations in order to keep the consistency of the whole manuscript. These models can only be compared if all of them consider exactly the same system under the same conditions. Therefore, the following assumptions will be used for comparison (1) Only radical desorption takes place in the system. No other chemical (propagation, termination, chain transfer) or physical (absorption, reabsorption) event will be considered. (2) There is no energy barrier for desorption. The rate of desorption is determined exclusively by diffusion of the radical in the particle. (3) The particle is spherical and homogeneous. The probability density function of the radicals inside the particles is uniform. (4) Any radical inside the particle will eventually desorb to the aqueous phase and never re-enter again. Therefore, in the case of models considering the mass-transfer equilibrium, the partition coefficient will be assumed to be zero (m ) 0). The different values of λ obtained for these conditions are compared in Table 2 (only for those models yielding the structure presented in eq 1). The value of λ ) 60 determined in the present work, which was confirmed by BD simulation, is 5-30 times larger than the values obtained in previous models. These differences can be explained by the particular assumptions considered in each case. An example is the use of Smoluchowski equation for describing the flux of radicals from the particles to the aqueous phase.10,11,14,28 The Smoluchowski equation cannot be used to describe radical desorption because it was derived to describe the rate of absorption of Brownian entities by an infinitely diluted sphere; therefore, the geometry of the system for which the Smoluchowski equation applies is completely different from the geometry of radical desorption. Use of the Smoluchowski equation underestimates the value of λ by a factor of 5. Some authors assumed the mean diffusion path of the radicals inside the particles to be the particle radius;8 this is equivalent to assuming that all the radicals are initially placed at the center of the particle. Comparing eqs 26 and 30 it can be deduced that for radicals homogeneously distributed inside the particles λ is 2.5 times larger than for radicals placed at the center of the

particle. An additional problematic assumption is the derivation of diffusion rates from Einstein’s equation of Brownian motion in only one dimension (eq 34).8,11,14,27,37,53 In reality, the motion of the radicals occurs in three dimensions (eq 47). The onedimensional Einstein’s equation gives a value of λ three times smaller than the three-dimensional equation. 〈r2 〉 ) 6Dpt

(47)

A similar value of λ ) 60 can be obtained following Grady’s mathematical treatment,27 when Brownian motion is considered in three dimensions and the correct value of 6/15 for the coefficient of equation IV.B.2-3 is used instead of the misprinted value of 6/5. 5.2. Equilibrium Radical Desorption. In section 4.1 some arguments were presented showing that the foundations of the equilibrium radical desorption rate coefficient based on macroscopic mass-transfer analysis may have been erroneous. However, radical desorption models derived from macroscopic mass transfer have succeeded in describing experimental results in the past.9,10,14 One plausible explanation is presented as follows. The rate coefficient of equilibrium radical desorption can be expressed as λDp

(

k*0 )

d2p Dp δw 1+m Dw δp

)

(48)

For a given set of experimental conditions it is possible to assume that m, Dp/Dw, and δw/δp are constant, and therefore k*0 )

λ*Dp

(49)

d2p

where λ* )

(

λ Dp δw 1+m Dw δp

)

(50)

Comparing eq 49 with eq 1 it is observed that both expressions are structurally identical and that they only differ in the value of the proportionality constant λ. A further comparison of eq 49 with eq 33 shows that when an energy barrier for desorption is considered it is possible to incorporate the effect of radical solubility in the simple desorption model without assuming the equilibrium condition. In fact, the energy barrier approach can be regarded as a more general model compared to the equilibrium approach because the former is valid whether the system has reached equilibrium or not. The corresponding expression is

( )

λ* ) λ exp -

( )

Edes Edes ) 60 exp kBT kBT

(51)

Comparative results between the simple desorption model with an energy barrier and the equilibrium model are presented in Table 3. The values of λ*/λ are obtained from eq 50 for several reported experimental conditions. The values of Edes are calculated from eq 51. Notice that this equation already considers the differences in the value of λ between the models. The results are summarized in Table 3. It is observed that the values of the energy barrier for desorption presented in Table 3 are reasonable. The hydrophobic monomers present higher desorption energy barriers than the hydrophilic monomers. The results obtained for styrene

9806 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 Table 1. Representative Models of Radical Desorptiona model

radical desorption formulation

Smith and Ewart (1948)

κ1ap 6κ1 k0 ) ) Vp dp

Ugelstad et al. (1969)3

k0 ) Harriott (1971)5

( π6 )

2⁄3

dp2

( )

K0 ) k*0 )

K0 ) k0 )

kfm[M]p + kft[T] +

ri(1 - nj) Nnj

njk*0 + kp[M]p 12DwDp

)

k*0

k0 obtained from Einstein’s diffusion equation in 1-D from the center of the particles

k0 kfm[M]p · NA k0+kp[M]p 8Dp dp2

Ugelstad and Hansen (1976)28

K0 ) k*0 )

(

φ: diffusion modulus

1⁄2

dp2(Dw + Dpm)

Friis and Nyhagen (1973)8

kfm k′p

k′p: propagation of chain-transfer-generated radicals

k*0

12DwDp dp (mDp + Dw) 2

( RTE )

k0 ) κ3 exp -

Nomura and Harada (1981)11

K0 ) zk*0 k*0 )

Chang, Litt and Nomura (1982)37

Dp

dp kt[R•]0 φ) 2 Dp

Harada et al. (1971)10

Brooks and Makanjuola (1981)51

κ2

remarks for Case 2 kinetics: k0 ≈ 0

2

k0 )

dp2(6Dw + mDp) 2Dp

, 2

dp

(

Nomura (1982)12

K0 ) k*0Z s

Z)

z is related to the degree of polymerization of the exiting radical

kfm kp 12DwDp

∑ j)1

(

k*0 ) dp2

k0 )

4Dp

5Dp

dp

dp2

, or k0 ) 2

kfm[M]p + kft[T]p kp[M]p kp[M]p

)

k*0nj + kp[M]p λDpDw λmDp 2Dw + 12

(

j

)

)

when the radicals are generated at the center, at the edge, or anywhere in the particle, respectively

4 e λ e 10

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9807 Table 1. Continued model Asua et al. (1989)

radical desorption formulation

14

∞ (1 - Pp,i+1)Ni 1 nj i)0 NT

∑

kdes ) kfm[M]p

∞

1 - (1 - Pw)

∑

remarks a correction was later presented by Asua,15 including the term Pw in the numerator

(1 - Pp,i+1)Ni NT

i)0

kt(n - 1) kp[M]p + VpNA Pp,n ) kt(n - 1) k*0 + kp[M]p + VpNA kp[M]w + ktw[R]w Pw ) kabsNTφww kp[M]w + ktw[R]w + NA 12DwDp k*0 ) 2 dp (mDp + 2Dw) Asua and de la Cal (1991)52

k0 ) Lacik et al. (1992)29

0eRe2

κ4 dpR

notice that for m f 0, k0* f ∞

kdes ) 2kfm[M]p K0 ) kfm[M]p k*0 )

Morrison et al. (1994)9

k0 )

k*0 k*0 + kp[M]p

12Dw mdp2 R ) 1.5 ( 1.2

κ5 dpR

Grady (1996)27,53

k0 obtained from random walk calculations (Gambler’s ruin) nc ) maximum chain length of radicals in equilibrium with aqueous phase

(nc - 1)k0 *

K0 ) kfm[M]p

(nc - 1)k0 + kp[M]p *

nc

k*0 ) k0

∑ 1 + 1m(n) n)1

k0 ) present model

20 DwDp 3dp2 Dw + 3Dp

kfm[M]pPw(1 - Pp) 1 - (1 - Pw)(1 - Pp) K0 ) kfm[M]p(1 - Pp) kdes )

∑k

i,p

Pp )

k*0 +

∑k

,

i,p

( ) i

k*0 ) k0 exp k0 ) 60

a

κ variables are undetermined constants.

Dp dp2

∑k

i,w

i

Edes kBT

Pw )

i

kabs +

∑k

i,w

i

9808 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 Table 2. Comparison of the Values of λ for Different Models model

λ

Ugelstad et al. (1969)3 Harada et al. (1971)10 Friis and Nyhagen (1973)8 Ugelstad and Hansen (1976)28 Nomura and Harada (1981)11 Chang, Litt, and Nomura (1982)37 Asua et al. (1989)14 Grady (1996)27,53 present theoretical model BD simulation results

1.54κ2 12 8 12 2 5 6 20/3 60 57.14

and Ugelstad’s models for radical desorption, they obtained the following expression for the net radical desorption rate coefficient (assuming no reabsorption of radicals) K0 ) kfm[M]p k*0 )

kdes ) ψs

(

ψsNp ψfNA 1ψpm ψsNp + kp,w[M]w + 2kt,w[R]w

)

(52)

where ψs )

2πDwdp Dw δs Dw 1 1+2 + dp Ds dp Dpm dp ψ coth √ψp - 1 2√ p 2 kfm,p[M]p ψf ) VpNADp

(

kp,p[M]p + 2kt,p ψp )

Dp

n-1 VpNA

)

(53)

(54)

(55)

where δs is the thickness of the surfactant layer, Ds is the diffusion coefficient of the radicals in the surfactant layer, and n is the number of radicals in the particle. The net desorption rate coefficient in this case is given by K0 ) ψs

ψfNA ψpm

(56)

Later, Thickett and Gilbert16 presented a modified Smoluchowski equation for diffusion of radicals through the surfactant layer. On the basis of this equation and Nomura’s and Hansen

k*0 + kp[M]p

(

12DsDw [M]w dp + 2δs 2 [M]p dpDs + 2δsDw d p

(22.92-27.03 kJ/mol) are comparable to the values reported by Lukhovitsky55 for the activation energy of transferring benzene molecules from a micelle into water (36.4-42 kJ/mol) and the values of interfacial energy between styrene and water in the presence of surfactants (29-30 kJ/mol) obtained from interfacial tension data.56 Additionally, Lansdowne et al.57 obtained activation energy values of 42 ( 10 kJ/mol for radical desorption in seeded emulsion polymerization of styrene initiated by γ radiolysis. Incorporation of an energy barrier for desorption is a more general approach which also describes the solubility effect of the monomer well without assuming macroscopic mass transfer through stationary films. Furthermore, the energy barrier assumption seems more realistic, more reliable, and physically more reasonable than an equilibrium radical concentration profile around individual particles. 5.3. Net Radical Desorption through a Surfactant Layer around the Particle. A further development in the theory of radical desorption was presented by Asua15 by considering the effect of a surfactant layer around the particles. He derived an analytical model based on the solution of Fick’s equation of mass transfer under steady-state conditions for calculation of the effective radical desorption rate. Reabsorption of radicals by the particle was assumed to follow Smoluchowski’s equation. He also considered a stagnant film surrounding the surfactant layer. His model can be summarized as follows (the original nomenclature was changed for consistency)

k*0

(57)

)

(58)

In order to compare the two models with the BD simulation method the conditions presented by Asua for studying the effect of the diffusion coefficient in the surfactant layer were used to determine the net radical desorption rate coefficient by theory (eqs 56 and 57) and BD simulation. The results are presented in Figure 12. Initially, since the energy barrier for desorption for this example is unknown, the BD simulation was performed in the absence of energy barriers to obtain the rate coefficient of simple desorption. Clearly, if no energy barrier is present, the rates of radical desorption are much higher than under equilibrium conditions. However, by setting the energy barrier to 5.5 kJ/mol, the agreement between both models and the simulation results is very good. It can also be concluded that there are not significant differences in the results obtained with the models of Asua and Thickett and Gilbert when no reabsorption is considered. This example shows once again that the idea of an energy barrier for desorption can be safely used to describe the kinetics of radical desorption in emulsion polymerization instead of the macroscopic mass transfer of radicals through stationary films between the particles and the continuous phase. 6. Conclusions Desorption of radicals from polymer particles plays a very important role in the determination of the extremely complex kinetics of emulsion polymerization. Several quite different models have been developed to estimate the rate of radical desorption. Although these models present evident differences, they have been able to describe experimental emulsion polymerization kinetics. Since the available experimental data do not provide enough sensitivity for discrimination of radical desorption models, a different approach is proposed in this paper based on simulation of the Brownian diffusion of the radicals inside the particles. Given that different authors have defined the process of radical desorption in different ways, the results obtained for different models are difficult to compare. We classify the desorption process into four different categories: simple, equilibrium, net, and effective radical desorption. Only the results for each type of desorption are comparative. In the case of simple desorption, we theoretically derived an expression for estimation of the simple rate coefficient of radical desorption. This expression is structurally similar to that of some previous models; however, they differ in the value of the proportionality constant. The values of the simple desorption rate coefficients in homogeneous spherical particles obtained by Brownian dynamics simulation were in better agreement with the theoretical model derived in section 3.1 than with the models previously proposed. Brownian dynamics simulation was also used to determine the simple desorption rate coefficient in particles stabilized by a surfactant layer. These examples demonstrate the usefulness and potential of BD simulation for estimation of desorption rate coefficients in heterogeneous polymerization systems.

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9809 Table 3. Equivalence between the Partition Coefficient and the Energy Barrier for Desorption Using Reported Experimental Conditions

d

system

ref

m

vinyl acetate at 50 °C vinyl acetate at 60 °C vinyl acetate at 60 °C vinyl chloride at 50 °C methyl methacrylate at 50 °C styrene at 50 °C styrene at 60 °C styrene + 20% MeOH at 60 °C

11 14 37 11 54 54 14 14

28 28 44.2 35 6.6 1350 1600 3480

a Assumed to be 1 when not considered in the model. Calculated from eq 51.

δw/δpa 1/6 1/2 1 1/6 1 1 1/2 1/2 b

Dw (× 10-9 m2/s)

Dpb (× 10-9 m2/s)

1.90 1.00 2.00 2.50 1.68 1.50 0.50 0.54

1.90 1.00 0.10 2.50 1.19 1.13 0.50 0.54

λ*/λ

c

0.1765 0.0667 0.3115 0.1463 0.1762 0.0010 0.0012 0.0006

Edesd (kJ/mol) 8.98 13.87 10.11 9.48 8.98 22.92 24.89 27.03

Assumed to be equal to the diffusion in water when not reported. c Calculated from eq 50.

such as the rate coefficients of radical desorption, radical absorption, particle aggregation, monomer uptake by particles, etc. Abbreviations

Figure 12. Effect of the diffusion coefficient in the surfactant layer on the net rate coefficient of radical desorption. Comparison between Asua’s (eq 56) and Thickett and Gilbert’s (eq 57) models and BD simulation results.

Derivation of equilibrium desorption rate coefficients based on the macroscopic mass transfer through stagnant films was critically discussed. It was demonstrated that some of the assumptions required by this model are not fulfilled in real emulsion polymerization systems and that the results obtained with this theory can also be explained considering an energy barrier for desorption caused by the difference in the chemical potential of the radical between the particle and the aqueous phase. The equations for calculating the net and effective rate coefficients of radical desorption were presented using an electrical circuit analogy to facilitate their comprehension. In this case, determination of desorption rate coefficients requires the coupling of Brownian dynamics (BD) simulation with the kinetic Monte Carlo (kMC) method. The rate coefficients obtained with this hybrid BD-kMC simulation were in very good agreement with the theoretical values. Computational methods based on Brownian dynamics simulation were successfully used to validate different models of radical desorption, offering higher levels of accuracy which could not be achieved using the currently available experimental methods. BD simulation methods are recommended when a detailed simulation of the events taking place in heterophase polymerization is required, especially for systems of increased geometrical complexity such as polydisperse systems, nonspherical particles, and systems with nonuniform molecular distribution inside the particles or in the continuous phase. Finally, BD simulation can be regarded as a complement to the traditional modeling of heterophase polymerization. BD simulation can be used, for example, to obtain better estimates of some of the kinetic parameters required by the traditional models,

dp ) particle diameter (m) D ) diffusion coefficient of radicals (m2/s) E ) energy (kJ/mol) Eabs ) energy barrier for absorption (kJ/mol) Edes ) energy barrier for desorption (kJ/mol) p ) Planck’s constant I ) number of initiator molecules IE ) electrical current analogy (radicals/s) IR• ) number of radicals derived from initiator decomposition Jr ) flux of radicals (mol/m2s) kabs ) rate coefficient of absorption of radicals by polymer particles (s-1) kB ) Boltzmann’s constant kdes ) rate coefficient of effective radical desorption (s-1) kfm ) rate coefficient of chain transfer to monomer (m3/mol · s) kfm ) rate coefficient of chain transfer to chain-transfer agent (m3/ mol · s) k0 ) rate coefficient of simple radical desorption (s-1) k0* ) rate coefficient of equilibrium radical desorption (s-1) kp ) rate coefficient of radical propagation (m3/mol · s) krgen ) rate of generation of radicals inside the particles (s-1) krin ) rate of appearance of radicals inside the particles (s-1) kt ) rate coefficient of radical termination (m3/mol · s) K0 ) rate coefficient of net radical desorption (s-1) m ) partition coefficient of the radicals between the particles and water [M] ) monomer concentration (mol/m3) nj ) average number of radicals per particle NA ) Avogadro’s number Np ) total number of particles in the dispersion P ) probability of reaction of a radical rdes ) rate of effective radical desorption (radicals/s) rˆ ) rate of radical desorption per unit volume (radicals/m3s) R• ) number of radicals derived from chain transfer to monomer [R•] ) concentration of radicals (mol/m3) RE ) electrical resistance analogy (s) Re ) Reynolds dimensionless number S ) number of molecules of the key reacting species Sc ) Schmidt dimensionless number Sh ) Sherwood dimensionless number t ) time (s) T ) system temperature (K) Tg ) glass-transition temperature (K) TR• ) number of radicals derived from chain-transfer agents [T] ) chain-transfer agent concentration (mol/m3) V ) volume of the dispersion (m3)

9810 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 ∆VE ) electrical voltage analogy (number of molecules) wp ) weight fraction of polymer in the particles Greek Letters δ ) thickness of the mass-transfer boundary layer (m) φ ) volume fraction η ) viscosity (Pa · s) κ ) undetermined constant λ ) constant parameter for simple radical desorption λ* ) constant parameter for equilibrium radical desorption µ ) chemical potential (kJ/mol) Fd ) probability density distribution (m-3) τdes ) desorption time (s) τdiff ) diffusion time (s) ψ ) group of variables Superscripts I ) radical with an initiator fragment end group T ) radical with a chain-transfer agent end group * ) equilibrium condition Subscripts h ) polymer particle index j ) degree of polymerization index p ) polymer particle r ) reaction index s ) surfactant layer w ) aqueous phase

Acknowledgment The authors gratefully acknowledge the Max Planck Society for financial support of this research as well as the reviewers for all their valuable suggestions to improve the manuscript. Supporting Information Available: Details of the generalized electrical circuit analogy. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Fitch, R. M. Polymer Colloids: A ComprehensiVe Introduction; Academic Press: London, 1997. (2) Smith, W. V.; Ewart, R. H. Kinetics of Emulsion Polymerization. J. Chem. Phys. 1948, 16, 592. (3) Ugelstad, J.; Mork, P. C.; Dahl, P.; Rangnes, P. A Kinetic Investigation of the Emulsion Polymerization of Vinyl Chloride. J. Polym. Sci. C 1969, 27, 49. (4) Nomura, M.; Harada, M.; Nakagawara, K.; Eguchi, W.; Nagata, S. The Role of Polymer Particles in the Emulsion Polymerization of Vinyl Acetate. J. Chem. Eng. Jpn. 1971, 4, 160. (5) Harriot, P. Kinetics of Vinyl Acetate Emulsion Polymerization. J. Polym. Sci. A-1 1971, 9, 1153. (6) Litt, M.; Patsiga, R.; Stannett, V. Emulsion Polymerization of Vinyl Acetate II. J. Polym. Sci. A-1 1970, 8, 3607. (7) O’Toole, J. T. Kinetics of Emulsion Polymerization. J. Appl. Polym. Sci. 1965, 9, 1291. (8) Friis, N.; Nyhagen, L. A Kinetic Study of the Emulsion Polymerization of Vinyl Acetate. J. Appl. Polym. Sci. 1973, 17, 2311. (9) Morrison, B. R.; Casey, B. S.; Lacik, I.; Leslie, G. L.; Sangster, D. F.; Gilbert, R. G.; Napper, D. H. Free Radical Exit in Emulsion Polymerization II. Model Discrimination via Experiment. J. Polym. Sci. A: Polym. Chem. 1994, 32, 631. (10) Harada, M.; Nomura, M.; Eguchi, W.; Nagata, S. Studies of the Effect of Polymer Particles on Emulsion Polymerization. J. Chem. Eng. Jpn. 1971, 4, 54. (11) Nomura, M.; Harada, M. Rate Coefficient for Radical Desorption in Emulsion Polymerization. J. Appl. Polym. Sci. 1981, 26, 17.

(12) Nomura, M. Desorption and Reabsorption of Free Radicals in Emulsion Polymerization. In Emulsion Polymerization; Piirma, I., Eds.; Academic Press: New York, 1982; p 191. (13) Nomura, M.; Yamamoto, K.; Horie, I.; Fujita, K.; Harada, M. Kinetics of Emulsion Copolymerization 2. Effect of Free Radical Desorption on the Rate of Emulsion Copolymerization of Styrene and Methyl Methacrylate. J. Appl. Polym. Sci. 1982, 27, 2483. (14) Asua, J. M.; Sudol, E. D.; El-Aasser, M. S. Radical Desorption in Emulsion Polymerization. J. Polym. Sci. A: Polym. Chem. 1989, 27, 3903. (15) Asua, J. M. A New Model for Radical Desorption in Emulsion Polymerization. Macromolecules 2003, 36, 6245. (16) Thickett, S. C.; Gilbert, R. G. Rate-controlling Events for Radical Exit in Electrosterically Stabilized Emulsion Polymerization Systems. Macromolecules 2006, 39, 2081. (17) Thickett, S. C.; Gilbert, R. G. Emulsion Polymerization: State of the Art in Kinetics and Mechanisms. Polymer 2007, 48, 6965. (18) Gillespie, D. T. Stochastic Simulation of Chemical Kinetics. Annu. ReV. Phys. Chem. 2007, 58, 35. (19) Tauer, K.; Kaspar, H.; Antonietti, M. Equilibrium Swelling of Colloidal Polymeric Particles with Water-insoluble Organic Solvents. Colloid Polym. Sci. 2000, 278, 814. (20) Bolton, C. E.; Green, N. J. B.; Harris, R.; Pimblott, S. M. Competition between Geminate Recombination and Reaction with a Macromolecule. J. Phys. Chem. A 1998, 102, 730. (21) Green, N. J. B. On the Simulation of Diffusion Processes close to Boundaries. Mol. Phys. 1988, 65, 1399. (22) Bluett, V. M.; Green, J. B. On the Competition between Scavenging and Recombination in Solutions of Macromolecules. J. Phys. Chem. A 2006, 110, 6112. (23) Hernandez, H. F.; Tauer, K. Brownian Dynamics Simulation of the Capture of Primary Radicals in Dispersions of Colloidal Polymer Particles. Ind. Eng. Chem. Res. 2007, 46, 4480. (24) Hernandez, H. F.; Tauer, K. Brownian Dynamics Simulation Studies on Radical Capture in Emulsion Polymerization. Macromol. Symp. 2007, 259, 274. (25) Rawlings, J. B.; Ray, W. H. The Modelling of Batch and Continuous Emulsion Polymerization Reactors 1. Model Formulation and Sensitivity to Parameters. Polym. Eng. Sci. 1988, 28, 237. (26) Einstein, A. Zur Theorie der Brownschen Bewegung. Ann. Phys. 1906, 19, 371. (27) Grady, M. C. Preparation of ω-unsaturated Oligo(Methyl Methacrylate) Macromer and its Application in Emulsion Polymerization: Key Learnings about Radical Desorption. Ph.D. thesis, Eidgeno¨ssische Technische Hochschule Zu¨rich (ETH), Zurich, 1996. (28) Ugelstad, J.; Hansen, F. K. Kinetics and Mechanism of Emulsion Polymerization. Rubber Chem. Technol. 1976, 49, 536. (29) Lacik, I.; Casey, B. S.; Sangster, D. F.; Gilbert, R. G.; Napper, D. H. Desorbed Free Radicals in Emulsion Polymerizations: Effect of Aqueous-phase Spin Trap. Macromolecules 1992, 25, 4065. (30) Kim, J. U.; Lee, H. H. Exit Process and Re-entry Rate in Emulsion Polymerization. Polymer 1996, 37, 1941. (31) Saslow, W. M. Electricity, Magnetism, and Light; Academic Press: Toronto, 2002. (32) Fujita, H. Diffusion in Polymer-diluent Systems. AdV. Polym. Sci. 1961, 3, 1. (33) Faldi, A.; Tirrell, M.; Lodge, T. P.; von Meerwall, E. Monomer Diffusion and the Kinetics of Methyl Methacrylate Radical Polymerization at Intermediate to High Conversion. Macromolecules 1994, 27, 4184. (34) Hidi, P.; Napper, D. H.; Sangster, D. F. Reversible Swelling of Poly(Vinyl Acetate) Latex Particles in Sodium Dodecyl Sulfate Solution. Macromolecules 1995, 28, 6042. (35) Vrentas, J. S.; Duda, J. L. Diffusion of Small Molecules in Amorphous Polymers. Macromolecules 1976, 9, 785. (36) Yasuda, H.; Lamaze, C. E.; Peterlin, A. Diffusive and Hydraulic Permeabilities of Water in Water-swollen Polymer Membranes. J. Polym. Sci. A-2 1971, 9, 1117. (37) Chang, K. H. S.; Litt, M. H.; Nomura, M. The Reinvestigation of Vinyl Acetate Emulsion Polymerization 1sThe Rate of Polymerization. In Emulsion Polymerization; Piirma, I., Ed.; Academic Press: New York, 1982; p 89. (38) Hiss, T. G.; Cussler, E. L. Diffusion in High Viscosity Liquids. AIChE J. 1973, 19, 698. (39) Griffiths, M. C.; Strauch, J.; Monteiro, M. J.; Gilbert, R. G. Measurement of Diffusion Coefficients of Oligomeric Penetrants in Rubbery Polymer Matrixes. Macromolecules 1998, 31, 7835. (40) Karlsson, O. J.; Stubbs, J. M.; Karlsson, L. E.; Sundberg, D. C. Estimating Diffusion Coefficients for Small Molecules in Polymers and Polymer Solutions. Polymer 2001, 42, 4915.

Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9811 (41) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: San Diego, CA, 2002. (42) Harmandaris, V. A.; Adhikari, N. P.; van der Vegt, F. A.; Kremer, K.; Mann, B. A.; Voelkel, R.; Weiss, H.; Liew, C.-C. Ethylbenzene Difussion in Polystyrene: United Atom Atomistic/Coarse Grained Simulations and Experiments. Macromolecules 2007, 40, 7026. (43) Hofmann, D.; Fritz, L.; Ulbrich, J.; Paul, D. Molecular Simulation of Small Molecule Diffusion and Solution in Dense Amorphous Polysiloxanes and Polyimides. Comput. Theor. Polym. Sci. 2000, 10, 419. (44) Masaro, L.; Zhu, X. X. Physical Models of Diffusion for Polymer Solutions, Gels and Solids. Prog. Polym. Sci. 1999, 24, 731. (45) Krywko, W. P.; McAuley, K. B.; Cunningham, M. F. Mathematical Modeling of Particle Morphology Development Induced by Radical Concentration Gradients in Seeded Styrene Homopolymerization. Polym. React. Eng. 2002, 10, 135. (46) Lemons, D. S.; Gythiel, A. Paul Langevin’s 1908 Paper ‘On the Theory of Brownian Motion’. Am. J. Phys. 1997, 65, 1079. (47) Chern, C. S. Desorption of Free Radicals in Semibatch Emulsion Polymerization of Methyl Acrylate. J. Appl. Polym. Sci. 1995, 56, 231. (48) Bird, R. B.; Stewart, W. E.; Lightfoot; E. N. Transport Phenomena; John Wiley & Sons: New York, 2002. (49) Gillespie, D. T. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. J. Comput. Phys. 1976, 22, 403. (50) Chen, J. C.; Kim, A. S. Brownian Dynamics, Molecular Dynamics and Monte Carlo Modeling of Colloidal Systems. AdV. Colloid Interface Sci. 2004, 112, 159.

(51) Brooks, B. W.; Makanjuola, B. O. Measurements of the Rate of Radical Desorption from Polymer Lattices during Emulsion Polymerization. J. Chem. Soc., Faraday Trans. 1 1981, 77, 2659. (52) Asua, J. M.; de la Cal, J. C. Entry and Exit Rate Coefficients in Emulsion Polymerization of Styrene. J. Appl. Polym. Sci. 1991, 42, 1869. (53) Grady, M. C.; Matheson, R. R. Small-molecule Exchange in Emulsion Polymerization. J. Comput.-Aided Mater. Design 1996, 3, 296. (54) Casey, B. S.; Morrison, B. R.; Maxwell, I. A.; Gilbert, R. G.; Napper, D. H. Free Radical Exit in Emulsion Polymerization. I. Theoretical Model. J. Polym. Sci. A: Polym. Chem 1994, 32, 605. (55) Lukhovitsky, V. I. On the Mechanism of the Processes of Sorption and Desorption of Radical Particles in Emulsion Polymerization. Acta Polym. 1986, 37, 384. (56) Chatzi, E. G.; Boutris, C. J.; Kiparissides, C. On-line Monitoring of Drop Size Distributions in Agitated Vessels. 2. Effect of Stabilizer Concentration. Ind. Eng. Chem. Res. 1991, 30, 1307. (57) Lansdowne, S. W.; Gilbert, R. G.; Napper, D. H.; Sangster, D. F. Relaxation Studies of the Seeded Emulsion Polymerization of Styrene Initiated by γ-Radiolysis. J. Chem. Soc., Faraday Trans. 1 1980, 76, 1344.

ReceiVed for reView February 22, 2008 ReVised manuscript receiVed September 9, 2008 Accepted September 25, 2008 IE800304T