Modeling of the Free-Radical Copolymerization Kinetics with Cross

Ind. Eng. Chem. Res. 2005, 44, 2823-2844

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Modeling of the Free-Radical Copolymerization Kinetics with Cross Linking of Vinyl/Divinyl Monomers in Supercritical Carbon Dioxide Iraı´s A. Quintero-Ortega,† Eduardo Vivaldo-Lima,*,† Gabriel Luna-Ba´ rcenas,‡ Juan F. J. Alvarado,§ Jose´ F. Louvier-Herna´ ndez,‡ and Isaac C. Sanchez| Departamento de Ingenierıá Quı´mica, Facultad de Quı´mica, Universidad Nacional Auto´ noma de Me´ xico (UNAM), Conjunto E, Ciudad Universitaria, 04510 Me´ xico D.F., Me´ xico, CINVESTAV, Unidad Quere´ taro, Libramiento Norponiente no. 2000, Frac. Real de Juriquilla, 76230 Quere´ taro, Me´ xico, Departamento de Ingenierıá Quı´mica, Instituto Tecnolo´ gico de Celaya, Celaya, Guanajuato 38010, Me´ xico, and Chemical Engineering Department, The University of Texas at Austin, Austin, Texas 78712

A mathematical model for free-radical copolymerization kinetics with cross linking of vinyl/ divinyl monomers in carbon dioxide at supercritical conditions was developed. The copolymerization of styrene and divinylbenzene is analyzed as a case study. The effects of the kinetic and physical parameters on monomer conversion, molecular-weight development, copolymer composition, appearance of the gelation point, gel fraction, and average cross-link density are studied. Model predictions show the expected trends, although the system is quite sensitive to pressure, which provides an interesting and promising way to tailor some of the polymer properties. 1. Introduction Supercritical carbon dioxide (scCO2) has become an attractive medium for polymerization processes because of its low toxicity, reasonably low cost, mild critical point (Tc ) 31.1 °C, Pc ) 73.8 bar), and environmentally benign nature.1,2 Many polymers have been synthesized in scCO2, including fluoropolymers, polysiloxanes, poly(methyl methacrylate), polystyrene, and polycarbonates, as reviewed elsewhere.3-6 Unfortunately, besides fluoropolymers and polysiloxanes, most high-molecularweight polymers do not show appreciable solubility in scCO2, thus reducing the applications of homogeneous polymerization to a very few materials.7-10 Because of the severe solubility limitation of many polymers in scCO2, heterogeneous polymerizations are carried out by precipitation, dispersion, or emulsion processes. The specific type of heterogeneous process will depend on the solubility of the monomers and initiators in scCO2. Most of the polymeric materials obtained in scCO2 are produced by heterogeneous processes.11-20 Supercritical fluids (SCF) can exhibit the best features of two worlds: they can have gaslike diffusivities and liquidlike densities. In the vicinity of the fluid’s critical point, its density is highly sensitive to modest changes in pressure or temperature. Higher SCF diffusivities have important implications in polymerization kinetics and in polymer processing (i.e., diminishing the “cage effect” associated to the initiator decomposition in freeradical polymerization processes). The resulting polymer can be isolated from the reaction medium by simple depressurization, resulting in a dry, solvent-free product. This technique eliminates drying procedures required in polymer manufacturing, offering a costeffective and very attractive technology. * To whom correspondence should be addressed. Tel.: +(5255)-5622-5256. Fax: +(5255)-5622-5355. E-mail: [email protected]. † Universidad Nacional Autońoma de Me´xico (UNAM). ‡ CINVESTAV. § Instituto Tecnolo´gico de Celaya. | The University of Texas at Austin.

The thermodynamic and transport properties of SCFs can be easily tuned by adjusting the pressure or temperature. This feature makes SCFs an attractive medium for polymerization reactions, an area that has been the focus of many published technical papers in the field. The literature on polymer chemistry in scCO2 is extensive and keeps growing, as evidenced by Kendall et al.’s review.4 However, the modeling of polymerization processes in fluids at supercritical conditions has not received much attention to date. To the best of our knowledge, the only two papers that report the comprehensive modeling of free-radical dispersion homopolymerization of methyl methacrylate, MMA, in scCO2 come from the groups of Kiparissides18 and Morbidelli.19 Although very few mathematical models on the polymerization of vinyl monomers in scCO2 are available, systematic studies of this type of process, from an engineering perspective, are starting to appear, as evidenced from the recent experimental study by Rosell et al.20 on the effect of mixing on polymerization rate and molecular-weight development. Cross-linked polymers (polymer networks) are very important in technology, medicine, biotechnology, agriculture, and other important areas. They are used as construction materials, paints and coatings, polymer glasses with high mechanical strength and high thermal stability, rubbers, ion-exchange resins and sorbents, insoluble polymer-supported reagents, controlled drugrelease matrixes, electronics and cabling, food packaging, sensors, “smart” materials, artificial organs, implants, superabsorbent materials, and so on.21,22 Poly(styrene-divinylbenzene) is a cross-linked polymer used for chromatographic applications and as a precursor for ion-exchange resins, among other uses. It is also a model system in the study of network formation via cross-linking free-radical copolymerization. The experimental study of commercial divinylbenzene (a mixture of meta and para isomers of divinylbenzene, ethylvinyl benzene, and other minor impurities) dispersion copolymerization in carbon dioxide at supercritical conditions has already been addressed by Cooper et al.23,24 They experimentally studied important aspects such as

10.1021/ie048922o CCC: $30.25 © 2005 American Chemical Society Published on Web 02/23/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005

the effects of the cross-linker ratio, monomer concentration, mechanical agitation, and stabilizer concentration on polymer yield, particle size, and surface area/porosity as well as the synthesis of terpolymers. The same group of Cooper et al.25,26 and Hebb et al.27 studied the synthesis of molded monolithic porous poly(methacrylate) polymers using scCO2 as the porogenic solvent. Although these last studies25-27 were not dispersion polymerization processes, they focused on the cross linking of di- and trimethacrylates in the presence of scCO2 using much higher monomer concentrations than those used in dispersion polymerizations and are, therefore, relevant to the Discussion section of this paper. Despite the great relevance of these first studies by Cooper et al.,23-27 more systematic kinetic studies, including experimental and theoretical studies on the effects of pressure and temperature on the gelation point and the homogeneity of the polymer network as well as a more comprehensive understanding of the phenomena determining the particle size distribution, are still needed. Because of the very important scientific and technological applications of polymer networks and the several scientific and technological challenges needed to be addressed in order to produce polymer networks from vinyl/divinyl copolymerization in scCO2 in a more effective way, the attention of our group was focused on this topic. The copolymerization of styrene and divinylbenzene (DVB) in scCO2 was chosen as a model system. The objective of our group in this contribution was to develop a sound mathematical model that can simultaneously predict, for the first time in this area, the overall conversion, copolymer composition, molecularweight development, appearance of the gelation point, gel fraction evolution, and average cross-link density as function of the system’s pressure and temperature.

2. Model Development 2.1. Reaction Scheme. The chemical system to be analyzed in this paper is free-radical copolymerization with cross linking of vinyl/divinyl monomers in scCO2. The complete reaction scheme, which includes the most important reactions that are considered to take place in such system, is summarized in Table 1. R*m,n,i,j in Table 1 is a polymer radical with m units of monomer 1 (M1) and n units of monomer 2 (M2) bound to the polymer chain, with the active center located in monomer unit i. Subscript j refers to the phase where the polymer molecule is located (i.e., 1 for the continuous phase, rich in solvent, or 2 for the dispersed phase, rich in polymer). Pm,n,j is a polymer molecule with m units of monomer 1 and n units of monomer 2, located in phase j. Subscript 3 denotes active centers not located in the chain ends of the primary polymer molecule, namely, active centers formed from propagation through pendant double bonds, or active centers located at the end of a branch attached to the primary polymer molecule. It is assumed that a polymer radical molecule can have no more than one active center (monoradical assumption). 2.2. Description of the Dispersion Polymerization Process. Many polymerization reactions in carbon dioxide at supercritical conditions are conducted under heterogeneous processes, either as precipitation, dispersion, or emulsion polymerizations, because of the inher-

Figure 1. Schematic representation of the dispersion copolymerization of vinyl/divinyl monomers in scCO2 with emphasis on the species present in each phase and on the stages of the process.

ent insolubility of most polymers in carbon dioxide. The specific type of heterogeneous process will depend on the solubilities of the monomers and initiators in carbon dioxide.4 The free-radical homopolymerization of styrene in carbon dioxide at supercritical conditions proceeds as a dispersion polymerization process starting as a single phase with the later appearance of a second phase. When divinylbenzene is added as the second monomer (in a major proportion), two phases are reported to be present from the beginning of the polymerization, and the copolymerization may proceed as an unstabilized suspension copolymerization or as an emulsion copolymerization, depending on whether a stabilizing agent is present.23,24 What Cooper et al.23,24 described as emulsion polymerization, in the case of commercial DVB polymerization, is indeed a dispersion polymerization process in which the appearance of the second phase takes place almost instantly. The appearance of the second phase takes place much more rapidly than in the styrene homopolymerization case because of the formation of branched and cross-linked polymer molecules that promote the generation of larger molecules in less time. Therefore, the model used here basically considers a dispersion copolymerization process with the possibility, as limiting cases, of homogeneous singlephase copolymerization (solution polymerization) and copolymerization in dispersed media, with no reactions taking place in the continuous phase (bead suspension copolymerization). Two phases are considered: a dispersed organic liquid/viscous phase and a continuous “fluid” phase. A free-radical dispersion polymerization is a heterogeneous process where latex particles are formed in the presence of an adequate stabilizer, with a reaction mixture that is homogeneous at the beginning of the polymerization. This process can be described as proceeding in three stages.28 Figure 1 shows a schematic representation of this situation, which is explained below. In Stage 1, primary radicals are formed from thermally promoted fragmentation of the initiator. These primary radicals rapidly react with monomer molecules to produce polymer chains that are insoluble in the continuous phase. Experimental data and theoretical calculations on the solubility of polystyrene oligomers in SCFs, including carbon dioxide, are available in Kumar et al.29 The aggregation of polymer chains

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2825 Table 1. Reaction Mechanism for the Free-Radical Copolymerization of Vinyl/Divinyl Monomers initiation

kdj

/ Ij 98 2Rin,j k1,j

/ / Rin,j + M1,j 98 R1,0,1,j

transfer to small molecules (either solvent or transfer agent)

k2,j

kzl,j

/ Rm,n,1,j + Zj 98 Pm,n,j

/ Rm,n,3,j + Tj 98 Pm,n,j + T/j

transfer to polymer

kfp1,j

/ / Rm,n,1,j + Pr,s,j 98 Pm,n,j + Rr,s,1,j

kz2,j


kfp2,j

kz3,j



kfp3,j


propagation

kft2,j

/ Rm,n,2,j + Tj 98 Pm,n,j + T/j kft3,j

/ / Rin,j + M2,j 98 R1,0,2,j

inhibition

kft1,j

/ Rm,n,1,j + Tj 98 Pm,n,j + T/j

k11,j

/ / Rm,n,1,j + M1,j 98 Rm+1,n,1,j

termination by disproportionation

ktd11,j

/ / Rm,n,1,j + Rr,s,1,j 98 Pm,n,j + Pr,s,j

k12,j


ktd13,j

k21,j


k22,j


k31,j


k32,j


/ / Rm,n,1,j + M2,j 98 Rm,n+1,2,j

ktd12,j

/ / Rm,n,2,j + M1,j 98 Rm+1,n,1,j

ktd21,j

/ / Rm,n,2,j + M2,j 98 Rm,n+1,2,j

ktd22,j

/ / Rm,n,3,j + M1,j 98 Rm+1,n,1,j

ktd23,j

/ / Rm,n,3,j + M2,j 98 Rm,n+1,2,j

ktd31,j


propagation through pendant double bonds (cross-linking)

/ kp31,j

ktd32,j

/ / / Rm,n,1,j + Pr,s,j 98 Rm+r+1,n+s,3,j


/ kp32,j

ktd33,j


/ / / Rm,n,2,j + Pr,s,j 98 Rm+r,n+s+1,3,j / kp33,j

/ / / Rm,n,3,j + Pr,s,j 98 Rm+r,n+s,3,j

termination by combination transfer to monomer

ktd11,j

/ / Rm,n,1,j + Rr,s,1,j 98 Pm+r,n+s

/ / Rm,n,1,j + M1,j 98 Pm,n,j + R1,0,1,j

kf11,j

/ / Rm,n,1,j + Rr,s,2,j 98 Pm+r,n+s

/ / Rm,n,1,j + M2,j 98 Pm,n,j + R0,1,2,j

kf12,j

/ / Rm,n,1,j + Rr,s,3,j 98 Pm+r,n+s

/ / Rm,n,2,j + M1,j 98 Pm,n,j + R1,0,1,j

kf21,j

/ / Rm,n,2,j + Rr,s,1,j 98 Pm+r,n+s

/ / Rm,n,2,j + M2,j 98 Pm,n,j + R0,1,2,j

kf22,j

/ / Rm,n,2,j + Rr,s,2,j 98 Pm+r,n+s

/ / Rm,n,3,j + M1,j 98 Pm,n,j + R1,0,1,j

kf31,j

/ / Rm,n,2,j + Rr,s,3,j 98 Pm+r,n+s

kf32,j

/ / Rm,n,3,j + Rr,s,1,j 98 Pm+r,n+s

/ / Rm,n,3,j + M2,j 98 Pm,n,j + R0,1,2,j

ktc12,j ktc13,j

ktc21,j

ktc22,j ktc23,j

ktd31,j ktc32,j

/ / Rm,n,3,j + Rr,s,2,j 98 Pm+r,n+s ktc33,j

/ / Rm,n,3,j + Rr,s,3,j 98 Pm+r,n+s

results in the formation of unstable polymer microdomains. The reaction mixture consists primarily of pure monomers, initiator, primary radicals, and oligomer radicals because the polymer concentration is less than its solubility limit. Therefore, in this stage, the polymerization can be described as a solution polymerization process. In Stage 2, because of the very limited stability of the microdomains, they rapidly aggregate to form primary polymer particles, also called domains. From this point on, the polymerization proceeds in two phases, namely, the polymer-rich phase and the continuous, solvent-rich phase. This stage goes from the time of appearance of the dispersed polymer phase to a fractional overall monomer conversion xc, at which the monomer concen-

tration in the continuous phase is negligible and eventually disappears. It is assumed that during this stage the rate of mass transfer of both monomers (vinyl and divinyl monomers) and solvent from the continuous phase to the polymer phase is very fast so that the later is kept saturated with monomer and CO2. The overall polymerization rate is given by the sum of the polymerization rates in each phase. Finally, in Stage 3, at higher overall monomer conversions (xc< x e 1), the polymerization proceeds mainly in the polymer-rich phase. The polymer particles are swollen with monomer and CO2, thus the monomer mass fraction in the polymer phase decreases as the total monomer conversion approaches a final limiting

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value. During this stage, diffusion-controlled phenomena become very important. Representing the dispersion polymerization process as proceeding in three stages has some implications regarding its mathematical treatment. If the system is modeled as starting as a single-phase polymerization, then the rigorous modeling of the formation of the second phase will not be an easy issue to handle because fulfillment of the initial conditions may lead to numerical difficulties. Moreover, the three-stage representation suggests imposing a second discontinuity (when going from stage 2 to 3), which may not be necessary because a continuous model should be able to capture the behavior of the system when monomer in the continuous phase has been fully consumed. Although Kiparissides et al.28 first proposed and used this threestage approach, which implied using two discontinuous transitions, a later contribution from the same group18 and the model approach used by Mueller et al.19 treat the system as consisting of a two-phase polymerization from the very beginning, which is reasonable because of the very fast formation of high-molecular-weight polymer, which will phase separate. In section 2.7 of this paper, we will explain why this three-stage approach was used in this contribution. 2.3. Overview of the Developed Kinetic Model. The mathematical model presented here is an extension to multiphase copolymerization, at high pressures, of the model for single-phase, low-pressure copolymerization with cross linking of vinyl/divinyl monomers developed by Vivaldo-Lima et al.30-32 The mathematical model developed by Vivaldo-Lima et al.30-32 is based on the Tobita-Hamielec33,34 model for cross-linking kinetics for the pre-gelation period, an improved version of the Marten-Hamielec model for diffusion-controlled kinetics in free-radical polymerization35 (which incorporates the recommendations of Zhu and Hamielec36 on the use of different number- and weight-average termination constants), and a simple phenomenological approach for the termination kinetic rate constant during the post-gelation period. (Although simple, this approach takes into account the unequal reactivity of vinyl groups and cyclization reactions.) The model consists of a set of ordinary differential and algebraic equations that describe the most important reactions that take place during the copolymerization for each phase according to the reaction scheme shown in Table 1, coupled to the algebraic equations describing the phase equilibria behavior of the system. The detailed kinetic and moment equations describing the behavior of the system during the pre- and postgelation periods as well as those dealing with diffusioncontrolled reactions and the pseudo-homopolymer approach are provided in sections 2.4-2.6 of this paper. The kinetic scheme can be treated as if it was a homopolymerization by making use of the pseudokinetic rate constants method developed by Hamielec and MacGregor.37 The method of moments is used to follow the molecular-weight development. Initiation, propagation, and termination reactions are considered to be diffusion-controlled and are modeled using a freevolume theory from the beginning of the polymerization. Two averages, number- and weight-average termination kinetic rate constants, are used to model the mechanism of bimolecular termination. The number-average termination kinetic rate constant, ktn, is used to calculate the polymerization rate and number-average molecular

weight. The weight-average termination kinetic rate constant, ktw, is used to calculate the weight-average molecular weight. These averages depend on polydispersity and conversion and are defined in such a way that no additional parameters are needed in the model. All diffusion-controlled reactions are modeled using a “series” structure for the effective kinetic rate constants, as opposed to the rather common “parallel” approach. The differences between these two modeling approaches are explained in detail in Vivaldo-Lima et al.35 The model equations can be solved using the steady-state hypothesis (SSH) for polymer radicals, but this is reliable only during the pre-gelation period. Cyclization reactions are modeled using the equations proposed by Tobita and Hamielec,33,34 although only average cyclization densities are calculated instead of the full density distributions. Likewise, only the average cross-linking density as a function of time is calculated. Tobita and Hamielec33,34 generalized Flory’s theory for the post-gelation period by using a cross-linking density distribution. Instead, in this paper the original FloryStockmayer equation for the calculation of the sol fraction is used, but the simplifying assumptions regarding the equal reactivity of double bonds, absence of cyclization, and independence of double bonds were removed. 2.4. Detailed Kinetic and Moment Equations for the Pre-Gelation Period. The kinetic and moment equations that describe the polymerization kinetics and molecular-weight development during the pre-gelation period are now presented. These equations are obtained from the corresponding material balances for each species, as dictated by the reaction mechanism, and by applying the pseudo-kinetic rate constants method (also known as the pseudo-homopolymer approach), as explained in section 2.3 of this paper. All symbols are defined in the Nomenclature section. Equations 1 and 2 describe the mass balances for overall amounts of initiator and inhibitor (if present), respectively. The first two terms on the right-hand side (RHS) of eq 1 correspond to the consumption of initiator in phase 1 (the continuous phase), and the last two correspond to the consumption of initiator in the dispersed phase. Likewise, the first term on the RHS of eq 2 corresponds to the consumption of inhibitor in the continuous phase, and the second one corresponds to the consumption of inhibitor in the dispersed phase.

d(V[I]) ) -kd1[I]1 - kft1[I]1Y0,1 - kd2[I]2 - kft2[I]2Y0,2 V dt (1) d(V[Z]) ) -kz1[I]1 - kz2[I]2Y0,2 V dt

(2)

In the case of monomer conversion, overall (vinyl plus divinyl monomer concentrations, considering both phases) and individual mass balances for each phase were derived. This was done for convenience purposes because some calculations in the cross-linking model required overall amounts of monomer and others required the total monomer conversion in a specific phase. Equation 3 shows the mass balance for overall monomer consumption in the two phases. The first term on the RHS corresponds to total monomer consumption in the continuous phase, and the second one corresponds to consumption in the dispersed phase. Equations 4 and

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2827

5 correspond to overall monomer consumption in the continuous and dispersed phases, respectively. It should be noted that the subscript on the conversion in eqs 4 and 5 refers to the phase and not to the monomer type.

cm1 cm2 dx ) (kp1 + kfm1) Y0,1 + (kp2 + kfm2) Y dt cm0 cm0 0,2

(3)

dx1 ) (kp1 + kfm1)Y0,1(1 - x1) dt

(4)

dx2 ) (kp2 + kfm2)Y0,2(1 - x2) dt

(5)

/

The amount of total unreacted monomer, in terms of mole fraction, in the reactive mixture (considering both phases) is obtained from eq 6. The mole fraction of each monomer type in eq 6 is defined in terms of the total amount of monomer and not in terms of all of the components of the system, as defined in eq 7. Subscripts 1 and 2 in eqs 6 and 7 refer to vinyl and divinyl monomer types, respectively. The second subscript, “j”, refers to the phase being considered, either continuous or dispersed.

[

]

(6)

M2,j f2,j ) ) 1 - f1,j M1,j + M2,j

(7)

F2 dx F22 dx2 F21 dx1 df2 ) + dt 1 - x dt 1 - x1 dt 1 - x2 dt M1,j f1,j ) M1,j + M2,j

F2,j ) • • • + k22φ2,j + k32φ3,j (k12φ1,j )f2,j • • • • • • (k11φ1,j + k21φ2,j + k31φ3,j )f1,j + (k12φ1,j + k22φ2,j + k32φ3,j )f2,j

(8) f20 - f2,j(1 - xj) xj

1 d(V[R1,f]) ) RI,f + (kfm,f[M]f + kft,f[P1,f])[R/f ] V dt / ] (kp,f + kfm,f)[Mf][R1,f / / ] - (ktd,f + ktc,f)[R/f ][R1,f ]-kft,f[Tf][R1,f / / + kfp,f)QI,f[R1,f ] (kp,f / ] -kz,f[Zf][R1,f / 1 d(V[Rr,f])

V

dt

(9)

Molecular-weight development during the pre-gelation period is described using the pseudo-kinetic rate constants method and the method of moments. The pseudo-

/ ] + kfp,fr[Pr,f][R/f ] + ) kp,f[Mf][R1-f r-1

/ s[Rr-s,f ][Ps,f] ∑ s)1

/ kp,f

/ ] (ktd,f + ktc,f)[R/f ][Rr,f / / / + kfp,f)Q1,f[Rr,f ] - kz,f[Zf][R1,f ] -(kp,f

(11)

Application of the method of moments to eqs 10 and 11, neglecting polymer radical consumption by inhibition, results in eq 12. If the steady-state hypothesis (SSH) on the rate of change of polymer radicals is used, then eq 12 can be replaced by eq 13. By expressing the rates of transfer to different molecules and termination by disproportionation, termination by combination, cross linking, and transfer to polymer relative to the rate of polymerization (i.e., using the terms defined in eqs 14 to 17), then eq 13 becomes eq 18. It should be noted that in going from eq 13 to eq 18, upon application of the SSH, the rate of initiation has been assumed to be equal to the rate of radical termination so that the initiation term in eq 18 is given in terms of βf, the ratio of the bimolecular radical termination rate to the polymerization rate.

1 d(V[Yi,f]) ) RI,f + (kfm,f[Mf] + kft,f[Tf])Y0,f + V dt kfp,fQi+1,fY0,2

∑ (j )Qj+1,fYi - j,f + kp,f[Mf]∑j)0(j )Yj,f j)1 i

/ + kp,f

i-1

i

i

-(kfm,f[Mf] + kft,f[Tf] + (ktd,f + ktc,f)Y0,f + kfp,fQ1,f)Yi,f (12)

∑ ( j)1 i

RI,f + (kfmf[M]f + kftf[T]f)Y0,f + kfpfQi+1,fY0,f + kp/ f Yi,f )

(10)

/ / / -(kp,f + kfm,f)[Mf][Rr,f ] - kft,f[Tf][Rr,f ]-

The instantaneous amount of divinyl monomer bound to the polymer chains, namely, the instantaneous copolymer composition, is obtained as the ratio of the rate of addition of divinyl monomer units to the polymer chains to the total polymerization rate (addition of vinyl and divinyl monomer units to the polymer chains), resulting in eq 8. The accumulated copolymer composition (i.e., the one measured with analytical techniques such as NMR) is obtained from the overall mass balance on one of the monomers of the system. For instance, the total amount of divinyl monomer (monomer 2) bound to the polymer chains is the result of subtracting the concentration of divinyl monomer in both phases at time t to the initial total amount of divinyl monomer, which results in eq 9.

F2,j )

kinetic rate constants used in this model are defined in section 2.5 of this paper. Derivation of the moment equations requires us to start from the detailed mass balance equations for the different polymer populations, considering all possible molecular sizes. Equations 10 and 11 show the resulting mass balance equations for living polymer radical of sizes 1 and r, respectively, with r g 2.

)

∑( i-1

)

i i Q Y + kpf[M]f Y j j+1,f i - j,f j j j)0

kfmf[M]f + kftf[T]f + (ktdf + ktcf)Y0,f + kfpfQ1,fY1,f

(13)

2828


τf )

Rtd,f + Rfm,f + Rfs,f + Rz,f Rp,f

(14)

Rtc,f Rp,f

(15)

k/pfqi,f kp,f

(16)

βf ) Cp/i,f )

kfp,fqi,f Cpi,f ) kp,f Yi,f )

[

∑ ( j)1 i

τf + βf + Cpi+1,f +

(17)

)

i-1

∑

() ]

i / i C Y + Y j pj+1,f i - j,f j)0 j j,f

(τf + βf + Cp1,f) (18) The mass balance for dead polymer molecules of size r is given by eq 19. Application of the method of moments to eq 19 results in eq 20. Equations 21 and 22 relate the definitions of the moments of the living and dead polymer populations to their corresponding concentrations. Number- and weight-average molecular weights can be calculated from the moments of the distribution, as shown in eqs 23 and 24. The molecular weight of the repeating unit, Mav, is defined in terms of the molecular weights of the two monomers and the copolymer composition, as defined in eq 25.

d(VPr,f) V dt

) [kfmf[M]f + kftf[T]f + ktdf[R/]f + ∞

1 r-1 / s[Ps]f][R/r ]f + ktcf [R/s ]f[Rr-s ]f 2 s)1 s)1 (k/pf + kfpf)r[Pr]f[R/]f (19)

kfpf

d(VQi,f) V dt

∑

∑

) RI,f + [kfmf[M]f + kftf[T]f]Yo,f +

∑ (j )Yj,fYi - j,f + kp,f/ ∑j)0(j )Qj+1,fYi - j,f j)0 i

1

ktcf

2

i

i

∑ ( j)0 i-1

- k/pfQi+1,fY0,f + kpf[M]f

i

)

i Y j j,f

(20)

∞

Yi,j )

∑ri[R•r]j

(21)

r)1 ∞

Qi,j )

ri[Pr]j ∑ r)1

qi,j )

VQi,j V0M0

(22)

Mn )

Y1,1 + Y1,2 + Q1,1 + Q1,2 M Y0,1 + Y0,2 + Q0,1 + Q0,2 av

(23)

Mw )

Y2,1 + Y2,2 + Q2,1 + Q2,2 M Y1,1 + Y1,2 + Q1,1 + Q1,2 av

(24)

Mav ) F1MW1 + F2MW2

(25)

To calculate the fractions of polymer radical types used to calculate the copolymer composition, molecular-

weight development, and pseudo-kinetic rate constants, the set of simultaneous algebraic equations represented by eqs 26 to 28 should be solved. This set of equations comes from the individual mass balances for each type of polymer radical and the use of the SSH on each of them. Some of the terms appearing in eqs 26 to 28 are defined in eqs 29 to 31. • • • k12f′2jφ1,j - (k21f′1j + k/23 f′3j)φ2,j + k32f′2jφ3,j )0

(26)

• • • (k12f′2j + k/13 f′3j)φ1,j - k21f′2jφ2,j - k31f′1jφ3,j ) 0 (27) • • • φ1,j + φ2,j + φ3,j )1

f′1j )

f′2j )

f′3j )

f1j(1 - xj) 1 - xj + (F2j - Fjaj - Fcj)xj f2j(1 - xj) 1 - xj + (F2j - Fjaj - Fcj)xj (F2j - Fjaj - Fcj)xj 1 - xj + (F2j - Fjaj - Fcj)xj

(28) (29)

(30)

(31)

The model presented here is capable of representing the intramolecular (primary) as well as intermolecular (secondary) cyclization reactions present in the copolymerization with cross linking of vinyl/divinyl monomers. From all of the pendant double bonds present in the system, it is assumed that some of them are wasted through primary cyclization reactions. Therefore, it is assumed that the fraction of pendant double bonds lost by primary cyclization is proportional to the fraction of divinyl monomer units in the polymer chains, as shown in eq 32. The proportionality constant, kcp, remains constant in bulk copolymerization, but it may change when a solvent is used.33,34 Because secondary cyclization involves at least two cross linkages to form cycles between two different molecules, the fraction of secondary cross-linked units is assumed to be proportional to the average cross-link density, as shown in eq 33. The average cross-link density results from a balance between added and consumed pendant double bonds at any given time, as represented in eq 34.

Fcp,j ) kcp,jF2,j

(32)

Fcs,j ) kcs,jFa,j

(33)

/ d[xjFa,j] kp,j [F2,j(1 - kcp,j) - Fa,j(1 + kcs,j)]xj dxj ) dt dt kp,j(1 - xj)

(34)

2.5. Pseudo-Kinetic Rate Constants and Diffusion-Controlled Effects. As explained in section 2.3 of this paper, the treatment of copolymerization can be simplified tremendously if the pseudo-kinetic rate constants method is used. All of the kinetic rate constants used in section 2.4 are indeed pseudo-kinetic rate constants. Their relationship with the individual kinetic constants shown in Table 1 is given by the appropriate definitions. Equations 35 to 42 define the kinetic rate constants for propagation, cross-linking, inhibition, transfer to monomer, transfer to a small molecule (other than the monomer), transfer to polymer, termination


by disproportionation, and termination by combination, respectively. 3

3

/ kp,j )

2

• kab,jφa,j fb,j ∑ ∑ a)1b)1

kp,j )

kp/ ∑ i)1

i3,j

3

∑ ∑ kfm i)1b)1 3

kfT,j )

2

2

∑ ∑ kfp i)1b)1

(38)

• φi,j fb,j

(39)

(40)

• • φi,j φb,j

(41)

• • φi,j φb,j

(42)

ib,j

3

∑ ∑ ktc i)1b)1

• φi,j Fb,j

ib,j

3

∑ ∑ ktd i)1b)1 3

ktc,j )

• φi,j fb,j

ib,j

3

ktd,j )

(37)

ib,j

∑ ∑ kfT i)1b)1 3

kfp,j )

• kzi,jφi,j ∑ i)1 2

ib,j

{[ ( {[ (

)]} )]}

1 1 Vf1 Vf0

0 exp - A2 k h tcnij,2 ) ktc nij

1 1 Vf2 Vf0

ktcwij,1 )

0 ktc wij

[ ] {[ ( Pn1

Pw1

x/2

x/2

exp - A2

1 1 Vf2 Vf0

)]}

+ ktcrd2 (46)

k h tcrd,1 ) c0rdkp1(1 - x1)

(47)

k h tcrd,2 ) c0rdkp2(1 - x2)

(48)

Propagation involves a reaction between a large and a small molecule, and a single average of the corresponding kinetic rate constant is adequate in this case for each phase, as shown in eqs 49 and 50. The initiator efficiency is allowed to change with the free-volume dependence, as shown in eq 51. Subscript j in eq 51 means that the efficiency of the initiator can be different for each phase. The fractional free volume for the continuous and dispersed phases is calculated using eqs 52 and 53, respectively.

[( [( [ (

)] )] )]

k h pij,1 ) k0pij exp -

1 1 Vf1 Vfcr

(49)

0 k h pij,2 ) kpij exp -

1 1 Vf2 Vfcr

(50)

fj ) f0 exp -Dj

1 1 Vfj Vf0

+ ktcrd1 (43)

1 1 exp - A1 Vf1 Vf0

+ ktcr2 (44)

)]}

+ ktcrd1 (45)

Vf1 )

∑i [0.025 + Ri(T - Tg )] V i

∑i [0.025 + Ri(T - Tg )] V

(52)

1

Vi,2

n

Vf2 )

(51)

Vi,1

n

In this paper, diffusion-controlled reactions are modeled using equilibrium free-volume theory. The chain-length dependence of the kinetic rate constants is important when the reaction involves two large macromolecules, such as termination (whose kinetic rate constant is kt(n, m), where n and m are the chain lengths of the reacting molecules). One simple way to account for the dependence of kt(n, m) on chain length is to use averages of kt(n, m).35,36 Bimolecular radical termination is modeled using number and weight averages of kt(n, m), which could be different for each phase, as shown in eqs 43 to 46. The first term in these equations accounts for translational termination, and the other term in the equations refers to “reaction diffusion” or “residual” termination. Residual termination is calculated using eqs 47 and 48 for each phase. The number average of kt(n, m) is used to calculate the polymerization rate and the numberaverage chain length (to calculate all zeroth- and firstorder moments). The weight average is used to calculate the weight-average chain length (second-order moments of the radical, dormant, and dead polymer species). 0 exp - A1 k h tcnij,1 ) ktc nij

Pw2

(36)

3

kfm,j )

[ ] {[ ( Pn2

(35)

φ•i,j(F2 - Fa - Fjc)j

kz,j )

0 ktcwij,2 ) ktc wij

i

(53)

2

2.6. Detailed Model Equations for the PostGelation Period. Modeling of the post-gelation period in a cross-linking vinyl/divinyl copolymerization is a challenging problem. Tobita and Hamielec generalized Flory’s theory for the calculation of the sol fraction.33,34 However, the expressions that they obtained were quite complex, and the information obtained with them was not much better than that obtained with Flory’s original model. Therefore, in this paper a simplified approach for the post-gelation period is used. It is assumed that gelation takes place only in the dispersed phase because the polymer produced in the continuous phase can reach only a critical size, before phase separating, to be part of the dispersed phase. Therefore, the following equations apply only to the dispersed phase. Equation 54 corresponds to Flory’s original equation for the calculation of the sol fraction in vinyl/divinyl copolymerization. If eq 54 is solved analytically, assuming that τ and β are chain length-independent, then eq 55 is obtained.30 Equation 56 indicates the relationship between the sol and gel fractions. It should be noted that eq 55 is implicit in Wg and therefore has to be solved numerically at each integration step of the ODEs that model the polymerization kinetics.

∫1∞(1 - FWg)r[rτ +

Ws(x) ) (τ + β)

]

β (τ + β)r2 e-(τ+β)r dr (54) 2

2830


Ws(x) )

(τ + β)(1 - FWg)e-(τ+β)

{

(

β(τ + β) ln(1 [ln(1 - FWg) - (τ + β)]2 FWg) - (τ + β - 1) + [ln(1 - Wg) - (τ + β)]τ β (55) [ln(1 - FWg) - (τ + β)]2 τ + (τ + β) 2

]})

[

Wg ) Wg(x) ) 1 - Ws(x)

(56)

To calculate the number- and weight-average chain lengths of the sol fraction during the post-gelation period, eqs 57 and 58 are used.30 Equations 59 to 65 provide definitions of the different terms needed to evaluate eqs 57 and 58.

P h sol n (x) )

P h sol w (x) )

P h sol n (x) )

P h sol wp(x) )

2P h sol np (x)

(57)

2 - F(x)P h sol np (x) P h sol wp(x)

(58)

1 - F(x)P h sol w (x)

W h s(x) β HG1 τ + HG1 2

[

]

HU (LG2 + HVG1G3) W h s(x) H)L+V

(59)

(60) (61)

L)

τ τ + β + Fj(x) W h g(x)

(62)

V)

β τ + β + Fj(x) W h g(x)

(63)

U)

G1 τ + β + Fj(x) W h g(x)

h g(x) Gi ) i - Fj(x) W

i ) 1, 2, 3

(64) (65)

2.7. Partition of Components between the Continuous and Dispersed Phases. Calculation of the partition of components among the phases in a dispersion polymerization process is basically a phase equilibria problem. It can be addressed from the point of view of a rigorous equilibrium problem, namely, equating the chemical potentials of both phases for each component and solving for the composition of the mixture or using a semiempirical approach. Our system consists of two phases: a carbon dioxiderich one (continuous) and a polymer-rich one (dispersed). The continuous phase contains CO2, vinyl monomer (M1), divinyl monomer (M2), and initiator (I). The dispersed phase contains polymer, M1, M2, I, and CO2. Strictly speaking, we have a fluid/liquid multicomponent, phase equilibria problem. Activities of the components in the fluid (CO2-rich) phase can be calculated using conventional theory for nonideal solutions of small molecules. For the dispersed phase, a first approach would be to extend the Flory-Huggins equation for the activity of large molecules in dilute solutions (a binary system) to a multicomponent nondilute situation. Another approach would be to use a more rigorous equation

of state, such as the Sanchez-Lacombe (SL) one, as Chatzidoukas et al.18 and Mueller et al.19 did for the MMA homopolymerization case. Strictly speaking, our system contains five components: CO2, M1, M2, initiator, and polymer. Although the system studied by Chatzidoukas et al.18 consisted of four components (only one monomer), they modeled it with the SL equation of state assuming that only three components were present (presumably initiator and monomer are considered to be a single component). Mueller et al.19 also used the SL equation of state, assuming that phase 1 contained CO2 and monomer and phase 3 contained CO2, monomer, and polymer. They used a simple interphase partition coefficient for the initiator.19 Implementation of the SL equation of state for three components, although not straightforward, is not too complicated. However, when more components are considered, the model gets more complicated. Given the complexity of a vinyl/divinyl copolymerization with cross linking, in this paper it was decided not to use the SL equation of state. As a matter of fact, the same first approach used by the Kiparissides group28 prior to their publication with the SL equation of state was followed here, and it is described below. 2.7.1. Overall Monomer Partitioning. This subsection is based on the approach proposed by Kiparissides et al.28 for dispersion homopolymerization on MMA in scCO2. However, the MMA concentration is replaced by the total monomer concentration (concentration of vinyl plus divinyl monomers) in each phase. Additional assumptions for the partitioning of initiator and the fraction of divinyl monomer from that overall concentration are made in this paper, as explained in subsection 2.7.2 below. Stage 1 (0 < x < xs). As described in section 2.2, this stage proceeds as a solution polymerization process, and it goes from the start of the polymerization until a limiting conversion xs, which is associated with the polymer solubility. This stage ends with the appearance of the second phase. The amount of total monomer (vinyl monomer plus divinyl monomer) in the continuous phase, during stage 1, is given by eq 66. Because there is no dispersed phase in stage 1, the amount of monomer in such phase is zero, as indicated in eq 67.

Gm1 ) Gm0(1 - x)

(66)

Gm2 ) 0

(67)

Stage 2 (xs < x < xc): Two-Phase Polymerization. This stage lasts from the appearance of the second phase until the monomer in the continuous phase is completely consumed. The total amount of monomer in the continuous and dispersed phases, during stage 2, is given by the corresponding material balances, shown in eqs 68 and 69, respectively.

(

Gm1 ) Gm0 1 - x -

)

x x xc(1 + K) 1 + K

( )

1 -1 xC Gm2 ) Gm0[x] 1+K

(68)

(69)

K in eqs 68 and 69 is defined as the ratio of the mass of CO2 to the mass of monomer in the dispersed phase.28


It should be noticed that if K , 1, which is reasonable, and x ) xc, then eqs 68 and 69 are transformed into eqs 70 and 71. Stage 3 (xc < x < 1). The polymerization proceeds only in the dispersed phase. The amount of total polymer in the continuous and dispersed phases, during stage 3, is now given by eqs 70 and 71.

Gm1 ) 0

(70)

Gm2 ) Gm0(1 - x)

(71)

Although xs and xc can in principle be calculated from mass balance considerations,28 in this paper they were considered to be free (adjustable) model parameters. To understand why this is convenient, it is useful to refer to a recent study by Mueller et al.19 on the reaction locus in the dispersion polymerization of MMA in scCO2. They considered two models representing two opposite extreme conditions, in terms of the interphase transport of active polymer chains. In what they called “the radical segregation model (RS)”, the active chains spend their whole life in the same phase where they have been initiated, which implies that the characteristic time for radical termination is much shorter than that for interphase mass transport.19 According to Mueller et al.,19 the model of Chatzidoukas et al.18 falls into this category. In “the radical partitioning model (RP)”, the rate of mass transport between polymer particles and the continuous phase is assumed to be infinitely fast for all species so that thermodynamic equilibrium is established at every time during the reaction. (An example of this situation is the desorption of radicals by chain transfer to monomer in water emulsion polymerization.) This situation corresponds to the case where the characteristic time for radical termination is large compared to that for interphase transport.19 After an evaluation of the two limiting models (immersed in a comprehensive mathematical model for two-phase MMA homopolymerization using the SL equation of state for the calculation of monomer partitioning), it was concluded that the experimental molecular weight was largely underestimated by the RS model and not fully satisfactorily determined by the RP model. This result means that the most comprehensive available models for the calculation of species partitioning in dispersion polymerization in scCO2 are not good enough for the reliable calculation of molecular-weight development in such a system. The conclusions obtained by Mueller et al.19 point to the fact that an intermediate model between RS and RP might be needed to obtain more meaningful and reliable predictions of the system performance. The equations used in this paper to calculate the partition of the components between the two phases correspond to the RS model. By allowing xs to be a “free parameter”, although certainly estimated from solubility experimental data, the fact that the radicals are not fully segregated is being considered in a rough yet practical way. However, parameter sensitivity analyses carried out in this study (section 3.2.3 of this paper) showed that the system is not very sensitive to the value of xs. Once again, what was wanted in this study was to have a good enough model for species partitioning yet not too complicated, given the fact that the kinetic system studied in this paper (nonlinear copolymerization with cross linking) is quite a bit more complex that the system analyzed by Chatzidoukas et al.18 and Mueller et al.19 (linear homopolymerization).

2.7.2. Initiator and Divinyl Monomer Partitioning. It was assumed that the ratio of initiator concentration to total monomer concentration is the same for both phases at any given time, which leads to eqs 72 and 73. In the case of the divinyl monomer, it was assumed that it is completely soluble with the vinyl monomer and, therefore, its mole fraction (with reference to total monomer and not total moles of all of the components in the system) is the same in both phases, as indicated in eq 74.

[I]1 )

[I] [M]2 +1 [M]1

(72)

[I]2 ) [I] - [I]1

(73)

f2,1 ) f2,2

(74)

3. Results and Discussion 3.1. Some Remarks on the Model Implementation, Solution, and Reference System. The model presented in this paper consists of 15 ordinary differential equations (ODEs), 1 for the overall initiator consumption, 1 for overall conversion, 2 for conversion in each phase, 6 for moments of the dead polymer, 3 in each phase, 1 for the divinyl monomer mole fraction, 2 for the cross-link density, 1 for each phase, 1 for the overall transfer agent, and 1 for overall inhibitor, 6 nonlinear algebraic equations for the different radical types, 3 in each phase, and a number of explicit algebraic equations, such as the ones for the moments of the living polymer population. The model was implemented in Fortran language. Subroutine LSODE was used to integrate the ODEs. The Adams Moulton method, with a very short time step, was chosen in LSODE because the use of the GEAR method caused numerical problems during the start up of the integration. A standard subroutine for the solution of systems of nonlinear algebraic equations was used to calculate the fractions of polymer radicals, for each phase, in every time step of the ODE solver. A conventional Newton-Raphson algorithm was used to calculate the sol fraction, in every step of the ODE solver, during the post-gelation period. The criterion to define the gelation point was the time when the weight-average chain length reached a value of 1010, which was used as the maximum chain length such that if a higher value was calculated by the model it was set to this limit. This limit on the weight-average chain length, which is determined by the value of the second moment of the dead polymer population, was implemented to avoid divergence of the ODE system. Once the gelation point was reached, the number- and weight-average chain lengths were calculated using eqs 57 and 58 instead of using the moments of the polymer distributions. However, calculation of the zeroth moments was still needed for mass calculations. The reference system used in our calculations was the polymerization of commercial DVB (DVB55 or DVB80 with 55 and 80 mol % DVB, respectively), namely, the copolymerization of ethyl vinyl benzene (EVB) and DVB, under the same conditions studied by Cooper et al.23,24 (i.e., T ) 65 °C, P ) 310 bar, [AIBN]0 ) 0.0974 mol L-1 (0.16 g), [M]0 ) 1.6025 mol L-1 (2 g), and the amount of CO2 needed to fill the 10 mL reactor used by them). For

2832


Table 2. Kinetic Parameters for the Free Radical Copolymerization with Cross Linking of EVB/DVB in scCO2 parameter, units

value

Vivaldo-Lima et al.30

f0, dimensionless

0.7

kd, s-1

kd ) 1.053 × 1015 exp -

k11, L mol-1 s-1

kp ) 107.630 exp

k22, L mol-1 s-1

( ) k22 k11

) 0.95,

m

0.18k11

Quintero-Ortega,38 corrected by pressure

30 660 RT

(

kft, L mol-1 s-1

reference

)

(-7770 RT )


k22 k11


( )

) 1.9

p


12 671 RT

kfm, L mol-1 s-1

kfm ) 2.31 × 106 exp -

kfp, L mol-1 s-1

0.0

ktc, L mol-1 s-1

ktc ) 1.223 × 108 exp -3586.81

ktd, L mol-1 s-1

kd ) 2.19 × 105 exp -


kft1, kft2, L mol-1 s-1 kft3, L mol-1 s-1

0.0133k11 2k11

Vivaldo-Lima et al.30 Vivaldo-Lima et al.30

(

Vivaldo-Lima et al.30 Vivaldo-Lima et al.30

[

1 (T1 - 333.15 )]

[ (13 T810)]

( ( ( (

(r1)m, dimensionless

(r2)m, dimensionless

(r1)p, dimensionless

(r2)p, dimensionless

(r1)mix, (r2)mix, dimensionless k31, L mol-1 s-1

) ) ) )

k11 k12

m

k22 k12

m

k11 k12

p

k22 k21

p


) 1.0


) 0.13


) 2.0


(ri)mix )

[np(ri)p + nm(ri)m] np + n m


( )

k32 ) r2 k31

)2

m

k*3i, L mol-1 s-1

/ (k3i )mix )

kcp, dimensionless kcs, dimensionless

0.25 0.0

/ / k3i k3i , k11 p k11

) 0.626 +

p

/ / [np(k3i )p + nm(k3i ) m] np + n m

x ) p, t

1.245 × 10-3 f20


Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30

modeling purposes, EVB was considered to have the same reactivity as styrene; therefore, the kinetic rate constants for styrene were used whenever the kinetic rate constants of EVB were needed. The kinetic, freevolume, and physical parameters used in the calculations presented in this paper are summarized in Tables 2 and 3. It is important to point out that the kinetic rate constants for propagation and termination listed in Table 2 were corrected by pressure using eq 74,40,41 with the activation volumes reported by Beuermann et al.40,41 The reference pressure, P0, in eq 74 is 1 bar, and kx0 is the kinetic rate constant at T and P ) 1 bar, with x ) p standing for propagation and x ) t standing for termination.

∆V * (P - P0) RT



( ) ( )( ) / k3i k11

k*3i, L mol-1 s-1


) 0.4

(k31)m ) 0.0067k11, (k31)p ) 0.109k11

k32, L mol-1 s-1

ln kx ) ln kx0 +

)

(75)

Besides correcting by pressure the values of the propagation and termination kinetic rate constants, using eq 74, the effect of pressure on the system was also taken into account by calculating the density of carbon dioxide at any given temperature and pressure with program CO2PAC, which uses Wagner’s equation of state.42 3.2. Parameter Sensitivity Analyses. 3.2.1. Kinetic Rate Constants. Once the model was implemented, parameter sensitivity analyses were carried out around the reference case described in section 3.1 with several objectives in mind. The first objective was to test the model implementation (i.e., to verify that the expected trends were adequately predicted by the model). The second was to determine which kinetic rate constants have more pronounced effects on the rate of polymerization, molecular-weight development, appearance of the gelation point, consumption of the sol fraction, evolution of the cross-link density, and evolu-

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2833 Table 3. Free Volume and Physical Parameters for the Free-Radical Copolymerization with Cross Linking of EVB/DVB in scCO2 parameter

value

reference

Vfcr, dimensionless Tgm1, °C Tgm2, °C Tgp, °C Tgsol, °C Rm1, °C-1 Rm2, °C-1 Rpol, °C-1 Rsol, °C-1 crd, L mol-1 D1 ) D2, dimensionless A1, dimensionless A2, dimensionless xs, dimensionless

0.036 -88.1 -90.0 93.5 -88.1 0.001 0.0008 0.00048 0.001 135 0.001 0.0 0.465 0.043

xc, dimensionless

0.4

K, ratio of mass of solvent to mass of monomer in the dispersed phase

0.0088

Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Vivaldo-Lima et al.30 Quintero-Ortega38 Quintero-Ortega38 Quintero-Ortega,38 estimated for homopolymerization Quintero-Ortega,38 estimated for homopolymerization Nilsson et al.39 It is assumed here that the solubility of CO2 in EVB/DVB is roughly the same as the solubility of vinyl chloride (VCM) in water and is independent of temperature and pressure.

tion of the copolymer composition. Finally, these analyses were intended to help us better understand the behavior of systems such as the one studied here. In all cases presented in this paper, the overall properties (i.e., properties of polymer molecules obtained in the continuous phase (mainly oligomers) and polymer molecules obtained in the dispersed phase (high-molecular-weight polymer)) are shown in the following figures. That does not mean that the model calculates only the overall properties because the degree of detail in the model allows us to monitor the behavior and properties of polymer molecules and concentrations of small-molecular-weight reactants in both phases, as

detailed in section 2 of this paper. If one were to measure experimentally the properties of the polymer produced by dispersion copolymerization in scCO2, then pressure would be released and the powder polymer would be retained so that one would not differentiate from polymer or remaining reactants coming from the continuous or the dispersed phases. That is why the overall properties are being shown here. Figure 2 shows the effect of the propagation kinetic rate constant of the vinyl monomer, kp11 (also shown as k11), on (a) the polymerization rate, (b) the molecularweight development of the sol fraction (the whole system during the pre-gelation period), (c) the consumption of the sol fraction (evolution or growth of the gel fraction), and (d) the cross-link density. It should be emphasized that all of the kinetic rate constants that are defined in terms of k11 in Table 2 are modified in the calculations shown in Figure 2, except the kinetic rate constant for propagation through pendant double bonds (cross linking). It is observed that changing k11 from 300 to 500 L mol-1 s-1 did not have any effect on the polymerization rate (Figure 2a), molecular-weight development (Figure 2b), consumption of the sol fraction (Figure 2c), or evolution of the average cross-link density (Figure 2d). However, if k11 was further increased to a value of 700 L mol-1 s-1, then the expected trends were obtained (i.e., an increase in k11 causes a faster polymerization rate and an anticipation of the gelation point, as observed in Figure 2a-c). It is interesting to observe that although a faster polymerization rate causes an anticipation of the gelation point, which was expected, it also causes a limiting value of the gel fraction that is reached, as shown in Figure 2c. This effect was observed experimentally by Vivaldo-Lima et al.31 The effects of the termination kinetic rate constant on the same responses analyzed before are shown in

Figure 2. Effect of the kinetic rate constant for the propagation of vinyl monomers, k11, on (a) the polymerization rate, (b) the numberand weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

2834


Figure 3. Effect of the kinetic rate constant for bimolecular termination, kt, on (a) the polymerization rate, (b) the number- and weightaverage chain lengths, (c) the gel fraction, and (d) the cross-link density.

Figure 3. The expected behavior was predicted adequately with the model. The higher the value of kt, the slower the polymerization proceeds (Figure 3a), the later the gelation point appears (Figure 3b and c), and the lower the value of the average cross-link density, as shown in Figure 3d. This makes sense because the bimolecular termination of free radicals stops the growth of the polymer molecules, consuming radicals that could be otherwise propagating, attacking pendant double bonds or creating more cross linkages by attacking pendant double bonds within the gel molecules or terminating free radicals trapped in the gel molecule. In the case of the kinetic rate constant for propagation through pendant double bonds (cross-linking reaction), kp*, it is observed in Figure 4a that the rate of polymerization is not significantly affected at low and intermediate conversions. This is so because the consumption of pendant double bonds by polymer free radicals does not consume new monomer units and, therefore, the total rate of polymerization is not being changed. At high conversions, some of the trapped monomer units may react with free radicals attached to the gel molecules, or some of the remaining primary radicals might attack pendant double bonds, thus somehow increasing the polymerization rate when this reaction is faster. As expected, when kp* is increased the gelation point occurs sooner, as shown in Figure 4b and c, because of the increased reactivity of the pendant double bonds, which are the main source of cross-linked polymer molecules. The increased reactivity of the pendant double bonds also promotes having more cross-linked units, therefore significantly increasing the average cross-link density as shown in Figure 4d. Primary cyclization reactions consume pendant double bonds. Because this reaction does not consume new monomer units, the polymerization rate should be

unaffected by this kinetic constant, which is confirmed in Figure 5a. By consuming pendant double bonds without increasing the size of the molecules, the primary cyclization will delay gelation, a situation confirmed in Figure 5b and c. However, under these reaction conditions, the effect of increasing kcp from 0 to 0.5 is not as pronounced as it is at ambient pressure.30 The consumption of pendant double bonds by primary cyclization causes a reduction in the number of cross-linked units, thus reducing the average cross-link density. This behavior is clearly observed in Figure 5d. Vivaldo-Lima et al.30 showed that adequate modeling of diffusion-controlled effects is very important in order to produce model simulations with predictive power in batch copolymerization with cross linking of vinyl/ divinyl monomers. The parameter in the model presented in this paper that accounts for diffusioncontrolled termination (the so-called autoacceleration effect) is the overlap parameter from the free-volume theory, A2 in eqs 44 and 46. The simulations shown in Figure 6 clearly show the importance of the autoacceleration effect in vinyl/divinyl dispersion copolymerization in scCO2. As expected, when the autoacceleration effect is more pronounced (when the value of A2 is higher), the polymerization rate is much faster, as shown in Figure 6a, and the gelation point occurs much sooner, as observed in Figure 6b and c. Once again, it is observed that a system with a high polymerization rate, although it reaches the gelation point sooner, does not fully consume its sol fraction, as shown in Figure 6c for the system with the highest autoacceleration effect. An important feature to notice, though, is that the cross-link density of the produced polymer network does not change very much when the autoacceleration effect is much stronger, as evidenced in Figure 6d.


Figure 4. Effect of the kinetic rate constant for propagation through pendant double bonds (cross linking), kp*, on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

Figure 5. Effect of the kinetic rate constant for primary cyclization, kcp, on (a) the polymerization rate, (b) the number- and weightaverage chain lengths, (c) the gel fraction, and (d) the cross-link density.

3.2.2. Physical, Formulation, and Process Parameters. As mentioned in section 2.7 of this paper, the partitioning of components between the continuous

and dispersed phases is calculated from partition coefficients and mass balances. A key parameter in this model regarding the partitioning of components is the

2836


Figure 6. Effect of the free-volume parameter, A2, for diffusion-controlled termination on (a) the polymerization rate, (b) the numberand weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

critical conversion, xs, which is related to the polymer solubility, and corresponds to the conversion value when the second phase appears. As observed by Cooper et al.,23,24 the appearance of the second phase in the polymerization of commercial DVB (copolymerization of EVB/DVB) in scCO2 occurs very fast, almost from the very beginning of the polymerization. That means that the critical conversion for the appearance of the second phase, xs, should be a very low value. Figure 7 shows the effect of changing xs on the behavior of the polymerization system. It is clearly observed that the system is fairly insensitive to the value chosen. Only when an unrealistically high value is used is an effect on the polymerization rate observed, as shown in Figure 7a. However, even with such a high value, the molecular-weight development, the gelation point, and the evolution of the gel fraction and the crosslink density profiles are practically unchanged, as observed in Figure 7b-d. This observation seems to justify the choice of the phase equilibrium model. Of course, using a more rigorous model, such as the Sańchez-Lacombe (SL) equation of state, and a more fundamental model for interphase transport of polymer radicals (i.e., a hybrid RS-RP19 model based on first principles) would have been preferred. Nonetheless, the important features of the vinyl/divinyl copolymerization situation are well represented with the simpler model. Figure 8 shows the effect of increasing the initial concentration of the initiator. As expected, the polymerization rate is increased (Figure 8a), lower molecular weights are obtained at low conversions, but MW increases much faster (Figure 8b), thus reaching the gelation point sooner, as observed in Figure 8b and c, but the cross-link density is unchanged. It should be noticed that adding too much initiator does not accelerate the reaction further (i.e., there exists an optimal

value of initiator initial concentration that maximizes the polymerization rate without adversely affecting the average cross-link density). However, it should be kept in mind that increasing the polymerization rate is tied to a high release of energy. If the heat removal capacity of the reactor is not good enough at the center of the reacting mass, then a temperature profile with a maximum at the center could cause an increase in pressure. Depending on the magnitude of the temperature increase, the corresponding pressure increase might reach unsafe operational conditions. This points to the need to improve the model by considering nonisothermal and nonisobaric operational conditions. Figure 9 shows the effect of the initial overall monomer concentration (amount of DVB added to the recipe). An interesting behavior of the conversion versus time profiles is observed in Figure 9a. At low conversions, the polymerization rate seems to be insensitive to the initial monomer concentration. As polymerization proceeds, the profile with the lowest monomer content proceeds a bit faster, but as times goes on, the case with the highest monomer content proceeds faster, followed by the intermediate value and the one with the least added monomer being the slowest at high conversions. This situation may have something to do with diffusioncontrolled (DC) effects. Having too many monomer molecules at the beginning, with the same number of primary radicals, might promote the production of shorter molecules (more in quantity but shorter in length). However, because there is cross linking going on coupled with DC effects, the situation reverses at some point. Figure 9b and c clearly shows that the gelation point is reached sooner when more monomer was added at the beginning. It is also interesting that the average cross-link density is fairly insensitive to the amount of initial monomer added to the system, as


Figure 7. Effect of the critical conversion for the appearance of the second phase, xs, on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

Figure 8. Effect of the initiator initial concentration, [I]0, on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

shown in Figure 9d. These results might point to the idea of adding more monomer to increase productivity. However, once again, one should proceed with caution because higher polymerization rates might be associated

with temperature and pressure increases that could reach unsafe operational conditions, as pointed out before in the analysis of the initiator initial concentration effect.

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Figure 9. Effect of the overall monomer initial concentration, [M]0, on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

The effect of the initial concentration of cross linker is shown in Figure 10. As expected, the higher the amount of cross-linker (DVB) in the system, the faster the polymerization will proceed (Figure 10a) and the sooner the gelation point will be reached (Figure 10b and c). It is also observed in Figure 10c that if the amount of added cross linker is low, although a polymer network is certainly produced, there might be significant sol content remaining in the system. Figure 10d shows the effect of DVB content on copolymer composition. It is clearly observed that if too much cross linker is present in the system the DVB content in the produced molecules in the early stages of polymerization will be dominant (i.e., the fraction of DVB units is close to 1). This situation will go on until the DVB monomer is consumed, and then the EVB monomer is consumed, mostly at the end. In the previous analyses, the copolymer composition profiles were not shown because the behavior was basically the same. Given the high amount of DVB present in the system (the commercial DVB80 was used as the reference formulation for most of the simulations presented in this paper), all of the profiles looked very similar to the one with the highest DVB content in Figure 10d. Figure 10e shows that the initial amount of DVB added to the system has a pronounced effect on the average cross-link density, which makes perfect sense because the source of cross linkages are the pendant double bonds coming from the divinyl monomer. Figure 10b shows predictions of the molecular-weight development. As observed in Figure 10c, the gelation point occurs sooner when more divinyl monomer is added at the beginning, but the anticipation of the gelation point is not directly proportional to the initial mole fraction of DVB because the effect is quite

notorious when going from 0.1 to 0.3 in DVB mole fraction but almost negligible when going from 0.3 to 0.6. One of the most interesting results obtained in this paper is related to the effect of pressure on the behavior of the polymerization and the properties of the produced polymer, shown in Figure 11. It is clearly observed that increasing pressure in the region well above the critical point of CO2 (300-500 bar) causes a modest increase in the polymerization rate. However, if pressure is increased in the region closer to the critical point (75200 bar), then the increase in the polymerization rate is remarkable (Figure 11a). Of course, this effect is also reflected in the appearance of the gelation point, as shown in Figure 11b and c. The gelation point is modestly anticipated when pressure is increased in the high-pressure range, but it can be easily tuned within a range of almost 2 h in the pressure range closer to the critical point of CO2. The amount of gel produced is almost 100% in most of the profiles analyzed (only in the profile at 75 bar is a limiting value of the gel fraction of approximately 95% observed). It is observed in Figure 11d that although the gelation point can be moved within a time period of 2 h (when the total reaction time is not higher that 6 h in the slowest system) the crosslink density does not change very much. Even though the average cross-link density did not change very much, the cross-linking density distribution (not calculated with our model) (i.e., the homogeneity of the polymer network) might change significantly because Cooper et al.25,26 and Hebb et al.27 experimentally found that the pore size and surface area of porous crosslinked poly(methacrylate) monoliths synthesized in scCO2 could be fine tuned by changing the density of carbon dioxide, namely, by changing the pressure of the


Figure 10. Effect of the cross-linker (DVB) initial mole fraction (with respect to total monomer content), f20, on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction (d) the copolymer composition (accumulated F2), and (e) the cross-link density.

polymerization system. The remarkable effect of pressure on the gelation point is also observed in the molecular-weight development profile (Figure 11b). This situation provides a promising way to change the production time significantly, moving to safer operational conditions, without significantly changing the properties of the produced polymer. Figure 12 shows the effect of temperature on the copolymerization of vinyl/divinyl monomers in scCO2. As expected, it is observed that increasing the temperature by 20 °C, from 60 to 80 °C, with P ) 310 bar, causes a significant but not impressive increase in the polymerization rate (Figure 12a), a modest anticipation (around 20 min) of the gelation point (Figure 12b and c), and a quite significant increase in the average crosslink density, which means producing a polymer network of significantly different quality. Even though the temperature effect is well captured through the Arrhenius dependence of the kinetic rate constants and the temperature dependence of the densities of the several ingredients present in the system

(especially the density of CO2), it should be emphasized here that the effect of temperature on the polymerization behavior and the polymer properties of the produced material could be underestimated. The reason this may be so is that the solubility of carbon dioxide in the mixture of monomers, K, was assumed to be independent of temperature (see last entry of Table 3). This is more a problem of not having enough sources of reliable data other than a model limitation. To study the effect of temperature on K qualitatively, we carried out simulations with values 1 order of magnitude smaller (K ) 0.001) and 1 order of magnitude higher (K ) 0.1) than the reference value (K ) 0.0088), using a value of T ) 60 °C to calculate the kinetic rate constants and the densities of the components of the reacting mixture. The results are shown in Figure 13. It is observed that there is basically no difference between the profiles of conversion versus time (Figure 13a), gel fraction versus time (Figure 13b), average cross linking versus conversion (Figure 13c), and chain-length development (Figure 13d) when the lower value of K is

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Figure 11. Effect of the operating pressure, P, on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

Figure 12. Effect of the operating temperature, T, on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

compared against the reference value. However, when the high value of K is used, the polymerization rate is significantly reduced (Figure 13a), the gelation point is delayed by approximately 50 min (Figure 13b and d), and the cross-linking density (Figure 13c) and the copolymer composition (not shown) are unaffected. These results confirm that although the effect of temperature on the polymerization rate and the gelation point is not as pronounced as the effect of pressure there might be some degree of underestimation in our calcula-

tions on the effect of temperature if the solubility of CO2 in the monomer is significantly higher than 0.01. 3.3. Comparison against Experimental Data. As evidenced from the literature review of section 1, there are very few studies of copolymerization with crosslinking of vinyl/divinyl monomers in scCO2. We found studies only from the group of Cooper.23,24 Although the studies of Cooper et al.23,24 on the copolymerization of commercial DVB (a mixture of EVB and DVB) provide valuable insight into this process, the kinetic data


Figure 13. Effect of the solubility of CO2 in the monomer, K, in the dispersed phase on (a) the polymerization rate, (b) the number- and weight-average chain lengths, (c) the gel fraction, and (d) the cross-link density.

sion from about 7 h, and the experimental data at about 8 h is a bit higher that 90% monomer conversion. This crude comparison gives an indication that the model predictions are not far from reality, an aspect that might be considered remarkable because no single parameter was fitted to match the data. 4. Concluding Remarks

Figure 14. Comparison of polymerization-rate model predictions against experimental data of Cooper et al.23,24 for EVB/DVB copolymerization at T ) 65 °C, P ) 310 bar, [AIBN]0 ) 0.0974 M, and [M]0 ) 1.6025 M in a 10-mL reactor.

reported by them is minimal in part because that was not the focus of their studies. They basically report the final yield after an imprecise number of hours of polymerization. Figure 14 shows one piece of such experimental data reported by Cooper et al.23,24 They reported the final yield after leaving the system to react overnight. Two data points linked by one line are used to represent the available data point because overnight could mean anything from 8 to about 10 h (or even more) depending on what time the system was left to polymerize the day before. The solid line is the predicted profile of conversion versus time under the same conditions reported by Cooper et al.23,24 and using kinetic rate constants and model parameters taken from the literature (Table 2). It is observed that our model predicts the total conver-

A detailed kinetic model for the dispersion copolymerization with cross-linking of vinyl/divinyl monomers in carbon dioxide under supercritical conditions was developed and applied to the copolymerization of EVB/ DVB (commercial DVB80). Parameter sensitivity analyses on the effects of the different kinetic rate constants on the polymerization rate, molecular-weight development, copolymer composition, appearance of the gelation point, and evolution of the cross-link density were carried out. The expected trends were reproduced with the model, although the magnitude of these effects may vary with respect to a system operated at ambient pressure and operating in batch or suspension polymerization. The effects of the recipe (cross-linker content, overall monomer content, and initiator concentration) and the operation conditions (temperature and pressure) on the studied responses mentioned above were carefully analyzed. It was found that the effect of pressure is particularly important. If the system is operated under the same conditions of the experimental studies reported in the literature (well above the critical point of CO2), then the changes that can be obtained in productivity and polymer properties by changing the process conditions and formulation are modest. However, if one moves to lower pressures, closer to the critical point of

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CO2, then the model predicts quite significant changes in the polymerization rate and appearance of the gelation point without severely changing the average cross-link density of the produced polymer network. It is important to carry out more systematic experimental studies on the copolymerization kinetics of vinyl/ divinyl monomers in scCO2, and it would be interesting to explore the polymerization experimentally at pressures closer to the critical point of CO2. Although the model presented here is sound and provides detailed and possibly reliable predictions of the polymerization kinetics, molecular-weight development, copolymer composition, and cross-link density, it is convenient to extend the model to nonisothermal and nonisobaric operational conditions to model more realistic processes. Using a more rigorous equation of state to calculate the partitioning of components among the phases seems convenient but is not crucial to obtaining more reliable predictions. The issues concerning the interphase mass transport of active polymer chains raised by Mueller et al.19 should be incorporated in more refined models of dispersion homo- and copolymerization in scCO2. Our group is already addressing some of these issues, and results will be communicated soon. Acknowledgment Financial support from the National Council for Science and Technology of Mexico (CONACyT) (Project IAMC U40259-Y) and from DGAPA-UNAM (Projects PAPIIT IN100702 and IX115404) is gratefully acknowledged. I.A.Q.-O. acknowledges the M. Eng. scholarships from CONACyT and DGAPA-UNAM and her present scholarships for Ph.D. studies from CONACyT and DGEP-UNAM. G.L.-B. acknowledges support from NSFCONACyT 39377-U and CONACyT 42728-Y. Nomenclature A1 ) effectiveness factor to account for overlap of free volume and separation of reactive radicals in phase 1 A2 ) effectiveness factor to account for overlap of free volume and separation of reactive radicals in phase 2 B ) effectiveness factor to account for overlap of free volume and separation of monomer/polymer radicals, assumed to be the same for both phases Cm1 ) concentration of monomer 1, mol L-1 Cm2 ) concentration of monomer 2, mol L-1 Cm0 ) initial overall monomer concentration, mol L-1 / Cpi,f ) defined in eq 16 Cpi,f ) defined in eq 17 Crd ) proportionality factor for reaction-diffusion termination constant, L mol-1 Dj ) effectiveness factor to account for overlap of free volume and separation of fragment radical molecules in phase j fj ) initiator efficiency in phase j f0 ) initial initiator efficiency f1,j ) relative vinyl monomer concentration (mole fraction) in phase j f2,j ) relative divinyl monomer concentration (mole fraction) in phase j f2 ) overall relative divinyl monomer concentration (mole fraction) F2,j ) instantaneous relative composition of monomer 2 in polymer in phase j F2,j ) accumulated copolymer composition (molar relative content of DVB in copolymer) in phase j f20 ) initial divinyl monomer concentration

Gm0 ) initial mass of overall monomer, g Gm1 ) mass of monomer 1, g Gm2 ) mass of monomer 2, g [I] ) initiator concentration, mol L-1 [I]1 ) initiator concentration in phase 1, mol L-1 [I]2 ) initiator concentration in phase 2, mol L-1 K ) solubility constant of solvent (carbon dioxide) in monomer mixture kcp ) proportionality constant between primary cyclization density and mole fraction of divinyl monomer bound in the polymer chains kcs ) proportionality constant between the average number of secondary cycles per cross link and the fraction of free pendant double bonds in the primary polymer molecule kd,j ) initiator decomposition kinetic rate constant in phase j, s-1 kfm,j ) pseudo-kinetic rate constant for chain transfer to monomer in phase j, L mol-1 s-1 kfp,j ) pseudo-kinetic rate constant for chain transfer to polymer in phase j, L mol-1 s-1 kfT,j ) pseudo-kinetic rate constant for chain transfer to a small molecule, L mol-1 s-1 kp,j ) pseudo-kinetic rate constant for propagation, L mol-1 s-1 / kp,j ) pseudo-kinetic rate constant for propagation through pendant double bonds, L mol-1 s-1 k h pij,m ) effective propagation kinetic rate constant between radical type i with monomer j in phase m, L mol-1 s-1 0 kpij ) intrinsic chemical kinetic rate constant for propagation, L mol-1 s-1 ktc,j ) pseudo-kinetic rate constant for termination by combination in phase j, L mol-1 s-1 ktcnij,m ) effective number average termination by combination kinetic rate constant between radical types i and j in phase m, L mol-1 s-1 k h tcwij,m ) effective weight average termination by combination kinetic rate constant between radical types i and j in phase m, L mol-1 s-1 0 ktcnij ) intrinsic chemical kinetic rate constant for number average termination by combination between radicals i and j, L mol-1 s-1 0 ktcwij ) intrinsic chemical kinetic rate constant for weight average termination by combination between radicals i and j, L mol-1 s-1 k h trcd,j ) reaction diffusion termination kinetic constant in phase j, L mol-1 s-1 ktd,j ) pseudo-kinetic rate constant for termination by disproportionation in phase j, L mol-1 s-1 kz,j ) pseudo-kinetic rate constant for inhibition in phase j, L mol-1 s-1 M ) total monomer [M]0 ) initial monomer concentration, mol L-1 M1,j ) monomer 1 in phase j M2,j ) monomer 2 in phase j M/1 ) monomeric radicals of type 1 M/2 ) monomeric radicals of type 2 [M]f ) total monomer concentration in phase f, mol L-1 Mav ) molecular weight of the repeating unit in the copolymer, g mol-1 MWi ) molecular weight of monomer i, g mol-1 PDI ) polydispersity index Pm,n,j ) polymer molecule with m units of monomer 1 and n units of monomer 2 in phase j; when a superscript asterisk is used in Table 1, it indicates that the attack to a dead polymer molecule by a polymer radical takes place at a pendant double bond Pn,j ) number-average chain length in phase j Pw,j ) weight-average chain length in phase j P h sol n (x) ) number-average chain length in the sol phase

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2843 Psol np (x) ) accumulated number average chain length for the primary polymer molecule in the sol phase Pr ) dead polymer molecules with chain length r [Pr,f] ) concentration of polymer with chain length r in phase f, mol L-1 Psol w (x) ) weight-average chain length in the sol phase Psol wp(x) ) accumulated weight-average chain length for the primary polymer molecule in the sol phase qi,j ) normalized i moment in phase j Qi,j ) moment i of the dead polymer distribution in phase j, mol L-1 R ) universal gas constant, cal mol-1 K-1 r1 ) reactivity ratio r2 ) reactivity ratio / [R1,f ] ) concentration of polymer radicals of size 1 in phase f, mol L-1 [R*]f ) total concentration of polymer radicals in phase f, mol L-1 Rfm,f ) reaction rate for chain transfer to monomer in phase f, mol L-1 s-1 Rfp,f ) reaction rate for chain transfer to polymer in phase f, mol L-1 s-1 Rft,f ) reaction rate for chain transfer to small molecules in phase f, mol L-1 s-1 / Rin ) primary radicals from initiator decomposition / Rm,n,i,j ) polymer radicals with m units of monomer 1 and n units of monomer 2, with the active center located on monomer unit i and in phase j Rp,f ) polymerization rate in phase f, mol L1 s-1 R/r ) polymer radical of size r Rtc,f ) reaction rate for termination by combination in phase f, mol L-1 s-1 Rtd,f ) reaction rate for termination by disproportionation in phase f, mol L-1 s-1 Rz,f ) reaction rate for inhibition in phase f, mol L-1 s-1 [Tf] ) concentration of small molecules in phase f, mol L-1 Tgi ) glass-transition temperature for species i, °C. V ) total volume, L V0 ) total initial volume, L V1 ) volume in phase 1, L V2 ) volume in phase 2, L Vi,1 ) volume of species i in phase 1, L Vi,2 ) volume of species i in phase 2, L Vf1 ) fractional free volume for phase 1 Vf2 ) fractional free volume for phase 2 Vfcr ) critical fractional free volume for glassy effect Wg(x) ) instantaneous gel fraction Ws(x) ) instantaneous sol fraction x ) general (overall) conversion x1 ) overall conversion in phase 1 x2 ) overall conversion in phase 2 xc ) critical conversion xs ) limit solubility conversion Yi,f ) moment i of the polymer radical distribution in phase f, mol L-1 [Z]f ) inhibitor concentration in phase f, mol L-1 Greek Letters Ri ) expansion coefficient for species i, °C-1 βf ) ratio of rate of termination by combination to rate of polymerization in phase f ∆V* ) activation volume, L mol-1 • φi,j ) mole fraction of radicals of type i in phase j φj ) volumetric fraction of species j FCO2 ) carbon dioxide density, g L-1 Fa,j ) cross-linking density in phase j Fcp,j ) primary cyclization density in phase j

τj ) ratio of chain transfer and inhibition rates to polymerization rate in phase j Fj ) accumulated cross-linking density Fjc,j ) cyclization density (primary + secondary) in phase j Fjcs,j ) additional secondary cyclization density in phase j

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Received for review November 5, 2004 Revised manuscript received December 16, 2004 Accepted December 23, 2004 IE048922O

Modeling of the Free-Radical Copolymerization Kinetics with Cross

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