Copolymerization of VDF and HFP in Supercritical Carbon Dioxide: A Robust Approach for Modeling Precipitation and Dispersion Kinetics

2.1. Phase Behavior

2.2. Reactions

2.3. Comments

The different behavior in terms of partitioning between radicals and other species (low-MW components and polymer chains) is due to the different ratio between the characteristic times of termination and diffusion. Depending on the reaction conditions, the timescale of radical termination may be comparable to that of interphase transport making the two processes competitive so that no equilibrium partitioning can be achieved.16, 26 On the other hand, the lifetime of VDF, HFP, initiator (I), CO₂ and dead chains, is comparable to the reaction duration and equilibrium conditions are easily established.

When bimodal MWDs were obtained for poly(VDF-co-HFP) synthesized in scCO₂, the low-MW peak, attributed to polymer formed in the continuous phase, is typically narrower than the high-MW peak.16-18 This evidence supports the assumption that chain transfer to polymer and diffusion limitations for the termination events are negligible in the continuous phase (assumption vii) since both events lead to broadening of the MWD.

2.4. Thermodynamic Modeling

No data about initiator partitioning are available. Therefore, equipartitioning of diethylperoxidicarbonate (DEPDC) was assumed at all operative conditions, i.e., equal to one, being the weight fraction of the initiator in phase i. The partitioning of monomers and CO₂ between V^sc and V^pa was modeled by the Sanchez-Lacombe (SL) molecular theory.33-35 In the following, the subscript c indicates a generic component (VDF, HFP, CO₂, or the amorphous copolymer fraction).

In the SL theory, molecules are s-mers randomly allocated in a lattice together with empty sites. The lattice is made of cells whose molar volume v* equals the volume of mers and holes. Each pure component c is characterized by six parameters fulfilling the following constraints:

((2))

((3))

Accordingly, only three parameters out of the six mentioned are truly independent. In Equation 2 and 3, ρ* is the mass density of the occupied sites (close-packed density), ε* the mer/mer interaction energy and T* and P* are the so-called characteristic pressure and temperature.

The mixing rules proposed by McHugh and Kroukonis33 have been used to evaluate the mixture properties. For each phase of known composition ω_c the mixing rules are:

((4))

((5))

((6))

((7))

((8))

((9))

Using such mixing rules, the density and chemical potential of each component in each phase are given by

((10))

((11))

where the following reduced variables have been introduced:

((12))

The fraction of crystalline polymer is assumed to be a linear function of the copolymer average composition according to the following equation:18

((13))

Such equation holds for . At larger HFP contents the copolymer is fully amorphous and X_cr = 0. The density of the crystalline phase, ρ^pc, has been evaluated as:

((14))

where ρ^0,pc is the density of the crystalline polymer at the reference temperature, T⁰, and at zero pressure. The values of the coefficients α_pc and β_pc as well as of the values ρ^0,pc and T⁰ were estimated by Pantani et al.36 Accordingly, for a system of given overall composition, we can estimate the pressure and the partitioning of all species by solving a system of equations involving mass balances for each component and the equality of the chemical potentials of VDF, HFP, and CO₂ in the two phases, V^sc and V^pa, together with initiator equipartitioning (m_ω,I = 1), polymer insoluble in the continuous phase (m_ω,pol = 0) and volume additivity:

((15))

2.5. Kinetic Scheme, Mass, and Population Balance Equations

In the following, the subscripts n,l = [1,∞] indicate the chain lengths and R_n and P_n the moles of the growing and terminated chains R_n and P_n of length n. Square brackets specify molar concentrations. The following kinetic scheme is applied to both phases:

((16))

((17a))

((17b))

((17c))

((17d))

((18))

((19))

Note that while indicates the active chains in phase i, denotes dead chains formed by termination events occurred in phase i (according to assumption iv, the chains are formed in the homogeneous phase, but they precipitate and accumulate in the dispersed phase).

With the classical pseudo-steady-state assumption for the growing chains, the mass balances for monomers and initiator and the PBEs for growing and dead chains are:

((20))

((21))

((22))

((23))

((24))

((25))

((26))

((27))

where fⁱ is the monomer mole fraction (based on monomers only), ψⁱ the initiator efficiency, and δ_n,1 the Kronecker delta function, defined as equal to 1 for n = 1 and equal to 0 otherwise. The variable θ in Equation 24 and 25 is a transport efficiency lumping all uncertainties in the evaluation of the product K_nA_p. θ was introduced in the model because the transport parameters required to estimate the transport coefficient of the radicals are known for the VDF homopolymerization but not for the copolymer case. Accordingly, θ is set to 1 for the VDF homopolymerization while a different value can be used for the copolymer case thus accounting for the effect of composition on the transport coefficients. For example θ is lower or higher than 1 if radical interphase transport is slower or faster compared to the PVDF case.

Equation 20–27 describe the copolymerization kinetics in a batch reactor when all terms containing the residence time, τ, are set to zero, or in a CSTR when the same terms are accounted for. In the latter case, the steady-state solution is obtained by integrating Equation 20–27 until a time larger than 5–10 τ, i.e., large enough to reach the steady state. The moments of order jth of the polymer chain length distributions are defined as:

((28))

and the moments of order jth of the polymer in the reactor are simply:

((29))

The average copolymer composition is calculated as:

((30))

Note that, since chain transfer to polymer is taking place in the pa-phase only, radicals R^pa give transfer reaction (cf. Equation 24 and 25) to the amorphous fraction (1–X_cr) of the total polymer formed in the pa phase () or to the polymer formed in the sc phase () but immediately precipitated.

3.1. Thermodynamic Parameters

To solve Equation 10 and 11, the values of three characteristic parameters for each component (, , ) as well as of two binary interaction parameters (η_cj and ζ_cj) for each possible pair of components are necessary. In the case of monomers and carbon dioxide, they were evaluated by pressure volume temperature (PVT) or vapor liquid equilibrium (VLE) data (cf. Supporting Information). The resulting values are listed in Table 1 and 2.

Table 2. SL binary interaction parameters (η, ζ).

	VDF	HFP	CO2	P[VDF-co-HFP]
				= 0	= 0.22
VDF	–	0.0661, 0.0018	0.0628, 0.0680	0, 0.1365	0,–a)
HFP		–	–0.0465, 0.011	–	0,–a)
CO2			–	0, 0.0603	0, 0

• a) Values of the monomer/polymer interaction parameters, ζ_VDF/Ppol and ζ_HFP/pol, at varying composition are shown in Figure 6 (inset).

The characteristic parameters of the amorphous polymer were evaluated as a function of copolymer composition, , using the approach proposed by Panayiotou and Oehmke,37 as shown in the Supporting Information. The values of the binary interaction parameters of the low-MW species/copolymer pairs could be estimated from sorption/swelling measurements of each component in the polymer. However, experimental data have been reported only for the pairs CO₂/PVDF,38-40 CO₂/P[VDF₇₈-co-HFP₂₂] (Supporting Information), VDF/PVDF and for the ternary system CO₂/VDF/PVDF.39, 40 About the SL CO₂/copolymer interaction parameters, a suitable dependence upon the copolymer composition is needed to well describe the experimental sorption profiles (cf. Supporting Information). Therefore, in the absence of experimental data covering the whole range of interest, we assumed a linear variation of the CO₂/copolymer interaction parameters with copolymer composition, between the values obtained at  = 0 and  = 0.22 (Supporting Information), and we used a linear interpolation between such values to estimate the interactions at intermediate copolymer compositions. About the VDF/PVDF pair, the value of η_VDF/PVDF was set to zero (as for ) and the value of ζ_VDF/PVDF was estimated using VDF/PVDF sorption data measured at T = 50 °C for the ternary system CO₂/VDF/PVDF at conditions very close to those selected for VDF dispersion polymerizations (i.e., CO₂/VDF ≈ 60/40_w/w).39 The results are shown in Figure 1; note that the experimental solubility of CO₂ in PVDF in the ternary system is accurately reproduced when using the value of the interaction parameters determined from the binary system (Supporting Information).

3.2. Kinetic Parameters: Continuous Phase

3.2.2. Propagation and Termination

Since the reaction invariably starts in the continuous phase, the experimental molecular weight measured at low conversion is representative of the reaction occurring in the sc phase.18 Accordingly, a set of low conversion experiments was carried out to evaluate cross propagation and termination rate constants in the continuous phase. Experimental conditions and main results are collected in Table 3, together with experiments previously reported in the literature.18 The resulting Mayo-Lewis plot is shown in Figure 2. In the same figure, composition data from three different literature sources are also reported as empty symbols. Our measurements are consistent with those obtained by other authors and all data are well described by the Mayo-Lewis equation using the following values of the reactivity ratios:

((31))

Table 3. Recipes and main characterization results for VDF–HFP low conversion copolymerization reactions in scCO₂ at T = 50 °C (experiments and product characterizations from Costa et al.18).

Entry	[VDF] + [HFP]	fHFP,0	[DEPDC]	Time	Xw	P0			PDI
	[mol · L−1]	[mol%]	[10−3 mol · L−1]	[min]	[%]	[MPa]	[mol%]	[kg · mol−1]
1a)	5.4	0	5.5	38	5.3	37.3	0	76.1	2.7
2a)	4.3	6.1	5.3	60	5.5	29.4	1.5	51.3	1.9
3a)	3.6	19.9	5.6	120	6.7	27.5	6.6	41.7	1.8
4b)	5.8	28.6	6.9	30	4.3	40.0	9.4	66.9	1.7
5a)	5.6	29.9	6.9	30	4.5	40.3	10.2	66.6	1.5
6b)	5.4	30.5	12.4	55	6.5	37.6	10.0	53.4	1.5
7a)	5.1	20.5	5.4	60	6.2	39.6	6.3	85.2	2.0
8b)	5.5	21.5	5.4	60	8.7	39.6	6.7	74.3	2.7
9a)	6.2	43.3	5.5	180	12.3	38.2	15.7	83.3	1.8
10a)	9.3	47.7	5.5	180	12.4	39.2	18.6	129.3	2.4

• a) With surfactant;
• b) without surfactant.

Such values are also in good agreement with those reported by other Authors based on the combination of ¹⁹F NMR and elemental analysis (r_VDF = 3.6–4.7 and r_HFP = 0–0.1 in Ahmed et al.28), of ¹⁹F NMR and Fourier-transform infrared (FT-NIR) spectra (r_VDF = 3.46–4.34 and r_HFP = 0.11 in Möller et al.43) and by ¹⁹F NMR alone (r_VDF = 5.1–3.6 and r_HFP = 0 in Beginn et al. and Tai et al.15, 22). Note that the zero value of the reactivity ratio of HFP is due to the fact that such monomer does not homopolymerize.25 Finally, assuming a value of  = 3000 L · mol⁻¹ · s⁻¹ (typical value; cf. Mueller et al.16), the values of the remaining kinetic parameters, and , can be estimated from the measured number-average molecular weights. When termination occurs only by combination the following equation applies:44

((32))

where [I], [VDF], and [HFP] are total molar concentrations, and MW_ru is the average molecular weight of the copolymer repeating unit, defined as:

((33))

Using the experimental data collected at very low monomer conversion (Table 3), Equation 32 was used to estimate the values of the ratio . Such values are shown in Figure 3 as a function of monomer feed composition for reactions carried out at initial pressures between P = 380 and 400 bar (symbols). In the same figure, the predicted values of the same quantity (see Appendix A) are plotted as continuous line. Using the ratio value at f_HFP,0 = 0, i.e., in the PVDF case, and the typical value of mentioned above, the rate coefficient of bimolecular termination in supercritical phase was estimated as  = 3.2 × 10⁹ L · mol⁻¹ · s⁻¹. Since it is well known that HFP is much less reactive than VDF, the cross propagation rate coefficient was constrained to be smaller than for the VDF homopolymerization ( ≤ ). Given this constraint, the best agreement which could be achieved with the experimental data is the one shown in Figure 3, obtained by setting the cross propagation coefficient equal to . The slight underestimation of the calculated values for the copolymer case can be due at least to two reasons: (i) experimental inaccuracies of the measured molecular weights from gel permeation chromatography (GPC) due to tailing at high molecular weight in the MWDs (cf. Costa et al.18); (ii) the experimental termination rate constant could be a (decreasing) function of the copolymer composition. On the other hand, the general agreement between calculated and experimental values is quite good and the approximation of composition-independent was retained. The values of the rate constants so far determined were considered valid at pressure of P = 400 bar.

The pressure effect on the rate constants is taken into account by the transition state theory:44

((34))

where ΔV* is the activation volume of the reaction. Assuming composition independent activation volumes and  = 0 (as found by Charpentier et al.41) one obtains:

((35))

Finally, for reactions at constant initiator concentration (such as the copolymerization reactions in CSTR at different pressures reported by Ahmed et al.14), the following equation can be worked out combining Equation 32 and 35:

((36))

Equation 36 was used to estimate the activation volume of the propagation step from available low conversion experimental data.14 Using a typical value of  = 15 × 10⁻⁶ m³ · mol⁻¹,16, 44  = −76 × 10⁻⁶ m³ · mol⁻¹ was calculated.

3.3. Kinetic Parameters: Dispersed (pa) Phase

3.3.1. Diffusion Coefficients

Diffusion coefficients in polymers are usually estimated through the Vrentas-Duda free-volume theory (VD-FV; cf. Achilias,45 Keramopoulos and Kiparissides,46 Mueller et al.,16, 26 Vrentas and Duda47). Such theory involves a very large number of adjustable parameters, whose evaluation becomes very difficult, if not impossible, for copolymers, where the composition dependencies must be considered. This problem was circumvented by using the recently proposed lattice free-volume (LFV) theory.48 In the frame of the LFV theory, the self-diffusion coefficient of the monomer A in a multicomponent polymeric system is expressed as:

((37))

where D_0,A is a prexponential coefficient, γ the overlap factor, and are the weight fractions of the different species in phase i (i = sc or pa). In Equation 37, ρ*, are the SL characteristic densities of the pure component (indicated by the subscript) and of the mixture, while ρⁱ is the density of phase i. For the generic component c with molecular weight MW_c* of its “jumping unit”, ξ_A/c represents the ratio of the molar volumes of the monomer A and of the jumping unit of c:

((38))

Experimental values of monomer diffusion coefficients at different T, P, and compositions are not available. Therefore, we used typical values of pre-exponential coefficient and overlap factor: D_0,VDF = D_0,HFP = 10⁻⁶ dm² · s⁻¹ and γ = 0.2.48 Note that, for low-molecular-weight components (monomers and CO₂), MW_c* = MW_c. About the polymer jumping unit size, its molecular weight can be related to the molecular weight of the polymer repeating unit ru. In this work, a constant value of the following ratio was assumed:

((39))

Such value was estimated in order to provide reasonable predictions of diffusion coefficient of VDF in CO₂-swollen PVDF: at 50 °C, under the operative conditions of Table 4 entry 1, the order of magnitude of was about 10⁻⁹ dm² · s⁻¹, thus consistent with the range of experimental values of low-MW species in PVDF.49

Table 4. Reaction recipes. In all cases T = 50 °C; P₀ = 380–400 bar for entries 1–5; P₀ = 133, 204, and 332 bar for entry 6. (Note that the adopted recipes for entries 1–4 are averaged on available experimental conditions from Costa et al.18).

Entry	[VDF] + [HFP]	fHFP,0	[DEPDC]
	[mol · L−1]	[%]	[10−3 mol  · L−1]
1	5.5	0.0	5.5
2	5.5	15.2	5.5
3	5.4	21.1	5.4
4	5.6	29.0	6.6
5	6.2	43.3	5.5
6	3.1	0.0	5.0

Equation 37 has been applied to both the polymer pa-phase and the continuous sc-phase. Notably, the values of most parameters are those previously estimated describing the phase behavior of the system. Accordingly, experimental values of the transport properties of the polymer-rich phase involved in the conventional VD-FV equation (cf. Keramopoulos and Kiparissides,46 Mueller et al.,16) are not needed any more: this represents a major step forward in the direction of simplifying the parameter evaluation process.

Finally, the semiempirical “universal” scaling law proposed by Griffiths et al.50 was used for the chain-length dependence of the self-diffusion coefficients:

((40))

where is the average diffusion coefficient of the monomers

((41))

Note that, in the case under examination, we have found that the copolymer composition exhibits a very limited drift. Moreover, the composition of the copolymer produced in each phase is practically the same.18 Therefore, the cumulative composition of the entire copolymer is used in Equation 41 instead of the instantaneous composition in phase i.

3.3.2. Radical Interphase Transport

The transport of active chains between the two phases was described through the classical two film theory, yielding the following expression of the overall mass transport coefficient referred to the supercritical phase:

((42))

where δ is the thickness of the diffusive boundary layer, m_n and are the (molar) partition and diffusion coefficients in phase i of radicals of length n, respectively. The radical partition coefficient is defined as:

((43))

The value of the boundary layer is assumed of the same order of magnitude of the particle radius of samples produced in dispersion reactions, i.e., δ = 0.5 µm (µm-sized particles). Because of the strong composition dependence of the solubility of fluorinated copolymers in scCO₂, m_n is expected to be a strong function of copolymer and monomer mixture compositions, as well as of the pressure. Nevertheless, since only data for relatively short PVDF chains in scCO₂ at one single value of both temperature and pressure (n < 40, T = 338 K, P = 150 bar) are available for m_n,30 the following values were used in all simulations:

((44))

((45))

Finally, the interphase area through which radical transport occurs was evaluated as:

((46))

The number of particles to be used in Equation 46, N_p, was estimated from scanning electron microscopy (SEM) pictures. Such estimates were affected by the largely different particle morphologies. Dispersion reactions at f_HFP,0 = 0–0.3 produced µm sized particles,18 with average particle concentration of N_p/V_R = 9.74 × 10¹³ L⁻¹. Precipitation reactions at 0 < f_HFP,0 < 0.3 produced irregularly shaped particles of average size 2–3 times larger,18 and therefore with particle concentrations about 10–30 times smaller. Accordingly, a particle concentration 20 times smaller was assumed for precipitation reactions leading to N_p/V_R = 4.87 × 10¹² L⁻¹. Finally, in all reactions producing irregular and compact structures (precipitations at f_HFP,0 = 0 and reactions with f_HFP,0 > 0.4; Costa et al.18) we set a value of particle concentration arbitrarily small and equal to N_p/V_R = 9.74 × 10¹⁰ L⁻¹, that is three order of magnitude smaller than the value used for dispersions providing µm sized spherical particles. Such choice is justified by the experimental observation that VDF precipitations produce aggregates at least ten times larger than the particles produced in dispersion reactions (cf. VDF precipitations in Charpentier et al.51, and VDF dispersions in Mueller et al.20)

3.3.3. Initiator Decomposition

The same value of dissociation rate constant as for the continuous phase was used. About the initiator efficiency, ψ^pa, diffusion limitations in the polymer-rich phase were taken into account following Shen et al.:52

((47))

3.3.4. Propagation and Termination

Diffusion limitations were taken into account by considering the following expression of the rate coefficients in the dispersed phase:16, 26

((48))

((49))

where N_AV is Avogadro's constant, σ the collision radius, r_nl the separation distance at which termination between radicals is instantaneous, a the root-mean end-to-end distance divided by the square root of the number of monomer units in the chain. and are the self-diffusion coefficients in the pa-phase of the growing chains of length n and l, respectively. In Equation 48 and 49, the first term reflects the reactivity among radicals in the absence of diffusion limitations (i.e., that determined for the continuous phase), whose effect is accounted for in the second term. More details about physical meaning and derivation of Equation 48 and 49 are available in the cited literature.

The values of a and of the collision radius, σ, are often quite similar:53 hence, a = σ has been used. Instead of approximating the size of the repeating unit averaging the Lennard-Jones diameter of the two monomers,54 a different evaluation approach was followed taking advantage of LFV theory. In fact, Costa and Storti48 have shown that the specific closed-packed volume obtained by the SL theory provides a reasonable estimation of the molecular volume. Therefore, assuming spherical geometry, the size of a low-MW component A is given by:

((50))

The reliability of Equation 50 was tested using known values of the parameters a, σ, and ρ*: for example for MMA, using  = 1074 g · L⁻¹,48 Equation 50 provides σ_MMA = a_MMA = 0.67 nm, a value very close to those reported in the literature (a_MMA = 0.69 nm and σ_MMA = 0.585 nm; Russell et al.53). For the system under examination, σ_VDF = 0.55 nm and σ_HFP = 0.61 nm have been estimated; being the two values similar, a single average value a = σ = 0.58 nm was used in all simulations.

Finally, considering that the reactions were carried out well above the glass transition temperature of the VDF–HFP copolymers (T = 50 °C vs. T_g < 0 °C),15 the flexible chain limit has been selected for r_nl:53, 55

((51))

where j_c,0 is the entanglement spacing for the pure polymer.

3.3.5. Chain Transfer to Polymer

The rate constant of chain transfer to polymer has been defined as:

((52))

where α is the ratio of chain transfer to propagation rate constants for PVDF. Dealing with copolymers, the composition has been introduced as a factor in Equation 52 because we assumed that only H atoms, present in the VDF units only, can be extracted by growing radicals. Apostolo et al.56 estimated a value of α = 3 × 10⁻⁵ for the emulsion copolymerization of VDF and HFP at T = 85 °C. Since there is evidence that temperature promotes branching formation,57 the value to be used to simulate reactions at T = 50 °C should be smaller. Accordingly, we set α = 10⁻⁶.

3.3.6. Tuning of Remaining Parameters

In this paragraph, the evaluation of all model parameters whose a priori prediction or estimation was not possible is discussed.

Let us consider the case of VDF homopolymerization first. In this case, all information related to monomer and radical solubility are available, by setting θ = 1 only the value of the entanglement spacing, j_c,0, remains to be evaluated. Such value was determined by comparison of the experimental and calculated MWD of a single VDF dispersion polymerization, whose recipe is provided in Table 4 entry 1. Under these conditions, the large number of µm-sized polymer particles provided large interphase area and the polymerization occurred mainly in the dispersed phase. The resulting experimental MWD was broad but monomodal (cf. Figure 4a). The MWDs calculated by the model when using j_c,0 = 50 and j_c,0 = 200 (typical upper and lower limiting values of j_c,0 as reported by Buback et al.55), are shown in Figure 4b as dashed and continuous lines, respectively. In both cases the model predictions underestimate the distribution broadening in the low-MW region. Decreasing j_c,0 values result in reduced termination rate constants (cf. Equation 49 and 51) thus shifting the whole MWD toward higher MW and increasing the disagreement in the low-MW region (dashed vs. continuous line). As a result, the largest value of the entanglement spacing, j_c,0 = 200, was selected as best value and kept constant in all simulations.

Let us now consider the copolymer case. The monomer/copolymer interaction parameters, η_A/pol and ζ_A/pol, mainly affecting monomer partitioning between the sc and pa phase, and the radical partition coefficient, m_n, affecting the radical transport rate (Equation 24, 25, 42), have to be evaluated as a function of the system composition. Indeed, several experimental evidences suggest that the thermodynamics of the system is heavily affected by the HFP content in the copolymer and by the composition of the fluid phase: the solubility of CO₂ in PVDF is much smaller than in poly[VDF₇₈-co-HFP₂₂] (Supporting Information), the cloud point pressure of PVDF in CO₂ is much larger than that of poly[VDF₇₈-co-HFP₂₂]29 and the solubility of the copolymer in the continuous phase is heavily affected by its HFP content.29, 58 Due to the difficulty of theoretically account for all these effects, a simplified approach aimed to minimize the number of adjustable parameters has been adopted, as detailed in the following.

To model monomer partitioning, η_A/pol was set to zero (like in the case of CO₂), and ζ_VDF/pol was used as the only fitting parameter. The remaining parameter ζ_HFP/pol was adjusted to obtain the same instantaneous composition of the monomers in the two phases according to experimental results described elsewhere.18 Therefore, just one thermodynamic parameter was actually fitted (ζ_VDF/pol).

About the radical partitioning, the same m_n values estimated for VDF homopolymerization were used. The effect of system composition on transport rates was accounted for through the adjustable quantity θ, the multiplying factor of the mass transport term in Equation 24 and 25. Accordingly, θ < 1 implies slower radical interphase transport compared to the PVDF case and vice versa.

Summarizing, two parameters only have to be determined by direct fitting of copolymer MWDs, θ and ζ_VDF/pol. Therefore, before comparing the model predictions to the experimental data, it is worth to analyze their impact on the model results in terms of MWD. At low conversion (Figure 5a), increasing θ from 0.1 (continuous line) to 1 (dashed line) at constant ζ_VDF/pol results in the disappearance of the mode at high molecular weight in the MWD. The larger amount of radicals transported into the pa-phase results in more polymer at smaller chain length, and the two modes merge. On the contrary, larger monomer concentration in the pa-phase (decreasing ζ_VDF/pol) at constant θ = 0.1 (dotted line vs. continuous line), results in more polymer with larger molecular weight. Therefore, because of the opposite effect the two parameters have on the amount and molecular weight of the copolymer produced in pa-phase (cf. Figure 5b), θ and ζ_VDF/pol can be unambiguously tuned in order to match amount and location of the experimentally observed high-MW tail or mode in the MWDs using reactions carried out at low conversion. Such parameter evaluation was performed for the reactions carried out at f_HFP,₀ = 0.21 and f_HFP,0 = 0.29 (Table 4 entries 3 and 4). In both cases, θ = 0.1 was estimated as best value. Such value was then used in all simulations at f_HFP > 0.

Finally, the monomer partitioning and the corresponding values of ζ_A/pol (inset) are shown in Figure 6. At increasing HFP content in the monomer mixture, and therefore in the produced copolymer, the solubility of the monomers in the sc phase increases. The specific values of ζ_VDF/pol used to simulate the copolymerization at different compositions (e.g., f_HFP,0 = 0.15 and 0.43 in Table 4 entry 2 and 5) were determined by interpolation or extrapolation and they are shown as empty symbols in Figure 6.

4.1. Effect of Monomer Composition

Let us consider first, copolymerization reactions carried out in the presence of surfactant, with 3 h reaction time (reaction recipes in Table 4 entries 1–5). Figure 7 and 8 show a good agreement between model (open symbols) and experimental data (filled symbols) in terms of conversion and copolymer composition, F_HFP, as a function of the initial monomer feed composition, f_HFP,0. For the same reactions, the experimental MWDs are shown as symbols in Figure 9-12, together with curves representing the model predictions. In all cases, the comparison is remarkably favorable. In particular, the gradual transition of the main MWD peak from high to low-MW region at increasing f_HFP,0 is correctly predicted. From the model perspective, such transition indicates that the relevance of the dispersed phase as reaction locus progressively decreases when the starting monomer mixture is made richer in HFP. This behavior is explained by the lower radical transport rate compared to that of the homopolymer (θ < 1) and by the reduction of the monomer concentration (Figure 6) in the pa phase as f_HFP,0 increases.

4.2. Effect of Reaction Time and Interphase Area

The time evolution of the experimental MWDs for precipitation and dispersion copolymerization are shown in Figure 13a and 14a at an intermediate value of composition (f_HFP,0 = 0.2), for which experimental data at small (precipitation) and large (dispersion) interphase areas but similar conversions and reaction times are available.18 At this HFP concentration in the reacting mixture, the solubility of the monomers in the dispersed phase is limited but still non-negligible. Accordingly, Figure 13a shows that for precipitation at small conversion, with a low amount of polymer present in the reactor, the lower molecular weight peak prevails against the high molecular weight shoulder in the MWD. Therefore, the homogeneous phase is the dominant reaction locus. On the other hand, at increasing reaction time and polymer hold-up, a distinct second peak becomes evident in the MWD, and the contribution of the polymer phase as reaction locus progressively increases, as clearly indicated by the relative increase of this second peak with time. Such MWD evolution with time was already observed for the precipitation of VDF.16 In the case of the dispersion reactions (Figure 14a), such evolution is similar but, because of the enhanced radical transport, the high-MW peak becomes the dominant one quickly, and the MWD at larger reaction time is still broad but monomodal. All these features are well predicted by the kinetic model, whose results are shown in Figure 13b and 14b. Even the drift of average composition is well captured (Figure 15). Note that having imposed equal monomer composition in the two phases, the calculated curves of copolymer composition for precipitation and dispersion reactions are identical.

4.3. Effect of Pressure

In a previous contribution on VDF precipitation reactions, Mueller et al.16 showed that pressure enhances the reaction rate, and slightly shifts the whole bimodal MWDs toward larger MWs. The same authors showed that when using a value of  = −0.025 L · mol⁻¹, that is a typical value for the activation volume of the propagation rate constant of many monomers, these effects were quite underestimated. In this work, a larger value of  = −0.076 L · mol⁻¹ was determined from experimental VDF–HFP precipitation data.14 Excellent predictions of the pressure effect on conversion and MWDs for VDF precipitations have been obtained, as shown in Figure 16 and 17, respectively. The enhanced reaction rates as well as the production of longer polymer chains at increasing pressure are quantitatively well reproduced. Therefore, despite the estimated is larger than typical literature values, the good agreement supports the reliability of such value for VDF-based copolymerization in scCO₂. A possible explanation of such apparent overestimation might be the large variation of the solvent properties with pressure under supercritical conditions, as previously observed by other investigators.59-61

6.1. Bimodality in the MWDs

The prediction ability of the proposed 2-loci model in terms of occurrence, location, and relative amount of the larger-MW mode in the complete MWD was remarkable in all cases (cf. Figure 10, 13, and 17). This is especially true when considering that the interface area, A_p, was not taken as an adjustable parameter as previously done in the case of VDF precipitation polymerization.16 Nevertheless, the model nicely predicted the monomer concentration effect on VDF surfactant free polymerization (Figure 18), as well as the effect of the increase of interphase area when, adding an effective surfactant, dispersion copolymerizations were performed, as shown in Figure 13 and 14. The fair comparison with the experimental MWDs is a further confirmation of a mechanistic picture involving two phases as possible reaction loci.

6.2. Monomer Partitioning

Ahmed et al.62 speculated that because of the preferential partitioning of the monomers in the scCO₂-rich phase and the resulting poor solubility of the monomers in the polymer phase, the propagation of radicals in the polymer phase should occur under starved conditions, and therefore with negligible polymerization rate. With this respect, two points must be addressed. First, the monomer solubility in the polymer phase is small but not negligible. For example, in the first minutes of reaction, the model predicts VDF concentration in the amorphous polymer phase of about 0.3 mol · L⁻¹ at the conditions given in Table 4 entry 1. Second, while the propagation rate constant is almost unaffected by diffusion limitations (Equation 48), the termination rate constant in pa-phase is orders of magnitudes smaller than that in sc phase, as shown in Figure 20. Therefore, such very low value counterbalances the low monomer solubility and makes possible a fast polymerization rate in pa phase.

About the values of the interaction parameters used for VDF/PVDF, note that they differ from those reported by Mueller et al.16 and Galia et al.39. The reason is that different sets of data were used to determine both the pure component and binary SL interaction parameters. Nevertheless, from the model perspective, the exact value of the interaction parameters is not crucial. Indeed, the important point is to describe correctly the monomer sorption in the polymer matrix. In Figure 1, experimental sorption data of VDF in PVDF are compared with profiles calculated with the SL EOS using parameters determined by Mueller et al.16 (dash-dotted curve) and those used in this study (continuous line). It can be observed that the two profiles are quite similar and both match with acceptable agreement the experimental data.

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