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Polymer Miscibility in Supercritical Carbon Dioxide: Free Volume as a Driving Force Jeﬀrey DeFelice and Jane E. G. Lipson* Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, United States ABSTRACT: In this article, we connect the experimental miscibility of several polymer/supercritical carbon dioxide (scCO2) mixtures with their pure component properties, such as free volume and interaction energy. We directly address the experimental observations that suggest free volume-rich polymers and those with weak polymer segment−segment interaction energies mix more favorably with scCO2. By applying our simple locally correlated lattice (LCL) theory to model the pressure−volume−temperature (PVT) behavior of the pure polymers and supercritical solvent, we obtain characteristic molecular parameters which are then used to predict the key physical properties of interest. We probe the underlying thermodynamic contributions (entropic and enthalpic) to the free energies of mixing and show that our LCL theory can explain the experimental miscibility ranking based solely on our characterization of the pure components. oxide) (PPO) is partially miscible with scCO2,9 and upon ﬂuorination to poly(hexaﬂuoropropylene oxide) (PFPO) the miscibility is signiﬁcantly enhanced.14 The introduction of local branching can also have a powerful eﬀect e.g. poly(methyl acrylate) (PMA), is partially miscible with scCO2, however, PMMA, which diﬀers from PMA only by the addition of a methyl group per monomer in the polymer backbone, is completely immiscible with scCO2.3 This diﬀerence in polymer architecture results in a startling change in the polymer glass transition temperatures (Tg), such that the Tg of PMMA is ∼100 K greater than that of PMA.3 In fact, the connection between scCO2 miscibility and polymeric glass transition temperature reﬂects a fundamental feature of what is driving miscibility, as we discuss further below. Few phase diagrams for polymer/scCO2 mixtures have been comprehensively mapped out, which limits analysis of their behavior. What has been suggested as a result of these studies, however, is that weakly polar pure polymer energetic interactions, such as in ﬂuoropolymers, and a signiﬁcant fraction of polymer free volume, as is found in poly(siloxanes), favorably inﬂuence polymer miscibility with scCO2.1,3,4,6,16 In this work, we test these hypotheses using our simple locally correlalated lattice (LCL) model, and to this end, we apply our theory to a series of polymer/scCO2 solutions for which at least some experimental data are available. We begin by ﬁtting our LCL model equation of state (EOS) to experimental pressure−volume−temperature (PVT) data in order to obtain values for the pure component characteristic parameters, which can then be used to calculate a number of physical properties. We examine trends in the pure polymer

1. INTRODUCTION Polymers that are extremely miscible with supercritical carbon dioxide (scCO2) generally fall into two categories: poly(siloxanes) and ﬂuoropolymers; few examples of other very miscible polymer/scCO2 mixtures are available.1−9 More common are those polymer/scCO2 combinations that fall into the “partially miscible” category, including a number of poly(n-alkyl acrylates) and poly(n-alkyl methacrylates),1,3,10 as well as a few other examples.1,3,6,7,9 In these instances, temperature (T) and pressure (P) are useful variables in terms of controlling the formation of a homogeneous ﬂuid phase. One such approach is to vary the pressure, whereby increasing P at a constant temperature favors mixing. Then at elevated pressures, T can be varied to control phase behavior. In this regime, partially miscible polymer/scCO2 mixtures exhibit upper critical solution temperature (UCST) type behavior; i.e., phase separation occurs when T is lowered at a constant pressure.1,2,11 In some cases, both UCST and lower critical solution temperature (LCST) type behavior were observed. For example, a PDMS/scCO2 solution phase separated when the temperature was either lowered (UCSTtype) or raised (LCST-type), thus bracketing an intermediate T-range over which the mixture was homogeneous.12,13 Finally, there are numerous cases of polymer/scCO2 mixtures for which miscibility has been investigated but not observed, even at high T and P. Some examples in this category, which include poly(methyl methacrylate) (PMMA), polystyrene (PS), and poly(acrylic acid) (PAA), involve polymers having rather close relatives that do exhibit partial miscibility.1−3,6,7 In addition to changing T and P, varying molecular properties such as the degree of polymerization1,2,6,7 or the degree of ﬂuorination of the polymeric component5,14−17 may shift a partially miscible mixture to either the completely miscible or immiscible region. For example, poly(propylene © XXXX American Chemical Society

Received: June 10, 2014 Revised: July 31, 2014

A

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energy (A) is derived. With an analytical expression for the free energy in hand all other thermodynamic quantities of interest can be derived. The Helmholtz free energy for a binary mixture of molecules types i and j is given by

segment−segment interaction energies and free volume, and discuss them in the context of experimental results from the literature regarding polymer/scCO2 mixture miscibility. Finally, by making use of connections we have previously developed that provide a direct link between pure component properties and thermodynamic contributions to the free energy of mixing, both entropic and enthalpic,18 we show that our theory is able to explain the miscibility ranking for the set of polymers studied here and thus leads to a predictive tool for other cases. The paper is organized as follows. Section 2 includes a brief description of the theoretical background, implementation of the LCL model equation of state, and the associated model parameters. In section 3, we provide an overview of our results and the trends that we have found between pure polymer segment−segment interaction energies, free volume, and experimental miscibility with scCO2. In section 4, we examine a number of speciﬁc cases in further detail, discussing our theoretical predictions for the pure component properties in the context of the experimental miscibility behavior. Turning to the mixed state, in section 5 we make predictions for the thermodynamic contributions to the free energies of mixing and illustrate how these predictions capture the experimentally observed miscibility trend. Finally, in section 6, we summarize and draw conclusions from the results presented in the previous sections.

A = Ni ln ϕi + Nj ln ϕj + Nh ln ϕh kBT ⎛ξ ⎞ ⎛ξ ⎞ Nq Nq j jz Nz ⎛ξ ⎞ j i iz ln⎜⎜ i ⎟⎟ + ln⎜⎜ ⎟⎟ + h ln⎜⎜ h ⎟⎟ + 2 2 2 ⎝ ϕi ⎠ ⎝ ϕh ⎠ ⎝ ϕj ⎠ ⎡ ⎤ ⎛ −ϵ ⎞ ⎛ −ϵij ⎞ Nq i iz ln⎢ξi exp⎜ ii ⎟ + ξj exp⎜ − ⎟ + ξh⎥ ⎢⎣ ⎥⎦ 2 ⎝ kBT ⎠ ⎝ kBT ⎠ −

Nq j jz 2

⎡ ⎤ ⎛ −ϵij ⎞ ⎛ −ϵjj ⎞ ln⎢ξi exp⎜ ⎟ + ξj exp⎜ ⎟ + ξh⎥ ⎢⎣ ⎝ kBT ⎠ ⎝ kBT ⎠ ⎦⎥

(1)

where Nh = (V /v) − Nri i − Nrj j ϕα = Nαrα /(Nri i + Nrj j + Nh) = Nαrαv /V ξα = Nαqα /(Nq i i + Nq j j + Nh) qαz = rαz − 2rα + 2

such that

2. THEORETICAL BACKGROUND AND IMPLEMENTATION This section provides a brief explanation of our theory, which is used to model the polymer melts, scCO2, and polymer/scCO2 mixtures. Our LCL model has been previously applied to a variety of polymer melts, solutions and blends,19−23 as well as small molecule ﬂuids and mixtures,24,25 although not under the extreme temperature and pressure conditions of interest here. While we provide in this section an overview of the fundamentals we do not give detailed derivations, as those may be found in numerous earlier studies.24−26 This work is the ﬁrst application of our theory to model a supercritical ﬂuid, although we note that similar theories have been applied to treat supercritical ﬂuids with some success.10,27−33 Past EOS modeling of polymer/scCO2 mixture behavior, notably the work of Tomasko et al.34,33 and Kiran et al.,12 have used theories such as Sanchez−Lacombe (SL) and statistical association of ﬂuids theory (SAFT).1 However, these approaches required a ﬁt to some experimental mixture data to determine a temperature-dependent binary interaction parameter. Here we provide a new perspective on polymer/scCO2 miscibility via our analyses of trends in the pure component properties and theoretical predictions about the relative thermodynamic contributions to the free energy of mixing. Our theoretical treatment is a lattice-based model for chain ﬂuids that incorporates the eﬀects of free volume (i.e., compressibility) and naturally accounts for the eﬀects of nonrandom mixing. A temperature dependent expression for the internal energy (U(T)) is obtained using results derived (via integral equation methods)25 for the temperature dependent nearest neighbor segment−segment conditional probabilities; thus incorporating “local correlations” as opposed to being solely mean ﬁeld-based. Making use of the Gibbs− Helmholtz relationship (U(T) = (d(A/T)/d(1/T))Ni,Nj,V) and integrating U(T) from an athermal reference state (using Guggenheim’s result),35 the expression for the Helmholtz free

α ∈ {i , j , h}

A is expressed as a function of the independent variables Ni, Nj, V, and T, which are, respectively: the numbers of molecules of components i and j, the total volume of the mixture, and the absolute temperature. We work on the simple cubic lattice, thus the lattice coordination number, z, is ﬁxed at z = 6; kB is the Boltzmann constant. Each component is described by three model parameters: ri (rj)the number of segments per molecule of type i (j), vthe volume per lattice site, and ϵii (ϵjj)the nonbonded segment−segment interaction energy between nearest-neighbor segments of types i−i (j−j). For mixtures, an additional parameter, ϵij, that characterizes the interaction between nearest-neighbor segments of type i and j is needed, and is discussed further below. Nh represents the total number of unoccupied lattice sites (h stands for “holes”); this value is ﬁxed by minimizing the system’s Gibbs free energy (G = U − TS + PV) at a given composition and set of {T,P} conditions, which determines the free volume. The total volume, V, is then the sum of the hardcore volume of the molecules, Niriv + Njrjv, and the free volume Nhv. Thus, the model deﬁnition of the free volume is V − Niriv − Njrjv (for chosen T and P), and it accounts for the portion of the total lattice volume that is made up of unoccupied sites. The amount of free volume therefore gives an indication of the compressibility of the system. The remaining deﬁnitions in eq 1 are as follows: ϕi is the volume fraction of sites of type i; ξi is a concentration variable that deﬁnes the fraction of nonbonded contacts ascribed to component i out of the total number of nonbonded contacts in the system, where, due to local connectivity, a molecule of type i has qiz nonbonded contacts. Recall that each component of the mixture is described by its three pure component model parameters r, v, and ϵ (note that in the following discussion, we have dropped the i or j subscript because we are discussing a component in the pure state). Here we introduce the route through which we characterize the pure B

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mixture must be known in order to quantify ϵij, or equivalently, g. Examples include: a lower or upper critical solution temperature20 or small angle neutron scattering (SANS) data (where the zero angle scattering intensity as a function of temperature is related to the second derivative of the free energy with respect to composition).36 The mixed interaction energy and pure component parameters may then be used to model the mixture behavior and calculate the associated changes in thermodynamic quantities upon mixing. It is often the case in both experimental and theoretical studies that, in addition to the Gibbs free energy of mixing (ΔGmix), the entropic (ΔSmix) and enthalpic (ΔHmix) contributions are of interest. A theoretical route to these quantities is valuable, as they are typically not available experimentally. In previous work we have tested the ability of our model in this area by showing that our predictions for the thermodynamic contributions to mixing compare well with experimental values determined via SANS data for a deuterated polystyrene/poly(vinyl methyl ether) blend and ΔHmix data for a polystyrene/polybutadiene blend.18 Further, it is useful to consider that the total entropy of mixing, ΔSmix, may be divided into two contributions: the “ideal” or combinatorial entropy of mixing ΔSideal mix (which is always positive) and the “excess” or noncombinatorial entropy ideal of mixing, ΔSexcess mix . ΔSmix accounts for the increase in molecular translational entropy associated with mixing, assuming V is ﬁxed. We choose37 to approximate this using the Flory− Huggins result for an incompressible mixture.38 The point is to separate the contribution to ΔSmix arising solely from combinatorial eﬀects; changing the particular form used does not signiﬁcantly change the results of our calculations. The expression for ΔSideal mix using the Flory−Huggins formalism is

component parameters, i.e., the EOS derived from our LCL theory, which is given by ⎛ k T ⎞ ⎛ 1 ⎞ ⎛ k Tz ⎞ ⎛ ϕ ⎞ ⎛ ∂A ⎞ P = −⎜ ⎟ = ⎜ B ⎟ ln⎜⎜ ⎟⎟ + ⎜ B ⎟ ln⎜ h ⎟ ⎝ ∂V ⎠ N , T ⎝ v ⎠ ⎝ ϕ ⎠ ⎝ 2v ⎠ ⎝ ξh ⎠ h ⎛ k Tzξ ⎞⎛ ξ(exp[−ϵ/kBT ]) − 1 ⎞ ⎟⎜ −⎜ B ⎟ ⎝ 2v ⎠⎝ ξ(exp[−ϵ/kBT ]) + ξh ⎠

(2)

The expression for the EOS shown in eq 2 follows from the thermodynamic relationship (P = −(∂A/∂V)N,T), where the Helmholtz free energy, A, is that of a single pure component. Note that although P is expressed in terms of the independent variable V in eq 2, V cannot be directly expressed as a function of P (as is typical with many theoretical EOS’s). Therefore, in applying the model, we use numerical root ﬁnding to determine V in situations where P is the known input variable. Furthermore, it is often more convenient to deﬁne these functions in terms of intensive variables. For example, the set of independent variables [Ni,Nj,V,T] can be reduced to the set [x,V̅ ,T] where x is the mole fraction of component i, V̅ = V/N is the intensive volume per molecule, and N = Ni + Nj is the total number of molecules. Correspondingly, one then calculates the intensive properties: A̅ = A/N, S̅ = S/N, etc. In this work, we deﬁne the intensive variables in a “per mass” basis, which is more convenient for polymers due to their high MW. In this case, the overbar notation signiﬁes any quantity per total mass, and the composition variable x is thus the mass fraction. The LCL theory is implemented as follows: The ﬁt of the EOS (eq 2) to experimental PVT data is optimized in order to determine values for the parameters r, v, and ϵ for each of the pure components. Because r scales linearly with MW, a MW diﬀerent from that of the experimental data can be modeled without changing v or ϵ, such that rnew = MWnew × (rold/MWold), which keeps the quantity r/MW constant. Although each pure component has three characteristic parameters, we describe mixtures using a single v throughout, such that all components ﬁt the same lattice. In fact, we ﬁnd that for polymer−small molecule mixtures, selecting a mixture v closer to that of the optimized small molecule v value, best preserves the agreement of the model with the PVT behavior of both pure components.22 Thus, in this work, v was ﬁxed at the value found for the scCO2, and only the parameters r and ϵ were optimized for each of the pure polymeric components (the T and P conditions for which are provided in the following section). Selecting the same v, as well as T and P ranges for ﬁtting for all of the pure components, establishes a consistent basis to compare the many systems investigated in this work. In addition to the result for the EOS, the Helmholtz free energy given in eq 1 leads to analytic expressions for the other thermodynamic properties of the mixture, including the internal energy (U = (∂(A/T)/∂(1/T)) N i , N j ,V ) and entropy (S = −(∂A/∂T)Ni,Nj,V).18,25 We now return to the interaction energy, ϵij, between nonbonded segments of type i and j. This mixed interaction parameter can be expressed as the geometric mean of the pure component parameters ϵii and ϵjj scaled by a factor g. ϵij = g(ϵiiϵjj)1/2

⎡ Nri i + Nrj j ⎤ ⎡ Nri i + Nrj j ⎤ ideal ⎢ ⎥ ΔSmix = Nk ⎥ + Nk i B ln⎢ j B ln ⎢⎣ Nrj j ⎥⎦ ⎣ Nri i ⎦ (4)

ΔSexcess mix

accounts for all remaining contributions to ΔSmix such as those from nonrandom mixing, nonzero ΔVmix, and compressibility (free volume). Using our theoretical expressions for the total and the combinatorial pieces we therefore ideal ﬁnd ΔSexcess mix = ΔSmix − ΔSmix .

3. OVERVIEW OF RESULTS Here we report the results of applying our LCL model to scCO2 and a series of polymers, listed along with their common acronyms in Table 1, for which experimental observations are available regarding their miscibility with scCO2. Such behavior is typically characterized using cloud point measurements, wherein the intensity of a light beam passed through the mixture is measured while conditions such as T and/or P are changed; a drop in intensity occurs at the “cloud point”, i.e. the point at which ﬂuid−ﬂuid phase separation occurs. We begin with the characteristic model parameters tabulated in Table 2, obtained from ﬁtting the EOS (eq 2) to experimental pure component PVT data.39,40 Examples of the agreement between the theoretical P−V isotherms and experimental data for scCO2 and PMMA are shown in Figure 1. It is important to note that all of the systems discussed were modeled over the same T and P ranges (420−450 K, 100−200 MPa). By selecting the same ranges for all species, we most accurately characterize one component relative to another and establish a consistent basis to make comparisons between them.

(3)

Just as pure component experimental PVT data are used to obtain values for r, v, and ϵ, some information about the C

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of free volume in a bulk sample of scCO2 is large in comparison to most polymers. We ﬁnd that these characteristics are captured in the ϵ parameter and % FV for scCO2 shown in Table 2. Note the weak |ϵ| value (1085.5 J/mol) and large % FV (28.0%) compared to the polymers listed in the table. That being the case, a weak pure polymeric segment−segment interaction energy and large fraction of free volume are desirable properties for maximizing the potential of polymer/ scCO2 miscibility, as discussed further below. We initiate our investigation of the relationship between pure component properties and miscibility with Figure 2, which presents a correlation between the energetic mismatch in the strength of the pure component interactions, |ϵpolymer − ϵscCO2|, and the degree of experimental polymer/scCO2 miscibility. The ﬁgure is divided into three regions, based on available experimental cloud point data,3,9,12,14 such that the polymers have been categorized as “no observed miscibility”, “partially miscible”, and “very miscible”. Notice how |ϵpolymer − ϵscCO2| decreases as miscibility increases. The polymers categorized under “no observed miscibility” are those for which two ﬂuid phases persist even under extreme conditions that favor mixing (high T and P). We predict that these have the greatest mismatch in energetics, as reﬂected by the values of |ϵpolymer − ϵscCO2|, and include PAA, PMMA, and PS (high MW). The results of Table 2 show that PAA is the most extreme member of this set, with an ϵ-diﬀerence of over 1300 J/mol. The other two members of the group have ϵdiﬀerences that are considerably smaller, although even for high molecular weight PS the ϵ-diﬀerence, being about 852 J/mol, is still about 9% larger than that for its nearest neighbor, PMA, which is in the “partially miscible category”. The majority of the polymers considered in the present work fall in the intermediate region, labeled “partially miscible”. For many of these cases only a single point of the coexistence curve has been reported, i.e. a single cloud point in the T-P plane for ﬁxed mixture composition, such that the mixture forms a homogeneous ﬂuid phase at higher pressures. The species that fall into this category are PPO, oligomeric PS (o-PS), the poly(n-alkyl acrylates), and the poly(n-alkyl methacrylates)

Table 1. Polymer Abbreviations acronym

full name

scCO2 PMA PEA PPA PBA PMMA PEMA PPMA PBMA PS PDMS PPO PFPOa PAA

supercritical carbon dioxide poly(methyl acrylate) poly(ethyl acrylate) poly(n-propyl acrylate) poly(n-butyl acrylate) poly(methyl methacrylate) poly(ethyl methacrylate) poly(n-propyl methacrylate) poly(n-butyl methacrylate) polystyrene poly(dimethylsiloxane) poly(propylene oxide) poly(hexaﬂuoropropylene oxide) poly(acrylic acid)

a PFPO, poly(hexaﬂuoropropylene oxide), is commonly known as Krytox.

Furthermore, the characterizations were carried out over the same T and P conditions for which the calculated properties discussed below were evaluated. As noted above, it has been suggested that weakly polar polymers and/or those with large free volume content tend to be most miscible with scCO2.1,3,4,6,16 Table 2 includes our values for |ϵpolymer|, the magnitude of the polymer segment− segment interaction energy (the actual numbers are negative), as well as our predictions for % FV, the percent of free volume at 435 K and 150 MPa for each polymer. We thus have the means to test the notion that the strength of the polymer segmental interactions and/or the amount of free volume inﬂuence the miscibility of a polymer with scCO2. The composition of many of the polymer/scCO2 mixtures considered in this work is heavily weighted in favor of the scCO2 (∼95 wt %). As a result, the miscibility of any polymer should strongly depend on the compatibility of its properties with those of scCO2. Carbon dioxide is a nonpolar molecule, and therefore exhibits very weak dipole−dipole interactions but is capable of intermolecular quadrupole interactions.1,2,6,9 Additionally, scCO2 is highly compressible, and the proportion

Table 2. Model Parameters and Calculated Properties for the Pure Components experimental miscibility with scCO2

component name

MW (g/mol)

r

v (mL/mol)

|ε| (J/mol)

|εpolymer − εscCO2| (J/mol)

% FV (435 K, 150 MPa)

− partially misciblea partially misciblea partially misciblea partially misciblea none observeda none observeda unknown partially misciblea none observeda partially misciblea partially miscibleb very misciblec very miscibled none observeda

scCO2 PMA PEA PPA PBA PMMA PEMA51 PPMA PBMA PS o-PSe PPO PFPO PDMS PAA

44.01 31 000 120 000 140 000 62 000 100 000 340 000 100 000 100 000 100 000 910 2000 2500 224 000 100 000

5.0 3571.9 14725.3 17946.4 8111.8 11694.8 12271.4 12688.4 13035.5 14435.7 123.4 272.7 138.8 29865.1 10000.2

6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5

1085.5 1867.8 1750.4 1744.4 1713.3 2028.8 1810.9 1766.1 1764.5 1937.1 1863.4 1688.0 1472.3 1432.3 2447.0

− 782.3 664.9 658.9 627.8 943.3 725.4 680.6 679.0 851.6 777.9 602.5 386.8 346.8 1361.5

28.0 10.3 11.2 11.3 11.5 9.3 10.7 11.1 11.1 9.9 10.5 11.8 13.9 14.2 7.2

a For experimental studies, see: McHugh et al.3 bFor experimental studies, see: Johnston et al.9 cFor experimental studies, see: Enick et al.14 dFor experimental studies, see: Kiran et al.12 eo-PS is oligomeric polystyrene.

D

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Figure 1. EOS P − V isotherms (blue curves) for scCO2 [left] 400, 450 K and PMMA [right] 423.25, 433.25, 442.95, 452.45 K, ﬁt to experimental data points (black circles).39,40

necessary).3 Therefore, the “very miscible” polymers exist as a single ﬂuid phase with scCO2 over a much larger range of T and P than the other mixtures. Note that the polymers in this category, PDMS and PFPO, have the closest energetic match to scCO2, with ϵ-diﬀerences that are just a quarter to a third that of PAA. Our analysis shows that |ϵpolymer − ϵscCO2| serves as a reasonable guide for predicting trends in polymer/scCO2 miscibility, which is to say that our theoretical, pure component characteristic parameter, ϵ, correlates well with experimental data on the mixtures. In other words, we are able to gain insight about polymer/scCO2 miscibility based only on the characterization of the pure component properties. We now turn to the role of polymer free volume in inﬂuencing polymer/scCO2 miscibility. As noted above, in contrast to most polymers, scCO2 has a relatively large fraction of free volume when comparing at the same T and P. In Figure 3, we show how our predictions for the percent free volume, % FV (calculated at 435 K and 150 MPa), correlates with the magnitude of the polymer−polymer interaction energies (|ϵpolymer|) for each of the polymer species in Table 2. Not shown in Figure 3 is scCO2, which is comprised of about 28% FV (at the same T and P), nearly double that of the highest % FV polymer species (PDMS). Before discussing the % FV trend with polymer/scCO2 miscibility, we will ﬁrst highlight the general connection between the ϵ parameter and the degree to which a polymer expands. We have noticed in this and previous studies18,36 that for LCST-type systems a large diﬀerence in thermal expansion coeﬃcients (α = (1/V)(dV/dT)) between components tracks with immiscibility. Indeed, for LCST-type mixtures at pressures closer to atmospheric P, an increase in pressure, which shifts the values of the thermal expansion coeﬃcients into closer agreement, promotes miscibility and raises the LCST. Recall that in our model, the free volume is deﬁned as the diﬀerence between the total volume, V, and the total hardcore volume, Nrv, of all N molecules. By deﬁnition, the free volume is the only contribution to the total volume that can be expanded. Therefore, % FV serves as a measure of how expanded a species is at a given T and P. We have also observed a relationship between α and the pure component interaction energy, ϵ, such that a weaker tendency for a polymer to expand is reﬂected by a larger |ϵ|, or stronger nonbonded segmental interactions.41 Putting these observations together explains the strong correlation we see in Figure 3 between % FV and |ϵpolymer|. In fact, these results are consistent

Figure 2. Comparison of |ϵpolymer − ϵscCO2| with experimental miscibility. The “no observed miscibility”, “partially miscible”, and “very miscible” regimes are separated according to the results of available experimental cloud point data. Inset: Enhanced view of the partially miscible region.

except PMMA. The inset of Figure 2 ranks the ordering of this set according to their |ϵpolymer − ϵscCO2| values. For both families, the poly(n-alkyl acrylates) and the poly(n-alkyl methacrylates), |ϵpolymer − ϵscCO2| decreases as the number of carbon atoms in the n-alkyl group of the acrylate/methacrylate species increases. Experimentally, within each of these families, the observed miscibility trend3 matches the trend in theoretical ϵ-diﬀerences. This relationship suggests an increase in miscibility as the length of the n-alkyl group increases, a trend we will explore further in section 4.1. Finally, the “very miscible” label refers to those polymers for which exceptionally favorable partial miscibility was observed; i.e., they achieved miscibility at temperatures and pressures much lower than the other partially miscible polymers. For example, PFPO is miscible with scCO2 at pressures above ∼11 MPa at 313 K.14 In comparison, a polymer from the “partially miscible” category would not be miscible with scCO2 at all, at this T and P (e.g., PBMA), or would only achieve miscibility at a signiﬁcantly higher P (e.g., for PMA, a P > 200 MPa would be E

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for polymers having low % FV. In contrast, when the polymer segment−segment interaction energies (|ϵ|) are small, the % FV is large, e.g. PDMS. This makes for a better % FV match with that of scCO2, which is consistent with experimental observations that report favorable mixing. In the next section, we provide more detail by examining a number of speciﬁc cases.

4. CASE STUDIES 4.1. Poly(n-alkyl acrylates) and Poly(n-alkyl methacrylates). Having data available for a series of poly(n-alkyl acrylates) and poly(n-alkyl methacrylates) provides the opportunity to observe how miscibility with scCO2 varies within each family of polymers and between analogous pairs across families (e.g., PMA and PMMA). We begin with the poly(n-alkyl acrylate) family, for which experimental cloud point data are available for poly(methyl acrylate) through poly(n-butyl acrylate).3 McHugh et al. experimentally investigated these systems, probing the eﬀect of lengthening the nonpolar acrylate tail on their miscibility with scCO2; all mixtures were 5 wt % of polymer. The “acrylate tail” refers to the n-alkyl chain that branches oﬀ of the acrylate repeat unit, as illustrated in Figure 4. McHugh et al. noted that at a ﬁxed T (e.g., 438 K), lower pressures were needed to reach the one-phase region as the acrylate tail length increased, thus concluding that increasing the acrylate tail length enhances poly(n-alkyl acrylate)/scCO2 miscibility. Brennecke et. al reached a similar conclusion in their investigation of ionic liquid (IL)/scCO2 miscibility, wherein they found that increasing the n-alkyl chain length of the cationic portion of the IL favorably inﬂuenced miscibility.42 Our tabulated parameters for the poly(n-alkyl acrylates) in Table 2 indicate a trend in both |ϵ| and % FV such that as the acrylate tail length increases, |ϵ| decreases, and % FV increases. The strength of the segment−segment interactions weaken upon tail lengthening because of the decreased polarity of the segments, and the % FV increases with increasing acrylate tail length because bulkier substituents lead to less eﬃcient chain packing. We ﬁnd that using ϵ and the % FV as a guide to predict miscibility with scCO2 for the poly(n-alkyl acrylates) yields the same trend as that exhibited by the experimental cloud point observations. For example, consider the trend in |ϵ| values: 1867.8, 1750.4, 1744.4, and 1713.3 J/mol for PMA, PEA, PPA, and PBA, respectively, which track exactly with the observed miscibility trend. At low temperatures (∼320−420 K), we note that the observed miscibility trend is reversed; i.e., as the acrylate tail length decreases, miscibility increases. McHugh et al. suggested that at lower temperatures the strength of the polymer−CO2 energetic interactions dominate the phase behavior. We agree that % FV diﬀerences will not reﬂect miscibility trends near a UCST. In order to explore this

Figure 3. Correlation between |ϵpolymer| and % FV, calculated for each polymer at 435 K and 150 MPa. The curve has been drawn to guide the eye of the reader. The plot shows a strong relationship between the magnitude of the ϵ parameter and free volume; i.e., large values of |ϵpolymer| result in small % FVs.

with those in recent work,41 which showed the same connection between % FV and |ϵ| for a number of polymers. This connection between ϵ, α, and % FV is important because it provides a link between the characteristic parameter, ϵ, and the extent to which a component is expanded, which allows for predictions about the compatibility of components. For example, in comparing blends of PS and of dPS (dPS = deuterated PS) with PVME (poly(vinyl methyl ether)) we related the diﬀerence in the respective miscibility to a ∼40 J/ mol separation in their |ϵ| values, with both α and |ϵ| for dPS being closer to the values for PVME.36 The related increase in compatibility of dPS with PVME is reﬂected in a 40 K increase in the blend LCST. Therefore, in conjunction with the polymer−scCO2 energetic mismatch, another important indicator of polymer/scCO2 miscibility is the amount of polymer free volume. Turning to Figure 3, we note that in cases where the polymer segment−segment interactions are strong (large |ϵ|), the % FV is low (e.g., PAA) leading us to expect poor compatibility between these species and a high % FV species like scCO2. Indeed a comparison with experimental observations on the mixtures indicates that no miscibility with scCO2 was observed

Figure 4. Chemical structures of the repeat units of the poly(n-alkyl acrylates). (From left to right) poly(methyl acrylate) (PMA), poly(ethyl acrylate) (PEA), poly(n-propyl acrylate) (PPA), and poly(n-butyl acrylate) (PBA). The repeat units become increasingly nonpolar as the length of the nonpolar “acrylate tail” increases. F

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and P (for example, at ∼400 K a pressure of ∼170 MPa is needed).3 It was reported that no miscibility with scCO2 was observed for PMMA and poly(ethyl methacrylate) (PEMA) up to 528 K, 255 MPa and 558 K, 255 MPa, respectively.3 Cloud point data for the poly(n-propyl methacrylate) (PPMA) system were not available, however, we have included characterization results for PPMA. The entries in Table 2 corresponding to the members of the poly(n-alkyl methacrylate) family indicate, again, that as the length of the acrylate tail increases, |ϵ| decreases and % FV increases; this matches what our theory predicted for the poly(n-alkyl acrylate) family. Therefore, we expect on theoretical grounds that PBMA would be the best match for scCO2 among the poly(n-alkyl methacrylates). This agrees well with the experimental observation that PBMA was the only poly(n-alkyl methacrylate) that exhibited partial miscibility with scCO2. At the other extreme in this family, the |ϵ| of PMMA is more than 250 J/mol larger and its % FV nearly 2% lower. The pure component properties of PMMA are strongly shifted in an unfavorable direction, i.e., away from those of scCO2 (see for example, Figure 3), and therefore suggest PMMA would be a signiﬁcantly poorer match with scCO2, consistent with the lack of miscibility observed experimentally. Experimental cloud point data for PPMA/scCO2 were unavailable, however, our analysis suggests partial miscibility with scCO2 given the proximity of its pure component properties to those of other polymers, such as PPA and PBMA (see Figures 2 and 3), for which partial miscibility has been recorded. Although McHugh et al. did not observe miscibility of PEMA/scCO2 up to 558 K and 255 MPa, our predictions suggest PEMA is likely to fall into the “partially miscible” category based on its |ϵ| and % FV. First, the energetic analysis suggests partial miscibility, since |ϵ| for PEMA is situated between the values for PMA and PEA, both of which are partially miscible with scCO2. In addition, we predict a relatively large % FV for PEMA, compared to PMMA, such that its value is quite close to that of PBMA, observed in Figure 5. Our ranking agrees with that from positron lifetime (PALS) experiments on poly(n-alkyl methacrylates),44 which were performed to probe the amount of free volume. PALS results indicated that PEMA has signiﬁcantly more free volume than PMMA between 420 and 450 K at 1 atm, and only slightly less than that of PBMA. Therefore, our placement of PEMA in Figure 2 seems reasonable, despite the reported absence of miscibility for the mixture with scCO2. Turning back to Figure 5, our results indicate the analogous trend for poly(n-alkyl methacrylates) as observed for the poly(n-alkyl acrylates), viz. an inverse relationship between the experimental Tg’s and our calculated free volumes. This provides additional support for clear links between Tg, % FV, and miscibility with scCO2. Finally, we compare analogous pairs of poly(n-alkyl acrylate) and poly(n-alkyl methacrylate) family members, the only structural diﬀerence being the additional methyl group along the polymer backbones of the poly(n-alkyl methacrylates). While this might appear to be a small structural modiﬁcation it strongly aﬀects pure component properties. Experimentally, members of the methacrylate family have Tg values between about 70 and 90 K higher than the acrylate family, and our theoretical results for the % FV (under the same T and P conditions) predict a decrease in the methacrylates ranging from about 4% to 10%, relative to the acrylates. In both

behavior more closely, we would need to obtain values for g that require correlation with experimental data, thus going well beyond the geometric mean approximation that appears to suﬃce near the LCST. McHugh et al. suggested that at high T the amount of polymer free volume more strongly inﬂuences miscibility than the strength of intermolecular interactions. They characterized the amount of polymer free volume indirectly via the experimentally measured glass transitions, using T − Tg. In general, a higher Tg signiﬁes a material with less intrinsic free volume; McHugh et al. concluded Tg and miscibility with scCO2 are inversely related for the series of poly(n-alkyl acrylates). In Table 3, we list the experimental Tg’s along with Table 3. Poly(n-alkyl acrylate) and Poly(n-alkyl methacrylate) Glass Transition Temperatures family poly(n-alkyl acrylates)

poly(n-alkyl methacrylates)

polymer

Tg (K)

% FVc (435 K, 150 MPa)

PMA PEA PPA PBA PMMA PEMA PPMA PBMA

282a 250a 236a 224a 378a 336a 306b 293a

10.3 11.2 11.3 11.5 9.3 10.7 11.1 11.1

a Experimental Tg’s for the poly(n-alkyl acrylates) and poly(n-alkyl methacrylates) reported in McHugh et al.3 bExperimental Tg of PPMA reported by Mark.43 cOur calculations for % FV.

our calculated % FVs for the poly(n-alkyl acrylates) and poly(nalkyl methacrylates) (discussed below). Note that within a family of polymers there appears to be an inverse relationship between Tg and the model values for % FV; this is illustrated further in Figure 5. First, consider the poly(n-alkyl acrylate)

Figure 5. Calculated % FVs and their corresponding experimental Tg’s for the poly(n-alkyl acrylates) and poly(n-alkyl methacrylates).

family: As the length of the acrylate tail increases, the side branch becomes increasingly bulky, the % FV increases and the experimental Tg values decrease, and miscibility with scCO2 becomes increasingly favorable. Next, consider the poly(n-alkyl methacrylate) family. Experimentally, only poly(n-butyl methacrylate) (PBMA) was found to exhibit even partial miscibility with scCO2 at high T G

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corresponding to 373 K and 52.4 MPa. At the same temperature, the cloud point for the PMA mixture was observed at ∼215 MPa. In order to gain insight about what is driving the more favorable miscibility of o-PS/scCO2 it is useful to consider the entropic and enthalpic contributions to the free energy of mixing, and we turn to this subject in section 5. 4.3. Poly(propylene oxide) and Poly(hexaﬂuoropropylene oxide). Experimental data suggest that polymer miscibility with scCO2 increases upon ﬂuorination. Systems for which this phenomenon has been observed include: poly(nalkyl acrylates) and poly(n-alkyl methacrylates),5,16 poly(vinyl esters),15,17 and polystyrenes.15 It has been proposed that the increased miscibility of ﬂuorinated polymers (ﬂuoropolymers) with scCO2 may be a result of speciﬁc scCO2−ﬂuoropolymer interactions and/or complex formation.5,15,45 Here we compare one such system for which cloud point data were available, poly(propylene oxide) (PPO) and its ﬂuorinated counterpart poly(hexaﬂuoropropylene oxide) (PFPO). The cloud point experiments on a 1 wt % PPO (2000 g/mol) mixture with scCO2 indicated miscibility at 303 K for pressures above 21.4 MPa.9 However, a 5 wt % PFPO (2500 g/mol) mixture with scCO2 was miscible at 313 K and only 11 MPa.14 These results suggest more favorable miscibility for PFPO/scCO2 than PPO/ scCO2 because a higher concentration of PFPO was miscible with scCO2 at lower P and comparable T.46 From Figure 2 (and Table 2) we see that |ϵ| is more than 200 J/mol larger for PPO than PFPO, which is the greatest disparity between any two chemically similar polymers considered in our work. Note that, in terms of |ϵ|, PPO’s nearest neighbor is PBA, which also exhibits only partial miscibility with scCO2. In comparison, |ϵ| for PFPO is closer to that of PDMS, which is very miscible with scCO2. A detailed discussion regarding PDMS/scCO2 miscibility is contained in the following section; here, we only refer to it as a reference point for the purpose of examining PFPO. Related to the large |ϵ| diﬀerence between PPO and PFPO, we ﬁnd that their % FVs (see Figure 3) diﬀer by more than 2%, with the weaker nonbonded interactions experienced in PFPO being reﬂected in a larger percentage of free volume, one that more closely matches that of scCO2. Recall the hypothesis mentioned above that speciﬁc scCO2− ﬂuoropolymer interactions and/or complex formation may be responsible for the increased miscibility of ﬂuoropolymer/ scCO2 mixtures. Experimental45 and theoretical investigations5 have probed both the existence and strength of these speciﬁc interactions in several ﬂuoropolymer/scCO2 systems. These studies suggest that although only weak ﬂuoropolymer−scCO2 complexes likely form in these mixtures, the eﬀect of ﬂuorination may well produce a small increase in the mixed interaction strength, and that the presence of speciﬁc interactions could have a signiﬁcant impact on polymer/ scCO2 miscibility. In our model, it is the magnitude of the mixed interaction parameter, ϵij, that captures the strength of the energetic interactions between mixture components. This must be characterized using experimental information about the mixture (see section 2) and, in fact, it is the dearth of such data that has motivated the focus on pure component properties in this paper. If mixture data were available such that the ϵij’s could be characterized, an ϵij for PFPO/scCO2 that is greater in magnitude than that of PPO/scCO2 would suggest the presence of more favorable, or in this case, speciﬁc mixed

categories the most dramatic diﬀerence is observed in the case of the shortest alkyl branch, i.e., PMMA relative to PMA. This is consistent with the signiﬁcant shift in experimentally observed miscibility recorded for PMA with scCO2 (partially miscible) relative to that for PMMA (immiscible). In this section we have made connections between experimentally observed miscibility behavior and the results of our theoretical characterization for two families of polymers. We have also shown additional links to experimental data: Our % FV predictions for the poly(n-alkyl methacrylates) agree well with PALS results on these systems. In addition, within families, at a ﬁxed choice of T and P, there is a strong correlation between % FV and the polymer glass transition. The latter is particularly intriguing, since our LCL model predictions for % FV make use of characteristic parameters obtained by ﬁtting melt PVT data, such that we take signiﬁcant care to remain tens of degrees away from the region of the glass transition. 4.2. PS: High and Low Molecular Weights. PS of MW greater than 100 000 g/mol does not exhibit any miscibility with scCO2 up to 498 K and 210 MPa.3 However, the oligomeric PS (o-PS) (MW < 1000 g/mol) mixture is miscible at 373 K and 52.4 MPa.2,3 It is not unusual to observe experimental changes in polymeric miscibility when the molecular weight changes by orders of magnitude; in this case we seek to determine whether the origin of the experimentally observed trend is contained in our theoretical characterization of the high- and low-MW PS samples. In fact, we do ﬁnd a diﬀerence in |ϵ| in going from oligomeric to polymeric PS molecular weight, as illustrated in Figure 2. This is consistent with earlier work using our LCL theory,20 in which we found that when the samples are oligomeric (below roughly several thousand g/mol) or very high molecular weight (on the order of 106 g/mol) it is important to have experimental data in the range of interest. On the other hand, the characteristic parameter values turn out to be extremely robust within a large swathe of intermediate molecular weight values. For example, we found that using the same ϵ and v values for PS ranging between 20 000 g/mol and 200 000 g/mol (along with a constant r/MW) allowed us to predict the experimentally observed shift in LCST and coexistence curve for a set of PS/poly(vinylmethyl ether) (PVME) blends.21 Here our results indicate that o-PS is a better energetic match with scCO2 than high MW PS by ∼70 J/mol. Examination of the experimental PVT data for the two PS samples reveals that o-PS expands more readily with increasing T,39 which implies that it has a larger α value. Recall the close correlation we ﬁnd between α and |ϵ|: the larger thermal expansion coeﬃcient for o-PS is likely due to the inﬂuence of chain ends. This translates into a smaller magnitude of |ϵ|, which signiﬁes somewhat weaker nonbonding interactions, hence better compatibility with scCO2. Turning to our other important metric, Figure 3 shows that we also predict a better match in % FV between o-PS and scCO2, relative to high MW PS. The location of o-PS in Figure 3 is closest to that of PMA, whose partial miscibility with scCO2 was discussed in the previous section. Indeed, we ﬁnd % FV and |ϵ| values for o-PS and PMA that are very similar: 10.5% and 1863.4 J/mol for o-PS, 10.3% and 1867.8 J/mol for PMA. However, in terms of their experimental miscibility with scCO2, a much higher P is needed to achieve miscibility for the PMA mixture than for the o-PS mixture, at the same T. Only one cloud point datum was reported for the o-PS mixture, H

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the mixture itself, predicting the eﬀects of changing temperature, pressure, and composition on the free energy, and related quantities such as the chemical potentials, and the enthalpy and entropy of mixing. In the case of polymer/scCO2 solutions we are faced with a signiﬁcant constraint, however, in that very limited mixture data are available, and certainly not enough to characterize the mixed state for our complete set of polymers. We therefore turn to some of the features of our theoretical model that have allowed us to deduce information about thermodynamic contributions to mixing even in the absence of having values for the parameter, g, that characterizes mixed interactions. In previous work18 we have found that for partially miscible polymer blends exhibiting a LCST, the g parameter serves largely to control ΔHmix. For example, we have shown that rescaling g from its geometric mean estimate (where g = 1; ϵij = (ϵiiϵjj)1/2) so as to produce up to a 25% shift in the mixed interaction energy produces a dramatic eﬀect on ΔHmix, but has relatively little impact on the results for ΔSmix. Conversely, we have discovered that for such mixtures it is the diﬀerence |ϵii − ϵjj| that strongly aﬀects ΔSmix, such that an increase in the ϵdiﬀerence results in a more negative, ΔSexcess mix while having very little eﬀect on ΔHmix. The net result is a more dominant and unfavorable excess contribution to the entropy of mixing. For example, we have predicted that for a PS/PVME-type polymer blend that modifying the ϵ-diﬀerence by up to 25% would produce a large ΔSexcess shift, while leaving ΔHmix values mix relatively unchanged. Given that we do not have the experimental data needed to determine g values for all of the solutions of interest, we shall proceed by using the geometric mean approximation throughout for g, since we need a numerical value in order to undertake further calculations. Under rare circumstances, using the geometric mean of the pure component values ϵii and ϵjj, such that g = 1, provides a good result for ϵij. For example, using the approximation of g = 1 for PE/alkane mixtures predicts the lower critical solution temperature very well, even capturing its dependence on the alkane chain length.22 While there are few mixtures for which g = 1 represents an optimal choice, some insight can still be gained by using this approximation for immiscible mixtures or mixtures for which no data are available. This means that our predictions will be focused on the entropy of mixing, with the goal of investigating the extent to which experimentally observed trends in miscibility can be correlated with this aspect of the mixture thermodynamics. Listed in Table 4 are the results of our calculations for TΔS̅mix in units of J/g for the case in which each solution is 5 wt % in polymer, except for PFPO (more on this below). TΔS̅mix was calculated using only the pure component parameters listed in Table 2, and g = 1 to model all of the polymer/scCO2 mixtures at a constant temperature (T = 435 K) and pressure (P = 150 MPa). There is a clear ordering from top (most favorable TΔS̅mix) to bottom (least favorable TΔS̅ mix), which tracks with the overall polymer/scCO 2 experimental miscibility trend. With our predictions for TΔS̅mix in hand, recall the similarity between our calculated % FV and |ϵ| values for o-PS and PMA: 10.5% and 1863.4 J/mol for o-PS, 10.3% and 1867.8 J/mol for PMA, despite cloud point data that indicated o-PS was more miscible with scCO2 than PMA (at 373 K, o-PS was miscible at P = 52.4 MPa, while P = ∼215 MPa was needed for the PMA mixture). The results of Table 4 provide an explanation: TΔS̅mix

interactions. This behavior would serve to enhance the miscibility of PFPO/scCO2 relative to PPO/scCO2 via the enthalpic contribution to the free energy of mixing, ΔHmix, because of the relationship between more favorable ϵij’s and more negative (favorable) ΔHmix values that we have observed previously.18 As we shall address below (section 5), even in the absence of such information we are able to predict the relative entropic contributions to the free energy of mixing, ΔSmix, based only on our characterization of the pure components. We will see that trends in the entropy of mixing provide a revealing picture within which to discuss the observed experimental miscibility behavior. 4.4. Poly(dimethylsiloxane) and Poly(acrylic acid). In this section, we compare the two polymers, poly(dimethylsiloxane) (PDMS) and poly(acrylic acid) (PAA), for which we calculate the largest discrepancy in pure component properties. The purpose is to connect their contrasting pure component characteristics and their respective experimental behavior with scCO2. Beginning with PDMS, we note that, in addition to the ﬂuoropolymers mentioned in the previous section, siliconbased polymers like PDMS also exhibit exceptional miscibility with scCO2.1,4,6,7,12 PDMS is miscible over a wide range of polymer MWs and at relatively low T and P. Cloud point data show that for a 5 wt % polymer mixture, PDMS/scCO2 (MW= 369 200 g/mol) is miscible at 400 K for pressures above ∼45 MPa.12 Note that this mixture exhibited both UCST-type and LCST-type phase behavior. Upon decreasing MW, no signiﬁcant enhancement in miscibility was observed until the MW was decreased to 93 700 g/mol, which shifted the PDMS/ scCO2 cloud point at 400 K down to ∼40 MPa. Of all the polymers considered in this work, we ﬁnd the smallest value of |ϵ| (1432.3 J/mol) and the largest % FV (14.2%) for PDMS. Of course, the two features are related, as has been noted already. This places PDMS closest to scCO2 in terms of both properties, which correlates well with its experimentally observed miscibility. Turning to PAA, this polymer is polar and hydrophilic, similar in chemical structure to that of PMA, but with a signiﬁcant functional group substitution; i.e., the single methyl group comprising the acrylate tail of the PMA repeat unit is replaced with a hydroxyl group. The substitution of the nonpolar methyl group with a polar hydroxyl group has a noticeable eﬀect on miscibility with scCO2; i.e., while PMA is partially miscible, studies on a 5 wt % PAA mixture with scCO2 did not exhibit miscibility even going up to the relatively extreme conditions of 545 K and 222 MPa.3 The strong dipole−dipole interactions and the ability of PAA to form hydrogen bonds do not favor miscibility in the presence of the weak intermolecular interactions of nonpolar scCO2. Our calculations reﬂect these strong PAA interactions, as its |ϵ| value (2447.0 J/mol) is the largest of the set of polymers listed in Table 2, and Figure 2 conﬁrms the signiﬁcant mismatch in |ϵ| values between PAA and scCO2. Furthermore, Figure 3 illustrates that the % FV of PAA is 7.2%, the lowest of all that we have calculated; the free volume of PAA is a poor match for that of highly compressible scCO2.

5. THERMODYNAMICS OF MIXING To this point we have correlated scCO2/polymer miscibility with trends and diﬀerences in pure component properties, alone. This is in contrast to the more direct approach one might employ using our LCL theory, which would be to model I

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Table 4. Ranking of TΔSm̅ ix with Experimental Miscibility experimental miscibility with scCO2 very miscible partially miscible

no observed miscibility

polymer

TΔS̅mix (J/g) (435 K, 150 MPa)

PDMS PFPOa PPO o-PS PBA PPA PBMA PPMA PEA PEMA PMA PS PMMA PAA

2.92 2.86 2.68 2.57 2.22 2.10 2.08 2.02 1.99 1.84 1.62 1.70 1.22 0.04

Figure 6. Total entropic contribution to the free energy of mixing, TΔS̅mix, is shown as a function of P (100−200 MPa) for ﬁxed T = 435 K and 5 wt % polymer. (Inset) In order to account for the signiﬁcantly higher density of PFPO relative to the other polymers, we plot TΔS̅mix such that all of the mixtures contain the same volume fraction of polymer (equivalent to that of PFPO at 5 wt %).

a TΔS̅mix was calculated such that the segment fraction of PFPO in solution was roughly the same as those of the other polymer/scCO2 solutions at 5 wt %; this corresponds to approximately a 10 wt % PFPO mixture with scCO2.

the free energy of mixing. As an aside, using the relationships of classical thermodynamics we therefore predict that the volume change upon mixing decreases as T increases, with P ﬁxed. Inclusion of the other polymers that exhibit partial miscibility would not change the trends observed; indeed, the other members of this class produced curves that created a band around the PBMA result (note that PBMA falls roughly in the middle of the “partially miscible” category in Table 4). In the lower half of the pressure range we predict that while the entropy of mixing is favorable for the very miscible polymers PDMS and PFPO, as well as for the partially miscible PBMA, there is an unfavorable entropic contribution to the free energy of mixing for PAA/scCO2, until intermediate pressures are reached, while at pressures above about 150 MPa entropic contributions favor all of the mixtures. The fact that the PFPO curve looks notably diﬀerent from the others requires some comment. We have chosen to plot these results for solutions “made up” in a manner analogous to experiment, i.e., as a weight percent. However, PFPO is extremely dense; it has a speciﬁc volume under ambient conditions of 0.5392 mL/g,39 which is smaller than that of most other polymers by a factor of about 2. This means that a 5 wt % sample will contain signiﬁcantly fewer moles of polymer than the equivalent solution for each of the other polymers, even though the PFPO sample in the solution studied experimentally had a much lower molecular weight (2500 g/mol). Using the weight percent protocol for comparing mixtures, and having one “rogue” polymer that is enormously more dense than the others, thus signiﬁcantly skews the ordering of ΔSideal mix (see eq 4), since that is the one thermodynamic quantity that is very dependent on how many molecules of each type are being mixed. As shown in Figure 6, this reveals itself in the calculations for the total TΔS̅mix through the unexpected positioning, and diminished slope, of the curve for the PFPO solution. If we redo the calculation for a series of solutions that are the same segment fraction of polymer in solution as a 5 wt % PFPO solution (which, assuming no volume change on mixing, would be equivalent to having the same volume fraction) then the ordering shifts to what would be expected given the molecular size diﬀerences; this is illustrated by the inset plot, where the PFPO solution is topmost, as expected, due to the

is more favorable for o-PS than PMA by nearly 1 J/g, a diﬀerence that essentially spans the set of 10 partially miscible polymers in the table. In fact, it is the ideal contribution, ΔSideal mix , to ΔSmix that is responsible for the more favorable TΔSm̅ ix of oPS than PMA, due to its oligomeric size. The stronger inﬂuence of ΔSideal mix for o-PS than PMA is made even more evident by comparing their excess entropic contributions, TΔSexcess ̅ ix , which m are similar in magnitude: −2.78 and −2.49 J/g for o-PS and PMA, respectively, at T = 435 K, P = 150 MPa. ΔSideal mix makes up the remaining contribution to ΔSmix (see Section 2) and accounts for the more favorable TΔS̅mix for o-PS, thus explaining the diﬀerence in experimental miscibility between o-PS and PMA. In addition, we have previously highlighted the experimental miscibility trend for the poly(n-alkyl acrylate) family of mixtures with scCO2 at high temperatures, where miscibility increased for increasing n-alkyl tail length, i.e. from PMA to PBA. The corresponding TΔSm̅ ix values from Table 4 are 1.62, 1.99, 2.10, and 2.22 J/g for PMA, PEA, PPA, and PBA, respectively. Thus, we ﬁnd that our LCL model predictions for TΔS̅mix exactly mirror the high temperature experimental miscibility trend at high-T for the poly(n-alkyl acrylate)/ scCO2 family of mixtures. Figure 6 shows the results of our calculations for TΔSm̅ ix for a selected subset of mixtures from Table 4: scCO2 mixed with, respectively, PAA, PBMA, PFPO, and PDMS. For clarity’s sake we use only the PBMA results to represent the partially miscible category. In the main ﬁgure, we plot the product TΔS̅mix, i.e., the total entropic contribution to the free energy, as a function of P over the range of 100−200 MPa, at a constant temperature T = 435 K in units of J/g for the case in which each solution is 5 wt % in polymer. There is a clear ordering of the curves from top (most favorable TΔSm̅ ix) to bottom (least favorable TΔS̅mix) over the entire pressure range (more about this below). As expected from the discussion above, we ﬁnd that this ordering is not sensitive to changes in g.47 A feature shared by all of the polymers is that an increase in the pressure encourages mixing by producing an increase in TΔS̅mix and therefore a more favorable entropic contribution to J

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same for all of the mixtures, then the result is as shown in the inset, and we see that the PDMS and the PFPO curves are essentially identical, reﬂecting their extremely close values of ϵ.

very large ideal entropy contribution from mixing an oligomeric sized sample with the scCO2. Having discussed the trend in ΔSmix, we further explore the entropic contribution to the free energy of mixing by calculating ΔSexcess mix , which accounts for all of the nonideal entropic eﬀects upon mixing; we determine this quantity by subtracting the ideal contribution from the total entropy of mixing. As summarized at the start of this section, we have previously observed in LCST blends that the magnitude of ΔSexcess mix tracks with the diﬀerence |ϵii − ϵjj|. In the present case, since one of the components is always scCO2 we would expect the ordering of ΔSexcess mix should match the ordering of polymer ϵvalues, such that as the magnitude of ϵ increases so does the magnitude of the unfavorable ΔSexcess mix . Figure 7 shows the excess

6. SUMMARY AND CONCLUSIONS The goal of this work was to provide a deeper understanding of the thermodynamic behavior of polymer/scCO2 mixtures by exploiting the ability of our simple locally correlated lattice (LCL) theory to make connections between experimental behavior and model characterization. In particular, given the signiﬁcant lack of data on the mixed systems, we have aimed toward pushing the boundaries of what we can predict using pure component analysis, alone. We ﬁnd that both the strength of the nonbonded pure polymer segment−segment interactions and the percent free volume track miscibility with scCO2. Using our theory we have calculated these quantities for a group of polymers that range from very miscible with scCO2, to those for which no miscibility was observed. We ﬁnd a correlation between the strength of pure component segment−segment interaction energies and free volume; weaker polymer segment−segment interaction energies exist in higher free volume species. The trend that we identiﬁed corresponds to the results of experimental studies that investigated the miscibility of these polymers with scCO2, and thus act as a guide to predict miscibility. The energetic and free volume diﬀerences between a polymer and scCO2 manifest in the entropic contribution to the free energy of mixing, ΔSmix. Using pure component characteristics, alone, we predict the most favorable entropic contributions to the free energy of mixing, ΔGmix, for the very miscible polymer/scCO2 mixtures, such as PDMS/scCO2; indeed, our predictions for the entire series mirrors experimental miscibility ﬁndings. This trend in ΔSmix was traced to the excess entropic contribution to the total entropy of mixing, ΔSexcess mix , which captures all of the nonideal entropic eﬀects of mixing, including the eﬀects of free volume and nonzero volume changes on mixing. The more direct route to characterize the mixed (here, polymer−scCO2) interaction energy would be to model the mixture itself; however, some experimental data that reﬂect the strength of these interactions would be required. Because these data were unavailable for most of the polymer/scCO2 mixtures of interest, we have relied on our parametrization of the pure components to make predictions about the mixtures. This meant that we could not accurately assess the role of the enthalpy of mixing, ΔHmix, in inﬂuencing polymer/scCO2 mixture miscibility. However, our analyses using pure component interaction strength and % FVs, as well as our theoretical predictions for ΔSmix exactly track with the experimentally observed miscibility behavior. We expect the correlations to prove useful for predicting the miscibility of other polymers with scCO2. Furthermore, recent work by Balsara et al.48 addressed the need to develop an understanding of the underlying thermodynamic behavior of multicomponent polymer/scCO2 mixtures. In these systems, a pure polymer or blend matrix absorbs scCO2, resulting in an inhomogeneous mixture, where pressure is typically varied to control solubility. The sorption of scCO2 can then result in desirable material properties, including: swelling, plasticization, and foaming. We direct the reader to notable review articles by Tomasko et al.49 and Spontak et al.50 that cover these areas of current interest, and to which our LCL model could be extended.

Figure 7. Excess entropic contribution, TΔS̅excess mix , to the total entropy of mixing is shown as a function of P (100−200 MPa) for ﬁxed T = 435 K and 5 wt % polymer. (Inset) Analogous to the inset of Figure 6, we plot TΔS̅excess mix such that the volume fraction of polymer in all of the mixtures are equal to that of a 5 wt % PFPO solution.

entropic contribution to the free energy of mixing, TΔSexcess ̅ ix for m the set of 5 wt % solutions, plotted as a function of P for a ﬁxed T = 435 K. The excess contribution, being negative, works against the ideal contribution, and serves to diminish the drive toward mixing. The ordering of |ϵ|-values (in J/mol) is 1432.28 (PDMS), 1472.32 (PFPO), 1764.52 (PBMA), and 2447.03 (PAA). First, we note that these predictions follow the same general behavior as the total entropies of mixing shown in Figure 6: as P increases, TΔS̅excess mix becomes less unfavorable; i.e., increasing P reduces the excess entropic penalty to the free energy of mixing. The least miscible polymer in this set, PAA, shows the most unfavorable excess entropy of mixing throughout the entire pressure range. Next is the partially miscible PBMA solution, the values for which are slightly less unfavorable than the PAA; the TΔS̅excess mix values of the other partially miscible mixtures that we have discussed in this work, but are not shown in the ﬁgure, also fall in this general region. The two miscible mixtures, PDMS and PFPO are at the top; if we credit the notion that the order of the curves should essentially track the order of the ϵ-values for the polymers then there appears to be some discrepancy, since these two polymers have ϵ-values that are very close, and yet the PFPO curve is signiﬁcantly higher than that for the PDMS. However, once again, this is related to the choice of using a constant wt % for all the mixtures, with one polymer being twice as dense as the rest. If, as for Figure 6, we do the same calculation keeping the volume fraction the K

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(30) Beckman, E. J.; Koningsveld, R.; Porter, R. S. Macromolecules 1990, 23, 2321−2329. (31) Ozkan, I. A.; Teja, A. S. Fluid Phase Equilib. 2005, 228−229, 487−491. (32) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Ind. Eng. Chem. Res. 2012, 51, 3832−3840. (33) Guo, Z.; Burley, A. C.; Koelling, K. W.; Kusaka, I.; Lee, L. J.; Tomasko, D. L. J. Appl. Polym. Sci. 2012, 125, 2170−2186. (34) Liu, D.; Li, H.; Noon, M. S.; Tomasko, D. L. Macromolecules 2005, 38, 4416−4424. (35) Guggenheim, E. A. Proc. R. Soc. London, Ser. A 1944, 183, 203. (36) White, R. P.; Lipson, J. E. G.; Higgins, J. S. Macromolecules 2010, 43, 4287−4293. (37) Another route to estimate this quantity would be to use Guggenheim’s expression,35 a slight modiﬁcation of the Flory− Huggins form. However, as noted in the text, changing the particular form used does not signiﬁcantly change the results of our calculations. (38) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (39) Zoller, P.; Walsh, D. Standard Pressure-Volume-Temperature Data for Polymers; Technomic Pub Co.: Lancaster, PA, 1995. (40) Mantilla, I. D.; Cristancho, D. E.; Ejaz, S.; Hall, K. R.; Atilhan, M.; Iglesias-Silva, G. A. J. Chem. Eng. Data 2010, 55, 4611−4613. (41) White, R. P.; Lipson, J. E. G. Macromolecules 2014, 47, 3959− 3968. (42) Muldoon, M. J.; Aki, S. N. V. K.; Anderson, J. L.; Dixon, J. K.; Brennecke, J. F. J. Phys. Chem. B 2007, 111, 9001−9009. (43) Mark, J. E. Physical Properties of Polymers Handbook, 2nd ed.; Springer: Berlin, 2007. (44) Kilburn, D.; Dlubek, G.; Pionteck, J.; Alam, M. A. Polymer 2006, 47, 7774−7785. (45) Dardin, A.; DeSimone, J. M.; Samulski, E. T. J. Phys. Chem. B 1998, 102, 1775−1780. (46) Critical compositions (ϕcrit’s) for polymer solutions are typically skewed toward solvent-rich values.23 For polymer/supercritical ﬂuid mixtures, ϕcrit is generally in the range of 3−15 wt % of polymer.1 Thus moving toward even lower compositions of polymer, i.e. lower wt %, favors mixing. However, the cloud point data for a 1 wt % PPO mixture with scCO2 and a 5 wt % PFPO mixture with scCO2 indicate that PFPO/scCO2 is miscible at a lower pressure and comparable temperature, despite the lower wt % of polymer in the PPO/scCO2 solution. (47) We tested this assertion by increasing the magnitude of g by 1% (g = 1.01) for all of the mixtures (results not shown). This is a considerable change in g given the range of g values that we have characterized for a number of polymer blends and solutions.20 However, we found no change in the order of the curves from that shown in Figure 6. (48) Inceoglu, S.; Young, N. P.; Jackson, A. J.; Kline, S. R.; Costeux, S.; Balsara, N. P. Macromolecules 2013, 46, 6345−6356. (49) Tomasko, D. L.; Li, H.; Liu, D.; Han, X.; Wingert, M. J.; Lee, L. J.; Koelling, K. W. Ind. Eng. Chem. Res. 2003, 42, 6431−6456. (50) Walker, T. A.; Frankowski, D. J.; Spontak, R. J. Adv. Mater. 2008, 20, 879−898. (51) PEMA exhibits partially crystalline behavior not present in the other poly(n-alkyl methacrylates).52 We undertake parametrization for PEMA over the temperature range of 420−450 K, to ensure that we are capturing the behavior of the polymer only in the melt phase, even at elevated pressures. (52) Walsh, D. J.; Ougizawa, T.; Tuminello, W. H.; Gardner, K. H. Polymer 1992, 33, 4793−4797.

AUTHOR INFORMATION

Corresponding Author

*(J.E.G.L.) E-mail: [email protected] Notes

The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS We appreciate the ﬁnancial support provided by the National Science Foundation (Grant No. DMR-1104658) and the Graduate Assistance in Areas of National Need (GAANN) fellowship program.

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