Ionic Liquid Solutions

Article pubs.acs.org/Macromolecules

Origins of Unusual Phase Behavior in Polymer/Ionic Liquid Solutions Ronald P. White and Jane E. G. Lipson* Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, United States

ABSTRACT: Recent experimental studies [Lee et al. Macromolecules 2012, 45, 3627−3633] on polymer/IL solutions of poly(ethylene oxide) (PEO) and 1-ethyl-3-methylimidazolium tetrafluoroborate ([EMIM][BF4]) show some striking trends in the phase boundaries. While most polymer solutions have critical compositions (ϕcrit) that are solvent-rich, these systems had phase diagrams that were either roughly symmetric (ϕcrit ≈ 50%), or, in other related systems, had ϕcrit shifted even more toward polymer-rich compositions. In this work, we carry out a modeling investigation to probe the origins of this shift in the location of the critical composition. There appear to be two underlying sources: The first is a significantly stronger cohesive energy density, and lowered free volume, for the pure solvent component relative to the pure polymer component. This manifests itself as a “role reversal” in the pure component energetic parameters, compared to those of more typical polymer solutions. The resulting unfavorable entropic effects in the mixture produce a coexistence curve shifted toward compositions that are richer in the component with the lower cohesive energy densityin this case the polymer component. The second factor is the aggregation of the ionic liquid component, effectively reducing (and repositioning) the ideal contribution to the entropy of mixing relative to what would result if the IL were fully dispersed. Evidence for IL aggregation is discussed in the context of a simple model to predict aggregation in a dielectric continuum. In agreement with experiment, both of these factors (role reversal in energetics and IL aggregation) serve to shift the critical composition to the more polymer-rich side of the phase diagram compared to what is found for more typical polymer solutions.

I. INTRODUCTION

In the present work, we use a modeling approach to investigate the thermodynamics and phase behavior observed in some recent experimental studies of polymer/IL solutions. In experiments carried out by Lee et al.7,8 on polymer/IL solutions of poly(ethylene oxide) (PEO) and 1-ethyl-3-methylimidazolium tetrafluoroborate ([EMIM][BF4]), there were some striking results observed in the phase boundaries. Some cloud point data taken from this work are shown in Figure 1. Notably, the phase diagrams for these systems are fairly symmetric with critical compositions (ϕcrit) around roughly 50% mass fraction. This is in strong contrast to what is normally observed in typical polymer solutions that show very solvent-rich ϕcrit values. Flory−Huggins (FH) theory9 has often been used as a source of modeling insight into the behavior of polymer solutions. One of its successes has been its ability to predict that

There has been considerable recent interest in ionic liquids (IL) due to their unique and useful properties.1−4 Important characteristics of IL’s include their low melting point and large liquid range, along with high thermal stability and a very low vapor pressure. These features make them an option in developing “green chemistry”. The large variety of possible combinations of cations and anions available also yields significant options for “tuning” the desired physical properties. IL’s thus have potentially wide application in mixtures and solutions; indeed, they have already shown a good ability to solvate a range of solutes. Included in these possibilities are IL mixtures with polymers, systems which suggest a unique spectrum of opportunities in processing and materials applications (see for example refs 5 and 6). In considering mixture applications for polymers in ionic liquids, it is of central importance to understand their miscibility behavior and therefore to analyze the fundamental underlying thermodynamics which is driving it. © XXXX American Chemical Society

Received: April 5, 2013 Revised: May 30, 2013

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dx.doi.org/10.1021/ma400710z | Macromolecules XXXX, XXX, XXX−XXX

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II. THEORETICAL BACKGROUND AND IMPLEMENTATION In this section, we briefly introduce the equation of state (EOS) which we will apply to the PEO/[EMIM][BF4] mixture. It is derived using a lattice-based theory for chain molecule fluids and is simple to implement, having been successfully applied to a range of small molecule mixtures as well as polymeric melts, blends, and solutions. Incorporated into the model are the important effects of free volume (the model is compressible) and nonrandom mixing. Unlike Flory−Huggins theory9 and similar theories that attempt to treat only the mixture properties relative to the pure states, the theory effectively models the experimental characteristics of the pure components. We have recently found that this can produce some insight on the expected nature of the mixture even in the absence of mixture data. We refer to refs 13 and 14 as sources for detailed derivations and background, as well as for recent examples and explanations on how to apply the model. In brief, the approach follows an integral equation formalism to derive temperature dependent nearest neighbor segment−segment probabilities. This leads to an expression for the internal energy (U) which is integrated from an athermal reference state to give the Helmholtz free energy (A); from the result for A all remaining thermodynamic properties can be derived. The analytic result for A, for the case of a binary mixture comprised of components i and j, is as follows. 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 ⎝ ϕh ⎠ ⎝ ϕi ⎠ ⎝ ϕj ⎠ ⎡ ⎤ ⎛ −ε ⎞ ⎛ −εij ⎞ Nq i iz ln⎢ξi exp⎜ ii ⎟ + ξj exp⎜ − ⎟ + ξh⎥ ⎥⎦ ⎢⎣ 2 ⎝ kBT ⎠ ⎝ kBT ⎠ ⎡ ⎤ Nq ⎛ −εij ⎞ ⎛ −εjj ⎞ j jz ln⎢ξi exp⎜ − ⎟ + ξj exp⎜ ⎟ + ξh⎥ ⎢⎣ ⎥⎦ 2 ⎝ kBT ⎠ ⎝ kBT ⎠

Figure 1. Experimental data taken from Lee et al., ref 7: cloud point measurements for PEO/[EMIM][BF4] solutions with PEO molecular weights of M = 20500 g/mol (squares), 4200 g/mol (circles), and 2100 g/mol (diamonds). The solutions phase separate at high temperature, i.e., exhibiting lower critical solution temperatures. These phase diagrams are unusual for polymer solutions, having critical compositions of roughly 50/50 mass fraction rather than solvent-rich critical compositions.

polymer solution critical compositions are strongly skewed toward values rich in the solvent component. However, for the case of the solutions in Figure 1, solvent-rich critical compositions are not being observed. As an alternative to FH theory other, somewhat more detailed, models are available. These are beginning to be applied to ionic liquids; for example, pressure−volume−temperature (PVT) behavior of pure ionic liquids has been modeled with the lattice fluid equation of state (EOS)10 and the modified cell model EOS.11 Some examples of EOS modeling for the case of mixtures containing ionic liquids have been reviewed in ref 12. In this work, we apply for the first time our own simple lattice-based theory to model polymer/IL solutions and contrast their behavior alongside our results for more standard polymer solutions. The goal is to explain what physical factors are driving the observed shift in phase separation behavior. By combining our theoretical equation of state (EOS) and available experimental data, we can extract insight, both by analyzing the characteristic molecular parameters and by calculating and comparing the key underlying thermodynamic properties. This allows us to assess and compare differences in the free volume and cohesive energy density characteristics of the IL and polymer components. Our theoretical approach also allows us to account for possible aggregation of the IL in the mixture, which we pursue using a simple treatment to predict ion pair aggregation in a dielectric continuum. In analyzing the ideal and excess contributions to the entropy of mixing, which is particularly revealing, we find both of these contributions have a significant effect on the coexistence behavior of polymer/IL solutions. The remainder of the paper is organized as follows: In section II, we give a brief theoretical background describing the equation of state and its associated microscopic parameters. In section III, we present and discuss our results, including analysis of pure component behavior, solution phase diagrams, and a close look at the entropy of mixing including its ideal and excess contributions. Along with the results for the present PEO/IL system, we will also provide corresponding results for more usual polymer solutions. We summarize and give our conclusions in section IV.

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

(1)

where α can be i, j, or h, and qh = rh = 1. In eq 1, A is expressed as a function of independent variables Ni, Nj, V, T which are, respectively, the numbers of molecules of components i and j, the total volume, and the absolute temperature. z is the lattice coordination number which is fixed at a value of 615 and kB is the Boltzmann constant. The key microscopic lattice parameters are v, the volume per lattice site, ri (rj), the number of segments per chain molecule of component i (j), εii (εjj), the pure component nonbonded segment−segment interaction energy between near neighbor segments of types i − i (j − j), and g, which defines the mixed interaction energy according to εij = g(εiiεjj)1/2. (g characterizes εij relative to the geometric mean value.) As mentioned above, the model is compressible, so the total volume, V, is comprised B


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of filled and empty lattice sites. Therefore, Nh is the total number of vacant sites (h stands for “holes”); this value increases with V for any given Ni, Nj. The remaining definitions in eq 1 are as follows: ϕα is the volume fraction of sites of type α (α ∈ {i, j, h}), and ξα is a concentration variable defining the fraction of nonbonded contacts ascribed to component α out of the total number of nonbonded contacts in the fluid, where, due to its bonded connections, a chain molecule has qαz nonbonded contacts. As noted above, the expression for the Helmholtz free energy, A[Ni, Nj, V, T], leads to all of the other thermodynamic properties. This includes for example, the entropy (S = −(∂A/ ∂T)Ni,Nj,V), the pressure (P = −(∂A/∂V)Ni,Nj,T), the internal energy16 (U = (∂(A/T)/∂(1/T))Ni,Nj,V), and the chemical potentials (μi = (∂A/∂Ni)Nj,V,T), which can all be obtained by straightforward differentiation of eq 1. The Gibbs free energy (G = U − TS + PV = A + PV) and the enthalpy (H = U + PV) follow from their definitions. It is noted that while P is a natural expression of the independent variable V, V cannot be directly expressed as a function of P (as is typical with many theoretical EOS’s). Therefore, in applying the model, it is common to use numerical root finding to determine V in situations where P is the known input variable (e.g., in calculating results for P = 1 atm), and following this, any of the other properties (also natural expressions of V) can then be straightforwardly evaluated. In practice, it is often convenient to work 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 = Ni/N 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, G̅ = G/N, H̅ = H/N, S̅ = S/N, and so on. Another option is to define the working system of intensive variables on a “per mass” basis (convenient for polymers); in this case, the overbar notation signifies any quantity per total mass, and the working composition variable x is thus the mass fraction. (Which system is used should be clear from the given units.) In polymer mixture modeling, one often wants to characterize partial miscibility behavior, such that there are two liquid phases, I and II, at a given T and given P (e.g., 1 atm). Because we are working with functions of [x, V̅ , T] and not [x, P, T], this leads to four equations which express the conditions for phase equilibrium, μiI = μiII, μjI = μjII, PI = 1 atm, PII = 1 atm (all for some given T); these can then be solved for the four unknowns, xI, xII, V̅ I, V̅ II. The compositions in each phase (xI, xII), obtained over a range of T, thus provide the information necessary to map out the theoretical phase diagram. Another important phase boundary, the spinodal, marks the metastability limit. This is described by two equations, (∂2G̅ /∂x2)T,P = 0, P = 1 atm, which can be solved for the two unknowns, x and V̅ , thus giving a spinodal point for a given T. The critical point is found by solving the three equations, (∂3G̅ /∂x3)T,P = 0, (∂2G̅ /∂x2)T,P = 0, P = 1 atm, for the three unknowns T, x, and V̅ , these thus being the critical temperature (UCST or LCST), critical composition, and critical volume. Other important quantities that will be discussed below are the thermodynamic mixing functions, which express the value of any particular thermodynamic property in the mixture relative to that for the corresponding amount of material in its pure unmixed state. (i.e., it is the change in that property on

going from unmixed to mixed.) For example, the enthalpy of mixing, ΔHmix, is defined as ΔHmix = H − NiH̅ pure − NjH̅ pure i j pure pure where H is that of the mixture, and H̅ i and H̅ j are the corresponding pure component properties at the same temperature and pressure; the intensive form (per mass, or per molecule) is thus ΔH̅ mix = H − xH̅ pure − (1 − x)H̅ pure i j . Similar definitions apply for all the other mixing functions, ΔSmix, ΔGmix, ΔVmix, etc. We find it particularly useful to break down the overall entropy of mixing, ΔSmix, into two contributions: “ideal” and “excess”. The “ideal entropy of mixing” (which is always positive) can be identified with the usual quantity calculated to describe the entropy change in Flory−Huggins theory. It accounts for the gain in translational entropy when chain molecules are allowed to spread over the entire mixture volume (and, it assumes that total combined volume remains fixed). In terms of the present model parameters we define17 this idealized quantity as ⎡ Nri i + Nrj j ⎤ ⎡ Nri i + Nrj j ⎤ ideal ⎢ ⎥ ΔSmix = Nk ⎥ + Nk i B ln⎢ j B ln ⎢⎣ Nrj j ⎥⎦ ⎣ Nri i ⎦ (2)

The “excess entropy of mixing” is given by excess ideal ΔSmix = ΔSmix − ΔSmix

(3)

and thus comprises all of the remaining contributions to the overall entropy change. It includes a number of important effects, such as that of free volume and compressibility (e.g., accounting for nonzero ΔVmix), as well as temperature dependence and nonrandom mixing. We have found ΔSexcess mix to be strongly connected to the difference in the model pure component energetic parameters (|εii − εjj|).14 The quantity, |εii − εjj|, speaks strongly for the difference in the cohesive energy density of the two pure components, and given its connection to ΔSexcess mix , it thus provides valuable a priori insight regarding the mixture. Beyond the absolute value, |εii − εjj|, it is also useful to note which component has the stronger ε, and which the weaker, as we explain here (for the first time) that the critical composition (ϕcrit) and skew of the phase diagram can be strongly influenced by their relative values. The tendency is for ϕcrit to be pushed toward compositions that are richer in the component with the weaker ε value (lower cohesive energy density). In order to model a mixture, one needs to obtain a full set of characteristic parameters. We follow a standard procedure (applied to other EOS’s as well) where we first fit the pure component parameters to experimental pure component data, and then the one remaining parameter, the mixed interaction parameter, g, is fit to a small amount of mixture data. This should be done with some care; as discussed below, the most consistent results are obtained by careful and consistent choice of the data ranges for the fitting. There are five parameters, ri, rj, v, εii, and εjj, which are derived from pure component fitting. Most commonly these are obtained by fitting the model pure component equation of state to experimental pressure−volume−temperature (PVT) data. For each pure component, there will thus be a resulting pure pure set of optimal values, rpure for component i, and, i , vi , and εii pure pure pure rj , vj , and εjj for j. In modeling the mixture, the ε parameters are applied directly from the pure component fits, i.e., εii = εpure and εjj = εpure ii jj . The other parameters in the mixture (ri, rj, v) are determined by the following rules. Note C


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that, while there is a v parameter for each pure component and vpure (vpure i j ), in modeling the mixture we must use a single v. Reasonable options are to set this value to that of one of the pure components or to a compromise value, such as the average, v = (vpure + vpure i j )/2. Here we do the former, choosing the mixture v to be that of the solvent. When the v is changed for one (or both) of the components, there needs to be an appropriate corresponding change in that component’s r parameter which compensates for this. Specifically, the r value is scaled such that the hard core molecular volume (rv) remains the same. Here we must adjust the r for the polymer component and given for example, a value rpure obtained from the pure component fitting (for some i particular molecular weight), we would then model a polymer molecule of the same molecular weight in the mixture with an r pure parameter value of ri = (rpure i vi /v). (Again, ri and v are the values used in the mixture.) Furthermore, in most cases, the molecular weight (M) will be different in the mixture compared to the value in the pure component PVT data/sample. (Moreover, even if they are the same, we are often interested in making predictions for other molecular weights.) So if M changes, we again make an appropriate change in the r parameter. For polymers, end effects can be ignored, and the r value is expected to be proportional to M. Therefore, at any old old single v, we use the scaling rnew = Mnew i i (ri /Mi ) where the , is the appropriate value for the desired Mnew in new r value, rnew i i old old the mixture, and ri and Mi are the previously obtained (PVT-fitted) values. As it turns out, this scaling is relevant to both the polymer component (because the PVT sample M is different than the mixture value), and the IL component (to reflect aggregation of ions, where the “effective M” is thus increased). The latter will be discussed in detail in section III. For each component, throughout the process, it is always true that rv/M = constant (the value of which depends on the component). This constant, along with ε, are thus the two key pieces of information obtained from a pure component fit; they carry over to the mixture regardless of changes in M or v. Changing from the component’s optimal v to a compromise value used for the mixture means that some of the agreement with the corresponding pure component physical properties will be sacrificed. However, scaling the r value at constant hard core volume (rv) as described above does preserve the agreement at low pressure. This follows from the fact that for polymers (over a very wide range of r, v) the V(T) curve at zero pressure depends only on ε, and on the product, rv, not on r and v separately. These points are discussed in more depth in our recent articles.13,14 Details on fitting the pure component PVT data are available in ref 13. Briefly, the fitting, which is weighted for best agreement at low pressure, is performed in two stages. In the first stage, the ε parameter and the molecular hard core volume rv are determined by fitting to just the low pressure data. In the second stage, the r and v parameters are separately resolved by fitting to higher pressure data, e.g., up to ∼100 MPa; this is carried out as a one-parameter fit for v (from which r then follows) using the ε and rv values determined in the first stage. A key point is that we choose data ranges for the fitting such that the midpoint of the temperature range for each component is the same. (e.g., data permitting, we try to keep the midpoint T’s the same to within 10 K or less.) It was shown in ref 13 that doing this is a crucial ingredient in obtaining a reliable and consistent parametrization for mixture modeling. It helps to compensate for the fact that this and many other theoretical

equations of state have an overly strong temperature dependence in the coefficient of thermal expansion.18 When the fitted data ranges are kept the same, we most accurately characterize the properties of one component relative to the other, furthermore these relative differences are preserved even outside of that range. There remains one parameter, the mixed interaction parameter, g, which is required to model the mixture. There is in general no way to determine its value (or equivalently, the value of εij) a priori using only pure component data. To obtain g, we require at least a single datum point on the mixture, such as a critical temperature, specifically, the LCST of one of the polymer solutions. Here, with the first five parameters (ri, rj, v, εii, εjj) from the pure component fitting held fixed, the g parameter is then tuned to a value that brings the theoretical critical solution temperature into agreement with that found from experiment.

III. RESULTS AND DISCUSSION As outlined in the section above, the first step in modeling this polymer/IL solution is to obtain a set of characteristic molecular parameters for the model equation of state. The molecular parameters for the PEO component were determined by fitting the model to PVT data from Zoller and Walsh;19 those for [EMIM][BF4] were obtained by fitting to PVT data in Taguchi et al. in ref 20. The one remaining parameter, g, characterizing the mixed energetic interaction, was fixed by fitting to the lower critical solution temperature, LCST ≈ 392 K, reported for the “PEO-20(−OH)”/[EMIM][BF4] system in ref 7, where the PEO molecular weight is M = 20500 g/mol. Table 1 lists the resulting PEO and [EMIM][BF4] pure component parameters. In addition to these values we have Table 1. Model Parametersa PEO [EMIM][BF4]

rb

v (mL/mol)c

−ε (J/mol)

3232 22.84

5.119 6.255

1857.4 2203.9

a Parameter values are listed out to a large number of significant figures to ensure that calculated properties can be reproduced with ample resolution. A change of 1 in the last digit of an ε parameter, for example, can shift a phase boundary by tenths of a degree. Phase boundaries can also be especially sensitive to the g parameter (values made available in the text further below). bThe r value listed for PEO corresponds to a molecular weight (M) of 20500 g/mol, which is the M value in one of the studied polymer/IL solutions; the r value for the PVT-fitted sample is 2917, thus corresponding to the PVT sample M = 18500 g/mol. The r value listed for [EMIM][BF4] corresponds to the molecular weight for one neutral ion pair (M = 197.97 g/mol); using this r/M for a single ion pair, IL aggregation in the mixture can be implemented, where the r value for aggregates is appropriately scaled up for larger effective aggregate M values. (See section III for details.). c Note that a single v is used to model the solution, and in that case, we use the v of the pure solvent. Therefore, the r value used for PEO in the solution is 2645.2; this is the appropriate scaled value that gives the same molecular hard core volume (rv) as for pure PEO.

found certain combinations to be useful, particular examples being the difference |εii − εjj|, as well as the product rv, which is the molecular hard core volume for that component. Subtracting this from the overall fluid volume gives an estimate of the free volume characteristics for each species. D


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|εPVME| by about 195 J/mol. In the upper panel of Figure 2 is the predicted phase diagram (g has been fixed solely using the

The important observations from the characterization of the two pure components are as follows: First, the magnitude of the εii value for [EMIM][BF4] is much greater than that for PEO, and thus the model calculations show that the solvent, in this case, has a much stronger cohesive energy density (U/V) compared to the polymer component. Here, |εIL − εPEO| ≈ 350 J/mol which, in the terms of the present EOS model, is certainly a sizable difference. Connected to this, one can also look at the difference in the free volume in the two pure components, choosing for example, the point T ≈ 390 K and P ≈ 20 MPa. For the IL at this T, P, the overall volume per mass (V̅ ) is ∼0.819 mL/g while the hard core volume per mass (rv/ M) is 0.7218 mL/g, so this gives a theoretical free volume of 11.9%. A similar calculation at the same T, P for PEO gives a free volume of 14.1%. Thus, the polymer has more free volume than the solvent. This unusual situation with the pure component energetic parameters (εii), and its resulting impact on cohesive energy density and free volume, is one of the keys to explaining the unique behavior of the PEO/[EMIM][BF4] system and its atypical critical composition. Given characteristics such as ionic interactions and vanishing vapor pressures, strong εii and (U/ V) values for other ILs might also be anticipated. We note in particular, that compared to more “standard” polymer solutions, for instance, ones having simpler hydrocarbon solvents, it is uncommon in our experience that |εsolvent| > |εpolymer|. For purposes of illustration we turn to a more typical polymer−solvent system; this will allow us to demonstrate what the effect of the εii parameters is on the mixture phase diagram and critical composition. Among the systems that we have modeled previously, polyethylene/decane (PE/C10) provides a good example for comparison.21 Here, as is usually the case, |εC10| for the solvent is less than |εPE| for the polymer (by ∼200 J/mol), and it is the solvent that has the larger relative amount of free volume. The model EOS gives a critical composition (for PE with M = 50000 g/mol) that is less than 3% in mass fraction of PE, and thus, it also shows the typical strong shift toward solvent-rich compositions that is usually observed for polymer solutions. We can roughly break down the driving forces which give rise to a solvent-rich critical composition into two basic contributions. One contribution is that of the ideal (combinatorial) entropy of mixing. It is useful to consider how it is affected by the different chain lengths of the two components. A quick calculation using the simple Flory−Huggins formula (ϕcrit = 1/(n1/2 + 1)) gives a critical composition of about 5% mass fraction PE.22 Our EOS model of course accounts for this ideal (combinatorial) effect as well, but it also includes the other important contribution, that of the energetics. The fact that our own model gives a critical composition of 3% rather than 5% (and thus better agreement with experiment) is due to the added effect of the energetics. In the terms of the model, having |εC10| < |εPE| is what shifts the critical composition even more toward the solvent-rich side of the composition axis. In general, the effect of disparate pure component energetic parameters produces unfavorable entropic effects that, in turn, lead to a skewing of the coexistence curve toward compositions that are richer in the component with the weaker |ε|. For example, the volume change on mixing gets skewed such that it is stronger (more negative) at that side of the composition range. As another illustrative example of the effect of the ε parameters on the mixture phase diagram, we consider the LCST-type PS/PVME blend. In this system, |εPS| is greater than

Figure 2. Example showing the effect of the pure component energetic parameters on the model phase diagram. Upper panel shows the model phase diagram for the polymer blend, PS/PVME (alongside experimental cloud point data (points) from ref 23). Solid curves show the binodal boundary, and dashed curves show the spinodal boundary. The lower panel shows the resulting hypothetical phase diagram (in particular, the impact on the critical composition) for a case where the energetic parameters for each component are exchanged (keeping all other parameters fixed).

critical temperature), which shows excellent agreement with the experimental data,23 including the location of the critical composition. (The diagram shows the binodal and spinodal temperature as a function of composition at P = 1 atm where MPS = 120000 g/mol and MPVME = 99000 g/mol.) In the lower panel is the hypothetical phase diagram that results if one swaps the ε parameter values of the two components. So, the “hypothetical PS component” now has the weaker |εii| of PVME, and the “hypothetical PVME component” gets the stronger |εii| of PS (all of the other parameters remain unchanged). There is a resounding shift in the critical composition on going from the actual PS/PVME system where ϕcrit = 27.5% PS, to the hypothetical system where ϕcrit becomes 70.1%. The above examples illustrate the notion that the critical composition moves toward compositions richer in the component with the weaker |εii|, and we expect that these observations will be useful in analyzing many general systems. In the case of the present PEO/[EMIM][BF4] system, this is indeed, exactly what is observed. That is, the observed shift in ϕcrit is toward the PEO end of the composition range and indeed, it is the PEO component that has the weaker |εii| when compared to the IL solvent component. The next step of the analysis is to incorporate another special feature of IL’s compared to “typical” solvents. Because of their strong Coulombic interactions, we consider the degree to which the IL ions might be aggregating. The possibility of nonnegligible aggregation is sensible, as it is expected that the ions E


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into one larger string comes from bringing together (effectively) two half-strength point charges. The equilibrium distribution of string (aggregate) sizes is determined by a balance of this energy of aggregation, against the entropic cost of localizing freely translating units (S ∼ ln V) into a much smaller region where the ions are in near-contact. The notion of string-like aggregation and the resulting exponential size distribution is well accepted in other contexts; in particular, Safran (ref 29) applies these ideas to the aggregation of wormlike (string-like) micelles. See chapter 8, section 2 of this reference for a framework for the basic ideas and formulation. In addition, ref 30 and references therein provide some simulation results describing characteristics of the observed aggregation (cluster-like, chain-like, etc.) in similar systems of simple ionic molecules. In our case, the inputs for this model include the dielectric strength of the medium, and the size and separation distance for the [EMIM][BF4] IL contact pair; these were estimated from values in refs 31 and 32 (dielectric constants) and ref 33 (separation distances). Note that given that dielectric constants decrease with T, going as 1/T to first approximation, the energy of aggregation will also increase with T, and therefore the corresponding Boltzmann factor, and thus the aggregation numbers will remain fairly independent of T. The results of these calculations do indeed indicate a significant degree of aggregation. (The aggregate numbers vary based on concentration, though we will approximate it in the EOS at a reasonable average constant.) We obtain average numbers of IL ions in the aggregates indicating that an aggregation factor, γ, in the range 8−16 (i.e., 8 to 16 neutral ion pairs) would be a reasonable choice. Within this range our EOS results are not strongly choice dependent (and in the Appendix, we provide more details on this). Taking into consideration both the model aggregation calculations, and the insight from the simulation studies, we choose γ = 10. Thus, the IL component r value will correspond to 10 neutral ion pairs. In this simple, but transparent, way we are now representing the incomplete dispersal of the IL in the mixture, and thus accounting for its impact on the ideal entropy of mixing. Indeed we find that recognizing the importance of some aggregation is necessary, as γ = 1 will not give correct phase separation behavior for a system with these cohesive energy density characteristics (i.e., these particular ε values). We can add further that while the EOS does not consider the detail of the actual shape of the aggregations (e.g., cluster-like, chain-like, etc.), these configurations should all be represented in the simplified lattice model chain-molecule picture. What is still most important in these thermodynamic calculations is to quantify how aggregations in any form affect the overall miscibility. As noted in section II, the final step in modeling the mixture is to optimize the mixed interaction energy parameter, g, so as to obtain agreement with a piece of experimental data on the mixture. Thus, we make use of the experimental LCST = 392 K (for PEO M = 20500 g/mol), and obtain a value of g = 1.00245. The finding that g > 1 indicates that there are favorable mixed interactions (stronger than the geometric mean value), and this seems reasonable, and perhaps even expected. We have encountered very favorable g values a number of times before in modeling mixtures involving polyethers. Having only hydrogen bond acceptors along the backbone, new hydrogen bonding possibilities can open up for the PEO component upon mixing, so favorable interactions appear that simply were

should stand to lose significant energetic interactions when separated by a nonionic matrix. This would lead to associated effects on the thermodynamic properties, which would be impacted by incomplete dispersal and nonuniform mixing. We consider here a picture in which the IL ions can organize into “clusters” or “strings” of successive near-contact pairs (more on this below). We have the ability to model the effect of this aggregation in the same way that we model a change in molecular weight of a polymer component. As noted in section II, the chain length parameter (r) must increase proportionally for any increased polymer molecular weight. If, on average, the IL ions are localized into aggregations of γ neutral pairs, then the “effective” molecular weight of the IL component is γMIL (where MIL is the molecular weight of a single neutral ion pair), and therefore, the “effective” r value for the polymer/IL solution is γrIL (where rIL is the chain length for a single neutral ion pair from the PVT fitting given in Table 1). Changes in chain length (r) have an important connection to the phase equilibria, in particular ΔSideal mix (see Figure 3 in ref 24 as an example which demonstrates this clearly when compared to experimental data). An important reminder from section II: It is principally r/M that is most important for the fitted PVT properties rather than just r, itself. Thus, on transitioning from the pure component modeling to the IL mixture modeling, we continue to maintain the relationship rv/M = constant; the hardcore volume, per total mass of the component, is always the same. There is evidence for IL aggregation from the literature. Recent simulation studies on IL/molecular solvent mixtures25,26 illustrate that aggregation plays an important role, as the IL clearly remains associated as the mixture becomes richer in the other component. These simulated IL’s were not the particular IL solvent ([EMIM][BF4]) studied here, however, the results provide important guidance. In one study,25 the IL component “[C4MIM][PF6]” was in a mixture with naphthalene, and in the other,26 the IL “[EMIM]/[NTf2]” was in chloroform. In the former study, as the IL became dilute, the authors described filamentous chains of the IL component comprised of sequential cation−anion contacts, as well as the formation of disjoint clusters of ions. There was also significant aggregation in the latter study; the authors speak of “extensive formation of neutral aggregates typically consisting of more than one single ion pair” and remark on examples of clusters of 11 ion pairs with lifetimes on the order of nanoseconds. Furthermore, it is noted that both studies compare to experimental data, e.g., conductivities, which show trends that are consistent with the observed IL aggregation. It should also be mentioned that ionic aggregation was recently observed in systems of PEO-based sulfonate ionomers with Li+, Na+, and Cs+ counterions (using X-ray scattering and dielectric spectroscopy measurements);27 these systems share some similarity with the present PEO/IL system, especially when one considers the fact that Cs+ is a large ion. The above simulation results provide guidance in choosing the value for the aggregation factor, γ. However, in addition to this, we can actually calculate some estimates for average aggregation numbers. Recently, Milner and co-workers have formulated a simple model to characterize the length distributions in ionic strings (publication upcoming28). The model predicts average numbers of ionic aggregates based on a picture where the + and − ions line up successively to form string-like assemblies in a continuum dielectric medium. The beneficial energy of joining the ends of two smaller ion strings F


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not available before, as is the case here, when paired with the IL. With the complete set of model mixture parameters we can proceed to calculate the mixture properties. In the lower panel of Figure 3 is the model phase diagram for the PEO/

Figure 4. Comparison of model EOS predictions for the entropy of mixing (TΔS̅mix, in J/g) as a function of composition, including a excess breakdown of the ideal (TΔS̅ideal mix ) and excess (TΔS̅mix ) contributions. In the upper panel are the model results for a typical polymer solution, PE/decane (PE M = 50000 g/mol, T = 560 K, P = 1 atm) that show that the overall ΔSmix is most unfavorable at solvent-rich compositions (minimum skewed strongly to the left). The lower panel shows the model results for the polymer/IL solution PEO/[EMIM][BF4] (PEO M = 20500 g/mol, T = 390 K, P = 1 atm). Marked are the strong shifts in both the ideal and excess contributions; together these push the minimum in the overall ΔSmix away from solvent-rich compositions.

Figure 3. Polymer solution phase diagrams showing binodal boundaries (solid curves) and spinodal boundaries (dashed curves) along with experimental data (points; from refs 7 and 34). The upper panel shows model results for a typical polymer solution, PE/decane. The lower panel shows the predicted model phase diagram for the polymer/IL solution PEO/[EMIM][BF4] (PEO M = 20500 g/mol), which shows a strong shift in ϕcrit toward compositions that are richer in the polymer component.

panel. (The quantities are given as a function of composition at P = 1 atm and at a temperature close to each system’s respective LCST. The composition variable in both cases is mass fraction of the polymer component.) Looking first at the overall ΔSmix curves, it is the location of the minimum that speaks strongly for the location of the critical composition. In other words, where the entropy of mixing is the most unfavorable is closer to where the onset of phase separation will first occur.35 Thus, the PE/decane solution has a minimum in ΔSmix that is strongly skewed to the solvent-rich side of the composition axis, and in contrast, the minimum in ΔSmix for the PEO/[EMIM][BF4] solution is instead shifted back toward the center of the composition range. The two contributions to the entropy of mixing determine the degree of skewing in the overall curve. In the case of PE/ decane, the ideal contribution ΔSideal mix has a maximum peaked on the polymer-rich side of the composition axis, which means (looking at it the other way) that its favorability is weaker on the solvent-rich side. At the same time, the solvent-rich side is also where ΔSexcess mix is most unfavorable, and thus, this particular excess combination of ΔSideal mix and ΔSmix results in the strong skewing of the overall ΔSmix making it unfavorable on that solvent-rich side. In the case of the PEO/[EMIM][BF4] solution, both ΔSideal mix ideal and ΔSexcess mix have shifted. The maximum in ΔSmix has moved (to a degree) away from the polymer-rich compositions, and at the same time, the minimum in ΔSexcess mix has shifted away from solvent-rich compositions. The combined effect on the overall

[EMIM][BF4] solution, which shows both the binodal and spinodal curves; for comparison, the PE/decane phase diagram is given in the upper panel. (Diagrams show temperature as a function of composition at P = 1 atm. The composition variable in both cases is mass fraction of the polymer component. Experimental data are from refs 7 and 34.) There is a very big qualitative difference between the two polymer solutions. The phase diagram for the PEO/[EMIM][BF4] solution shows indeed, that there is a strong shift in critical composition toward values richer in the polymer component. We obtain a ϕcrit value of 26%, which falls short of the experimentally observed ϕcrit ≈ 50%,7 but still, the model is clearly demonstrating a strong departure from, for example, the value of 2.9% for PE/decane, which is representative of the usual solvent-rich compositions seen for most polymer solutions. In order to understand this shift of the critical composition, we turn to the thermodynamics of mixing. In particular, we find the entropy of mixing (ΔSmix) to be very revealing, especially when it is broken down into the separate ideal and excess excess contributions (ΔSideal mix and ΔSmix ). In the lower panel of Figure ideal excess 4 are ΔSmix , ΔSmix , and the overall ΔSmix for the PEO/ [EMIM][BF4] solution, and again for comparison, these same quantities are shown for the PE/decane solution in the upper G


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entropy of mixing is such that the location of its minimum (i.e., where the entropy is most unfavorable) is now associated with a more polymer-rich composition. We can look in more detail at what specifically, has contributed to the changes in each of these entropic contributions. In the case of the PEO/[EMIM][BF4] ΔSexcess mix curve, we can make a connection to the trends that were discussed earlier regarding the ε parameters. When the |ε| of one component is weaker than that of the other, the result is to push unfavorable entropic effects (specifically, ΔSexcess mix ) in the direction of mixture compositions richer in component with the weaker |ε|. In this case that component is the polymer, PEO. Thus, we see here that the strong cohesive energy density of the IL (strong |εIL|) leads to a situation where there is a role reversal in the energetic parameters making |εIL| > |εPEO|, and this then excess for PEO/[EMIM][BF4] that leads to a model ΔSmix approaches its most unfavorable values at compositions that are much more rich in the polymer component compared to the more typical PE/decane system (where ΔSexcess mix was clearly skewed toward solvent-rich compositions). Looking at ΔSideal mix for PEO/[EMIM][BF4], we see that the other important connection is the effect of IL component aggregation, reflected in the effective increase in the r parameter for the IL. This causes the maximum in ΔSideal mix to shift more toward the center of the composition range (i.e., it shifts away from the more usual polymer-rich side such as the case for PE/ excess decane). Thus, the impact both on ΔSideal mix and on ΔSmix serve to make the overall ΔSmix more symmetric and thus the critical composition as well.

approximated the effect of this aggregation in the EOS model by introducing an aggregation factor (γ), which we estimate to have a value corresponding to 10 neutral ion pairs (a value that is consistent with the simple model calculations and the simulation results as our guide). The association of the IL leads to a reduction in the usual favorable impact of the ideal contribution to the entropy ideal of mixing, ΔSideal mix . At polymer-rich compositions, ΔSmix is reduced more strongly, and thus, this further shifts the critical composition toward these more polymer-rich compositions. We anticipate that stronger cohesive energy and the potential for aggregation should be an expected characteristic for all IL’s, and therefore there is the possibility of observing similar general behavior in other polymer/IL systems. The degree to which this occurs will certainly be system dependent, being linked to the extent to which cohesive energies are imbalanced and the tendency of the chosen IL to aggregate in the particular medium, etc. We expect that future modeling efforts should to lead to insight on additional IL mixtures.

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APPENDIX

Testing the Observed Trends in the Model Parameters

In the discussion in the main text, we identified two key effects driving the unusual shifted critical composition for this polymer/IL solution. These conclusions have been supported by the results for the model phase diagram in Figure 3, demonstrating that this trend in phase behavior is indeed captured by the theory. In this appendix, the philosophy is to explore the realm of the slightly hypothetical. Recall our observations from the main text regarding how the model parametrization is different for this polymer/IL system compared to more typical polymer solutions (observations which have led to the noted effects excess ideal on ΔSmix and ΔSmix ). We now attempt to push our conclusions further by strengthening and weakening these effects within the model. We undertake this through changes that will enhance (or diminish) what has been observed in the model characteristic parameters, in particular, (1) the very strong |εIL| compared to |εPEO| and (2) the scaled up value of rIL representing IL aggregation. We consider first the strength of |εIL|, the energetic parameter for the IL component (giving |εIL| > |εPEO|, with large |εIL − εPEO|), and test what happens to the model phase diagram when we make this parameter even stronger relative to that for PEO. Specifically, we try an increase in |εIL| by about 250 J/mol and then refit the g parameter to match the experimental LCST for PEO M = 20500 g/mol; all other parameters are left the same as those in Figure 3, including the rIL, which is kept at the same scaling of γ = 10 where rIL = γrILpure. Figure 5 shows the resulting phase diagram, which has now become strikingly similar to the experimental phase diagram (the cloud point data); the phase boundary is flatter than previously predicted, and the critical composition is now at ∼50% mass fraction. The optimized g value (1.00860) has increased compared to that in Figure 3 (1.00245), likely a reflection of the importance of the expected hydrogen bonding between the IL and the PEO. The results in Figure 5 for the other PEO/IL systems, for PEO molecular weights of M = 4200 g/mol and 2100 g/mol are particularly notable. These use the same g value as that for the LCST-fitted M = 20500 g/mol system. Therefore, they are predictions, which are obtained following the appropriate

IV. SUMMARY AND CONCLUSIONS Using our simple model for polymer mixtures, we have identified key features that help to explain the unusual phase behavior of some polymer/ionic liquid solutions, relative to their nonionic counterparts. In particular, our studies on PEO/ [EMIM][BF4] lead us to predict the experimentally observed critical composition behavior, while simultaneously revealing an explanation for why the ϕcrit values for this system are so much richer in the polymer component than those typically observed for other polymer solutions. In our analysis, we have found trends in several properties, including contributions to the entropy of mixing (ΔSideal mix and ), and related patterns in the characteristic molecular ΔSexcess mix parameters, which have proven to be particularly revealing. We identify two key factors that appear to be driving the shift in ϕcrit for this polymer/IL solution toward composition values that are richer in the polymer component: 1 Compared to the polymer, the IL component has a significantly stronger cohesive energy density and less free volume. This is manifested in our EOS model in the form of an unusual “role reversal” in the energetic parameters (where now, |εIL| > |εPEO|, with large |εIL − εPEO|). The result is a shift in unfavorable entropic effects, in particular, ΔSexcess mix , to compositions that are richer in the component with the lower cohesive energy density (the polymer), which has the effect of shifting the critical composition in that direction. 2 The second factor involves aggregation of the IL component within the mixture. Evidence for this comes both from our supporting calculations using a simplified model for aggregation of ions in a dielectric medium and from results of some recent simulation studies. We have H


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ACKNOWLEDGMENTS We appreciate the financial support provided by the National Science Foundation (Grant No. DMR-1104658). We would also like to acknowledge Scott T. Milner for suggesting the application of the string-like aggregation model, as well as very helpful conversations with Tim Lodge.

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Figure 5. Polymer solution phase diagrams showing binodal boundaries (solid curves) and spinodal boundaries (dashed curves) for a hypothetical test case where |εIL| is increased to a value of 2450 J/ mol. Three PEO/IL systems are shown corresponding to PEO molecular weights of M = 20500, 4200, and 2100 g/mol. A g value of 1.00860 is obtained by fitting to the M = 20500 g/mol system; the model results for the other systems are predictions using that g value. Experimental data taken from Lee et al.7

change in the rPEO value to reflect the new molecular weight, and they are in excellent agreement with the experimental data. It should further be noted that the accuracy of these predictions is better than that obtained for corresponding predictions using the parameters in Figure 3. We interpret the above test as further evidence for the importance of the underlying effects linked to the strong |εIL| compared to |εPEO|. In a similar way, we have also tested the other key effect, that of IL aggregation, by considering what happens when we increase the value of rIL from its value in Figure 3; as before, when we make this change we also refit g to the M = 20500 g/mol LCST, with all other parameters remaining fixed (i.e., no change to |εIL|). As one might expect, this has the effect of increasing the critical composition. For instance, at γ = 32, we obtain a critical composition of ∼50% mass fraction (similar to Figure 5) and an improvement (flattening) in the shape of the phase diagram. However, we find that the predictions for the PEO/IL systems with M = 4200 and 2100 g/mol are not nearly as good as obtained when |εIL| was increased, as shown in Figure 5. These observations serve as additional support for the significance of the effects due to the strong |εIL|; the experimental behavior can not be rationalized from the point of view of aggregation alone. On the other hand, the effect of aggregation also appears to be a verifiable necessary ingredient; this becomes clear upon testing decreases in rIL (decrease γ) while trying to compensate with increased |εIL| values. For example, our model calculations indicate that the quality of the phase diagrams begin to drop off for γ below 5, regardless of the |εIL|. We conclude from these tests that both the strong cohesive energy density of the IL (|εIL| > |εPEO|, with large |εIL − εPEO|), and the IL aggregation, are contributing synergistically to the observed experimental behavior.

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REFERENCES

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(28) Lu, K.; Maranas, J. K.; Milner, S. T. Manuscript in preparation. (29) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Westview Press: Boulder, CO, 2003. (30) Zhang, R.; Jha, P. K.; Olvera de la Cruz, M. Soft Matter 2013, 9, 5042. (31) Porter, C. H.; Boyd, R. H. Macromolecules 1971, 4, 589. (32) Krossing, I.; Slattery, J. M.; Daguenet, C.; Dyson, P. J.; Oleinikova, A.; Weingartner, H. J. Am. Chem. Soc. 2006, 128, 13427. (33) Tsuzuki, S.; Tokuda, H.; Mikami, M. Phys. Chem. Chem. Phys. 2007, 9, 4780. (34) Kodama, Y.; Swinton, F. L. Br. Polym. J. 1978, 10, 191. (35) Technically, in LCST-type systems, it is strong positive curvature in ΔS̅mix, i.e., positive (∂2S̅/∂x2)T,P, that determines the phase separation. See, for example: Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths: London, 1982. However, we can still discuss here in the simpler terms of ΔS̅mix, noting that positive (∂2S̅/∂x2)T,P is still connected to unfavorable ΔS̅mix, at least when viewed relative to neighboring compositions.

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Ionic Liquid Solutions

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