How Pure Components Control Polymer Blend Miscibility

Nov 2, 2012 - In this modeling investigation we show that two quantities, connected to the molecular characterization parameters, serve as separate â€...
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How Pure Components Control Polymer Blend Miscibility Ronald P. White and Jane E. G. Lipson* Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, United States

Julia S. Higgins Department of Chemical Engineering and Chemical Technology, Imperial College, London, SW7 2AZ, U.K. ABSTRACT: Fundamental insight regarding what drives polymer blend immiscibility/miscibility requires understanding the enthalpic and entropic contributions to the free energy of mixing. In this modeling investigation we show that two quantities, connected to the molecular characterization parameters, serve as separate “controls” on these thermodynamic mixing functions. The g parameter is defined as g = εij/(εiiεjj)1/2, with ε representing a segment−segment interaction energy, which may be obtained using minimal data on the mixture (for example, a phase separation temperature), appears to be correlated with the enthalpy of mixing. Characterization of the pure components, for example by fitting equation of state data, yields εii values. In this work we present evidence that |εii − εjj| controls the entropy of mixing. Furthermore, by analyzing the separate ideal and excess contributions to the entropy of mixing, we demonstrate that it is the excess contribution in particular that is strongly influenced by the value of |εii − εjj|, becoming increasingly unfavorable as |εii − εjj| increases. This sheds further light on a correlation noted in recent work of ours [White; et al. Macromolecules 2012, 45, 1076−1084], that for LCST-type blends an increasingly favorable εij (meaning an increasing value of g) is needed to offset a greater mismatch in εii and εjj values (meaning an increasing |εii − εjj| difference) in order to maintain even partial miscibility. Given the importance of the excess entropy of mixing in driving miscibility, especially for LCST-type blends, we conclude that knowledge of |εii − εjj| can lead to a degree of a priori insight in assessing the mixture thermodynamics in the absence of any mixture data.

I. INTRODUCTION In this article we confront some of the challenging issues in understanding polymer blends from a modeling perspective. Polymeric mixtures offer a potentially rich source of possibilities for desirable material properties. However, understanding the behavior of a blend, for instance, predicting its properties, a priori, based on just the properties of two proposed components has proven to be much more elusive than is the case for simple small molecule mixtures. Key thermodynamic quantities, such as the changes in enthalpy and volume as a result of mixing, are very difficult to obtain experimentally for polymeric systems. Modeling them has proven equally challenging, due not only to the difficulty of correctly capturing the physics but also to the difficulty in then obtaining an adequate parametrization. Many polymer mixtures are immiscible, and most of the systems that do mix still show phase separation (partial miscibility) when conditions are changed, e.g., upon a change in temperature, pressure, molecular weight, or composition. Characterizing, and potentially predicting, this behavior is of central practical interest. The strong tendency for phase separation can be traced to the fact that a key driving force for mixing, the ideal (or combinatorial) entropy change on mixing, ΔSideal mix , is very small © 2012 American Chemical Society

for polymers. Therefore, even in cases where the components are fairly similar, small energetic and topological incompatibilities in the mixture can cause immiscibility, simply because ΔSideal mix is too weak to overcome them. This presents a daunting challenge for modeling as small changes in mixed model energetic parameters can cause large changes in the model phase boundaries, thus requiring a higher parameter resolution than is reasonably obtainable by theoretical approximation. As a consequence of this, mixture data must be used to complete the model parametrization. Note that this means in order to model a blend, one requires that an experiment has already been performed on that blend. The fact that mixed energetic parameters cannot be obtained a priori is one of the outstanding issues in polymeric mixture modeling as it limits the potential for true model prediction. Certainly, there are still many properties that can be predicted once the complete model parametrization is obtained, and frequently, these are properties that are difficult to obtain experimentally, but clearly it would be especially powerful if Received: August 29, 2012 Revised: October 23, 2012 Published: November 2, 2012 8861

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equation approach is used to determine temperature-dependent nearest-neighbor segment−segment probabilities. This leads to an expression for the internal energy (U) which can then be integrated from an athermal reference state to give the Helmholtz free energy (A), and from this all remaining thermodynamic properties can be derived. Shown below is the analytic result for A, for the case of a binary mixture comprised of components i and j.

one could say something about the mixture based on pure component information only. Over the years a number of modeling approaches have been developed for the study of polymers and their mixtures. Many of these fall into a category of physics-based equation of state approaches which are derived on a basis of statistical thermodynamics and therefore include some of the key molecular level characteristics of the polymeric fluid.1−19 Examples include SAFT-related approaches,3−5 lattice-based models,6−13 and cell-based models.14−18 Among the equation of state theories capable of describing the range of miscibility behavior, we have focused on a simple analytic lattice model20−22 (developed by one of us) which we have successfully applied to a range of blend problems as well as polymer solutions and small molecule fluids and mixtures (see for example refs 23−28). As noted above, the difficulty in obtaining mixed energetic parameters for polymeric systems has posed challenges in polymer blend modeling (see for example refs 29−33.) However, in a recent article34 we summarized interesting correlations between the energetic parameter characterizing the mixture and the difference in the two pure component energetic parameters. Because the latter are determined from pure component data alone (without the need for data on the mixture), this is a potentially important connection. The focus in ref 34 was primarily to describe the model parametrizations (fitting) for a large sample of experimental blends and then to present only the general trends in the resulting characteristic molecular parameters. In the present work, we look at these correlations more closely and focus on the underlying enthalpic and entropic driving forces in the model blend behavior. A goal is to clarify how these thermodynamic quantities relate to the parameters involved in the observed correlations. In some important cases (e.g., LCST-type systems) we can show that the separate thermodynamic driving forces (enthalpy and entropy) appear to align individually with particular parameters. Furthermore, we will show how the difference in the pure component energetic parameters can be useful at least as a “rough predictor”, providing an approximate way to gauge the impact of the excess entropy of mixing. 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, along with suggestions on how to implement the theory in a way that gives the most consistent results. In section III, we present and discuss our results, starting first by verifying some model predictions for enthalpic and entropic properties against experimental data. Then we take an in-depth look at the thermodynamics of mixing for a set of representative blends including a breakdown of the contributions coming from the enthalpy of mixing and both the ideal and excess entropy of mixing. We then demonstrate how the model parameters are linked to these quantities. We summarize and give our conclusions in section IV.

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)

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

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, and 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 6,35 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 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 mentioned 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 energy36 (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

II. THEORETICAL BACKGROUND AND IMPLEMENTATION In this section we briefly introduce the equation of state which we apply in the modeling of polymeric melts and blends. It is based on a lattice model for chain molecule fluids, and it incorporates both the effects of free volume (the model is compressible) and the effects of nonrandom mixing. In the derivations (see details in refs 20−22, 27, and 28), an integral 8862

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

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 used should be clear from the given units.37) In polymer blend modeling one often wants to characterize partial miscibility behavior, where 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 , μIj = μIIj , PI = 1 atm, and PII = 1 atm (all for some given T); these can then be solved for the four unknowns, xI, xII, V̅ I, and 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 and 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, and 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 thermodynamic quantities that will be featured in the results below are the mixing functions. These quantities 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 = pure H − NiH̅ pure − NjH̅ pure i j , where H is that of the mixture and H̅ i and H̅ pure are the corresponding pure component properties at j the same temperature and pressure; the intensive form (per mass or per molecule) is thus ΔH̅ mix = H̅ − xH̅ ipure − (1 − x)H̅ pure j . Similar definitions apply for all the other mixing functions, ΔSmix, ΔGmix, ΔVmix, etc. One particularly insightful form of analysis which can be elucidated through modeling is 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.38 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 define39 this idealized quantity as

(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 the effects of free volume and compressibility (e.g., accounting for nonzero ΔVmix) as well as temperature dependence and nonrandom mixing. Below we will show how ΔSexcess is strongly connected to the difference in the model mix pure component energetic parameters. In order to model a blend, one needs to obtain a full set of characteristic parameters. We focus first on the parameters which characterize the pure components, keeping in mind that the ultimate goal is to model a mixture. The one remaining parameter, g, that characterizes the mixture is discussed further below. This procedure of first fitting pure component parameters to pure component data and then the mixed interaction parameter to a small amount of blend data follows a standard practice used in many of the earlier references cited above. This should be done with some care, however, and as discussed below, the most consistent results are obtained by careful and consistent choice of the data ranges for the fitting. The five parameters ri, rj, v, εii, and εjj are most commonly obtained by fitting the model pure component equation of state to experimental pressure−volume−temperature (PVT) data.40 For each pure component, there will thus be a resulting set of pure pure optimal values, rpure for component i and, rpure i , vi , and εii j , pure pure vj , and εjj for component j. In modeling the blend 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 blend (ri, rj, v) are determined by the following rules. Note that, while there is a v parameter for each pure component (vpure and vpure i j ), in modeling the mixture we use a single v. We usually set this to a compromise value, such as the average, v = (vpure + vpure i j )/2. To compensate for this change in the v parameter, there needs to be a corresponding change in the r parameters. This is done by scaling the r value for each component such that the hardcore molecular volume (rv) remains the same. If, for example, we obtained a value rpure for some particular molecular weight, i we would then model a molecule of the same molecular weight pure in the blend with an r parameter value of ri = (rpure i vi )/v. (Again, ri and v are the values used in the blend.) Commonly the molecular weight (M) will be different in the blend compared to the value in the pure component 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 single v, we use the scaling ri = Mi(rold i /Mi ), where ri is the appropriate value for the desired Mi and rold and Mold are the i i previously obtained values. Altogether, these scalings follow rv/M = constant. This constant and ε are thus the two key pieces of information obtained from a pure component fit; they carry over to the blend regardless of changes in M or v. Changing from the component’s optimal v to a compromise value used for the blend means that some of the agreement with the corresponding pure component physical properties 8863

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high T and phase separate on cooling (exhibiting an upper critical solution temperature (UCST)) and those which are miscible at low T and phase separate on heating (exhibiting a lower critical solution temperature (LCST)). As outlined above, for most of these systems we first obtained the pure component parameters by fitting to pure component PVT data and then obtained the mixed interaction energy (as characterized by the parameter g) by fitting to a single datum point for the blend, viz., its critical temperature (UCST or LCST). The pure components in the studied systems are first summarized in Table 1, which gives the

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 (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. Conversely, it follows that fitting only to zero pressure data will produce a unique value for ε and a unique value for rv, but not r and v. In view of the fact that the v value is changed on going to the model blend, the above observations have lead us to believe that it is important in our fitting to weight for best agreement with low-pressure data because this is the agreement that is still guaranteed to be preserved. Furthermore, the relevant experimental predictions and comparisons for the blend are most often for the low-pressure behavior anyway. The pure component fitting procedure was described in detail in ref 34. Briefly, the fitting is performed in two stages where, in the first stage, the ε parameter and the molecular hard core volume rv are determined by fitting to just the lowpressure data. In the second stage, the r and v parameters are separately resolved by fitting to higher pressure data say, up to ∼100 MPa; this is posed as a one-parameter fit for v (from which r then follows) using the ε and rv values determined in the first stage. Importantly, 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 keep the midpoints T’s the same to within 10 K or less). It was shown in ref 34 that doing this is a crucial ingredient in obtaining a reliable and consistent parametrization for blend 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.41 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 blend. As discussed in the Introduction, 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 (UCST or LCST). Here, with the first five parameters (ri, rj, v, εii, εjj) from the pure component fitting held fixed, the g parameter is then tuned (fit) to a value that brings the theoretical critical solution temperature into agreement with that found from experiment.

Table 1. Table of Polymer Acronymsa acronym

full name

ref PVT data

PS dPS PB PIB PEP PMS PI PVME

polystyrene deuterated polystyrene polybutadiene polyisobutylene poly(ethylene-co-propylene) alternating copolymer poly(α-methylstyrene) polyisoprene poly(vinyl methyl ether)

42b 43c 42 44 44 45 46 47

a

The table contains the acronyms used in this article and the corresponding full polymer names. The column on the right contains the corresponding experimental references for PVT data which were used to obtain pure component parameters for each species. See ref 34 for more details. bSeparate parametrizations were performed for two different molecular weights, M = 110 000 and 910 g/mol. cSANS data for the dPS/PVME blend (see Table 2) were used to obtain pure component parameters for dPS which were then used to predict PVT data in ref 43.

polymer acronyms along with corresponding experimental references for PVT data. The characteristic parameters obtained for each blend are given in Table 2. As mentioned in the Introduction, the critical temperature and phase boundaries for polymer blends can be very sensitive to the mixed interaction parameter (g), and thus we typically report it to 5 or 6 decimal places. Also included in Table 2 is the relevant experimental information including molecular weights, critical solution temperatures, and references. In ref 34 more details are available describing the parametrization of these as well as a number of other blends. Much insight can be gained by a study of the separate enthalpic and entropic contributions to polymer blend behavior. However, this type of information (e.g., changes on mixing) is often difficult to obtain experimentally for polymeric systems. It is thus a key area where the application of a theoretical EOS model can potentially be very valuable. However, even for a model theory, it is one thing to correctly capture an overall free energy; much more challenging is to resolve accurately the associated separate enthalpic and entropic contributions. We first verify that the model is resolving these driving forces by comparing with some available experimental data. In Figure 1 are results for the enthalpy change upon mixing (ΔHmix) as a function of composition for a UCST-type system (PS/PB). Shown are the model predictions (curve) compared alongside the corresponding experimental data54 (points) at T = 343 K and P = 1 atm. Bearing in mind that the model was parametrized based on different properties (PVT data and a critical solution temperature for a different experimental PS/PB

III. RESULTS AND DISCUSSION As mentioned in the Introduction, a goal of the present work is to model the separate entropic and enthalpic thermodynamic driving forces in polymer blends and to analyze the ways in which the theory captures this behavior as it is manifested in the underlying microscopic parameters. Furthermore, we attempt to rationalize the correlations in these parameters which were observed over the large set of polymer blends studied in our previous paper.34 We will present detailed results here for several systems that serve as representative examples of the fundamental behavior that can be observed in polymer blends. (These systems are a subset of those in ref 34.) Two basic types of blends are considered, based on the nature of their partial miscibility behavior: those which are miscible at 8864

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Table 2. Blend Parameters blend: i/j

refa

Mi/Mjb (g/mol)

PS/PB PS/PI PS/PMS PIB/PEP dPS/PVMEc PS/PVME

48 49 50 51 52 53

1344/2585 2286/2802 49000/56400 38600/59900 255000/99000 120000/99000

Tc (K) 360 391 469 313 425 394

UCST UCST UCST LCST LCST LCST

ri/rj

v (mL/mol)

−εii/−εjj (J/mol)

|εii − εjj| (J/mol)

g

151.31/335.42 233.22/315.59 4561.3/4973.9 3925.6/6460.6 26929/11419 13652/11419

8.0000 8.8280 9.6573 10.137 7.6670 7.6670

2042.5/1960.3 2042.5/1813.7 2405.1/2403.4 2239.1/2032.9 2106.0/1948.7 2144.3/1948.7

82.2 228.8 1.7 206.1 157.3 195.6

0.996 415 0.998 684 0.999 809 1.000 860 1.000 980 1.001 317

a The column lists the experimental references for the blend. Experimental references for the corresponding pure component PVT data used to obtain the pure component parameters are given in Table 1. See ref 34 for more details. bMolecular weights (M) are given as weight-averaged molecular weights (Mw). cModel parametrization was obtained by fitting to SANS data; the corresponding model LCST is 428 K, which is close to the experimental value (425 K).

Figure 1. Enthalpy of mixing (ΔH̅ mix) in J/g for a UCST-type blend: comparison of experimental data for PS/PB with the corresponding model EOS prediction. The experimental data (points) are from ref 54 for a system with molecular weights of 1010 g/mol (PS) and 2350 g/ mol (PB). The corresponding model prediction (curve) is based on a parametrization derived from fitting pure component PVT data (ref 42) along with a fit (to obtain g) to the UCST value in ref 48. See also details in Table 2.

Figure 2. Temperature dependence of the second derivative of the Gibbs free energy for an LCST-type blend: comparison of experimental SANS data for dPS/PVME (points, ref 52) with the corresponding model EOS results (curve). The plot shows (∂2(G̅ / kBT)/∂x2)T,P as a function of 1/T at P = 1 atm and a composition of 50/50 mass fraction; the enthalpic component (∂2(H̅ /kB)/∂x2)T,P is the slope (where here x is the mole fraction of repeat units and intensive properties are per total number of repeat units). See also details in Table 2.

blend), the prediction for ΔHmix is fairly good, being within roughly 20%. We next compare enthalpic and entropic mixture properties for the case of an LCST-type blend (dPS/PVME), here by consideration of the temperature dependence of the second derivative of the Gibbs free energy with respect to composition (∂2G̅ /∂x2)T,P. Shown in Figure 2 is a plot of (∂2(G̅ /T)/∂x2)T,P [= (∂2(ΔG̅ mix/T)/∂x2)T,P] as a function of 1/T; the slope of this plot is the enthalpic contribution, (∂2H̅ /∂x2)T,P [= (∂2ΔH̅ mix/∂x2)T,P], which is closely related to ΔH̅ mix itself.55 It is seen that the model results for (∂2H̅ /∂x2)T,P are in good agreement with the experimental SANS data, as the slope of the model curve is close to the average slope of the data over the entire temperature range (though the data do show some curvature not captured by the theory). In addition to (∂2H̅ / ∂x2)T,P, it can also be verified from this figure that the model is giving the correct entropic contribution, (∂2S̅/∂x2)T,P = (1/T)[(∂2H̅ /∂x2)T,P − (∂2G̅ /∂x2)T,P]. This follows because there is not only agreement in the slope, (∂2H̅ /∂x2)T,P, but also in the values of (∂2(G̅ /T)/∂x2)T,P itself over the temperature range. The model results in Figure 2 were technically not a prediction, but rather a fit to the data. However, it should also be noted that the associated parameters resulting from this fit were then successfully applied43 to predict both pure component PVT data for dPS and the critical solution temperature (and the complete phase diagram). Given the

quality of those predictions which showed excellent agreement with experimental data, had we reversed the order of the procedure and fitted/parametrized to those latter properties instead, we would then regenerate (as a prediction) nearly the same model curve for (∂2(G̅ /T)/∂x2)T,P in Figure 2. Below we will discuss in more detail specifically what is required in the parameters in order to match (∂2(G̅ /T)/∂x2)T,P, not only at a single temperature, but over a range of temperature, that is, to obtain agreement in not only the free energy but also its enthalpic and entropic components. The results from the comparison of theory and experiment in Figures 1 and 2 give us some confidence that we can now proceed, choosing any of our parametrized blend systems, and analyzing them with a thorough breakdown of the thermodynamics of mixing. As noted in section II, it is insightful to further break down the overall entropy of mixing into its ideal and excess contributions, with ΔSideal mix describing the idealized entropy change from mixing the two components (e.g., assumes ΔVmix = 0) and ΔSexcess describing the often very mix important additional contributions coming from the effects of free volume and how it is impacted by the energetics. Results are shown in Figure 3 for the thermodynamic quantities of mixing as a function of composition for a typical 8865

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parameters from our previous paper34 are summarized in Table 2.) These results are in the miscible region just below the LCST (313 K). In contrast to UCST-type blends, LCST-type blends typically have enthalpy and (overall) entropy of mixing that are both negative. ((∂2S/̅ ∂x2)T,P and (∂2H̅ /∂x2)T,P must be positive at the LCST.) A further contrast is that here we see the excess contribution ΔSexcess is the most important part of the mix entropy of mixing. Indeed, given that the enthalpy of mixing is negative (which is favorable) and that the ideal contribution ΔSideal mix is positive (which is also favorable), it is only the unfavorable excess entropy of mixing which can drive the system toward immiscibility. Therefore, for LCST-type systems, ΔSexcess mix plays a central role, and this is the general case for such blends. In some UCST-type systems the excess entropy of mixing can play an important role. Our modeling shows that the PS/PI system is an example of this. The results for the mixing thermodynamics are shown in Figure 5, given at T = 400 K in

Figure 3. Model EOS predictions for the thermodynamics of mixing for a UCST-type blend: PS/PMS. Properties are given as a function of composition at T = 480 K and P = 1 atm. Shown are ΔG̅ mix (Gibbs free energy of mixing), ΔH̅ mix (enthalpy of mixing), and TΔS̅mix (the overall entropy of mixing, multiplied by T). Also shown are TΔS̅ideal mix and TΔS̅excess mix , which are respectively the ideal and excess contributions to the overall entropy of mixing (multiplied by T). Units are J/g. See also details in Table 2.

UCST-type system, PS/PMS at T = 480 K. (The parameters for this system were obtained previously in ref 34, and they are summarized here in Table 2.) For UCST-type systems, the enthalpy and entropy of mixing are typically both positive (as (∂2S̅/∂x2)T,P and (∂2H̅ /∂x2)T,P must be negative at a UCST). The results shown correspond to the miscible region just above the UCST (469 K). Here ΔHmix and TΔSmix are comparable in strength, and ΔGmix is close to showing an inflection. Looking at the breakdown of entropic contributions, we also note that for this system the ideal entropy of mixing ΔSideal mix (the shortdashed curve) makes a stronger contribution to the overall entropy of mixing than ΔSexcess mix . The excess contribution is, however, by no means negligible. In Figure 4 we show the thermodynamics of mixing for a typical LCST-type blend, PIB/PEP, at T = 305 K. (Again the

Figure 5. Model EOS predictions for the thermodynamics of mixing for a UCST-type blend: PS/PI. Properties are given as a function of composition at T = 400 K and P = 1 atm. Shown are ΔG̅ mix (Gibbs free energy of mixing), ΔH̅ mix (enthalpy of mixing), and TΔS̅mix (the overall entropy of mixing, multiplied by T). Also shown are TΔS̅ideal mix and TΔS̅excess mix , which are respectively the ideal and excess contributions to the overall entropy of mixing (multiplied by T). Units are J/g. See also details in Table 2.

the miscible region just above the UCST = 391 K. Here ΔSexcess mix is strong enough compared to ΔSideal mix to reduce the overall ΔSmix considerably, though as typical for a UCST-type system, it is still net positive. The blend is, as a result of this excess contribution, less miscible than might be expected from a knowledge of the enthalpy of mixing alone. Figures 3−5 illustrate results for a fixed temperature; it is also interesting to view the results for the mixing thermodynamics over a range of temperature. These are shown in Figure 6 for the case of an LCST-type system, PIB/PEP, at a fixed composition of 50/50 mass fraction. ΔSexcess mix is much stronger than ΔSideal mix over the entire temperature range (200−600 K), thus making the overall ΔSmix unfavorable over all T. The results in the figure also clearly demonstrate the typical temperature-dependent behavior for an LCST-type blend (a system that is miscible at lower T but becomes immiscible at higher T). Here as T increases the unfavorable TΔSmix begins to dominate ΔHmix, thus driving up ΔGmix and ultimately causing phase separation.

Figure 4. Model EOS predictions for the thermodynamics of mixing for an LCST-type blend: PIB/PEP. Properties are given as a function of composition at T = 305 K and P = 1 atm. Shown are ΔG̅ mix (Gibbs free energy of mixing), ΔH̅ mix (enthalpy of mixing), and TΔS̅mix (the overall entropy of mixing, multiplied by T). Also shown are TΔS̅ideal mix and TΔS̅excess mix , which are respectively the ideal and excess contributions to the overall entropy of mixing (multiplied by T). Units are J/g. See also details in Table 2. 8866

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other hand, the points are fairly scattered, and there appears to be no such correlation, with the one possibly very interesting exception of a small group of blends all containing very similar chemical components; these appear to line up with the LCSTtype blends at very small |εii − εjj|. Here we will try to rationalize the source of this correlation by looking carefully at how g and |εii − εjj| affect the thermodynamic properties of model LCST-type systems. In Figure 8, we revisit the experimental data for (∂2(G̅ /T)/ ∂x2)T,P as a function of 1/T for the LCST-type dPS/PVME

Figure 6. Model EOS predictions for the thermodynamics of mixing for an LCST-type blend: PIB/PEP. Properties are given as a function of temperature at a fixed composition of 50/50 mass fraction and P = 1 atm. Shown are ΔG̅ mix (Gibbs free energy of mixing), ΔH̅ mix (enthalpy of mixing), and TΔS̅mix (the overall entropy of mixing, multiplied by excess T). Also shown are TΔS̅ideal mix and TΔS̅mix , which are respectively the ideal and excess contributions to the overall entropy of mixing (multiplied by T). Units are J/g. See also details in Table 2.

We are now in a position to make some connections between the characteristic molecular parameters and the mixture thermodynamics. In particular, we will investigate what seems to be almost separate control by certain parameters over the enthalpic or entropic characteristics of the blend. As noted in the Introduction, we recently34 characterized a large sample of experimental blends and reported some interesting correlations between the mixed energetic parameter, g (= εij/(εiiεjj)1/2), and the (absolute value of the) difference between the two pure component energetic parameters, |εii − εjj|. These trends are shown in Figure 7. For LCST-type blends (diamonds) there appears to be a clear correlation such that g increases as |εii − εjj| increases: g tends to be larger for blends where the pure component |εii − εjj| is larger. For UCST-type blends, on the

Figure 8. Temperature dependence of the second derivative of the Gibbs free energy. Shown is a comparison of model EOS results (P = 1 atm and 50/50 mass fraction) for the hydrogenated PS/PVME blend (solid curve) alongside experimental SANS data for the deuterated dPS/PVME blend (points, ref 52). (See also details for both blends in Table 2.) Also shown are hypothetical model results where the PS/ PVME model is altered by changing only a single parameter, in one case (long-dashed curve) by changing the g parameter only, and in the other case (short-dashed curve) by changing only the value of |εii − εjj| (in particular the PS εii). The hypothetical curves aim to fit the dPS/ PVME data; however, agreement can only be obtained at a single temperature (chosen here to be the spinodal point).

blend (which was originally shown in Figure 2). Here, however, we ask the question: what do the model results for the similar (hydrogenated) PS/PVME blend look like plotted against these dPS/PVME data? Furthermore, if one starts with the PS/ PVME model parameters, what kinds of changes need to be made to the parameters to get a match to the data? In particular, what if hypothetically one were allowed to change only a single parameter, g or |εii − εPVME|? (∂2(G̅ /T)/∂x2)T,P for the PS/PVME blend is shown as the solid curve in Figure 8. It lies below the data for the dPS/ PVME blend, indicating the lower miscibility of PS/PVME compared to dPS/PVME. Model calculations show that even though ΔHmix is lower (more favorable) for PS/PVME, ΔSmix is much less unfavorable for the case of dPS/PVME. Now, we imagine changing just one parameter keeping all the others fixed. If we increase the g parameter (long-dashed curve), we can increase the miscibility of the system, and as shown, this raises the (∂2(G̅ /T)/∂x2)T,P curve. However, it is not possible to match all of the data points over varied temperature. The curve shown (for g = 1.001 497) matches the spinodal point, but not the data at the other temperatures. A similar situation is encountered (short-dashed curve) for the case where we

Figure 7. Correlations in the mixture interaction parameter, g, with the difference in the pure component interaction parameters (|εii − εjj|) for the 25 blend systems studied in ref 34. UCST-type blends are given by triangles and LCST-type blends by diamonds. 8867

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change |εii − εjj|, which is done here by changing only the εii parameter for the PS component (εjj = εPVME remains fixed). Again, it is seen that we can increase the miscibility (raise the (∂2(G̅ /T)/∂x2)T,P curve), but agreement cannot be obtained over the full range of temperature. The curve shown (for εii = −2132 J/mol) gives a match at only the spinodal point. As it turns out, a simultaneous change is required in both g and |εii − εjj| (= |εii − εPVME|) in order to capture the characteristics of the dPS/PVME blend. In fact, inspection of the dPS/PVME parameters in Table 2 shows that g is actually lower than for PS/PVME; this alone would serve to decrease the miscibility (lower the curve in Figure 8). However, the concomitant decrease in |εii − εPVME| counters this effect strongly enough to bring about a net increase in miscibility and a match to the data. The fact that agreement could not be obtained through the adjustment of only a single parameter is linked to the fact that both the enthalpic and entropic contributions to (∂2(G̅ /T)/ ∂x2)T,P have changed on going from PS/PVME to dPS/PVME. It is interesting to note how when |εii − εjj| was changed in Figure 8, there was very little change in the slope of the plot. Apparently, despite the fact that an energetic parameter was changed, the model results still give essentially the same value for (∂2H̅ /∂x2)T,P; that is, it appears that we only changed (∂2S̅/ ∂x2)T,P. Below we will now further clarify the effect of each parameter by looking directly at their respective effects on the entropy and enthalpy of mixing. We will use again the model PS/PVME blend as a test system, but we note that the effects demonstrated in this example are typical of all model LCST-type blends. We look first at the difference in the pure component energetic parameters |εii − εjj|. To test its effect on the thermodynamics of mixing, we start with results for the experimental PS/PVME blend and consider a hypothetical change in the εii for PS such that |εii − εjj| is decreased by about 25%. (All other parameters are kept fixed.) Shown in Figure 9 are the effects of this parameter change on both the enthalpy and entropy of mixing: a very strong effect is seen for the case of ΔSmix and only a small effect for ΔHmix, in agreement with the observation regarding Figure 8. Next, in Figure 10, we test the effect of the g parameter. Starting again from the PS/PVME experimental blend, we make a hypothetical change in g, decreasing it by such an amount that there is a 25% change in the difference between εij and the corresponding geometric mean value. (Again this is done keeping all other parameters fixed.) The figure clearly shows the effect of g is strongest on ΔHmix. Figures 9 and 10 therefore illustrate that |εii − εjj| and g are acting almost as “separate controls” on the model entropy and enthalpy of mixing, respectively. In the case of |εii − εjj|, it is important to note the direction of the impact on the mixture behavior. Specifically, an increase in |εii − εjj| makes ΔSmix increasingly unfavorable (ΔSmix becomes lower). Furthermore, as only a change in the r parameter can bring about a change in ΔSideal mix (for a given composition), the effect of |εii − εjj| is wholly on the excess contribution to the entropy of mixing. Its domain in determining the mixture behavior therefore lies in such effects as the influence of free volume, compressibility, and thermal expansivity. Indeed, we find strong connections between the pure component ε parameters and the corresponding coefficients of thermal expansion (α). An unfavorable mismatch in the pure component α values can be indicated by a large difference in

Figure 9. Effect of |εii − εjj| on the enthalpy (ΔH̅ mix) and entropy of mixing (TΔS̅mix) for a typical LCST-type blend. Shown are the results for the model PS/PVME system at T = 390 K and P = 1 atm (solid curves). (See also blend details in Table 2.) The dashed curves are the hypothetical model results that are obtained when the value of |εii − εjj| (in particular the PS εii) is altered keeping all remaining parameters fixed. Here |εii − εjj| has been decreased by roughly 25% from the starting PS/PVME model system. A strong effect on the entropy of mixing is observed with only a small effect on the enthalpy of mixing.

Figure 10. Effect of the g parameter on the enthalpy (ΔH̅ mix) and entropy of mixing (TΔS̅mix) for a typical LCST-type blend. Shown are the results for the model PS/PVME system at T = 390 K and P = 1 atm (solid curves). (See also blend details in Table 2.) The dashed curves are the hypothetical model results that are obtained when the value of g is altered, keeping all remaining parameters fixed. Here g has been decreased by an amount such that g − 1 is decreased by roughly 25% compared to the starting PS/PVME model system. A strong effect on the enthalpy of mixing is observed with only a small effect on the entropy of mixing.

the pure component ε’s, and this brings about an unfavorable contribution to ΔSexcess mix . One can imagine the way in which |εii − εjj| manifests itself in blend behavior by considering the behavior in polymer solutions, even when the components are chemically similar (e.g., alkanes and polyethylene). Here we can comfortably ignore the effect of the mixed interaction (g) (since g = 1 works very well for PE/alkane solutions) and just roughly envision the 8868

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qualitative behavior which is LCST-type partial miscibility. Our modeling of these types of polymer solutions23 reveals large values of |εii − εjj| that follow from the large mismatches in the compressibility and thermal expansivity behavior of the pure components. The larger the |εii − εjj|, the less miscible the mixture (lower LCST). These kinds of factors, which are rooted in the differences in free volume behavior, are what |εii − εjj| speaks for when we are dealing with polymer blends, and it stands aside from the further implications of any chemical similarities or dissimilarities of the components. Though the |εii − εjj| values might often be smaller in blends than in the case of some polymer solutions, the difference is still very important because the ideal entropy of mixing is smaller for blends, and thus small differences in |εii − εjj| can be very influential on the overall miscibility behavior. It is important to recognize that g may exert some influence on ΔSexcess as well. (This is visible in Figure 10.) Though its mix effect on ΔSexcess mix is weaker than that of |εii − εjj|, especially for LCST-type systems, g can still play an important role. For instance, we find that the g parameter is directly connected to any positive ΔSexcess mix or, relatedly, a positive volume change on mixing. In such cases, g is always less than unity. These are common situations for UCST-type systems. With that said, here we only focus on the stronger connection of g with ΔHmix. We have been drawing correlations between LCST-type behavior and our model |εii − εjj| parameter, exploring the link between energetic differences in the pure components, and free volume, compressibility, and ΔSexcess for the mixture. What mix appears to be a different origin for partial miscibility arises from work by Freed, Dudowicz, and co-workers using the lattice cluster theory (LCT).6,7 They have traced LCST-type blend behavior to an unfavorable contribution to the free energy that is purely entropic in nature.56,57 This term quantifies the entropic cost of packing different blend components together by accounting for the local structural (topological) differences between the two. No analogue of the LCT structural term appears in our model. Thus, one might ask: Which mechanism is driving the LCST-type phase separation behavior? Is it topology or free volume/compressibility? We have come to the opinion that these are intimately connected. Thus, although we do not have explicit terms in our EOS that estimate the effect of molecular topology, we believe that the local structural effects (for example, on packing) are embedded in the PVT data to which we parametrize our model. The clearest example is for the case of polyolefins and their blends. Table 3 summarizes results for a set of polyolefins. In it, we compare the LCT topological parameter (taken from ref 56) alongside our pure component model energetic parameter (εii) and our predictions for the pure component free volume (the overall volume minus the hard core volume, rv). PE, the least branched of the polyolefins, has the smallest value (1.0) for the topological parameter from ref 56, and PIB, the most branched polyolefin in the series, has the largest value (1.75). Similarly, our model gives the smallest |εii| of 2013 J/mol (and largest free volume of 16.6%) for PE and the largest |εii| of 2239 (and smallest free volume of 13.9%) for PIB. In fact, all five polyolefin components line up (least branched to most branched) with a systematically increasing topological parameter, and at the same time, they also line up in order in terms of increasing |εii| (and decreasing free volume). In ref 56, a large (squared) difference in the topological parameter in the two blend components gives a large unfavorable contribution to the entropy of mixing, making it

Table 3. Comparison of Polyolefins: Parameters and Free Volume Characteristics pure componentsa PE PEP hhPP PP PIB examples: blend pairingsc PEP/PE PEP/hhPP PIB/PEP PIB/hhPP

ref 56 structural parameterb 1.00 1.20 1.33 1.33 1.75

|εii| (J/mol)c 2013 2033 2058 2066 2239

free vol (%)d 16.6 16.3 16.0 15.9 13.9

diff in ref 56 parameter

|εii − εjj| (J/mol)

diff in free vol (%)

type phase behavior

0.20 0.13 0.55 0.42

20 25 206 181

0.3 0.3 2.4 2.1

UCST UCST LCST LCST

a

Polymer acronyms: PE (polyethylene), PEP (poly(ethylene-copropylene)), PP (polypropylene), hhPP (head-to-head polypropylene), and PIB (polyisobutylene). bParameter values taken from Table 1 in ref 56. cDetails describing the parametrization of these polyolefins and their blends are available in ref 34. dFree volume is computed at the conditions, T = 450 K and P = 1 atm. It is defined in the present EOS model as the overall system volume minus the volume occupied by the segments, i.e., “free volume” = V − Nrv for any pure component fluid of N molecules each having r segments per molecule; “% free volume” = 100 × (V − Nrv)/V.

negative and leading to the possibility of LCST-type behavior. Likewise, in our EOS, the unfavorable entropy of mixing occurs when there is a growing difference in the εii parameters (growing |εii − εjj|) and, connected to that, a growing mismatch in the pure component free volumes. Notice therefore (Table 3) that in polyolefin blend pairings where the energetic (or topological) parameter differences are large (e.g., PIB paired with any of the others) LCST-type behavior typically occurs. Furthermore, comparing the two LCST-type blends PIB/PEP and PIB/hhPP, we note that the latter blend is the more miscible one (has a higher LCST), and this is clearly reflected by the fact that |εii − εjj| (and the pure component free volume difference) has decreased, leading to an entropy of mixing that is more favorable (i.e., less unfavorable) than in the former blend. As a last point we observe that polyolefin blend pairings having a small |εii − εjj| (e.g., PEP/PE and PEP/hhPP) are thus not subjected to a strong unfavorable contribution to the entropy of mixing, and this leads directly to a prediction for UCST-type behavior. The effect that |εii − εjj| has on the excess entropy of mixing, particularly for LCST-type blends, and the connection of g with ΔHmix, brings us back to Figure 7, which illustrates the correlation betweeen g and |εii − εjj|. It appears, at least for these LCST-type systems (again, systems for which ΔSexcess mix is very important), that we can imagine each of these two quantities as acting in an orthogonal “component direction” one component “controlling” the entropy and the other the enthalpy. Increasing |εii − εjj| acts in the direction of decreasing entropy of mixing, which is unfavorable. However, increasing g acts in the direction of decreasing the enthalpy of mixing, which is favorable. Thus, the overall correlation moves in a direction such that the free energy remains somewhat stabilized by compensating enthalpic and entropic effects. The appearance of this compensating relationship among the parameters seems sensible if the system is to show any chance for miscibility. Imagine, on the other hand, the opposite 8869

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correlation, where g instead decreases with |εii − εjj|. Here there would be small g values occurring at large |εii − εjj|, which would be unfavorable both entropically and enthalpically, and there would be little chance for any polymeric system (with its small ΔSideal mix ) to show even partial miscibility. For a system with large |εii − εjj| to be at least partially miscible, there must be something favorable in ΔHmix (e.g., favorable mixed interactions) in order to overcome what is expected to be a very unfavorable ΔSexcess mix . We thus can see why the correlation in Figure 7 works for LCST blends. There the very small ideal entropy of mixing contributions means the systems are essentially a balance of the two terms controlled by g and |εii − εjj|. However, in the UCST blends we may have three sizable contributions to mixing, and the excess term seems to play the role of a “wild card” sometimes adding to the ideal term and sometimes opposing it. The simple correlation thus no longer holds.

interaction may be tolerated, leading to a partially miscible system where a UCST would be anticipated.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We appreciate the financial support provided by the National Science Foundation (J.E.G.L. and R.P.W., Grants DMR0804593 and DMR-1104658) and EPSRC (J.S.H. and J.E.G.L.). J.E.G.L. also benefited from the hospitality of those in the Chemical Engineering Department at Imperial College, where some of this work was carried out.

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IV. SUMMARY AND CONCLUSIONS In this work we explore how choice of blend components determines the potential for miscibility. We find that certain model parameters act almost as separate “controls” on the enthalpy and entropy of mixing. Specifically, the mixed energetic parameter, g, is strongly correlated with ΔHmix, while the difference in the pure component energetic parameters, |εii − εjj|, most strongly influences the excess entropy of mixing. Indeed, we have found that the larger the |εii − εjj| difference, the more unfavorable is ΔSexcess mix . This insight helps to rationalize the positive, linear correlation between g and |εii − εjj| that we had recently34 deduced, based on the behavior of a large set of LCST blends. On the basis of the new work presented here, we now make the connection that a growing mismatch in pure component characteristic energies leads to an increasingly unfavorable contribution to the entropy of mixing. This must be compensated by a growing enthalpically favorable increase in the mixed interaction, as reflected in g, in order that partial miscibility continue to be observed. A growing disparity in pure component energies that is not balanced by a stronger mixed interaction will lead to a decrease in miscibility. On the other hand, for UCST systems, where the excess contribution may be large and may either enhance or detract from the ideal entropic term, the situation is more complicated. The fact that the source of unfavorable entropic effects can be anticipated by simple inspection of |εii − εjj| is very important because all that is required to obtain |εii − εjj| is knowledge of pure component information. This provides an avenue to make at least some approximate a priori predictions regarding a potential blend of interest. Thus, while work clearly remains in terms of accurate miscibility predictions, we propose there is significant utility in using |εii − εjj| as a “rough predictor”. For instance, making the connection between changes in |εii − εjj| and increasingly unfavorable contributions excess to ΔSexcess mix provides the ability to calculate estimates for ΔSmix . Thus, we venture to generalize that if |εii − εjj| is very large (e.g., approaching 300 J/mol or more), we expect the system to be totally immiscible, unless there is reason to anticipate a strongly favorable mixed interaction between the components. For smaller |εii − εjj| values the potential for partial miscibility, such as is found in LCST-type systems, is enhanced. Finally, in cases where the pure component energies are very similar, one might expect that the ideal contribution of the entropy of mixing is likely to dominate. In that scenario, a very small unfavorable

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(34) White, R. P.; Lipson, J. E. G.; Higgins, J. S. Macromolecules 2012, 45, 1076. (35) Using other values of z (e.g., z = 8 or 10) will cause the optimal values of the other parameters to change but will not appreciably change the overall quality of the fitted properties. (36) In the original derivations, U was arrived at first (via integral equation theory), and then A was obtained. So, from the point of view of the theory, taking the derivative is not necessary for obtaining U, though it does indeed “regenerate” it. (37) Another system of intensive variables (used in Figures 2 and 8) is “per total number of polymer repeat units”, with the composition variable (x) being the mole fraction of species i repeat units. (38) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (39) We define ΔSideal mix such that there is no dependence on free volume; the free volume per segment is assumed the same before and after mixing (i.e., ΔVmix = 0). Any effects associated with a system’s nonzero ΔVmix are part of ΔSexcess mix . We believe the present simple definition adequately serves for what most would picture as an ideal . It captures the increase in idealized contribution for ΔSmix translational entropy as one point on each chain molecule (say, the center of mass) gains access to the larger mixture volume. This contribution must therefore be proportional to the number of molecules (not the number of segments), and thus its relative importance (per total mass of sample) becomes weaker as the size of the molecules increases. (40) Another option is to fit to SANS data, which was done here in one case for the dPS/PVME blend. Details are available in ref 43. (41) Patterson, D. Macromolecules 1969, 2, 672. (42) Zoller, P.; Walsh, D. Standard Pressure-Volume-Temperature Data for Polymers; Technomic Pub. Co.: Lancaster, PA, 1995. (43) White, R. P.; Lipson, J. E. G.; Higgins, J. S. Macromolecules 2010, 43, 4287. (44) Krishnamoorti, R.; Graessley, W. W.; Dee, G. T.; Walsh, D. J.; Fetters, L. J.; Lohse, D. J. Macromolecules 1996, 29, 367 . Tabulated values for these data were made available to us by D. J. Lohse. (45) Callaghan, T. A.; Paul, D. R. Macromolecules 1993, 26, 2439. (46) Rudolf, B.; Kressler, J.; Shimomai, K.; Ougizawa, T.; Inoue, T. Acta Polym. 1995, 46, 312. (47) Ougizawa, T.; Dee, G. T.; Walsh, D. J. Macromolecules 1991, 24, 3834. (48) Rostami, S.; Walsh, D. J. Macromolecules 1985, 18, 1228. (49) Rudolf, B.; Cantow, H.-J. Macromolecules 1995, 28, 6586. (50) Lin, J.-L.; Roe, R.-J. Macromolecules 1987, 20, 2168. (51) Krishnamoorti, R.; Graessley, W. W.; Fetters, L. J.; Garner, R. T.; Lohse, D. J. Macromolecules 1995, 28, 1252. (52) Shibayama, M.; Yang, H.; Stein, R. S.; Han, C. C. Macromolecules 1985, 18, 2179. (53) Beaucage, G.; Stein, R. S.; Hashimoto, T.; Hasegawa, H. Macromolecules 1991, 24, 3443. (54) Chong, C. L. Ph.D. Thesis, Imperial College, London, 1981. (55) Provided that the curve ΔH̅ mix vs x has no point of inflection, then the sign of ΔH̅ mix is determined by the sign of (∂2 H̅ /∂x2)T,P (positive (∂2H̅ /∂x2)T,P implies negative ΔH̅ mix; negative (∂2H̅ / ∂x2)T,P implies positive Δ H̅ mix). Further, if ΔH̅ mix vs x is also roughly quadratic, then for any given composition ΔH̅ mix will be roughly proportional to the value of (∂2H̅ /∂x2)T,P; i.e., ΔH̅ mix ≈ −(1/2)x(1 − x)(∂2H̅ /∂x2)T,P. See: Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths: London, 1982. (56) Freed, K. F.; Dudowicz, J. Macromolecules 1998, 31, 6681. (57) Dudowicz, J.; Freed, K. F. Macromolecules 2000, 33, 9777.

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dx.doi.org/10.1021/ma3018124 | Macromolecules 2012, 45, 8861−8871