Free Volume, Cohesive Energy Density, and Internal Pressure as

Jun 4, 2014 - In this paper, we illuminate and predict trends in polymer miscibility using our model understanding of pure component properties...
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Free Volume, Cohesive Energy Density, and Internal Pressure as Predictors of Polymer Miscibility Ronald P. White and Jane E. G. Lipson* Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, United States ABSTRACT: In this paper, we illuminate and predict trends in polymer miscibility using our model understanding of pure component properties. We introduce a new metric that maps a bulk property to a theoretical characteristic parameter, viz. the percent free volume of a polymer melt as a function of the strength of its segment−segment (nonbonded) interaction energy. To this end we apply our simple Locally Correlated Lattice (LCL) model, first to characterize several dozen polymers via pressure−volume−temperature (PVT) data, and then to calculate properties that cannot be directly determined via experiment, such as percent free volume and cohesive energy density, and we rank all of the polymers in terms of these properties. We reveal strong correlations between bulk behavior and theoretical pure component character, and use those correlations to motivate a discussion of what drives polymer miscibility.

I. INTRODUCTION Can polymer miscibility be predicted? In this paper we present a picture of how the potential for miscible behavior is embedded in a polymer’s characteristic properties, as revealed (in part) through both the percent free volume and the cohesive energy density (or, its doppelganger, the internal pressure). There are aspects of mixture thermodynamics that lie outside of this emphasis, for example the strength of the mixedsegment interaction energy. However, cumulative understanding acquired from numerous studies of polymer blends and solutions has lead us to appreciate the utility of monitoring contributions to the entropy of mixing as a relatively powerful diagnostic tool. In this area we have found strong correlations between that quantity and the pure component energetic mismatch.1 Going even further, we have observed that the relative size of the mismatch correlates with whether a blend phase separates as T is raised (larger mismatch connecting with lower critical solution, LCST, behavior) or lowered (smaller mismatch correlating with upper critical solution, UCST, behavior).2 We are thus motivated to continue pushing our simple Locally Correlated Lattice (LCL) model in terms of developing metrics for how to rank polymeric compatibility, and that is the goal for this work. One of the early examples of a pure component property being correlated to mixture behavior is the cohesive energy density (CED), which is the magnitude of the internal energy per volume (|U|/V). It has been applied in regular solution theory3 in terms of the solubility parameter (δ), defined as the square root of the cohesive energy density, δ = (|U|/V)1/2. The δ values for the two components, i and j, are used to calculate the energy change on mixing, which produces ΔUmix ∝ (δi − δj)2, where the resulting form follows from the geometric mean approximation for mixed energetic interactions. The model © 2014 American Chemical Society

predicts that the more of a mismatch there is between the pure component solubility parameters (or, cohesive energy densities) the more unfavorable the energy change on mixing. A limitation of this approach (noting the form of ΔUmix) is that it can only predict positive (unfavorable) energies of mixing, and furthermore, it assumes no volume change on mixing; i.e., it does not account for compressibility. Casting this in terms of the Flory−Huggins interaction parameter, χ = a + b/T (where a is the entropic part and b/T is the enthalpic part), the solubility parameter approach allows only for positive b ∝ (δi − δj)2, and, the entropic part, “the excess entropy”, is neglected entirely (a = 0). This is in conflict with experimentally determined a and b values for many polymer blends. As an example, Table 4.3 in ref 4 shows b values that can be positive or negative, and the entropic part, a, is usually of comparable strength to b/T (and often stronger). However, in spite of these drawbacks, the simple solubility parameter approach has still been successful in a number of cases for small molecules (where there is positive ΔUmix, and small/negligible ΔVmix). For polymers, its applicability has been demonstrated for a number of (but not all) polyolefin blends.5,6 As noted, the utility of the solubility parameter approach is less clear for most other polymer mixtures, where such a simplified treatment is not usually able to capture the essential physics. Attempts at improvements have been made by trying to break the solubility parameter down into more detailed individual contributions,7 but there are significant limitations to incorporating only “positive” mixed interactions, and attempts to allow for negative interactions often come at the expense of a Received: March 16, 2014 Revised: May 8, 2014 Published: June 4, 2014 3959

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volume, energy, and so onall of these now having a clear definition within the model framework. For example, with molecular parameters describing the molecular size at hand, the free volume can be defined as the total system volume, minus the volume excluded by the molecular hard cores. Meaningful comparisons can then be made, provided these definitions are applied consistently. The simple lattice model we use in this paper (LCL) has been successfully applied to a range of polymer blend problems,1,2,14 as well as polymer solutions15,16 and small molecule fluids and mixtures.17,18 Of course, a number of modeling approaches have been developed over the years for the study of polymers and their mixtures. Included among these are molecular level physics-based equation of state approaches which are derived on a basis of statistical thermodynamics and therefore incorporate some of the key molecular characteristics of the polymeric fluid.19−38 Examples of these include SAFTrelated approaches,22−24 cell-based models,25−29 lattice-based models,30−36 and other lattice-hole theories.37,38 Among the equation of state theories capable of describing the range of miscibility behavior, we have found our model to be increasingly fruitful in terms of leading us toward deeper insights and broader applicability, and it is this potential that we explore more fully here. As noted above, we explore here a range of polymers. There have been other works that have reported on the model characterization of a large set of polymers, for instance, the compilation in ref 19. Some of the goals in these works were to document the characterization parameters, or compare theories (compare quality of fits) or, in other cases (e.g., ref 37), to compare the polymer properties. We share some of these goals in the present work. This is the first time, for our particular theory, where we report such an extensive and consistent characterization and comparison of polymer pure component properties.39 There are choices in how to perform the data fitting for the pure component characterizations, and we have made an effort here to fit all of the polymers as consistently as possible (e.g., over similar data ranges), as we have found this to be extremely important if one then wants to make meaningful property comparisons. Thus, while some of our goals are to illuminate correlations between bulk behavior and our own model’s molecular parameters, we believe that the present analysis is relevant beyond this particular theory. The results include a ranking of polymers with respect to a number of properties of general interest and impact, including free volume, cohesive energy density, internal pressure, and our discussion also involves an analysis for some selected systems as to when each of these properties might be playing a key role. The remainder of the paper is organized as follows. In section II, we provide a brief background on the model theory and how it is applied, along with a description of some of the properties to be analyzed. In section III, we present a broad overview of the results, and in section IV, a more detailed discussion of selected topics is given. A summary and concluding comments are given in section V.

further increase in the number of model parameters. (refs 8 and 9 provide a recent perspective and background on these approaches for polymers.) Regardless of whether or not a simple solubility parameter mixture model is appropriate, the cohesive energy density of each pure component remains a potentially insightful property to analyze. Indeed, it is reasonable to expect that CED values, combined with other factors, may well influence and contribute to the behavior of many mixtures and thus provide some rough a priori insight. Related to the CED is the internal pressure, Pint (defined in the following section), which has the appealing feature of being accessible via PVT data. As we will show, Pint and the CED contain significantly overlapping information about the behavior of a component; indeed, when calculated using van der Waals-type equations of state the two are identical. We will discuss this further below. A pure component property that has become of particular interest to us is the polymer’s free volume which, as we show below, does not necessarily track with the cohesive energy density (e.g., low free volume does not always translate into a high cohesive energy density, and vice versa). The concept of free volume is connected to various aspects of physical behavior, for example, it provides a link between thermodynamic and transport properties. It plays a strong role in diffusion, such as the diffusion of small molecules within a polymer matrix, or a polymer’s own self-diffusion, and is therefore also connected to viscosity. (Examples showing how model free volume values can be incorporated into predictive calculations for diffusion and viscosity can be found in refs 10 and 11.) Free volume is clearly important in any system where compressibility plays a key role, and is thus often discussed in the context of the polymer glass transition (see examples of model free volume applications in refs 12 and 13). We note a close connection between a system’s free volume and its coefficient of thermal expansion, and expect it to be a factor in any compressible mixture where the so-called “equation of state effects” are important; indeed, mismatches in free volume may sometimes drive mixture behavior. All of these observations have lead us to the conclusion that assessing and comparing this quantity for pure components could be a worthwhile endeavor. Furthermore, an advantage of modeling the free volume and the cohesive energy density is that these quantities can be evaluated using data from just the pure components, which is often much more readily available than data on mixtures. The concept of properties like internal energy (or energy density) and free volume are useful constructs in the description and comparison of different systems. However, while they are attractive because they are seemingly tangible and physically intuitive, they are not always experimentally accessible. Enthalpy of vaporization measurements can sometimes provide a route to the internal energy (and thus the cohesive energy density) but these measurements cannot be made on nonvolatile species like polymers. Free volume is not directly measurable; indeed, it first needs to be clearly defined before it can be extracted from experimental data. Here, our LCL model theory provides the needed formalism, having a framework that is based on energetic interactions, molecular size, and distance metrics, producing a unified interconnected set of thermodynamic properties. In its application, our theory links accessible experimental data on a system of interest (e.g., PVT data) to underlying molecular parameters. Following this characterization (a data fit), we can then predict values for free

II. THEORY AND IMPLEMENTATION Background on Model Equations and Molecular Parameters. A short description of the Locally Correlated Lattice (LCL) model and a few of the key expressions are covered briefly here. We refer to refs 1 and 2 as sources for detailed derivations and background, as well as for recent examples and applications. In brief, the theory is a lattice-based 3960

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data (PVT data) so as to determine values for r, v, and ε. The characteristic parameters can be analyzed and compared across a range of pure components, and subsequently be used to predict, and compare, other properties for the system, such as the free volume. Other thermodynamic quantities of interest are connected with the internal energy (eq 3), including the cohesive energy density and internal pressure (discussed further below). A key motivation for the comparison of pure component properties involves the implications for mixture behavior (e.g., in polymer blends). The extension of thermodynamic relationships given above to the case of mixtures is straightforward. A key benefit of our approach is that the pure component molecular parameters obtained from pure component characterization are transferable for modeling the mixture. Shown below is the result for A, for the case of a mixture composed of two components i and j.

model for a compressible fluid of chain-like molecules. Compressibility arises because a fraction of the lattice sites are vacant; this is the “free volume” which is so important in the discussion of the results below. We employ an integral equation approach in deriving temperature dependent nearest neighbor segment−segment probabilities. These probabilities thus incorporate local correlations, as opposed to being wholly mean field-based. This leads to an expression for the internal energy (U) which is integrated (using the Gibbs−Helmholtz relationship) from an athermal reference state to give the Helmholtz free energy (A). From the result for A, all remaining thermodynamic properties can be derived. For a single pure component, the model expression for the Helmholtz free energy, A, is derived to be ⎛ Nqz ⎞ ⎛ ξ ⎞ A ⎟ ln⎜ ⎟ = N ln ϕ + Nh ln ϕh + ⎜ ⎝ 2 ⎠ ⎝ϕ⎠ kBT +

⎛ Nhz ⎞ ⎛ ξh ⎞ ⎛ Nqz ⎞ ⎜ ⎟ ln⎜ ⎟ ln[ξ exp[ − ε / k T ] + ξ ] ⎟−⎜ B h ⎝ 2 ⎠ ⎜⎝ ϕ ⎟⎠ ⎝ 2 ⎠ h

⎛ξ ⎞ Nq A i iz ln⎜⎜ i ⎟⎟ = Ni ln ϕi + Nj ln ϕj + Nh ln ϕh + 2 kBT ⎝ ϕi ⎠ (1)

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

with the folllowing definitions: Nh = (V /v) − Nr ; ϕ = Nrv/V ;

+

ϕh = Nhv/V ;

qz = rz − 2r + 2; ξ = Nq/(Nq + Nh); ξh = Nh/(Nq + Nh)

Substituting the definitions into eq 1, it is seen that A is a function of independent variables N, V, T which are, respectively, the number of molecules, the total volume, and the absolute temperature. z is the lattice coordination number which is fixed at a value of 6,40 and kB is the Boltzmann constant. The key microscopic lattice parameters are v, the volume per lattice site, r, the number of segments per chain molecule, and ε, the nonbonded segment−segment interaction energy between near neighbor segments. In the definitions, Nh is the number of vacant lattice sites (“h” stands for “holes”) and V/v is the total number of lattice sites. ϕ is the volume fraction of segments, and ϕh is the volume fraction of vacant sites. qz is the total number of possible nonbonded contacts available to a single chain molecule, which follows by subtracting the (2r−2) bonded contacts. ξ and ξh are thus “concentration variables” which express fractions of nonbonded contacts, for segments and vacancies respectively, out of the total number of possible nonbonded contacts. As mentioned above, the other thermodynamic properties can be derived from A[N,V,T]. Two of the most important are the pressure, P, and the internal energy, U;41 these are given by

−

Nq j jz 2

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

(4)

with the following definitions: Nh = (V /v) − Nri i − Nrj j

ϕm = Nmrmv /V ξm = Nmqm /(Nq i i + Nq j j + Nh) qmz = rmz − 2rm + 2

where m can be i, j, or h, and qh = rh = 1. In eq 4, A is expressed as a function of the set of independent variables [Ni, Nj, V, T] where Ni and Nj are the numbers of molecules of components i and j. From A[Ni,Nj,V,T], all of the other mixture properties can be derived through standard thermodynamic relations. The definitions for Nh, ϕm, ξm, and qm, follow similarly from the pure component case above where now, m represents any one of the three possible types of lattice sites, m ∈ {i, j, h}. The r parameters for components i and j are denoted by ri and rj respectively, and similarly for the ε parameters, εii and εjj, represent the pure component nonbonded segment−segment interaction energy between near neighbor segments of types i−i (j−j). In the blend there is also the mixed interaction, εij, which we usually express in terms of the g parameter which defines the mixed interaction energy according to εij = g(εiiεjj)1/2. (g characterizes εij relative to the geometric mean value.) While the optimized v parameter is typically different for each pure component, we use a single v parameter to model the mixture; this value is typically chosen as a compromise between the two pure component values, whose r values are then renormalized so that the product, rv, the hardcore volume, remains constant. Thus, the pure component ε values, and hard-core volumes, rv, are unchanged from the

⎛ k T ⎞ ⎛ 1 ⎞ ⎛ k Tz ⎞ ⎛ ϕ ⎞ ⎛ ∂A ⎞ P = −⎜ ⎟ = ⎜ B ⎟ ln⎜⎜ ⎟⎟ + ⎜ B ⎟ ln⎜ h ⎟ ⎝ ∂V ⎠ N , T ⎝ v ⎠ ⎝ ϕ ⎠ ⎝ 2v ⎠ ⎝ ξh ⎠ h ⎛ k Tzξ ⎞⎧ ξ(exp[−ε/kBT ] − 1) ⎫ ⎬ ⎟⎨ −⎜ B ⎝ 2v ⎠⎩ ξ exp[−ε /kBT ] + ξh ⎭

Nq j jz

(2)

and ⎛ ∂(A /T ) ⎞ ⎛ Nqz ⎞⎢ εξ exp[−ε /kBT ] ⎥ ⎟⎢ =⎜ U=⎜ ⎥ ⎟ ⎝ ∂(1/T ) ⎠ N , V ⎝ 2 ⎠⎣ ξ exp[−ε /kBT ] + ξh ⎦ (3)

The equation for the pressure (eq 2) provides an important link to the experimental system. For example, it can be fit to pure component experimental pressure−volume−temperature 3961

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Table 1. Polymer Characterization Results: Molecular Parametersa acronym

full name

r/Mw (mol/kg)

v (mL/mol)

−ε (J/mol)

⟨T⟩ fit (K)

ref

PEA PEO PEMA PVA PE PVME PB PEP PVF hhPP PMA dPS PVC PECH PC PS SAN PPO PIB PMMA TMPC PMS PES PAN

poly(ethyl acrylate) poly(ethylene oxide) poly(ethyl methacrylate) poly(vinyl acetate) polyethylene poly(vinyl methyl ether) polybutadiene poly(ethylene-co-propylene) poly(vinylidene fluoride) head-to-head polypropylene poly(methyl acrylate) deuterated polystyrene poly(vinyl chloride) polyepichlorohydrin polycarbonate polystyrene poly(styrene-co-acrylonitrile) poly(phenylene oxide) polyisobutylene poly(methyl methacrylate) tetramethyl bisphenol A polycarbonate poly(α-methyl styrene) poly(ether sulfone) poly(acrylonitrile)

112.47 149.45 127.82 122.88 138.25 111.53 134.80 118.24 93.23 115.90 118.86 100.09 129.49 90.54 118.09 115.29 122.87 103.42 113.22 110.71 88.54 104.73 99.24 121.62

7.2276 5.4156 6.2985 6.2787 7.7962 7.9296 7.6468 9.1379 5.9197 9.1022 6.3718 8.0023 5.0090 7.4799 6.3724 7.5621 6.9214 7.9638 9.0454 6.9576 9.3484 8.0947 6.7126 6.6925

1889.6 1899.7 1917.9 1922.9 1930.4 1946.4 1951.3 1955.6 1976.1 1979.9 1998.8 2018.8 2022.3 2082.8 2104.6 2136.4 2164.0 2166.1 2172.5 2177.7 2286.0 2362.9 2588.7 3047.2

423 423 438 427 433 424 424 437 473 437 428 422 425 403 433 423 423 503 436 424 518 483 522 423

47 47 47 47 47 48 47 6b 47 6b 47 49 47 19 47 47 47 47 47 47 50 44 47 47

The table contains the results from pure component polymer characterization via fitting to PVT data. The resulting molecular parameters are r, the number of segments per chain molecule, v, the volume per lattice site, and ε, the segment−segment nonbonded interaction energy. The acronyms used in this article are given along with the corresponding full polymer names, and references for the experimental PVT data, and the average temperature (⟨T⟩) of the data range used for the fit. bTabulated values for these data were made available to us by D. J. Lohse.

a

properties of one component relative to the other. An important observation from ref 2 is that the ε parameter for all polymers tends to drift in the same direction as the fitting range changes (|ε| increases with midpoint T of fitting range). Thus, the relative differences in the polymer properties are preserved even outside of that range. Our focus on consistency is key since we want to make meaningful comparisons over a large set of polymers. For example, we want to be able to single out any particular polymer and see how it ranks in various properties of interest compared to all the rest of the polymers. Therefore, similar to making sure that any two components in a blend have been fit consistently, in an analysis such as this it would be optimal for all of the polymers to be fit over the same temperature range. Inspection of available polymer melt data shows that we can get the most polymers fit over a similar range, by targeting a temperature range midpoint of about 425 K and so this is what we have done, but there are some limitations due to lack of data, or limited ranges for the melt (e.g., high Tg’s), and so on. In the fits, the width of the data range was typically about 80 degrees K; the pressure range of the data was usually from 0 to 100 MPa. Only data points corresponding to the melt state are incorporated in the fit. Free Volume, Cohesive Energy Density, and Internal Pressure. For small molecules, the cohesive energy density (CED) has been defined as the energy of vaporization per unit volume of the liquid (the energy of the liquid measured relative to the gas phase, divided by the volume of the liquid). Using a model theory the analogous quantity is the (absolute value of the) internal energy divided by the volume, |U|/V, where U is given by eq 3. While the CED can be determined

initial, single component, characterization. More details on the implementation of the model for mixtures is available in refs 1 and 2. Model Fitting. In order to obtain the characteristic molecular parameters, r, v, and ε, we fit the model pressure equation (eq 2) to corresponding experimental pressure− volume-temperature data (PVT data) such that the best agreement with the data is reached, i.e., the sum of squared deviations is minimized.42 To get the most modeling insight, we have found that it is very important to be consistent with the data range used in the fitting. For example, when we obtain pure component parameters for the purpose of modeling a blend, we make it a priority to fit each of the two pure components over similar temperature ranges. It is best, whenever possible, to try to match the midpoint of the temperature ranges to within 10 K or less. (The breadth of the range is less important than the midpoint.) Following this procedure helps to compensate for the fact that best-fit parameter values may vary with the temperature range, an effect which is traceable to the fact that this, and many other theoretical equations of state, have an overly strong temperature dependence in the coefficient of thermal expansion.43−45 In ref 2, we discuss, using examples, how matching the fitting ranges is an important ingredient in obtaining a reliable and consistent parametrization for blend modeling, since our route yields values that we take to be constant, not temperature- or composition-dependent functions. That is, we avoid the drawbacks of introducing additional parameters and aim to get the most out of the existing physically meaningful parameters. Keeping the fitted data ranges consistent allows us to characterize most accurately, the 3962

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29 polymers, along with the full polymer name, the polymer acronym, the references to the source of PVT data,6,19,44,47−50 and the midpoint (average) temperature of the data range used in the fit.42 In Table 1, the polymers are ordered according to the strength of their ε value, a parameter we have found to be very revealing. For example, we have shown recently1,2 that there is a strong connection between the magnitude of the difference in the ε’s of two components (|εii − εjj|) and the resulting mixture behavior. We have found that UCST-type blends often have relatively small |εii − εjj| values, while those for LCST-type blends are typically significantly larger. Connected to this, we have discovered that the model excess entropy of mixing becomes increasingly unfavorable as |εii − εjj| increases; an unfavorable entropy of mixing is an important driver of LCSTtype behavior. Additionally, we have also shown how the “direction” of this |εii − εjj| difference (which ε is stronger) affects the critical solution composition, or the “skewness” of the phase diagram.16 In an upcoming manuscript, we also demonstrate how |εii − εjj| is strongly connected to miscibility behavior in polymer−super critical CO2 solutions.51 In accumulating results for an increasingly broad set of polymers, we became intrigued by the growing correlation we observed between the ε parameter and a polymer’s free volume. In Figure 1, we plot the theoretical prediction for percent free volume as a function of that polymer’s ε value (for high molecular weight polymers). A meaningful comparison means

experimentally from enthalpy of vaporization measurements for small molecules this is not possible for nonvolatile molecules (polymers). As noted in the Introduction, the polymer community often makes use of a related quantity, the solubility parameter (δ), which is given by the square root of the CED. Where CED values are not available an alternative means to define the solubility parameter is to use in its place the internal pressure, Pint, a property that is accessible from pressure− volume-temperature (PVT) measurements. Pint =

⎛ ∂U ⎞ ⎛ ∂S ⎞ ⎛ ∂P ⎞ ⎜ ⎟ = T⎜ ⎟ − P = T⎜ ⎟ −P ⎝ ∂V ⎠ N , T ⎝ ∂V ⎠ N , T ⎝ ∂T ⎠ N , V

Model values for Pint can be calculated using any of these derivative relationships, e.g., taking the derivative of eq 3. As noted, the internal pressure is related to cohesive energy density; indeed, for simple van der Waals model liquids the two quantities are the same. This follows from the fact that for a van der Waals fluid, U is proportional to 1/V, thus making (∂U/ ∂V)T and |U|/V the same, both proportional to 1/V2. For many simple nonpolar small molecule liquids, the CED and Pint have been shown experimentally to have fairly similar values,46 though there can be significant differences between the two for more complex molecules. See ref 46, for example, for a discussion of some of the distinctions between CED and Pint. Compared to simpler van der Waals models our model energy result (eq 3), derived via integral equation theory, is more complex. For example, while the van der Waals energy depends on volume only, the present model energy depends on both volume and temperature, and the resulting model CED is different from its Pint. We will discuss both values in the present work. Another property that will be featured here is the polymer free volume. The concept is useful and physically appealing, and we shall explore its connection to a number of other important physical properties. In terms of the present theory the definition for free volume for a pure component is, Vfree = V − Nrv, viz., the overall volume (a function of the given temperature and pressure) minus the total hard core volume. The hard-core volume (Nrv), which is independent of temperature and pressure, is the excluded volume occupied by all of the segments on all the molecules. In the results below we will report the percent free volume, which is simply 100 × Vfree/V. Once a characterization has been performed and the molecular parameters obtained, we can thus report this property for the given polymer, for a given T and P. Obviously, the free volume changes as the overall V changes with T and P, i.e., holes/vacancies increase with increase in T or decrease in P. Therefore, when we compare the free volume of different polymers we ensure they are at the same T and P.

III. OVERVIEW OF RESULTS Model characterization and analysis are powerful tools for obtaining insight into the nature of different polymers, and provide a route toward a deep understanding of what drives miscibility. We have accumulated significant evidence1,2,14−18 to support the assertion that fitting our LCL equation of state to experimental PVT data produces physically meaningful molecular parameters. Therefore, a simple way to compare polymers is to compare the molecular parameters resulting from these characterizations. In this paper we aim to show that such comparisons, involving only pure component properties, have the potential to be predictive. In Table 1, we list values for

Figure 1. Free volume of each polymer as a function of its corresponding ε parameter value. All free volume values are computed for the conditions, T = 425 K and P = 1 atm. Polymer acronyms are defined in Table 1. The placement of the acronym labels in the figure is consistent with the relative ranking in free volume, i.e., if one label is higher than another, that polymer has a higher free volume. Note that PAN is outside the plot range to allow better view of the other polymers. For PAN, |ε| = 3047 J/mol, and percent free volume =7.67. Topics marked on the right side of the figure are discussed explicitly in the text. 3963

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that all free volumes are computed under the same conditions which we choose to be, T = 425 K and P = 1 atm. A strong correlation is clearly evident: polymers with the weakest interaction per nonbonded segment contact have the most free volume, and as the magnitude of the ε value increases, the free volume smoothly decreases. Looking at the broader overall trends in Figure 1 some sensible correlations are immediately apparent. For example, a lot (but not all) of the polyolefins have relatively high free volume and weak ε. However, there are a few somewhat more polar polymers like PEA and PEO that also have high free volume and weak ε. On the other end of the spectrum, polymers like poly(ether sulfone) (PES) and poly(acrylonitrile) (PAN) have low free volume and strong ε. (Note that PAN (|ε| = 3047 J/mol, % free volume =7.67) is outside of the plot range, for clarity in the figure.) More explicit discussion points regarding the results in Figure 1 will be covered below, and we will point to this figure often, as it contains much valuable information. Percent free volume is not measured directly, however it is related to the coefficient of thermal expansion, α = (1/V)(∂V/ ∂T)P, which is experimentally accessible. In prior work on specific systems (e.g., dPS vs PS) we have commented on the apparent connection between α and ε. Taking this a step further, Figure 2 shows each polymer’s free volume as a

Figure 3. Polymers ranked in terms of cohesive energy density (CED) (left panel), and polymers ranked in terms of internal pressure (Pint) (right panel). All CED values and Pint values are computed for the conditions, T = 425 K and P = 1 atm. Polymer acronyms are defined in Table 1. Values on right side of right panel are from ref 6; see discussion in text.

panel, and internal pressure values (Pint = (∂U/∂V)T) are given in the right panel. Again, to make the comparison the most meaningful, all of the CED and Pint values are computed at T = 425 K and P = 1 atm. Looking at the broader trends, we note that the nonpolar polyolefins tend to have the weaker CED and Pint values compared to most of the other polymers, while polymers having strong CED and Pint include PVC, PES, and PAN. Note that the CED value for PAN is so large (799 J/mL) that it is off the scale of that plot, although it is included on the Pint plot. Further observations based on this figure are discussed below, including a more detailed analysis of results for the polyolefins. As noted above, our LCL theory yields different results for the CED and internal pressure, in contrast to the situation for van der Waals type models. That the two quantities, while reflecting similar aspects of a given system, are distinct46 is indicated by comparing the two panels in Figure 3, and illustrated more clearly in Figure 4, which shows internal pressure values plotted against CED. (All are computed at T = 425 K and P = 1 atm.) The correlation is linear, with significant scatter. Both CED and Pint values have been used extensively in the literature as routes to a polymer’s solubility parameter, which has commonly been applied as a guide to miscibility. While we have not found the solubility parameter formalism to be very illuminating for polymer mixtures, we believe there is a role to be played for using either CED or Pint as one in a set of diagnostic guides. We pursue this line of reasoning below. Given the reasonably strong relationship between cohesive energy density and internal pressure, one might now ask whether either are related to the free volume. A reasonable hypothesis is that low free volume (and strong |ε|) would imply a strong CED, and vice versa. In fact, as Figure 5 shows, mapping the percent free volume against CED yields a scatter plot, with no correlation evident; the same is found in plotting percent free volume against Pint. Why the free volume and the related ε parameter do not always track with the CED can be understood by considering how a strong ε value for segment−

Figure 2. Free volume of each polymer as a function of its corresponding coefficient of thermal expansion value (α). All free volume values and α values are computed for the conditions, T = 425 K and P = 1 atm. Polymer acronyms are defined in Table 1

function of its corresponding α value. (In all cases, both the free volume and the α value are computed for the conditions T = 425 K and P = 1 atm.) A positive correlation is to be expected, since it is the f ree volume, not the hard-core contribution, that expands with increasing T. Still, the strength of the correlation is striking, being almost linear, and roughly going to zero intercept. Thus, if an experimentally measured α value for one polymer is found to be twice the value of another polymer (at the same T and P), then one can conclude that the former polymer has roughly twice the free volume. In addition to the free volume, there are other important measures by which the polymers can be compared, for instance, according to cohesive energy density, and the related internal pressure. These results are shown in Figure 3, where cohesive energy density values (CED = |U|/V) are ranked in the left 3964

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IV. CASE STUDIES We now take a closer look at some of the free volume results in Figure 1 and demonstrate the kinds of insight can be gained by inspection of the free volume and the ε parameter, as well as via consideration of the cohesive energy density and internal pressure. Molecular Topology and Free Volume and Their Relation to Tg. We begin by observing that small structural (molecular topological) variations within families of similar polymers are manifested in the free volume properties. For example, consider the addition of a single methyl group to the backbone in each polymer repeat unit. This is the case upon going from poly(methyl acrylate) (PMA) to poly(methyl methacrylate) (PMMA), or, from polystyrene (PS) to poly(αmethyl styrene) (PMS), or, from polyethylene (PE) to poly(ethylene-co-propylene) (PEP). In all of these examples adding a methyl group to the backbone leads to a decrease in the free volume (increase in |ε|). These changes are highlighted by the inset in Figure 1 labeled “topic 1”. In addition to the three marked examples, there is PEA to PEMA, which also shows a drop in free volume. Going farther, we find that the drop in the model free volume (and ε) translates into an increase in the experimentally observed glass transition temperature (Tg). For example, the Tg values for PMA and PMMA are, respectively, 283 and 378 K; for PEA and PEMA, 249 and 338 K; for PS and PMS, 373 and 441 K. In all of these cases, adding a methyl group to the repeat unit leads to an increase in the experimental Tg, with a concomitant decrease in model free volume and increase in model |ε|. Our preliminary results indicate that among polymers that are chemically similar, those with the lower free volume (larger |ε|) are associated with higher Tg values. Broadening the discussion we find that for our entire polymer set there is a noisy, but still fairly obvious, correlation between percent free volume (thus |ε|) and Tg. In addition to ε and the associated PVT properties there are other important contributions affecting Tg such as the effect of chain stiffness. (For example, see ref 52, which shows how quantities such as fragility and stiffness energy are correlated with free volume at Tg.) We plan to expand the range of systems as more experimental data are uncovered and analyzed. Analysis of Polyolefins: A New View of Polyisobutylene (PIB). Next we move to a discussion of the polyolefins. Above, we noted how CED and/or Pint have been applied to calculate mixture behavior within the solubility parameter formalism (Hildebrand/regular solution-type approaches). Graessley and co-workers demonstrated that this formalism was applicable for many polyolefin blends.5,6 In their work, estimates for the energy of mixing were calculated (proportional to (δi − δj)2) using PVT data, where Pint was used as the route to δ (=Pint1/2). Experimental mixture results show that many pairs of polyolefins do behave in a “regular” fashion; i.e., reasonable correlations may be obtained using solubility parameters. However, some pairs of polyolefins show “irregular mixing”, where the solubility parameter formalism cannot work; these results cannot be explained on the basis of Pint or CED. The irregular polyolefin blends typically contain poly(isobutylene) (PIB) as one of the components. (Note that, compared to other polymers, PIB has also been shown to exhibit peculiar dynamic behavior.53) We have characterized PIB here, as well as several other polyolefins (PE, PEP, hhPP),

Figure 4. Internal pressure (Pint) of each polymer as a function of its corresponding cohesive energy density (CED). All Pint and CED values are computed for the conditions, T = 425 K and P = 1 atm. Polymer acronyms are defined in Table 1.

Figure 5. Free volume of each polymer plotted against its corresponding cohesive energy density (CED). All free volume and CED values are computed for the conditions, T = 425 K and P = 1 atm. Polymer acronyms are defined in Table 1

segment interactions can be offset in systems that have fewer segments per overall system volume. If the optimal value for v, the parameter that characterizes lattice site size, is large then a chain of a given molecular weight will require fewer lattice sites (segments) than it would on a lattice having a smaller spacing size, and therefore fewer segment−segment interactions contribute. The percent free volume tracks smoothly with |ε|; however, the internal energy depends not only on the strength but also on the number density (probability), of segment− segment contacts. The fact that free volume and CED (or Pint) are not strongly correlated is important. Each quantity is linked to important characteristics and could serve as a driver for im/miscibility, depending on the situation. Our evidence that these properties are not redundant leads to an independent pair of tools for understanding and predicting mixture behavior; we will turn to some specific examples in the section that follows. 3965

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volume with the blending partner (Figure 1) that plays a large role in controlling miscibility, a trend that results for the CED or Pint (Figure 3) completely fails to capture. Effect of Deuteration. In prior work, we have drawn a connection between a substantial difference in |εii − εjj|, and thus α values, being correlated with LCST behavior; PS/PVME is one such example. In addition, it has been shown experimentally60,61 that deuteration of the PS component in the PS/PVME blend (PS/PVME to dPS/PVME) produces a significant increase in blend miscibility, with the dPS/PVME blend having an LCST value about 40 K higher than PS/PVME (for comparable molecular weights). In a recent paper probing the effect of deuteration we had connected this shift in going from hPS to dPS to a significant change in α-values, with the dPS α being closer to that of PVME than the hPS α, thus driving a higher LCST. Turning here to “topic 3” in the results of Figure 1, we see that of the three polymers PVME has the highest free volume and weakest |ε|. We also find that the effect of shifting from PS to dPS leads to a sizable increase in free volume (lowering of |ε|), making the value for dPS much closer to the value of the blend partner PVME. The trend in blend miscibility is embedded in this model pure component property. Not only is dPS better than PS in matching PVME in terms of free volume, inspection of Figure 3 also shows a better match in terms of CED and Pint. The CED and Pint of PVME is not particularly strong, being fairly close to that of the polyolefins, while PS shows considerably stronger values. Notice how dPS, on the other hand, shows a weaker CED and Pint than PS, making it closer to that of PVME. It is sensible to think that the move to a better match in CED and Pint for the case of dPS with PVME is also something that could only help increase the blend miscibility. Deuteration of PS to dPS decreases the mismatch in both, free volume, and in CED (or Pint). Other Cases of Interest. There are additional observations that can be made based on the polymer property rankings: Poly(acrylonitrile) (PAN) represents an extreme among the polymers we have studied so far, in having both a very low free volume and strong ε parameter, as well as very strong CED and Pint values. Looking at Pint in particular (Figure 3), it is interesting to compare the number for PAN (520.5 J/mol) with that of PS (363.1 J/mol), and note that SAN, a copolymer of PAN and PS, has a value (401.7 J/mol) that falls in between the two homopolymers. This particular SAN sample was composed of 25% PAN and 75% PS, and a 25/75 weighted average of the Pint values of PAN and PS yields a value of 402.4 J/mol, almost identical to that obtained directly from the analysis of the copolymer. The weighted average of CED values gives 523.9 J/ mol as compared to the observed SAN value of 481.7; this is not as close, but it still roughly reflects the combination of the two homopolymers. In contrast, one would not have been able to anticipate the free volume of SAN with a similar line of reasoning. The free volumes of PS and SAN (Figure 1) are fairly close (13.62% and 13.33% respectively) and nowhere near the very different 7.67% for PAN. From this one might surmise that, while PAN units are evidently rather closely packed, the presence of PS units must serve a disruptive role, increasing the free volume in a nonlinear fashion. Another illustrative case involves the blends PS/PMMA and PMS/PMMA. PS and PMMA are only sparingly miscible, showing UCST-type phase separation for smaller (oligomeric) molecular weights, while being completely immiscible at larger molecular weights. The addition of a methyl group to PS,

and it is illuminating to compare our results with those of the earlier studies. As the Pint values in Figure 3 show, we have obtained very similar results for the polyolefins compared to those in ref 6. The ref 6 values (diamonds), which are for a similar temperature (440 K), are offset to the right of our results (triangles) in the figure. Within the solubility parameter formalism energies of mixing are proportional to (δi − δj)2 and must always be positive (unfavorable); they become increasingly unfavorable as the mismatch in Pint (or CED) increases. Note that with a positive energy of mixing one can reliably conclude that the type of phase separation will be UCST-type.54 Now, consider the two blend pairings, hhPP/ PEP and PE/PEP. They both exhibit UCST-type phase separation, but there is a smaller difference in Pint between hhPP and PEP than there is between PE and PEP. This implies a less unfavorable energy of mixing (smaller (δi − δj)2) for the hhPP/PEP blend, and experiments have in fact shown55,56 that hhPP/PEP is more miscible (i.e., lower UCST value for comparable molecular weights) than the PE/PEP blend. On the other hand, there is PIB, whose behavior has proven to be difficult for other theoretical approaches to capture. Figure 3 shows that all polyolefins tend to have fairly low values for Pint (and CED) compared to most other polymers, and PIB is no exception to this. Thus, any application of the solubility parameter formalism would predict PIB mixing behavior analogous to the other polyolefin mixtures, viz., positive energy of mixing proportional to (δi − δj)2, and partial miscibility behavior (at large enough Mw) that would be of the UCSTtype. However, experimentally, PIB-containing blends show LCST-type phase separation; its peculiar miscibility cannot be captured by trends in Pint or CED. Turning to the free volume results, marked as “topic 2” in Figure 1, provides a route to another correlation, and here PIB is strikingly different. Its free volume is much lower (stronger |ε|) than any of the other polyolefins, being notably low even on the scale of the entire polymer set. This is also consistent with the observations that PIB has very low gas permeability.53 The pairing of PIB with any other polyolefin component will thus lead to a significant mismatch in pure component free volumes, or equivalently, a mismatch in coefficients of thermal expansion (Figure 2), which also implies a mismatch in the model pure component ε parameters (large |εii − εjj|). This produces the so-called “equation of state effects” which are captured in our model mixture calculations. The large |εii − εjj| results in a negative volume change on mixing, and concomitantly, a negative entropy change on mixing;1 the result is mixture instability as the temperature increases. In other words, based on our analysis, we would predict LCSTtype phase separation behavior, consistent with the observed “irregular” mixture behavior of PIB. We can also consider the corollary, viz. how a reduction in the |εii − εjj| mismatch can serve to increase the mixture miscibility via a less unfavorable (more favorable) entropy of mixing. Both the PEP/PIB and hhPP/PIB blends exhibit LCST-type phase separation, but the results of Figure 1 show that the difference in free volume is a little bit larger for PEP relative to PIB than it is for hhPP relative to PIB. As it turns out experiments do indicate57,58 that PEP/PIB is somewhat less miscible than hhPP/PIB (lower LCST for same Mw). Further, PE/PIB is less miscible still, 59 and Figure 1 shows correspondingly, an even larger free volume mismatch. In explaining the behavior of PIB, it is evidently a mismatch in free 3966

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forming poly(α-methyl styrene) (PMS), serves to increase its CED and Pint bringing the values closer to those for PMMA. The reduced mismatch in properties between PMS and PMMA is reflected in experimental results that show PMS/PMMA is significantly more miscible than PS/PMMA. All of this seems to make sense, but then we turn to the results of Figure 1 that show the free volumesand ε valuesare more of a mismatch in PMS/PMMA. How to reconcile the apparently different trends? In fact, there is an additional experimental complexity: the PMS/PMMA blend is of the LCST-type, not the UCSTtype observed in low Mw PS/PMMA. Now the free volume results make sense, as they reflect the correlation already discussed, viz. that larger |εii − εjj| and free volume differences track with LCST behavior, smaller differences with UCST behavior. The former is associated with increasingly unfavorable contribution to the entropy of mixing, which destabilize the mixture as the temperature increases. Finally, we note that there are some noticeable “clumps” in the CED and Pint values (Figure 3). For example, SAN, PMMA, and PMA are all nearby in both Pint and CED, with PVA also close in Pint, and PVF also close in CED. Picking pairs from among these pure components there are actually quite a few examples of blends that are at least partially miscible, examples including combinations such as PMA/PMMA, SAN/PMMA, PMA/PVA, and PMA/PVF.

in a fundamental way to underlying influences on the excess entropy of mixing. We believe there is real promise in advancing our ability to discern miscibility trends via analysis of pure component properties, alone. However, we do not want to suggest that experimental data on mixtures will no longer be required in order to understand polymer miscibility. We are very much aware that in blend thermodynamics small contributions can be powerful drivers of behavior. There are limits to what can be learned from pure components; we do not believe those limits have yet been reached.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We appreciate the financial support provided by the National Science Foundation (Grant No. DMR-1104658). REFERENCES

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V. SUMMARY AND CONCLUSIONS There are few trustworthy guides through the landscape of polymer miscibility. In this paper we suggest a promising candidate, the percent free volume of a pure component melt. This is a theoretical quantity, closely related to an experimentally accessible one (the thermal expansion coefficient), and calculated using characteristic parameters that are obtained via fit of our LCL equation of state to pure component data (here we use pressure−volume−temperature data). Indeed, there is an extremely strong correlation between a polymer’s percent free volume and one of these parameters, viz. ε, the strength of its nonbonded nearest-neighbor interaction energy. We have illustrated this connection through analysis of several dozen polymers; it is important to emphasize that the characterizations were done over a consistent temperature- and pressure-range, which is key in terms of being able to make comparisons between polymers. We find that the percent free volume is a metric unlike others that have been commonly used, for example the cohesive energy density, CED, or internal pressure, Pint. A plot of the first against either of the latter two is, in fact, a scatter plot. Here we note that while CED and Pint values for a polymer are often closely related they are distinguishable in our LCL theory, which is not the case in van der Waals type theories, wherein they provide the same information. With our explorations using percent free volume our goal is to add a dimension to polymer characterization, instead of trying to replace one. Using these tools we have presented several “case studies”. One involves an explanation of the anomalous behavior of polyisobutylene in the realm of polyolefin miscibility; another focuses on the effect of adding local branching to the glass transition and free volume trends. We also discuss trends that highlight a connection between the increasing disparities between the percent free volume (hence ε) of two possible blending partners and the likelihood that partial miscibility will shift from producing a UCST to an LCST. We find this relates 3967

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