Connecting Theory and Experiment To Understand ... - ACS Publications

Article pubs.acs.org/jced

Connecting Theory and Experiment To Understand Miscibility in Polymer and Small Molecule Mixtures Jane E. G. Lipson* and Ronald P. White Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, United States ABSTRACT: In this article, we discuss applications of our locally correlated lattice (LCL) theory to problems involving small molecules (e.g., simple alkanes), ionic liquids, and polymer melts and blends. The theory employs a compressible lattice model (segments and vacancies) combined with integral equation-based nearest neighbor segment−segment probabilities (“local” correlations) leading to analytic solutions (e.g., a simple three-parameter equation of state) that are applicable for modeling the thermodynamic properties of fluids and fluid mixtures in both gas and liquid states of aggregation. The theory incorporates a physically meaningful set of molecular parameters that are transferable and that have been shown both to capture and to predict fundamental thermodynamic behavior. The model can be used to study the full spectrum of thermodynamic properties, ranging from pressure−volume−temperature behavior to phase equilibria for pure and mixed systems. The applications discussed in this paper, which encompass both a review of prior results as well as new work, involve extensive connections with experimental data as well as a new system for ranking polymers as a route to predictions regarding miscibility. these, one starts with a model fluid of chain-like molecules (molecules with multiple segments), and the goal is to characterize the fluid such that fundamental features such as the balance between the entropy of the chain molecules and the attractions between the segments are effectively captured. In the case of SAFT-related models, perturbation methods such as those of Barker and Henderson,21 or others, are typically applied to handle the effect of attractions between the segments (hard spheres are perturbed into attracting spheres). Chain-like structure is also incorporated via perturbation of free segments into chains (e.g., Wertheim’s approach22). In some cases (e.g., PCSAFT), the former step proceeds by utilizing integral equation results for the structure of hard-sphere chains23 as a starting point. Cell-based approaches for chain molecules aim to separate out the chain molecule’s relevant intermolecular “external” degrees of freedom (via a parameter to be fit) and then to simplify the chain segment’s intermolecular environment into a mean field picture, wherein the segment resides in a “cell” created by surrounding segments, thus leading to tractable (separable) independent partition functions. In lattice-based approaches, the entropy of the chain molecules is typically handled by using the lattice to enumerate chain configurations (to various degrees of approximation). For the energy, the interactions are “added up” using various approximations for the probability for

I. INTRODUCTION In this paper, we both review highlights and present recent results using our locally correlated lattice (LCL) theory for modeling fluids and fluid mixtures. We have applied this approach to liquids ranging from alkanes to ionic liquids to a large array of polymers. We have predicted liquid−vapor coexistence envelopes, liquid−liquid phase diagrams with phase boundaries containing both upper- and lower-critical solution temperatures (U/LCST), and numerous pure component properties. We have studied the effects on miscibility from deuteration and from local branching as well as from manipulating the more commonly encountered variables of polymer molecular weight, temperature, and pressure. Most recently, we have been exploring how different metrics characterizing pure polymers can be used to “rank” components, with the goal of introducing new tools to predict miscibility trends in the absence of mixture data. This field is crowded with approaches that vary wildly in terms of their fundamental physical underpinnings, their ability to capture fluid/mixture behavior using a minimal number of physically motivated and well-defined variables, and their success in terms of transferring insights from one experimental set of conditions to another. It is incumbent, therefore, on theorists to demonstrate just how well their methods meet these stringent requirements; such is another of the goals of this paper. Over the years a number of theoretical equation-of-state (EOS)-based treatments have been developed for fluids of polymers and small molecules.1−20 Examples of these include SAFT-related approaches,4−6 cell-based models,7−11 latticebased models,12−18 and other lattice-hole theories.19,20 In all of © 2014 American Chemical Society

Special Issue: Modeling and Simulation of Real Systems Received: March 25, 2014 Accepted: July 14, 2014 Published: July 28, 2014 3289

dx.doi.org/10.1021/je5002818 | J. Chem. Eng. Data 2014, 59, 3289−3300

Journal of Chemical & Engineering Data

Article

⎛ Nqz ⎞ ⎛ ξ ⎞ A ⎟ln⎜ ⎟ = N ln ϕ + Nh ln ϕh + ⎜ ⎝ 2 ⎠ ⎝ϕ⎠ kBT

segments to be near neighbors. When the goal is to allow for varying amounts of free volume as the system expands/ contractsthat is, for the system to be compressiblethen one of the “components” will be holes, or lattice vacancies. Our own model falls into the category of compressible, lattice-based theories and more details will be given below. An intrinsic feature of our LCL approach (and some of the others) is that fundamental properties (energy, entropy, etc.) are formulated in terms of physically meaningful microscopic parameters. Such parameters appear in all theoretical treatments, demonstrating that the parameters actually contain the physical information ascribed to them requires analyzing a significant number of systems. The result of many such applications using our model is that we now have a route to understanding and predicting properties that are not easily accessible via experiment. In what follows we will cover some theoretical background and details of LCL model implementation in Section II. In Section III, we move to results and discussion of particular systems. A number of examples are covered, from small molecules to polymers, demonstrating the different types of properties that can be analyzed with the model. We provide a summary and concluding comments in Section IV.

+

⎛ Nhz ⎞ ⎛ ξh ⎞ ⎛ Nqz ⎞ ⎜ ⎟ln⎜ ⎟ln[ξ exp[ − ε / k T + ξ ]] ⎟−⎜ B h ⎝ 2 ⎠ ⎜⎝ ϕ ⎟⎠ ⎝ 2 ⎠ h (1)

with definitions Nh = (V /v) − Nr ϕ = Nrv /V

ϕh = Nhv /V

qz = rz − 2r + 2 ξ = Nq/(Nq + Nh)

ξh = Nh/(Nq + Nh)

where z is the lattice coordination number which is fixed at a value of 6 (see below) and kB is the Boltzmann constant. If the segments in a molecule were separated, they would be capable of experiencing a maximum of rz interactions. Accounting for chain connectivity, qz is the total number of possible nonbonded contacts available to a single chain molecule. This follows by subtracting the (2r − 2) bonded contacts from rz. ξ and ξh are, thus, concentration variables that reflect the local connectivity experienced by a component. As noted above, Nh is the number of holes, expressed as the total number of sites (V/v) minus the total number of segments (Nr). ϕ is the hardcore volume fraction. ϕh is the free volume fraction. Substituting these definitions into eq 1, it is seen that A is expressed as a function of the independent variables, [N,V,T], and thus starting from A[N,V,T], all of the other thermodynamic properties can be derived through standard relationships and definitions, for example, , the pressure, P = −(∂A/∂V)N,T, the enthalpy, H = U + PV, the Gibbs free energy, G = A + PV, and so on. In application, it is also convenient to work in terms of all intensive variables, that is, A̅ expressed as a function of [V̅ ,T] where A̅ = A/N and V̅ = V/N. For units, we suggest the following system. With v in mL/mol of lattice sites, ε in J/mol of segment−segment interactions, and kB correspondingly replaced by the gas constant R, and then taking V̅ in mL/mol of molecules and T in K, then all computed energies will be in J/mol of molecules and pressure in MPa. The pressure, P, obtained by taking the volume derivative of the Helmholtz free energy (eq 1) is given by

II. COMPUTATIONAL METHODS In this section, we will briefly cover some of the key expressions for the locally correlated lattice (LCL) theory and discuss its application. We refer to refs 24−27 for more background, including additional recent examples and detailed explanations regarding how to apply the model. As noted above, the LCL model utilizes a lattice-based description for a fluid of chain-like molecules. We consider N chain molecules, in a volume, V, at absolute temperature, T. There are three fundamental molecular parameters: r, the number of segments per molecule, v, the volume per lattice site, and ε, the nonbonded segment−segment interaction energy between near neighbor segments. Recall that this is a compressible lattice model (essential for modeling pressure− volume−temperature, or “PVT”, properties); therefore, there will be a number, Nh, of the lattice sites that are vacant (where “h” stands for “holes”). The overall volume can thus increase (at fixed N) with an increase in the number of vacancies (Nh), specifically, V = Nrv + Nhv, where Nhv is the “free volume”, and Nrv is the “hard-core” volume. (Nr is the total number of segments.) The derivations (see for example ref 26) follow an integral equation formalism, the goal of which is to obtain the temperature- and density-dependent segment−segment nearest neighbor probabilities. In other words, we consider the conditional probability that a site will be occupied or vacant, given that a neighboring site is occupied (or vacant). Therefore, these probabilities incorporate “local correlations”, as opposed to being wholly mean field-based. Expressions for the probabilities then lead to the result for the internal energy (U); this is then integrated (using the Gibbs−Helmholtz relationship, U = [∂(A/T)/∂(1/T)]N,T) from an athermal reference state (at 1/T = 0) in order to obtain the result for the Helmholtz free energy, A = U − TS, where S is the entropy. The expression for A is as follows:

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

(2)

Again, substituting in the definitions for ξ, ξh, and ϕh above, eq 2 is, thus, a relatively easy-to-apply three-parameter equation of state that is applicable to both liquids and gases; it shows a van der Waals loop at intermediate densities (below Tc), and in the limit of low density, it reduces to the ideal gas law. Although P is a natural expression of the independent variable V, V cannot be directly expressed as a function of P (as is typical with many theoretical equations of state). 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 once this V is found, any of the other properties (also natural expressions of the independent variable V) can then be straightforwardly evaluated. In application, it is convenient to code the model 3290



Article

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

(define the functions for A, P, etc.) in commonly available mathematical/computational software; this can then be used for numerical root finding, fitting, solving simultaneous equations, defining other thermodynamic properties through derivative relationships, and so on. PVT data are some of the more common types of data that are available, and the three molecular parameters, r, v, and ε, for the system are often obtained by a fit of eq 2 to a PVT surface. Alternatively, in cases where PVT data are lacking, there is perhaps an even simpler route to the parametrization. This can be done with a single molar volume measurement (or equivalently specific volume), a single value for α = (1/ V)(∂V/∂T)N,P, and a single value for κ = −(1/V)(∂V/∂P)N,T. (In terms of the natural working independent variables for the theory, α and κ can be computed from the relations, (∂V/ ∂T)N,P = −(∂P/∂T)N,V/(∂P/∂V)N,T and (∂V/∂P)N,T = 1/(∂P/ ∂V)N,T.) The three parameters (r, v, ε) can be obtained by the (numerical) solution of three simultaneous equations: one equation specifying the model pressure at that experimentally measured molar volume (e.g., P = 1 atm), and the other two equations specifying the model α and model κ to be the experimentally measured values. Only the three parameters r, v, ε are optimized in the manner described above; the lattice coordination number, z, is kept fixed at z = 6. Using other values of z (e.g., z = 8 or 10) will cause the optimal values of the r, v, and ε parameters to change, but it will not appreciably change the overall quality of the fitted properties. Further, although the values of r, v, and ε will change with z, the combined molecular quantities such as qzε (the molecular energy at maximum/close packing), and the product rv (the molecular hard-core volume) are less sensitive to z. Much insight can be obtained by comparing the molecular parameters, but when comparing them (e.g., comparing the ε values), this should always be done at a single choice of z. The expression for the internal energy, U, is given by

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

−

Nq ⎛ −εij ⎞ ⎛ −εjj ⎞⎤ j jz ⎡ ln⎢ξiexp⎜ + ξh⎟⎥ ⎟ + ξjexp⎜ ⎢⎣ 2 ⎝ kBT ⎠ ⎝ kBT ⎠⎥⎦

(4)

with definitions Nh = (V /v) − Nri i − Nrj j ϕm = Nmrmv /V qmz = rmz − 2rm + 2

ξm = Nmqm/(Nq i i + Nq j j + Nh)

where m can be i, j, or h, and qh = rh = 1. It is seen (substituting in the definitions) that A is expressed as a function of the independent variables [Ni,Nj,V,T], and thus, this is the key expression from which all of the other thermodynamic properties can be derived. The above definitions for Nh, qm, ϕm, and ξm are analogous to the pure component case above, where now in the mixture, “m” denotes any one of the three possible types of lattice sites, m ∈ {i, j, h}. Here, for the mixture, the r parameters for each component are denoted ri and rj, and the corresponding i−i and j−j nonbonded energy are denoted by εii and εjj. In mixtures, there is also the mixed interaction energy εij; in very limited circumstances, this can be successfully approximated from the geometric mean approximation, but in other cases, it must be treated as a parameter to be fitted to mixture data. For polymer blends, the latter case is typical, wheredue to the very delicate balance between energetics and the weak entropic drive to mix large moleculessmall adjustment relative to the geometric mean is necessary. In these cases, we introduce the mixture parameter, g, which determines the mixed interaction energy according to εij = g(εiiεjj)1/2

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

(5)

When the geometric mean approximation applies, g = 1. Examples of mixture data that can be used to fit g are critical solution temperatures (e.g., UCST’s or LCST’s) and SANS data (second derivative of the free energy). For small molecule components, there is also the possibility of data for vapor-phase partial pressures and enthalpy or volume changes on mixing; model expressions for these quantities are discussed below. In modeling the mixture, there can be only a single v parameter, and so this is often taken to be a compromise between the v’s of two pure components. In order to use the same v for pure and mixed states we rescale the ri and rj values by adjusting the pure component-fitted r’s such that the molecular hard-core volume (rv) remains the same in the model mixture, on the new lattice, as it was in the pure component fit. Thus, in modeling the mixture, the ε parameter and the molecular hard-core volume are unchanged from the initial, single component, characterization. More details on the LCL model implementation for mixtures can be found in refs 24 and 25. Important for mixtures are the “changes on mixing” defined relative to the corresponding amounts of pure components at the same T and P. For the enthalpy, ΔH̅ mix = H̅ − xH̅ pure − (1 i − x)H̅ jpure, and similarly for the other properties. The

(3)

As noted above, integration of U = [∂(A/T)/∂(1/T)]N,T is the route to the expression for A. The internal energy, and the related cohesive energy density (U/V), are good examples of physically insightful properties that may not always be experimentally accessible. Further examples include the entropy (S) and, for mixtures, the entropy change of mixing, along with a breakdown of its ideal and excess contributions. A powerful aspect of a model theory is that such quantities can be calculated once molecular parameters have been determined via characterization using whatever data are available. Another key aspect of working with an effective theory is that a pure component’s molecular parameters are transferable for purpose of modeling the mixture. Again, the key expression is the Helmholtz free energy, and for the case of two components, i and j, it is given by 3291



Article

(∂ 2G̅ /∂x 2)T , P = (∂(μi − μj )/∂x)T , P

composition variable, x = Ni/N, is the mole fraction of component i, where for the mixture N = Ni + Nj. The overbar again signifies an intensive quantity; H̅ = H/N in the mixture; H̅ pure is H/N for i in its pure state, and so forth. Alternatively for i polymeric systems (as opposed to small molecule systems), it can be convenient to define the intensive units as “per mass” and using mass fraction as the composition variable. In modeling phase equilibrium, we make use of our theoretical result for the chemical potential, μ. In binary mixtures μi = (∂A/∂Ni)Nj,T,V = (∂G/∂Ni)Nj,T,P, again where the Gibbs free energy is G = A + PV = H − TS. The chemical potential reduces to simply μ = (∂A/∂N)V,T = (∂G/∂N)P,T = G̅ for the case of a pure component, whereas for the binary mixture, the expression for μ is as follows:

= (∂(μi − μj )/∂x)T , V̅ + (∂(μi − μj )/∂V̅ )T , x (∂V̅ /∂x)T , P = (∂(μi − μj )/∂x)T , V̅ − (∂(μi − μj )/∂V̅ )T , x (∂P /∂x)T , V̅ /(∂P /∂V̅ )T , x (7)

Note that finding the point where (∂G̅ /∂x)T,P is zero defines the so-called “spinodal boundary”. This is described by two equations, (∂2G̅ /∂x2)T,P = 0, P = 1 atm (or other specified pressure), which can be solved for the two unknowns, x and V̅ , thus giving a spinodal point for a given T. The critical point is found by solving the three equations, (∂3G̅ /∂x3)T,P = 0, (∂2G̅ / ∂x2)T,P = 0, P = 1 atm, for the three unknowns T, x, and V̅ , these thus being the critical temperature (UCST or LCST), critical composition, and critical volume. Also important are the enthalpic and entropic contributions to (∂2G̅ /∂x2)T,P: (∂2H̅ /∂x2)T,P and (∂2S̅/∂x2)T,P. In differentiating twice with respect to composition, the pure component terms in the mixing functions drop out, and therefore, (∂2G̅ /∂x2)T,P = (∂2G̅ mix/∂x2)T,P, (∂2H̅ /∂x2)T,P = (∂2H̅ mix/∂x2)T,P, and so forth. Indeed, there is even a strong association of the second derivatives to the mixing functions themselves: Given that the mixing function expressed as a function of composition (at constant T, P) has no change in curvature, then, if the second derivative of a property is positive then the mixing function must be negative and vice versa. Mixtures that have an upper critical solution temperature (UCST) (ones that phase separate as T is decreased) must have negative (∂2H̅ /∂x2)T,P and (∂2S̅/∂x2)T,P at the UCST, and therefore, typically these systems have positive ΔH̅ mix and ΔS̅mix. Following similar arguments, LCST-type systems (phase separating upon increase in T) often have negative ΔH̅ mix and ΔS̅mix.28

μi = kBT ln(ϕi) − rk i BT ln(ϕh) − (ri − 1)kBT ⎛qz⎞ ⎛ξ ⎞ ⎛ rz ⎞ ⎛ ξ ⎞ + kBT ⎜ i ⎟ln⎜⎜ i ⎟⎟ − kBT ⎜ i ⎟ln⎜⎜ h ⎟⎟ ⎝ 2 ⎠ ⎝ϕ ⎠ ⎝ 2 ⎠ ⎝ϕ⎠ i

h

⎛qz⎞ ⎛z⎞ + kBT ⎜ ⎟(ri − qi) − kBT ⎜ i ⎟ln[ξie−εii / kBT ⎝2⎠ ⎝ 2 ⎠ ⎛ zξ ⎞ ⎛ + ξje−εij / kBT + ξh] − kBT ⎜ i ⎟⎜⎜(ri − qi) ⎝ 2 ⎠⎝ +

qie−εii / kBT − ri ξie−εii / kBT + ξje−εij / kBT

⎞ ⎛ zξ ⎞ ⎟ − kBT ⎜ j ⎟ ⎝ 2 ⎠ + ξh ⎟⎠

⎛ ⎞ qie−εij / kBT − ri ⎜(ri − q ) + ⎟ ⎜ i ξie−εij / kBT + ξje−εjj / kBT + ξh ⎟⎠ ⎝

(6)

Equation 6 defines the chemical potential, μi, for component i. μj for component j follows from exchanging the “i” and “j” labels in eq 6. Phase boundaries are mapped out by solving for the conditions where the chemical potential is the same in each phase, and in the case of mixtures, this condition is applied for each component. More specifically, to treat a binary mixture with two phases I and II, at a given T and given P (e.g., 1 atm) (and noting that we work 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, PII = 1 atm, (all for some given T); these can then be solved for the four unknowns, xI, xII, V̅ I, V̅ II. The compositions in each phase (xI, xII) obtained over a range of T, thus provide the information necessary to map out the theoretical phase diagram. Another important set of mixture properties are described by taking the second derivatives with respect to composition, x, at constant T, P. For example, a stability condition for the mixture is that the second derivative of the Gibbs free energy,

III. RESULTS AND DISCUSSION We have theoretical expressions for a self-consistent and interconnected set of thermodynamic properties. In order to characterize a fluid or system of interest, we match the relevant model expression to data for some thermodynamic property. Using the parameter values thus obtained allows us to calculate other properties of interest; in other words, we rely on our parameters being transportable from one set of circumstances to another. This significantly broadens our power because it permits access to quantities that are not easily accessible experimentally, for example, the entropy of mixing, the pressure-dependence of any thermodynamic function of interest, as well as (particularly for polymers) the enthalpy and volume changes on mixing. We now consider a series of applications of LCL theory to a range of systems. We begin with a simple small molecule fluid: heptane. Pressure−volume−temperature data (PVT data) are an example of experimental results that are available for a wide variety of systems. We use eq 2 for the model pressure and the molecular parameters (r, v, ε) are adjusted until best agreement with the data is reached; in this case, we use experimental PVT data for heptane29 in its liquid state. The results of the fit are shown in Figure 1 in the form of pressure−volume isotherms over a range of 200 K, which shows that the theory (curves) is able to fit the data (points) fairly well. From this fit, we obtain

(∂ 2G̅ /∂x 2)T , P , be positive. It can be shown that the first

derivative (∂G̅ /∂x)T , P is simply (μi − μj), and thus, the second derivative can be obtained from 3292



Article

Figure 3. Model predictions (curve) for pure heptane vapor pressure as a function of temperature and comparison with experimental data (points); data are from ref 30.

Figure 1. Model fit to pure heptane pressure−volume−temperature data (heptane liquid state). Model results are shown as curves plotted in the form of pressure−volume isotherms at temperatures of 311 K, 344 K, 378 K, 411 K, 44 K, 478 K, and 511 K. Experimental data (points) are from ref 29. Resulting fitted model parameter values: r = 12.971, v = 9.712 mL/mol, ε = −1569.3 J/mol.

would provide enough information to model the mixture. In refs 26 and 27, we characterized a number of n-alkanes (butane, heptane, decane, tridecane, hexadecane) by fitting to the corresponding pure component properties.31 The theory was then applied to predict the binary mixture properties of several combinations of alkanes. No mixture data were fit in this work; the simple geometric mean combining rule was used for the mixed interaction energy, that is, in eq 5, g = 1. Resulting predictions for the mixed system liquid−vapor coexistence properties are shown in Figure 4 where the predicted phase boundaries are plotted in the form of temperature−composition phase envelopes (at constant pressure). Shown are the results for the butane−heptane mixture and the butane−decane mixture, each at a pressure of 100 psi and at 200 psi. The model curves agree well with the

the molecular parameters for heptane: r = 12.971, v = 9.712 mL/mol, ε = −1569.3 J/mol. The PVT-fitted heptane parameters may now be used in any of the other theoretically derived thermodynamic expressions. For example, to predict the liquid−vapor phase equilibria, the parameters are used to calculate the model chemical potential, μ (where as noted above, μ = G/N for a single component, and G = U + PV − TS). We then solve for the densities of the liquid and vapor phases coexisting at any given temperature such that μ (and P) are equal in each phase. The predicted liquid−vapor coexistence curve (line) for hexane is plotted in Figure 2,

Figure 2. Model predictions (curve) for pure heptane liquid and vapor coexistence densities (coexistence temperature plotted against the liquid and vapor density values). The right side (higher densities) corresponds to the equilibrium liquid, and the left side (lower densities) corresponds to the equilibrium vapor. Experimental data (points) are from ref 30. Figure 4. Model predictions (curves) for liquid−vapor equilibrium in alkane mixtures based on parametrization of pure component properties only. Shown are temperature−composition phase boundaries (T−x phase envelopes) at constant pressure for the butane− heptane mixture and the butane−decane mixture. Four T−x phase envelopes are shown; the two at the bottom, are for the butane− heptane mixture at P = 100 psi (bottom-most) and 200 psi, and the envelopes above these are for the butane−decane mixture at P = 100 psi and 200 psi. The upper curve in each envelope corresponds to the mixed vapor at dew point, and the lower curve corresponds to the mixed liquid at bubble point. Experimental data (points) are from refs 32 and 33; figure adapted from ref 27.

showing excellent agreement with the experimental data30 (points). The corresponding predictions for the equilibrium vapor pressure as a function of T are shown in Figure 3 also showing close agreement with the experimental data.30 Figures 1, 2, and 3, provide one example of how the LCL model parameters are transferable, having been fit to one property and then used to predict other properties. A long-held ambition is the notion of extending this principle to mixtures, wherein characterization of pure component properties, alone, 3293



Article

equation of state (curves) provides a good fit to the experimental data37 (points). The molecular parameters from this PS fit are r = 12 514, v = 7.667 mL/mol, ε = −2144.3 J/ mol. As noted above, these parameter values can now be applied to model and predict other properties for PS, for example, to model its behavior in a polymer blend or solution or to calculate other pure component properties, such as the free volume and cohesive energy density. Examples of these applications will be given further below. The physical significance of the molecular parameters can be illustrated by comparing the parametrizations for polystyrene and heptane. Both the r parameter (segments per molecule) and the molecular hardcore volume, rv (r multiplied by the volume per site/segment (v)) are much larger for PS than the corresponding values for the much smaller heptane molecule: r = 12 514 and rv = 95 945 mL/mol for PS (molecular weight = 110 000 g/mol) vs r = 12.971, rv = 125.97 mL/mol for heptane (molecular weight =100.2 g/mol). In the case of polymers, it is not uncommon to find a variety of data collected on different samples, having different (average) molecular weight. Rather than having to fit each such sample, we have exploited the fact that the r parameter scales linearly with molecular weight; we will return to this point below. To this point, we have discussed application of the theory to both small and very large molecules (polymers), all of which are hydrocarbons. However, we are not restricted to this kind of chemical simplicity; indeed, some of our recent efforts have involved ionic liquids (IL’s). Figure 7 shows the model fit to

experimental data32,33 (points). Furthermore, in addition to accurate prediction of liquid−vapor coexistence properties, this same set of model parameters also does well in predicting the PVT properties for the mixed systems. This is demonstrated in Figure 5 which shows the model predictions (curves) for

Figure 5. Model predictions (curves) for pressure−volume-temperature behavior (PVT) behavior for butane−decane mixtures based on parametrization of pure component properties only. Results are shown as pressure−volume isotherms at 344 K. The curves from left to right correspond to mixture compositions (mole fractions of butane) of xbutane = 1 (pure butane), 0.8358, 0.6617, 0.4643, 0.1787, and 0 (pure decane). Experimental data (points) are from refs 34−36; figure adapted from ref 27.

pressure−volume isotherms (at 344 K) in the butane−decane mixture at varied mixture compositions, again, compared to the experimental mixture data34 (points). (Pure butane and decane data shown are from refs 35 and 36.) Having discussed examples of fitting and prediction for smaller molecules (alkanes), we now move to some more complex fluids. Much of the work in our laboratory has been applied to polymeric systems. As discussed above, we begin by characterizing each polymeric species of interest using available experimental data. The most common (but not the only) route is to utilize experimental PVT data for the pure polymer in its melt state to obtain the molecular characterization parameters (r, v, ε). An example of this is provided in Figure 6, which gives PVT behavior in the form of volume−temperature isobars for polystyrene (PS). As for small molecules, the theoretical

Figure 7. Model fit to ionic liquid pressure−volume-temperature data: 1-ethyl-3-methylimidazolium tetrafluoroborate ([EMIM][BF4]). Model results are shown as curves plotted in the form of volume− temperature isobars at pressures of 10 MPa, 20 MPa, 30 MPa, 40 MPa, 50 MPa, and 60 MPa (from top to bottom). Experimental data (points) are from ref 38; figure adapted from ref 39. Resulting fitted model parameter values: r = 22.84, v = 6.255 mL/mol, ε = −2203.9 J/ mol.

PVT data for an IL, [EMIM][BF4] (1-ethyl-3-methylimidazolium tetrafluoroborate) (data from ref 38). The molecular parameters resulting from this fit were applied in ref 39 to model a polymer/IL solution (PEO/[EMIM][BF4]), where our theoretical predictions for the entropy of mixing provided a source of insight in explaining some unusual features of the phase diagram for that system. Pressure−volume−temperature data are not the only route to the molecular characterization parameters. For example, the approach in Figures 1, 2, and 3 could have been done in reverse, that is, if the liquid−vapor equilibrium data for heptane had been used for fitting, then the resulting parameters would

Figure 6. Model fit to polystyrene pressure−volume−temperature data (polystyrene liquid/melt state). Model results are shown as curves plotted in the form of volume-temperature isobars at pressures of 0 MPa, 20 MPa, 40 MPa, 60 MPa, 80 MPa, and 100 MPa (from top to bottom). Experimental data (points) are from ref 37; figure adapted from ref 42. The polystyrene molecular weight is 110 000 g/mol. Resulting fitted model parameter values: r = 12 514, v = 7.667 mL/ mol, ε = −2144.3 J/mol. 3294



Article

the trend in free energy but also the enthalpic contribution, which means that results for the entropic piece may also be reliably extracted. As noted above, we characterized dPS by parametrizing to mixture data (with the parameters for the other component fixed). Although pure component PVT data for deuterated polymers are rare, we were able to arrange for data to be collected on a sample (details in ref 42), which presented us with a unique opportunity to test our ability to characterize dPS using SANS data. We did so by using the SANS-derived dPS parameters to generate a prediction for the PVT properties. Thus, in Figure 9 we compare our prediction for V(T) at

have been similar to those quoted earlier, and these lead to a good prediction of the PVT surface. Yet another way to obtain the molecular characterization parameters is to utilize experimental mixture data. The following case provides an example. Small angle neutron scattering (SANS) has become a popular technique for analyzing polymer blends.40 One of the things that can be probed by SANS are the concentration fluctuations in mixtures, and this leads to an experimental quantification of the second derivative of the Gibbs free energy with respect to composition and can thus be connected to the model expression in eq 7. In Figure 8, we show the theoretical

Figure 9. Predicted pure component PVT behavior for deuterated polystyrene (dPS) and comparison with the actual experimental data (points; see ref 42). Shown is the volume as a function of temperature at atmospheric pressure. Also shown for comparison is corresponding model V(T) curve for regular (hydrogenated) polystyrene (denoted in the figure as “hPS”) along with the experimental data (points; from ref 37); figure adapted from ref 42.

Figure 8. Model fit to mixture data: temperature dependence of the second derivative of the Gibbs free energy for the deuterated polystyrene (dPS)/polyvinyl methyl ether (PVME) polymer blend. The experimental values (points) are derived from small angle neutron scattering (SANS) data taken from ref 41; figure adapted from ref 25. (See also ref 42 for details.) The plot shows (∂2(G̅ /kBT)/∂x2)T,P as a function of 1/T at P = 1 atm for a blend with a composition of 50/50 mass fraction. (Here, x denotes mole fraction of polymeric repeat units (monomers) and the intensive Gibbs free energy is per total number of repeat units in the mixture.) The model dPS energetic parameter (ε = −2106.0 J/mol) as well as the mixed interaction parameter (g = 1.000 980) were determined from this fit to mixture data while the model PVME parameters were fixed at their values determined from a pure component PVT data fit. For comparison, the dashed curve shows the model prediction for (∂2(G̅ /kBT)/∂x2)T,P for the regular hydrogenated polystyrene/PVME blend, which was characterized from PVT data for each component and for g, cloud point data for the mixture (details in ref 25).

atmospheric pressure (solid line) with the experimental results (points); as the figure shows, there is outstanding agreement. The results below show the theoretical fit (dashed line) to analogous experimental data (points) for hPS. Although the difference in the V(T) curves for dPS and PS appears to be relatively small our theoretical model captures this difference. Indeed, we note that the difference only “appears to be small” because, in fact, in this distinction between the two polymers originates the large shift in LCST in going from hPS/PVME to dPS/PVME. With almost identical molecular weights for the h/ d PS, and the same PVME molecular weight, the LCST for the dPS blend is about 40 K higher than for the hPS blend. Making a further comparison, the dashed line in Figure 7 shows our theoretical prediction for (∂2(G̅ /kBT)/∂x2)T,P for the hPS/ PVME blend, based on pure component parameters determined via PVT fitting and mixed interaction parameter, g, from mixture cloud point data. The predicted results for hPS/PVME illustrate the decreased miscibility (larger regime of negative values) compared to dPS/PVME. Furthermore, given that the two lines have different slopes, the separate enthalpic and entropic contributions will also be different. Indeed, our analysis reveals that although the enthalpy of mixing for hPS/ PVME is somewhat more favorable than for dPS/PVME, the entropy of mixing for the dPS blend is much less unfavorable, and this is what drives its enhanced miscibility. Even when there are reasonably favorable interactions between the unlike segments, polymer blends are apt to phase separate due to the very small entropic drive for large

fit (solid blue line) to experimental data determined via SANS (ref 41) for the 50/50 mass fraction mixture of deuterated polystyrene (dPS) and poly(vinyl methyl ether) (PVME) over a temperature range of about 300 K to 430 K. In fact, this is only a partial f it (see ref 42 for details) because the parameters for PVME had already been determined using PVT data for that pure melt. In matching theory to experiment for the dPS/ PVME mixture, we optimized the value of only two parameters: the pure component ε for dPS, and the mixed interaction parameter, g. The plot shows (∂2(G̅ /kBT)/∂x2)T,P as a function of 1/T, therefore the slope of this plot is the enthalpic contribution, (∂2(H̅ /kBT)/∂x2)T,P. As the figure shows, the slope of the model curve is in good agreement with the slope of the data on average (though the model does show somewhat less curvature), which means that the theory captures not only 3295



Article

10, were 22 752, 5802, and 2321 for the PS Mw’s of 200 000, 51 000, and 20 400 g/mol, respectively. The agreement with the experimental data is very strong. The theoretical predictions capture very effectively both the significant increase in miscibility (phase boundaries pushed to higher T) upon each decrease in PS molecular weight, and also the shift in the critical compositions (degree of skewness in the phase diagrams). It is not uncommon for cloud point measurements to lie between the spinodal and binodal, and this can also be seen in the figure. Thermodynamic quantities such as ΔHmix, ΔSmix, ΔVmix can provide tremendous insight regarding the driving forces in mixture behavior; however, for polymeric systems they are challenging to measure experimentally. Given that such data are not typically available, generating theoretical predictions for these quantities is an important modeling application. Here, we cover case studies for a UCST-type and an LCST-type blend, with results shown in Figures 11 and 12, respectively (details in ref 25).

molecules to mix. Thus, one useful application of a theory is mapping miscibility in the form of solution phase diagrams. As described in Section II, the phase boundary (the binodal) is determined by solving for the set of coexisting compositions (and overall phase densities for the chosen P) such that the chemical potentials (μi = (∂A/∂Ni)Nj,T,V = (∂G/∂Ni)Nj,T,P) for each polymer species are equal in each of the two phases. The other boundary noted in Section II is the spinodal, which marks the meta-stability limit, and is calculated by finding the set of concentrations that satisfy the condition (∂2(G̅ /kBT)/∂x2)T,P = 0. The spinodal and the binodal converge at the critical point. A series of polymer blend phase diagrams (details in ref 43), including both the binodals (solid lines) and spinodals (dashed lines) is given in Figure 10 for several PS/PVME blends (from this point on we drop the “h” in front of “PS”).

Figure 10. Predicted phase diagrams for polystyrene (PS)/poly(vinyl methyl ether) (PVME) blends (P = 1 atm). Phase diagrams are of the LCST-type (miscible at low T, immiscible at high T). Solid curves show the binodal boundary and dashed curves the spinodal boundary. Experimental data (points) are from ref 44; figure adapted from ref 43. Three phase diagrams are shown for three different PS molecular weights (Mw) of 200 000 g/mol (lower diagram), 51 000 g/mol (middle diagram), and 20 400 g/mol (upper diagram). The molecular weight of PVME was the same in all cases, Mw = 51 500 g/mol. All pure component parameters were obtained by fitting to PVT data. To obtain the mixed interaction parameter, g, only one datum point has been fit to mixture data, that being LCST value for the PS (Mw = 200 000 g/mol)/PVME blend.

Figure 11. Model predictions for changes in thermodynamic properties upon mixing for a UCST-type blend: PS/PMS. Results 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), TΔS̅mix (the overall entropy of mixing, multiplied by T). Also excess shown are TΔS̅ideal mix and TΔS̅mix , which are the ideal and excess contributions to the overall entropy of mixing. The overbars indicate intensive properties, where here, the units are Joules per gram of mixture. (Figure from ref 25.)

In the figures, we show ΔHmix, TΔSmix, and ΔGmix. We also exploit the fact that having a theoretical model makes it possible excess to break down and analyze separately, TΔSideal mix and TΔSmix , the ideal and excess contributions to the overall entropy of mixing. The “ideal entropy of mixing”, ΔSideal mix , (which is always positive) accounts simply for the gain in translational entropy experienced by each molecule when allowed to spread out over the entire mixture volume (in our theoretical calculation of this, we keep the total combined volume fixed, i.e., ΔVmix = 0.) The “excess entropy of mixing”, ΔSexcess mix , comprises all of the remaining contributions to the overall entropy of mixing; this includes a number of important effects traceable to free volume and compressibility (e.g., accounting for nonzero ΔVmix) as well as effects from nonrandom mixing. In Figure 11, the thermodynamic mixing functions are shown as a function of composition for a typical UCST-type blend, in this case consisting of polystyrene (PS) and poly(αmethylstyrene) (PMS) at a temperature of 480 K. The pure component parameters (r, v, ε) were obtained via PVT fitting

The results in Figure 10 demonstrate the ability of the theory to predict the effect of changes in molecular weight on the phase boundary. The three sets of curves correspond to three different PS molecular weights, 200 000, 51 000, and 20 400 g/ mol. (The molecular weight of the PVME was the same in all cases, 51 500 g/mol.) Only a single mixture datum point was used (the LCST value for the blend with PS Mw = 200 000 g/ mol) to obtain the mixed interaction parameter, g. All of the other parameters were obtained via pure component PVT data fits. Although there are three sets of experimental data,44 representing the three PS molecular weights, in applying our theory we fit one PVT data set for PVME, and only one PVT data set for PS (shown in Figure 6)and, in fact, that data set did not correspond to any of the molecular weights associated with the experimental results in Figure 10. Those data37 were for a molecular weight of 110 000 g/mol and gave r = 12 514, and thus r/Mw = 0.113 764. Using this value for r/Mw the r values used to generate the three model predictions in Figure 3296



Article

show in Figure 13 a sensitivity test on ΔSmix for a typical LCSTtype polymer blend, here the PS/PVME system (see ref 25 for

Figure 12. Model predictions for changes in thermodynamic properties upon mixing for an LCST-type blend: PIB/PEP. Results 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), TΔS̅mix (the overall entropy of mixing, multiplied by T). Also excess shown are TΔS̅ideal mix and TΔS̅mix , which are the ideal and excess contributions to the overall entropy of mixing. The overbars indicate intensive properties, where here, the units are Joules per gram of mixture. (Figure from ref 25.)

Figure 13. Effect of the pure component energetic parameters on the entropy and enthalpy of mixing. Shown by the solid curves, are the results for the model PS/PVME system (see ref 25) at T = 390 K and P = 1 atm. The dashed curves show the effect of a hypothetical change to one of the pure component energetic parameters (keeping all other parameters fixed) such that the difference, |εii − εjj|, is reduced to 25% of the original value. A strong effect on the entropy of mixing is observed, with little effect on the enthalpy of mixing. (Figure from ref 25.)

for each component and the mixed interaction parameter, g, obtained by fitting the experimental critical solution temperature determined from cloud point measurements (see ref 25 for details). The results show that the enthalpy and entropy of mixing are both positive, and this is expected for UCST-type systems (and connected to the requirement that both (∂2S̅/ ∂x2)T,P and (∂2H̅ /∂x2)T,P must be negative at a UCST). At 480 K, these results correspond to the miscible regime for the system, just above the UCST value of 469 K. Here, ΔHmix and TΔSmix are comparable in strength and ΔGmix is close to showing an inflection. Inspecting the breakdown of entropic contributions, we note that ΔSideal mix makes a stronger overall contribution to the overall ΔSmix than does ΔSexcess mix . In Figure 12, the thermodynamic mixing functions are shown for a typical LCST-type blend, polyisobutylene (PIB) and poly(ethylene-co-propylene) (PEP). Results are calculated for T = 305 K, in the miscible regime just below the LCST (313 K). Again, the pure component parameters were obtained via PVT data, and g was obtained by fitting the experimental LCST value determined from cloud point measurements (see ref 25). In contrast to UCST-type systems, LCST-type blends typically have enthalpy and (overall) entropy of mixing that are both negative (connected to the requirement that both (∂2S/̅ ∂x2)T,P and (∂2H̅ /∂x2)T,P must be positive at an LCST). In further contrast, note that the excess contribution (ΔSexcess mix ) is the most important part of the overall entropy of mixing; this quantity plays a central role in LCST-type systems: Noting how the ideal contribution, ΔSideal mix , is always positive (favorable), and that ΔHmix is negative (also favorable), it is only the excess unfavorable ΔSmix that can drive the system toward immiscibility. Theoretical predictions for ΔSmix and the corresponding breakdown of its ideal and excess contributions are one of the strongest examples of how a model theory can provide a source of insight in the analysis of mixture behavior, especially from the point of view of predicting potential mixture miscibility or immiscibility a priori, based only on information provided by characterization of the pure components. To illustrate this, we

details); we have obtained similar results for the other LCST blends we have studied. The dashed curve in the figure shows what happens when, hypothetically, one of the pure component energetic parameters is changed such that the magnitude of the difference in the two pure component energetic parameters (|εii − εjj|) is reduced by 25% (all other parameters have been kept fixed). Although there is little change in ΔHmix, the model predicts a very striking effect on ΔSmix: it increases (becomes less unfavorable) as εii and εjj become more similar (smaller |εii − εjj|). It is the excess contribution, ΔSexcess mix , in particular that changes strongly with changing |εii − εjj|. On the other hand, were we to change the mixed interaction parameter, g, over a span of typical values for polymer blends, then we find a significant change in ΔHmix, with only a small change in ΔSmix.25 The importance of these observations lies in the fact that, although fixing a value for g requires mixture data, ΔSmix appears to be rather insensitive to significant changes in its value. Hence, we have a route to a rough a priori estimate for ΔSmix and a breakdown of its ideal and excess contributions via knowledge of εii and εjj alone, and this requires only pure component information. We have found pure component properties to be a rich source of insight, and not just for LCST polymer blends. For example, over the course of studying several dozen blends, we have found that the averaged magnitude of |εii − εjj| is significantly larger for cases in which partial miscibility is reflected as LCST behavior, relative to UCST cases. For the former, the averaged ε-difference is roughly 150 J, whereas for the latter, it is roughly 50 J. Putting this together with the discussion above, we see that for the LCST blends a significant mismatch in ε-values is reflected in a significantly unfavorable ΔSmix, whose influence is magnified by the factor of T in its contribution to ΔGmix. For UCST blends the ε-difference is 3297



Article

smaller, as is the magnitude of the (now overall favorable) ΔSmix term. Among the properties we have been exploring most recently is the free volume, which is a meaningful physical construct, but requires a model theory framework to help define it. In a lattice model, the free volume is defined as Vfree = V − Nrv, the total system volume minus the excluded volume occupied by the Nr segments. We find that the free volume is directly connected to the pure component ε parameter; comparing a wide variety of polymers over a common T and P range we find that as the magnitude of ε increases, the percent free volume decreases. Thus, free volume mismatches will impact the entropy of mixing for a proposed pair of components, as discussed above. Another property of interest is the cohesive energy density (CED), which (indirectly) has a long history in the literature45−47 through its connection with the solubility parameter, δ. The CED is defined as |U|/V, and δ = (CED)1/2. These quantities are not directly experimentally accessible for large, nonvolatile, molecules. However, a theory such as ours yields predictions for the CED as well as for the related quantity, the internal pressure, Pint (=(∂U/∂V)T). Theories of the van der Waals type produce identical results for Pint and the CED; in our case, the two are not identical, although there is fairly strong overlap. In Figure 14, we provide a ranking for over two dozen polymers on three “scales”: Pint, CED, and % Free Volume. Values for these quantities were calculated in all cases under the same set of conditions: T = 425 K and P = 1 atm. In the context of one of the examples discussed above, note the closer proximity of dPS than of PS to PVME, using all three metrics, signifying that the dPS is a more closely matched blending partner for PVME. It is also interesting to consider how the various quantities might dif fer in terms of their indication that two polymers are well suited to blend. One example involves the blend poly(ethylene oxide) PEO and poly(methyl methacrylate) PMMA; although it is miscible over an appropriate temperature and composition range, the blend has been observed to exhibit two, distinct glass transition temperatures.48 According to Figure 14, we see that PEO and PMMA are very close on the CED scale, and reasonably close on the Pint scale, reflecting their relative compatibility. However, the two polymers are almost at opposite ends of the % free volume scale, reflecting their dramatically different glass transition temperatures (Tg), which are roughly 230 K for PEO vs 378 K for PMMA.E13 Another example involves poly(vinylidene fluoride) (PVF) and PMMA; these two are almost coincident on the CED scale and are quite close on the Pint scale. However, they are well separated on the % free volume scale; indeed, their disparity on this scale is directly correlated not only with the with their significant difference Tg values, 233 K for PVF compared to 378 K (PMMA),49 but also a large ε-difference, which is reflected in the LCST-type nature of this blend.50 Here, we should note that there are complexities in the last two examples: the tacticity of PMMA can make a difference, (we used data for an atactic sample), and both PEO and PVF may be at least partially crystalline in their respective blends (we characterized the melt, in both cases).

Figure 14. Polymers ranked in terms of free volume, cohesive energy density (CED), and internal pressure (Pint). All values are computed for the conditions, T = 425 K and P = 1 atm. Polymer acronyms are as follows. PEA, poly(ethyl acrylate); PEO, poly(ethylene oxide); PEMA, poly(ethyl methacrylate); PVA, poly(vinyl acetate); PE, polyethylene; PVME, poly(vinyl methyl ether); PB, polybutadiene; PEP, poly(ethylene-co-propylene); PVF, poly(vinylidene fluoride); hhPP, headto-head polypropylene; PMA, poly(methyl acrylate); dPS, deuterated polystyrene; PVC, poly(vinyl chloride); PECH, polyepichlorohydrin; PC, polycarbonate; PS, polystyrene; SAN, poly(styrene-co-acrylonitrile); PPO, poly(phenylene oxide); PIB, polyisobutylene; PMMA, poly(methyl methacrylate); TMPC, tetramethyl bisphenol A polycarbonate; PMS, poly(alpha-methyl styrene); PES, poly(ether sulfone).

behavior, cohesive energy density, and percent free volume to phase equilibria, including predictions for liquid−vapor phase diagrams for pure and mixed systems, as well as liquid−liquid phase diagrams for blends. LCL theory yields expressions for a self-consistent and interconnected set of thermodynamic properties, in which application to particular systems of interest produces a small set of physically meaningful characteristic parameters. We find that obtaining a set of such parameters via fitting the relevant theoretical expression (e.g., the equation of state) to one property (e.g., pressure−volume−temperature data) yields results that are transferable, in that they allow us to generate predictions for a wide range of other properties. In this work, we highlight theoretical predictions for properties that provide considerable physical insight, yet are difficult to obtain experimentally. Examples include the percent free volume, and the (excess) entropy of mixing. Indeed, access to such properties depends both upon a theoretical route to their definition and also on the ability to calculate them using characteristic parameters that truly reflect the underlying physical nature of the components of interest. We also explore the extent to which a thorough understanding of pure component behavior can lead to insight regarding miscibility. We believe that the theoretical tools discussed in this paper have the potential to increase substantively our ability to

IV. SUMMARY AND CONCLUSIONS In this article, we have presented applications of our LCL model spanning a wide array of fluids and fluid mixtures, ranging from simple, small molecules to polymer melts and blends. We are interested in properties ranging from PVT 3298



Article

(19) Utracki, L. A. Statistical thermodynamics evaluation of polymerpolymer miscibility. J. Polym. Sci., Part B: Polym. Phys. 2004, 42, 2909− 2915. (20) Utracki, L. A.; Simha, R. Statistical thermodynamics predictions of the solubility parameter. Polym. Int. 2004, 53, 279−286. (21) Barker, J. A.; Henderson, D. What is Liquid - Understanding States of Matter. Rev. Mod. Phys. 1976, 48, 587−671. (22) Wertheim, M. S. Fluids with Highly Directional Attractive Forces 0.1. Statistical Thermodynamics. J. Stat. Phys. 1984, 35, 19−34. (23) Chiew, Y. C. Percus-Yevick Integral-Equation Theory for Athermal Hard-Sphere Chains 0.2. Average Intermolecular Correlation-Functions. Mol. Phys. 1991, 73, 359−373. (24) White, R. P.; Lipson, J. E. G.; Higgins, J. S. New Correlations in Polymer Blend Miscibility. Macromolecules 2012, 45, 1076−1084. (25) White, R. P.; Lipson, J. E. G.; Higgins, J. S. How Pure Components Control Polymer Blend Miscibility. Macromolecules 2012, 45, 8861−8871. (26) White, R. P.; Lipson, J. E. G. Chain fluids: Contrasts of theoretical and simulation approaches, and comparison with experimental alkane properties. J. Chem. Phys. 2009, 131, 074109. (27) White, R. P.; Lipson, J. E. G. Fluid mixtures: Contrasts of theoretical and simulation approaches, and comparison with experimental alkane properties. J. Chem. Phys. 2009, 131, 074110. (28) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths: London, 1982. (29) Nichols, W. B.; Reamer, H. H.; Sage, B. H. Phase Equilibria in Hydrocarbon Systems - Volumetric Behavior of Normal-Heptane. Ind. Eng. Chem. 1955, 47, 2219−2221. (30) Smith, B. D.; Srivastava, R. Thermodynamic Data for Pure Compounds: Part A Hydrocarbons and Ketones; Elsevier: Amsterdam, 1986. (31) The heptane parameters in ref 26 are similar to those shown here; the only difference is that the ref 26 parameters came from fitting to pure component PVT data and pure component coexistence data simultaneously. (32) Kay, W. B. Liquid-vapor equilibrium relations in binary systems - n-Butane−n-heptane system. Ind. Eng. Chem. 1941, 33, 590−594. (33) Reamer, H. H.; Sage, B. H. Phase Equilibria in Hydrocarbon Systems. Phase Behavior in the n-Butane-n-Decane System. J. Chem. Eng. Data 1964, 9, 24−28. (34) Reamer, H. H.; Sage, B. H.; Lacey, W. N. Phase Equilibria in Hydrocarbon Systems - Normal-Butane Decane System in the Condensed Region. Ind. Eng. Chem. 1946, 38, 986−989. (35) Olds, R. H.; Reamer, H. H.; Sage, B. H.; Lacey, W. N. Phase equilibria in hydrocarbon systems - Volumetric behavior of n-butane. Ind. Eng. Chem. 1944, 36, 282−284. (36) Reamer, H. H.; Olds, R. H.; Sage, B. H.; Lacey, W. N. Phase equilibria in hydrocarbon systems - Methane-decane system. Ind. Eng. Chem. 1942, 34, 1526−1531. (37) Zoller, P.; Walsh, D. Standard Pressure-Volume-Temperature Data for Polymers; Technomic Pub Co.: Lancaster, PA, 1995. (38) Taguchi, R.; Machida, H.; Sato, Y.; Smith, R. L., Jr. HighPressure Densities of 1-Alkyl-3-methylimidazolium Hexafluorophosphates and 1-Alkyl-3-methylimidazolium Tetrafluoroborates at Temperatures from (313 to 473) K and at Pressures up to 200 MPa. J. Chem. Eng. Data 2009, 54, 22−27. (39) White, R. P.; Lipson, J. E. G. Origins of Unusual Phase Behavior in Polymer/Ionic Liquid Solutions. Macromolecules 2013, 46, 5714− 5723. (40) Higgins, J. S.; Benoit, H. C. Polymers and Neutron Scattering; Oxford Science Publications: Oxford, U.K., 1994. (41) Shibayama, M.; Yang, H. J.; Stein, R. S.; Han, C. C. Study of Miscibility and Critical Phenomena of Deuterated Polystyrene and Hydrogenated Polyvinyl Methyl-Ether) by Small-Angle NeutronScattering. Macromolecules 1985, 18, 2179−2187. (42) White, R. P.; Lipson, J. E. G.; Higgins, J. S. Effect of Deuterium Substitution on the Physical Properties of Polymer Melts and Blends. Macromolecules 2010, 43, 4287−4293.

leverage a limited amount of experimental data to a greater understanding of the properties of liquids and their mixtures, particularly those involving a polymeric component.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: 603-646-3946. Funding

We gratefully acknowledge the financial support provided by the National Science Foundation, grant DMR-1104658. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Rodgers, P. A. Pressure Volume Temperature Relationships for Polymeric Liquids - A Review of Equations of State and their Characteristic Parameters for 56 Polymers. J. Appl. Polym. Sci. 1993, 48, 1061−1080. (2) Kang, J. W.; Lee, J. H.; Yoo, K. P.; Lee, C. S. Evaluation of equations of state applicable to polymers and complex systems. Fluid Phase Equilib. 2002, 194, 77−86. (3) Song, Y. H.; Hino, T.; Lambert, S. M.; Prausnitz, J. M. Liquidliquid equilibria for polymer solutions and blends, including copolymers. Fluid Phase Equilib. 1996, 117, 69−76. (4) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Saft Equation-Of-State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31−38. (5) Paricaud, P.; Galindo, A.; Jackson, G. Recent advances in the use of the SAFT approach in describing electrolytes, interfaces, liquid crystals and polymers. Fluid Phase Equilib. 2002, 194, 87−96. (6) Sadowski; G. Modeling of Polymer Phase Equilibria Using Equations of State. In Polymer Thermodynamics: Liquid PolymerContaining Mixtures; Enders, S., Wolf, B. A., Eds.; Springer: Berlin, Germany, 2011; 389−418. (7) Prigogine, I.; Trappeniers, N.; Mathot, V. Statistical Thermodynamics of R-Mers and R-Mer Solutions. Discuss. Faraday Soc. 1953, 15, 93−125. (8) Prigogine, I.; Bellemans, A.; Mathot, V. The Molecular Theory of Solutions; North-Holland: Amsterdam, 1957. (9) Flory, P. J.; Orwoll, R. A.; Vrij, A. Statistical Thermodynamics of Chain Molecule Liquids.I. Equation of State for Normal Paraffin Hydrocarbons. J. Am. Chem. Soc. 1964, 86, 3507−3514. (10) Dee, G. T.; Walsh, D. J. Equations of State for Polymer Liquids. Macromolecules 1988, 21, 811−815. (11) Dee, G. T.; Walsh, D. J. A Modified Cell Model Equation of State for Polymer Liquids. Macromolecules 1988, 21, 815−817. (12) Dudowicz, J.; Freed, K. F.; Madden, W. G. Role of MolecularStructure on the Thermodynamic Properties of Melts, Blends, and Concentrated Polymer-Solutions - Comparison of Monte-Carlo Simulations with the Cluster Theory for the Lattice Model. Macromolecules 1990, 23, 4803−4819. (13) Freed, K. F.; Dudowicz, J. Influence of monomer molecular structure on the miscibility of polymer blends. Phase Behavior of Polymer Blends 2005, 183, 63−126. (14) Panayiotou, C.; Vera, J. H. Statistical Thermodynamics of R-Mer Fluids and their Mixtures. Polym. J. 1982, 14, 681−694. (15) Panayiotou, C. The QCHB model of fluids and their mixtures. J. Chem. Thermodyn. 2003, 35, 349−381. (16) Kleintjens, L. A.; Koningsveld, R. Liquid-Liquid PhaseSeparation in Multicomponent Polymer Systems 0.19. Mean-Field Lattice-Gas Treatment of the System N-Alkane-Linear-Polyethylene. Colloid Polym. Sci. 1980, 258, 711−718. (17) Sanchez, I. C.; Lacombe, R. H. Elementary Molecular Theory of Classical Fluids - Pure Fluids. J. Phys. Chem. 1976, 80, 2352−2362. (18) Lacombe, R. H.; Sanchez, I. C. Statistical Thermodynamics of Fluid Mixtures. J. Phys. Chem. 1976, 80, 2568−2580. 3299



Article

(43) Higgins, J. S.; Lipson, J. E. G.; White, R. P. A simple approach to polymer mixture miscibility. Philos. Trans. R. Soc., A 2010, 368, 1009− 1025. (44) Nishi, T.; Kwei, T. K. Cloud Point Curves for Polyvinyl MethylEther) and Monodisperse Polystyrene Mixtures. Polymer 1975, 16, 285−290. (45) Hildebrand, J. H., Prausnitz, J. M., Scott, R. L. Regular and Related Solutions; Van Nostrand Reinhold Company: New York, NY, 1970. (46) Dack, M. R. J. Importance of Solvent Internal-Pressure and Cohesion to Solution Phenomena. Chem. Soc. Rev. 1975, 4, 211−229. (47) Graessley, W. W. Polymer Liquids and Networks: Structure and Properties; Garland Science, Taylor and Francis Group: New York, 2004. (48) Lodge, T. P.; Wood, E. R.; Haley, J. C. Two calorimetric glass transitions do not necessarily indicate immiscibility: The case of PEO/ PMMA. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 756−763. (49) CRC Handbook of Chemistry and Physics, 93rd ed.; Haynes, W. M., Ed.; CRC Press, Taylor and Francis Group: Boca Raton, FL, 2012. (50) Sasaki, H.; Bala, P. K.; Yoshida, H.; Ito, E. Miscibility of PVDF/ PMMA Blends Examined by Crystallization Dynamics. Polymer 1995, 36, 4805−4810.

3300


Connecting Theory and Experiment To Understand ... - ACS Publications

Recommend Documents