Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in

Article Cite This: J. Phys. Chem. B 2018, 122, 5458−5465

pubs.acs.org/JPCB

Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in Random Porous Media: TPT of Wertheim and Scaled Particle Theory Taras V. Hvozd,† Yurij V. Kalyuzhnyi,*,† and Peter T. Cummings‡ †

Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii St., Lviv 79011, Ukraine Department of Chemical and Biochemical Engineering, Vanderbilt University, Nashville, Tennessee 37235-1604, United States

‡

S Supporting Information *

ABSTRACT: Extension of Wertheim’s thermodynamic perturbation theory and its combination with scaled particle theory is proposed and applied to study the liquid−gas phase behavior of polydisperse hard-sphere square-well chain fluid confined in the random porous media. Thermodynamic properties of the reference system, represented by the hardsphere square-well fluid in the matrix, are calculated using corresponding extension of the second-order Barker− Henderson perturbation theory. We study effects of polydispersity and confinement on the phase behavior of the system. While polydispersity causes increase of the region of phase coexistence due to the critical temperature increase, confinement decreases the values of both critical temperature and critical density making the region of phase coexistence smaller. This effect is enhanced with the increase of the size ratio of the fluid and matrix particles. The increase of the average chain length at fixed values of polydispersity and matrix density shifts the critical point to a higher temperature and a slightly lower density.

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HS fluid quenched at equilibrium, have been performed.9 These include grand canonical Monte Carlo (GCMC)10−14 and Gibbs ensemble Monte Carlo (GEMC)15 simulation studies of Lennard-Jones (LJ) fluids and a grand-canonical transitionmatrix Monte Carlo (GC-TMMC) study of square-well (SW) fluids.16 Theoretical studies of confined LJ fluid were performed using replica Ornstein−Zernike (ROZ) formalism7,17−19 and its two-density version,20−22 generalized version of the van der Waals approach,23 and Barker−Henderson (BH) perturbation theory.24 In the latter study recently proposed extension of the scaled particle theory (SPT) for the HS fluid in the HS matrix23,25−27 was used to describe corresponding reference system. Similar description of the reference system was utilized to study phase behavior of the network-forming fluid28 and polydisperse Yukawa HS fluid29 confined in the HS matrix using corresponding extensions of Wertheim’s thermodynamic perturbation theory (TPT)30−33 and high-temperature approximation (HTA), respectively. The major goal of the present paper is to extend and apply the scheme developed in our previous study29 to describe the phase behavior of polydisperse HS square-well chain fluid confined in random porous media. We will consider here

INTRODUCTION Fluids and fluid mixtures confined in porous media are attracting substantial theoretical and experimental interest due to a multitude of applications in industry, technology, and engineering.1 For development and improvement of new and existing technologies knowledge and understanding of the phase behavior of confined fluids are very important. In particular, there are many industrial processes, such as catalysis, adsorption separation, filtration, and purification, where porous materials are used as adsorbents. Understanding of a wide range of surface and interfacial phenomena on a molecular level2,3 can be achieved using liquid-state theoratical approach. Phase behavior of fluids confined in porous media is quite different from that in the bulk. For example, liquid−gas phase diagrams of 4He,4 N2,5 and mixture of isobutyric acid and water6 confined in dilute silica gel are much narrower than those in the bulk, with the critical points shifted toward the lower temperatures. There is large variety of the models used to represent porous media of different type. One of the most popular models, which is widely used in theoretical and computer simulation studies, is “quenched-annealed” model of Madden and Glandt.7,8 In this model porous media is represented by the disordered matrix of the obstacles, formed by the matrix particles quenched in equilibrium configuration. Over the last few decades several computer simulations and theoretical studies of the liquid−gas phase behavior of fluids confined in the matrix of hard-sphere (HS) obstacles, which are randomly placed in a configuration of © 2018 American Chemical Society

Special Issue: Ken A. Dill Festschrift Received: November 29, 2017 Revised: April 11, 2018 Published: April 15, 2018 5458

DOI: 10.1021/acs.jpcb.7b11741 J. Phys. Chem. B 2018, 122, 5458−5465

Article

The Journal of Physical Chemistry B

obstacles of diameter σ0 and the total number-density ρ0 (or packing fraction η0 = πρ0σ03/6).

polydispersity in the chain length. Similar, as in ref 29, porous media is represented by the matrix of randomly distributed hard spheres. Properties of the reference fluid is described using recently developed23,25−27 extension of the SPT.34−36 The properties of the chain fluid are described using second-order BH perturbation theory and Wertheim’s first-order TPT for associating fluids. In the framework of the theory outlined our model belongs to a class of “truncatable free energy” models,37 that is, Helmholtz free energy of our system is determined by the finite number of the moments of the chain length distribution function. This feature of the model allows us to follow the method developed earlier38−40 and formulate the phase equilibrium conditions in terms of these moments. We calculate and present the full phase diagram of the model (including binodals, cloud and shadow curves,37,38 and chainlength distribution functions in coexisting phases) and analyze its behavior with respect to the changes in polydispersity, density of the matrix, and size ratio of the fluid and matrix particles. The paper is organized as follows: in the next two sections we introduce the model and present a corresponding extension of the TPT of Wertheim and BH perturbation theory. Our numerical results for the phase behavior of the model are presented and discussed in the fourth section, and in the fifth section we collect our conclusions.

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THEORY To describe the thermodynamic properties of the model, we use the first-order TPT of Wertheim. Helmholtz free energy A for chain molecules in the framework of Wertheim’s approach is given by32,33,43−45 βA βAid βAM βAchain = + + V V V V

Here ⎧ ⎫ ⎡ ⎤ ⎪ ⎪ βAid ⎢∑ f (m)ln(ρ(m)Λ3(m))⎥ − 1⎬ = ρ⎨ ⎪ ⎪ ⎥⎦ V ⎩⎢⎣ m ⎭

THE MODEL We consider polydisperse square-well chain fluid confined in a matrix with HS obstacles. We are using Madden−Glandt version of the model for the porous media;7,8 that is, the matrix is represented by HS fluid quenched at equilibrium. We consider fully flexible tangent square-well chain fluid. Fluid molecules consist of a chain of m tangentially bonded attractive HS monomeric segments of diameter σ interacting through the following interparticle pair potential

1 ⎛ α + 1⎞ ⎜ ⎟ α! ⎝ m 0 ⎠

⎡ ⎛ α + 1⎞ ⎤ mα exp⎢ −⎜ ⎟m ⎥ ⎢⎣ ⎝ m0 ⎠ ⎥⎦

(6)

and expression for the pressure P of the system can be calculated invoking the following general relation: βP = β ∑ ρ(m)μ(m) − m

βA V

(7)

Monomer Contribution. To calculate thermodynamic properties of the fluid of monomer segments confined in the random porous media we use combination of the second-order BH perturbation theory46−49 and SPT.23,24,28 According to the second-order BH theory, Helmholtz free energy for square-well fluid in the matrix is given by

(1)

βAHS βA1 βA 2 βAM = + + V V V V

(8)

where AHS is the free energy of the HS fluid confined in the HS matrix. The corresponding expressions for Helmholtz free energy AHS, pressure PHS, and chemical potential μHS are obtained using recently proposed version of the SPT, so-called the SPT2b1 approximation23,24,28 (see also the Supporting Information). The contribution to the HS free energy due to square-well (SW) interaction is represented by A1 and A2. For A1 we have

α+1

(2)

Here m0 is the average chain length, and α is a polydispersity controlling parameter, I = 1 + 1/α, where I is the polydispersity index. Because of our choice of the form of the distribution function f(0)(m) it is normalized, that is:

βA1 = −2πρ2 β ϵ ∑ ∑ mf (m)m′f (m′) V m m′

∑ f (0) (m) = 1 m

∂ ⎛ βA ⎞ ⎜ ⎟ ∂ρ(m) ⎝ V ⎠

βμ(m) =

where ϵ is the depth of the square-well potential, and λ is a parameter that defines the range of the attractive part of the potential. We consider square-well chain fluid with polydispersity in the chain length m and assume the following form for the chain-length distribution function of the parent phase f(0)(m) = F(0)(m)/∑lF(0)(l), where for F(0)(m) we chose the Schultz− Zimm distribution41,42 F (0)(m) =

(5)

is the ideal gas Helmholtz free energy of a mixture, ρ(m) = ρf(m) is the number density of chain molecules with length m, and Λ(m) is the thermal de Broglie wavelength of species m, AM is the excess free energy due to the monomer segments, and Achain is the contribution due to the formations of the chains of monomers. To calculate the rest of the thermodynamic properties of the system we use standard thermodynamic relations; for example, differentiating the Helmholtz free energy A (4) with respect to the density we get the expression for the chemical potential

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⎧∞ , 0 < r ≤ σ ⎪ U M(r ) = ⎨− ϵ, σ < r ≤ λσ ⎪ ⎩ 0, r > λσ

(4)

(3)

The fluid is characterized by the temperature T (or β = (kBT)−1, where kB is the Boltzmann’s constant) and the total molecular number density ρ. Note that the packing fraction η = πρσ3∑mmf(m)/6 represents a reduced segment density ρs = ρ∑mmf(m). The matrix is characterized by the stationary HS

×

∫σ

λσ

gHS(r , η , η0)r 2 dr

(9)

where gHS(r,η,η0) is the HS radial distribution function (RDF) in the matrix. We approximate gHS(r,η,η0) using Percus−Yevick (PY) RDF for HS fluid50 at some effective value of the packing 5459


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The Journal of Physical Chemistry B fraction ηeff, which depends on the characteristics of the matrix, that is: (PY) gHS(r , η , η0) = gHS (r , ηeff )

Chain Contribution. On the basis of the TPT of Wertheim,30−33 Chapman et al.44,45 derived expression for the free energy contribution due to the formation of a chain of m monomers. It is expressed in terms of the contact value of the radial distribution function of the reference system. Taking into account that in our case the reference system is represented by the square-well hard-sphere fluid in the matrix, we have

(10) 29

We follow the scheme developed in our previous study and calculate ηeff using relation (10) at r = σ. The value of the HS RDF gHS(σ,η,η0) is calculated using SPT.28 ηeff = 1 +

1 (SPT) + 4gHS (σ ,

⎡ × ⎣⎢1 −

1+

(SPT) + gHS (σ , η , η0) =

βAchain = −ρ ∑ f (m)(m − 1)ln(y SW (σ , ηeff )) V m

η , η0)

(SPT) + 24gHS (σ ,

⎤ η , η0) ⎦⎥

In the framework of Wertheim’s TPT approach our model belongs to the class of the TFE models too, and expression for Helmholtz free energy (18) is reduced to

(11)

1 3 (η + η0τ ) + ϕ0 − η 2 (ϕ0 − η)2

βAchain = −(L1 − L0)ln(y SW (σ , ηeff [L1])) (19) V SW where y (σ,ηeff[L1]) is the contact value of the cavity pair correlation function for a system of square-well monomers in the matrix.

(12)

is the contact value of the radial distribution function of HS fluid in a matrix obtained in the framework of the SPT,28 where τ = σ/σ0 and for HS matrix ϕ0 = 1 − η0. The approximation (10), suggested for gHS(r,η,η0), appears to be reasonably accurate.29 For the second-order term A2 we use here the macroscopic compressibility approximation46−49

y SW (σ , ηeff [L1]) = g SW (σ , ηeff [L1])e−β ϵ

×

∫σ

Here g (σ,ηeff[L1]) is obtained using first-order hightemperature expansion about HS reference system in the matrix:49 (PY) g SW (σ , ηeff [L1]) = gHS (σ , ηeff [L1]) + β ϵg1(PY)(σ , ηeff [L1])

(PY) gHS (r , ηeff )r 2 dr

(21)

(13)

which in case of square-well HS in the matrix is reduced to

where KHS = [∂ρs/∂(βPHS)]T is the isothermal compressibility of the HS fluid in HS matrix and is obtained using SPT:

(PY) g SW (σ , ηeff [L1]) = gHS (σ , ηeff [L1]) +

⎡ η /ϕ0 η/ϕ 1 + +a K HS = ⎢ 2 ⎢⎣ 1 − η /ϕ0 (1 − η /ϕ0) (1 − η /ϕ0)3 ⎤ ⎥ + 2b 4 (1 − η /ϕ0) ⎥⎦ (η /ϕ0)

a1SW = 2πρs

Here expressions for ϕ and for coefficients a and b, which define the porous media structure, are presented in the Supporting Information. In the framework of the second-order BH perturbation theory thermodynamnical properties of our model are defined by the finite number of the moments of the distribution function f(m). We have the following moments: L1 = ρ ∑ mf (m) = ρs

m

m

∫σ

λσ

βA 2 = −πL12β 2 ϵ2K HS V

(PY) gHS (r , ηeff [L1])r 2 dr

∫σ

λσ

(15)

(16)


(17)

The analytical expressions for βμ (m), and βPSW are presented in the Supporting Information. g(PY) HS (r,ηeff[L1]),

λσ

(PY) U SW(r )gHS (r , ηeff [L1])r 2 dr

∫σ

λσ


(23)

Thus, substituting expression for the contact value of the cavity pair correlation function (20) into expression for Helmholtz free energy (18) or (19) we obtain expression for the free energy of polydisperse chain molecules in the matrix. The analytical expressions for gSW(σ,ηeff[L1]), βμchain(m), and βPchain are presented in the Supporting Information. Phase Equilibrium Conditions. We consider polydisperse square-well chain fluid with chain length polydispersity confined in random porous media, which is characterized by the parent density ρ(0) and parent-phase distribution function f(0)(m). We assume that at a certain temperature T, the system separates into two daughter phases, with the densities ρ(1) and ρ(2), and two daughter distributions f(1)(m) and f(2)(m). Conservation of the total number of the particles, total number of the particles of each species m, and the normalization condition for the distribution function together with the equality of the pressure and chemical potential of particles of the same species m in the coexisting phases form the phase equilibrium conditions. Taking into account that our model belongs to the class of TFE models and following the scheme developed earlier these conditions can be expressed in terms of the set of finite number

which represent the total molecular number density and segment number density, respectively. Taking into account the above assumptions, expressions for the free energy (9) and (13) are reduced to βA1 = −2πL12β ϵ V

∫σ

= −2πL1ϵ

(14)

L 0 = ρ ∑ f (m) = ρ ,

SW β ⎡ ∂a1SW λ ∂a1 ⎤ ⎢ ⎥ − 4 ⎣ ∂η 3η ∂λ ⎦ (22)

where

−1

2

(20)

SW

βA 2 = −πρ2 β 2 ϵ2K HS ∑ ∑ mf (m)m′f (m′) V m m′ λσ

(18)

SW

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Figure 1. Vapor−liquid phase diagrams of monodisperse square-well fluid in T* vs ρ* = ρσ3/(1 − η0) coordinate frame at different matrix packing fraction η0 and different parameter λ obtained by first-order BH TPT (dotted blue line), second-order BH TPT (solid black line), GE-MC51 (red ●, left panel) and GC-TMMC16 (red ●, intermediate and right panels).

Figure 2. Vapor−liquid phase diagrams of polydisperse square-well chain fluid with chain-length polydispersity in the random porous media in T* vs ρ* = η coordinate frame at polydispersity index I = 8.1, size ratio of the fluid and matrix particles τ = 1, and different values of matrix packing fraction η0 for different the average chain length m0 = (a) 4, (b) 8, and (c) 16. The phase diagrams include cloud (solid black line) and shadow (dotted blue line) curves and critical binodals (dashed black lines). The large filled circle denotes the position of the critical points. The small red filled circles and the dotted red line correspond to GE-MC52 and our theoretical results, respectively, for the monodisperse system I = 1.

of equations for the moments of the chain-length distribution function in the coexisting phases

and from the normalization condition for the distribution function in the first or second phase:

L1(γ ) = ρ(γ ) ∑ mf (0) (m)W (γ )

∑ f (0) (m)W (γ)(m , T , ρ(0) , ρ(1) , ρ(2) , L1(1), L1(2)) = 1,

m

m

(0)

(1)

(2)

× (m , T , ρ , ρ , ρ ,

L1(1) ,

L1(2)),

γ = 1, 2

γ = 1 or γ = 2 (24)

Note that the distribution function in phase γ is given by

where γ denotes number of phase and (γ )

(γ )

(0)

(1)

(2)

ρ W (m , T , ρ , ρ , ρ , (0)

=

(2)

ρ (ρ

(0)

(ρ

L1(1) ,

f (γ ) (m) = f (0) (m)W (γ )(m , T , ρ(0) , ρ(1) , ρ(2) , L1(1) , L1(2))

L1(2))

(29)

(1)

− ρ )[1 − δ1γ + δ1γ exp(β Δμ12 )] (1)

(0)

− ρ ) − (ρ

(2)

− ρ )exp(β Δμ12 )

To avoid unnecessary repetition we refer the readers to the original publications,38−40 where detailed description of this scheme is presented.

(25)

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β Δμ12 = β[μex(2)(m , T , ρ(2) , L1(2)) − μex(1) (1)

× (m , T , ρ ,

L1(1))]

RESULTS AND DISCUSSION In this section we present results of our numerical calculations. Monodisperse Square-Well Fluid in the Matrix. To validate the accuracy of the theoretical predictions for the properties of the reference system, represented by monodisperse square-well fluid, we compare theoretical phase diagrams of the system in the bulk and confined in the matrix against corresponding phase diagrams, obtained using Gibbs ensemble Monte Carlo (GE-MC)51 and grand-canonical

(26)

(γ) (γ) Here μ(γ) ex (m,T,ρ ,L1 ) is the value of the chemical potential of the molecule of the length m in the phase γ in excess to its ideal gas value. And two more equations are obtained from the equality of the pressure in the coexisting phases:

P(1)(T , ρ(1) , L1(1)) = P(2)(T , ρ(2) , L1(2))

(28)

(27) 5461


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Figure 3. Vapor−liquid phase diagrams of polydisperse square-well chain fluid with chain-length polydispersity in the random porous media in T* vs η coordinate frame at polydispersity index I = 8.1, different size ratio of the fluid and matrix particles (τ = 2/3 and τ = 1/2) and different values of matrix packing fraction η0 for different average chain length m0 = (a) 4, (b) 8, (c) 16. The phase diagrams include cloud and shadow curves and critical binodals. Notation is the same as in Figure 2.

Figure 4. Two-phase coexisting chain-length distributions of polydisperse square-well chain fluid with chain-length polydispersity (I = 8.1, m0 = 8) on the critical binodal in the bulk (η0 = 0, panel a) at lower (T*2 = 2.2, blue lines) and higher (T*1 = 2.45, red lines) temperatures and in the random porous media (η0 = 0.1) for τ = 1 (panel b) at lower (T2* = 1.3, blue lines) and higher (T1* = 1.55, red lines) temperatures, and for τ = 2/3 (panel c) at lower (T*2 = 0.9, blue lines) and higher (T*1 = 1.15, red lines) temperatures. The black thick solid line shows parent f(0)(m) distribution, the thin solid lines correspond to the liquid phases, and the dotted lines denote the vapor phases.

transition matrix MC (GC-TMMC)16 methods, respectively. In Figure 1 we present the phase diagrams for monodisperse square-well fluid in the bulk (panel a) and in the random porous media at two different matrix packing fractions (η0 = 0.05 and η0 = 0.1, (b) and (c) panels, respectively) and at three different potential range parameters (λ = 1.5; 1.75; 2). Here, the monomer density and temperature are expresseded in reduced units, that is, ρ* = ρσ3/(1 − η0) and T* = kBT/ϵ, and we consider the case of equal size of the chain monomers and matrix particles; that is, τ = 1. In general, agreement between theory and simulation is good for the system in the bulk and slightly less accurate in the matrix. This decrease in the accuracy is caused by slightly less accurate predictions for the structure and thermodynamic properties of the reference hard-sphere fluid confined in the matrix. In all cases predictions of the second-order BH approach is slightly more accurate than the first-order. Here, in agreement with the experimental studies4−6 increase in the matrix packing fraction from η0 = 0 to η0 = 0.1 causes decrease of phase coexistence region moving it in the direction of lower temperatures and lower densities. Note that the accuracy of the theoretical predictions for the structure and thermodynamics of the BH reference system (HS fluid confined in the HS matrix), which are needed to calculate

the properties of the reference system, has been tested in our previous study29 and in refs 23, 25, and 26, respectively. Comparison of the theoretical and computer simulation predictions for gHS(r,ηeff) showed that agreement between theory and simulation is good for the equal size of the chain monomers and matrix particles (τ = 1) and slightly less accurate for different sizes (τ = 1/2). Polydisperse Square-Well Chain Fluid in the Matrix. Next we proceed to discuss results of our phase-diagram calculations for polydisperse square-well chain fluid in the matrix. These results are presented in Figures 2−4. In all cases studied we consider the model with polydispersity index I = 8.1, width of the square-well λ = 1.5, and terminate chain-length distribution function at m = 100. Using the larger termination chain length does not change our results. In Figures 2 and 3 we show the phase diagrams for the model with three different values of the average chain length, that is, m0 = 4, 8, 16 (panels (a), (b), and (c), respectively) and at three different values of the matrix packing fraction, that is, η0 = 0, 0.1, 0.2 (Figure 2) and two different size ratios of the fluid and matrix particles, that is, τ = 2/3, 1/2 (Figure 3) for η0 = 0.1. In addition, in each panel we present comparison of the phase diagrams for monodisperse square-well chain fluid of the length m0 = 4 5462


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The Journal of Physical Chemistry B (panel (a)), m0 = 8 (panel (b)), and m0 = 16 (panel (c)) in the bulk, calculated using our theory and obtained by Escobedo and de Pablo52 via the GEMC simulation method. In general, our results are in a good agreement with computer simulation results.52,53 For the system in the bulk (η0 = 0) increase of the polydispersity index from I = 1 to I = 8.1 shifts the critical point to the higer temperature and slightly lower monomer density (or lower packing fraction). Similar effects were observed for polydisperse Yukawa hard-sphere chain fluids.40 Note that now the phase diagram is represented by the cloud and shadow curves, which do not coincide with the binodals.37,38 We also present the critical binodals, which intersect with cloud and shadow curves at the critical point. The system with the same degree of polydispersity (I = 8.1) but confined in the matrix with packing fractions η0 = 0.1 and η0 = 0.2 has substantially smaller region of coexistence, and its critical point moves to the lower values of the temperature and packing fraction at increasing matrix packing fraction (Figure 2). A decrease of size ratio of the fluid and matrix particles from τ = 1 to τ = 2/3 and, moreover, τ = 1/2 (or increase of sizes of the matrix particles) for η0 = 0.1 makes these effects even more pronounced; that is, the corresponding phase diagram is shifted toward the region of still smaller temperatures and smaller packing fractions (Figure 3). Thus, the phase behavior of our model is defined by the competition between polydispersity effects and effects of the confinement; that is, polydispersity increases the region of the phase coexistence, and confinement causes it to decrease. Comparison of the phase diagrams for the models with different average chain length shows that polydispersity has a tendency to dominate at larger values of m0; that is, increase of m0 causes their shift in the direction of higher temperatures and slightly lower densities. Finally in Figure 4 we present chain-length distribution functions of the coexisting phases of the model with I = 8.1 and m0 = 8 in the bulk and in the matrix with different values of τ for the points located on the critical binodal together with distribution function of the parent phase at two values of the temperature: T1* and T2* (T1* > T2*). In all cases Tcr* − T1* ≈ 0.06, and ΔT* = T1* − T2* = 0.25. Similarly, as in the case of the bulk fluids,54,55 at both values of the temperature the larger molecules fractionate into the liquid phase, and smaller molecules fractionate into the gas phase. This effect is more pronounced at lower temperatures and is enhanced for the fluid confined in the matrix with larger size of the hard-sphere obstacles.

In general, our theoretical predictions appear to be in a reasonable agreement with computer simulations predictions. Further improvement of the theory can be achieved using second-order TPT for associating fluids56 in combination with better description of the structure of the reference system, for example, using appropriately extended SPT approach due to Boublik.57 This is planned for a future work. We have studied the effects of the chain-length polydispersity, porous media, and average chain length on the phase behavior of the model. The critical binodals and the cloud and shadow curves have been obtained at different degree of polydispersity, different matrix density, and different average chain length. The increase of the average chain length and/or pylydispersity causes the increase of the critical temperature and the slight decrease of the critical density. With the increase of the matrix density, both the critical temperature and the critical density decrease, and the region of the phase coexistence decreases. Increasing the size ratio of the fluid and matrix particles enhances this effect. This competition between polydispersity effects and effects of the confinement defines the phase behavior of system, that is, polydispersity increases the region of the phase coexistence, and confinement causes it decrease. Distribution functions of the coexisting phases on critical binodals for different confinement parameters have been determined at higher and lower temperatures. In all cases the larger molecules fractionate into the liquid phase, and smaller molecules fractionate into the gas phase.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.7b11741. Analytical expressions for the structure and thermodynamic properties needed for the phase diagram calculations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Taras V. Hvozd: 0000-0002-0156-1753 Notes

■

The authors declare no competing financial interest.

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CONCLUSIONS In this paper we propose extension of the first-order thermodynamic perturbation theory of Wertheim using corresponding extension of the second-order Barker−Henderson perturbation theory and in combination with scaled particle theory apply it to study the phase behavior of polydisperse HS square-well chain fluid with chain-length polydispersity confined in the random porous media. We are using Madden−Glandt version of the model for the porous media; that is, the matrix is represented by the hard-sphere fluid quenched at equilibrium. To evaluate the accuracy of our theoretical predictions, we compare them with corresponding computer simulation predictions for the phase diagram in the two limiting cases. We consider monodisperse hard-sphere square-well fluid confined in the random porous media and monodisperse square-well hard-sphere chain fluid in the bulk.

ACKNOWLEDGMENTS This work was supported in part by FP7 EU IRSES Project No. 612707 “Dynamics of and in Complex Systems”.

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REFERENCES

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Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in

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