Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in

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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 J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b11741 • Publication Date (Web): 15 Apr 2018 Downloaded from http://pubs.acs.org on April 16, 2018

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The Journal of Physical Chemistry

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 of the National Academy of Sciences of Ukraine 1 Svientsitskii St., Lviv, Ukraine 79011 ‡Department of Chemical and Biochemical Engineering, Vanderbilt University, Nashville, TN 37235-1604, USA E-mail: [email protected]

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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 behaviour 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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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 level 2,3 can be achieved using liquid state theoratical approach. Phase behaviour of fluids confined in porous media is quite different from that in the bulk. For example, liquid-gas phase diagrams of 4 He, 4 N2 5 and mixture of isobutyric acid and water 6 confined in dilute silica gel, are much narrower than those in the bulk, with the critical points shifted towards 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 simulation 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 HS fluid quenched at equilibrium, have been carried out. 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 transition-matrix Monte Carlo (GCTMMC) study of square-well (SW) fluids. 16 Theoretical studies of confined LJ fluid was carried out using replica Ornstein-Zernike (ROZ) formalism 7,17–19 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 matrix 23,25–27 was used to describe corresponding 3



reference system. Similar description of the reference system was utilized to study phase behaviour of the network-forming fluid 28 and polydisperse Yukawa HS fluid 29 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 study 29 to describe the phase behaviour of polydisperse HS square-well chain fluid confined in random porous media. We will consider here polydispersity in the chain length. Similar, as in 29 porous media is represented by the matrix of randomly distributed hard spheres. Properties of the reference fluid is described using recently developed 23,25–27 extension of the SPT. 34–36 The properties of the chain fluid is 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 i.e. 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 earlier 38–40 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 chain length distribution functions in coexisting phases) and analyse its behaviour 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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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 i.e. the matrix is represented by HS fluid quenched at equilibrium. We consider fully-flexible tangent squarewell chain fluid. Fluid molecules consist of a chains of m tangentially bonded attractive HS monomeric segments of diameter σ interacting through the following interparticle pair potential    ∞ ,    U M (r) = − ,      0 ,

0 λσ

,

where is the depth of the square-well potential and λ is a parameter, which define 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 P distribution function of the parent phase f (0) (m) = F (0) (m)/ l F (0) (l), where for F (0) (m) we have chosen the Schultz-Zimm distribution, 41,42

F

(0)

α+1 α+1 1 α+1 α (m) = m exp − m . α! m0 m0

(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. Due to our choice of the form of the distribution function f (0) (m) it is normalized, i.e. X

f (0) (m) = 1.

(3)

m

The fluid is characterized by the temperature T (or β = (kB T )−1 , where kB is the Boltzmann’s constant) and the total molecular number density ρ. Note that the packing fraction

5



η = πρσ 3

P

m

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mf (m)/6 represents a reduced segment density ρs = ρ

P

m

mf (m). The

matrix is characterized by the stationary HS obstacles of diameter σ0 and the total numberdensity ρ0 (or packing fraction η0 = πρ0 σ03 /6).

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 by 32,33,43–45 βA βAid βAM βAchain = + + . V V V V

(4)

Here βAid =ρ V

(" X

# 3

)

f (m) ln(ρ(m)Λ (m)) − 1

,

(5)

m

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, e.g. differentiating the Helmholtz free energy A (4) with respect to the density we get the expression for the chemical potential ∂ βA , βµ(m) = ∂ρ(m) V

(6)

and expression for the pressure P of the system can be calculated invoking the following

6


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general relation:

βP = β

X

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 theory 46–49 and SPT. 23,24,28 According to the second-order BH theory, Helmholtz free energy for squarewell fluid in the matrix is given by βAHS βA1 βA2 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 approximation 23,24,28 (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 XX βA1 = −2πρ2 β mf (m)m0 f (m0 ) V m m0

Z

λσ

gHS (r, η, η0 )r2 dr ,

(9)

σ

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 fluid 50 at some effective value of the packing fraction ηef f , which depends on the characteristics of the matrix, i.e. (P Y )

gHS (r, η, η0 ) = gHS (r, ηef f ).

(10)

We follow the scheme developed in our previous study 29 and calculate ηef f using relation

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(10) at r = σ. The value of the HS RDF gHS (σ, η, η0 ) is calculated using SPT. 28

ηef f = 1 +

1 (SP T )

4gHS

(σ + , η, η0 )

(SP T )

(σ + , η, η0 ) =

q (SP T ) 1 − 1 + 24gHS (σ + , η, η0 ) .

(11)

3 (η + η0 τ ) 1 + , φ0 − η 2 (φ0 − η)2

(12)

Here

gHS

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 ), appeares to be reasonably accurate. 29 For the second-order term A2 we use here the macroscopic compressibility approximation 46–49 XX βA2 = −πρ2 β 2 2 K HS mf (m)m0 f (m0 ) V m m0

Z

λσ

(P Y )

gHS (r, ηef f )r2 dr ,

(13)

σ

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

K

HS

1 η/φ η/φ0 (η/φ0 )2 = + + a + 2b 1 − η/φ0 (1 − η/φ0 )2 (1 − η/φ0 )3 (1 − η/φ0 )4

−1 .

(14)

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:

L0 = ρ

X

f (m) = ρ,

L1 = ρ

m

X

mf (m) = ρs ,

m

8


(15)

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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 = −2πL21 β V

Z

λσ

(P Y )

gHS (r, ηef f [L1 ])r2 dr,

(16)

σ

βA2 = −πL21 β 2 2 K HS V

λσ

Z

(P Y )

gHS (r, ηef f [L1 ])r2 dr .

(17)

σ

(P Y )

The analytical expressions for gHS (r, ηef f [L1 ]), βµSW (m) and βP SW are presented in the Supporting Information.

Chain contribution Based on 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 X βAchain = −ρ f (m)(m − 1) ln(y SW (σ, ηef f )) . V m

(18)

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 βAchain = −(L1 − L0 ) ln(y SW (σ, ηef f [L1 ])) , V

9


(19)


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where y SW (σ, ηef f [L1 ]) is the contact value of the cavity pair correlation function for a system of square-well monomers in the matrix.

y SW (σ, ηef f [L1 ]) = g SW (σ, ηef f [L1 ])e−β .

(20)

Here g SW (σ, ηef f [L1 ]) is obtained using first-order high temperature expansion about HS reference system in the matrix: 49 (P Y )

(P Y )

g SW (σ, ηef f [L1 ]) = gHS (σ, ηef f [L1 ]) + βg1

(σ, ηef f [L1 ]) ,

(21)

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

g

SW

Z

λσ

(σ, ηef f [L1 ]) =

(P Y ) gHS (σ, ηef f [L1 ])

β ∂aSW λ ∂aSW 1 1 + − , 4 ∂η 3η ∂λ

(22)

where

aSW 1

= 2πρs

U

SW

(P Y ) (r)gHS (r, ηef f [L1 ])r2 dr

Z = −2πL1

σ

λσ

(P Y )

gHS (r, ηef f [L1 ])r2 dr . (23)

σ

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 g SW (σ, ηef f [L1 ]), βµchain (m) and βP chain 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 parentphase 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) , ρ(2) and two daughter distributions

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f (1) (m), 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 of equations for the moments of the chain length distribution function in the coexisting phases

(γ)

L1 = ρ(γ)

X

(1)

(2)

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

(24)

m

where γ denotes number of phase and (1)

(2)

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

ρ(0) (ρ(2) − ρ(1) )[1 − δ1γ + δ1γ exp(β∆µ12 )] , (ρ(0) − ρ(1) ) − (ρ(0) − ρ(2) ) exp(β∆µ12 ) (2)

(1)

(2) (1) (1) β∆µ12 = β[µ(2) ex (m, T, ρ , L1 ) − µex (m, T, ρ , L1 )]. (γ)

(25)

(26)

(γ)

Here µex (m, T, ρ(γ) , 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: (2)

(1)

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

(27)

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

(1)

(2)

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

(28)

m

Note, that the distribution function in phase γ is given by (1)

(2)

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

11


(29)


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

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 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, i.e., ρ∗ = ρσ 3 /(1 − η0 ) and T ∗ = kB T / and we consider the case of equal size of the chain monomers and matrix particles, i.e. 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 studies 4–6 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

12


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needed to calculate the properties of the reference system, has been tested in our previous study 29 and in, 23,25,26 respectively. Comparison of the theoretical and computer simulation predictions for gHS (r, ηef f ) have shown 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).

η0=0

3

(a)

η0=0.05

3

λ=2

2.5

(b)

η0=0.1

3

2.5

(c)

2.5 λ=2

1.5

2

2

λ=2

Τ∗

λ=1.75

Τ∗

2 Τ∗

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


λ=1.75

1.5 λ=1.5

1

1

0.5

0.5

1.5 λ=1.75

λ=1.5

1 λ=1.5

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ρ∗

0.5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ρ∗

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ρ∗

Figure 1: Vapor-liquid phase diagrams of monodisperse square-well fluid in T ∗ vs ρ∗ = ρσ 3 /(1 − η0 ) coordinate frame at different matrix packing fraction η0 and different parameter λ obtained by 1-order BH TPT (dotted blue line), 2-order BH TPT (solid black line), GEMC 51 (red filled circles, left panel) and GC-TMMC 16 (red filled circles, intermediate and right panels)

Polydisperse square-well chain fluid in the matrix Next we proceed to discuss results of our phase diagram calculations for polydisperse squarewell chain fluid in the matrix. These results are presented in figures 2, 3 and 4. In all cases studied we consider the model with polydispersity index I = 8.1, width of the squarewell λ = 1.5 and terminate chain length distribution function at m = 100. Using the larger termination chain length do 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, i.e. m0 = 4, 8, 16 (panels a, b and c, respectively) and at three different values of the matrix packing fraction, i.e. η0 = 0, 0.1, 0.2 (figure 2) and two different size ratio of the fluid and 13



matrix particles, i.e 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 (panel a), m0 = 8 (panel b) and m0 = 16 (panel c) in the bulk, calculated using our theory and obtained by Escobedo and de Pablo 52 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 shift 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 conside 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, i.e. the corresponding phase diagram is shifted towards the region of still smaller temperatures and smaller packing fractions (figure 3). Thus the phase behaviour of our model is defined by the competition between polydispersity effects and effects of the confinement, i.e. polydispersity increases the region of the phase coexistence and confinement causes it decrease. Comparison of the phase diagrams for the models with different average chain length shows that polydispersity have a tendency to dominate at larger values of m0 , i.e. increase of m0 causes their shift in the direction of higher temperatures and slightly lower densities. Finally in Fig. 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

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2 Τ∗

I=1 η0=0

2.5

I=1 η0=0

2

1.5

1.5 1

1.5 1

η0=0.2

η0=0.2 0.5

I=1 η0=0

2

η0=0.1

η0=0.1 1

(c)

η0=0

η0=0

2.5

η0=0

3

(b)

Τ∗

2.5

3

(a)

Τ∗

3

0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 η

η0=0.1

η0=0.2

0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 η

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 η

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 = 4, 8, 16 (a, b and c panels respectively). 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-MC 52 and our theoretical results respectively for the monodisperse system I = 1.

2

I=1 η0=0

1.5 1 0.5

1.5 η0=0.1 τ=2/3 η0=0.1 τ=1/2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 η

2.5

I=1 η0=0

2

(c)

η0=0

η0=0

2.5

η0=0

3

(b)

Τ∗

2.5

3

(a)

I=1 η0=0

2 Τ∗

3

Τ∗

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.5 η0=0.1 τ=2/3

1 0.5

η0=0.1 τ=1/2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 η

η0=0.1 τ=2/3 1 0.5

η0=0.1 τ=1/2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 η

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 = 4, 8, 16 (a, b and c panels respectively). The phase diagrams include cloud and shadow curves, and critical binodals. Notation is the same as in Fig. 2.

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phase at two values of the temperature: T1∗ and T2∗ (T1∗ > T2∗ ). In all cases Tcr∗ − T1∗ ≈ 0.06, and ∆T ∗ = T1∗ − T2∗ = 0.25. Similar, 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. 0.4

0.4 (a)

0.35

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0 0

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30

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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 (T2∗ = 2.2, blue lines) and higher (T1∗ = 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 (T2∗ = 0.9, blue lines) and higher (T1∗ = 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 vapour phases.

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, i.e. the matrix is represented by the hard-sphere fluid quenched at equilibrium. To 16


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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. In general, our theoretical predictions appeare to be in a reasonable agreement with computer simulations predictions. Further improvement of the theory can be achieved using second-order TPT for associating fluids 56 in combination with better description of the structure of the reference system, e.g. 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, 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 behaviour of system, i.e. 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.

17



Supporting Information Available The Supporting Information is available free of charge • Supp Inf.pdf: We present analytical expressions for the structure and thermodynamic properties needed for the phase diagram calculations.

Acknowledgement This work was supported in part by FP7 EU IRSES project No. 612707 "Dynamics of and in Complex Systems".

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Figure 5: TOC Graphic

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Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in

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