Solution of reference interaction site model for

Solution of reference interaction site model for mixtures of shortchain polyatomic molecules RongSong Wu, Lloyd L. Lee, and Jeffrey H. Harwell Citation: The Journal of Chemical Physics 91, 4254 (1989); doi: 10.1063/1.456805 View online: http://dx.doi.org/10.1063/1.456805 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/91/7?ver=pdfcov Published by the AIP Publishing

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.15.115.227 On: Wed, 12 Mar 2014 14:52:26

Solution of reference interaction site model for mixtures of short-chain polyatomic molecules Rong-Song Wu, Lloyd L. Lee, and Jeffrey H. Harwell School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019

(Received 10 April 1989; accepted 9 June 1989) Mixtures of chain molecules-monomers, dimers, trimers, and tetramers-are studied using the soft interaction site model. The site-site Ornstein-Zernike equations are solved using the Percus-Yevick closure. The site-site potential is of the Lennard-Jones 12-6 type. The method of solution, based on the efficient algorithm of Labik and employing Newton-Raphson accelerations, is shown to be fast, accurate and stable; it also shows good convergence behavior even with inaccurate initial estimates. New symmetrical properties among the atom-atom pairs are used to simplify the Jacobian matrix of solution. Pure as well as mixture systems are investigated. Comparison with simulation data of Balion et al. and Massobrio et al. is made. The structure is qualitatively described by the integral equations. The internal energy is well predicted by the reference interaction site model calculations. I. INTRODUCTION

The interaction site model (ISM) I,Z has been used, with reasonable degree of success, to model the interaction forces between polyatomic molecules, such as methane,3 benzene,4 hydrogen fluoride, 5 and acetonitrile. 6 The theories built around these models for structural and thermodynamic information are called the ISAs (interaction site approximations 7 ). The common ones in use are the reference interaction site approximationS (RISA) for hard-core interactions and soft ISAs for soft-core interactions, where Percus-Yevick-like or hypernetted chainlike closures are used. Recently, these methods have been extended to mixtures. For example, perturbation theory has been formulated to simulate mixtures of argon + nitrogen and argon + oxygen systems, 9 as well as for hard spheres/homonuclear hard dumbbells. 10 Later on, Enciso et al. extended the soft ISA to N z-02, CO2ethane, and CCI4 , II as well as SiCLc TiCI4 , SnClc TiC14,12 CClc SnCI4 ,13 and polar dumbbells/nonpolar dumbbells mixtures. 14 Bohn et al. 15 have studied the quadrupolar dumbbell mixtures. In addition, polymeric structure has been studied in terms oflSM. 16 In this paper, we are interested in applying ISM to mixtures of hydrocarbonlike molecules, such as methane-ethane, methane-butane, and propane-butane. The purpose is threefold: first to test the accuracy of soft ISAs for such mixture systems, since simulation data l7 are available; second, interesting symmetry properties in the Jacobian matrices of solution are exhibited by such systems that could be exploited in efficient numerical programming; third, the results could be used to test the group-contribution ideas in chemical engineering for calculating equilibrium properties of mixtures with similar function groups. 18 An important contribution to our understanding of the equilibrium properties of molecular fluids has come from the theoretical study of ISM. In this model, the interaction potential between two molecules is represented as a sum of interactions between sites on different molecules, which can be written as u(1,2) =

L uap(r). ap

4254

J. Chern. Phys. 91 (7), 1 October 1989

(1.1 )

The site-site potential u aP (r) can then be modeled by, e.g., a hard-sphere or Lennard-Jones 12-6 potential. Most theoretical methods are based on the site-site Ornstein-Zernike (SSOZ) equation originally proposed by Chandler and Andersen. 8 The SSOZ equation can be written in a matrix form H(r) = W*C*W(r)

+ W*C*pH(r),

( 1.2)

where "*,, indicates convolution. The elements ofH(r) and C(r) are the site-site total and direct correlation functions, respectively. W (r) contains the intramolecular correlations

defined by _ wap (r) =8ap 8(r)

+

(l-8 ap ) 2

8(r -laP),

(1.3)

41TI ap where laP is the distance between sites a and f3 in the same molecules. Note that wap(r) = 0, when the sites are in different molecules. The SSOZ equation simply relates the total correlation function haP (r) to the direct correlation function caP (r). Asecond closure relation is needed to yield solutions. For hard-core fluids, Chandler and Anderson s proposed

=0 gaP (r) = 0

for r>dap , for r < d ap ,

Cap(r)

( 1.4a) (l.4b)

where gaper) is the site-site pair correlation function and gaP ( r) = haP ( r) 1. d ap is the diameter of hard spheres.

+

For soft-core fluids, two of the commonly used closures are the Percus-Yevick (PY) closure cap (r) = (1

+ raP (r»( exp [ k~ ~ uap (r) ]

-

1)

(1.5)

and the hypernetted chain (HNC) closure cap(r) = exp

(-=..!. kBT

uap(r)

+ raP (r») -

raper) -

1, ( 1.6)

where raper)

= hap(r)

- caP(r).

( 1.7)

The combination of the SSOZ equation and an approximate closure is called RISA. The indirect correlation function

0021-9606/89/194254-11$02.10

@ 1989 American Institute of Physics

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.15.115.227 On: Wed, 12 Mar 2014 14:52:26

Wu, Lee, and Harwell: Solution of RISM for mixtures

r a/3 (r) has been used in the numerical solution of integral equations by Gillan's method 19 and Labik's method,20 because it is a continuous function of r. We have a smooth transition across the molecular core. Due to the approximate nature of the model, there are inherent anomalies in RISA,21 but RISA can lead to accurate values for the internal energy and mean-square force for a number of molecular fluids,22.23 and can give qualitative information for the fluid structure. Recently, modifications 24--26 have appeared to remedy RISA. Conventional methods for solving the site-site OZ (SSOZ) integral equation for pure fluids and/or mixtures are as follows: (i) The variational procedure. Lowden and Chandler2? used a variational procedure, then Morriss, Perram, and Smith28 used the factorization method to solve the SSOZ equation for hard-core molecules. (ii) The Picard methods. The common iterative technique of successive Fourier transforms used for the OZ equation was applied to solve SSOZ equations. 29.3o (iii) Recursive algorithm. This method was used by Hirata and Rossky31 and Pratt,32 etc. (iv) Gillan's method. 19 A technique combining the standard direct iterations and Newton-Raphson acceleration technique was developed recently as the most powerful method in use, as in Monson 22 for homonuclear diatomics fluids and in Enciso for homonuclear molecule fluid mixtures (e.g., N 2-02 , CO 2-ethane, and CC14,11 SiCLc TiC1412 ). Morriss 33 generalized it to mixture systems with charges. Recently, Labik, Malijevsky, and Vonka 20 developed a new numerical method for solving the OZ equation for pure spherical molecule systems. Due to its efficient convergence and low sensitivity to the choice of initial estimates, we generalize the Labik method here to the solution ofSSOZ equations for mixtures of chain molecules. We apply the method to methane-ethane-like (monomer-dimer), methane-butane-like (monomer-tetramer), and propane-butane-like (trimer-tetramer) systems. The site-site correlation functions are compared with simulation results when available. II. FORMULATION OF RISM FOR MIXTURES OF CHAIN MOLECULES A. Molecular models of chain molecule

three methyl groups connected by a C-C bond, with bond angle 112.15°. Butane will be treated as a rigid molecule with four methyl groups connected by a C-C bond, with bond angle 112.15°. The distribution of rotational conformers, distinguished by their dihedrals angle as the gauche form and the trans form, will be treated by the method of Hsu. 34 We have chosen the same parameters for all sites in these oligomers. Therefore, the potential parameters between different oligomers are also the same as in pure components. The Lennard-Jones 12-6 potential was chosen for the site-site potential. The size and energy parameters and bond lengths used are u = 3.5 A, E = 419 J/gmol, C-C bond length = 1. 54 A. These parameters are the same as in the simulation data of Massobrio et al. l ? (Note: these parameters were taken from the literature and did not necessarily correspond to best fits to experimental values.) Some other sets of parameters, where simulation data were available, were also used in order to check the accuracy of the integral equations; these will be indicated when appropriate. Both the PY and HNC closures were tested in this study; the PY theory gave better results. Thus, all the results reported in this paper are from PY calculations. B. Fourier transformed SSOZ equations

Making a Fourier transformation (FT) of the SSOZ equations into the k space, we obtain A

AA

A.A

A

H= WC(1- WC)-IW AA

A

= [(1 - WC) -

J] W,

I -

(2.2.1)

where the matrix elements of B,C, and Ware

1 00

A (H)a/3 =41T - (PaP/3) 1/2

k

0

(C) a/3 = 41T (PaP/3) 1/2

k

A

A

(W)a/3

= OJ a/3(k) =

(00

Jo

drrsm(kr)h a/3(r),

(2.2.2)

drr sin (kr)c a/3 (r),

(2.2.3 )

•

sin kla/3

(2.2.4)

kl a/3

The density factor PaP/3 is included in the FT of the correlation functions for convenience in numerical programming. The Jacobian is needed in the Newton-Raphson cycle; its elements can be derived by33:

Jh a /3

J

-a~ = -J~ CAY

The systems studied in this paper include pure (methanelike) monomers, (ethanelike) dimers, (propanelike) trimers, and (butanelike) tetramers, as well as mixtures simulating methane--ethane, methane-butane, and propanebutane systems. The potential parameters chosen for the pair potentials conform to the simulation data of Massobrio et al.17 for oligomers and are not parameters that reproduce "exactly" the properties of these short chain hydrocarbons. So the study presented here is for a test of the validity of the model chain molecules, not for real substances. A methanelike molecule can be modeled as a single spherical particle. Ethane can be modeled as a dumbbell particle, consisting of two methyl groups connected by a fixed c-c bond. Propane will be treated as a rigid molecule with

4255

CAY

=

LS (1- WC)as A

L (1 s

A

_

I

A

(W)s/3

AA

AA

WC) v~ I [ (1 AA

A

WC) -

A

I(

A

W) LA Ws/3

AA

A

= [(1- WC)-IW]aA [(1- WC)- IW lv/3

(2.2.5 )

A

A

A

A

= (1' + C + W)aA (1'+ C + W)v/3

- 8 aA 8 v/3'

(2.2.6)

J. Chern. Phys., Vol. 91, No.7, 1 October 1989 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.15.115.227 On: Wed, 12 Mar 2014 14:52:26

4256

Wu, Lee, and Harwell: Solution of RISM for mixtures

Simplifications of the SSOZ equations and the corresponding jacobian matrix is possible for each specific case because of molecular and matrix symmetries. This leads to greater efficiency in computer calculations. Techniques for matrix reduction and Jacobian simplification based on these simplifications are given in Appendix A. C. SSOZ integral equations for alkanelike molecules In our study, for methanelike molecules, the OZ integral equation theory for spherical molecular fluid is applied. For ethanelike molecules, the SSOZ integral equation is used for dimers. Due to the symmetry of the dimers, the matrix SSOZ equation can be reduced to a single equation

A-I],

h=(1+W)[ 1 (2.3.1 ) 2 1 - le(1 + w) where W = sin kl I kl; I is the bond length. For propanelike molecules, the SSOZ integral equations are applied to trimers. Due to the symmetry of molecules, we have only three distinct pairs: the end-end, inner-inner, and inner-end pairs of groups. There are also three distinct pair correlation functions: (gEE' gIl' gEl)' describing the fluid structure. The matrix SSOZ equation can be reduced to three single equations, which can be written in FT space as:

-1 + wz) + -1- [1-(1 + wz) +,cnwp , hEE = --(1 2 detp 2 (2.3.2a) A

A

A

A

A

]

wherew B = (WI +W2)2- (1 +w l )(1 +W3) and det B = 1 - 2[C EE (1 + 3 ) + cn (1 + WI)

w

+ 2cE dw i +

w

2 )]

+ 4WB (CEICE1

CnCEE )·

-

Following HSU,33 we specify for conformers

W3

= W31 %, + w3g ( 1 - X,),

(2.3.4)

where WI = sin kl/kl l ,

II is the bond length,

W2 = sin kl21 k12, Iz is the distance between the inner-outer group, W31 = sin kl,l kl" I, is the distance between the outer-outer group of the trans conformer,

w

3g = sin klglklg, Ig is the distance between the outer-outer group of the gauche conformer, X, is the trans conformer mole fraction.

The matrix SSOZ equations for mixtures systems can also be reduced to a much fewer number of distinct site-site correlation functions. For example, the number of distinct pairs in methane-ethane-like mixtures is three; for methanebutane-like mixtures, six; and for propane-butane-like mixtures, ten. The explicit forms of site-site correlation functions for the first two systems are shown in Appendix B. III. NUMERICAL PROCEDURES A. Numerical methods for the SSOZ equation

The indirect correlation function is defined as

(2.3.2b)

(3.1)

(2.3.2c)

The Fourier transforms of form, are

r a{3 ( r)

and

Ca{3 (

r),

in discrete

where WI = sin kl/kl l ,

II is the bond length,

w2 =

(3.2a)

sin klzlkl2, 12 is the distance between the end-end group,

detp = 1 - [2C EE (1 +

wp

w

2)

+ 4c EI WI + cn ]

+4Wp(CEICEI -CnCEE ), = (wI)z - (1 + 2 )!2.

w

XSin(;

For butanelike molecules, the SSOZ integral equations are written for tetramers; just as the propane matrix reduced to three equations, the matrix SSOZ equation for butane can be reduced to three equations

A

ij).

A

j= 1,2, ... ,N - 1, A.

(3.2b)

A

A

r a{3.j = r a{3 (k), Ca{3.j = Ca{3 (kj ). We use r a{3.j and Ca{3.j instead of r a{3 (kj ) and ca{3 (k j ), so that the special case about k = 0 does not arise. The inverse FT are ~here

I:l.k

= 2rrj (PaP{3)

ra{3.j

_ 1/2 N -

I

A

•

(1T ..)

/~1 ra(:/j sm N

l] ,

(3.3a)

i = 1,2, ... ,N - 1, C

a{3.;

I:l.k -1/2 N - I (1T 2rrj (PaP{3) j~1 Ca{3j sm N A

=

i

= 1,2, ... ,N -

•

..)

l] ,

1,

where ra{3.; = ra{3(r;), ca{3.; = ca{3(r j ) and I:l.rtl.k = r; = il:l.r, k j =jl:l.k.

(3.3b)

1TIN, (3.4)

The Fourier transform of the SSOZ matrix, in discrete form, is J. Chern. Phys., Vol. 91, No.7, 1 October 1989 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.15.115.227 On: Wed, 12 Mar 2014 14:52:26

Wu, Lee, and Harwell: Solution of RISM for mixtures A

AA

AA

A

H j = ~Cj(l- ~Cj)-I~.

(3.5)

The rank of the matrix is (m X m) with

=I

m

number of sites in a molecule of species c;.

c,

As a measure of convergence of the NR cycles, we defined the variable 5 as

5=

[1;(PaP8)-'~(~

uP,; ) , ] 112

I

The closure relation can be written as

c af3,; =

c~f3,i

+ ¢~f3,i (raf3,i -

~f3,; ).

(3.7)

The FT of the above equation is C af3J =

C~f3J

j=

(3.6)

c af3.; =!(raf3,i)'

where! is the chosen closure. A first-order expansion of function c af3,; about point C~f3,i is

N-I

+ 4rrt:..r(PaPf3) 1/2 I ¢~f3,ir;(raf3,; - ~f3,;) i= I

(3,8)

(Papf3)

-II(raf3J)

(3.13)

2

af3

I

When 5 got below 0.01, we corrected the fine part of r af3J' where} > M, in the FT space, by a direct iterative procedure. Then we inverted the FT and used the new r af3,i as inputs for ~f3,; in Eq. (3.7) and the entire procedure was repeated. The computational flow sheet is shown in Table I. Since the initial estimates we used were often very inaccurate, the NR scheme might diverge at intial stages unless we restricted the changes of r af3,j within certain limits. In order to avoid divergence, we set A. 1 M A.

t:..r af3,j < -

Max [r af3.j ]. 2 j= 1

with

¢~f3,; =

f3 (:!a ) _ r af3 Y"".• -

Substituting r we get

af3.l

(3.9)

.

y;,,,., and r~f3.' for

r af3,i

and ~f3, i> respectively, ( 3.10)

Also, we compared the NR convergence numbers 5 from successive cycles; if no improvement was observed, we interrupted the NR cycle and switched to direct iterations to get better estimates, then entered the NR cycle again. As a measure of the total accuracy of the solution at the end of each refinement cycle, we checked the quantity 1], defined by N-I

where

1] =

Caf3,j/ = ~ Nil ¢~f3,; sin(~ if )sin(~ ij) N

=~

N

;=1

Ni

I

N,=I

N

¢~f3,; {cos[~ i(l- })] - cos[~ i(l +})]}. N

N

(3.11 )

By substituting Eqs. (3.10) and (3.11) for C af3,j in the matrixequations (3.5), we arrive at a set ofm 2 (N - 1) nonlinear equations. After linearizing these equations, we get t:..r af3J -

I[ a~af3J Nil ac;. . ;'v

A

= r:f3J -

A

VJ

r af3J ,

(c;,vJ/t:..r;.V.l)]

/= I

}= 1,2, ... ,N - 1,

(3.12)

where ar af3/ac;,vJ and r:f3J are calculated at C af3j from Eq. (3.8). The number of simultaneous equation~ are 2 m 2 (N - 1) with m (N - 1) unknown variables t:..r af3J'

}=

4257

1,2, ... ,N - 1.

A.

Analogous to Labik's method, the r af3 (k) can be divided into a coarse part and afine part by choosing an integer M such that when}> M, the value ofr af3 (k) rapidly decreases with increasing k; thus the solution of the set of linearized equations (3.12) will depend primarly on r af3 (k) at small k values. Owing to structural symmetry, the number of pairs of sites m 2 can also be reduced to a small number for efficient calculation. For example, in the tetramer system studied here,m = 4, and m 2 = 16, but we can reduce this to three terms only: gEE' gIl' and gEl' By choosing a suitable grid size, in this case t:..r = 0.025, the number of steps on the coarse part is M = 16-32, with N = 256-512. We solved r af3 (k) for the coarse part (namely,}