Adsorption of fluids in confined disordered media from inhomogeneous replica
Ornstein-Zernike equations
Orest Pizio
1
and Stefan Sokolowski
2
1
Instituto de Quı ´mica de la Universidad Nacional Auto ´noma de Me ´xico, Circuito Exterior, Coyoaca ´n 04510, Distrito Federal, Mexico
2
Department for the Modelling of Physico-Chemical Processes, Faculty of Chemistry, Marie Sklodowska Curie University,
200-31 Lublin, Poland
Received 12 February 1997
The density distribution and pair distribution function for a fluid adsorbed in a slitlike pore filled with a
quenched hard-sphere fluid have been investigated using the inhomogeneous replica Ornstein-Zernike equa-
tions. The one-particle and two-particle functions are related via Born-Green-Yvon equation and the inhomo-
geneous Percus-Yevick approximation is used. The effect of the matrix is to lower the amount of adsorbed
fluid in the entire pore at low chemical potentials. For high values of the fluid chemical potential, layering of
adsorbed fluid is observed. The solution of the problem is important for gel-exclusion chromatography and in
separation science. S1063-651X9752007-1
PACS numbers: 68.45.-v, 61.20.Gy
The problem of describing the adsorption of fluids in ran-
dom porous media has received much interest recently. Im-
portant experimental observations 1–4, and relevant theo-
retical developments 5–15 have been concerned with
homgeneous systems. Adsorption of fluids in confined inho-
mogeneous disordered media, however, is of much interest in
practice for gel-exclusion chromatography and in separation
science. This problem is qualitatively more difficult to deal
with than the homogeneous case. To our best knowledge,
there was only one attempt to consider inhomogeneous fluids
adsorbed in disordered porous media 14. The inhomoge-
neous replica Ornstein-Zernike IROZ equations, supple-
mented by either Born-Green-Yvon BGY, or Lovett-Mou-
Buff-Wertheim LMBW equation for density profiles DPs,
have been proposed to study adsorption of a fluid near a
plane boundary of a disordered matrix. The theory has not
been followed by any numerical solution. In this paper our
main goal is to propose a simple model for adsorption of
fluids in confined disordered porous media and to solve it.
Let us consider the adsorption of a fluid consisting of
particles m , in a slitlike pore of the width H . The pore walls
are normal to the z axis and the pore is centered at z =0. The
fluid m , i.e., the matrix, in the pore is in an equlibrium with
the bulk fluid with the chemical potential
m
. The fluid is
then characterized by the DP
m
( z ), and by the inhomoge-
neous pair correlation function h
mm
(1,2). Due to external
factors, the structure of the fluid becomes quenched at a state
determined by
m
, and a confined porous medium is
formed.
Now, we would like to investigate adsorption of another
fluid, f , in the pore filled by the matrix. The fluid f , in the
bulk has the chemical potential
f
; at equilibrium the ad-
sorbed fluid f reaches the DP
f
( z ) and is characterized by
the inhomogeneous correlation function h
ff
(1,2). The matrix
and fluid species are denoted by the subscripts 0 and 1 5.
We assume the interactions between particles and between
particles and pore walls in a simple form choosing both spe-
cies as hard spheres of unit diameter:
U
ij
r =
, r 1
0, r 1
; U z =
, z 0.5| H -1 |
0, otherwise
,
1
where i , j =0,1.
The evaluation of the matrix structure is irrelevant to the
procedure below. The structure of a quenched inhomoge-
neous matrix is obtained from the inhomogeneous Ornstein-
Zernike OZ2 equation 14,16
h
00
1,2 -c
00
1,2 =
d 3
0
z
3
c
00
1,3 h
0
3,2 , 2
supplemented by the LMBW equation for the DP,
ln
1
z
1
z
1
+
U z
1
z
1
=
d 2 c
00
1,2
0
z
2
z
2
, 3
and the second order Percus-Yevick PY2 closure,
y
00
1,2 =1 +h
00
1,2 -c
00
1,2 . 4
In Eq. 4, y
00
(1,2) is the inhomogeneous cavity distribution
function CDF. The solution of Eqs. 2 – 4 yields
0
( z )
and h
00
(1,2) such that the one-particle CDF y
0
( z ), y
0
( z )
=
0
( z )exp
U(z), outside the pore tends to its limiting
value, determined by the configurational chemical potential,
y
0
( z → ) =exp(
0
). In fact, the structure of a matrix
may be intentionally prepared experimentally given some
functions
0
( z ) and h
00
(1,2).
The IROZ equations, representing the essence of the pro-
cedure, are 14
h
10
1,2 -c
10
1,2 =
d 3
0
z
3
c
10
1,3 h
00
3,2
+
d 3
1
z
3
c
c ,11
1,3 h
10
3,2 ,
5
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