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The structure of a sticky interface
17 February 1995
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 233 (1995) 420-423
The structure of a sticky interface M.F. Holovko, E.V. Vakarin, Yu.Ya. Duda Institute for the Physics of Condensed Matter, Suientsitskoho
I, 290011 Lviv, Ukraine
Received 24 October 1994
Abstract Sticky interface properties are studied. The influence of bulk dimerization on the density profile and adsorption coefficient is investigated using the associative version of the Henderson-Abraham-Barker equation. It is shown that the dimerization process in the interfacial region can be treated as a second adsorbed layer formation. It is also shown that beyond the second adsorbed layer the structure of the system is similar to the hard-wall case.
1. Introduction The treatment of real fluid-solid interfaces in statistical mechanics requires taking into account contributions from both the liquid and the solid side. The most common approach to this problem starts from the model in which the solid is represented by a hard wall [l-3]. The wall-particle correlations are usually described by the Henderson-AbrahamBarker equation [l-3]. Due to its simplicity this approach has been extensively used for the investigation of charged as well as uncharged interfaces [l-4]. A more realistic model of the surface, which allows the following of the influence of the wall on the adsorbed layer formation, has been proposed in Ref. [5]. In this model the surface was represented by a hard wall with surface adhesion. In order to mimic the structure of a real liquid the bulk phase was modelled by hard spheres or a sticky hard spheres system. However, the assumption that the intermolecular interaction is of the same nature as the wall-molecule one looks somewhat restricted. The purpose of this Letter is to investigate the influence of dimerization in the bulk on the structure of the interfacial region. The surface is treated as a Elsevier Science B.V. SSDI 0039-6028(94)01480-9
wall with surface adhesion and the bulk phase is modelled by the system of dimerizing hard spheres, introduced by Wertheim 161. This model of the interface permits the description of the formation of the second adsorbed layer as the associative effect. Wall-particle correlations are described by the associative version of the Henderson-Abraham-Barker equation derived in Ref. [7]. Solving this equation we find the number density profile and analyze its behavior at the various values of the adhesive parameter A, and dimerizing constant K. On the basis of this solution we obtain analytical expressions for the contact value of the wall-particle correlation function and the adsorption coefficient, and investigate their dependence versus the model parameters. We also obtain the density of the second adsorbed layer. In some particular cases our results reproduce the well-known ones.
2. Model and integral equation We start with the model consisting hard spheres, introduced by Wertheim
of dimerizing [6] and a hard
421
M.F. Holovko et al. /Chemical Physics Letters 233 (1995) 420-423
wall with surface adhesion. potential is of the form U(r)
= %(r)
+ u,,(z)
The total intermolecular
+ KV( x) + 4(x),
(1)
where U,,(r) is the hard spheres potential, r = / r2 I, I and o is the diameter of the hard sphere, UhS(~) =x,
r
U,,(r)
rap.
=O,
g’(x) (2)
U,,(z)
is the associative part modelling a directed chemical bond which is realized due to the existence of an attractive site embedded into each molecule. The position of the site is given by the vector d(0), u,,,(z)
=O,
U,,,(z)
= LI”,
function, ( p I is the matrix containing as elements the number density p and number density of ‘monomers’ pa. The indexes describe unbonded (0) and bonded (1) states of a given particle. From the partial functions we find the total correlation function in the form [6,10]
z G 1.
v-2d
where d=
Uhw(X) =o,
x2
f&(X)
x+.
where xg = pa/p is the fraction of undimerized particles. In this Letter we propose the closure relations constructed as an analog of the PY closure for the bulk problem,
(9)
Idl.
(4)
c,ow
=m,
The adhesive
L(x)
is the Mayer function
for the adhesive
=A,+(x-+o).
(11)
Using Baxter’s factorization scheme and closure relations we find the equation involving Baxter’s functions qij(r) [6,7],
(5) is described
x)] = h,6( x - to)
in the form [9]
+ 1,
(6)
where x is the distance to the wall and A, is the stickiness parameter. ‘Wall-particle’ pair correlations are described by the associative analog of the Henderson-Abraham-Barker equation, which has been derived in Ref. [7] for the hard-wall problem, + c/ lm
Xc$(r’)Plmh,a(
dr’ Ir--‘I),
(7)
where h,,(x) is the wall-particle ‘pair’ correlation function, which is related to the corresponding ‘binary’ function gjO(x), =$0(x)
W)
$r,
interaction
=cjo(r)
= [~,(X)Y,o(X)]8(~~-X):
where f,,(x) interaction,
g,,,(x)
g,,(x)
(8)
(3)
The last condition ensures that trimers and higher polymers cannot be formed. U,,,(x) is the hard-wall potential,
hj,(r)
+&,&)~
z>l
Here z = ) r2 + d(R,) - rl - d(R,) 1is the site separation, R are the orientational variables, and U,, is the depth of the potential well. The value 1 satisfies the steric saturation condition
exp[ -PC&(
=&K)(x)
-a,.“.
c,,(x) is the direct correlation function for which (7) is the definition, c,b,(r) is the bulk direct correlation
-2nC/x-g’2 Im 0
drqi/(r)p,,g,a(x-r)
= glb( to),
(12)
where gP”(ia) is the contact without surface adhesion.
value for the system
3. Density profile, contact value and adsorption We solve (12) using the scheme of Perram [8]. The results for the density profiles are plotted in Fig. 1. One can observe a discontinuity at distance x = $a. This can be explained by the associative interaction between the particles placed near the wall. Therefore we define the second adsorbed layer as that consisting of particles placed at a distance x = zu by means of an associative bond with the first adsorbed layer. The value of the discontinuity Sy’&)
=y’($a-)
-yt($a+),
M.F. Holouko et al. /Chemical
422
Physics Letters 233 (1995) 420-423
duces the contact value for dimerizing near the smooth wall, y’($r)
hard spheres
= gb($).
Neglecting the association in the bulk we get the contact value for the hard spheres near the adhesive wall 151,
(16) In order to calculate the surface density excess it is convenient to separate the adhesive contribution to the correlation function 0.5
1.5
X
2.5
Fig. 1. The density profiles at A, = 0.1 without dimerization (dashed) and at the same Aa for the totally dimerized system (solid). (a) The number density r) = 0.1; (b) the number density 77= 0.2.
Y,,(X)
=g,O,(x)
g’(x)
=g;(x)
+Gia(x,
A”),
+G’(x,
Here we introduced
A,,).
(17)
the value
Gt( x, A,) = G,,( X, A,) + G,,(x, The adsorption can be calculated
from the set (12) in the form
r=
r”
coefficient
+ p ,‘l\Gt(
A,)$
is given by
s) + pA,cry’($),
(18)
Sy’( ;Cr> = +A,_$ y,,( ;a>. The density of p&y’(fg). Using contact values
the second adsorbed layer is (12) and (9) we obtain for the
where r ” is the adsorption coefficient onto the hard smooth wall, G’(s) is the Laplace transform of G’(x, Ao>. In order to find G’(s) we perform the Laplace transformation of (131 and use (17) Gio( S) - 27F C qir( ‘) P,mGtno( lm = 2nA0
The solutions y,,(+c+)
= &(fo)
y’(&)
=
+
[’
2nAo
poaqoo(O)l
x
[l - 2nA,~rpq,,(O)(l
do(h))
+ +A&)]
-I,
1 - 2nA,crpq,,(O)(l
where x,’ = 4nu( tion of dimerized
3x9Yoo(b) go(P) + ix,’ goo(h) . Y'(3U)
p~/p)qll(0) particles.
+ $A,x,‘)
’
(15)
= 1 -x,” is the fracAt A, = 0 (15) repro-
t t
+
(20)
If A, = 0, we have r = r ‘. When xi = 0 we get the result for hard spheres near an adhesive wall [5],
r=r;+
g;(h) + ;Aodo(h+,’
the result into (18) we
get r=r"+puA,
{g~o(~a)2~Aoapoq,,(O) -
(19) Solving this set and replacing
$ox,‘) (14)
=
-t~0)Cq~1(s)~l,y,o(~cr). lm
_
+2~A,p,c+q,,(O)g,0,(3~)
1 - 2lTAovpqoo(O)(l+
Y,o(hq
ev(
are given by
‘)
PUA,
1-
2npmAo
qoo(O)
+
Aorl(l1+ 7(6A,
7) - 1)
.
(21)
M.F. Holooko et al. /Chemical
The values xf and x,’ are functions of the bulk number density 71 and dimerizing constant K, determining the strength of association [6,7].
4. Conclusion Studying a dimerizing hard spheres system near an adhesive wall we have calculated the number density profile, contact value of the wall-particle correlation function and the adsorption coefficient. We have investigated the behavior of these quantities with respect to changing the model parameters: bulk density 7, stickiness constant h,, dimerizing constant K (or fraction of undimerized particles x:1. As can be seen from Fig. 1, for the undimerized case, surface adhesion diminishes the contact value ~‘(+a> and leads to a peak at a distance x = $a. This proves that in the region between x = $o and x = $(r we have an excess of unstuck particles near the first adsorbed layer. Beyond this region the behavior of the density profile is the same as in the hard wall case. It is shown that in a totally dimerized system we have a discontinuity instead of a peak. The last observation allows us to assume that this discontinuity is caused by an associative site interaction which fixes a given particle at distance x = $a by the formation of a dimer. In the case of a
Physics Letters 233 (1995) 420-423
423
monomer-dimer mixture we could treat this discontinuity as a bound of the second adsorbed layer, because the associative sites of the first adsorbed layer are the centers of the second adsorption. The density of the second adsorbed layer is proportional to the fraction of dimers.
Acknowledgement The research described in this publication made possible, in part, by Grant No UlJOOO.
was
References [ll D. Henderson, F.F. Abraham and J.A. Barker, Mol. Phys. 31 (1976) 1291. [2] L. Blum and G. Stell, J. Stat. Phys. 15 (1976) 439. [3] L. Blum and D. Henderson, J. Chem. Phys. 74 (1981) 1902. [4] M.F. Holovko, 0.0. Pizio and Z.B. Halych, Electrochim. Acta 36 (1991) 1715. [5] J.W. Perram, E.R. Smith, Proc. R. Sot. A 353 (1977) 193. [h] M.S. Wertheim, J. Stat. Phys. 35 (1984) 19; 35; J. Chem. Phys. 85 (1986) 2929. [7] M.F. Holovko and E.V. Vakarin, Mol. Phys. in press. [8] J.W. Perram, Mol. Phys. 30 (1975) 1505. [9] R.J. Baxter, J. Chem. Phys. 52 (1970) 4559. [lo] M.F. Holovko and Yu.V. Kalyuzhnyi, Mol. Phys. 73 (1991) 1145.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 233 (1995) 420-423
The structure of a sticky interface M.F. Holovko, E.V. Vakarin, Yu.Ya. Duda Institute for the Physics of Condensed Matter, Suientsitskoho
I, 290011 Lviv, Ukraine
Received 24 October 1994
Abstract Sticky interface properties are studied. The influence of bulk dimerization on the density profile and adsorption coefficient is investigated using the associative version of the Henderson-Abraham-Barker equation. It is shown that the dimerization process in the interfacial region can be treated as a second adsorbed layer formation. It is also shown that beyond the second adsorbed layer the structure of the system is similar to the hard-wall case.
1. Introduction The treatment of real fluid-solid interfaces in statistical mechanics requires taking into account contributions from both the liquid and the solid side. The most common approach to this problem starts from the model in which the solid is represented by a hard wall [l-3]. The wall-particle correlations are usually described by the Henderson-AbrahamBarker equation [l-3]. Due to its simplicity this approach has been extensively used for the investigation of charged as well as uncharged interfaces [l-4]. A more realistic model of the surface, which allows the following of the influence of the wall on the adsorbed layer formation, has been proposed in Ref. [5]. In this model the surface was represented by a hard wall with surface adhesion. In order to mimic the structure of a real liquid the bulk phase was modelled by hard spheres or a sticky hard spheres system. However, the assumption that the intermolecular interaction is of the same nature as the wall-molecule one looks somewhat restricted. The purpose of this Letter is to investigate the influence of dimerization in the bulk on the structure of the interfacial region. The surface is treated as a Elsevier Science B.V. SSDI 0039-6028(94)01480-9
wall with surface adhesion and the bulk phase is modelled by the system of dimerizing hard spheres, introduced by Wertheim 161. This model of the interface permits the description of the formation of the second adsorbed layer as the associative effect. Wall-particle correlations are described by the associative version of the Henderson-Abraham-Barker equation derived in Ref. [7]. Solving this equation we find the number density profile and analyze its behavior at the various values of the adhesive parameter A, and dimerizing constant K. On the basis of this solution we obtain analytical expressions for the contact value of the wall-particle correlation function and the adsorption coefficient, and investigate their dependence versus the model parameters. We also obtain the density of the second adsorbed layer. In some particular cases our results reproduce the well-known ones.
2. Model and integral equation We start with the model consisting hard spheres, introduced by Wertheim
of dimerizing [6] and a hard
421
M.F. Holovko et al. /Chemical Physics Letters 233 (1995) 420-423
wall with surface adhesion. potential is of the form U(r)
= %(r)
+ u,,(z)
The total intermolecular
+ KV( x) + 4(x),
(1)
where U,,(r) is the hard spheres potential, r = / r2 I, I and o is the diameter of the hard sphere, UhS(~) =x,
r
U,,(r)
rap.
=O,
g’(x) (2)
U,,(z)
is the associative part modelling a directed chemical bond which is realized due to the existence of an attractive site embedded into each molecule. The position of the site is given by the vector d(0), u,,,(z)
=O,
U,,,(z)
= LI”,
function, ( p I is the matrix containing as elements the number density p and number density of ‘monomers’ pa. The indexes describe unbonded (0) and bonded (1) states of a given particle. From the partial functions we find the total correlation function in the form [6,10]
z G 1.
v-2d
where d=
Uhw(X) =o,
x2
f&(X)
x+.
where xg = pa/p is the fraction of undimerized particles. In this Letter we propose the closure relations constructed as an analog of the PY closure for the bulk problem,
(9)
Idl.
(4)
c,ow
=m,
The adhesive
L(x)
is the Mayer function
for the adhesive
=A,+(x-+o).
(11)
Using Baxter’s factorization scheme and closure relations we find the equation involving Baxter’s functions qij(r) [6,7],
(5) is described
x)] = h,6( x - to)
in the form [9]
+ 1,
(6)
where x is the distance to the wall and A, is the stickiness parameter. ‘Wall-particle’ pair correlations are described by the associative analog of the Henderson-Abraham-Barker equation, which has been derived in Ref. [7] for the hard-wall problem, + c/ lm
Xc$(r’)Plmh,a(
dr’ Ir--‘I),
(7)
where h,,(x) is the wall-particle ‘pair’ correlation function, which is related to the corresponding ‘binary’ function gjO(x), =$0(x)
W)
$r,
interaction
=cjo(r)
= [~,(X)Y,o(X)]8(~~-X):
where f,,(x) interaction,
g,,,(x)
g,,(x)
(8)
(3)
The last condition ensures that trimers and higher polymers cannot be formed. U,,,(x) is the hard-wall potential,
hj,(r)
+&,&)~
z>l
Here z = ) r2 + d(R,) - rl - d(R,) 1is the site separation, R are the orientational variables, and U,, is the depth of the potential well. The value 1 satisfies the steric saturation condition
exp[ -PC&(
=&K)(x)
-a,.“.
c,,(x) is the direct correlation function for which (7) is the definition, c,b,(r) is the bulk direct correlation
-2nC/x-g’2 Im 0
drqi/(r)p,,g,a(x-r)
= glb( to),
(12)
where gP”(ia) is the contact without surface adhesion.
value for the system
3. Density profile, contact value and adsorption We solve (12) using the scheme of Perram [8]. The results for the density profiles are plotted in Fig. 1. One can observe a discontinuity at distance x = $a. This can be explained by the associative interaction between the particles placed near the wall. Therefore we define the second adsorbed layer as that consisting of particles placed at a distance x = zu by means of an associative bond with the first adsorbed layer. The value of the discontinuity Sy’&)
=y’($a-)
-yt($a+),
M.F. Holouko et al. /Chemical
422
Physics Letters 233 (1995) 420-423
duces the contact value for dimerizing near the smooth wall, y’($r)
hard spheres
= gb($).
Neglecting the association in the bulk we get the contact value for the hard spheres near the adhesive wall 151,
(16) In order to calculate the surface density excess it is convenient to separate the adhesive contribution to the correlation function 0.5
1.5
X
2.5
Fig. 1. The density profiles at A, = 0.1 without dimerization (dashed) and at the same Aa for the totally dimerized system (solid). (a) The number density r) = 0.1; (b) the number density 77= 0.2.
Y,,(X)
=g,O,(x)
g’(x)
=g;(x)
+Gia(x,
A”),
+G’(x,
Here we introduced
A,,).
(17)
the value
Gt( x, A,) = G,,( X, A,) + G,,(x, The adsorption can be calculated
from the set (12) in the form
r=
r”
coefficient
+ p ,‘l\Gt(
A,)$
is given by
s) + pA,cry’($),
(18)
Sy’( ;Cr> = +A,_$ y,,( ;a>. The density of p&y’(fg). Using contact values
the second adsorbed layer is (12) and (9) we obtain for the
where r ” is the adsorption coefficient onto the hard smooth wall, G’(s) is the Laplace transform of G’(x, Ao>. In order to find G’(s) we perform the Laplace transformation of (131 and use (17) Gio( S) - 27F C qir( ‘) P,mGtno( lm = 2nA0
The solutions y,,(+c+)
= &(fo)
y’(&)
=
+
[’
2nAo
poaqoo(O)l
x
[l - 2nA,~rpq,,(O)(l
do(h))
+ +A&)]
-I,
1 - 2nA,crpq,,(O)(l
where x,’ = 4nu( tion of dimerized
3x9Yoo(b) go(P) + ix,’ goo(h) . Y'(3U)
p~/p)qll(0) particles.
+ $A,x,‘)
’
(15)
= 1 -x,” is the fracAt A, = 0 (15) repro-
t t
+
(20)
If A, = 0, we have r = r ‘. When xi = 0 we get the result for hard spheres near an adhesive wall [5],
r=r;+
g;(h) + ;Aodo(h+,’
the result into (18) we
get r=r"+puA,
{g~o(~a)2~Aoapoq,,(O) -
(19) Solving this set and replacing
$ox,‘) (14)
=
-t~0)Cq~1(s)~l,y,o(~cr). lm
_
+2~A,p,c+q,,(O)g,0,(3~)
1 - 2lTAovpqoo(O)(l+
Y,o(hq
ev(
are given by
‘)
PUA,
1-
2npmAo
qoo(O)
+
Aorl(l1+ 7(6A,
7) - 1)
.
(21)
M.F. Holooko et al. /Chemical
The values xf and x,’ are functions of the bulk number density 71 and dimerizing constant K, determining the strength of association [6,7].
4. Conclusion Studying a dimerizing hard spheres system near an adhesive wall we have calculated the number density profile, contact value of the wall-particle correlation function and the adsorption coefficient. We have investigated the behavior of these quantities with respect to changing the model parameters: bulk density 7, stickiness constant h,, dimerizing constant K (or fraction of undimerized particles x:1. As can be seen from Fig. 1, for the undimerized case, surface adhesion diminishes the contact value ~‘(+a> and leads to a peak at a distance x = $a. This proves that in the region between x = $o and x = $(r we have an excess of unstuck particles near the first adsorbed layer. Beyond this region the behavior of the density profile is the same as in the hard wall case. It is shown that in a totally dimerized system we have a discontinuity instead of a peak. The last observation allows us to assume that this discontinuity is caused by an associative site interaction which fixes a given particle at distance x = $a by the formation of a dimer. In the case of a
Physics Letters 233 (1995) 420-423
423
monomer-dimer mixture we could treat this discontinuity as a bound of the second adsorbed layer, because the associative sites of the first adsorbed layer are the centers of the second adsorption. The density of the second adsorbed layer is proportional to the fraction of dimers.
Acknowledgement The research described in this publication made possible, in part, by Grant No UlJOOO.
was
References [ll D. Henderson, F.F. Abraham and J.A. Barker, Mol. Phys. 31 (1976) 1291. [2] L. Blum and G. Stell, J. Stat. Phys. 15 (1976) 439. [3] L. Blum and D. Henderson, J. Chem. Phys. 74 (1981) 1902. [4] M.F. Holovko, 0.0. Pizio and Z.B. Halych, Electrochim. Acta 36 (1991) 1715. [5] J.W. Perram, E.R. Smith, Proc. R. Sot. A 353 (1977) 193. [h] M.S. Wertheim, J. Stat. Phys. 35 (1984) 19; 35; J. Chem. Phys. 85 (1986) 2929. [7] M.F. Holovko and E.V. Vakarin, Mol. Phys. in press. [8] J.W. Perram, Mol. Phys. 30 (1975) 1505. [9] R.J. Baxter, J. Chem. Phys. 52 (1970) 4559. [lo] M.F. Holovko and Yu.V. Kalyuzhnyi, Mol. Phys. 73 (1991) 1145.