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Pair distribution function in polar fluid-hard wall systems. Long range components
Physica A 158 (1989) 633-648 North-Holland, Amsterdam
PAIR DISTRIBUTION FUNCTION IN POLAR FLUID-HARD WALL SYSTEMS. LONG RANGE COMPONENTS M.-E. B O U D H - H I R International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy Received 24 January 1989
Using the diagrammatic technique we prove that the total correlation function (TCF) for dipolar fluid systems in front of an ideal wall has four long range components. The first term is identical to the long range component of the TCF in homogeneous systems. The second is a longitudinal term characterizing two non-adsorbed particles. It has the same symmetry as the dipole-image interaction. The third describes the correlation between adsorbed and nonadsorbed particles. The last one is a transversal term; it is relativ÷, to two adsorbed particles. One can note that the arising of the dipole-image interaction symmetry without introducing this potential in the Hamiltonian.
1. Introduction Considerable efforts have beem made to understand polar fluid systems [1-7]. In homogeneous systems, many important thermodynamic and structure properties are now well known. For example, in dipolar fluid the TCF's asymptotic behaviour is proportional to the dipole-dipole interaction [8, 9]. The purpose of this paper is to examine the long range components of the TCF, in dipolar fluids, in front of a structureless hard wall. The long range component is used here in the sense that it decreases as the dipole-dipole interaction. In homogeneous systems, the only distance intervening is the distance -_,---_.-o.. :_ .L.. "-~ "long between the two particles and so there ib no amu~gmty., t,,,- meaning ... range component". In the presence of a wall. the knowledge of this parameter only is not sufficient and consequently the positions of the particles with respect to the surface should be given. Using the statistical mechanics concepts, we Frove that, for dilute dipolar fluids, the TCF contains two long range components which have, respectively. the symmetry of the dipole-dipole and dipole-image interactions. Away from the low density limit, we find four long range terms. The first characterizes the 0378-4371/89/$03.50 © Elsevier Science Publishers B.¥: (North-Holland Phvsics Publishin~ Division)
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M.-E. Boudh-tlir / Pair distribution function
homogeneous fluids. The second, having the symmetry of the dipole-image interaction, is related to the correlation between two non-adsorbed particles. The third gives the correlation between adsorbed and non-adsorbed particles. The last one is a transversal term describing two adsorbed particles. It should be noted that dipole-image potential symmetry arises, though this interaction has not been introduced a priori in the Hamiltonian.
2. Modelling and general developments The dipolar fluid-wall system is studied as a limit case of a binary mixture. We consider two species of particles denoted by w (wall) and d (dipole). The first has a hard core diameter trw and a density p,,,. The parameters characterizing the second one are o- and p. In this system the particles are assumed to be interacting by the following pair-potentials: Vww(i, j ) = vwa(i, j ) = yaw(i, j) = v*(rij ) .
(1)
Vdd(i, j)= v*(rij ) + w(i, j).
(2)
Here, v* denotes the hard sphere potential, and w designates the dipolar part of the interaction dipole-dipole. They are given, respectively, by the equations for r~j ~<(o'i + % ) / 2 . (3) for % > (o-~ + o"i)/2,
w(i, j)= ( - ix"/rij)tt(r # - 6 )(3~i - r#pj- P i j - / t i •/Ttj),
(4)
where tr, is the hard core diameter of the particles of species i;/2; and ~;j are unit vectors, H is the Heaviside step-function (i.e., H(x)= 1, for x > 0 and 0 otherwise), and 6 is a cutoff parameter. A .,i~dioi,a,,~_ ._....,~. choice• of the ,.,~u,,""'""~parameter can accelerate the convergence of the calculations and make them easier to perform. Because of the hard core potential, the value of the dipolar interaction inside the core does not modify the physical properties of the system [9]. Nevertheless, the correlation functions may be sensitive to the particular cutoff-prameter choice when an approximation is introduced. This is due to the fact that the phase space includes all the space regions (including the interior of the core) and to the coupling via the Ornstein-Zernike equation [10]. One therefore takes 6 going to zere [11-13]. This choice will be adopted here.
M.-E. Boudh-Hir / Pair distribution function
635
In this system the probability to find, at the same time, three particles, the first of species w, the two others of species d, at the points 0, 1, and 2 respectively can be written as pwdd(O, 1, 2 ) = pw(P/4"rr)2gwdd(O, 1, 2)
= pw(P/4"n')2gwd(O, 1)gwd(O, 2)g(1,2) exp(c(O, 1, 2)).
(5)
gwd and g are the hard sphere-dipole and the dipole-dipole pair distribution functions (PDF) respectively; and exp(c(0, 1, 2)) is a coupling term. Its graphical expansion is given by Stell [14]. As in the case of the one-particle distribution function (OPDF) [15], the limit fluid in front of a hard wall is obtained using the following procedure:
lim [ lim [pwdd(O, 1,2)/Pwl} = (pg(1)/4"tr)(pg(2)/4~r)g(1, 2)
trw---~ ~ I. pw--~O
x exp(c(0/1; 2)).
(6)
When the first limit i~ (6) is taker, the only particle of species w, remaining in the development, is denoted by 0; consequently, g becomes the PDF for pure, homogeneous dipolar system. Taking the second limit, (pg~,j(O, i)/4v) gives the OPDF, p(i), i = 1 cr 2, tor a dipolar fluid in front of an ideal wall. Thus the PDF for a dipolar fluid against a hard wall may be written as g(1; 2) = g(1, 2) exp(c(O / 1; 2)).
(7)
For the sake of simplicity, the origin of the coordinates is chosen in such a way so that the accessible volume coincides with the half-space z ~ 0. The comma and the semi-colon between the arguments are used to distinguish the assumed known homogeneous system property and its analogue for the system under consideration. The function c is given by c ( O / 1 ; 2 ) = s u m of all distinct connected diagrams free o1~ articulation points; consisting of: three non-adjacent root points, O, of species w, the two others, 1 and 2, of species d, (p/4rr) field points, and fdd and fwd bonds, such that the three root points do not form an articulation triple
(8) In this development, the root and the field points are denoted r~y the white and
M.-E. Boudh-Hir / Pair distribution fimction
636
the black circles, respectively. The solid and the dashed lines represent the f, ld and the f*,~ bonds. These functions are related to the pair-pontentials by
Ad(i, j)= exp(--flVdd(i, j ) ) - 1,
(9a)
fwd(O, i ) = exp(-[3vwe(O, i ) ) - 1.
(9b)
Before examining the more general case, let us consider a dilute and moderately polar fluid, and look for the asymptotic behaviour of the PDF. In this way, we will gain some insight as how to implement the study of the general case.
3. Dilute fluids At low density and for weak dipolar moment, the asymptotic behaviour of g(1;2) in (7) is given by the contributions of its first graphs in which fdd is replaced by ~b defined as follows:
~b(i, j ) = - 3 w ( i , j ) .
(lo)
Thus g(l; 2) at the first order in density becomes g(1; 2) = 1 + h(1;2) (11) The dashed cir"le represents the quantity (pf*(O, i)/4-rr). The contribution, i2(1,2), of the second graph in (11), can be deduced from the calculations done in homogeneous systems (see for instance ref. [9]). Introduciag y = (4"rr3pl.t2/9), we get i2(1,2) = - Y 6 ( 1 , 2 ) + (4~r13y/3)6(r,2)~ , • #2.
(12)
The arising singularity has no contribution, at low density, to the asymptotic behaviour of the TCF. We will see that, as in the case of homogeneous systems [9], this term becomes important when the density ~ncreases. The contribution of the third graph, i3(1;2), i:; calculated using the two dimensional Fourier transform:
637
M.-E. Boudh-Hir I Pair distribution fi, nction
i3(l;2)=(-p/4rr) f
d3(h(l.3),b(3,2)
z3
X f dz 3 f d~{'~3 ~(1, 3)~(3, 2) ,
(13)
in which d g ] 3 refers to the solid angle element characterizing the orientation of the third dipole. K and R~z are two dimensional vectors; and ,~ denotes the two dimensional Fourier transform of 4'. It can be written as [16] 4,(i, j ) = / ~ , "
{
(-2"tr~/K)exp(-K]zj-
x (ilK, K sgn(zj -
z,I)
zi))t(-iK , - K
+ ( 4 r r / 3 / 3 ) 6 ( z j - z~)
1 0
s g n ( z j - z~)) • 0tj.
(14)
-
Here an x-row vector and its transposed x-column vector are denoted by x and x t respectwely. It is clear that the two particles 1 and 2 play a symmetrical role and then z~ may be assumed greater than or equal to z 2. By integration over z 3 and g2~, i s ( l; 2) becomes
i3(1;2)=(3y/Z){(llZ~)2 f
dK e x p ( - i1¢ - R~2)(-2rr~/K
)
x e x p ( - K(z I + z2))#,-(iK,= 3y~b(1, 2 " ) / 2 .
(15)
Here 2* is the image of the dipole 2 (i.e., if the dipole 2 is characterized by its moment la2 =(tz2,, ~2,,, g 2 : ) a n d its position vector r 2 = (x 2, y:. z2), then 2* is defined by its moment 0t* = (/z2~, ix2,, -ta~:) and its position r*_,= (x2, Y2, - z2)).
Inserting (12) and (15) in (11), the asymptotic oehaviour of ~.he TCF is given by
h(i; 2) r~7._.,:~( 1 -- y ) ~ ( l, 2) + 3y4~( 1 , 2 * ) / 2 .
(,16)
!n this equation, the first term in the right..hand side is the exact limit of TCF for homogeneous dipolar system, h ( l , 2 ) at low density and for large dis-
638
M.-E. Boudh-Hir / Pair distribution function
tances. One obtains lira f lim h(1, 2)} = lim ((e - 1)/3y)(4~(1,2)/e).
p-'*O I, r12-'~°
p--~O
(17)
The second one is in good agreement with the well known image potential (see for instance ref. [17]). Indeed, we have lim ( ( e - 1)/e(e + 1))~b(1, 2") = 3y~b(1, 2")/2.
p'-*O
(18)
At low density, the wall action consists of the arising of the dipole-image interaction.
4. Dense fluids
We now turn to the general case of dense polar fluids. At high density, other diagrams must be taken into account and the pair-potential should be replaced by a renormalizcd interaction. Several methods may be used. Here, the dipole-dipole and the wall-dipole f functions will be eliminated in favour of the TCFs, h, in homogeneous dipolar fluid and its analogue wall-dipole, h*. For simplicity, the indiccs dd and wd will be omitted. Therefore the function c in (8) will be
c(O/1; 2 ) = sum of all the distinct connected diagrams free of: articulation points, and pair of articulation points; consisting of: three non-adjacent root points (0, 1 and 2), p/4~ field points, h and h* bonds, such that the root points do not form an articulation triple (19) The solid and th : dashed lines denote the h and the h* bonds respectively. Following Hoye nncl Stoll [Cll t h o TCF ;- r,,,,~,~.,.,,,.~,,,~ n;..~,~.. ~.,:.~ .... ,~_~ :~ introduced as
p6(i,
j)/4rr +
(p/4.tr)Zh(i, j)
= P(i, j) + ( {pS(i, k)/4w + (p/4~r)2h(i, k)}tb(k, l)P(l, j) dk dl. 3
(20)
P(i, j) is given by the sum of all the TCF's graphs non-singly connected with th
M.-E. Boudh-Hir 1 Pair distribution .hmction
639
bonds and so, it decreases at least as the square of the dipole-dipole interaction, h(i, j) may be developed in tt~c following form:
h(i, j ) = C(i, j) + s(i, ]) + ' . ' ,
(21a)
C(i, j)= Adp(i, j ) ,
(21b) ^
s(i, J) = (E/4~r)8(r#)~t, . lai .
(21c)
The other components in this expausion are regular terms decreasing, for large distances, faster than the dipolar potential. The coefficients A and E are given by
A
=
((e - 1)/3y)Z/e,
E =4~(B + 2 z ( e - 1)2/3ey)/p,
(22)
(23)
B and z being related to s by the following equation: ( e - 1)/(e + 2) = y(1 + B/3)
(24) Using eq. (20), the wall-dipole TCF can then be written as
h*(i) = f * ( i ) + (p/4rr) f dk f*(k)h(k, i) .
(25)
The argument 0, denoting the wall, is omitted. This function is - 1 inside the wall, and a regular function outside it, having a Y,ong range component going as (a + bcos 20i)/z ~ as zi goes to infinity [18]. Now we have to select the graphs giving a long range contribution to the TCF.
4. I. Long rmlge contribution graphs Since we are interested in the long range components of h(1; 2), the asymptotic behaviour of h(1, 2) being known, we should classify the graphs of c. The y-ordering procedure [10, 19-21] can be used. We have indeed h ( # i, k r , , r u ) = y~h(#,, ~j, 7r u) + C(T").
t"'6}
M.-E. Boudh-ltir I Pair distribution function
640
The use of cq, (25) allows us to express h* as a function of h and then to classify the graphs of c in (19). Considering a given diagram of c, and let m and n be the numbers of its h bonds and (p/4rr) field points, respectively. The order of this diagram contribution, when only h's regular part is retained, is 7 3tin-,). Therefore the contribution of this part to the asymptotic behaviour of c is given by the graphs for which m - n = 1 (i.e., the chain graphs of h bonds having their field points connected to the wall by f* bonds). Only the step part of h* contributes to the asymptotic behaviour of the TCF wall-dipolar fluid systems. Concerning the singular part of h, we have to examine the convolution of s with C in (21), i.e.,
f dk s(i, k)(ph*(k)/4-a)C(k, j) = A(E/4~) f dk 6(r,k)~ , • ~tk(ph*(k)/4~)dp(k, j).
(27)
Two different cases should be considered: First, if the point i is a field point, it should be inside the wall. Indeed, in the opposite case, due to the decrease of h*(i), the convolution of the result of (27) with a C function will not give a long range contribution. Taking into account this condition, eq. (27) leads to
f dks(i, j)(ph*(k)/4~r)C(k, j)=(-pE/12~r)C(i, j).
(28)
Consequently, in this case, an s bond can be considered as a field point weighted by ( p El* / 12~). Second, if the point i is a root point (1 for example), it must be outside the wall. Because of h*(k) and s(i, k) in (27), the particle 1 should be on the surface, to obtain a long range term (i.e., particle 1 is adsorbed). In this case, the position (outside the wall) of the particle i (which is identical to 1) is not so important, because we do not have to integrate with respect to its z-coordinate. Using the fact that h*(i) may be expanded in spherical harmonics,
h*(i)= h*(z~, O~) -:C
= ~ h>,(..ilY:,,.,, t l = l}
eq. (27) now takes the form
(29)
M.-E. Boudh-Hir / Pair distribution function
fdks(l,k)(ph*(k)/4rOC(k, 'I = (E/4"a') ~'~
dk~(r,k)~
641
j) , • ~k{Ph2*(Zk)Yz,,.o(Ok,
q~k)/4~r}C(k, /).
n=O
(30)
Because of the orthogonality properties, only the two first terms in (29) give a non-zero contribution. Using the fact that particle 1 is adsorbed and thus it may be assumed having the same position as its image (but not the same moment vector), eq. (30) gives after an elementary manipulation:
fdks(1, k)(ph*(k)/4~r)C(k, j) =
(pE/12ar)(4~r) -'/2 {[hg(zt)
-
h*(zt)/lOlC (1, 2) (31)
+ 3h*(z~)C(l*,2)/lO}.
It follows that the total long range contribution to c ( 0 / l ; 2) is given by the chain graphs of h bonds and field points weighted by (ph*/4rr). From the above calculations, h(i, i) can bc replaced by its two first comps, nents in (21); while f*(i) is substituted for h*(i) if i is not connected to one of the root points (1 or 2) by s bonds. The field points connected to 1 or 2 by s bonds are outside the wall and, because of the symme~i v of h*, only thc two first terms in the development (29) contribute. Now a single chain of h bonds will give chains of C a n d / o r s bonds. It is then useful to find the equivalent s* of the chains of s bonds defined as:
s*(i, j) = s(i, j) + f dk s(i, k)(ph*(k)/4~r)s(k, j) + f dk dis(i, k)(ph*(k)/4~r)s(k, l)(ph*(l)/47r)s(l, j) + " " t"
= s(i. j) + ) dk s*(i, k)(ph*(k)/4~r)s(k,
j),
(32i
in which, for the reasons evoked above h* is replaced by its step part if the s bond chains do not contain a root point (1 or 2). In the opposite case, the field points are outside the wall and h* must be replaced by its two first components. Two different terms are then obtained. Let us denote them by s~ and s* respectively.
M.-E. Boudh-Hir / l'air distribution function
642
4.2. Sum o f the s bond chains We first consider the case of chains containing no root point. It is easy to prove that the form taken by s~(i, ]) in (32) can be expressed as
s~(i, ])=
(-oE/12"tr)"s(i, j) t! : 0
= (1/(1 + p E / 1 2 ~ ) ) s ( i , ] ) .
(33)
If the point i is identical to one of the root points (e.g., 1), the form taken by eq. (32) could be quite different from (33) for two reasons: the solution depends on the contact value of the wall-particle correlation function which is an anisotropic function. Therefore some modifications in the symmetry as in (31) should be expected. For instance, the second term of this series is given by f d4 s(1,4)(ph*(4)/4rt)s(4, 3) = (p/4"tr)(E/4rr) 2
'f
d4 6(r,4)Ii
, •
m(h2.(z4) L.,,.,,(O~, ~P4) ) a ( r ~ , )
^ /L4" #3,
rl = 0
(34)
which can be written in the form ¢,
i d4 s(1,4)(ph*(4)/4~r)s(4, 3) - (pE/47r)(4~r)- '"" { [ h i*, ( z 3 ) / 3 - V-5h*(z3)/3Ols(1,3 ) + [V'Sh~z3)/lOls(l*,3)}.
(35)
One can note the appearance of the moment of the dipole 1 image. It is clear that the only function that one can obtain is s. Evidently, s can depend on the moment of the dipole 1 or on its image moment. We therefore anticipate the solution, s*l, of eq. (32) in the form s T ( l , 3) = p i z ~ ) s ( 1 , 3 )
+
q(z,)s(l*, 3~.
(36)
Inserting (36) in (32), we obtain p ( z , ) = 1 + (pE/12w)(4~r) -''~-lhi,(z3) * -
V-Sh*(z3)/lO]p(z3)
+ (pEI4"r;)(4"rr)I~-[x/3h*,(z~)/lO]q(z:,),
(37a)
M.-E. Boudh-Hir / Pair distribution fimction
64?
q(z~) = (pE/12rr)(4~)-"'-[h~(za)- V~h~;(z~)/lOlq(z,) + (oE/4~)(4"tr)-"Z[V~h~(z3)/10lp(z3).
(37b)
Therefore the solutions of this system can be written as
p(z3) = {I1
(pE/12"tr)(4~)
-
-~'2
* - 3) (ho(,:
V~h*E(Z3)/lO)l"
/ 10} -~
-- ( p E / 4 ~ ) 2 ( a , r t ) - l / 2 V r ' 5 h * ( z 3 )
x {1 -
-
(pE/12~r)(4~r)-~'2[h~(z3)-V3h*,.(z3)/lO]},
(38a)
q(z3) = (pE/4'rr){ 1 - (pE/12'rr)(4"rr)-'/2[h~(z3) - V5h*:(z3)/lOl}- 'p(z~) . (3Sb)
4.3. Calculations of the c asymptotic behaviour In agreement with eqs. (28) and (31), the long range part, I( 1; 2), of c( 1; 2) is given by the sum of the chain graphs consisting of C and s q] bonds, and ph*/4at field points, such that each chain which may or not be ended with s7 bonds has, at least, two bonds; one of them is of type (_ (h* being approximated as above). Now according to (28) and (33), an s~ bond can be replaced bv a field point weighted by (pEf"/12~)/(1 + pE/12w). Therefore, I(1; 2) will be simply siven by l ( 1 ; 2 ) = s u m of the chains graphs consisting of: C bonds and ( p f * / 4"rr)[1 + E/3(1 + pE/12ar)] field points, such that each of these chains, which may or not be ended with s*~ bonds, should have at least two bonds = (~-~-o
+
~
+ c~-~-~-~
+
• • .) + ( ~,z
+ ~ z - o
+
• • •
• .. + o--~o+ c,~--~-~z + . - - ) + (o-~--~-~z+ o--,~-~--~¢ + ---) I
-_-
i
1o(1,
2t) X
4
.
~-) - r
fll
I . " ~ +
n. t ~
:
~ .
I,, I~ and 1: refer, respectively, to the sum of graphs with zero, one and t\~o s bonds on their ends. The solid and the dashed lines represents the C and the s ~ i bonds, respectively. The dashed and the crossed circles are the field points having the respective weights (pf*/4~r)[1 + E/3(1 + pE/12rr)] and ph*/4v. It is clear that the three components of I in (39) satisfy the follov~ing integral equations"
M.-E. Boudh-Hir / Pair distribution function
644
1~(1; 2) = (d3sT(l,3)(ph*(3)/4~r)[C(3,2) + 1o(3;2)1 + f d3 [C(1, 3) + lo(1;3)](oh*(3)/4"tr)s'~(3,2 ) ,
(40a)
/2(1;2) = f d3d4 s";(1,3)(oh*(3)/4~r)[C(3, 4) + •0(3; 4)1 x (ph*(4)/4ar)sT(4 , 2).
(40b)
Let us start by the calculation of the component I o. The contribution of the 2) may be deduced from (15) in which/z 2 is first graph of this series, t"t~)(1; o replaced by A/z"-- and y by Aey, where e is [1 + E/3(1 + pE/12~)], t.~t)(1;2 o ) becomes
itot)(1; 2) = 3A2ey$(1; 2*)/2 =3AeyC(1;2*)/2.
(41)
The contribution of the other graphs are calculated by induction. The two dimensional Fourier transform is used. One can prove that the nth element of this series (i.e., which has n field points) contributes
i(,,) ,, tt l ; Z ) = 3 ( A e y / 2 ) " C ( 1 ; 2 * ) .
(42)
Therefore, lo(1; 2) is given by Io(1;2) = ~] 3(Aey/2)~C(1,2 *) n=l
= [ 3 A e y / ( 2 - Aey)]C(1,2*) .
(43)
Because of the nature of the bonds reaching the root points (C), the particles 1 and 2 can be far from the surface. I o describes two non-adsorbed particles while I~, that we will calculate now, is relative to two adsorbed-non-adsorbed t ...........
...,~,,,~
1~(1; 2) =
~..,,,!
tt~u
~-rJl,
,~t. t .
l,"rvoj
U~k.Ullt~,
f 03 [p(z3)s(1, 3) ~ q(z3)s (1., 3)l(oh*(3)/4~r) x [C(3, 2) + ( 3 A e y / ( 2 - Aey))C(3,2*)l + f d3 [C(1,3) + ( 3 A e y / ( 2 - Aey))C(1,3*)l(ph*(3)/4.rr ) x [p(z3)s(3,2 ) + q(z3)s(3,2*)l "
(44)
M.-E. Boudh-Hir / Pair distribution function
645
The calculations may be performed in the same manner as in (30). Taking into account the symmetry properties of dipolar potential (i.e., C(i, j*)= C(i*, j) and C(i*, ]*)= C(i, ])) 1~ can then be written: I , ( 1 ; 2 ) = a,(z,, z2)C(1,2 ) + b,(7.1, zz)C(1,2*) ,
(45)
where the functions a, and b ! are defined by:
a,(z,, z2) = (oE112"~)(4~) -'`2 x { p ( z , ) ( h ~ ( z , ) - h~(z,)) + p(z2)(h;(z2) - h~(z2) ) + 3 q ( z , ) h * ( z , ) / l O + 3q(zz)h~(z2)/lO + ( 3 A e y / ( 2 - aey))[q(z,)(h~(zl) - h*(z,)) + q(Zz)(h;(z2)- h~(z2) ) + 3p(z,)h*(z,)/lO + 3p(z2)h
(z2)/lOl},
(46a)
b , ( z , , z2) = (pE[12"rr)(4"rt) -1/2
× ( q ( z , ) ( h ~ ( z , ) - h~(z,))+ q(z._)(h;(z.)- h~(z.)) + 3p(z,)h2(z,)/lO+ 3p(z2)h*.(..2)/1() , * + (3Aey/(2 - A e y ) ) [ p ( z , ) ( h ; ( z z ) - h*._(z,)) + p ( z 2 ) ( h ; ( z 2 ) - h~(z2) ) + 3q(z,)h~(z,)/lO + 3q(z2)h~(zz)/lO]}.
(46b)
Let us now consider the last term in which intervene two-adsorbed particles. Inserting (36) and (43) in (40b), 12 becomes f I_.(1; 2) = J d3d4[p(z3)s(1.3 ) + q(z3)s(l*.3)](ph~'(3)/4v) x [C(3, 4) + ( 3 A e y / ( 2 - Aey))C(3, 4*)](ph*(4)/4v)
x [p(z4)s(4.2)+ q(z4)s(4.2*)].
(47)
Using the same metho( as before, L_ takes the form
12(1;2)=a2(z,,z2)C(1.2)+ b2(z,.z2)C(1.2*). in which the coefficients a2(z 1, z . ) and b ( z ~ . . ~ ) are given, rcspcct~ e l .
(48t b~
M. -E. Boudh- Hir I Pair distribution function
646
a,(~,, z,) = (pEl12n)*(1/47~) X {i&h)
- h;(z,))(h;r(z,) - h;(Z2))
x WJPW
+ dm(z*))
+ 3(Pwd~,) x
+
P%)vm~*)
&,)&2))
-
W2))
+
w&)(4(q)
+ 9h~(z,)h~(z*)(P(z,)P(z,)
+ q(q)q(z,))/lOO
+ (3Aey@
- h;(q))@;@,)
- Aey))[(h;(z,)
x
Moh2)
+
3(PWP(Z,)
+
h;(z*))]llO
-
- h;(z,))
dZ,)P(Z*))
+ dz,)q(z,))
x [h:(z,)(h;(z,)
- 4~2))
+ gmz,
P(Zl
)W*)(
)dz*)
+ h;(z,)(h;(z,) +
q(q
-
)p(z2>
h;(z,))]llO
/loo)]}
)
(49)
b,(z,, z2) = (pE/12$*(1/4n) x {(P(Z&(Z*)
+ &)P(z*))vG(~,)
x
vG(z2)
+
3( P(z* )P(Z*) + &)dz,
-
- W,))
WZ2))
x [h;(q)@;(q)
- hT(z,)) _ .
+9(P(z,)&*)
)) + h;(z,)(h&)
- h;(z,))]/lo
+ ~~~l)P(~,))~~~~)~~(~)~~~~ - Aey))K Pk, )P(Z*)
+ Weyl(2
+ dz,h(z,))(hXz,)
+3(P(z,)dz*)
- W,NVG(z,)
+ dZ,)P(Z*))
X [hjl:k)(h$(zz)
+%PWP(Z,)
- W2))
- W2))
+ h;(z,)(h$(z,)
- h;(q))]/10
+ q(~,)q(z3))~~~(z~)~~T(Z,)/loo]}. _
(50)
According to eqs. (7’), (21), (43) and (48) the TCF has therefore the following asymptotic behaviour: h(1;2)
‘l’-X C&2)+
1,,(1;2)+
= (1 + a,(+,
z2) + a,(z,,
I,(l;2)+
Z*(1;2)
z2j)C( 1,2)
‘((3Ae~f(2_Aey))+b,(z,,z,)+
b2(zl,~2))C(1,2*j.
(51)
M.-E. Boudl~-Hir / Pair distribution function
647
The functions a~ and b~ are non-zero only if, at least, one of the particles is on the surface. These two particles must be adsorbed to have non-zero functions a 2 and b 2.
5. Concluding remarks
The purpose of this paper has been to examine the TCF asymptotic behaviour, in dipolar fluid, in front of a hard wall. It has been shown that" First, at low density this function contains two long range terms. The first, characterizing the homogeneous phase, is proportional to the dipolar potential• The second is due to the wall and has the symmetry of the dipole-image interaction. Second, for dense fluids, two other terms arise. They describe the adsorbedadsorbed and adsorbed-non-adsorbed particle correlations. The symmetries that one can have are those of dipole-dipole and dipole-image potentials• In the two first components, only the wall-particle steric-part correlation contributes (i.e., h* can be replaced by f * ) . The other part gives a non-zero contribution to the last terms when, at least, one particle is adsorbed. Consequently, when the hard wall is replaced by a soft one, only 11 and 12 should be sensitive to this change. In this case, the wall-particle pair distribution function tends continuously to zero in the neighborhood of the wall, and so it is reasonable to expect that the magnitude of these functions decreases.
Acknowledgements
The financial support and the facilities provided by ICTP, Trieste (Italy), are thankfully acknowledged.
References ~.
Vll~d~l,,71
•
./'-~,IIII.
LII~.III
.
d~.r~.
.rL;
X, I 1 . . ~ r ]
.~r~..
I21 J.G. Kirkwood, J. Chem. Phys. 4 (1936) 592. [3] J.G. Kirkwood, J. Chem. Phys. 7 (1939) 911. [41 J. Yvon, Actualites Scientifiques et Industrieiles No. 543 (Hermann, Paris, 1937). [5] D.W. Jepsen and H.L. Friedman, J. Chem. Phys. 38 (1963) 846. [61 D,W. Jepsen, J. Chem. Phys. 44 (1966) 774. [7] D.W. Jepsen, J. Chem. Phys. 45 (1966) 709. [8] G. Nienhuis and J.M. Deutch, J. Chem. Phys. 56 (1972) 1819. [9] J.S. H0ye and G. Stell, J. Chem. Phys. 61 (1974) 562. [10] G. Stell, in" Modern Theoretical Chemistry, B.J. Berne, cd. (Plenum, New York, 1977).
648
[Ill [121 [la] [14]
M.-E. Boudh-Hir I Pal, distributioa function
J.D. Ramshaw, J. Chem. Phys. 55 (1971) 1763. J.D. Ramshaw, J. Chem. Phys. 66 (1977) 3134. J,D. Ramshaw, J. Chem. Phys. 70 (1979) 1577. G. Stell, in: The Equilibrium of Classical Fluids, H.L. Frisch and J.L. Lebowitz, eds. (Benjamin, New York, 1964). [151 D. Henderson, F.F. Abraham and J.A. Barker, Molec. Phys. 31 (1976) 1291. [161 A,L. Nichols Ill, Ph.D. Thesis (University of California, Berkeley, 1983). [171 L. Landau and E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1969). 118] M.-E. Boudh-Hir, Physica A 158 (1989) 619, this volume. [191 P.C. Hemmer, J. Math. Phys. 5 (1964) 75. [201 J.L. Lebowitz, G. Stell and S. Baer, J. Math. Phys. 6 (1965) 1282. [21] H.C. Andersen (See ref. [10]).
PAIR DISTRIBUTION FUNCTION IN POLAR FLUID-HARD WALL SYSTEMS. LONG RANGE COMPONENTS M.-E. B O U D H - H I R International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy Received 24 January 1989
Using the diagrammatic technique we prove that the total correlation function (TCF) for dipolar fluid systems in front of an ideal wall has four long range components. The first term is identical to the long range component of the TCF in homogeneous systems. The second is a longitudinal term characterizing two non-adsorbed particles. It has the same symmetry as the dipole-image interaction. The third describes the correlation between adsorbed and nonadsorbed particles. The last one is a transversal term; it is relativ÷, to two adsorbed particles. One can note that the arising of the dipole-image interaction symmetry without introducing this potential in the Hamiltonian.
1. Introduction Considerable efforts have beem made to understand polar fluid systems [1-7]. In homogeneous systems, many important thermodynamic and structure properties are now well known. For example, in dipolar fluid the TCF's asymptotic behaviour is proportional to the dipole-dipole interaction [8, 9]. The purpose of this paper is to examine the long range components of the TCF, in dipolar fluids, in front of a structureless hard wall. The long range component is used here in the sense that it decreases as the dipole-dipole interaction. In homogeneous systems, the only distance intervening is the distance -_,---_.-o.. :_ .L.. "-~ "long between the two particles and so there ib no amu~gmty., t,,,- meaning ... range component". In the presence of a wall. the knowledge of this parameter only is not sufficient and consequently the positions of the particles with respect to the surface should be given. Using the statistical mechanics concepts, we Frove that, for dilute dipolar fluids, the TCF contains two long range components which have, respectively. the symmetry of the dipole-dipole and dipole-image interactions. Away from the low density limit, we find four long range terms. The first characterizes the 0378-4371/89/$03.50 © Elsevier Science Publishers B.¥: (North-Holland Phvsics Publishin~ Division)
634
M.-E. Boudh-tlir / Pair distribution function
homogeneous fluids. The second, having the symmetry of the dipole-image interaction, is related to the correlation between two non-adsorbed particles. The third gives the correlation between adsorbed and non-adsorbed particles. The last one is a transversal term describing two adsorbed particles. It should be noted that dipole-image potential symmetry arises, though this interaction has not been introduced a priori in the Hamiltonian.
2. Modelling and general developments The dipolar fluid-wall system is studied as a limit case of a binary mixture. We consider two species of particles denoted by w (wall) and d (dipole). The first has a hard core diameter trw and a density p,,,. The parameters characterizing the second one are o- and p. In this system the particles are assumed to be interacting by the following pair-potentials: Vww(i, j ) = vwa(i, j ) = yaw(i, j) = v*(rij ) .
(1)
Vdd(i, j)= v*(rij ) + w(i, j).
(2)
Here, v* denotes the hard sphere potential, and w designates the dipolar part of the interaction dipole-dipole. They are given, respectively, by the equations for r~j ~<(o'i + % ) / 2 . (3) for % > (o-~ + o"i)/2,
w(i, j)= ( - ix"/rij)tt(r # - 6 )(3~i - r#pj- P i j - / t i •/Ttj),
(4)
where tr, is the hard core diameter of the particles of species i;/2; and ~;j are unit vectors, H is the Heaviside step-function (i.e., H(x)= 1, for x > 0 and 0 otherwise), and 6 is a cutoff parameter. A .,i~dioi,a,,~_ ._....,~. choice• of the ,.,~u,,""'""~parameter can accelerate the convergence of the calculations and make them easier to perform. Because of the hard core potential, the value of the dipolar interaction inside the core does not modify the physical properties of the system [9]. Nevertheless, the correlation functions may be sensitive to the particular cutoff-prameter choice when an approximation is introduced. This is due to the fact that the phase space includes all the space regions (including the interior of the core) and to the coupling via the Ornstein-Zernike equation [10]. One therefore takes 6 going to zere [11-13]. This choice will be adopted here.
M.-E. Boudh-Hir / Pair distribution function
635
In this system the probability to find, at the same time, three particles, the first of species w, the two others of species d, at the points 0, 1, and 2 respectively can be written as pwdd(O, 1, 2 ) = pw(P/4"rr)2gwdd(O, 1, 2)
= pw(P/4"n')2gwd(O, 1)gwd(O, 2)g(1,2) exp(c(O, 1, 2)).
(5)
gwd and g are the hard sphere-dipole and the dipole-dipole pair distribution functions (PDF) respectively; and exp(c(0, 1, 2)) is a coupling term. Its graphical expansion is given by Stell [14]. As in the case of the one-particle distribution function (OPDF) [15], the limit fluid in front of a hard wall is obtained using the following procedure:
lim [ lim [pwdd(O, 1,2)/Pwl} = (pg(1)/4"tr)(pg(2)/4~r)g(1, 2)
trw---~ ~ I. pw--~O
x exp(c(0/1; 2)).
(6)
When the first limit i~ (6) is taker, the only particle of species w, remaining in the development, is denoted by 0; consequently, g becomes the PDF for pure, homogeneous dipolar system. Taking the second limit, (pg~,j(O, i)/4v) gives the OPDF, p(i), i = 1 cr 2, tor a dipolar fluid in front of an ideal wall. Thus the PDF for a dipolar fluid against a hard wall may be written as g(1; 2) = g(1, 2) exp(c(O / 1; 2)).
(7)
For the sake of simplicity, the origin of the coordinates is chosen in such a way so that the accessible volume coincides with the half-space z ~ 0. The comma and the semi-colon between the arguments are used to distinguish the assumed known homogeneous system property and its analogue for the system under consideration. The function c is given by c ( O / 1 ; 2 ) = s u m of all distinct connected diagrams free o1~ articulation points; consisting of: three non-adjacent root points, O, of species w, the two others, 1 and 2, of species d, (p/4rr) field points, and fdd and fwd bonds, such that the three root points do not form an articulation triple
(8) In this development, the root and the field points are denoted r~y the white and
M.-E. Boudh-Hir / Pair distribution fimction
636
the black circles, respectively. The solid and the dashed lines represent the f, ld and the f*,~ bonds. These functions are related to the pair-pontentials by
Ad(i, j)= exp(--flVdd(i, j ) ) - 1,
(9a)
fwd(O, i ) = exp(-[3vwe(O, i ) ) - 1.
(9b)
Before examining the more general case, let us consider a dilute and moderately polar fluid, and look for the asymptotic behaviour of the PDF. In this way, we will gain some insight as how to implement the study of the general case.
3. Dilute fluids At low density and for weak dipolar moment, the asymptotic behaviour of g(1;2) in (7) is given by the contributions of its first graphs in which fdd is replaced by ~b defined as follows:
~b(i, j ) = - 3 w ( i , j ) .
(lo)
Thus g(l; 2) at the first order in density becomes g(1; 2) = 1 + h(1;2) (11) The dashed cir"le represents the quantity (pf*(O, i)/4-rr). The contribution, i2(1,2), of the second graph in (11), can be deduced from the calculations done in homogeneous systems (see for instance ref. [9]). Introduciag y = (4"rr3pl.t2/9), we get i2(1,2) = - Y 6 ( 1 , 2 ) + (4~r13y/3)6(r,2)~ , • #2.
(12)
The arising singularity has no contribution, at low density, to the asymptotic behaviour of the TCF. We will see that, as in the case of homogeneous systems [9], this term becomes important when the density ~ncreases. The contribution of the third graph, i3(1;2), i:; calculated using the two dimensional Fourier transform:
637
M.-E. Boudh-Hir I Pair distribution fi, nction
i3(l;2)=(-p/4rr) f
d3(h(l.3),b(3,2)
z3
X f dz 3 f d~{'~3 ~(1, 3)~(3, 2) ,
(13)
in which d g ] 3 refers to the solid angle element characterizing the orientation of the third dipole. K and R~z are two dimensional vectors; and ,~ denotes the two dimensional Fourier transform of 4'. It can be written as [16] 4,(i, j ) = / ~ , "
{
(-2"tr~/K)exp(-K]zj-
x (ilK, K sgn(zj -
z,I)
zi))t(-iK , - K
+ ( 4 r r / 3 / 3 ) 6 ( z j - z~)
1 0
s g n ( z j - z~)) • 0tj.
(14)
-
Here an x-row vector and its transposed x-column vector are denoted by x and x t respectwely. It is clear that the two particles 1 and 2 play a symmetrical role and then z~ may be assumed greater than or equal to z 2. By integration over z 3 and g2~, i s ( l; 2) becomes
i3(1;2)=(3y/Z){(llZ~)2 f
dK e x p ( - i1¢ - R~2)(-2rr~/K
)
x e x p ( - K(z I + z2))#,-(iK,= 3y~b(1, 2 " ) / 2 .
(15)
Here 2* is the image of the dipole 2 (i.e., if the dipole 2 is characterized by its moment la2 =(tz2,, ~2,,, g 2 : ) a n d its position vector r 2 = (x 2, y:. z2), then 2* is defined by its moment 0t* = (/z2~, ix2,, -ta~:) and its position r*_,= (x2, Y2, - z2)).
Inserting (12) and (15) in (11), the asymptotic oehaviour of ~.he TCF is given by
h(i; 2) r~7._.,:~( 1 -- y ) ~ ( l, 2) + 3y4~( 1 , 2 * ) / 2 .
(,16)
!n this equation, the first term in the right..hand side is the exact limit of TCF for homogeneous dipolar system, h ( l , 2 ) at low density and for large dis-
638
M.-E. Boudh-Hir / Pair distribution function
tances. One obtains lira f lim h(1, 2)} = lim ((e - 1)/3y)(4~(1,2)/e).
p-'*O I, r12-'~°
p--~O
(17)
The second one is in good agreement with the well known image potential (see for instance ref. [17]). Indeed, we have lim ( ( e - 1)/e(e + 1))~b(1, 2") = 3y~b(1, 2")/2.
p'-*O
(18)
At low density, the wall action consists of the arising of the dipole-image interaction.
4. Dense fluids
We now turn to the general case of dense polar fluids. At high density, other diagrams must be taken into account and the pair-potential should be replaced by a renormalizcd interaction. Several methods may be used. Here, the dipole-dipole and the wall-dipole f functions will be eliminated in favour of the TCFs, h, in homogeneous dipolar fluid and its analogue wall-dipole, h*. For simplicity, the indiccs dd and wd will be omitted. Therefore the function c in (8) will be
c(O/1; 2 ) = sum of all the distinct connected diagrams free of: articulation points, and pair of articulation points; consisting of: three non-adjacent root points (0, 1 and 2), p/4~ field points, h and h* bonds, such that the root points do not form an articulation triple (19) The solid and th : dashed lines denote the h and the h* bonds respectively. Following Hoye nncl Stoll [Cll t h o TCF ;- r,,,,~,~.,.,,,.~,,,~ n;..~,~.. ~.,:.~ .... ,~_~ :~ introduced as
p6(i,
j)/4rr +
(p/4.tr)Zh(i, j)
= P(i, j) + ( {pS(i, k)/4w + (p/4~r)2h(i, k)}tb(k, l)P(l, j) dk dl. 3
(20)
P(i, j) is given by the sum of all the TCF's graphs non-singly connected with th
M.-E. Boudh-Hir 1 Pair distribution .hmction
639
bonds and so, it decreases at least as the square of the dipole-dipole interaction, h(i, j) may be developed in tt~c following form:
h(i, j ) = C(i, j) + s(i, ]) + ' . ' ,
(21a)
C(i, j)= Adp(i, j ) ,
(21b) ^
s(i, J) = (E/4~r)8(r#)~t, . lai .
(21c)
The other components in this expausion are regular terms decreasing, for large distances, faster than the dipolar potential. The coefficients A and E are given by
A
=
((e - 1)/3y)Z/e,
E =4~(B + 2 z ( e - 1)2/3ey)/p,
(22)
(23)
B and z being related to s by the following equation: ( e - 1)/(e + 2) = y(1 + B/3)
(24) Using eq. (20), the wall-dipole TCF can then be written as
h*(i) = f * ( i ) + (p/4rr) f dk f*(k)h(k, i) .
(25)
The argument 0, denoting the wall, is omitted. This function is - 1 inside the wall, and a regular function outside it, having a Y,ong range component going as (a + bcos 20i)/z ~ as zi goes to infinity [18]. Now we have to select the graphs giving a long range contribution to the TCF.
4. I. Long rmlge contribution graphs Since we are interested in the long range components of h(1; 2), the asymptotic behaviour of h(1, 2) being known, we should classify the graphs of c. The y-ordering procedure [10, 19-21] can be used. We have indeed h ( # i, k r , , r u ) = y~h(#,, ~j, 7r u) + C(T").
t"'6}
M.-E. Boudh-ltir I Pair distribution function
640
The use of cq, (25) allows us to express h* as a function of h and then to classify the graphs of c in (19). Considering a given diagram of c, and let m and n be the numbers of its h bonds and (p/4rr) field points, respectively. The order of this diagram contribution, when only h's regular part is retained, is 7 3tin-,). Therefore the contribution of this part to the asymptotic behaviour of c is given by the graphs for which m - n = 1 (i.e., the chain graphs of h bonds having their field points connected to the wall by f* bonds). Only the step part of h* contributes to the asymptotic behaviour of the TCF wall-dipolar fluid systems. Concerning the singular part of h, we have to examine the convolution of s with C in (21), i.e.,
f dk s(i, k)(ph*(k)/4-a)C(k, j) = A(E/4~) f dk 6(r,k)~ , • ~tk(ph*(k)/4~)dp(k, j).
(27)
Two different cases should be considered: First, if the point i is a field point, it should be inside the wall. Indeed, in the opposite case, due to the decrease of h*(i), the convolution of the result of (27) with a C function will not give a long range contribution. Taking into account this condition, eq. (27) leads to
f dks(i, j)(ph*(k)/4~r)C(k, j)=(-pE/12~r)C(i, j).
(28)
Consequently, in this case, an s bond can be considered as a field point weighted by ( p El* / 12~). Second, if the point i is a root point (1 for example), it must be outside the wall. Because of h*(k) and s(i, k) in (27), the particle 1 should be on the surface, to obtain a long range term (i.e., particle 1 is adsorbed). In this case, the position (outside the wall) of the particle i (which is identical to 1) is not so important, because we do not have to integrate with respect to its z-coordinate. Using the fact that h*(i) may be expanded in spherical harmonics,
h*(i)= h*(z~, O~) -:C
= ~ h>,(..ilY:,,.,, t l = l}
eq. (27) now takes the form
(29)
M.-E. Boudh-Hir / Pair distribution function
fdks(l,k)(ph*(k)/4rOC(k, 'I = (E/4"a') ~'~
dk~(r,k)~
641
j) , • ~k{Ph2*(Zk)Yz,,.o(Ok,
q~k)/4~r}C(k, /).
n=O
(30)
Because of the orthogonality properties, only the two first terms in (29) give a non-zero contribution. Using the fact that particle 1 is adsorbed and thus it may be assumed having the same position as its image (but not the same moment vector), eq. (30) gives after an elementary manipulation:
fdks(1, k)(ph*(k)/4~r)C(k, j) =
(pE/12ar)(4~r) -'/2 {[hg(zt)
-
h*(zt)/lOlC (1, 2) (31)
+ 3h*(z~)C(l*,2)/lO}.
It follows that the total long range contribution to c ( 0 / l ; 2) is given by the chain graphs of h bonds and field points weighted by (ph*/4rr). From the above calculations, h(i, i) can bc replaced by its two first comps, nents in (21); while f*(i) is substituted for h*(i) if i is not connected to one of the root points (1 or 2) by s bonds. The field points connected to 1 or 2 by s bonds are outside the wall and, because of the symme~i v of h*, only thc two first terms in the development (29) contribute. Now a single chain of h bonds will give chains of C a n d / o r s bonds. It is then useful to find the equivalent s* of the chains of s bonds defined as:
s*(i, j) = s(i, j) + f dk s(i, k)(ph*(k)/4~r)s(k, j) + f dk dis(i, k)(ph*(k)/4~r)s(k, l)(ph*(l)/47r)s(l, j) + " " t"
= s(i. j) + ) dk s*(i, k)(ph*(k)/4~r)s(k,
j),
(32i
in which, for the reasons evoked above h* is replaced by its step part if the s bond chains do not contain a root point (1 or 2). In the opposite case, the field points are outside the wall and h* must be replaced by its two first components. Two different terms are then obtained. Let us denote them by s~ and s* respectively.
M.-E. Boudh-Hir / l'air distribution function
642
4.2. Sum o f the s bond chains We first consider the case of chains containing no root point. It is easy to prove that the form taken by s~(i, ]) in (32) can be expressed as
s~(i, ])=
(-oE/12"tr)"s(i, j) t! : 0
= (1/(1 + p E / 1 2 ~ ) ) s ( i , ] ) .
(33)
If the point i is identical to one of the root points (e.g., 1), the form taken by eq. (32) could be quite different from (33) for two reasons: the solution depends on the contact value of the wall-particle correlation function which is an anisotropic function. Therefore some modifications in the symmetry as in (31) should be expected. For instance, the second term of this series is given by f d4 s(1,4)(ph*(4)/4rt)s(4, 3) = (p/4"tr)(E/4rr) 2
'f
d4 6(r,4)Ii
, •
m(h2.(z4) L.,,.,,(O~, ~P4) ) a ( r ~ , )
^ /L4" #3,
rl = 0
(34)
which can be written in the form ¢,
i d4 s(1,4)(ph*(4)/4~r)s(4, 3) - (pE/47r)(4~r)- '"" { [ h i*, ( z 3 ) / 3 - V-5h*(z3)/3Ols(1,3 ) + [V'Sh~z3)/lOls(l*,3)}.
(35)
One can note the appearance of the moment of the dipole 1 image. It is clear that the only function that one can obtain is s. Evidently, s can depend on the moment of the dipole 1 or on its image moment. We therefore anticipate the solution, s*l, of eq. (32) in the form s T ( l , 3) = p i z ~ ) s ( 1 , 3 )
+
q(z,)s(l*, 3~.
(36)
Inserting (36) in (32), we obtain p ( z , ) = 1 + (pE/12w)(4~r) -''~-lhi,(z3) * -
V-Sh*(z3)/lO]p(z3)
+ (pEI4"r;)(4"rr)I~-[x/3h*,(z~)/lO]q(z:,),
(37a)
M.-E. Boudh-Hir / Pair distribution fimction
64?
q(z~) = (pE/12rr)(4~)-"'-[h~(za)- V~h~;(z~)/lOlq(z,) + (oE/4~)(4"tr)-"Z[V~h~(z3)/10lp(z3).
(37b)
Therefore the solutions of this system can be written as
p(z3) = {I1
(pE/12"tr)(4~)
-
-~'2
* - 3) (ho(,:
V~h*E(Z3)/lO)l"
/ 10} -~
-- ( p E / 4 ~ ) 2 ( a , r t ) - l / 2 V r ' 5 h * ( z 3 )
x {1 -
-
(pE/12~r)(4~r)-~'2[h~(z3)-V3h*,.(z3)/lO]},
(38a)
q(z3) = (pE/4'rr){ 1 - (pE/12'rr)(4"rr)-'/2[h~(z3) - V5h*:(z3)/lOl}- 'p(z~) . (3Sb)
4.3. Calculations of the c asymptotic behaviour In agreement with eqs. (28) and (31), the long range part, I( 1; 2), of c( 1; 2) is given by the sum of the chain graphs consisting of C and s q] bonds, and ph*/4at field points, such that each chain which may or not be ended with s7 bonds has, at least, two bonds; one of them is of type (_ (h* being approximated as above). Now according to (28) and (33), an s~ bond can be replaced bv a field point weighted by (pEf"/12~)/(1 + pE/12w). Therefore, I(1; 2) will be simply siven by l ( 1 ; 2 ) = s u m of the chains graphs consisting of: C bonds and ( p f * / 4"rr)[1 + E/3(1 + pE/12ar)] field points, such that each of these chains, which may or not be ended with s*~ bonds, should have at least two bonds = (~-~-o
+
~
+ c~-~-~-~
+
• • .) + ( ~,z
+ ~ z - o
+
• • •
• .. + o--~o+ c,~--~-~z + . - - ) + (o-~--~-~z+ o--,~-~--~¢ + ---) I
-_-
i
1o(1,
2t) X
4
.
~-) - r
fll
I . " ~ +
n. t ~
:
~ .
I,, I~ and 1: refer, respectively, to the sum of graphs with zero, one and t\~o s bonds on their ends. The solid and the dashed lines represents the C and the s ~ i bonds, respectively. The dashed and the crossed circles are the field points having the respective weights (pf*/4~r)[1 + E/3(1 + pE/12rr)] and ph*/4v. It is clear that the three components of I in (39) satisfy the follov~ing integral equations"
M.-E. Boudh-Hir / Pair distribution function
644
1~(1; 2) = (d3sT(l,3)(ph*(3)/4~r)[C(3,2) + 1o(3;2)1 + f d3 [C(1, 3) + lo(1;3)](oh*(3)/4"tr)s'~(3,2 ) ,
(40a)
/2(1;2) = f d3d4 s";(1,3)(oh*(3)/4~r)[C(3, 4) + •0(3; 4)1 x (ph*(4)/4ar)sT(4 , 2).
(40b)
Let us start by the calculation of the component I o. The contribution of the 2) may be deduced from (15) in which/z 2 is first graph of this series, t"t~)(1; o replaced by A/z"-- and y by Aey, where e is [1 + E/3(1 + pE/12~)], t.~t)(1;2 o ) becomes
itot)(1; 2) = 3A2ey$(1; 2*)/2 =3AeyC(1;2*)/2.
(41)
The contribution of the other graphs are calculated by induction. The two dimensional Fourier transform is used. One can prove that the nth element of this series (i.e., which has n field points) contributes
i(,,) ,, tt l ; Z ) = 3 ( A e y / 2 ) " C ( 1 ; 2 * ) .
(42)
Therefore, lo(1; 2) is given by Io(1;2) = ~] 3(Aey/2)~C(1,2 *) n=l
= [ 3 A e y / ( 2 - Aey)]C(1,2*) .
(43)
Because of the nature of the bonds reaching the root points (C), the particles 1 and 2 can be far from the surface. I o describes two non-adsorbed particles while I~, that we will calculate now, is relative to two adsorbed-non-adsorbed t ...........
...,~,,,~
1~(1; 2) =
~..,,,!
tt~u
~-rJl,
,~t. t .
l,"rvoj
U~k.Ullt~,
f 03 [p(z3)s(1, 3) ~ q(z3)s (1., 3)l(oh*(3)/4~r) x [C(3, 2) + ( 3 A e y / ( 2 - Aey))C(3,2*)l + f d3 [C(1,3) + ( 3 A e y / ( 2 - Aey))C(1,3*)l(ph*(3)/4.rr ) x [p(z3)s(3,2 ) + q(z3)s(3,2*)l "
(44)
M.-E. Boudh-Hir / Pair distribution function
645
The calculations may be performed in the same manner as in (30). Taking into account the symmetry properties of dipolar potential (i.e., C(i, j*)= C(i*, j) and C(i*, ]*)= C(i, ])) 1~ can then be written: I , ( 1 ; 2 ) = a,(z,, z2)C(1,2 ) + b,(7.1, zz)C(1,2*) ,
(45)
where the functions a, and b ! are defined by:
a,(z,, z2) = (oE112"~)(4~) -'`2 x { p ( z , ) ( h ~ ( z , ) - h~(z,)) + p(z2)(h;(z2) - h~(z2) ) + 3 q ( z , ) h * ( z , ) / l O + 3q(zz)h~(z2)/lO + ( 3 A e y / ( 2 - aey))[q(z,)(h~(zl) - h*(z,)) + q(Zz)(h;(z2)- h~(z2) ) + 3p(z,)h*(z,)/lO + 3p(z2)h
(z2)/lOl},
(46a)
b , ( z , , z2) = (pE[12"rr)(4"rt) -1/2
× ( q ( z , ) ( h ~ ( z , ) - h~(z,))+ q(z._)(h;(z.)- h~(z.)) + 3p(z,)h2(z,)/lO+ 3p(z2)h*.(..2)/1() , * + (3Aey/(2 - A e y ) ) [ p ( z , ) ( h ; ( z z ) - h*._(z,)) + p ( z 2 ) ( h ; ( z 2 ) - h~(z2) ) + 3q(z,)h~(z,)/lO + 3q(z2)h~(zz)/lO]}.
(46b)
Let us now consider the last term in which intervene two-adsorbed particles. Inserting (36) and (43) in (40b), 12 becomes f I_.(1; 2) = J d3d4[p(z3)s(1.3 ) + q(z3)s(l*.3)](ph~'(3)/4v) x [C(3, 4) + ( 3 A e y / ( 2 - Aey))C(3, 4*)](ph*(4)/4v)
x [p(z4)s(4.2)+ q(z4)s(4.2*)].
(47)
Using the same metho( as before, L_ takes the form
12(1;2)=a2(z,,z2)C(1.2)+ b2(z,.z2)C(1.2*). in which the coefficients a2(z 1, z . ) and b ( z ~ . . ~ ) are given, rcspcct~ e l .
(48t b~
M. -E. Boudh- Hir I Pair distribution function
646
a,(~,, z,) = (pEl12n)*(1/47~) X {i&h)
- h;(z,))(h;r(z,) - h;(Z2))
x WJPW
+ dm(z*))
+ 3(Pwd~,) x
+
P%)vm~*)
&,)&2))
-
W2))
+
w&)(4(q)
+ 9h~(z,)h~(z*)(P(z,)P(z,)
+ q(q)q(z,))/lOO
+ (3Aey@
- h;(q))@;@,)
- Aey))[(h;(z,)
x
Moh2)
+
3(PWP(Z,)
+
h;(z*))]llO
-
- h;(z,))
dZ,)P(Z*))
+ dz,)q(z,))
x [h:(z,)(h;(z,)
- 4~2))
+ gmz,
P(Zl
)W*)(
)dz*)
+ h;(z,)(h;(z,) +
q(q
-
)p(z2>
h;(z,))]llO
/loo)]}
)
(49)
b,(z,, z2) = (pE/12$*(1/4n) x {(P(Z&(Z*)
+ &)P(z*))vG(~,)
x
vG(z2)
+
3( P(z* )P(Z*) + &)dz,
-
- W,))
WZ2))
x [h;(q)@;(q)
- hT(z,)) _ .
+9(P(z,)&*)
)) + h;(z,)(h&)
- h;(z,))]/lo
+ ~~~l)P(~,))~~~~)~~(~)~~~~ - Aey))K Pk, )P(Z*)
+ Weyl(2
+ dz,h(z,))(hXz,)
+3(P(z,)dz*)
- W,NVG(z,)
+ dZ,)P(Z*))
X [hjl:k)(h$(zz)
+%PWP(Z,)
- W2))
- W2))
+ h;(z,)(h$(z,)
- h;(q))]/10
+ q(~,)q(z3))~~~(z~)~~T(Z,)/loo]}. _
(50)
According to eqs. (7’), (21), (43) and (48) the TCF has therefore the following asymptotic behaviour: h(1;2)
‘l’-X C&2)+
1,,(1;2)+
= (1 + a,(+,
z2) + a,(z,,
I,(l;2)+
Z*(1;2)
z2j)C( 1,2)
‘((3Ae~f(2_Aey))+b,(z,,z,)+
b2(zl,~2))C(1,2*j.
(51)
M.-E. Boudl~-Hir / Pair distribution function
647
The functions a~ and b~ are non-zero only if, at least, one of the particles is on the surface. These two particles must be adsorbed to have non-zero functions a 2 and b 2.
5. Concluding remarks
The purpose of this paper has been to examine the TCF asymptotic behaviour, in dipolar fluid, in front of a hard wall. It has been shown that" First, at low density this function contains two long range terms. The first, characterizing the homogeneous phase, is proportional to the dipolar potential• The second is due to the wall and has the symmetry of the dipole-image interaction. Second, for dense fluids, two other terms arise. They describe the adsorbedadsorbed and adsorbed-non-adsorbed particle correlations. The symmetries that one can have are those of dipole-dipole and dipole-image potentials• In the two first components, only the wall-particle steric-part correlation contributes (i.e., h* can be replaced by f * ) . The other part gives a non-zero contribution to the last terms when, at least, one particle is adsorbed. Consequently, when the hard wall is replaced by a soft one, only 11 and 12 should be sensitive to this change. In this case, the wall-particle pair distribution function tends continuously to zero in the neighborhood of the wall, and so it is reasonable to expect that the magnitude of these functions decreases.
Acknowledgements
The financial support and the facilities provided by ICTP, Trieste (Italy), are thankfully acknowledged.
References ~.
Vll~d~l,,71
•
./'-~,IIII.
LII~.III
.
d~.r~.
.rL;
X, I 1 . . ~ r ]
.~r~..
I21 J.G. Kirkwood, J. Chem. Phys. 4 (1936) 592. [3] J.G. Kirkwood, J. Chem. Phys. 7 (1939) 911. [41 J. Yvon, Actualites Scientifiques et Industrieiles No. 543 (Hermann, Paris, 1937). [5] D.W. Jepsen and H.L. Friedman, J. Chem. Phys. 38 (1963) 846. [61 D,W. Jepsen, J. Chem. Phys. 44 (1966) 774. [7] D.W. Jepsen, J. Chem. Phys. 45 (1966) 709. [8] G. Nienhuis and J.M. Deutch, J. Chem. Phys. 56 (1972) 1819. [9] J.S. H0ye and G. Stell, J. Chem. Phys. 61 (1974) 562. [10] G. Stell, in" Modern Theoretical Chemistry, B.J. Berne, cd. (Plenum, New York, 1977).
648
[Ill [121 [la] [14]
M.-E. Boudh-Hir I Pal, distributioa function
J.D. Ramshaw, J. Chem. Phys. 55 (1971) 1763. J.D. Ramshaw, J. Chem. Phys. 66 (1977) 3134. J,D. Ramshaw, J. Chem. Phys. 70 (1979) 1577. G. Stell, in: The Equilibrium of Classical Fluids, H.L. Frisch and J.L. Lebowitz, eds. (Benjamin, New York, 1964). [151 D. Henderson, F.F. Abraham and J.A. Barker, Molec. Phys. 31 (1976) 1291. [161 A,L. Nichols Ill, Ph.D. Thesis (University of California, Berkeley, 1983). [171 L. Landau and E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1969). 118] M.-E. Boudh-Hir, Physica A 158 (1989) 619, this volume. [191 P.C. Hemmer, J. Math. Phys. 5 (1964) 75. [201 J.L. Lebowitz, G. Stell and S. Baer, J. Math. Phys. 6 (1965) 1282. [21] H.C. Andersen (See ref. [10]).