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The potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres
Chemical Physics 134 (1989) North-Holland, Amsterdam
19-30
THE POTENTIAL OF AVERAGE FORCE BETWEEN TWO IONS AT INFINITE IN A SOLVENT OF POLARIZABLE HARD SPHERES
DILUTION
Fabian GRqDZKI Institute ofphysics, Warsaw University, Bialysfok Branch, 41 Lipowa street, 15-424 Bia/ystok, Poland Received
29 September
1988
The potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres is considered. The ions are treated as hard spheres with point charges and the solvent molecules are treated as polarizable hard spheres. The diameter of ions may be different than the diameter of polarizable particles. The potential of average force rises near contact much more steeply than the primitive model would permit and also has maxima and minima vanishing at larger distances, in semiquantitative agreement with the similar results of Patey and Valleau. The oscillating character of the potential of average force is due above all to the oscillations of the radial distribution function between the ion and polarizable hard sphere.
1. Introduction The interactions of ions at infinite dilution in a dipolar or polarizable solvent were considered by Bellemans and Stecki [ 11, Stecki [ 2 ] and independently by Jepsen and Friedman [ 3 1. McDonald and Rasaiah [ 41 determined the potential of average force for two ions in a Stockmayer fluid by a Monte Carlo simulation. Patey and Valleau [ 5 ] determined the potential of average force for two ions in a dipolar solvent also by Monte Carlo simulation. They obtained that the potential of average force rises near contact much more steeply than the primitive model would permit and also has maxima and minima vanishing at larger distances. By the primitive model we mean that the solvent-averaged effect may be represented by treating it as a continuum dielectric with a dielectric constant t. In this paper we consider the potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres. The ions are treated as hard spheres with point charge q< or q,, and diameter uII, and the solvent molecules are treated as a polarizable hard spheres with diameter app. In section 2 we present a general theory of the potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres. In section 3 we consider the short-range potential of average force between two ions. In section 4 we consider the long-range potential of average force between two ions. In section 5 we present the numerical calculations of the potential of average force between two ions at infinite dilution and discuss our results.
2. The potential of average force between two ions at infinite dilution in a solvent of pohuizable hard spheres Let us consider a mixture of three species of molecules in grand canonical ensemble (one and two designed as a solute and the other designed as a solvent). The radial distribution function for solute molecules (ions) at fixed locations in the solution is defined by
x
s
) d( l)...d(K,) exp(-P@~I+K2+~+c+q
d( l)...d(K,)
0301-0104/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
B.V.
d( l)...d(L),
(1)
F. Grqdzki /Force
20
potential
between ions at inJinite dilution
where the superscripts r and rl indicate the kind of ion 1 and 2, z, and K, are appropriate activities and numbers of ions (solute molecules), y and L are appropriate activity and number of solvent molecules, QKi +K2+L+C+,, is the energy of interaction of K, + K2 + L + <+ r] particles, p= 1/kT, .5( z,, z2, y, V, T) is the grand partition function, and p, are the appropriate concentrations of ions in the mixture. Now let us define the potential of average force between two ions [ 61 immersed in a solvent of polarizable hard spheres -pt@(
1,2)=
lim lng”( ;I .zz-0
We now introduce
1, 2).
(2)
the activity coefficients
7, equal to
y, = lim +%, z,.rr-OPI
(3)
yz = lim ??. z,.r-2-op2
(4)
The coefficients
y, # 1 because yf 0. From eq. ( 1) we have
&-o~“‘(~’2)=Z(0, Let us introduce
C II” j exp( -jDDr.+c+V) i;;‘V, T) ,a L!
the quantities
d( l)...d(L).
(5)
Zoo,Z, and .Z2 defined earlier [ 7-101,
-=“o(Y,V, T) =%(A 0, Y, v, T),
3 ( 1; qr,Y, K T) =
and similarly
(6)
& j exp( -P&+J
d(l ).-d(L),
for Z’/( 1; q,,, y, V, T). Then
-/31#~(1,2)=lnZ$~-lnE~
+ln y, +ln yz =ln(Z~~/~o)-ln(Z~/~o)-ln(~~/~~,),
where we used the definition
of y, given by Friedman
-pWf( -BkY?(l,
(7)
[ 61. We also define the quantities
(9) W, and W, as follows:
1; qe)=ln(Zf/Eo),
(10)
2; qt(, 4~)=ln(~5~l~o),
(11)
where W, is the work of putting in one ion into the solvent and W, is the work of putting solvent and keep them at a distance r<,,.Then we have [ 1,2,7] W’“(l,2)=Wrq(l,2;q,,q,,)-wF(1;4~)-W?(2;4,). Let us introduce
a parameter
in two ions into the
(12)
A in such a way that
WC? 1, 2; 2) = WP( 1,2; &?e, 141) - Wf( 1; J.q,) - W?(2; A&j).
(13)
Then we obtain (14)
F. Grqdzki /Force potential between ions at infinite dilution
The potential of average force @( 1, 2; 0) is due to short-range eq. (13) we have
interactions
21
and we denote it by w$( 1, 2). From
~~(1,2)=w~~(1,2;0,O)-Wf(l;O)-W1(2;0),
(15)
i.e. w$‘Jis the potential of average force between discharged ions. The second term on the right-hand ( 14) is due to long-range interactions and we denote it by I&( 1, 2 ). So
side of eq.
(16) Then from eq. ( 14) we have V”(l,2)=V~~(l,
2)+wP(l,
2),
(17)
i.e. the potential of average force between two ions may be divided into a short-range part and a long-range part. In the next sections we will consider independently the short-range and long-range contributions to the potential of average force between two ions.
3. The short-range potential of average force between two ions in a solvent of polarizable hard spheres The energy of interaction
@L+t;+Vis equal to
9 L+C+?l=$@ +@P,
(18)
where @zqrepresents the short-range interactions, i.e. with qc= qq= 0 and @F represents tions. In our case the @ represents the interaction of two hard spheres with diameter with diameter aPP The quantity Wp( 1, 2; 0, 0) is equal to
the long-range interacan and L hard spheres
(19) where @ represents the short-range interactions tion for short-range interactions. The quantity Wj ( 1; 0) is equal to
between solvent particles, n; is a two-particle
where @$represents the short-range interaction of one ion and L solvent particles, tion function for short-range interactions. Similarly we may write for WY ( 1; 0). Fromeqs. (15), (19)and(20)wehave exp ( - j@zq) = lim Z,,z2_0,;(
ni(l, 2) l)nY(2)
=z,fljfn_ogF(lY 2)y
distribution
n S is a one-particle
func-
distribu-
(21)
F. Grqdzki /Force potential between ions at infinite dilution
22
or -j?@=
lim lng$“( 1, 2). Z,.;2 -*o
(22)
In our case gzrl( 1, 2) is the radial distribution hard spheres with diameter ai,.
function
of mixture of hard spheres, where in position
1 and 2 are
4. The long-range potential of average force between two ions at infinite dilution The quantity
W$q(A) is equal to
-pw~~(n)=ln[~~“(~)/~~],
(23)
where E{“(A) = 5: $
I exp( -A’/M@ -/M.$“)
d( 1 )...d(L).
(24)
Then
aWgg) an
a
kT ma;i
=-kTzlnE$V(l)=-
tE2(h,(n)
exp(-A2j?@$V-/BD$U) The long-range equalto [11,12]
energy of interaction
d(l)...d(L)=2A(@$V),.
of two ions and L polarizable
(25) particles with isotropic polarizabilities
is
(26)
~S”=qcq,llr~~-~crIE,lTIAl[E,l, where
[&I =4<[N,+q,[sl,.
(27)
Here [S] c and [S] 1 are L-dimensional
vectors
(28)
[Sl, = N,I> with every element an ordinary
3-component
vector. The Sir are defined by
where P,, = r,, /r,e = ( r, - rc ) /r,< and r,<= )rlc I. The tensor [A ] is equal to
[Al={[Il-aIT])-‘=
C {~[TII”, n
where [T] is an L-dimensional is given by
(30)
tensor with [T] lj= T (i, j) ( 1 -d,,), where S,j is the Kronecker 6. The tensor T (i, j)
T(i,j)=r;3(3?,jt,,-I).
(31)
From eqs. (26 ) and (27 ) we obtain (32)
F. Grqdzki /Force potential between ions at in$nite dilution [S]T[A]
where we used the equality (~%>n=qeq~lr~~-cr(q~q,(
The quantity
[S],=
[S]T[A]
[S],.
23
So
~~l~~~l~~l,~~+t~~~~~l:~~l~~l,~~+t~~~~~l;r~~l~~l,~~~~
(33)
Wf(A) is equal to
-BwF(n)=ln[aF(n)/~~],
(34)
where Zf(A)=
F 2
I exp( -A’@$-~@$)
d(l)...d(L).
(35)
Then awf(n) aA
a
kT
=-kTzlnEf(A)=--
=&F$j
a,(n)7
~~exp(-I’P~-B~)d(l)...d(L)32n(~pS>i,
where @$is the long-range @i=
G,(A)
interaction
(36)
-tw:[W[Al[W,.
Fromeqs.
(16), 1
wF(lJ)=
(25),
(37) (33),
(36) and (37) weobtain I
il a= 7
j24+#4~h-<@sh-
-w,p
j =( [WA1
0
Denoting
LES
[%)A~.
(38)
0
the average value on the right-hand
P~~(~,~;I)~([~IT[AI[SI~)I= =
particles and is equal to [ 13,14 I
of one ion with polarizable
side of eq. ( 38 ) by P”, we may write
( [Sl~n~oWIY’[~l,)A
S&~‘L’(rc, r,,, 1, 2, .... L;A)
where q CL)are the reduced distribution may be expressed by [ 17 ] qCL)(r,, rV, 1,2, .... L; A) = lim II ,Z2-0
B=( 1, 2, . . .. L) &,d(
functions
l)...d(L),
in the solvent containing
nL+*(r.+rq, 1,2, .... L; J-1 n2(q,
rs;
A)
(39) two discharged
ions [ I%16
1. They
(40)
’
functions in the grand canonical ensemble. The quantity BL( 1, where nr+*(rC, rq, 1, 2, .... L; A) are distribution 2 , .... L) symbolizes the sum of all chains beginning at the ion r, touching each of the L points at least once and ending at the ion v. Because we have nL+*(rC, rtl, 1,2, ....L; 2)
=w2pf;gL+2(rc,
n2(rC, rV; ~)=plp2g2(rb
rv;
rv,
L2,
.-.,
k
(41)
Ah
(42)
A).
So we obtain tfcL)(rc,rq,
1,2, .... L;A)=pb
lim 21,12-0
gL+2(rc, rq, 1,2, .-.,GA) s2(q,
rq;
A)
(43) ’
24
F. Grqdzki /Force potential
between ions at inJhte dilution
The functions g, in eq. (43) are dependent on activity y. By a topological reduction of the graphs in these functions we pass from activity y to density pp. Since every graph in BL ( 1, 2, .... L) appears with its proper symmetry number 1, it is possible to carry out the usual operations with the whole classes of graphs [ 181. In order to make the formalism practically useful, a way of selecting graphs is needed. We first consider the graphs which give us simple chains as follows ?
P”‘(~rl; 1) = -
j” s$(C
L=l
x...xppg$p(L-l,
L;i)
1; A)S:, p,g4p( 1, 2; A) aT(l, cuT(L-
l,L)ppgyV(L,
2) ppgYp(2, 3; 1) aT(2,
q;A)S,,d(
3)
l)...d(L).
(44)
Eq. (44) is obtained by assuming the superposition approximation for the function gI.+2, which is a rather crude approximation, at any value of the density in the liquid range (but we have not a better approximation). We also put in eq. (39) g$‘“( r; A) = 1, if there is no T bond between polarizable particles. In eq. (44) we changed sign, because SC, = -S,s. One of the simplest approximations we obtain by putting A=O, i.e. the average is calculated in the hard-sphere reference system. Let us define the Fourier transforms ^ p(k)= jg$(r)sexp(ik-r)dr=4niR g$(r)j,(kr) dr=ik^a(k), (45) s 0
e(k)=
j gP”(r)T(r)
exp(ik*r)
7 r-‘&P(r)
dr= -4rrT,(k)
j,(kr)
dr= -T,(k)
b(k),
(46)
0
where To(k) E 3&- I and the functions g2 are taken for A= 0. The superscripts #I or pp in the radial distribution functions indicate the kind of particle in position r, and the kind of particle in position rz. Then we have [ 10,171 Ps’J(rCV;0) = - J, xgjZq(rL,)SId,
1
=
(2K)3PP
P;
I
sS”(raWfl
gTP(r12)
=p
2) s4p(r2~)
aT(Z
[a(k) I’ s 1+2ap,b(k)
exp( -ik*r)
= (a~,)~
aTG--
1. L)
[a(k) I2 ’ 1+2ap,b(k)
dk.
(47)
0) is equal to (48)
.
We next consider the graphs which we can obtain by decorating R =PP s aT(
3)...gSp(rL-l.I.)
d( 1 )...d(L)
From eq. (47) one can see that P”(k;
Pl( k; 0)
aT(l,
1,2)
p,gP”(
1,2)
j g$‘p(r)r-6(21+To)
aT(2,
of simple chains with R graphs, where
1) dr2
dr2 =81c(ap,)‘lT
r-‘gyp(r) 0
Then we obtain following series for Peq (rev; 0 ):
dr.
(49)
F. Grqdzki /Force potential
Pcq(rtq; 0) = x
I
between ions at infinite dilution
25
L Lz,nio n R”(&P”
0
g$(r,)Sl&
g4P(rlz)~T(1,
2) gYP(rz3)
aT(2,
3)...g4P(r,-I,L)(YT(L--I,L)
xgYYrLq)SLq d( 1).-d(L)
=-$&go(:)
Rn(pp)L-n
= --
(2:)3 J,
(pp+R)L j
SF(k) [aG(k)lL-‘F(k)
F(k) [aG(k)lL-‘F(k)
PP+R F(k)[l-a(p,+R)G(k)]-‘p(k) =--(2x)3 s Using the appendix
exp( -ik*r)
exp( -ik*r)
exp(-ik*r)
dk
dk
(50)
dk.
we have
PCq(rc7; 0) =-P~~P(a)[l-~(pp+R)b(k)]-r
I-
3a(p +R)b(k) l+2aFP +R)h(k)i%
F(k) exp(-ik-r)
dk.
(51)
P
Introducing
eq. (45 ) we obtain (52)
From eq. ( 52) we have
mk; 0) = [a(k) I2
+2a;P++“,,,(,) .
1
To calculate the dielectric we obtain
@V; 0) =q
e
constant,
we consider the Fourier transform
(53) of I&~( 1, 2; A). From eqs. ( 38 ) and ( 53 )
4q
(54)
For k+O we have @yA;O)=
4”q* k2
‘t
11
Then the dielectric
‘=
1 +&rcu(p, +R) 1-$cu(p,+R)
l-
47=Q, +R) l+$ccu(p,+R)
constant
4n > = kzEqCq”
(55)
is given by the van Vleck [ 2,19 ] expression
*
One can also decorate the chain graphs in other ways and then the dielectric constant obtained by Wertheim [ 18 ] and Hoye and Stell [ 20 1. The numerical results will be discussed in the section 5.
(56) will be closer to the results
5, Results and discussion
The distribution functions in a solvent containing two discharged ions are in our case the hard-sphere distribution functions. One approximation for small densities is obtained by putting gp(r)=O,
for r
=I,
forr>Okp,
(57)
= 1,
for r>gp,
(58)
and
with Q,= ( CT~~+Q,)/~, where a,, is the hard-sphere diameter of a solute particle and gP1)is the hard-sphere diameter of a solvent particle. From eqs. (45 ) and (46) we have
b(k,p,)=473:
f 0
r-‘@‘(r)j2(kr)
dr.
(60)
So for small densities we obtain [ 1O]
(629 and P”&(k; 0) is equal to
From eq. (6 1) we see that for small P = ka,, the quantity a(&?, p*) R+1/k*, so in fig. 1 we plotted the function A(k*,pg)=k*a(k*,p*,) forp:==p,a&,=0.8 and for ob=oPP. The less oscillating curve is the function a( k*, pz) from eq. f 6 1 f , which is independent of the density of the polariz~bie particles. The second curve was calcuiated with g$( r; p,) obtained by the method proposed by Throop and Bearman [ 2 11. For large value of A? the maxima and minima of A (ky, pz 9 are at the same values of k*, but the amplitudes of oscillations obtained from eq. (59 ) are several times those amplitudes obtained from eq. (6 1). The values of A (0, p:) obtained from eq. (59 ) and eq. (6 1) are the same and are equal to 47~.We also calculated a(k) for p;*,= 0.8 and values of rr&f oPP,but these are not shown. The shape of appropriate curves is similar to those in fig. 1. In fig. 2 we plotted the function b(k*, p*,) for & ~~0.8 and for c~rp=a,,~ The function b(k+, p;) from eq. (62) is the less oscillating curve and is independent of the density of the polarizable particles. The second curve was calculated from eq. (60) with gpP( r; p,) obtained by a method proposed by Throop and Bearman. From both equations we have b( 0, &) =4n/3 and for large values of k the oscillations are in phase, but the amplitudes of oscillations calculated from eq. (60) are greater than corresponding amplitudes of oscillations caiculated from eq. (62 ) I We also calculated hf IL+,PI”; f for the same values of & but the diameter of ions was different from the
21
F. Grqdzki /Force potential between ions at infinite dilution
A(K"9"l 'P 10
Fig. l.Thef.mctionA(P, &)=k*a(k*,p;) for&=O.S.Theions and the polarizable particles have the same diameters. (----_) fromeq. (59), (---) fromeq. (61).
Fig. 2. The function b(k*, p;) for p; =0.8. The ions and the polarizable particles have the same diameters. (--) from eq. (60), (---) fromeq. (62).
diameter of polarizable particles. The shape of the appropriate curves is similar as in fig. 2. In fig. 3 the function da(k*; 0) = a*k**pB(k*; 0) is plotted for the same values of parameters as those in fig. 1 and in fig. 2 for o*=a/a& =0.16 and R calculated from eq. (49) with g$‘*(r; p,) obtained by the method proposed by Throop and Bearman and equal to 0.2275. The quantity Ba(O; 0)=16rc2~(pp+R)/ [ 1 + $~(p, + R) ] and is the same for the discussed approximations here. For large values of k the oscillations are in phase but the amplitudes of oscillations are larger if d”(k*; 0) is calculated from eq. (53) in comparison with calculations from eq. ( 63 ). The calculations for different diameters of ions and polarizable particles give similar appropriate curves. From eq. (38) we have 1 q@(l 92)=!k!c r0!
q
s
2;IP”( 1,2; 1) d;i.
(64)
0
6-
-----b
-60
: 1
Fig. 3. The function ba(k*; 0) for p: =0.8 and LU*=O. 16. The ions and the polarizable particles have the same diameters. (---) fromeq. (53), (---) fromeq. (63).
2
3
L .*
Fig. 4. The long-range part of the potential of average force between two ions in hard-sphere polarizable particles. The ions and polarizable particles have the same diameters. (-) from eqs. (53) and (66), (- - -) from eqs. (63) and (66), (-.--) from the primitive model.
F. Grqdzki /Force
28
In this paper we consider
wi”(l, 2) = -4&l r&II
-4&&a
potentialbetween ions at infinite dilution
one of the simplest approximations,
s
211P”( 19 2.0) 1
i.e.
d/l.
(65)
0
Then we have cc ,,@(1,2)=
7
--qcqaoP&(l,
2; o)=
y
- %
j @(k,p,)j&kr,,)
?
dk.
(66)
0
In fig. 4 we plotted the long-range part of the potential of average force calculated for Pq@,,/a,, = - 188.0 on the basis of the primitive model with e given by eq. (56) or by numerical integration of eq. (66) for ions and polarizable particles of the same diameters. The Simpson rule was used with a cutoff of k* which had no effect in the studied range of rtq. The function prfl( k*; 0) from eq. (63) gives too low values of @(r* ) near r* = r/ a,, = 1. The best curve was obtained with the radial distribution function of hard spheres obtained by the method proposed by Throop and Bearman and used to calculate a( Kc, pX ), h( k*, pz) and P”“( k*; 0) and then the potential of average force between two ions. We see that @(r*) oscillates on the curve representing the primitive model in a way similar to the potential of average force obtained by the Monte Carlo method of Patey and Valleau [ 51 for the dipolar solvent. It rises, near contact, much more steeply than the primitive model would permit and shows maxima and minima vanishing at larger distances, in semi-quantitative agreement with similar results of Patey and Valleau. The oscillating character of the potential of average force is due above all to the oscillations of the radial distribution function between the ion and the polarizable hard sphere. In our case the dielectric constant t= 7.6 whereas Patey and Valleau had TV 7.8, which is close enough for our results to be comparable. In fig. 5 we plotted the potential of average force between ions if a diameter of ions is equal to half the diameter of the polarizable particles. The appropriate curves are similar to those in fig. 4 but the minima and maxima of the oscillating potential of average force are displaced to smaller values of r. Also the absolute values of the w$“( a;,) are larger because we took a, < a,,. In fig. 6 we plotted the short-range part of the potential of average force between ions, when ions and polar-
i 1
2
I
3
r*
I
Fig. 5. The potential of average force between two ions in hardsphere polarizable particles. The diameter of ions is equal to half the diameter of the polarizable particles. (----) from eqs. (53) and (66), (---) from eqs. (63) and (66), (-,-) from the primitive model.
r* Fig. 6. The short-range part of the potential of average force between two ions. The ions and the polarizable particles have the same diameters.
F. Grqdzki/Force potential betweenions at infinite dilution
Fig. 7. The sum of short-range part and long-range part of the potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres. The ions and the polarizable particles have the same diameters. (-) from eqs. (53), (66) and (22), (---) fromeqs. (63), (66) and (22), (-.-) from the primitive model and eq. (22).
: -6O- / 1
2
3
29
L
r*
izable particles have the same diameters. The distribution functions of the hard spheres were calculated by the method proposed by Throop and Bearman [ 2 11. In fig. 7 we plotted the sum of the short-range part and the long-range part of the potential of average force between ions, when ions and polarizable particles have the same diameters, i.e. fig. 7 is obtained by summing the results presented in fig. 4 and in fig. 6. We can see a very small contribution of the short-range interactions to the potential of average force between ions. The simplest case, i.e. the ideal gas model of solvent was considered earlier [ 7,8,22,23].
Acknowledgement The author would like to thank Professor J. Stecki for a helpful discussion.
Appendix Let us determine
the matrix A-‘, where A= [I -BG (k) 1. From eq. (46) we have
A=I-bB(I-3~~)=(1-bB)I+3bB~~=(l-bB)
I+ $+ (
&(k)[I+D(k)U]. >
(A.1)
In general for a square matrix M of order 3 x 3 we have [ 24 ] det(I+M)=l+tr(M)+
C det(M,)+det(M), n
where M, is the submatrix of M obtained det(M)=Oanddet(M,)=Oforn=1,2,3.S0
by crossing out of nth row and nth column
det(I+M)=l+tr(M)=l+D. To calculate the elements
(A.2) of matrix M. In our case
(‘4.3) of the matrix A-’ we use also the equality
det(l+N)=l+tr(N)+det(N),
(A.4)
where N is a square matrix of order 2 x 2. Then we obtain
A-l&+
-+).
(A.51
30
F. Grqdzki / Forcepotentlal between ions at infinite dilutlon
References [ 1 ] A. Bellemans and J. Stecki, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 9 ( 196 1) 339, 343, 349. [2] J. Stecki, Advan. Chem. Phys. 6 (1962) 413. [3] D.W. Jepsen and H.L. Friedman, J. Chem. Phys. 38 ( 1963) 846. [4] I.R. McDonald and J.C. Rasaiah, Chem. Phys. Letters 34 (1975) 382. [ 51 G.N. Patey and J.P. Valleau, J. Chem. Phys. 63 ( 1975) 2334. [ 61 H.L. Friedman, A Course in Statistical Mechanics (Prentice-Hall, Englewood Cliffs, 1985). (71 F. Gradzki, Acta Phys. Polon. A 61 (1982) 405. [S] F. Grgdzki, Acta Phys. Polon. A 63 (1983) 707. [9] F. Gradzki, Acta Phys. Polon. A 67 (1985) 631. [IO] F. Gradzki, Acta Phys. Polon. A 71 (1987) 591. [ 111 M. Mandel and P. Mazur, Physica 24 (1958) 116. [12]M.S.Wertheim,Mol.Phys.26(1973)1425;33(1977)95;34(1977)1109;36(1978)1217:37(1979)83. [ 131 F. Gradzki, Chem. Phys. 125 (1988) 247. [ 141 F. Gqdzki, Chem. Phys. 130 ( 1989) 167. [ 151 J. Stecki, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 9 ( 1961 ) 429. [ 161 J. Stecki, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 9 ( 1961) 435. [ 171 F. Gradzki, J. Sol. Chem. 17 ( 1988) 1101. [18]M.S.Wertheim,Mol.Phys.25(1973)211. [ 191 J.H. van Vleck, J. Chem. Phys. 5 (1937) 556. [ 201 J.S. Hoye and G. Stell, J. Chem. Phys. 73 ( 1980) 46 1. [21] G.J. Throop and R.J. Bearman, J. Chem. Phys. 42 (1965) 2408; 42 (1965) 2838. [22] F. Gradzki, Acta Phys. Polon. A 58 (1980) 9. [ 231 F. Gradzki, Acta Phys. Polon. A 59 ( 198 1 ) 123. [ 241 H.L. Friedman. J. Chem. Phys. 76 ( 1982) 1092.
19-30
THE POTENTIAL OF AVERAGE FORCE BETWEEN TWO IONS AT INFINITE IN A SOLVENT OF POLARIZABLE HARD SPHERES
DILUTION
Fabian GRqDZKI Institute ofphysics, Warsaw University, Bialysfok Branch, 41 Lipowa street, 15-424 Bia/ystok, Poland Received
29 September
1988
The potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres is considered. The ions are treated as hard spheres with point charges and the solvent molecules are treated as polarizable hard spheres. The diameter of ions may be different than the diameter of polarizable particles. The potential of average force rises near contact much more steeply than the primitive model would permit and also has maxima and minima vanishing at larger distances, in semiquantitative agreement with the similar results of Patey and Valleau. The oscillating character of the potential of average force is due above all to the oscillations of the radial distribution function between the ion and polarizable hard sphere.
1. Introduction The interactions of ions at infinite dilution in a dipolar or polarizable solvent were considered by Bellemans and Stecki [ 11, Stecki [ 2 ] and independently by Jepsen and Friedman [ 3 1. McDonald and Rasaiah [ 41 determined the potential of average force for two ions in a Stockmayer fluid by a Monte Carlo simulation. Patey and Valleau [ 5 ] determined the potential of average force for two ions in a dipolar solvent also by Monte Carlo simulation. They obtained that the potential of average force rises near contact much more steeply than the primitive model would permit and also has maxima and minima vanishing at larger distances. By the primitive model we mean that the solvent-averaged effect may be represented by treating it as a continuum dielectric with a dielectric constant t. In this paper we consider the potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres. The ions are treated as hard spheres with point charge q< or q,, and diameter uII, and the solvent molecules are treated as a polarizable hard spheres with diameter app. In section 2 we present a general theory of the potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres. In section 3 we consider the short-range potential of average force between two ions. In section 4 we consider the long-range potential of average force between two ions. In section 5 we present the numerical calculations of the potential of average force between two ions at infinite dilution and discuss our results.
2. The potential of average force between two ions at infinite dilution in a solvent of pohuizable hard spheres Let us consider a mixture of three species of molecules in grand canonical ensemble (one and two designed as a solute and the other designed as a solvent). The radial distribution function for solute molecules (ions) at fixed locations in the solution is defined by
x
s
) d( l)...d(K,) exp(-P@~I+K2+~+c+q
d( l)...d(K,)
0301-0104/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
B.V.
d( l)...d(L),
(1)
F. Grqdzki /Force
20
potential
between ions at inJinite dilution
where the superscripts r and rl indicate the kind of ion 1 and 2, z, and K, are appropriate activities and numbers of ions (solute molecules), y and L are appropriate activity and number of solvent molecules, QKi +K2+L+C+,, is the energy of interaction of K, + K2 + L + <+ r] particles, p= 1/kT, .5( z,, z2, y, V, T) is the grand partition function, and p, are the appropriate concentrations of ions in the mixture. Now let us define the potential of average force between two ions [ 61 immersed in a solvent of polarizable hard spheres -pt@(
1,2)=
lim lng”( ;I .zz-0
We now introduce
1, 2).
(2)
the activity coefficients
7, equal to
y, = lim +%, z,.rr-OPI
(3)
yz = lim ??. z,.r-2-op2
(4)
The coefficients
y, # 1 because yf 0. From eq. ( 1) we have
&-o~“‘(~’2)=Z(0, Let us introduce
C II” j exp( -jDDr.+c+V) i;;‘V, T) ,a L!
the quantities
d( l)...d(L).
(5)
Zoo,Z, and .Z2 defined earlier [ 7-101,
-=“o(Y,V, T) =%(A 0, Y, v, T),
3 ( 1; qr,Y, K T) =
and similarly
(6)
& j exp( -P&+J
d(l ).-d(L),
for Z’/( 1; q,,, y, V, T). Then
-/31#~(1,2)=lnZ$~-lnE~
+ln y, +ln yz =ln(Z~~/~o)-ln(Z~/~o)-ln(~~/~~,),
where we used the definition
of y, given by Friedman
-pWf( -BkY?(l,
(7)
[ 61. We also define the quantities
(9) W, and W, as follows:
1; qe)=ln(Zf/Eo),
(10)
2; qt(, 4~)=ln(~5~l~o),
(11)
where W, is the work of putting in one ion into the solvent and W, is the work of putting solvent and keep them at a distance r<,,.Then we have [ 1,2,7] W’“(l,2)=Wrq(l,2;q,,q,,)-wF(1;4~)-W?(2;4,). Let us introduce
a parameter
in two ions into the
(12)
A in such a way that
WC? 1, 2; 2) = WP( 1,2; &?e, 141) - Wf( 1; J.q,) - W?(2; A&j).
(13)
Then we obtain (14)
F. Grqdzki /Force potential between ions at infinite dilution
The potential of average force @( 1, 2; 0) is due to short-range eq. (13) we have
interactions
21
and we denote it by w$( 1, 2). From
~~(1,2)=w~~(1,2;0,O)-Wf(l;O)-W1(2;0),
(15)
i.e. w$‘Jis the potential of average force between discharged ions. The second term on the right-hand ( 14) is due to long-range interactions and we denote it by I&( 1, 2 ). So
side of eq.
(16) Then from eq. ( 14) we have V”(l,2)=V~~(l,
2)+wP(l,
2),
(17)
i.e. the potential of average force between two ions may be divided into a short-range part and a long-range part. In the next sections we will consider independently the short-range and long-range contributions to the potential of average force between two ions.
3. The short-range potential of average force between two ions in a solvent of polarizable hard spheres The energy of interaction
@L+t;+Vis equal to
9 L+C+?l=$@ +@P,
(18)
where @zqrepresents the short-range interactions, i.e. with qc= qq= 0 and @F represents tions. In our case the @ represents the interaction of two hard spheres with diameter with diameter aPP The quantity Wp( 1, 2; 0, 0) is equal to
the long-range interacan and L hard spheres
(19) where @ represents the short-range interactions tion for short-range interactions. The quantity Wj ( 1; 0) is equal to
between solvent particles, n; is a two-particle
where @$represents the short-range interaction of one ion and L solvent particles, tion function for short-range interactions. Similarly we may write for WY ( 1; 0). Fromeqs. (15), (19)and(20)wehave exp ( - j@zq) = lim Z,,z2_0,;(
ni(l, 2) l)nY(2)
=z,fljfn_ogF(lY 2)y
distribution
n S is a one-particle
func-
distribu-
(21)
F. Grqdzki /Force potential between ions at infinite dilution
22
or -j?@=
lim lng$“( 1, 2). Z,.;2 -*o
(22)
In our case gzrl( 1, 2) is the radial distribution hard spheres with diameter ai,.
function
of mixture of hard spheres, where in position
1 and 2 are
4. The long-range potential of average force between two ions at infinite dilution The quantity
W$q(A) is equal to
-pw~~(n)=ln[~~“(~)/~~],
(23)
where E{“(A) = 5: $
I exp( -A’/M@ -/M.$“)
d( 1 )...d(L).
(24)
Then
aWgg) an
a
kT ma;i
=-kTzlnE$V(l)=-
tE2(h,(n)
exp(-A2j?@$V-/BD$U) The long-range equalto [11,12]
energy of interaction
d(l)...d(L)=2A(@$V),.
of two ions and L polarizable
(25) particles with isotropic polarizabilities
is
(26)
~S”=qcq,llr~~-~crIE,lTIAl[E,l, where
[&I =4<[N,+q,[sl,.
(27)
Here [S] c and [S] 1 are L-dimensional
vectors
(28)
[Sl, = N,I> with every element an ordinary
3-component
vector. The Sir are defined by
where P,, = r,, /r,e = ( r, - rc ) /r,< and r,<= )rlc I. The tensor [A ] is equal to
[Al={[Il-aIT])-‘=
C {~[TII”, n
where [T] is an L-dimensional is given by
(30)
tensor with [T] lj= T (i, j) ( 1 -d,,), where S,j is the Kronecker 6. The tensor T (i, j)
T(i,j)=r;3(3?,jt,,-I).
(31)
From eqs. (26 ) and (27 ) we obtain (32)
F. Grqdzki /Force potential between ions at in$nite dilution [S]T[A]
where we used the equality (~%>n=qeq~lr~~-cr(q~q,(
The quantity
[S],=
[S]T[A]
[S],.
23
So
~~l~~~l~~l,~~+t~~~~~l:~~l~~l,~~+t~~~~~l;r~~l~~l,~~~~
(33)
Wf(A) is equal to
-BwF(n)=ln[aF(n)/~~],
(34)
where Zf(A)=
F 2
I exp( -A’@$-~@$)
d(l)...d(L).
(35)
Then awf(n) aA
a
kT
=-kTzlnEf(A)=--
=&F$j
a,(n)7
~~exp(-I’P~-B~)d(l)...d(L)32n(~pS>i,
where @$is the long-range @i=
G,(A)
interaction
(36)
-tw:[W[Al[W,.
Fromeqs.
(16), 1
wF(lJ)=
(25),
(37) (33),
(36) and (37) weobtain I
j24+#4~h-<@sh-
-w,p
j =( [WA1
0
Denoting
LES
[%)A~.
(38)
0
the average value on the right-hand
P~~(~,~;I)~([~IT[AI[SI~)I= =
particles and is equal to [ 13,14 I
of one ion with polarizable
side of eq. ( 38 ) by P”, we may write
( [Sl~n~oWIY’[~l,)A
S&~‘L’(rc, r,,, 1, 2, .... L;A)
where q CL)are the reduced distribution may be expressed by [ 17 ] qCL)(r,, rV, 1,2, .... L; A) = lim II ,Z2-0
B=( 1, 2, . . .. L) &,d(
functions
l)...d(L),
in the solvent containing
nL+*(r.+rq, 1,2, .... L; J-1 n2(q,
rs;
A)
(39) two discharged
ions [ I%16
1. They
(40)
’
functions in the grand canonical ensemble. The quantity BL( 1, where nr+*(rC, rq, 1, 2, .... L; A) are distribution 2 , .... L) symbolizes the sum of all chains beginning at the ion r, touching each of the L points at least once and ending at the ion v. Because we have nL+*(rC, rtl, 1,2, ....L; 2)
=w2pf;gL+2(rc,
n2(rC, rV; ~)=plp2g2(rb
rv;
rv,
L2,
.-.,
k
(41)
Ah
(42)
A).
So we obtain tfcL)(rc,rq,
1,2, .... L;A)=pb
lim 21,12-0
gL+2(rc, rq, 1,2, .-.,GA) s2(q,
rq;
A)
(43) ’
24
F. Grqdzki /Force potential
between ions at inJhte dilution
The functions g, in eq. (43) are dependent on activity y. By a topological reduction of the graphs in these functions we pass from activity y to density pp. Since every graph in BL ( 1, 2, .... L) appears with its proper symmetry number 1, it is possible to carry out the usual operations with the whole classes of graphs [ 181. In order to make the formalism practically useful, a way of selecting graphs is needed. We first consider the graphs which give us simple chains as follows ?
P”‘(~rl; 1) = -
j” s$(C
L=l
x...xppg$p(L-l,
L;i)
1; A)S:, p,g4p( 1, 2; A) aT(l, cuT(L-
l,L)ppgyV(L,
2) ppgYp(2, 3; 1) aT(2,
q;A)S,,d(
3)
l)...d(L).
(44)
Eq. (44) is obtained by assuming the superposition approximation for the function gI.+2, which is a rather crude approximation, at any value of the density in the liquid range (but we have not a better approximation). We also put in eq. (39) g$‘“( r; A) = 1, if there is no T bond between polarizable particles. In eq. (44) we changed sign, because SC, = -S,s. One of the simplest approximations we obtain by putting A=O, i.e. the average is calculated in the hard-sphere reference system. Let us define the Fourier transforms ^ p(k)= jg$(r)sexp(ik-r)dr=4niR g$(r)j,(kr) dr=ik^a(k), (45) s 0
e(k)=
j gP”(r)T(r)
exp(ik*r)
7 r-‘&P(r)
dr= -4rrT,(k)
j,(kr)
dr= -T,(k)
b(k),
(46)
0
where To(k) E 3&- I and the functions g2 are taken for A= 0. The superscripts #I or pp in the radial distribution functions indicate the kind of particle in position r, and the kind of particle in position rz. Then we have [ 10,171 Ps’J(rCV;0) = - J, xgjZq(rL,)SId,
1
=
(2K)3PP
P;
I
sS”(raWfl
gTP(r12)
=p
2) s4p(r2~)
aT(Z
[a(k) I’ s 1+2ap,b(k)
exp( -ik*r)
= (a~,)~
aTG--
1. L)
[a(k) I2 ’ 1+2ap,b(k)
dk.
(47)
0) is equal to (48)
.
We next consider the graphs which we can obtain by decorating R =PP s aT(
3)...gSp(rL-l.I.)
d( 1 )...d(L)
From eq. (47) one can see that P”(k;
Pl( k; 0)
aT(l,
1,2)
p,gP”(
1,2)
j g$‘p(r)r-6(21+To)
aT(2,
of simple chains with R graphs, where
1) dr2
dr2 =81c(ap,)‘lT
r-‘gyp(r) 0
Then we obtain following series for Peq (rev; 0 ):
dr.
(49)
F. Grqdzki /Force potential
Pcq(rtq; 0) = x
I
between ions at infinite dilution
25
L Lz,nio n R”(&P”
0
g$(r,)Sl&
g4P(rlz)~T(1,
2) gYP(rz3)
aT(2,
3)...g4P(r,-I,L)(YT(L--I,L)
xgYYrLq)SLq d( 1).-d(L)
=-$&go(:)
Rn(pp)L-n
= --
(2:)3 J,
(pp+R)L j
SF(k) [aG(k)lL-‘F(k)
F(k) [aG(k)lL-‘F(k)
PP+R F(k)[l-a(p,+R)G(k)]-‘p(k) =--(2x)3 s Using the appendix
exp( -ik*r)
exp( -ik*r)
exp(-ik*r)
dk
dk
(50)
dk.
we have
PCq(rc7; 0) =-P~~P(a)[l-~(pp+R)b(k)]-r
I-
3a(p +R)b(k) l+2aFP +R)h(k)i%
F(k) exp(-ik-r)
dk.
(51)
P
Introducing
eq. (45 ) we obtain (52)
From eq. ( 52) we have
mk; 0) = [a(k) I2
+2a;P++“,,,(,) .
1
To calculate the dielectric we obtain
@V; 0) =q
e
constant,
we consider the Fourier transform
(53) of I&~( 1, 2; A). From eqs. ( 38 ) and ( 53 )
4q
(54)
For k+O we have @yA;O)=
4”q* k2
‘t
11
Then the dielectric
‘=
1 +&rcu(p, +R) 1-$cu(p,+R)
l-
47=Q, +R) l+$ccu(p,+R)
constant
4n > = kzEqCq”
(55)
is given by the van Vleck [ 2,19 ] expression
*
One can also decorate the chain graphs in other ways and then the dielectric constant obtained by Wertheim [ 18 ] and Hoye and Stell [ 20 1. The numerical results will be discussed in the section 5.
(56) will be closer to the results
5, Results and discussion
The distribution functions in a solvent containing two discharged ions are in our case the hard-sphere distribution functions. One approximation for small densities is obtained by putting gp(r)=O,
for r
=I,
forr>Okp,
(57)
= 1,
for r>gp,
(58)
and
with Q,= ( CT~~+Q,)/~, where a,, is the hard-sphere diameter of a solute particle and gP1)is the hard-sphere diameter of a solvent particle. From eqs. (45 ) and (46) we have
b(k,p,)=473:
f 0
r-‘@‘(r)j2(kr)
dr.
(60)
So for small densities we obtain [ 1O]
(629 and P”&(k; 0) is equal to
From eq. (6 1) we see that for small P = ka,, the quantity a(&?, p*) R+1/k*, so in fig. 1 we plotted the function A(k*,pg)=k*a(k*,p*,) forp:==p,a&,=0.8 and for ob=oPP. The less oscillating curve is the function a( k*, pz) from eq. f 6 1 f , which is independent of the density of the polariz~bie particles. The second curve was calcuiated with g$( r; p,) obtained by the method proposed by Throop and Bearman [ 2 11. For large value of A? the maxima and minima of A (ky, pz 9 are at the same values of k*, but the amplitudes of oscillations obtained from eq. (59 ) are several times those amplitudes obtained from eq. (6 1). The values of A (0, p:) obtained from eq. (59 ) and eq. (6 1) are the same and are equal to 47~.We also calculated a(k) for p;*,= 0.8 and values of rr&f oPP,but these are not shown. The shape of appropriate curves is similar to those in fig. 1. In fig. 2 we plotted the function b(k*, p*,) for & ~~0.8 and for c~rp=a,,~ The function b(k+, p;) from eq. (62) is the less oscillating curve and is independent of the density of the polarizable particles. The second curve was calculated from eq. (60) with gpP( r; p,) obtained by a method proposed by Throop and Bearman. From both equations we have b( 0, &) =4n/3 and for large values of k the oscillations are in phase, but the amplitudes of oscillations calculated from eq. (60) are greater than corresponding amplitudes of oscillations caiculated from eq. (62 ) I We also calculated hf IL+,PI”; f for the same values of & but the diameter of ions was different from the
21
F. Grqdzki /Force potential between ions at infinite dilution
A(K"9"l 'P 10
Fig. l.Thef.mctionA(P, &)=k*a(k*,p;) for&=O.S.Theions and the polarizable particles have the same diameters. (----_) fromeq. (59), (---) fromeq. (61).
Fig. 2. The function b(k*, p;) for p; =0.8. The ions and the polarizable particles have the same diameters. (--) from eq. (60), (---) fromeq. (62).
diameter of polarizable particles. The shape of the appropriate curves is similar as in fig. 2. In fig. 3 the function da(k*; 0) = a*k**pB(k*; 0) is plotted for the same values of parameters as those in fig. 1 and in fig. 2 for o*=a/a& =0.16 and R calculated from eq. (49) with g$‘*(r; p,) obtained by the method proposed by Throop and Bearman and equal to 0.2275. The quantity Ba(O; 0)=16rc2~(pp+R)/ [ 1 + $~(p, + R) ] and is the same for the discussed approximations here. For large values of k the oscillations are in phase but the amplitudes of oscillations are larger if d”(k*; 0) is calculated from eq. (53) in comparison with calculations from eq. ( 63 ). The calculations for different diameters of ions and polarizable particles give similar appropriate curves. From eq. (38) we have 1 q@(l 92)=!k!c r0!
q
s
2;IP”( 1,2; 1) d;i.
(64)
0
6-
-----b
-60
: 1
Fig. 3. The function ba(k*; 0) for p: =0.8 and LU*=O. 16. The ions and the polarizable particles have the same diameters. (---) fromeq. (53), (---) fromeq. (63).
2
3
L .*
Fig. 4. The long-range part of the potential of average force between two ions in hard-sphere polarizable particles. The ions and polarizable particles have the same diameters. (-) from eqs. (53) and (66), (- - -) from eqs. (63) and (66), (-.--) from the primitive model.
F. Grqdzki /Force
28
In this paper we consider
wi”(l, 2) = -4&l r&II
-4&&a
potentialbetween ions at infinite dilution
one of the simplest approximations,
s
211P”( 19 2.0) 1
i.e.
d/l.
(65)
0
Then we have cc ,,@(1,2)=
7
--qcqaoP&(l,
2; o)=
y
- %
j @(k,p,)j&kr,,)
?
dk.
(66)
0
In fig. 4 we plotted the long-range part of the potential of average force calculated for Pq@,,/a,, = - 188.0 on the basis of the primitive model with e given by eq. (56) or by numerical integration of eq. (66) for ions and polarizable particles of the same diameters. The Simpson rule was used with a cutoff of k* which had no effect in the studied range of rtq. The function prfl( k*; 0) from eq. (63) gives too low values of @(r* ) near r* = r/ a,, = 1. The best curve was obtained with the radial distribution function of hard spheres obtained by the method proposed by Throop and Bearman and used to calculate a( Kc, pX ), h( k*, pz) and P”“( k*; 0) and then the potential of average force between two ions. We see that @(r*) oscillates on the curve representing the primitive model in a way similar to the potential of average force obtained by the Monte Carlo method of Patey and Valleau [ 51 for the dipolar solvent. It rises, near contact, much more steeply than the primitive model would permit and shows maxima and minima vanishing at larger distances, in semi-quantitative agreement with similar results of Patey and Valleau. The oscillating character of the potential of average force is due above all to the oscillations of the radial distribution function between the ion and the polarizable hard sphere. In our case the dielectric constant t= 7.6 whereas Patey and Valleau had TV 7.8, which is close enough for our results to be comparable. In fig. 5 we plotted the potential of average force between ions if a diameter of ions is equal to half the diameter of the polarizable particles. The appropriate curves are similar to those in fig. 4 but the minima and maxima of the oscillating potential of average force are displaced to smaller values of r. Also the absolute values of the w$“( a;,) are larger because we took a, < a,,. In fig. 6 we plotted the short-range part of the potential of average force between ions, when ions and polar-
i 1
2
I
3
r*
I
Fig. 5. The potential of average force between two ions in hardsphere polarizable particles. The diameter of ions is equal to half the diameter of the polarizable particles. (----) from eqs. (53) and (66), (---) from eqs. (63) and (66), (-,-) from the primitive model.
r* Fig. 6. The short-range part of the potential of average force between two ions. The ions and the polarizable particles have the same diameters.
F. Grqdzki/Force potential betweenions at infinite dilution
Fig. 7. The sum of short-range part and long-range part of the potential of average force between two ions at infinite dilution in a solvent of polarizable hard spheres. The ions and the polarizable particles have the same diameters. (-) from eqs. (53), (66) and (22), (---) fromeqs. (63), (66) and (22), (-.-) from the primitive model and eq. (22).
: -6O- / 1
2
3
29
L
r*
izable particles have the same diameters. The distribution functions of the hard spheres were calculated by the method proposed by Throop and Bearman [ 2 11. In fig. 7 we plotted the sum of the short-range part and the long-range part of the potential of average force between ions, when ions and polarizable particles have the same diameters, i.e. fig. 7 is obtained by summing the results presented in fig. 4 and in fig. 6. We can see a very small contribution of the short-range interactions to the potential of average force between ions. The simplest case, i.e. the ideal gas model of solvent was considered earlier [ 7,8,22,23].
Acknowledgement The author would like to thank Professor J. Stecki for a helpful discussion.
Appendix Let us determine
the matrix A-‘, where A= [I -BG (k) 1. From eq. (46) we have
A=I-bB(I-3~~)=(1-bB)I+3bB~~=(l-bB)
I+ $+ (
&(k)[I+D(k)U]. >
(A.1)
In general for a square matrix M of order 3 x 3 we have [ 24 ] det(I+M)=l+tr(M)+
C det(M,)+det(M), n
where M, is the submatrix of M obtained det(M)=Oanddet(M,)=Oforn=1,2,3.S0
by crossing out of nth row and nth column
det(I+M)=l+tr(M)=l+D. To calculate the elements
(A.2) of matrix M. In our case
(‘4.3) of the matrix A-’ we use also the equality
det(l+N)=l+tr(N)+det(N),
(A.4)
where N is a square matrix of order 2 x 2. Then we obtain
A-l&+
-+).
(A.51
30
F. Grqdzki / Forcepotentlal between ions at infinite dilutlon
References [ 1 ] A. Bellemans and J. Stecki, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 9 ( 196 1) 339, 343, 349. [2] J. Stecki, Advan. Chem. Phys. 6 (1962) 413. [3] D.W. Jepsen and H.L. Friedman, J. Chem. Phys. 38 ( 1963) 846. [4] I.R. McDonald and J.C. Rasaiah, Chem. Phys. Letters 34 (1975) 382. [ 51 G.N. Patey and J.P. Valleau, J. Chem. Phys. 63 ( 1975) 2334. [ 61 H.L. Friedman, A Course in Statistical Mechanics (Prentice-Hall, Englewood Cliffs, 1985). (71 F. Gradzki, Acta Phys. Polon. A 61 (1982) 405. [S] F. Grgdzki, Acta Phys. Polon. A 63 (1983) 707. [9] F. Gradzki, Acta Phys. Polon. A 67 (1985) 631. [IO] F. Gradzki, Acta Phys. Polon. A 71 (1987) 591. [ 111 M. Mandel and P. Mazur, Physica 24 (1958) 116. [12]M.S.Wertheim,Mol.Phys.26(1973)1425;33(1977)95;34(1977)1109;36(1978)1217:37(1979)83. [ 131 F. Gradzki, Chem. Phys. 125 (1988) 247. [ 141 F. Gqdzki, Chem. Phys. 130 ( 1989) 167. [ 151 J. Stecki, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 9 ( 1961 ) 429. [ 161 J. Stecki, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 9 ( 1961) 435. [ 171 F. Gradzki, J. Sol. Chem. 17 ( 1988) 1101. [18]M.S.Wertheim,Mol.Phys.25(1973)211. [ 191 J.H. van Vleck, J. Chem. Phys. 5 (1937) 556. [ 201 J.S. Hoye and G. Stell, J. Chem. Phys. 73 ( 1980) 46 1. [21] G.J. Throop and R.J. Bearman, J. Chem. Phys. 42 (1965) 2408; 42 (1965) 2838. [22] F. Gradzki, Acta Phys. Polon. A 58 (1980) 9. [ 231 F. Gradzki, Acta Phys. Polon. A 59 ( 198 1 ) 123. [ 241 H.L. Friedman. J. Chem. Phys. 76 ( 1982) 1092.