Point of departure is an extended. Kirkwood-Smoluchowski. (K-S) equation for the pair- correlation function (PCF) which is applicable to a rather large range of ...
Physica 145A (1987) 361-407 North-Holland, Amsterdam
PAIR-CORRELATION FUNCTION OF A FLUID UNDERGOING A SIMPLE SHEAR FLOW: SOLUTION OF THE KIRKWOOD-SMOLUCHOWSKI EQUATION H.-M.
KOO
and S. HESS
Institut fiir Theoretische Physik, Technische Universitiit Berlin PN7-1, D-loo0 Berlin 12, Fed. Rep. Germany
Received
9 March
Hardenbergstr.
36,
1987
Point of departure is an extended Kirkwood-Smoluchowski (K-S) equation for the paircorrelation function (PCF) which is applicable to a rather large range of shear rates. A solution procedure for the K-S equation in a stationary plane Couette-flow (simple shear flow) is outlined where the vorticity of the flow field is taken into account exactly and a perturbation calculation is made with respect to the deformation rate associated with the symmetric traceless part of the velocity gradient field. The first order solution is obtained explicitly and discussed in details. To obtain specific results, the intermolecular potential is assumed to consist of a hard core and a soft attractive force. Especially for hard spheres we obtain a completely explicit expression for the PCF, which is displayed graphically for some typical cases. It shows the ellipsoidal distrotion of the PCF (or the structure factor) observed by experiments”‘) at lower shear rates and a twisted distortion at higher shear rates.
1. Introduction The local structure of a fluid is described by the pair-correlation (PCF) g; its spatial Fourier transform determines the static structure a nonequilibrium situation, the local structure differs rium radial distribution. Experiments on suspensions
function factor. In
from the usual equilibof strongly interacting
spherical particles and nonequilibrium molecular dynamics simulation?) in model fluids have demonstrated the shear-flow-induced distortion of the paircorrelation function and its relation to the viscous flow behavior. This article is devoted to a calculation of these nonequilibrium phenomena starting from a kinetic equation for g(r). An explicit solution is given and presented graphically for a fluid of hard spheres. The equilibrium pair-correlation function (EPCF) g, is determined by the canonical distribution. In nonequilibrium, the PCF g may be deduced from the dynamical laws which describe the movements of the individual molecules
362
H.-M.
microscopically, Liouville’s
departure. K-S
e.g. from the BBGKY”,‘,”
S. HESS
) hierarchy
which follows
from the
equation.
In section equation
KOO AND
2 we will introduce
for
the
pair-distribution
Homogeneity
equation
an extended
we mean
Kirkwood-Smoluchowski
function
an equation
R,;t) as the point
p”‘(R,,
of the fluid system which
is assumed.
F in the extended
K-S
is valid for the extended
equation
of
By the “extended”
shear rates. In the stationary Couette-flow the extended K-S equation5,*) become of the same form. If the mean force
(K-S)
range
of
and the traditional
is approximated
by that in
equilibrium F eqand it is represented in terms of the EPCF g,,, the extended K-S equation is non-autonomous in the sense that it contains no information about the EPCF g,,, although it is trivially satisfied by g,,. In order to obtain a complete solution for g, we should know the EPCF g,,, which is to be calculated otherwise. In the appendix we introduce the Ornstein-Zernike’-‘*) theory for the density-density correlation and an approximation for g,,. Any kind of intermolecular potential may be introduced, but in this work the intermolecular potential is assumed to consist of a hard core and a weak soft (attractive) force which are spherically symmetric. The hard core provides us with a convenient boundary condition and certain advantages in calculations. The crucial point in obtaining the interesting solution for the PCF g is the perturbation calculation of the extended K-S equation with respect to the “deformation” which is the symmetric part of the shear-rate tensor. In the earlier perturbation calculations”.5) of the K-S equation the “whole” shearrate tensor was taken as a perturbation parameter, which is to show only the linear effect of shear rate in the first-order perturbation calculation. In sections 2 and 4 the extended K-S equation for g is solved in the first order of the deformation, where the effect of vorticity, which is the antisymmetric part of the shear-rate tensor, is taken into account exactly. Even this first-order solution shows interesting nonlinear effects. A nonlinear dependence of g on the shear
rate was obtained
the moment method13) damping term 14).
previously
and through
by solving
the K-S
the relaxation-time
equation
through
approximation
for the
In section 5, the solution for g is depicted graphically. Qualitatively phenomena observed in computer simulations”‘) and in experiments colloidal
2. Kinetic
dispersions’)
the with
are recovered.
equation
J.G. Kirkwood derived distribution function f(*'(R, _~_1_ I ,r. 1w.
a generalized
Fokker-Planck
equation
for
,R2,u, , u2;t), where u, and u, are the velocities ..
.
.
.
_
.. .
.
‘I\
-
the of .
PAIR-CORRELATION FUNCTION OF A FLUID
363
generalized Fokker-Planck equation for f(‘) one can, in turn, derive an equation for the configurational pair-distribution function pC2’(R,, R2) defined by P
c2):=
fc2) d3u, d3u2 .
(2-l)
It reads7)
+
di$l &,.(- k, $,(‘ip(‘)) +pc2’(F;- mtii) I
= 0,
(2.2)
I
where
u; =
a z
ui + ui -
-!u, . dRi
(2.3)
The temperature and the flow velocity at Ri are denoted by Ti and ui(i = 1,2) respectively. The friction coefficient 5 is assumed to be isotropic and constant, k, is the Boltzmann’s constant, and Fi is the mean force acting on a molecule at Ri in the presence of another at Rj (i # j = 1,2). Kirkwood et al.“) derived the equation (2.2) without the inertial term mlii. The equation for pc2) is similar to the Smoluchowski equation for the oneparticle distribution function p(l) in configuration space. So we call the eq. (2.2) the extended Kirkwood-Smoluchowski (K-S) equation for p(*! The difference of the extended K-S equation (2.2) from the K-S equation derived by Kirkwood et al. is the appearance of the inertial term mri, and stems from the different kind of linearization in deriving the corresponding equations: in deriving the extended K-S eq. (2.2) the linearization with respect to the “peculiar” flow velocity
(u;)(~) -
ui = j-
(v; -
u;)fc2)d3v, d3u2
(2.4)
is made. Notice that average in (2.4) still depends on the positions R, and R,. In deriving the traditional K-S eq. (without the inertial term mzii) the linearization with respect to the flow velocity ui (i = 1,2) is introduced. In the latter situation one should be restricted to small flow velocities, while the extended K-S eq. (2.2) may be applied to quite a large value of flow velocities7). Moreover, it is physically reasonable that the accelerated molecule should see practically the effective force (F; - mti,) in eq. (2.2) in the system
364
H.-M.
In order
g@,,
to obtain
an usuable
Rz) := pY&
we assume
The
4)
equation
for the PCF g(R,,
R2) defined
>
p(“(R,)
=: no.
upon
the relative
(2.6) coordinate
R: = R, - R2 only, (2.7)
flow for
this
homogeneous
system
is the
plane
Couette-flow
by 0
u;=R;y
,
0
0
y= i y 0 0, i 0 0 0
(2.X)
where y is the shear rate independent of Ri (i= 1,2). The mean force Fi acting on a molecule at r, may be approximated equilibrium’),
F_SF”q=-k I
T
Llng
B I’ aR#
I
Substitution K-S
by
(25)
= g(R).
simplest
defined
S. HESS
the fluid to be homogeneous,
the PCF depends
g(R,,
AND
R2)I{P”‘(R,)P”‘(R*)}
T, = T, =: T,, , Then
KOO
of eqs.
(2.9)
I’ .
(2.5-2.9)
into
by that in
eq. (2.2)
equation
for the PCF g:
$g+r.
i y-!+ - i -vg+v*j=o,
leads
to the following
extended
(2.10) j=
-?(Vg
+ T,‘gVw)
,
where w := -knT,,ln g, is the potential of mean force in equilibrium in the configurational relative pair space. The relative coordinate vector r is scaled by an effective diameter a; some abbreviations and notations are introduced, r:=
Rlu, V: =
alar,
T,:=k,T,,
~=cdD,
D:=2k,T,,l{. (2.11)
PAIR-CORRELATION
FUNCTION OF A FLUID
365
The relaxation time coefficient 7 and the diffusion coefficient D are introduced phenomenologically, and here we are not concerned with the calculation of the friction coefficient 5. In the eq. (2.10) j( r ) is the excess probability current density4-‘) in the configurational relative pair space. The equation of continuity for the number density p”)(R,) has been used in order to get rid of time derivatives of p(“(R,). In deriving eq. (2.10) the terms involving the center-of-mass coordinate R,, = 2-‘(R, + R,)l CTvanish under the assumption that g = g(r) is independent of R,, in the plane Couette-flow, and F, + F, g 0. The time derivative term of the shear-rate tensor in eq. (2.10) stems from the inertial term mzi, = $/at + ui - dujldRj in eq. (2.2), where the term nonlinear in flow velocity ui - au,laR, vanishes identically in the plane Couetteflow. It should be underlined that this nonlinear term disappears not through the linearization but due to the geometrical property of the plane Couette-flow, whereas in the traditional K-S equation the nonlinear term does not appear at all because the linearization in the flow velocity is introduced from the beginning of the derivation of the equation. The time derivative of the shear-rate tensor dylat vanishes also in the stationary Couette-flow and the eq. (2.10) becomes of the same form as the traditional K-S equation for PCF. But the qualitative features of the extended and the traditional K-S equation in a stationary Couette-flow are somewhat different from each other in their “hidden histories” of linearization and in the shear rate range of their applicability: As mentioned before, the former is derived through the linearization in the peculiar flow velocity (u - Us) and may be applied to quite a large shear rate, while the latter is restricted to small shear rates and flow velocities. In a strongly oscillating Couette-flow the time derivative of shear-rate tensor drldt is expected to give some significant contribution. We now introduce an intermolecular potential of the following type:
z@/(T) = 7Jh(RI(T)+ u,(RI(T) )
(2.12)
where u, is any (attractive) soft force. The hard-sphere potential uh may represent the strongly repulsive force part of an intermolecular potential effectively, where the effective hard sphere diameter depends both upon the temperature and the density1s-‘6). A s we shall see in the following, the decomposition (2.12) of the potential is quite advantagous.
H.-M.
366
3. Perturbation
expansion
and reduction The
g =
AND
of the
S. HESS
in the deformation
of the system of differential
deviation
characterized
KOO
nonequilibrium
by P or x according
parameter,
boundary
conditions,
equations pair-correlation
function
from
g,, is
to
&,(l + x>=go+ gY2+> (3.1) -112
x
=
go
v’.
Next, the shear-rate and its antisymmetric
tensor is decomposed part 7, as follows’.”
into its symmetric
traceless
part 3/c,
(3.2)
The first term (symmetric traceless part) is associated with the deformation effect of the shear flow on the fluid system and the second term (antisymmetric part) causes a rotation. Since the deformation parameter yd and the rotational parameter -y, in eq. (3.2) are, in principle, two different physical quantities, they should be considered as independent ones. Thus each one of them can be an independent perturbation parameter. Here we take only the deformation parameter yd as the perturbation parameter in solving the differential equation (2.10), since -yr can be taken into account parameters are of equal magnitude, Yd = -Yr =
where
In plane
Y/2,
y is the shear
Substitution
exactly.
Couette-flow
both
(3.3) rate.
of eqs. (3.1-2)
into eq. (2.10)
leads to
(3.4) where A is the Laplace operator, respect to r, and $ is the azimuthal
the prime denotes the differentiation with angle in polar coordinates. Notice that the
PAIR-CORRELATION
time dependence
FUNCTION OF A FLUID
367
of shear rate is not excluded. The function V= V(r) is defined
by (35) For the stationary
Couette-flow
one has
aylat=o, a$lat=o.
(3.6)
The inhomogeneous term in the r.h.s. of eq. (3.4) is associated with the deformation parameter y,, only. This inhomogeneous term is small over the whole range of the radial variable r except for near the hard core. The perturbation expansion of the equation with respect to yd seems to be appropriate, although the convergence of the expansion is simply assumed, (3.7) On substituting eqs. (3.6-7) into eq. (3.4) and comparing the perturbation coefficients {I+!J~: n = 0, 1,2, . . .} in each power of (7yd), one obtains the following differential equations for { &} :
(L + (vr)aia+)+l
=
4(xyW(gA’2)’ + $tbo,
(L + (7-yr>a/a+)~n+, =Z&bn, for nZ1, where A := xalay+ yaiax-
&=A+V,
Next, each 1cI, is expanded azimuthal angle 4 as follows: CCI, =
2
!Prm)(r,0)
*=--m
T;‘(xY/~)w’
again in Fourier-series
ei2@ ,
for integer
Eqs. (3.8) obey the property
(3.10).
for n L 0 .
(3.8) with respect
n 2 0,
where use is made of the symmetry of the Couette-flow, (ci,(I, 8, 4) = +n(r, 8, 4 + 77-),
.
to the
(3.9) i.e. (3.10)
H.-M. KOO AND S. HESS
368
We substitute following
the
differential
{Wrm)}
in polar
Fourier
expansion
equations
(3.9)
into
(2, + Y_) L, + i(2m7y,))YPyI’“‘(Y,
0) = (a,,,, - 6,,,
_
and obtain
the
functions
(3.11a)
0) = 0,
i(m
(3.8)
perturbation
coordinates:
(L, + r-*15, + i(2m7y,))!P~~2”)(r,
+
eqs.
for the two-dimensional
1 _
,)(-i
~)QV~f’“p2’ (r, 0) + i(m + 1 + fi)?Pif”’
(L, + rp2 LH + i(2mry,))?P~~~)(r, + i(m + 1-t fi)!Pj12m+2)(r,
sin%)y(gX”)’
+*)(Y, 0) ,
(3.11b)
0) = i(m - 1 - A)Wj,z’71m”(r, 0) 0) ,
for n 2 1 ,
(3.11c)
where L, := Y2(dl&)(r2
aiar)
Le := sin-‘8(~/~8)(sin A := -(4T,)-‘rw’ 6,n,:=l
or 0,
+ V(r) ) 0 a/~?(?) - 4m*/sin*O ,
(3.11d)
sin20 + 2-l sin28 r dldr + 4-l sin 26 a/J0 for m = II
or
m # n
respectively
,
.
is self-adjoint The differential operator (i, + r -‘L,) on the 1.h.s. of (2.11a-c) with the pertaining boundary conditions’) and the remaining term is a constant: if x (cf. (3.1)) would have been used instead of W, this would not be the case. The hierarchy of differential equations (3. lla-c) But through the hermitian property of the differential with the pertaining
boundary
simple system of equations for the first or the second
conditions
looks very complicated. operator (2, + r-‘&)
for Pj12”‘)(r, O), it can be reduced
to a
in which only a few members of !P!2’n)(r, 0) survive order perturbation theory (n = 1,2).
Since our model potential of intermolecular force consists of a hard core and a weak soft (attractive) force, the relative pair space is practically enclosed by two boundary surfaces: the hard sphere surface of radius u and the infinite sphere. We now examine the excess probability current density j, which is introduced in eq. (2.10), on these boundary surfaces in order to find boundary conditions. When two reference molecules are separated by a large distance, the presence of one molecule is ignored by the other, and this implies that the excess probability current density in the configurational relative pair space vanishes at the infinite surface8.5),
PAIR-CORRELATION
FUNCTION OF A FLUID
iiir j = 0 .
369
(3.12a)
The hard cores of any pair of molecules may not overlap one another. The molecules rebounds from each other elastically at the point of contact, and so the radial component of the excess probability current density should vanish at r = 1 in the reduced relative pair space5); ( j)radia, = 0
(3.12b)
at r = 1 .
On substituting the expression for j from (2.10) into the eqs. (3.12a, b), we can write the boundary conditions in terms of the 2-dimensional perturbation functions {!Py”‘(r, 0)}:
!& (goa(&F* ?PYrn))l,r) = ~&l((2T*)-1w’?Pj;2”’ + #PY%Jr)
=0 (3.13a)
and g,d(kC2
T(*“))/ar n
= g~‘2((2T,)-1w’!P~“)+
13?Pr”)lar) = 0,
at r = 1 , (3.13b)
where eqs. (3.1,7,9) for !Pr”), W: = - T,ln g,, and the boundary condition lim,,, g,,(r) = 1 have been used. The super- and the subscript m and rz are integers restricted to n 2 12ml. The boundary conditions (3.13a, b) are the mixed form of homogeneous Dirichlet and Neumann type”). Let G be the 2-dimensional Green’s function pertaining to the differential operator (i, + r-*iO). Then it can be shown that the operator (i, + r-‘LO) becomes hermitian if its eigenfunctions or the Green’s function G satisfy the same boundary conditions as eqs. (3.13a, b) for !Prm)(r, 0), /u+m_ (g, a(g,“*G)lar)
= li$(2T,)-‘w’G
+ aG/ar)
=0
(3.14a)
and &I
a(gfY*G)ldr
= gi'*((2T,)-'w'G
+ dG/dr) = 0,
at r = 1 .
(3.14b)
Since the eigenvalues of the hermitian operator i,. + r-‘&) are real and the number i(2mq) is purely imaginary, the solution of the following homogeneous differential equation,
(It, + r-*& + i(2mry,))!P = 0
H.-M.
370
is trivial,
i.e.
differential
?#J= 0. This
equations
lpm)
=
0
implies
@W I
= 0 ,
if m#-l,l,
@W 2
= 0 ,
if m # -2,0,2,
pm4
=
if m#-n,-n+2,.
0
n
In particular,
,
the
S. HESS
following
reduction
of the system
ol
(3.1 la-c):
for all integer
>
0
KOO AND
one
obtains
m ,
(3.15)
. . . ,n-2,n.
for the zeroth-,
the first,
and the second-order
perturbation
(3.16)
The result &, = 0 is consistent with the fact that the deviation factor I/J should be zero when there is no shear flow. As we have seen up to now, the infinite Fourier-series for +!I,,,,has been remarkably simplified. Moreover, as will be shown later explicitly for the first-order perturbation, lJIyrn)(y, 0) and pL-2m)(r, 0) are complex conjugates of each other, and so +!I,,becomes a real function in a natural way.
4. Solutions
Since
we
have
derived
the
differential
equations
(3.11a-c)
for
the
2-
eqs. (3.15), we dimensional perturbation functions { qyn(2m)} with the reduction, can now solve them systematically. In this work, only the equation of firstorder perturbation (3.11b) will be treated. The higher-order perturbations may be obtained hierarchically by using the equations (3.11c), where the same mathematical formalism is applied as in the case of first-order perturbation calculation which already reveals some interesting physical features. The extended K-S equation (2.10) for g in plane Couette-flow is not autonomous in the sense that it says nothing about the equilibrium paircorrelation function g,,, although g = g,, is the trivial solution. The EPCF g,, is to be calculated otherwise. The Ornstein-Zernike equation for density ) and its approximation are introduced in appendix in order to correlation9-“.5 calculate g,, .
PAIR-CORRELATION
FUNCTION OF A FLUID
371
Firstly, the general solution of first-order perturbation will be obtained. Secondly, this general solution will be shown explicitly by substituting the explicit form of g, into it. Especially the solution for the hard-sphere system will be obtained in its complete explicit form. 4.1. The general solution The starting point in this section dimensional perturbation function,
(L, + r-‘L,
+
ki)W’,2”’(r,
is eq. (3.11b)
0) = (-im
for the first-order
sin’e)r( gA’2)’ ,
2-
(4.1)
ki = eim’4 )2m~--y,1”~for m = +l, where ~~y,( = y/2 in plane Couette-flow) is assumed to be positive, so that m = sign(mry,) for m = +-l(the relaxation time
with
coefficient r > 0), and V0(2m)= 0 from the reduction of equations (3.15). Eq. (4.1) represents a pair of conjugate equations. So, as already mentioned, ?Py’ is the complex conjugate of .rY.(l-”and vice versa. We also introduce the following equation for the 2-dimensional Green’s function pertaining to eq. (4.1): (i, + rm2i, + ki)G(r,
Bjp, 8,; m) = (r2 sin O)-‘8(r - p)6(8 - 0,) ,
(4.2)
where the right-hand side is the a-function expression with the proper weighting. By using this Green’s function the solution of (4.1) can be written formally as follows:
p, 0,; m)(-im 1
sin2B,)p3{ g:“(p)}’
sin 0, de, dp
0
,
(4.3)
where the homogeneous solution does not appear, for the differential operator (i, + rm2.&) is hermitian and ki = i(2mryr) is purely imaginary, as mentioned in the last section, and the radial variable p in the integral runs over the range 15 r < 00 because of the particular ansatz for intermolecular potential (2.12). Now, our task is reduced to the calculation of the 2-dimensional Green’s function G(r, flip, 0,; m) in (4.2). The left-hand side of the eq. (4.2) is separable, and J$ := sin-‘8(dlM)(sin
0 a/&9) - 4m2/sin2B ,
is just the Legendre differential
operator
defined in (3.11d) ,
whose eigenfunctions
and the corres-
372
H.-M. KOO AND S. HESS
ponding -l(l+
eigenvalues l),
over
are
respectively.
the whole
range
the
That
associate the
Legendre
eigenfunctions
of 0(0 5 8 5 rr), requires
functions
Pf”‘(cos 0)
and
PF” (cos fI) should
be finite
I to be an integer
larger
equal to 12ml. The Green’s function G(r, 81 p, 0,; m) may be expanded eigenfunctions of angular part as follows:
or
in its
r G(r, 81p, 8,; m) =
2
(T,(v(p;
m)(21+
l)([ - 2m)!/{2(1+
2m)!}
142m
x
Pf” (cos 0) Pf”(cos
0, )) ,
(4.4)
where the nondiagonal terms do not appear, since the diagonalized by the eigenfunctions, and the normalization the following orthogonality relation:
Green’s function is factor comes from
n
J
PT”(cos
B)PfT”(cos
f3) sin 8 d0 = {2(1 + 2m)!}{(21
+ 1)(1-
2m)!}-‘a,,.
.
0
The coefficient T,(rlp; satisfies the following (4.4) into (4.2):
(E + V(r))c(rlp;
m) in (4.4) is the l-dimensional Green’s function which equation obtained by substituting G(r. 81 p, f3,; m) from
m) = re’6(r
- p) ,
(45)
where i := d21dr’ + 2F’dlar
+ kz,, - 1(1+ 1)~~’
is the spherical Bessel differential operator which stems from L,- defined in (3.11d). Now, our problem is further reduced to solving the linear ordinary differential equation
(4.5).
Since
V(r)
in (4.5)
is quite
a complicated
function,
it is
desirable to consider this function as a perturbation and obtain an approximate solution through perturbation expansion. But the function V(r) contains some singularity at r = 1. So it is not appropriate to take it (including the singularity) as a perturbation. If one does it, nevertheless, one meets a severe difficulty with the boundary condition (BC) for T,(rlp; m) at r = 1. (Since the BC for G at r = 1, (3.14b), contains the singular effect of the mean force w’, the BC of the l-dimensional Green’s function T,(rlp; m) and so that of the unperturbed solution Ty(rlp; m) at r = 1, (4.7). should also contain it. If the whole part of V(r) including the singularity is taken as a perturbation. the unperturbed
PAIR-CORRELATION
373
FUNCTION OF A FLUID
solution Tp(rlp; m) of eq. (4.7), which is obtained by neglecting V(r), won’t contain the singular effect of the mean force w’. But this result is not consistent with the BC at r= 1 (3.14b).) Fortunately the singularity of the function V(r) at r = 1 may be cancelled out by employing the singular effect of the differential operator b at r = 1 in (4.9, where use is to be made of the boundary condition for r,(rlp; m) at r = 1, which is to be derived from the BC for G at r = 1 (3.14b),
(16+ W)rl(hJ; = ({-V(r)
+ k;
m>),=, - l(l+ l)r-* + V(r)}r,(rlp;
= ({k:, - l(l+ l)W,(+;
m)),,,
m)),=, .
Thus the equation (4.5) may be rewritten in the following way:
(6 + V(r)}l;(rlp; m) = {fi” + V,(r)}r/(rlp;
m) =
r-%(r -
p)
)
where B+V, &
.if r=l, if r>l, (4.6) if r=l if 1->1:
v”(r)‘=(“v;,,
The new function V,(r) defined above contains no singularity and is small in the whole range of definition (r 2 l), and so may be taken suitably as a perturbation to solve the equation (4.6). The equation for the unperturbed Green’s function r:(ll p ; m) is
B,rp(rlp;
m) =
r-*qr- p)
(4.7>
)
where the superscript “0” means “unperturbed with respect to the function Vo(r)“. Then one can write the linear inhomogeneous integral equation for T,(rl~; m) introducing ry(rlp; m) as follows18): r T,(~/P; m) = Tp(rIp; m) - 1 rp(rlr,; 1+
m)V,(r,)T,(r,Ip;
This can be solved by iteration in the following way:
m)r:dr,
.
(4.8)
H.-M. KOO AND S. HESS
374
z
r:(rlP;
m) = rp(rlp;
m) - 1 T:‘(rlr,;
m)V,(r,)T:(r,Ip;
(4.9)
m)rT dr, ,
and so on, where the superscripts “0, 1,2, . . .” denote “the zeroth-, . . order perturbation”, respectively, and the point k](l) = 0. The
starting
point
C(rlp; m>of(4.7),
of the
iterative
which satisfies
solutions
the similar
is the
the first-, the secondr, = 1 is missing
unperturbed
BC’s to (3.14a,
for
solution
b) at r = CCand
1,
? := {B,(r; m)(dldr)A,(
r; m) - A,(r; m)(dldr)B,(r;
m)>,=,
is the Wronskian of the two functions at r = p. The solution (4.13) satisfies the BC’s (4.10a, b) both at r = COand 1 automatically, if the first derivative of Tp(rlp; m) at r = p is defined by
@~dr)~%b; m>= {p24B,,
A,; P))-'A,(p;
m)(dldr)B,(r; m) ,
for r = p. Since the operator &, for r = 1 in (4.6) is different from that for r > 1 by V, the l-dimensional Green’s function ry(rlp; m) or the two functions A, and B, should be calculated separately for the two different cases: a) r > 1 and b) r=l. a) Calculation of the unperturbed sphere (r > 1).
1 -dim. Green’s
function
outside the hard
The solution A, of the spherical Bessel equation (3.11a) can be immediately determined by the corresponding BC, (4.12a), as follows: A,(r;
m) = h,((3-m~)(~,~)
.
(4.14a)
The general solution of the spherical Bessel equation B,(r; m) = hj”(k,r)
+ C,,,hi2’(k,r)
.
(4.11b) is (4.14b)
The r.h.s. of (4.14a and b) are the spherical Hankel functions of first (if m = 1) or second (if m = -1) kind, and their linear combination, respectively. The proportional constant in each solution of (4.14a, b) is not needed, since it is canceled in (4.13). The coefficient C,,, cannot be determined by the BC for B,. (4.12b), since
H.-M. KOO AND S. HESS
376
the solution excluded functional
is restricted and
the
form from that at
We now (4.14b).
(4.14b)
in (4.14b) should
may be rewritten
(jr)n.2m
where
:=
the boundary r =
r =
is different
1
the coefficient
of the excess
near the boundary
1 is
in its
r =
probability
1 is needed.
C,,,
in
current The radial
as follows: z
j, = -r-‘g,,
1:
at
to determine
examination
density j, introduced in (2.10), component of j in (2.10),
r >
by (4.14b).
condition
a deeper
region
of (4.11b)
1 given
r >
find a new
To this end
to the
solution
a( g,“‘iJ)
ldr
=
-112 -7-lgo
d{g,
use has been
made
z
C 2 n:,“,=-Z
!P( ,P
)(r,
((7~~)”
O)}
e’2’n’(
j,),,
2m)
ldr ,
of the expansions
(3.7 and 9) and the component
doesn’t appear, since Vf:,rm’ = 0 for (i&n, (3.12b) forj at r = 1 was j, = 0 or equivalently values of yd. Since g,,(r) is discontinuous at r = r = 1, this boundary condition at r = 1 implies
all m. The boundary condition ( jr),l.2,,l = 0 at r = 1 for arbitrary 1 and its derivative is singular at neither j, = 0 nor ( jr)n,2,X = 0 at
as can be seen in the above expression. We should find some proper physical assumption in order to derive the condition for j, at r = l+. What kind of condition for j, can be given at r = l’? Now the radial component of the excess probability current density, j, may be decomposed into the hard core and the soft force contribution in the following way: r=l+,
j, = j,"+ j;', z
x
j,“.“Cr.0, 4) = ,,=, C C ((q)” e’2m’(j,Y(r,e),z.2n,) m=-r
(j,“(r, e>),,2m:=
-7
-‘g,(r)
j+Jd(G(r,
O[r,,
0,;
m)lg~‘*(r)}ldr
1 0 (4.15) xRr(r,,
sin 8, de, dr, ,
0,) r
(jS(r,e))n,2m := -T-lgJr) XRT(r,,
7-7
II a{G(r, dr,, 0,) sin 8, de, dr, ,
0,; m)
/g:“(r)}
ldr
PAIR-CORRELATION
FUNCTION OF A FLUID
377
where
R;(r, 0) := &,, - S,,_,)(-i
sin%)r( gA’2(r))f ,
Rr(r,0) := i(m - 1 - fi)Pfme2)(r,
0) -t-i(m + 1 + fi)Pym+2)
,
for nZ2, which are the inhomogeneous terms in the differential equations for the 2-dim. perturbation functions 9:“’ (3.11b and c). In (4.15) the corresponding 2-dim. Green’s function, G, defined by the equation (4.2), has been used. The radial component of excess probability current density represented in terms of the Green’s function G in (4.15) may be interpreted as the sum of the flowing effect due to the source (inhomogeneous term) Rr at (rl, 0,) on the flow of molecules at (r, e), where the relation between the source and the flowing effect is given by the Green’s function. The source term Rr(r,, 0,) contains the effect of the mean force at (ri, 0,) as can be seen in its definition in (4.15). This mean force has singularity at rl = 1. But, as one can see in (4.15), the soft force contribution jf does not contain the singular effect of the source term. We will see later in b) that the Green’s function G is proportional to g:“(r) at r = 1, and so no singularity arise from the both integrals in (4.15). Thus not only the sum of jr and js but also each of them vanishes at r = 1 seperately, since each of them is in proportion to g,,(r) which is zero at r = 1, as shown in (4.15). We now assume that the soft force contribution to the radial component of the excess probability current density goes to zero continuously as r- 1. In other words the soft force contribution to the radial component of the excess probability current density should vanish in the very vicinity of the hard core in the relative pair space, lim jS(r, 0, 4) = 0 .
(4.15a)
r-1+
The condition (4.15a) is based on the assumption of continuity of js for r 2 1. We call it “the secondary boundary condition at r = l’“, since it is related to the boundary condition of j, at r = 1. It is noted that j: need not be continuous at r = 1 and so j, need not, too. The secondary boundary condition (4.15a) is equivalent to
for all the allowed integers 171and IZ, as may be inferred from (4.15). On substituting the expansion of G( r, 81r, , 0,) (4.4) into the last expression for of (4.15), the above refined secondary boundary condition at r = li (is),,,,
H.-M.
378
may be rewritten
more
explicitly
KOO AND
S. HESS
as follows: cc g,(r)
c ((21+ /z12m~
2m)!} +;”
I)(1 - 2m)!
(cos 0)
lr
x
i 0
Pf”’ (cos e,)RX(r,,
13,) sin 8, d0, dr,)
= 0
Since for 1+ = r < r, &(rIr,;
m) = W~‘,B,(r; X (A,(r,;
from
(4.8)
and (4.13),
lim g,(r)(dldr){ r+1+
m)
m>-
the above B,(r;
Ar(r2; m)h(r2)C(r21r,;m)r;dr,) , condition
m) igb”(r)}
may be written
= 0
(4.15a)’
This is the secondary boundary condition at r = 1+ written function B,(r; m). Here the following condition:
JC(r, j 4 W,(r,
; m) -
as
in terms
of the
dr,) dr, f0 JA,(r2; m>Vo(r,)&(r,Ir,; m>rg
is assumed. Otherwise, from the both equations stated above eq. (4.15a)’ the nth derivative (d/dr)“{ js(r, 0)} = 0 at r = 1+ for all n 20, which could be satisfied only in an unnatural way. By making use of the secondary boundary condition at r = 1’ (4.15a)‘, the coefficient C,, , in (4.14B) into (4.15a)’ leads to (2T,))‘w’(l+){hj”(k,) + ((dldr){hj”(k,r)
or
+ C,~,hjz’(k,r)} + C,,~,hj2’(k,r)}),.,,
+= 0 ,
PAIR-CORRELATION FUNCTION OF A FLUID c m,l
= _cu)/p mJ
m,l
379
2
(4.15b)
with C$
= (2~,)-‘w’(l+)hj”(k,)
C:,; = (27-J’w’(l+)hj*‘(k,)
+ {(dldr)hj”(k,r)},,,+
,
+ {(dldr)hj2’(k,r)}r=1+
,
where w’(l+) = ml+ dw(r)ldr
= -T,
b) Calculation of the unperturbed
!lF+g;(r)Ig,(r)
.
1 -dim. Green’s function at the hard sphere
(r = 1).
Since the pair-correlation function g,(r) the quantities such as g,( 1) /g,(l) = O/O, well defined. So go(r) at r = 1 is considered of such a function g,(r; A) with parameter
vanishes discontinuously at r = 1, gA(1) - g;(l) = CO- 30, etc. are not in the sense of definition as a limit A,
go(r) = pi_ g&i A) T and such indefinite quantities are understood gr,(l)/g,(l) g:(l)
- g:(l)
:= F,m_g,(l; A)lg,(l; := !ym{&,(k
in the following way:
A) = 1,
A) - g;(l;
A)) = 0,
where g,(l; A) is assumed to become infinitesimally small for large A but nonzero, and gA(l; A) to become large for large A but finite. The r-dependence of the functions at r = 1 will be shown in order to perform algebra between them, as mentioned above. One particular solution at r = 1 of the homogeneous equation (4.11a or b) may be found to be g:“(r). Another independent solution at r = 1 is easily determined”) as
d’*(r)i f-*gi’(p) dp . Since the function B, should satisfy the BC at r = 1 (4.12b), it is to be proportional to g:‘*(r) at r = 1. The function A, may be expressed in general as a linear combination of the two independent solutions, where it is noticed that the solution for A, at r = 1 need not and cannot satisfy the BC at r = m (4.12a). Thus one has the following appropriate solutions for A, and B, at r = 1:
H.-M.
380
A,(r;
m) = ay.‘g;,“(r)
B,(r; m) =
KOO
+ c+g,,
p”,‘gi”(r) ,
AND
l’*(r)
S. HESS
1 Fzg,‘(d
(4.16a)
dp 7
at r = 1
(4.16b)
The constants (~‘;l,‘, cr;“, and pm.’ are determined in the following way. On the one hand, one represents or defines the first derivative of B, at r = 1 in terms of a delta-function: Since g:“(r) = 0 at r = 1 and g:,” = g,!,“(l+) # 0 at r = l’,
one has from
Bi(r; m) =
(4.16b)
pm,‘{g;‘*(r)}’
= p”.‘g:,‘*(l+)8(r
m) = a~“{g~,‘2(r)}r
cannot
be
(n > i)
of glf(r)A,(r; and taking
{ g;f(r)A,(r;
use
has
pm2g,y’(p)dp
of a delta-function because delta-function representation
m) at r = 1 for II > i. On multiplying
the first derivative,
m)}’
(4.16a)
by g:(r)
+“*( 1 +)S(r
+ a;“{
- 1) ,
g:;+“*(r)}’
at r = 1 ,
made
(4.16d)
of the fact that g:;-“‘(l) = 0 for rz > 4, and {$r,:‘*(r)}’ = g;f+“*(l’)t?(r - 1) at r = 1, since g:+“’ = 0 .at r = 1 and for n > i. The value of the integral at r=l+ go = gu“+“2(1+)#0 lrp-*g,‘( p) dp at r = 1 is independent of the “hidden” parameter A introduced in the beginning of this paragraph and so is identically zero at r = 1. On the other hand, since the functions g:fA, and B, jump from their zero at r = 1 values at r = 1 to their nonzero values at r = l+, their first derivatives may be written
been
of the term of the first
one obtains
= a’,““{ g:“‘*(r)}’
= ay”git where
+ a~“‘{g,!~“(r)}‘/
well defined in terms We examine rather the
derivative
(4.16~)
;,,lr-2g~112(r)
+ff
g;“*(r).
at r = 1 .
of A, in (4.16a)
The first derivative
A;(r;
- 1) ,
in terms
of delta-function
as follows:
PAIR-CORRELATION
FUNCTION OF A FLUID
-M%-)A,(r;ml>’ = g;f(l+)A,(l+;m)s(r -
381
1)) at r=l.
(4.16e)
&(r; m) = B,(l+; m)s(r - 1) ) By comparing eqs. (4.16~ and d) with eqs. (4.16e) one may determine values of (Y;,’ and pm,‘, a;*‘=
A,(l+;
m)/g;“(l+)
= hj(3-“)‘2)(k,)/g~‘2(1+),
pm.! = B,(l’;
m)/g;‘*(l+)
= {hj”(k,)
the
(4.16f) + c,,,hj*‘(k,)}lg~‘*(l+)
)
where the values of A, and B, at r = l+ are substituted from eqs. (4.14a, b). The remaining constant a:” can be determined in such a way that the Wronskian of A, and B, at r = 1 may have the same value as that at r = l’,
A(B,, A,; 1) = A(B,, A,; 1’) > or
P
m,l
m,l a2
=
wm,,
(4.&d
3
or (Y;,’ = W,.,Q?“~ ) where W,,,, is defined in eq. (4.13) and may now be calculated by using eqs. (4.14a and b) as follows: W,,, = r2A(& A,; r) = (-2ilk,)C~I,-““2’lC~,~
.
(4.16h)
The function V,(r) of eq. (4.6) is related to the long-range behavior of the equilibrium correlation function. Since the inhomogeneous term in the equation for *(12@(r, 0) (4.1) contains the effect of the long-range behaviour of g, and one can see how this term affects ?Pyrn) (r, 0) in the integral representation (4.3), one does not lose the first-order effect of the long-range behaviour of g, on F(12m)(r, O), 1‘f one uses the unperturbed Green’s function given by (4.13) in the integral (4.3). More generally speaking, one obtains the effect of the long-range behavior of g, on ?Pym)(r, 0) to the (n + 1)th order, if one employs the nth order approximation of one dimensional Green’s function r;l from (4.9) in the integral (4.3). In this section the unperturbed Green’s function ry only will be considered to avoid the mathematical complication, which still preserves the first-order effect of the long-range behavior of g,.
H.-M. KOO AND S. HESS
382
The unperturbed 4 in (3.4)
G”(r,
two-dimensional (4.13)
qp,
2 Tj)@(p; /Z12W/
q; m) =
This can be substituted zeroth-order
function
m)(21+
1)(1-
by replacing
2m)!{2(1+
2m)!} -’ (4.17)
0,) .
8)P;“(cos
again in (4.3) to obtain
approximation
is obtained
as follows:
x P;“(cos
bation
Green’s
by r:’ from
for the deviation
the following factor
expression
with respect
for the
to the pertur-
V,,(Y):
=
!zrn,((-i
m)P~"(COS 0)
Jr:)(rlPi m)(gi’2(p))‘p” dp I
x(21
+ 1)(1 - 2m)!{2(1
+ ~wz)!}~
J 0
Pf”‘(cos
0,) sin38, de,)
Since
(21+
= the above
1)(1-
2m)!{2(1
+ 2m)!} -’
I
P~“‘(cos 0,) sinjO, do,
6,,,(%,,/3 + f%n.-l>>
(4.18)
for “9\2’pZ)(v, 0) may be simplified
expression
“P~2m)(r, 0) = (-i
x
m)P~“‘(cos
~%IP;
I
= (-i
13)(6,,,/3
+ 86,,,,_,)
Ng:‘2(~))‘~3
~2/3)(6,~.,
dp
+ G,,~,)P~(cos
0)
x x I r;(rl
P; m)(d,‘“(
P))‘P’
dp .
as follows:
PAIR-CORRELATION
Substituting
(4.13) for ri(rl~;
383
m) into the above equation leads to
o!P(12m)(r,0) = (-i m/3)(6,,, x(A,(r;
FUNCTION OF A FLUID
+ G,,_,)P~(cos
0)Wj$
m) [ B,(p; W’(g%))’
dp
(4.19) where W,,,,2 is defined in (4.13), and the relation P;“(x)
= (-1)“(1-
m)!{(Z + m)!}_‘P:(x)
for integers m,l
)
(I 2 Irnl) ,
has been used. On substituting (4.19) into the reduced expression for I+$, (3.16), one obtains finally the following solution for I/+ in the zeroth-order approximation with respect to the perturbation V,(r), but without losing the effect of the long-range behavior of go, as mentioned before: $y(r, 8, 4) = C oP(12m)(r, 0) ei2+ m=-C1 = (i/3)Pz(cos
e”““(-m/W,.,) nl=*c1
0) C
(4.20) The function (gA’*( p))’ contains the singular effect of the repulsive force at the effective hard core. Since the molecules do “see” the singular force, this effect should be taken into account. In a more natural way one can obtain this singular force effect, if one approximates the repulsive core of the effective potential w(r) by a precipitous logarithmic wall, i.e. w,,,,(r) = -A In r, and takes the limit A+ CCafter calculation, although this method includes much more mathematical complications than the hard-core approach. One can single in the similar way as before: since out the singular effect from (g:‘*(r))’ g;‘*(r) =
0,
1
g;/*(l+)
,
if r=l, if r=l+,
384
H.-M.
one obtains
the following
=
S. HESS
expression:
1
if r = 1 ,
- 1) ,
g~“(1+)6(r
(bnN
KOO AND
(&y2@.))’
)
if r>l,
and similarly
&x4 = {(d’2G9)2Y = 2g;‘2w(g~‘2(r))’ =
g,(l+)a(r-
1))
i &Kr) 7 For
the
sake
of simplicity
of the
if r = 1, if r>l.
formalism
the
(4.21)
following
notations
are
introduced:
ji((3-r)i?)(ky) = Y
1
h;‘3-“‘2’(k,r)r3(g;,‘2(r))’
d,.
,
(4.22a)
V
t(k,)=1j2(k,r)r3(g”:,‘Z(r))’ dr ,
(4.22b)
$k,) =1n,(~,r)r’(&“(r))’ dr ,
(4.22~)
,
wheres=?l,
t=tl;y,z?l;
i”(Y) = ‘d1+ >
7
for r = 1 and
g,,(r)
for r > 1
(4.22d)
It is noted that g”;(l): = g;(l+) in the integrands of eqs. (4.22a-c). The deviation of nonequilibrium pair-correlation function from equilibrium is gox and x was defined by x = g,‘12 Cc,in (3.1). Thus the perturbation coefficient of x corresponding to I,!J,~may be written as P .-
xx .--
g?‘*& >
(4.23)
where the sub- and the superscript, n and p, mean the orders of perturbations with respect to the deformation parameter and the perturbation function V,,(r) in (4.6), respectively. Substitution of (4.20) into (4.23) leads to
PAIR-CORRELATION
xy=(i/3)Pt(cos
f3) 2
FUNCTION OF A FLUID
385
e”““(-s/W,,,)
s=*l
-1’2W42(~; X(g0
s)
j-
B,(P;
4~“(g;‘*(d)
dp
1
(4.20)'
where the factor g,1’2( r ) is combined with the terms in the bracket. This factor is singular at r = 1. Thus the factor gil’* (r) gives some finite contribution at r = 1 when it is multiplied by a quantity proportional to g:‘*(r), where the multiplication is to be understood in the sense of the limit of the “hidden” parameter, as already mentioned. Since the functions A,,B,,and g, are discontinuous at r = 1, the function xy is to be calculated seperately for the two different regions: r = 1 and I > 1. On substituting (4.16a, b) for A, and B, at r = 1, and on making use of (4.21) for the delta-function expression of g;(r) = g,(l+)a(r - l), into the above representation for *y, one obtains the following expressions for the two different regions: For r = 1
xy=(i/3)Pl(cos
/3) C e i2”m(-siW~,2)(,;~2/ ,P2g;‘2(p)p3( S=?l (1)
g;‘2(p))’ dp
P +p*
{a;” + as,’
i
i
T-2d(4 dT}g~‘2(p)p3(g~‘2(P))’ dp
1
(11 m
I
+p
A,(P;
4p3(d’2(pN’
dp)
If
= (i/3)Pi(cos
0) C
ei2”‘(-s/W,
*)
s=z1
x
(2-1(Y;.2p”‘2g,(l+) + 2-‘ps~2cz;~2g0(1+) Cc
+ps,*
I 1+
A,(P;
4p3(d2(pN’dp)
H.-M.
386
= (i/3)Pi(cos
0)
KOO
AND
S. HESS
C e’2S’(-s/IV,,2) s=5l
I
(~~;~2p~y~*go(l+) + pa.* ALP; ~P’(~‘*(PNdp) ;
x
(4.20a)’
I
and for r>
xy=(i/3)PG(cos
0)
C e’2S’(-sIW,,7) 5=t,
X(~c3+42(~;
d[
1
P”,‘s~‘*(p)p3(gt’*(p))’
dp
llJ r +
i
Up;
4p3(d’2b))’
dp
1+
+
s>j”A,(P; G3(d,‘*(d)
-“*W,(r; go
= (i/3)Pi(cos
+B,(r;
s>
6)
i
c ei2”~(-s/WS,2)g~“2(r)(2~‘~“~2g,,(l+)A2(r; .y=tl
A,(P;
4&d,‘*bN’
-B,(r; ~1 A,(P;
X:)(r, 0, qb) = (i/3)Pi(cos
dp
s)p3(gA’2(p))’ dp) ,
where the relation J” = ;+ One substitutes (4r16f \) ,;i . 3 B,, into (4.20a, b)‘, and makes
+
(4.20b)’
m.7 for ‘kl has been used. ,P ’ and Wn12, and (4.14a, b) for A, and uke of the notatidn and definition (4.22a, d),
f3) C e’*“~{(-i/2)sk,Cjf~lC,~j~~“~‘*~} ,=a,
Xg”~1’2(r)(2-‘(1
s)
dp
+A,@; 4 i B,(P;4&d”(d)
x {hi”
dp)
+ S,.,)g:‘*(l+) C,,2h~'(k,)}h~'3~""2'(k,r)
PAIR-CORRELATION
+ c~,,hl”(k,r)}~(‘-~,‘?‘(k,)
+{p(k,r)
+( p
387
FUNCTION OF A FLUID
c,,2~2’(k,)}h:‘3-~,‘z’(k,r)
‘(k,) +
- {hy’(k,r) + cs,2hy’(k,r)}$(3-s)‘2)(ks)) . Since { C~f;/C;~,‘-sV) }({z+k,)
+ c~,2~“(k,)}h:“-~“z’(k,r)
- {h$“(k,r) + c~,2~i2’(k,r)}~(‘-“‘2’(k,)) = h~‘(k,r)$“(k,)
- h:“(k,r)$2’(k,)
)
the above expression becomes ~y(r, 8, 4) = (1/6)Pi(cos
d)gil”(r)
c
ei2sCks. ((s/2)(1 + S,,r)gA’2(l’)
s=*l x { c~yz~“(k,)
+s{
-
Cjyzy’(k,r)
Cjllh~‘(k,)}h~(“-“““(k,r)
-
lcyi2)
Cjt:h:z’(k,r)}1;1”3-““2’(k,)/Cjll-”’”’
+sh:Z’(ksr)~l’(ks)
-
shy’(k,r)~*‘(k,))
)
where the definition C, r: = - C~~~/C~~/from (4.15b) has been used. This expression may be further simplified if one makes use of the following complex conjugate relations: h;“(k+) = hf'(k_)
k.=k+=k_, -
, (4.24)
-
C(1) : = c(:; 2 = c(2j 2 = : c”’ + and the result of the calculation: C!“@‘(k)
- C$%;“(k)
= ((2T,)-‘w’(l+)h;“(k)
+ {(dldr)h;“(kr)},=,)h~)(k) + {(dld~)h~‘(kr)},=,)/z~l’(k), = - kA(h;“, hr’; k ) ,
- ((22-J’w’(l+)hp)(k) from def. (4.15b)
from the def. of Wronskian
(4.13)
= - k(-2ilk2) = -2ilk.
(4.25)
H.-M.
388
Thus
one can write
KOO
the expression
~y(r, 8, 4) = (1/3)P:(cos
AND
S. HESS
for xy as follows:
O)g”,“*(r)
Re(-e’*@‘F(k,
r)) ,
with F(k, r) = i(1 + 6,,,)g~‘2(l+)h:1’(kr)lC(:’ +k{C’:‘h~‘(kr)
- C1Z’hl”(kr)}t;l(“(k)/C:‘:’
+kh:i’(kr)$*‘(k)
)
- khF’(kr$“(k)
(4.26a)
where k and C’:‘, and S;, are defined by (4.24) and (4.22d), respectively. The expression (4.26a) is pertaining to seeing the limiting behavior of x; for large values of r or k. To see the limiting behavior of xy for small k the following
equivalent
expression
X:)(r, 8, qb) = (1/3)P~(cos
for x: may be employed: O)g”,y”*(r) Re(-e’*@‘F(k,
r)) ,
with F(k, r) = i(1 + 6,,,)g:‘*(l’)h:“(kr)lC’:’ +k{(C’:’
- C’,“)j?(kr)
+2ikn,(kr)j(k) where
C(:,*), j(k),
respectively,
b(k),
- i(C’:’
- 2ikj,(kr)k(k) and S;, are defined
+ C~‘)nx(kr)}$“(k)/C’:’ , by (4.15b,
(4.26b) 24, 22b, 22c, and 22d),
and
C(l) + + C’,” = T*‘w’(l’)j,(k) c(1) _ c(Z) = i(T*‘w’(l+)n,(k) + +
+ 2{(dldr)j2(kr)},=,
,
+ 2{(dldr)n,(kr)},,,).
The two alternative expressions (4.26a, b) are the general solutions for the deviation factor in the first-order perturbation theory and the deviation is given function has by (v&dr)xy:‘(r, 6 4). Th e real property of pair-correlation been fulfilled in a natural way. The effect of the long-range behavior of g,, is concealed mainly in the integral H, ““‘(k), which is independent of r. It should be noticed that the deviation factor xy contains the long-range effect of g, even for a small value of r because of the r-independent integral, i.e. the long-range behavior of pair-correlation function in equilibrium affects both the short-range
PAIR-CORRELATION
FUNCTION OF A FLUID
389
and the long-range behavior in nonequilibrium. The viscosity coefficient, which is mainly determined by the short-range behavior of nonequilibrium paircorrelation function, is thus closely related to the long-range behavior of g,, and the delicate behaviors of viscosity coefficient (e.g., its functional dependence on the correlation length) won’t be explained with any oversimplified functional form of g,(r) which doesn’t describe the long-range behavior*). The solution (4.26a or b) shows an interesting functional dependence of the perturbation function ,Y: on k(:= k, = e’“‘4(2ry,)1’2) defined in (4.1,24), which is the result of the perturbation expansion in the deformation parameter. 4.2. Solutions with the improved
O-Z
approximation
of equilibrium
pair-
correlation function
In the expression (4.26a) all the terms are in their explicit forms except for the integrals of the form (4.22a)
=
j-
h~‘3-‘)‘2)(k,p)p3{g~‘2(P)}’ dp >
1+
for t=+l
and
s=+l,
which will be calculated by using the improved O-Z approximations for g, represented in the appendix by (A.2,lO). Since the spherical Hankel function of second order in the integrand is hi’3-f)‘2)(zs)
= i t exp(itz,)(z,’
+ i3tzi2 - 3.~:~) ,
with z, = k,r , the above integral may be rewritten
Z?(3-f)‘z)(ks) 1
as follows:
= itkr3 i eifksp(kzp2 + i3tk,p - 3){ g;“(p)}
dp
.
(4.27)
1+
For the analytic execution may be introduced:
of the integral (4.27) the following approximation
H.-M. KOO AND S. HESS
390
g;“(r)
where
the
= {1+ s’(Y)}“2
= 1 + s’(r)/2
El+s’(r)/2,
for r>l,
+ .. (4.28)
assumption
[s’(r)1 1, where the corresponding for the most values of r except for the mation is g,“2(r) z 1. Th is is acceptable short range near the effective hard core. So the approximation (4.28) is reasonable in the sense of average when it is used in an integration. The
2-‘g,“2(r){s’(r)}’
improved O-Z approximation By using (4.28) the integral
for S’(T) is given by (A.lO). (4.27) may be approximated
as follows:
, I=
ilk,’
I
exp{itk,p}(kzp’
+ i3tk,%p- 3){s’(p)/2}’
(4.29)
dp ,
where the subscript “A” means the approximation for g:,“, (4.28), and the constant term in (4.28) has vanished on taking the first derivative. The approximated integral (4.29) can be calculated by replacing s> in the integrand by its improved O-Z approximation (A.lO). Since S”(Y) = 2
(U,lr) eVr(‘-“(vj - 1 /r) ,\
the approximated
integral
rozZ$y3~r’:“(k,~)
becomes
= (it/2)k.f3
$u,
em”’ / e(y+‘rkQ)p(kfp2 + i3tk,vp - 3) I+
x (v, - 1 ip) lp dp . After
some
calculation
one
obtains
the
following
result
for the
integral
above:
roz$~3-‘)‘2)(k,y) = (i/z)ki’
2
u,
ep”‘(e’“rti’k”‘{
- 3trm’ f tv,(3 + v[T)
I
+ik,(l
- v,r) - tvy(4 + v,~)(v/ + itk,)-’
+tvy(v, + itk,))2} +e(“J+1’k‘){3t - fv,(3 + v,) - ik,(l
- v,)
+tvF(4
+ itk,,))2})
+ v,)(v, + itk,))’
- tv:(u,
,
(4.30a)
PAIR-CORRELATION
FUNGI-ION OF A FLUID
391
where “IOZ” means the improved O-Z approximation for s(r). Especially, if r+ 03with the signs of t = +1 and s = + 1, the first term in the angular bracket of (4.30a) vanishes, since Re(ik+r) = Re{i eimi4(27yr)“‘r} < 0, and one has the following r-independent
loZ$)(k) = (i/2)km3
approximated
integral of the form:
eik $ u,(3 - ~~(3+ v!) - ik(1 - v,)
+ vy(4 + v,)(v, + ik))’ - v~(v, + ik))2) ,
(4.30b)
where k := k,. If the expression for xy of the second type (4.26b) is desired, one can make use of the following relations: loz.$(k,) IoZtA(ks)
= 2-‘(‘ozI$a”(k,)
+ ‘“‘pa”)
= (2i)-1(10zZ$‘(k,)
,
(4.3Oc)
- ‘oz$~)(k,)),
which may be derived from the definitions (4.22a, b, c). With (4.30a-c) showing the integrals performed by using the improved O-Z approximation of s’(r) and (4.28) one can write the deviation factor instead of (4.26a,b) as follows: O)g”il’*(r) Re( - ei2”Fo(k, r)) ,
‘OZXy(r, 0, C#J)= (1/3)P:(cos with
F”(k, r) = i(1 + a,,,)(1 + s:/2)h$“(kr)lC’:’ + k{ Cy’hr’(kr)
- C!Z’h:“(kr)}‘O’~a’)(k)
+ kh~‘(kr)‘OZZ$‘(k)
/Cy’
- kh~)(kr)loZ$Z(~‘(k),
or equivalently, ‘oZxT(r, 8, 4) = ( f)P:(cos8)g”,“2(r)
with
Re( - ei2’Fo(k, r)) ,
(4.31a)
H.-M.
392
F”(k,
r) = i(l + a,,,)(1 +
k{(C+(l) -
S. HESS
+ sr/2)hi”(kr)lC(:) C(+2))j2(kr)
+ 2iknz(kr)““{,(k) where
KOO AND
-
i(Cy’
+ C”‘)n,(kr))‘““z$‘(k)
- 2ikjZ(kr)‘OZt,(k)
,
/c’:’ (4.31b)
the superscript
“IOZ” denotes the improved O-Z approximation of “A” means the approximation for gd” (4.28). The S’(I) and the subscript function s”, is defined by (4.22d). The factor (1 + SF/~) with the definition
s: :=SP(lf)
= &](lf)
(4.31c)
- 1
stems from the approximation g,!,“(l+) z 1 + SF/~, which was introduced in (4.28). This approximation for gi”(l+) seems to be unreasonable, since S:‘(T) is no more small at r = l+. The reason for the approximation is the following. The correct effect of the factor g:,‘*(r) at r = 1 was singled out from the integrals in (4.20)’ and written correctly in (4.26a, b), where the integrals also preserve the function g,!,‘*(r) for r > 1 in its correct form in their integrands. But the function g:“(r) in th e integrals in (4.31a, b) is approximated by (4.28). So it is reasonable to take the factor gb”(l’), which is singled out, also in its approximated form written in (4.31a, b), for some cancellation of similar terms may occur between the singled-out and the integral terms. This possibility of cancellation could be seen more naturally if one had performed the approximation of the function g;)‘*(r) from the beginning in the integrals of the original representation given by (4.20), which was avoided to derive first the correct expressions (4.26a, b). A completely explicit expression for the deviation factor ““xy of PCF for hard spheres may be obtained, if one makes use of the improved O-Z approximation Percus-Yevick
s&,(r) equation
given by (A. IS), for hard spheres’“)
where the exact solution of the is used to determine the parame-
ters U,‘S and z+‘s. Thus the expressions (4.31a, b) become those for spheres, when the density correlation function s> in the integral (4.29)
jl((‘-“i’)(k,)
G
hard
$-)(kr)
I
is replaced by s&,(r) u2 from (A.15,17,20) 4.3.
of (A. 18) and the corresponding are substituted.
values
of
V, ,
v2, u, , and
Remarks on an alternative notation of the PCF
With
the first-order
solution
for deviation
factor
(4.26a
or b) we can write
PAIR-CORRELATION
FUNCTION OF A FLUID
393
the PCF according to (3.1; 3.7; 4.23) as follows: g=
go(l + x>=ET”{1 + e--Yci)x;~
= go -
(q,/3)go&1’2
Re{Pz(cos
0) e’*‘F(k, r)} .
If we introduce the spherical harmonics of the type17) YL,(8, 4) = Pr (cos 0) cos(m4) and Yz, (0, 4) = PT( cos 0) sin(@), and define F,( y,, r) = Re{F(k, r)} and F,(-y,, r) = Im{F(k, r)} with k = eirr’412Ty,j1’2from (4.1,24), the above expression becomes
g(r) = g,(r) + g+b,r>G2@,44 + g-(7, r)y12(e,4) ,
(4.32)
where
with yd = y, = y/2 in plane Couette-flow. The equation (4.32) is the result of the first-order perturbation calculation in the deformation parameter and we have no zonal spherical harmonics of second order Y&(0, 4) = Pi< cos O), but in the second-order perturbation calculation this component of spherical harmonics contributes, as can be seen in (3.16, llc-d). The disapperance of the components of the spherical harmonics Y E;Oin (4.32) is due to the symmetry of the Couette-flow described by (3.10). The expression (4.32) corresponds to the following tensorial expansion of PCF with the special geometry*.*‘):
g(r) = g”(y, 4 + g+(y,r>(G) + 2-‘gm(xr)(22 - jY) +gO(y,r)(E-l/3)+...,
(4.33)
where x^,F, and z^are the Cartesian components of unit coordinate vector. The radial functions go(r), g+(y, r), and g-(y, r) in (4.32) correspond to the expansion coefficients g’, g+, and g- m ’ (4.33) respectively. The expansion coefficients in (4.33) depend not only on r but also on the shear rate y in general. The first-order solution (4.32) contains no shear-rate dependence of the scalar term. In the second-order perturbation calculation, however, the scalar term will depend on the shear rate and go will be nonzero, as can be deduced from (3.16, llb-d).
H.-M.
394
KOO
AND
S. HESS
Integrals over the radial functions g’, g’, g-, and g” yield the isotropic p_ and po, pressure p,, the shear pressure p+, and the pressure differences respectivelyz5). The solution for g(r) eq. (4.32) will show not only the ellipsoidal
distortion
coefficients
gi
and
but also a twisted distortion have different functional
in the x-y plane, since the form from each other and
g-
g’lgvaries section.
as r changes,
5. Discussion
and graphical
which
will be displayed
graphically
in the
next
representations
Since the solutions for the deviation complicated to apprehend the PCF, desirable. As already mentioned, (4.30a, b), is completely explicit,
factor (4.26a, b) or (4.31a, a graphical representation
the expression for when the improved
b) are too is highly
‘ozx~, (4.31a) with O-Z approximation
&YH( = g,(r) - I f or r > 1) for a hard sphere fluid given by (A. 18) is used and to it are substituted. The radial the values of u,, u2, vi, and V, belonging distribution function g, is shown in fig. 1 for the densities corresponding to the volume fractions 0.1, 0.3, 0.4, 0.4628, and 0.52. The viscosity diverges7.2h) at phase transition. the value 0.524. . . in accord with the expected fluid-solid The following nonequilibrium results are for volume fraction 0.46, i.e. a density
of about
9/10
of the solidification
density.
LINE
TYPE
DENSITY 0.5200 0.4626
g(rl
0.4000 0.3000 0.1000
Fig. 1. The equilibrium PCF for hard spheres the O-Z equation (A.18) _
corresponding
to the fourth-order
approximation
of
PAIR-CORRELATION
FUNCTION OF A FLUID
395
The hard-sphere fluid shows a remarkable structural similarity to real liquids2156) an d realistic model fluids**) (e.g. L-J fluid), especially in higher densities. Thus we demonstrate the solution (4.31a) for hard spheres graphically and discuss its configuration comparing with experiments. The only unknown parameter in (3.31a) is the relaxation time coefficient T appearing in k := k, = 12~y,[*‘*exp(ir/4) from (4.1,24), which is associated with the shear rate y. Thus we introduce the dimensionless shear rate my and perform the numerical calculations for the various values of ry instead of y. The relaxation time coefficient T may be determined for a certain density by comparing the Newtonian viscosity from experiments with the theoretical one which is calculated by using the solution’) (4.31a). Figs. 2a, b are the contour and the 3 - dimensional diagram of the equilibrium pair-correlation function (EPCF) in the x-y plane. The spherically symmetric configuration of the EPCF is responsible for the Debye-Scherrer rings observed in the X-ray scattering from pure liquids20~21) or in the light scattering from colloidal liquids’). If, however, the liquid system is sheared with the shear rate not so large, a typical ellipsoidal distortion of the pair-correlation (PCF) appears, as shown in the figs. 3a, b. A direct observation of such an ellipsoidal distortion is possible by means of computer simulations’). In real experiments one can observe the structure factor which represents the scattered intensity and is the Fourier transform of the PCF. Ackerson and Clark’) observed the ellipsoidal distortion of the structure factor which shows a complimentary configuration of PCF. By complimentary configuration’) we mean the following: if one assumes the PCF within the Stokes-Maxwell approximation”2”4’23), i.e. g(r) = g,(r - ~~~xylr), the corresponding structure factor becomes S(k) = S,(k + ~Oyk,k,lk), where S,(k) is the structure factor in equilibrium, k the wave vector, and 7. a phenomenological relaxation time. So, in this simple assumption the main axis of the ellipsoid of S(k) IS rotated from that of g(r) by 90”. Thus the ellipsoidal configurations of PCF in figs. 3a, b are compatible with the rotated ellipsoidal configurations of the structure factor or the scattered light intensity observed by Ackerson and Clark’). In figs. 4a, b and figs. 5a, b we see also the distortions of PCF in higher shear rates, which are no more of ellipsoidal form but of twisted form. These twisted forms, of course, are symmetric about the origin due to the symmetry of Couette-flow (3.10). Computer graphs based on nonequilibrium molecular dynamics also show this twisted structure at higher shear rates*) provided that higher rank tensorial contributions*‘) to the distorted PCF are disregarded. The first maxima in figs.4a, b are rotated towards the x-axis more than those in figs. 3a, b. In general, the main axis of the ellipsoidal distortion rotates towards the x-axis as the shear rate increases in the region of lower shear rates.
H.-M.
396
KOO
AND
S. HESS
a
Fig. 2. (a) Contour lines represent the 0.05. The number Three-dimensional
diagram for the equilibrium PCF for hard spheres on the x-y plane. The thick integer values of the PCF and the thin lines correspond to the increasing by of particles per the hard-sphere volume is x = 0.46. Maximum value is 4.12. (b) picture corresponding to fig. 2a.
PAIR-CORRELATION
a
FUNCTION
OF A FLUID
397
'-OIPECT.
Fig. 3. (a) Contour diagram for the PCF for hard spheres on the x-y plane in the Couette-flow of shear rate ry = 2. One sees the ellipsoidal distortion. The reduced density is x = 0.46. Maximum value is 5.38. (b) Three-dimensional picture corresponding to fig. 3a.
H.-M.
398
KOO
AND
S. HESS
b Fig. 4. (a) Contour diagram for PCF for hard spheres on the x-y plane in the Couette-flow of a intermediate shear rate q~ = 5. One sees a twisted distortion. The reduced density is x = 0.46. Maximum value is 6.15. (b) Three-dimensional picture corresponding to fig. 4a.
PAIR-CORRELATION
a
FUNCTION
OF A FLUID
399
X-DIRECT.
b Fig. 5. (a) Contour diagram for PCF for hard spheres on the x-y plane in the Couette-flow of a high shear rate 7y = 10. One sees a strongly twisted distortion. The reduced density is x = 0.46. Maximum value is 7.79. (b) Three-dimensional picture corresponding to fig. 5a.
H.-M. KOO AND S. HESS
400
The relations
between
be examined
in a following
6. Concluding
the distortion
of PCF and the viscosity
coefficients
will
article.
remarks
In this article, function has been
the shear-flow-induced distortion of the pair-correlation calculated starting from the extended K-S equation. To
obtain explicit analytic solutions spheres, a number of assumptions
which were displayed graphically for hard and approximations were introduced. Since
we have been working in a well defined expansion scheme, the solution can be amended by going to the next order in a perturbation calculation. There are three separate points for further improvements: i) consideration of the perturbing term
V,(r)
inclusion
of an attractive
of eq.
(4.8,9) force
f or the calculation for a completely
of Green’s function; explicit solution; and
ii) iii)
calculation of the terms of second order in the deformation rate -yd. The last point will lead to an effect of the shear flow on the scalar part g” of the PCF, and will imply a nonzero second rank tensorial contribution of the type proportional to Y&(0, 4) in eqs. (4.32,33) and also fourth rank contributions. These are the features which have been observed at high shear rates’-‘). feel a study of this point should have highest priority in an extension present work.
So we of the
been applied to the The theory developed so far can, and has indeed, nonlinear viscous behavior, in particular the non-Newtonian viscosity and the (first) normal pressure difference. Dramatic effects are found at densities close to solidification26.7 ). Details will be presented in a forthcoming publication. An extension of the present work to anisotropic fluids consisting nonspherical particles is desirable and seems to be feasible2”).
of oriented
Acknowledgement One of the authors (H.-M. Koo) wishes to thank DAAD for the financial support during the time when the most part of this work was performed in Erlangen.
Appendix A.l. The Omstein-Zernike and its approximation
theory for equilibrium
density-correlation function
PAIR-CORRELATION
FUNCTION OF A FLUID
401
For a homogeneous fluid system in equilibrium the Ornstein-Zernicke (O-Z) theory for density fluctuation’) reads as follows: s(r) = c(r) + n
.I
s(r - r’)c(r’)
d3r’
(A.la)
or in its Fourier-transform
s”(q)= c”(q)+ +Mq)
>
(A.lb)
where n is the reduced molecular number density (per the volume u”). The functions s(r) and c(r) are named, density correlation and direct correlation function, respectively, and s”(q) and E(q) are their Fourier transforms respectively. The density correlation function s(r) is called usually “correlation function” without the preceding noun “density” which is here added to distinguish s(r) from the equilibrium pair-correlation function g,,(r). The density correlation function s(r) is related to g,(r) by
s(r)= g,(r) - 1
64.2)
and the former is introduced by Ornstein and Zernike’) to explain the influence of density deviation in one volume element on the state in another, and the latter is the measure of frequency that one finds one molecule at r in the presence of another at the origin in the reduced relative pair space. The direct correlation function c(r) is of very short range (about the range of intermolecular forces)‘) and its contribution to the integral in eq. (A.la) is significant only for small values of r’, which implies that one may expand s(r - r’) in the integrand of eq. (A.la) in power series of r’ and take a few lowest orders to obtain a reasonable approximation. Thus eq. (A.la) is expanded as follows: s(r) = c(r) + nc,s(r)
+ (n/6)c,V2s(r)
+ (n/120)c,V2V2s(r)
+ ... ,
where C2n
:=
I
r”‘c(r) d3r ,
for n=0,1,2,.
..,
(A-3)
and especially c0 = C(O) . In (A.3) the integrals containing the odd-power missing for the symmetry reasons (isotropy).
factor in their integrands are
H.-M.
402
KOO
AND
S. HESS
A.2. An improved O-Z approximation As frequently made, one can neglect the fourth and the higher derivatives’O-“,’ ) in eq. (A.3) t o o b tain the following second-order differential equation for s(r): V’s(r) - v’s(r) = -6c(r)l(nc,)
,
(A.4) LJ*: = 6( 1 - nco) l(nc,)
.
Outside the range of intermolecular forces the inhomogeneous term in the right hand side of (A.4) is very small and practically the homogeneous solution, satisfying the boundary condition at infinity, is a good approximation for large r. Since for the moment the direct correlation function c(r) is not known and we have seen no way to calculate cg and c2, the constant v* is a phenomenological parameter to be determined under some conditions. Mostly and sometimes tacitly y2 is considered as a positive constant”-‘*) and immediately the homogeneous solution of (A.4) is expressed by s>(r) = (u/r) e-“(rm’) ,
with Y B 0,
VW
where s>(r) := s(r) outside the hard core (r > 1). For the negative or the complex values of v*, however, the ansatz (A.5) is not appropriate for the homogeneous solution of (A.4). In fact, v* becomes negative, if one calculates it by making use of the Percus-Yevick equation for hard spheres’). In order to improve (A.4) one can take the next higher-derivative term further in (A.3) and obtain the fourth-order partial differential equation s(r) = c(r) + nc,s(r)
+ (n/6)c2Vzs(r)
Eq. (A.6) may be rearranged
On comparing (A.6)’ with (A.6), algebraic equations:
a’p’
= -120(1
- cy’s(r)} = -{12Ol(nc,)}c(r).
(A.6)’
CY’and p’ should satisfy the following
)
- nco)l(nc,)
(A.6)
to see the solutions more easily as follows:
V2{V2S(r) - a*s(r)} - p’{V’s(r)
(Yz+ p’ = -2OcJc,
+ (nl120)c,V2V2s(r).
= -120/(n’T.K,c,).
PAIR-CORRELATION
FUNCTION OF A FLUID
403
The solutions for (Y* and /3* are (y* = -lOc,lc,
+ {(10c,/c,)*
+ 120/(n2T*K,c,)}1’2
)
p* = -10c,/c,
- {(10c,/c,)*
+ 120/(n2T*K,c,)}“2
)
(A.7)
where K, is the isothermal nT*K,
compressibility6,‘*)
= 1+ ns”(0) = (1-
and
ncO)-’ .
WV
Since s(r) is a radial function and the eq. (A.6) is symmetric with respect to permutation of (Y and p, the general form of the homogeneous solution of (A.6) is
Ii (u,lr) exp{v,(r- 1)) ,
(A.9)
I=1
where V,‘Sare the values of + (Yand + j3 from (A.7). The density correlation function is restricted to the boundary condition at infinity, i.e. s(r)--, 0 (or g,(r) --, 1) as r+ w. The terms which don’t satisfy this boundary condition should be abandoned in (A.9) (e.g. the terms with positive values of Re(v,)). Thus we denote the homogeneous solution of eq. (A.6) pertaining to the boundary condition at infinity, as follows: s’(r)
= $
(A. 10)
(24,/r) exp{ v,(r - 1)) .
The parameters U[‘Sand y’s are to be determined by some physical conditions, e.g. pressure condtion, compressibility condition5-6), etc., which will be made in the following for the Percus-Yevick (P-Y) equation for hard spheres as a special case. A.3.
The case of the P-Y
equation for hard spheres
The P-Y equation is exactly solved for hard-sphere system’“) and the function y,(r) defined by y,(r) = go(r) exp{u,(r)lT,} is given for the range r 1, the differential equation (A.6)’ becomes homogeneous for r > 1 and may be written as {V’(V’ - a*) - p2(V2 - a2)}s,tH(r)
= 0,
for r > 1 ,
(A. 13)
where “PYH” means the P-Y equation for hard spheres. The expressions for (Y and /3 (A.7) is general. Now the special case of the P-Y HS equation (P-Y equation for hard spheres) will be considered. In the case of the P-YHS equation the integral c2,, defined in (A.3) can be calculated for n = 0, 1,2 by using (A.ll, 12) as follows: 1
C”
c(r) d3r = -49r
:=
= n-r{1
I 0
yz (r)r* d3r
- (1+ 2x)‘/(l
-x)“} I
c2
:=
I
r*c(r) d3r = -47r
= -(r/20)(16
y,’ (r)r4 d3r
I 0
(A. 14)
- 11x + 4x2)1(1
- x)” ,
1 :=
I
r4c(r) d3r = -4~
= -(n/140)(80
I 0
yz (r)r’ d3r
- 72x + 12x’ + 7xX)/(1
- x)” .
PAIR-CORRELATION
FUNCTION OF A FLUID
These values of cO, c2, and cq may be substituted explicit expressions for (Y2 and p*,
fx2= {K(x)
+ iL’12(x)} /M(x) ,
p 2 = {K(x)
- iL “‘(x)} /M(x) ,
405
into (A.7), which gives the
K(x) = -70(16 - 11x +4x2) , L(x) = (3500/x)(64
(A. 15)
- 160x + 528x* - 535x3 + 184x4) ,
M(x) = 80 - 72x + 12x2 + 7x3 ,
where the polynomial in the square root, L(x), is always positive for all the allowed values of x(0 s x Z 7~1’2/6)) and (Y2 and p 2 are complex numbers with the negative real parts and the nonzero imaginary parts. So the solution of (A.13) may be written as (u/r)
emcrm’)+ (i/y)
ep(‘-‘) + (B/r)
eeacrm’)+ (B/r) e-‘(‘-‘)
,
(A. 16)
where (Y= fi from (A.15), and the solution becomes real. Since (2nn + 7r/2} < arg CY~ < (2nr + n} and (2n7r + r} < arg p* < (2n7~ + 3~/2} for n = 0, +1, +2,. . . . . . ) the arguments of the square roots of a2 and p2 are confined to the following regions: (57r/4) -=carg (Y< (37~12); (r/4)
< arg(-a)
< (n/2), (A.17)
(7~12) < arg p < (39~/4); (3~/2) < arg(-p)
< (77r/4) ,
where 0 5 arg z < (27r) for any complex number z # 0. The last two terms in (A.16) blow up for large Yand should be excluded. Thus the solution of (A.13) pertaining to the boundary condition at infinity is (A.18) where u, = U2, and V, = (Y and V, = p = 6 are given by (A.15) with the restrictions (A. 17). In order to determine the two coefficients u1 and u2 we introduce the compressibility and the pressure equatioirm6), nT, K, = 1 + n
I
s(r) d3r
H.-M.
406
KOO
AND
S. HESS
and p = nT, respectively. (A&
- (n2/6)
On substituting
11, 12) we obtain u,(vl
-2
ru’(r)g,,(r)
d'r ,
the eq. (A.18)
the following
- ~1’) + u2(v,
-2
- v,‘)
into (A.19)
and using
for U, and u2:
= (nT,K,.
+8x
- 1)/(24x)
+ 28x + x2) /( 1 + 2~)~ ,
(x/24)(34
u1 + u2 = a, + a, + a3 - 1 = (X/2)(5 solutions
for s&u(~)
equations
=
whose
(A.19)
-2X)/(1
- x)2 )
are
The values of (Y and p are given by (A.15) with (A.17). In calculating the integrals in (A.19) use has been made of the following special properties of the pair-correlation function for hard spheres: S(T) = -1 (or g,(r) = 0) for r 5 1 and with u,&, = u; exp(-u,/T,)y pyH(r) given by (A.18)
g,, corresponding
is displayed
for various
References 1) N.A. Clark and B.J. Ackerson, Phys. Rev. Lett. 44 (1980) 1005. B.J. Ackerson and N.A. Clark. Physica 118A (1983) 221. 2) D.J. Evans, H.J.M. Hanley and S. Hess, Physics today 37 (1984) S. Hess and H.J.M. Hanley, Phys. Lett. A 98 (1983) 35. 3) S. Hess, J. de Physique 46 C3 (1985) 191. 4) J.G. Kirkwood, J. Chem. Phys. 14 (1946) 180.
26.
to the densities
fourthin fig. 1
PAIR-CORRELATION
5) S.A. 6) 7)
8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28)
Rice and P. Gray,
The Statistical
FUNCTION
Mechanics
OF A FLUID
of Simple Liquids
(Interscience,
407
New York,
1965). R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975). H.-M. Koo, Analytic solution for the nonequilibrium pair-correlation function and the non-Newtonian viscosity coefficients of simple liquids, Thesis, University of ErlangenNiirnberg, Fed. Rep. Germany, 1985, unpublished. J.G. Kirkwood, F.P. Buff and MS. Green, J. Chem. Phys. 17 (1949) 988. L.S. Ornstein and F. Zernike, K. Akademie van wetenschappen te Amsterdam - Proc. 17 (1914) 793. L.S. Ornstein and F. Zernike, Physik. Zeitschr. XIX (1918) 134. F. Zernike, K. Akademie van wetenschappen te Amsterdam - Proc. 18 (1916) 1520. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, Oxford, 1971). S. Hess, Phys. Rev. A 22 (1980) 2844. J.F. Schwarzl and S. Hess, Phys. Rev. A 33 (1986) 4277. J.D. Weeks, D. Chandler and H.C. Andersen, J. Chem. Phys. 54 (1971) 5237. H.C. Andersen, J.D. Weeks and D. Chandler, Phys. Rev. A 4 (1971) 1597. P.M. Morse and H. Feshbach, Methods of Theoretical Physics I and II (McGraw-Hill, New York, 1953). E.N. Economou, Green’s Functions in Quantum Physics (Springer, Berlin, 1979). M.S. Wertheim, Phys. Rev. Lett. 10 (1963) 311. E. Thiele, J. Chem. Phys. 39 (1963) 474. J.E. Thomas and P.W. Schmidt, J. Chem. Phys. 39 (1963) 2506. A.J. Greenfield and J. Wellendorf, Phys. Rev. A 4 (1971) 1607. L. Verlet, Phys. Rev. 165 (1968) 201. W.T. Ashurst and W.G. Hoover, Phys. Rev. A 11 (1975) 658. R.W. Zwanzig, J.G. Kirkwood, K.F. Stripp and I. Oppenheim, J. Chem. Phys. 21 (1953) 2050. S. Hess and H.J.M. Hanley, Phys. Rev. A 25 (1982) 1801. H.-M. Koo and S. Hess, Z. Naturforsch. 42a (1987) 231. H.J.M. Hanley, J.C. Rainwater and S. Hess, Phys. Rev. A (1987). D. Baals and S. Hess, Phys. Rev. Lett. 57 (1986) 86.
