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Simple model for the calculation of the coefficient of self-diffusion in a liquid
Volume 56A, number 5
PHYSICS LETTERS
19 April 1976
SIMPLE MODEL FOR THE CALCULATION OF THE COEFFICIENT OF SELF-DIFFUSION IN A LIQUID R.M.YULMETYEV Kazan State Pedagogical Institute, Kazan, USSR Received 6 February 1976 The coefficient of self-diffusion in three-dimensional classical liquid is computed approximately from the hierarchy of kinetic equations for the time-correlation functions (TCF).
The coefficient of self-diffusion D5 of a molecule can be obtained from the TCF ir(t)/of momentump1(t) at time t ofa chosen particle ma fluid byD5= (kT/m)lim~~+oii(z), where ~(z) = f~’dtexp( zt)ir(t), where m is the mass of a particle and kT the thermal energy. Here we shall obtain the formula forD5 from the smallz behavior of the hierarchy of the equations for TCF. Zwanzig was the first to succeed in deriving a kinetic equation from the Liouville equation [1]: ir ‘(t) = —f~drK(r)ir(t r), where the memory function is K(r)
=
P=1
EW=PEP,
~
=
iL,
2=~~ P1V1
j~=1v1u(I,i)V~1,
II,
H is the projection operator, L is the ordinary Liouville operator and u (i, f) is the interparticle potential. If we use the identity [2] exp{t(A +B)} = exp(tA) + fdu exp{(t
u)A}B exp{u(A+ B)),
for the arbitrary operators A and B, we can obtain an expansion ofK(t): 2f(t)+ K(t) = c~.,
t ~II(— 1)~~fdti 00
tfl 1
...
0
2 =
0
dtnlr’(t
t1)ir’(t1
t2)
...1r’(tn
1
0
2 F1 = —~1>1V1 N exp (iJ2t)F1)/(F1>, u(l,j) is the total force on a chosen particle. the Laplace transform of K(t) in the form
21)/
where w
one can write
Now
f dt2 f
2f(z)+z K(z)ow
~I~f1 z~(z))’~1.
(1)
n1
Then forz -~-+Owe obtaink0(z) ~w2f(z) and ii(z) ~{z+w2f(z)}~. FollowingZwanzig [1] we kinetic equation for the TCFf(t):f’(t) = f~dr V(T)f(t r). where V(r) =
Q = 1 —R,
R=
may write the
387
Volume 56A, number 5
PHYSICS LFTTERS
19 April 1976
R is the projection operator. We can now give an expansion of the new memory function V(r) = ~2~(~)
tl
7~
00
tfl
( l)~ fdt
+ ~
1 fdt2
0
f
...
0
~
dt~~’(r t1)~’(t1 t2)
...
~‘(t~ 1
0
where ~2_
(cI~)
(~jexp{iEt}41)
~(t)-
,
(Ft)
1 N [(p1 ~i_~1~
<~I~)
p1)~1jv1u(1,~.
m 2~p(z)} 1•
Hence one in cana similar find in fashion the limitone z —~ +0,construct f(z) =~zthe + l2third equation for the TCF ~p(T) given by Finally can ~‘(t)
fdr.G(r)~(t
T),
where G(T)=a2g(r)+~(
l)n+lfdt 1fdt2...f
2_ KF~),
1
2viu0,1)
=~
t2)...g’(t~ 1
g(t)—
o
r
dt~g’(r t1)g’(t1
k~1[~lV
~1(pi1vi)
1u(l,j)V1
/+k)(~ñ1~
V1u(l, k).
Combining all these eqs. we can write for z +0 2g(z)}1, ~(z) =~ ~2 g z (2) (z2 + w2) [z + u2g(z)] + ~(z) {z + o A few remarks can be made. From the definition of the frequency parameters w, ~1 and a it is clear that one can write Q2 = (kT/ml~),a2 = (kT/ml~),and the correlation 1engthl~and 1~are related asl~ ~ ~l0 ~ One obtains also a ~ 1013 sec An essential feature of the analysis presented here is the fact, that the memory function g(T) is defined by the space-time ‘correlations of particles only and the momenta correlations are noneffective. Hence one can write, in our approximation g(t) f(t) and ~(z) ~f(z) in kinetic equations for the TCF. The solutions for z +0 can, on the basis of the discussion, be divided into two parts: ~p(z) ~z+ a2f(z)} 1, f(z) = ~ + ~2 ~ + a2f(z)j 1) 1and the solution defined by —-
~.
—~
f(z)—(2za2)
if (z2+~22 a2)+\4~2+&
u2)2÷4z2a2}
Now one can write~(z)in the formf(z) (1/2~l2){ z +~/z2+4~l2}since ~ a. The last expression forf(z) is equivalent to the nonlinear equation: f’(t) ~l2f~drf(r)f(t r)which has the solution [3] in the formf(t) = 2Ji 1J 2(2 ~2t), where Ji2 (x) z + v/~~~21) du u 2(u) and .12(u)one is the Bessel function. The Laplace transformir (z) is given 2/2~22)[ 1• Thus finds for the coefficient of self-diffusion by ir(z) =fz + (w 00
00
388
Volume 56A, number 5
D5= (kT/m) z~O~(z) =
PHYSICS LETTERS
~T.~
19 April 1976
3kT m 1/2 (4~nfdr r2g(r)v~u(r))
x (4xn!drr2g(r)[(u1~(r))2+2r_2(u?(r))2] +n2fdr’dr” [3u (r )u (r )~2u (r )r r’ ~ r’ ~
2
,
u (r )
u (r )
,
~_~2)]g3(r’,rU,r”
ri))
1J2
where n 00 N/V is the number density, g(r) and g3 are the radial and three-particle distribution functions. Equation (3) gives the correct temperature dependence: D5 {g(a)}~,where g(a) is the RDF at contact. the com~puter 3 we get 4irn From f~drr2g(r) V~u(r) experiments by Rahman [4] for liquid argon 94.4 K and and the density g/cm I .lOX iO~ erg/cm2.Assuming as in [5] thatataT 00 = 3.418A depth1.374 of the potential well c 1.71 X 10 14erg
we can estimate D~ 2.07 Xl 0 5cm2/sec, the experim&ital Ar data [5] for this case is 2.14 X iO~cm2/sec.
References [1] R. Zwanzig in Lectures in Theor. Phys., eds.W.E. Britten, B.W. Downs and J. Downs (Interscience Publ. Inc., New York, 1961) Vol. 3,p. 106. [2] R. Kubo, J. Phys. Soc. Japan 12 (1962) 570. [31 I.S. Gradshteyn and l.M. Ryzhik, Table üf integrals, series and products (Academic Press, New York, 1965). [4] A. Rahman, J. Chem. Phys. 45 (1966) 2585. [5] J. Naghizadeh, S.A. Rice, J. Chem. Phys. 36 (1962) 2710.
389
PHYSICS LETTERS
19 April 1976
SIMPLE MODEL FOR THE CALCULATION OF THE COEFFICIENT OF SELF-DIFFUSION IN A LIQUID R.M.YULMETYEV Kazan State Pedagogical Institute, Kazan, USSR Received 6 February 1976 The coefficient of self-diffusion in three-dimensional classical liquid is computed approximately from the hierarchy of kinetic equations for the time-correlation functions (TCF).
The coefficient of self-diffusion D5 of a molecule can be obtained from the TCF ir(t)
=
P=1
EW=PEP,
~
=
iL,
2=~~ P1V1
j~=1v1u(I,i)V~1,
II,
H is the projection operator, L is the ordinary Liouville operator and u (i, f) is the interparticle potential. If we use the identity [2] exp{t(A +B)} = exp(tA) + fdu exp{(t
u)A}B exp{u(A+ B)),
for the arbitrary operators A and B, we can obtain an expansion ofK(t): 2f(t)+ K(t) = c~.,
t ~II(— 1)~~fdti 00
tfl 1
...
0
2 =
0
dtnlr’(t
t1)ir’(t1
t2)
...1r’(tn
1
0
2 F1 = —~1>1V1 N exp (iJ2t)F1)/(F1>, u(l,j) is the total force on a chosen particle. the Laplace transform of K(t) in the form
21)/
where w
one can write
Now
f dt2 f
2f(z)+z K(z)ow
~I~f1 z~(z))’~1.
(1)
n1
Then forz -~-+Owe obtaink0(z) ~w2f(z) and ii(z) ~{z+w2f(z)}~. FollowingZwanzig [1] we kinetic equation for the TCFf(t):f’(t) = f~dr V(T)f(t r). where V(r) =
Q = 1 —R,
R=
may write the
387
Volume 56A, number 5
PHYSICS LFTTERS
19 April 1976
R is the projection operator. We can now give an expansion of the new memory function V(r) = ~2~(~)
tl
7~
00
tfl
( l)~ fdt
+ ~
1 fdt2
0
f
...
0
~
dt~~’(r t1)~’(t1 t2)
...
~‘(t~ 1
0
where ~2_
(cI~)
(~jexp{iEt}41)
~(t)-
,
(Ft)
1 N [(p1 ~i_~1~
<~I~)
p1)~1jv1u(1,~.
m 2~p(z)} 1•
Hence one in cana similar find in fashion the limitone z —~ +0,construct f(z) =~zthe + l2third equation for the TCF ~p(T) given by Finally can ~‘(t)
fdr.G(r)~(t
T),
where G(T)=a2g(r)+~(
l)n+lfdt 1fdt2...f
2_ KF~),
1
2viu0,1)
=~
t2)...g’(t~ 1
g(t)—
o
r
dt~g’(r t1)g’(t1
k~1[~lV
~1(pi1vi)
1u(l,j)V1
/+k)(~ñ1~
V1u(l, k).
Combining all these eqs. we can write for z +0 2g(z)}1, ~(z) =~ ~2 g z (2) (z2 + w2) [z + u2g(z)] + ~(z) {z + o A few remarks can be made. From the definition of the frequency parameters w, ~1 and a it is clear that one can write Q2 = (kT/ml~),a2 = (kT/ml~),and the correlation 1engthl~and 1~are related asl~ ~ ~l0 ~ One obtains also a ~ 1013 sec An essential feature of the analysis presented here is the fact, that the memory function g(T) is defined by the space-time ‘correlations of particles only and the momenta correlations are noneffective. Hence one can write, in our approximation g(t) f(t) and ~(z) ~f(z) in kinetic equations for the TCF. The solutions for z +0 can, on the basis of the discussion, be divided into two parts: ~p(z) ~z+ a2f(z)} 1, f(z) = ~ + ~2 ~ + a2f(z)j 1) 1and the solution defined by —-
~.
—~
f(z)—(2za2)
if (z2+~22 a2)+\4~2+&
u2)2÷4z2a2}
Now one can write~(z)in the formf(z) (1/2~l2){ z +~/z2+4~l2}since ~ a. The last expression forf(z) is equivalent to the nonlinear equation: f’(t) ~l2f~drf(r)f(t r)which has the solution [3] in the formf(t) = 2Ji 1J 2(2 ~2t), where Ji2 (x) z + v/~~~21) du u 2(u) and .12(u)one is the Bessel function. The Laplace transformir (z) is given 2/2~22)[ 1• Thus finds for the coefficient of self-diffusion by ir(z) =fz + (w 00
00
388
Volume 56A, number 5
D5= (kT/m) z~O~(z) =
PHYSICS LETTERS
~T.~
19 April 1976
3kT m 1/2 (4~nfdr r2g(r)v~u(r))
x (4xn!drr2g(r)[(u1~(r))2+2r_2(u?(r))2] +n2fdr’dr” [3u (r )u (r )~2u (r )r r’ ~ r’ ~
2
,
u (r )
u (r )
,
~_~2)]g3(r’,rU,r”
ri))
1J2
where n 00 N/V is the number density, g(r) and g3 are the radial and three-particle distribution functions. Equation (3) gives the correct temperature dependence: D5 {g(a)}~,where g(a) is the RDF at contact. the com~puter 3 we get 4irn From f~drr2g(r) V~u(r) experiments by Rahman [4] for liquid argon 94.4 K and and the density g/cm I .lOX iO~ erg/cm2.Assuming as in [5] thatataT 00 = 3.418A depth1.374 of the potential well c 1.71 X 10 14erg
we can estimate D~ 2.07 Xl 0 5cm2/sec, the experim&ital Ar data [5] for this case is 2.14 X iO~cm2/sec.
References [1] R. Zwanzig in Lectures in Theor. Phys., eds.W.E. Britten, B.W. Downs and J. Downs (Interscience Publ. Inc., New York, 1961) Vol. 3,p. 106. [2] R. Kubo, J. Phys. Soc. Japan 12 (1962) 570. [31 I.S. Gradshteyn and l.M. Ryzhik, Table üf integrals, series and products (Academic Press, New York, 1965). [4] A. Rahman, J. Chem. Phys. 45 (1966) 2585. [5] J. Naghizadeh, S.A. Rice, J. Chem. Phys. 36 (1962) 2710.
389