- Email: [email protected]
Analytical equations for the compressibility factor of a two-dimensional square-well fluid
Volume 72A, number 2
PHYSICS LETTERS
25 June 1979
ANALYTICAL EQUATIONS FOR THE COMPRESSIBILITY FACTOR OF A TWO-DIMENSIONAL SQUARE-WELL FLUID S. SOKOLOWSKI Department of Theoretical Chemistry, Institute of Chemistry UMCS, 20031 Lublin, Nowotki 12, Poland Received 13 March 1979
Analytical approximations for the compressibility factor of a two-dimensional square-well fluid, resulting from perturbational theories are presented.
The recently most successful description of the structure and equilibrium properties of simple threedimensional liquids can be obtained from the perturbation theories developed mainly in the last years [1]. The application of those theories for two-dimensional fluids, and consequently for the description of two-
dimensional mobile adsorption of a gas on a solid was recently reported by Steele [2] and by the present author [3]. The purpose of this note is to derive analytic approximations for the two-dimensional cornpressibility factor Z of a fluid of particles interacting according to the square-well potential: u(r)oe, =
—e,
r
‘~
r ~ Xa,
(1)
—
—~,
FHD =
—
4 ~
1
2y
api
I
apJHDa r2 LikT—i
—~
—
, i xg~~(x)dx , (3)
the reference system. Eq. (3) results from the secondorder perturbational theory and its application for a three-dimensional fluid is instantaneous. The computation of the integral from eq. (3) requires the knowledge of the radial distribution function for the reference system, and this function can be easily obtained from -
(2)
dimensional hard-disk fluids actually there are no analytical solutions of the PY equation, and consequently the numerical application of eq. (3) is much more complicated. Looking for some other theoretical possibiities the author has decided to apply the approximation introduced by Ponce and Renon [81 for threedimensional fluids. For this purpose we consider the integral from eq. (3) and rewrite it in the following equivalent form:
F
F
i~ i xg~~r~(x)dx• xh(x)dx+ i xdx— i xh(x)dx, -‘ 1 1 1 X (4)
J
h=g~~—l. 128
a y
aY
where T kT/e,y = pira~/4,p is the number density, and g~~(x) is the radial distribution function (rdf) for
r.—a.
The result of the BH theory for the Helmholtz free energy Fcan be wntten as follows:
f xg~ff~(x)dx
analytical [5,6] (PY) Percus—Yevick or numerical equation.[7] In contrast, solutions for of the two-
The starting point of our considerations is the theory of Barker and Henderson (BH) [4], according to which the potential (1) can be treated as the addition of the reference hard-disk potential uHD(r), and the perturbation u 1 (r): u(r) = UHD(r) + u1(r), uHD(r) = oo, r < a, ~
—
NkT
.
r> ~
=
F
Volume 72A, number 2
PHYSICS LETTERS
The first integral in eq. (4) can be evaluated from the compressibility equation, the third integral, however, is much smaller than the remaining two. Thus as a fair approximation one may use:
f
xg~x)
=
[i
/
kT B
1/
15
~
+
HD
A~2 _._—~
.
~
(~)]
I. kT ap
<
p
I A
1~)HD]
—
25 June 1979
(5)
/ / ii
as which was derived neglecting the last term of eq. (4). Thus, accepting for the compressibility factor of a two-dimensional hard-disk fluid the result of the scaled particle theory [91, ZHD = 1/(1 y)2, from eq. (3) we obtain the following expression:
I/
d
A •
—
Z = ZHD
(y/T*)A(1 +0.5B/T*)_ cL2s(l/T*2)C, (6)
—
________________
where A
0.4
(—2 + 3y
=
B = (1
—
4y
—
—
y3)(l
3y2)(l
+y)2 + 2X2 —
,
y)2(y + l)2
(7)
,
C—y 2 (1 —y) 3 (14.—6y—6y 2 —2y 3 )(l +y)— 4 —
Q6
________________
y
0.5 Y
0
Fig. 1. (A) The comparison of exact (circles) and approximated (lines) values of the integral I = J~’xg~j~(x) dx. (B) The comparison of the pressures calculated according to eq. (6)
(lines) with those obtained eq. (8) (white circles). The solidlinewascalculated forfrom T*0.5,thedash_dottedline
-
= 0.8, and the dashed line for T” 1. The black circles denote the results obtained from the two-dimensional Barker— Henderson equation and from the PY radial distribution function of the reference system (for T~= 0.8)- All computations were performed for X 1.5.
for T”
The last equation gives the analytic approximation for the pressure of the considered fluid in the limit of the BH theory. The higher-order corrections to the BH equation (3) were considered by Praestgaaed and Toxvaerd [10]. They have shown that the BH approximation can be used in any order of the expansion and the resulting series can be summed to obtain an equation involving only the rdf of the reference system. Application of this theory (eq. (33), ref. [10]) and the approximation (5) lead to the following equation for the compressibility factor: ~
— —
r
HD + 0. [1 —
1
1 j~p\
krkapJDj
jap~ 0 5 y ~ kr~apJ~~
+ 2 A2 1
—
‘8
~ I ~\ —
kT~ aP)HD,Y1
—
(3y
—
—
x)] + (3x
2y2)(l —y)2
—
=
~2)(l
T*_l
—
x)2
-
Unfortunately, the lack in the literature of the appropriate computer simulation data makes examination of the derived equations impossible. Thus we have confmed our numerical computations to the examination of approximation (5) and to the comparison of both equations (6) and (8). Fig. 1A compares the exact (computed from the PY rdf) and computed
-
1kT ay
x (1
In [x(1 —y)/y(l
‘~
2 ~
~‘1 1
‘~
(ap\
~
where y~is the root of the following equation:
Yi
‘
~
from eq. (5) values of the integral (4). Fig. 1B, however, presents the pressures calculated from eqs. (6) and (8). We note that at T~>0.8 andy ~ 0.7 the pressures calculated from eq. (8) are practically the same as those obtained from eq. (6). The deviations are larger at lower temperatures. 129
Volume 72A, number 2
PHYSICS LETTERS
References [1] H.C. Andersen, D. Chandler and J.E. Weeks, Adv. Chem. Phys., Vol. 34 (Academic Press, New York, 1976) p.105. [21 W.A. Steele, J. Chem. Phys. 65 (1976) 5256. [3] W. Rudzittski and S. Sokolowski, Acta Phys. Pol. A53 (1978) 201. [4] J.A. Barker and D. Henderson, J. Chem. Phys. 47 (1967) 2856.
130
[5] [6] [7] [8]
25 june 1979
E. Thiele, J. Chem. Phys. 39 (1963) 474. W.R. Smith and D. Henderson, Mol. Phys. 19 (1970) 411. J.W. Perram, Mol. Phys. 30(1975)1505. L. Ponce and H. Renon, J. Chem. Phys. 64 (1975) 638. [9] E. Helfand, H.L. Frisch and J.L. Lebowitz, J. Chem. Phys. 34 (1961) 1034. (10] E. Praestgaaed and S. Toxvaerd, J. Chem. Phys. 51(1969) 1895.
PHYSICS LETTERS
25 June 1979
ANALYTICAL EQUATIONS FOR THE COMPRESSIBILITY FACTOR OF A TWO-DIMENSIONAL SQUARE-WELL FLUID S. SOKOLOWSKI Department of Theoretical Chemistry, Institute of Chemistry UMCS, 20031 Lublin, Nowotki 12, Poland Received 13 March 1979
Analytical approximations for the compressibility factor of a two-dimensional square-well fluid, resulting from perturbational theories are presented.
The recently most successful description of the structure and equilibrium properties of simple threedimensional liquids can be obtained from the perturbation theories developed mainly in the last years [1]. The application of those theories for two-dimensional fluids, and consequently for the description of two-
dimensional mobile adsorption of a gas on a solid was recently reported by Steele [2] and by the present author [3]. The purpose of this note is to derive analytic approximations for the two-dimensional cornpressibility factor Z of a fluid of particles interacting according to the square-well potential: u(r)oe, =
—e,
r
‘~
r ~ Xa,
(1)
—
—~,
FHD =
—
4 ~
1
2y
api
I
apJHDa r2 LikT—i
—~
—
, i xg~~(x)dx , (3)
the reference system. Eq. (3) results from the secondorder perturbational theory and its application for a three-dimensional fluid is instantaneous. The computation of the integral from eq. (3) requires the knowledge of the radial distribution function for the reference system, and this function can be easily obtained from -
(2)
dimensional hard-disk fluids actually there are no analytical solutions of the PY equation, and consequently the numerical application of eq. (3) is much more complicated. Looking for some other theoretical possibiities the author has decided to apply the approximation introduced by Ponce and Renon [81 for threedimensional fluids. For this purpose we consider the integral from eq. (3) and rewrite it in the following equivalent form:
F
F
i~ i xg~~r~(x)dx• xh(x)dx+ i xdx— i xh(x)dx, -‘ 1 1 1 X (4)
J
h=g~~—l. 128
a y
aY
where T kT/e,y = pira~/4,p is the number density, and g~~(x) is the radial distribution function (rdf) for
r.—a.
The result of the BH theory for the Helmholtz free energy Fcan be wntten as follows:
f xg~ff~(x)dx
analytical [5,6] (PY) Percus—Yevick or numerical equation.[7] In contrast, solutions for of the two-
The starting point of our considerations is the theory of Barker and Henderson (BH) [4], according to which the potential (1) can be treated as the addition of the reference hard-disk potential uHD(r), and the perturbation u 1 (r): u(r) = UHD(r) + u1(r), uHD(r) = oo, r < a, ~
—
NkT
.
r> ~
=
F
Volume 72A, number 2
PHYSICS LETTERS
The first integral in eq. (4) can be evaluated from the compressibility equation, the third integral, however, is much smaller than the remaining two. Thus as a fair approximation one may use:
f
xg~x)
=
[i
/
kT B
1/
15
~
+
HD
A~2 _._—~
.
~
(~)]
I. kT ap
<
p
I A
1~)HD]
—
25 June 1979
(5)
/ / ii
as which was derived neglecting the last term of eq. (4). Thus, accepting for the compressibility factor of a two-dimensional hard-disk fluid the result of the scaled particle theory [91, ZHD = 1/(1 y)2, from eq. (3) we obtain the following expression:
I/
d
A •
—
Z = ZHD
(y/T*)A(1 +0.5B/T*)_ cL2s(l/T*2)C, (6)
—
________________
where A
0.4
(—2 + 3y
=
B = (1
—
4y
—
—
y3)(l
3y2)(l
+y)2 + 2X2 —
,
y)2(y + l)2
(7)
,
C—y 2 (1 —y) 3 (14.—6y—6y 2 —2y 3 )(l +y)— 4 —
Q6
________________
y
0.5 Y
0
Fig. 1. (A) The comparison of exact (circles) and approximated (lines) values of the integral I = J~’xg~j~(x) dx. (B) The comparison of the pressures calculated according to eq. (6)
(lines) with those obtained eq. (8) (white circles). The solidlinewascalculated forfrom T*0.5,thedash_dottedline
-
= 0.8, and the dashed line for T” 1. The black circles denote the results obtained from the two-dimensional Barker— Henderson equation and from the PY radial distribution function of the reference system (for T~= 0.8)- All computations were performed for X 1.5.
for T”
The last equation gives the analytic approximation for the pressure of the considered fluid in the limit of the BH theory. The higher-order corrections to the BH equation (3) were considered by Praestgaaed and Toxvaerd [10]. They have shown that the BH approximation can be used in any order of the expansion and the resulting series can be summed to obtain an equation involving only the rdf of the reference system. Application of this theory (eq. (33), ref. [10]) and the approximation (5) lead to the following equation for the compressibility factor: ~
— —
r
HD + 0. [1 —
1
1 j~p\
krkapJDj
jap~ 0 5 y ~ kr~apJ~~
+ 2 A2 1
—
‘8
~ I ~\ —
kT~ aP)HD,Y1
—
(3y
—
—
x)] + (3x
2y2)(l —y)2
—
=
~2)(l
T*_l
—
x)2
-
Unfortunately, the lack in the literature of the appropriate computer simulation data makes examination of the derived equations impossible. Thus we have confmed our numerical computations to the examination of approximation (5) and to the comparison of both equations (6) and (8). Fig. 1A compares the exact (computed from the PY rdf) and computed
-
1kT ay
x (1
In [x(1 —y)/y(l
‘~
2 ~
~‘1 1
‘~
(ap\
~
where y~is the root of the following equation:
Yi
‘
~
from eq. (5) values of the integral (4). Fig. 1B, however, presents the pressures calculated from eqs. (6) and (8). We note that at T~>0.8 andy ~ 0.7 the pressures calculated from eq. (8) are practically the same as those obtained from eq. (6). The deviations are larger at lower temperatures. 129
Volume 72A, number 2
PHYSICS LETTERS
References [1] H.C. Andersen, D. Chandler and J.E. Weeks, Adv. Chem. Phys., Vol. 34 (Academic Press, New York, 1976) p.105. [21 W.A. Steele, J. Chem. Phys. 65 (1976) 5256. [3] W. Rudzittski and S. Sokolowski, Acta Phys. Pol. A53 (1978) 201. [4] J.A. Barker and D. Henderson, J. Chem. Phys. 47 (1967) 2856.
130
[5] [6] [7] [8]
25 june 1979
E. Thiele, J. Chem. Phys. 39 (1963) 474. W.R. Smith and D. Henderson, Mol. Phys. 19 (1970) 411. J.W. Perram, Mol. Phys. 30(1975)1505. L. Ponce and H. Renon, J. Chem. Phys. 64 (1975) 638. [9] E. Helfand, H.L. Frisch and J.L. Lebowitz, J. Chem. Phys. 34 (1961) 1034. (10] E. Praestgaaed and S. Toxvaerd, J. Chem. Phys. 51(1969) 1895.