Indian Journal of Pure & Applied Physics
Vol. 44, August 2006, pp. 612-615
Thermodynamic properties of repulsive hard-core Yukawa fluid
Rajat Srivastava, Dhermendra Kumar Dwivedee & K N Khanna
Department of Physics, V S S D College, Kanpur
Received 22 December 2005; revised 19 April 2006; accepted 28 April 2006
A simple analytical expression of the mean spherical solution (Mean spherical approximation – high temperature
expansion) to the Ornstien-Zernike equation for repulsive hard core Yukawa fluid is examined. The thermodynamic and
structural properties of purely repulsive hard-core Yukawa particles in the fluid state are determined by employing this
model and compared with simulation data. The results show that the present model illustrates the remarkable improvement
of internal energy per particle over existing theoretical values.
Keywords: Thermodynamic properties, Yukawa fluid, Mean spherical approximation
1 Introduction
In recent years, the Yukawa hard sphere system has
received considerable attention due to its analytical
availability and simplicity in solving Ornstein-Zernike
integral equation under the mean spherical
approximation (MSA). Most of the work have been
done to model the attractive hard core Yukawa fluid.
It has been widely used for different fluid systems
including simple liquids, colloids, micelles and microemulsion, dense plasmas and C60 molecular
systems1–10. The repulsive hard core Yukawa system
(RHCY) has been recently introduced to describe
structure and phase behaviour of charge stabilized
colloids11,12 and dusty plasma13,14. The system of
repulsive hard core Yukawa (RHCY) spheres whose
inter particle potential energy is represented by:
U(r)
/ kT = ∞r < σ
∈ exp[−λ (r / σ − 1]
=
if r > σ
r /σ
… (1)
where λ is a screening length for the Yukawa tail and
Є is the strength of the interaction energy.
There are several theoretical models of attractive
Yukawa particles9,15-20 using MSA as the reference.
The MSA solution enables us to readily obtain the
thermodynamic and structural properties such as
radial distribution function (RDF), direct correlation
function (DCF) and equation of state of Yukawa fluid.
Henderson et al.21 proposed a perturbation
approximation expanded up to fifth order in terms of
inversion temperature in MSA approximation. The
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new versions of perturbation expansion by Duh and
Mier-Y-Teran19 and Shukla20 are very promising to
overcome the problems of full MSA. The validity of
these attractive Yukawa particles models have not
been tested and examined for repulsive Yukawa
particles. In the present work, we extend the model
proposed by Davies et al.16 for attractive Yukawa
system to model the repulsive hard core Yukawa
(RHCY) particles.
2 Theory
Henderson et al.21 proposed the analytical
expression of Helmholtz free energy applicable to
monomer Yukawa fluid based on the high
temperature expansion (HTE) method:
A = A0 + βA1 + β2A2 + …
… (2)
The mean-repulsive energy A1 for RHCY can be
written as:
∞
A1 = 2π ρЄ
∞
∫ gHS(r )
σ
exp [−λ (r-1)] r dr
… (3)
σ
where ρ = N / V is the density of a monomer Yukawa
fluid. The corresponding MSA solution of Eq (3) is:
A1=α / Φ
… (4)
where
α=
L (λ )
2
λ (1 − η) 2
and
φ=
S (λ ) + 12ηL(λ ) exp(−λ )
λ 3 (1 − η) 2
SRIVASTAVA et al.: THERMODYNAMIC PROPERTIES OF REPULSIVE HARD–CORE YUKAWA FLUID
with the coefficients L(λ) and S(λ) defined as:
L(λ) = (1 + η/2) λ + (1 + 2η)
S(λ) = (1 – η)2λ3 + 6 η (1 – η)λ2
+ 18 η2λ - 12 η(1+2 η)
… (5)
… (6)
The second order term in MSA in Eq. (2) is given
by:
A2 =
613
Further, Tang and Lu22 proposed a simplified
expression of the radial distribution function (RDF),
which is supposed to be an accurate expression. We
also compute g(σ) using this expression. In the MSA,
it can be written as:
g(σ) = ghs(σ) exp (g1(σ)/ T)
… (10)
where
3η
… (7)
λφ4
g hs (σ) =
Similarly, the radial distribution function at contact
g(σ) is given by:
g(σ) = ghs (σ) + βє g1 (σ) +…
… (8)
where the first order perturbation term of radial
distribution function can be solved as :
g1(σ) = - 1/Φ2
… (9)
(1 − η / 2)
(1 − η)3
and g1(σ) = - 1/Φ2
The equation of state for monomer Yukawa fluid can
be written as:
Z = Ze +
η∂A1 1 η∂A2 1
+
∂η T *
∂η T *2
… (11)
Table 1— Comparison of repulsive MSA-HTE theoretical values with Monte Carlo Simulation results for λ= 1.8. Th1 are the values
obtained from Eq. (8) and Th2 are the values obtained from Eq. (10). Th represents the present work obtained by employing Eq (11) for Z
and Eq. (12) for U*
T*
1.0
1.5
2.0
ρ*
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Th1
g(σ)
Th2
MC
Z
Th
MC
Th
MC
.357
0.709
1.070
1.459
1.899
2.423
3.078
3.936
0.619
0.912
1.226
1.576
1.986
2.486
3.123
3.966
0.750
1.014
1.304
1.635
2.029
2.518
3.145
3.982
0.521
0.716
0.963
1.273
1.664
2.162
2.810
3.673
0.677
0.878
1.126
1.432
1.815
2.303
2.938
3.788
0.772
0.973
1.217
1.519
1.896
2.377
3.005
3.847
0.540
0.754
1.017
1.344
1.738
2.234
2.874
3.741
0.692
0.908
1.162
1.480
1.857
2.355
2.979
3.850
0.783
0.992
1.214
1.558
1.928
2.413
3.049
3.878
1.854
2.810
3.938
5.287
6.907
8.870
11.292
14.370
1.643
2.392
3.289
4.377
5.706
7.353
9.441
12.169
1.540
2.182
2.962
3.917
5.100
6.590
8.512
11.067
1.802
2.809
4.004
5.407
7.047
9.006
11.403
14.466
1.619
2.394
3.323
4.439
5.772
7.432
9.498
12.251
1.526
2.183
2.975
3.960
5.143
6.639
8.570
11.099
0.457
1.043
1.709
2.425
3.178
3.957
4.762
5.590
0.493
1.087
1.748
2.455
3.197
3.970
4.769
5.594
0.512
1.109
1.767
2.469
3.207
3.976
4.773
5.596
0.482
1.070
1.728
2.436
3.182
3.960
4.765
5.595
0.504
1.098
1.754
2.457
3.198
3.972
4.772
5.600
0.517
1.114
1.769
2.470
3.208
3.978
4.778
5.602
U*
INDIAN J PURE & APPL PHYS, VOL 44, AUGUST 2006
614
Table 2—Comparison of repulsive MSA-HTE theoretical values with Monte Carlo results for λ = 3.0. Th1 and Th2 and Th are the same
as shown in Table 1
T*
1.0
1.5
2.0
ρ*
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Z
Th1
g(σ)
Th2
MC
Th
MC
Th
MC
0.308
0.630
0.977
1.363
1.809
2.343
3.011
3.882
0.587
0.860
1.164
1.513
1.926
2.433
3.078
3.931
0.726
0.975
1.398
1.588
1.984
2.478
3.112
3.955
0.496
0.662
0.878
1.158
1.521
1.996
2.628
3.481
0.655
0.834
1.059
1.344
1.710
2.184
2.810
3.655
0.753
0.935
1.162
1.448
1.813
2.284
2.905
3.745
0.503
0.681
0.911
1.210
1.589
2.063
2.717
3.574
0.661
0.844
1.084
1.375
1.745
2.233
2.871
3.723
0.756
0.948
1.186
1.476
1.850
2.322
2.945
3.793
1.578
2.269
3.127
4.204
5.562
7.287
9.504
12.414
1.460
2.028
2.744
3.650
4.807
6.298
8.250
10.868
1.403
1.909
2.550
3.370
4.425
5.799
7.619
10.092
1.532
2.244
3.152
4.286
5.688
7.420
9.659
12.552
1.439
2.016
2.759
3.689
4.862
6.372
8.336
10.953
1.391
1.904
2.564
3.398
4.471
5.848
7.667
10.145
0.224
0.532
0.906
1.331
1.801
2.310
2.855
3.436
0.248
0.565
0.938
1.359
1.822
2.325
2.865
3.442
0.261
0.582
0.954
1.373
1.833
2.333
2.870
3.445
0.236
0.541
0.907
1.328
1.795
2.305
2.855
3.440
0.253
0.566
0.935
1.353
1.817
2.323
2.867
3.448
0.262
0.581
0.951
1.368
1.830
2.332
2.875
3.453
Fig. 1—Comparison of the present theoretical values of the
compressibility factors for λ = 1.8 and λ = 3.0 with MC values
U*
Fig. 2—Comparison of the present theoretical values of the
internal energies for λ = 1.8 and λ = 3.0 with MC values. For
clarity of the results the Y-axis for T* = 1.5 and T* = 2 is raised
for 1 and 2 units, respectively
SRIVASTAVA et al.: THERMODYNAMIC PROPERTIES OF REPULSIVE HARD–CORE YUKAWA FLUID
where Z e =
(1 + η + η2 - η3 )
(1-η)3
The internal energy per particle is defined as:
U* =
2A
U
∂ ( A / NkT )
=
= A1 − 2
N∈
∂ (1/ T *)
T*
… (12)
where N is the number of particles and U is the total
energy.
3 Results and Discussion
Tables 1 and 2 present the theoretical results for the
radial distribution function at contact g(σ),
compressibility factor and internal energy as a
function of temperature T* and density ρ* of the
repulsive Yukawa fluid at λ = 1.8 and λ=3.0,
respectively. MC values are the Monte Carlo
simulation data of Cochran and Chiew23. In the
calculations of g (σ), Th1 are the values obtained from
the first order perturbation theory (Eq. 8) and Th 2 are
the values obtained from simplified exponential
approximation (SEXP) method (Eq. 10). Clearly, the
SEXP method predicts better agreement between
theory and Monte Carlo (MC) results for g (σ). Eq. (8)
predicts higher values than Monte Carlo data and thus
repulsive hard core Yukawa system predicts different
nature than attractive Yukawa potential. The values of
g (σ) in the attractive Yukawa potential obtained by
first order perturbation predict sufficiently lower
values in comparison to MC data and hence the
perturbation expansion is expanded up to fifth
order19,20. Compressibility factor results presented in
Tables 1 and 2 show good agreement with MC results
across the full density range ρ* for λ=1.8 and 3.0 and
comparable with other theoretical values23. It can be
seen that compressibility factor decreases with
increasing T* as the repulsive contact potential energy
decreases with T*. The internal energy shown in
Tables 1 and 2 indicate the remarkable improvement
overall the theoretical values displayed in Tables 1
and 2 of reference23. It is also worthwhile to mention
that the present results of the compressibility factor
and internal energy are better than that of an
analytical method based on inverse temperature
expansion (ITE) as computed by Cochran and
615
Chiew23. It may be due to the inclusion of high-order
terms approximated in ITE as also pointed out by
Tang et al.18. Thus, we see that the present model
predicts good results of compressibility factor, and
internal energy. The results of the compressibility
factors and internal energies are also compared for λ =
1.8 and 3.0 in Figs 1 and 2, respectively along with
the MC values. It can be seen that both
compressibility factors and internal energies decrease
with increasing value of λ and the decrement
increases with increasing density. It is due to decrease
in inter particle potential energy with increasing value
of λ.
Acknowledgement
One of the authors DKD is thankful to CSIR, New
Delhi for financial assistance.
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