A single-theory approach to the prediction of solid–liquid and liquid–vapor
phase transitions
Payman Pourgheysar
Department of Chemical Engineering, Amirkabir University of Technology, Tehran Polytechnique, Tehran,
Iran
G. Ali Mansooria)
Department of Chemical Engineering, University of Illinois at Chicago, Chicago, Illinois 60607-7000
Hamid Modarress
Department of Chemical Engineering, Amirkabir University of Technology, Tehran Polytechnique,
Tehran, Iran
~Received 20 February 1996; accepted 26 August 1996!
For simultaneous prediction of solid–liquid and liquid–vapor phase transitions it has been
customary to apply two different theories for solid and fluid phases. A single-theory approach will
be desirable to answer many of the fundamental problems of molecular theory and their relationship
with macroscopic behavior of the matter. Based on a modified version of the cell model of statistical
mechanics, a single-theory approach for simultaneous prediction of solid–liquid and liquid–vapor
phase transitions is presented here. In developing this theory the order–disorder transition is
considered as the essential feature of the fusion and a new function for the potential energy field
inside a single-occupancy cell is derived. By reporting the variations of total pressure of the
macroscopic system with respect to temperature and volume the nature of the various phase
transitions in the system are evaluated and discussed. Variations of the radial distribution function
of the molecules in the system with intermolecular distance, temperature, and volume are reported
for various phases of matter. © 1996 American Institute of Physics. @S0021-9606~96!50245-X#
I. INTRODUCTION
Phase transitions and their predictions are of prime importance in the study of matter and they constitute the cornerstone of many processes encountered in the science and
technology.1–11 Of the three phase transitions in macroscopic
systems consisting of simple molecules, solid–liquid,
liquid–vapor, and vapor–solid, the latter two are well understood and the related predictive statistical mechanical models
have found numerous practical applications. There has been
a wealth of research activity in the development of predictive
models for solid–liquid transitions ~both melting and freezing! since the mid-sixties and it has resulted in advancement
of a number of quantitatively accurate predictive
models.1–5,7–10 Among these the density functional theory
has attracted a great deal of attention in recent years.2,3
In predicting simultaneously solid–liquid and liquid–
vapor phase transitions it has been necessary to use two different theories to describe the solid and fluid phases, separately. What has been missing from the scene is a singletheory approach through which one can predict all the three
phases of matter and the related phase transitions. The
single-theory approach presented here is for the purpose of
achieving this goal.
The cell theory is shown to be able to establish, in a
typical way, a connection between liquid and dense gas
states.1,14–16 Also, some success has been achieved to do the
same in the case of solid and liquid phases using the cell
a!
Author to whom correspondence should be addressed.
9580
J. Chem. Phys. 105 (21), 1 December 1996
theory.12,14 Considering the above facts we recognize the cell
theory to be appropriate for predicting solid–liquid and
liquid–vapor phase transitions, simultaneously. The simplicity of the mathematical formulations involved in the singleoccupancy cell theory allows us to pay more attention to the
various features of the model and their relationships to the
physical phenomenon of phase transitions. This purpose
could not be generally achieved with the use of the other
more sophisticated theories of statistical mechanics.
II. DEVELOPMENT OF THE THEORY
In the development of a cell model capable of predicting
all the three phase transitions ~solid–liquid, liquid–vapor,
solid–vapor!, for simplicity the attention will be focused on
the basic single-occupancy version of the cell model. Thus,
let us imagine the volume space occupied by particles partitioned into a set of Wigner–Seitz cells corresponding to a
face-centered cubic lattice ~fcc! and each cell is supposed to
contain exactly one particle. The total intermolecular potential energy of the system, accepting the pairwise additivity
assumption, is then given by
U5
( (
1<i, j<N
uij ,
~1!
where N is the total number of molecules in the system and
u i j is the pair intermolecular potential energy function between molecules (I) and ( j) of the system.
The classical partition function of this system according
to the cell theory is given by
0021-9606/96/105(21)/9580/8/$10.00
© 1996 American Institute of Physics
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Pourgheysar, Mansoori, and Modarress: Prediction of phase transitions
Z5l 23N
E E
DN
•••
D1
e 2 b U dr1 •••drN ,
~2!
in which: l5[h /(2 p mkT)] 1/2, b51/(kT), and the integration over the position vector ~rI ! is taken over cell (I) ~i.e.,
Di and not over the whole volume!. Position vector ~rI ! denotes the position of the molecule in cell (I) relative to the
center of the cell.
To make possible the evaluation of the many-body integral in Eq. ~2!, we seek to approximate (u i j ) by the form
2
u i j ~ r i ,r j ! .w ji ~ r i ! 1w i j ~ r j ! ,
w ji ~ r i ! 5 21 u i j ~ 0,0! 1 @ u i j ~ r i ,0! 2u i j ~ 0,0!# .
~4!
The approach based on this choice is designated as the usual
‘‘free-volume’’ version of the cell theory.
Let us now assume that the only important interaction is
the one between the first-nearest-neighbor molecules. Then,
let us define
u i~ r i ! [
(j
w ji ~ r i ! 5
(j
1
(j @ u i j ~ r i ,0! 2u i j ~ 0,0!# .
~5!
Where the summation is taken over the first-nearestneighbors of molecule (I), and
N
U[
( u i~ r i ! .
i51
~6!
By substitution of Eq. ~6! in Eq. ~2! we obtain
)
E
SE
e 2 b u i dri
N
Z5l
23N
i51
5l 23N
Di
Di
e 2 b u i dri
D
~7!
N
~8!
,
where we have assumed that the function u i (r i ) for different
cells is the same.
Let us now rewrite Eq. ~8! as the following:
FE F (
H (
F (
SE H (
Z5l 23N
Di
exp 2 b
3exp 2 b
j
Di
1
2
u i j ~ 0,0!
G
J G
N
@ u i j ~ r i ,0! 2u i j ~ 0,0!# dri
5l 23N exp 2 b N
3
j
exp 2 b
j
j
1
2
F
Z5l 23N exp 2 b N
u i j ~ 0,0!
G
3
SE H
Di
~9!
J D
N
.
~10!
exp 2 b
u ik ~ 0,0!
G
(j @ u i j ~ r i ,0! 2u i j ~ 0,0!#
J D
N
dri
~11!
~12!
in which
U s5
and
( (
l<i, k<N
u ik ~ 0,0! ,
E H
Di
exp 2 b
(j @ u i j ~ r i ,0! 2u i j ~ 0,0!#
J
dri ,
~13!
where ~n f ! is called the free-volume integral.1,15
Evaluation of the individual terms in Eq. ~12!, the freevolume integral ~n f ! and the static potential energy (U s ), are
the essential problem in construction of the model which will
be addressed next in this report.
A. Evaluation of the free-volume integral (nf )
The integration in ~n f ! is to be performed over the volume of the cell which its shape, according to the Wigner–
Seitz lattice model for fcc lattice, is a dodecahedron. By
using the smearing approximation1,16 we can, for convenience, approximate the potential field inside the cell with a
spherically symmetric function depending only on the distance of the molecule from the center of its cell (R i ). Then
~n f ! can be calculated by averaging the potential energy over
a sphere of radius (R i ) centered at the cell center. The cell
field derived in this way is identical with that derived by
regarding the neighboring molecules as ‘‘smeared’’ with uniform probability distribution over the surfaces of concentric
spheres of appropriate radius. Also, by regarding this radial
symmetry of the potential function, we can, for simplicity,
approximate the actual cell with a sphere which its radius
(R c ) is obtained from setting the volume per molecule ~n
5V/N! equal to the volume of this sphere @(V) is the total
volume and (N) is the total number of molecules in the
system#. Therefore we will have
nf5
@ u i j ~ r i ,0! 2u i j ~ 0,0!# dri
( (
1<i, k<N
5l 23N e 2 b U s ~ n f ! N ,
nf5
u i j ~ 0,0!
1
2
The first exponential term in Eq. ~10! refers to the static
potential energy (U s ) of the system, and it can be modified
to take into account the interactions of molecule (I) with
molecules other than the first-nearest-neighbors of (I) by
extending the range of summation to include all the molecules (k) other than (I). Therefore, Eq. ~10! can be written
as
~3!
which would reduce the multiple integral in Eq. ~2! to a
product of a single integrals. One possible choice of the
functions (w ji ) is
9581
5
E
E
Rc
0
Rc
0
H
4 p R 2i exp 2 b
(j @ u i j ~ r i ,0! 2u i j ~ 0,0!#
4 p R 2i e 2 bc i ~ r i ! dR i ,
J
dR i ~14!
~15!
in which (R i ) is the absolute value of (r i ), 34 p R 3c 5n, and
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Pourgheysar, Mansoori, and Modarress: Prediction of phase transitions
c i ~ r i ! 5 ( @ u i j ~ r i ,0! 2u i j ~ 0,0!# .
~16!
j
For fcc lattice, the distance between the first nearest neighbor
lattice points ~a 1! is related to ~n! by the relation (a 31 5& v ).
Thus,
a 31
4
p R 3c 5
3
&
or
S D
Rc
.0.55.
a1
c ha
i ~ ri!5
It should be pointed out that the smearing approximation
is actually valid at the high density region. At lower densities
the increase of molecular interspace makes its accuracy a bit
doubtful.
Another noticeable concept, in relation to the assumptions made here, is the ‘‘communal entropy.’’1,15,16 As the
temperature approaches infinity, Eq. ~12! takes the form
Z T→` 5l 23N
SD
V
N
c1
2
0
@ R 2i 1a 21 22a 1 R i cos~ u !#
2 w ~ a 1 ! % sin~ u ! d u ,
G S D
2
Ri
1 2
a w 9 ~ a 1 ! 1O
2 1
a1
a 1w 8~ a 1 ! 1
4
.
~18!
This expansion converges quite rapidly at high densities,
where the potential energy well can be represented as a harmonic function.
At this stage we choose the Lennard-Jones intermolecular potential energy function to represent w(x):
w LJ~ x ! 54 e
FS D S D G
s
x
s
x
12
2
6
,
where s and e are the Lennard-Jones potential parameters.
Therefore, c ha
i (r i ) given by Eq. ~18! becomes
.
E $w A
p
S DF
c1 Ri
3 a1
N
At this limit, the kinetic energy of each molecule is sufficient
to make the effect of intermolecular forces unimportant as it
is also the case in the ideal gas state. On the other hand, by
comparing the above equation with the partition function of
an ideal gas @i.e., Z ideal5l 23N (V N /N!)#, it reveals that the
only difference between the two expressions is in the appearance of the factor ~1/N!! in ~Z ideal! instead of ~1/N! in
~Z T→` ! which leads to the appearance of an additional term
in entropy formula called the ‘‘communal entropy’’.16 Communal entropy is due to the unlocalization of ideal gas molecules, so that for a molecule in gaseous state, the whole
volume is available for its movement. The simplest way to
make these two partition functions identical, is entering the
factor (e N ) in Eq. ~12! @notice that N!.(N/e) N when N→`,
see Ref. 14#. It can be shown that communal entropy has no
effect on evaluating the pressure which is the only macroscopic property of the system that will be evaluated, here. In
the liquid state, this feature is arisen as the possibility of
multiple occupancy of the cells. However, the modifications
made to take into account this effect do not produce any
noticeable improvement over the single-occupancy version
of the cell model considered here. In the solid state, because
of the closed-packing of the molecules, this problem is not
arisen at all. Therefore, we can ignore the existence of
‘‘communal entropy’’ and multiple-occupancy effects in our
calculations.
Then with the assumptions of single-occupancy cells,
smearing approximation and consideration of only the firstnearest-neighbors interactions in calculation of the freevolume, the following expression for c i (r i ) will be
derived:1,14–16
c i~ r i ! 5
(x), and (c I ) is the number of first-nearest-neighbors of a
molecule ~for example for fcc lattice c I 512!. Equation ~17!
can be solved by a Taylor series expansion in terms of (R i ,).
For small values of (R i ,) it can be approximated by an equation which we designate as its ‘‘harmonic approximation’’
15
@ c ha
i (r i ) #
0,R i ,0.55,
~17!
where w(x) is the function representing the pair interaction
potential energy between every two molecules at the distance
c ha
i ~ ri!5
S DF
F
2
c1 Ri
3 a1
516e y 2i
8 ~ a1!1
a 1 w LJ
G
1 2
a w9 ~a !
2 1 LJ 1
G
16.5 7.5
2
1O ~ y i ! 4 ,
n *4 n *2
~19!
where ~n*5n/s3! is the dimensionless volume per cell,
(y i 5R i /a 1 ) is the dimensionless distance of a molecule from
its cell center and (a 31 5& v ).
Let us now consider the physical limitation imposed on
every molecule in its cell when it is in the solid and liquid
states. In such states of matter, every molecule is confined in
its cell and, effectively, the expression for c ha
i (r i ) will assume the following modified form, c mha
(r
),
by
taking into
i
i
account the repulsive forces at the cell wall,
c mha
i ~ ri!5
H
F
G
16.5 7.5
2
,
n *4 n *2
0.5,y i .
16y 2i e
`,
0<y i <0.5 .
~20!
Using the above effective modified form of c mha
i (r i ) the expression for the free-volume will assume the following form:
nf5
E
0.5
0
mha
4 p y 2i e 2 bc i
or, in dimensionless form
n *f 5
nf
54 p & n *
s3
E
0.5
0
~ri!
~21!
dy i ,
F
y 2i exp 2
S
16y 2i 16.5 7.5
2
T * n *4 n *2
DG
dy i .
~22!
Equation ~22! can be used to calculate the free-volume integral at various dimensionless temperatures and volumes for
the fcc lattice. Similar expressions can be derived for other
lattice structures and/or for multi-occupancy cell models.
B. Evaluation of the static potential energy ( U s )
Suppose the number of molecules (c i ) and distances
(a i ) of the successive neighboring shells with respect to a
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Pourgheysar, Mansoori, and Modarress: Prediction of phase transitions
S D
]n *f
]n *
central molecule in terms of the corresponding parameters
for the first-nearest-neighbor shell (c 1 ,a 1 ) is given by
c i 5n i •c 1
a i 5d i •a 1 ,
and
i51,2,3,... .
54 p &
N,T
Thus, (U s ), after replacing (u ik ) by wLJ , becomes
U s5
((
1<i,k<N
5
( (
i51 k51 ~ kÞi !
N
5
(
i51
4e
c
2 1
3exp
1
u ~ 0,0!
2 ik
FS D ( S D ( G
s
a1
12 `
s
122
a
di
1
ni
i51
6 `
ni
d 6i
i51
~23!
For the fcc lattice we can show that
3exp
and
S D
n 1 51, n 2 51/2, n 3 52, n 4 51, n 5 52, n 6 52/3,... .
]n *f
]n *
As a result, it can be shown that
(
i51
`
ni
.1.01
d 12
i
Therefore,
N
U s5
(
i51
and
F S D
s
24e 1.01
a1
524N e
F
(
i51
ni
d 6i
G
S DG
~24!
~25!
C. Evaluation of the total pressure
By using Eqs. ~21! and ~25! in the partition function, Eq.
~12!, we can evaluate the macroscopic properties of the system such as the total pressure
] ln Z
]V
5kT
N,T
F
H F S
5 2
P *5
]~ 2bUs!
] ln n f
1N
]V
]V
DG
]
1.01 1.205
kT ]n f
24e
2
1
]n
4 n *4 2 n *2
n f ]n
Thus, in dimensionless form
F
0
S
E
0.5
0
G S D
1.01 1.205
Ps3
T * ]n *f
524
52
3 1
e
n*
n*
n *f ]n *
S D
D GJ
S D
D GJ
y 2i ~ 16y 2i !
16y 2i 16.5 7.5
2 2
T
n4
n
4p&
T*
2
0.5
y 2i ~ 16y 2i !
S
16y 2i 16.5 7.5
2 2
T
n4
n
D GJ
66
n *4
15
n *2
dy i .
~29!
S
54 p & G1
N,T
G REP2G ATT
T*
D
~30!
~31!
Finally, substitution of Eqs. ~30! and ~31! in Eq. ~28! gives
6
1.01 1.205
2
.
4 n *4 2 n *2
S D
2
E
16y 2i 16.5 7.5
T
n4 n2
n *f 54 p & n * G.
By replacing Eqs. ~22! and ~25! into Eq. ~12! one can calculate the partition function, and from that, all the macroscopic
properties of the system. In what follows evaluation of the
total pressure of the system is reported through which one
can demonstrate the prediction of phase transitions.
P5kT
4p&
T*
HF S
and
.1.205.
s
21.205
a1
12
HF
y 2i exp
We designate the three integrals appearing in the above
equation as (G), ~G REP!, and ~C ATT!, respectively. Thus,
( ]n *f / ]n * ) N,T and ( n *f ) can be written in terms of these
three integrals as
d i 5 Ai
`
HF
3dy i 2
.
0.5
0
3dy i 1
u ik ~ 0,0!
N
N
E
9583
J
G
N,T
.
~26!
~27!
N,T
,
S
D F
S
DG
1.01 1.205
T*
1 G REP2G ATT
11
.
52
3 1
n*
n*
n*
T*
G
~32!
Equation ~32! is plotted for various dimensionless temperature ~T *! values as isothermal ~P *2n*! curves in Figs. 1~a!,
1~b!, and 1~c!. According to Fig. 1~a! for the isotherms
T *50.9, 1.5, 2, 3, we observe two phase transitions ~solid–
liquid and liquid–vapor!. As we increase the temperatures
~T *57 and 10!, Fig. 1~b!, we observe only the liquid–vapor
phase transition and at higher temperatures ~T *.12! no
phase transitions are observed. In the other hand, at much
lower temperatures ~T *50.1 and 0.2! the resulting isotherms
in Fig. 1~c! could indicate the existence of only one phase
transition which can be interpreted as the solid–vapor phase
transition.
According to Figs. 1~a!, 1~b!, and 1~c! the present model
is capable of predicting both solid–liquid and liquid–vapor
phase transitions, simultaneously at intermediate temperatures. This kind of capability is unique to the present theory,
and to our knowledge, no other single theory has been able
to predict these transitions together. In order to elucidate
further on the capability of this model we also report the
radial distribution functions of various phases of matter
~solid, liquid, and vapor! which can be predicted by this
single theory approach.
P * 524
~28!
N,T
where (T * 5kT/ e ) is the dimensionless temperature and
( P * 5 P s 3 / e ) is the dimensionless pressure. But, according
to Eq. ~22! we have
D. Evaluation of the radial distribution function (RDF)
For the modified harmonic approximation model proposed here, Eq. ~20!, we can calculate the RDF using the
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Pourgheysar, Mansoori, and Modarress: Prediction of phase transitions
FIG. 1. ~a!–~c! Variation of the dimensionless pressure P * vs n* at different
values of dimensionless temperatures as calculated by the present model.
following expression originally proposed by Corner and
Lennard-Jones based on the single-occupancy cell model13
g ~ Y , n * ,T * ! 5
`
1
(
16p &G 2 Y k51
mha
3e 2 bc i
~yi!
dy i
ck
dk
E
E E
`
dV
0
1/2
u V2Y u
1/2
u V2d k u
mha
y j e 2 bc j
yi
~y j!
dy j ,
~33!
where, according to Fig. 2, parameter (Y 5D/a 1 ) is the dimensionless distance between any two molecules ~i and j!
belonging to two different single-occupancy cells and parameter ~V5S/a 1! is the dimensionless distance of molecule (i)
from the center of the cell ( j). Parameter (k) refers to the kth
shell and uV2d k u and uV2Y u must be less than 1/2, otherwise
the two last corresponding integrals in Eq. ~33! become zero.
This guarantees the existence of a molecule in the neighborhood.
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Pourgheysar, Mansoori, and Modarress: Prediction of phase transitions
9585
By replacing Eq. ~20! in Eq. ~33! we can calculate the
RDF of molecules in a substance which obeys the singleoccupancy cell model. In Figs. 3~a!, 3~b!, and 3~c!, the RDFs
of solid, liquid, and vapor phases for various dimensionless
temperatures and volumes are reported.
In calculating the RDF, we have considered up to the 6th
nearest-neighbor shells which leads to the conclusion that the
~V! values greater than ~A61 21.2.95! are not effective on the
RDF. Therefore, we have substituted the required quantities,
mha
such as c mha
i (r i ) or c j (r j ) by Eq. ~20!, and we have replaced ~`! in the limit of the first integral of Eq. ~33! by
~3.00! ~which is the rounded value of 2.95! and then, we
have made numerical calculations. Details about the RDF
calculations and other related matters are discussed below.
III. DISCUSSION
FIG. 2. The intermolecular distance notations used to define the variables in
the radial distribution expression ~RDF!, Eq. ~33!.
As it is observed in Fig. 1~a!, at relatively moderate ~T *!
ranges, there are two S-shaped sections in each isotherm corresponding to a low and a high ~n*!, which can be attributed
FIG. 3. ~a!–~c! Variations of the RDF in different phases ~solid, liquid, and
vapor! with the dimensionless distance between any two molecules (Y ) at
different values of T * and n*.
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9586
Pourgheysar, Mansoori, and Modarress: Prediction of phase transitions
FIG. 4. ~a! and ~b! Effect of the repulsive ~REP! and attractive ~ATT!
parameters on the shape of the P * – n* isotherms and the occurrence of
phase transitions.
to solid–liquid and liquid–vapor phase transitions, respectively. If we apply the Maxwell’s criterion of phase equilibrium to these two S-shaped sections, we will find that the
ratio of equilibrium volume of the low density state to the
dense state for liquid–vapor transition is much greater than
the corresponding value for the solid–liquid case, which is
qualitatively in agreement with the experimental evidence.
Also, the increase in equilibrium pressure with temperature
in the solid–liquid case is much greater than the corresponding value for liquid–vapor transition, in agreement with the
experimental observation.
On Fig. 1~a!, it is observed that the equilibrium pressure
of liquid–vapor transition predicted by the present cell
model is negative. This negative-pressure liquid–vapor
phase transition is maintained even at high temperatures,
where the real matter according to the experimental observation, must be in its supercritical state. It should be pointed
out that at relatively high temperatures, Fig. 1~b!, the pressure can become positive. This deficiency of the present
model is an immediate result of the amplified attractive interaction effect of c mha
i (y i ) in the neighborhood of the cell
wall ~i.e., y i .0.5!. This increases the probability of molecules remaining in the condensed state, and that breaking
down this aggregated structure will need sufficient thermal
energy. We also notice the negative triple point pressure,
Fig. 1~c!, predicted by this model.
According to Fig. 1~b!, we observe a liquid–vapor critical point at rather high temperatures ~T *;10!. By investigating the isotherms in Fig. 1~a!, it seems that there is also a
critical point corresponding to solid–liquid phase transition.
As ~T *! increases, equilibrium dimensionless volume of
saturated liquid ~obtained from Maxwell’s criterion! decreases while for saturated solid it increases until they reach
at their critical point where they become identical. So far, up
to the pressures for which experiments can be performed, it
is shown that the decrease of saturated-liquid-volume with
~T *! is correct but saturated-vapor-volume must decrease,
also. It should be mentioned that while liquid–vapor critical
point can exist, there is no experimental evidence in support
of the existence of the solid–liquid critical point.
As it can be seen from Figs. 1~a!, 1~b!, and 1~c!, for
small values of ~n*!, which correspond to the solid state, the
RDF is noticeable only at the shell positions, otherwise it
becomes negligible. For large distances between the neighboring shells and the central molecule ~large y values!, the
RDF becomes smooth and approaches unity ~not shown
here! as we reach at the bulk or macroscopic density domain.
By increasing ~T *!, variations in the RDF will be further
smoothed out which is another way of saying the solid crystal becomes imperfect ~i.e., the disorder increases!. When
~n*! takes still larger values ~i.e., moving toward the liquid
domain!, the RDF takes an oscillatory form with damping
amplitude, typical of the liquid state representing short-range
order and long-range disorder, so that some peaks of the
RDF at the position of the neighboring shells of lattice points
in the previous case ~solid! disappear. This is a typical behavior observed during melting that can be attributed to
order–disorder phenomena. In the gaseous region ~i.e., at
relatively high n*!, the RDF, after the first-nearest-neighbor
position, becomes nearly unity in agreement with the theory
of RDF. Therefore, as a final result, variations of the RDF,
obtained on the basis of this new model can represent solid–
liquid and liquid–vapor transitions.
There is no special physical significance to coefficients
~16.5 and 7.5! appearing in Eq. ~20! other than the fact that
J. Chem. Phys., Vol. 105, No. 21, 1 December 1996
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Pourgheysar, Mansoori, and Modarress: Prediction of phase transitions
they result from the Lennard-Jones model potential. In order
to study the effect of variations in these coefficients on the
phase transitions we replace them by ~REP! and ~ATT! notations representing repulsive and attractive energy parameters, respectively. Therefore, we rewrite Eq. ~20! in a more
general form
c mha
i ~ y i!5
H
16y 2i e
`,
F
G
REP ATT
2
,
n *4 n *2
0<y i <0.5,
0.5,y i .
~34!
As it can be seen in Fig. 4~a!, increasing the value of
~ATT! ~i.e., attractive contribution of the pair interactions!
compared with ~REP! ~repulsive ones!, promotes the occurance of solid–liquid transition at higher pressures. It also
causes the equilibrium pressure of vaporization to increase as
expected. It can be also observed that increasing ~REP!,
compared with ~ATT!, has the reverse effect as shown in Fig.
4~b!.
IV. CONCLUSION
In predicting solid–liquid and liquid–vapor phase transitions it has been customary to use two different statistical
mechanical models, one for the solid state and another for
the fluid state. The modified cell model introduced in this
report has the capability of simultaneously predicting, even
though qualitively, solid–liquid and liquid–vapor phase transitions of molecular fluids using a single statistical mechanical model. It is based on a modified harmonic cell potential
function which is an approximation to the single-occupancy
cell theory originally proposed by Lennard-Jones and
9587
Devonshire.1,14–16 The proposed statistical mechanical approach has the capability of being improved to predict all the
phase transitions ~solid–liquid, liquid–vapor, and vapor–
solid! more quantitatively. Such improvements can be accomplished, possibly, by application of the present approach
in reformulation of the more advanced cell models in which
there are the possibilities of multi-occupancy and/or vacancy
of cells, as well as the possibility of variations of the number
of nearest neighbors as a result of changes of state or phase
transitions.
J. A. Barker, J. Chem. Phys. 63, 632 ~1975!.
C. Rascon, G. Navascues, and L. Mederos, Phys. Rev. B 51, 14899
~1995!.
3
S. M. Osman, J. Phys. Cond. Matter 6, 6965 ~1994!.
4
D. Kuhlmann-Wilsdorf, Phys. Rev. 140, 1599 ~1965!.
5
G. A. Mansoori and F. B. Canfield, J. Chem. Phys. 51, 4967 ~1969!.
6
Molecular Based Study of Fluids, edited by J. M. Haile and G. A.
Mansoori Adv. Chem. Series 204 ~1983!.
7
Y. Kato and N. Nagaosa, Phys. Rev. B: Condensed Matter. 47, 2932
~1993!.
8
M. Hirami, J. Chem. Phys. 99, 8290 ~1993!.
9
D. A. Young, J. Chem. Phys. 98, 9819 ~1993!.
10
S. Doniach, T. Garel, and H. Orland, J. Chem. Phys. 105, 1601 ~1996!.
11
M. R. Ekhtera and G. A. Mansoori ~unpublished!.
12
J. A. Barker, Proc. R. Soc. London Ser. A 240, 265 ~1957!.
13
J. Corner and J. E. Lennard-Jones, Proc. R. Soc. London Ser. A 178, 401
~1941!.
14
J. E. Lennard-Jones and A. F. Devonshire, Proc. R. Soc. London Ser. A
163, 53 ~1937!; 169, 317 ~1939!; 170, 464 ~1939!.
15
J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of
Gases and Liquids ~Wiley, New York, 1964!, Chap. 4, Secs. 5,6,7.
16
T. L. Hill, An Introduction to Statistical Thermodynamics ~Dover, New
York, 1986!, Chap. 16.
1
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J. Chem. Phys., Vol. 105, No. 21, 1 December 1996
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