Nonmonotonic pressure as a function of the density in a fluid without attractive forces
A simple result for the pressure of a hard sphere fluid that was developed many years ago by Rennert is extended in a straightforward manner by adding additional terms that are of the same form as Rennert's formula. The resulting expression is moderately accurate but its accuracy does not neces...
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irk-123456789-1208452017-06-14T03:03:19Z Nonmonotonic pressure as a function of the density in a fluid without attractive forces Henderson, D. A simple result for the pressure of a hard sphere fluid that was developed many years ago by Rennert is extended in a straightforward manner by adding additional terms that are of the same form as Rennert's formula. The resulting expression is moderately accurate but its accuracy does not necessarily improve as additional terms are included. This expression has the interesting consequence that the pressure can have a maximum, as the density increases, which is consistent with the freezing of the hard spheres. This occurs solely as a consequence of repulsive interactions. Only the Born-Green-Yvon and Kirkwood theories show such behavior for hard spheres and they require the numerical solution of an integral equation. The procedure outlined here is ad hoc but is, perhaps, useful just as the popular Carnahan-Starling equation for the hard sphere pressure is also ad hoc but useful. Простий результат для тиску плину твердих сфер, який був отриманий багато рокiв тому Реннертом, розширено в простий спосiб шляхом додавання членiв, якi мають такий же вигляд як формула Реннерта. Результуючий вираз є посередньо точним, але його точнiсть не обов’язково покращиться, якщо включити додатковi члени. Цiкавим наслiдком отриманого виразу є те, що тиск може мати максимум, коли густина зростає, що узгоджується iз твердненням твердих сфер. Це вiдбувається виключно як наслiдок короткодiйних взаємодiй. Лише теорiї Борна-Грiна-Iвона i Кiрквуда показують таку поведiнку для твердих сфер i вони потребують числового розв’язку iнтегрального рiвняння. Процедура, окреслена тут є ad hoc, але можливо є корисною такою ж мiрою, як i популярне рiвняння Карнагана-Старлiнга для тиску твердих сфер, яке є також ad hoc, але корисним. 2013 Article Nonmonotonic pressure as a function of the density in a fluid without attractive forces / D. Henderson // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43001:1-4. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 05.20.-y, 05.20.Jj, 64.10.+h, 64.30.+t, 64.70.Hz DOI:10.5488/CMP.16.43001 arXiv:1312.3547 http://dspace.nbuv.gov.ua/handle/123456789/120845 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A simple result for the pressure of a hard sphere fluid that was developed many years ago by Rennert is extended in a straightforward manner by adding additional terms that are of the same form as Rennert's formula. The resulting expression is moderately accurate but its accuracy does not necessarily improve as additional terms are included. This expression has the interesting consequence that the pressure can have a maximum, as the density increases, which is consistent with the freezing of the hard spheres. This occurs solely as a consequence of repulsive interactions. Only the Born-Green-Yvon and Kirkwood theories show such behavior for hard spheres and they require the numerical solution of an integral equation. The procedure outlined here is ad hoc but is, perhaps, useful just as the popular Carnahan-Starling equation for the hard sphere pressure is also ad hoc but useful. |
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Henderson, D. |
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Henderson, D. Nonmonotonic pressure as a function of the density in a fluid without attractive forces Condensed Matter Physics |
| author_facet |
Henderson, D. |
| author_sort |
Henderson, D. |
| title |
Nonmonotonic pressure as a function of the density in a fluid without attractive forces |
| title_short |
Nonmonotonic pressure as a function of the density in a fluid without attractive forces |
| title_full |
Nonmonotonic pressure as a function of the density in a fluid without attractive forces |
| title_fullStr |
Nonmonotonic pressure as a function of the density in a fluid without attractive forces |
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Nonmonotonic pressure as a function of the density in a fluid without attractive forces |
| title_sort |
nonmonotonic pressure as a function of the density in a fluid without attractive forces |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/120845 |
| citation_txt |
Nonmonotonic pressure as a function of the density in a fluid without attractive forces / D. Henderson // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43001:1-4. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hendersond nonmonotonicpressureasafunctionofthedensityinafluidwithoutattractiveforces |
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2025-07-08T18:43:14Z |
| last_indexed |
2025-07-08T18:43:14Z |
| _version_ |
1837105357485244416 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 4, 43001: 1–4
DOI: 10.5488/CMP.16.43001
http://www.icmp.lviv.ua/journal
Nonmonotonous pressure as a function of the
density in a fluid without attractive forces
D. Henderson¤
Department of Chemistry and Biochemistry, Brigham Young University, Provo UT 84602–5700
Received April 10, 2013, in final form May 18, 2013
A simple result for the pressure of a hard sphere fluid that was developed many years ago by Rennert is ex-
tended in a straightforward manner by adding the terms that are of the same form as the Rennert’s formula. The
resulting expression is moderately accurate but its accuracy does not necessarily improve as additional terms
are included. This expression has the interesting consequence that the pressure can have a maximum, as the
density increases, which is consistent with the freezing of hard spheres. This occurs solely as a consequence of
repulsive interactions. Only the Born-Green-Yvon and Kirkwood theories show such a behavior for hard spheres
and they require a numerical solution of an integral equation. The procedure outlined here is ad hoc but is,
perhaps, useful just as the popular Carnahan-Starling equation for the hard sphere pressure is also ad hoc but
useful.
Key words: partition function, equation of state, pressure, hard sphere fluid, freezing transition
PACS: 05.20.-y, 05.20.Jj, 64.10.+h, 64.30.+t, 64.70.Hz
1. Introduction
For a number of years, the author has been intrigued by a result obtained by Rennert [1] for a hard
sphere fluid. For this fluid, the interaction potential, u(R
12
), for a pair of spheres whose centers are lo-
cated at r
1
and r
2
, vanishes if the separation of the centers, R
12
Æ jr
1
¡r
2
j, exceeds their diameter, d . Since
the spheres are hard, they cannot overlap and u(R) is infinite ifR Ç d . To keep the discussion simple, here
the hard spheres are all assumed to be of the same diameter. Hard spheres are important because the
‘structure’ of a simple liquid or dense gas, such as argon, is, apart from the small perturbing effect of the
attractive dispersion forces, determined by hard sphere interactions [2].
The connection between the pressure of a fluid of N molecules and their interactions is provided by
the configurational partition function,Q
N
,
Q
N
Æ
1
N !
Z
exp
£
¡¯U (r
1
, . . . ,r
N
)
¤
dr
1
, . . . ,dr
N
, (1)
where ¯Æ 1/kT , and k and T are the Boltzmann constant and temperature, respectively. The interaction
energyU is assumed to be the pairwise additive sum of the pair potentials,
U (r
1
, . . . ,r
N
)Æ
N
X
iÇ jÆ1
u(r
i j
). (2)
The challenge is to determine the configurational partition function, since once it is known, the den-
sity dependent part of the Helmholtz function, A, and the pressure, p , for a given value of the density,
½ ÆN/V , where V is the volume, can be obtained from
A Æ¡kT lnQ
N
(3)
¤E-mail: doug@chem.byu.edu
© D. Henderson, 2013 43001-1
http://dx.doi.org/10.5488/CMP.16.43001
http://www.icmp.lviv.ua/journal
D. Henderson
and
p Æ¡
�A
�V
. (4)
Since N is enormous (of the order of Avogradro’s number), this can be done only approximately.
2. Rennert’s method
Rennert considers a collection of systems that have n hard spheres, where n is a variable. He defines
q
n
Æ
Q
NÅ1
VQ
N
, (5)
Q(0)Æ 1. From this it follows that
p
½kT
Æ 1¡½
�
�½
·
1
N
Z
N
0
lnq
n
dn
¸
. (6)
Reasoning by analogy to one dimension, Rennert suggests the ansatz
lnq
n
Æ
³
1¡
nv
d
V
´
¡²
nv
d
V ¡nv
d
(7)
from which, after some algebra, it follows that
p
½kT
Æ
²
1¡½v
d
¡
1¡²
½v
d
ln(1¡½v
d
), (8)
where v
d
and ² are the parameters to be chosen. In one dimension, Rennert chooses v
d
Æ d Æ b, where
B
2
Æ b is the correct second virial coefficient in one dimension, and ² Æ 0. The choice v
d
Æ d is sensible
because for ½d Æ 1, the hard particles fill space and a singularity is to be expected. The virial coefficients
for one dimensional hard particles that result from equation (8) are
B
n
Æ
1Å (n¡1)²
n
d
n¡1
. (9)
They are correct with ²Æ 0. The resulting equation of state is
p
½kT
Æ
1
1¡ y
, (10)
which, for hard particles in one dimension, is exact.
Rennert also applied equation (8) in three dimensions (hard spheres). With the choices, v
d
Æ B
2
Æ
b Æ 2¼d
3
/3 and ² Æ 8¼/3
p
2¡1, equation (8) yields the correct second and third virial coefficients for
hard spheres. As is seen in figure 1, Rennert’s procedure gives fair results for hard spheres. In summary,
for hard particles, using equation (8) and choosing v
b
so that ½
b
is the density at close packing and ²
yields the correct second virial coefficient, exact results are obtained in one dimension and fair results
are obtained in three dimensions.
3. Extension
A reasonable extension of equation (7) is
lnq
n
Æ ln
³
1¡
nv
d
V
´
¡
1
X
iÆ1
²
i
µ
nv
d
V ¡nv
d
¶
i
, (11)
which, neglecting the terms in the sum for i È 4 leads to the following extension of equation (8)
p
½kT
Æ¡
1
½v
d
ln(1¡½v
d
)Å
4
X
iÆ1
a
i
²
i
, (12)
43001-2
Nonmonotonous pressure as a function of the density in a fluid without attractive forces
where
a
1
Æ
1
½v
d
ln(1¡½v
d
)Å
1
1¡½v
d
, (13)
a
2
Æ¡
2
½v
d
ln(1¡½v
d
)¡3
1
1¡½v
d
Å
1
(1¡½v
d
)
2
, (14)
a
3
Æ
3
½v
d
ln(1¡½v
d
)Å
11
2(1¡½v
d
)
¡
7
2(1¡½v
d
)
2
Å
1
(1¡½v
d
)
3
, (15)
and
a
4
Æ¡
4
½v
d
ln(1¡½v
d
)¡
25
3(1¡½v
d
)
Å
23
3(1¡½v
d
)
2
¡
13
3(1¡½v
d
)
3
Å
1
(1¡½v
d
)
4
. (16)
Keeping only ²
1
yields Rennert’s result. Each of the ²
i
can be chosen to give B
iÅ1
, which are known to
high order [2]. The result for B
i
that follows from the above result has a pleasing form. It is
B
i
/v
i¡1
b
Æ
1
i
Ų
1
i ¡1
i
Ų
2
(i ¡1)(i ¡2)
i
Ų
3
(i ¡1)(i ¡2)(i ¡3)
2i
Ų
4
(i ¡1)(i ¡2)(i ¡3)(i ¡4)
6i
Å¢¢ ¢ . (17)
The sum is terminated when a negative term is encountered. If one wishes to add an additional term
in equation (12), the previously determined values of ²
i
remain unchanged. In one dimension, ²
1
Æ 1
and the additional ²
i
Æ 0. In three dimensions, ²
1
Æ 4.92384391, ²
2
Æ 2.80081908, ²
3
Æ¡0.918729979 and
²
4
Æ¡0.218154034.
Figure 1. (Color online) Pressure of hard spheres as a function of density. The points give the simulation
results taken from reference 2; the circles and squares give the simulation results for the fluid and solid
branches, respectively. The solid curve gives the results obtained from equation (8). The dashed and
dot-dashed curves give the results obtained from equation (12) with ²
i
included for i É 3 and i É 4,
respectively.
Results for p/½kT as a function of ½ for hard spheres (three dimensions) are given in figure 1 when
the series is terminated after ²
1
, ²
3
, and ²
4
. The results for the case where only ²
1
is included because
this is Rennert expression. In this case, the pressure is monotonous. Results for the cases when the se-
ries terminates after ²
2
are not included because the pressure is monotonous. The displayed curves are
43001-3
D. Henderson
compared with the results of computer simulations taken from [2]. The simulation curves consist of two
branches, one for the fluid branch and one for the solid branch, which terminates at close packing. The
agreement is quite reasonable. However, the most intriguing feature is not the numerical accuracy of this
procedure that, for the fluid branch, is not quite as good as that of the ad hoc Carnahan-Starling [3] ex-
pression but the fact that with terms through ²
3
or ²
4
included in equation (12), there is amaximum in the
pressure at approximately the location of the transition from the fluid to the solid phases. Better results
are given with ²
4
included but neglecting the contribution of ²
4
does a better job of locating the transi-
tion. It is reasonable to regard this maximum as indicative of the freezing of the hard sphere fluid, since
a system with a negative compressiblity would expand with an increase of pressure, which is unphysical.
The Carnahan-Starling expression does not give any indication of the presence of this transistion and, in
fact, is unphysical at very high densities since it continues to give results for densities past close packing.
To be sure, the procedure reported here is ad hoc. There is no guarantee that including more ²
i
will con-
tinue to give better results or even a maximum pressure. In fact, this is the situation in two dimensions
where neglecting ²
4
yields a maximumwhile including ²
4
does not. The point of this article is to point out
that this procedure does give an indication of a phase transition in a hard sphere fluid. To the author’s
knowledge, beyond the more complex Born-Green-Yvon [4–6] and Kirkwood [7] approximations, this is
the only theory to do so. This paper might probably point the way to a simple non-empirical description
of the hard sphere transition and be useful in this regard.
Acknowledgements
It is a pleasure to enthusiastically acknowledge the lifetime achievements by Myroslav Holovko and
my friendship with him.
References
1. Rennert P., Z. Naturforsch, 1967, 22, 981.
2. Barker J.A., Henderson D., Rev. Mod. Phys., 1976, 48, 587; doi:10.1103/RevModPhys.48.587.
3. Carnahan N.F., Starling K.E., J. Chem. Phys., 1969, 51, 635; doi:10.1063/1.1672048.
4. Born M., Green H.S., P. R. Soc. London A-Conta., 1946, 188, 10; doi:10.1098/rspa.1946.0093
5. Born M., Green H.S., P. R. Soc. London A-Conta., 1947, 191, 168; doi:10.1098/rspa.1947.0108.
6. Yvon J., Actualitiés Scientifiques et Industrielles, 1935, 203.
7. Kirkwood J.G., Maun E.K., Alder B.J., J. Chem. Phys., 1950, 18, 1040; doi:10.1063/1.1747854.
Немонотонний тиск як функцiя густини у плинi без
притягальних сил
Д. Гендерсон
Вiддiл хiмiї та бiохiмiї, Унiверситет Брайхем Янг, Прово, США
Простий результат для тиску плину твердих сфер, який був отриманий багато рокiв тому Реннертом, роз-
ширено в простий спосiб шляхом додавання членiв, якi мають такий же вигляд як формула Реннерта.
Результуючий вираз є посередньо точним, але його точнiсть не обов’язково покращиться, якщо вклю-
чити додатковi члени. Цiкавим наслiдком отриманого виразу є те, що тиск може мати максимум, коли
густина зростає, що узгоджується iз твердненням твердих сфер. Це вiдбувається виключно як наслiдок
короткодiйних взаємодiй. Лише теорiї Борна-Грiна-Iвона i Кiрквуда показують таку поведiнку для твердих
сфер i вони потребують числового розв’язку iнтегрального рiвняння. Процедура, окреслена тут є ad hoc,
але можливо є корисною такою ж мiрою, як i популярне рiвняння Карнагана-Старлiнга для тиску твердих
сфер, яке є також ad hoc, але корисним.
Ключовi слова: статистична сума, рiвняння стану, тиск, плин твердих сфер, перехiд тверднення
43001-4
http://dx.doi.org/10.1103/RevModPhys.48.587
http://dx.doi.org/10.1063/1.1672048
http://dx.doi.org/10.1098/rspa.1946.0093
http://dx.doi.org/10.1098/rspa.1947.0108
http://dx.doi.org/10.1063/1.1747854
Introduction
Rennert's method
Extension
|