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Cell-cluster theory of the liquid state IV
Salsburg, Z . W . Cohen, E. G . D . Rethmeier, B. C. 1957
Phys~ca X X l I I 407--422
C E L L - C L U S T E R T H E O R Y O F T H E L I Q U I D S T A T E IV A F L U I D OF H A R D S P H E R E S
by Z. W. SALSBURG*), E. G. D. COHEN, B. C. RETHMEIER and J. DE BOER (Instituut voor theoretische Physica, Universiteit van Amsterdam, Nederland) Synopsis • The cell-cluster theory of the liquid state, developed in two previous papers 1) 2), is applied here to a fluid of hard elastic spheres. The corrections introduced by considering cell-clusters of two cells are investigated. The double cell partition function is calculated in an approximate way and the influence on the entropy and the equation of state is studied. The results of the cell-cluster theory for the equation of state are compared with the recent Monte-Carlo calculations for a hard sphere fluid by R o s e n b l u t h and R o s e n b l u t h s).
§ 1. Introduction. Aside from the intrinsic theoretical interest, involved in the study of a fluid composed of hard elastic spheres, this hypothetical model is often considered to be a good starting point for the study of the thermodynamic properties of the liquid state. In particular also the relative ease of the calculations makes it an appropriate test case for proposed improvements of the theory of liquids. For that reason, we will apply the cell-cluster theory to a system of hard spheres in this publication. However, in discussing a fluid of elastic spheres, one is hampered by the fact that a fluid of elastic spheres does not actually exist, which makes it impossible to obtain direct experimental data for comparison. Theories could therefore only be compared amongst each other without knowing the true behaviour of a system of hard spheres at higher densities. This drawback of the present model is somewhat overcome by the recent Monte Carlo calculations, made by R o s e n b l u t h and R o s e n b l u t h 1), on the Maniac in Los Alamos. The results of these calculations which represent a direct numerical evaluation of the desired canonical average for a system of a limited number of molecules, takes the place in a certain sense of the ,experimental" data of a fluid of elastic spheres. These data are therefore the most suited for comparison with theoretical calculations made for a fluid of elastic spheres. *) Present address: Rice Institute, Texas, U.S.A.
-- 407 --
408
Z. W. SALSBURG, E. G. D. COHEN, B. C. R E T H M E I E R AND J. DE BOER
§ 2. The cell-cluster theory. The cell-cluster theory of the liquid state developed in two previous publications (12) and 118) of this series) is an extension of the cell theory of the liquid state proposed originally b y Lennard Jones. In Lennard-Jones' theory each molecule is supposed .to move, independent of the motions of the other molecules, in a cell, of volume V / N and in a potential field due to the neighbouring molecules. For the purpose of calculating the potential field in such a cell, all the other molecules are placed in the centers of their cells, which are taken to be the sites of a close packed cubic structure. In the cell-cluster theory a systematic extension of this theory is given, in which not only the motion of a single molecule in a single cell, but also the motion of two or more molecules in cell-clusters of two or more neighbouring cells is taken into account in a systematic way. The configurational partition function of l molecules in a cell-cluster of l neighbouring cells is written as: Q~..~ -- ( l ! ) - z f . . . f W ~ . . . ~ ( r l ... rz) d r l ... drz
(1)
Here W~...~(rl ... rz) is the configurational probability density for finding the molecules 1 ... l in a definite configuration rl ... rz in the cell-cluster ... ;t. They move in a potential field composed of two parts: (1) the mutual interaction of the l molecules and (2) the interaction of these l molecules with the remaining N - - l molecules, all being placed in the centers of their cells. This probability density is normalized b y making it equal to one for a configuration in which the l molecules inside the cell-cluster are also placed in the l cell centers. In terms of these configurational integrals Q~, Q~#, etc. of 1, 2 etc. molecules one now defines functions of the type: t
t
t
Vl = v~ = Q~ t
(2a) t
t
... etc .... as explained in detail in I and II'. The partition function m a y now be expressed in terms of these vz functions (comp. I and II) as:
ZN =
/(Z.J.)
,v
' [Z{m.}
IIz,k
(3)
where we have written wl~ ---- vzk/v11. Here the summation is carried out over all possible divisions of the system of N cells in cell-clusters. If mzk is the number of cell-clusters of l cells and type k in a particular division, g({m~k}) is the number of different divisions, which are geometrically possible corresponding to a given set {m~k}. As Z(~ "J') is the partition function, one obtains with Lennard-Jones' theory, based on. single cells only, the factor between [ ] represents the correction factor, which results from the cellcluster theory. The combinatorial problem of calculating g({mz~}) could be solved ex-
409
A F L U I D OF H A R D S P H E R E S
actly only in the one-dimensional case (See II % b u t for the case of more dimensions an approximate expression has been given, based on "the assumption, t h a t as a first approximation one could only consider single cells and cell-clusters of two neighbouring cells. U n d e r this assumption the following result was obtained with the m e t h o d of the m a x i m u m term: ZN = Z ~ "J'l [(1 -- ~)z/2/(1 -- z~)] 2v (4) where k, the fraction of the m a x i m u m possible n u m b e r of double ceils (equal to {zN), which contribute to the m a x i m u m term in the partition function, is given b y : = (2zw2 + 1 -- ~/4(z -- 1)92 + l)/2(z2w2 + 1) (5) F r o m this partition function one obtains for the Helmholtz' free energy, the pressure and the e n t r o p y the following expressions:
F
= F (L'J') - - N k T
i n (1
" x),/
-
2
(6)
z{l + ¢z ( 09, p = p~L.J,I+ kT 2(1 -- a$)(1 -- z$) dw2 \---~--v/r da~ ( a w z ~ (1--~)z/2 z{1 "F ( z - - 2 ) ~ } S = S IL'J') + N k T 2(1 -- k)(1 -- z$) dwz \-~-V/T + N k in 1 -- z$
i7) (8)
which will form the basis of the calculations of the next section.
§ 3. Equation o/state o / a fluid o/hard elastic spheres. According to the principles of the cell-cluster t h e o r y given in the previous section the corrections on the free volume, or single cell t h e o r y are determined b y the q u a n t i t y : l
- -
l
l
92 = O•B Q QbQ=Qfl
t
t 2
1 = Q=fl/Q= -
1
(9)
from which, according to the equations (5)-(8) the corrections on the free energy, the pressure and the e n t r o p y can be evaluated in first approximation. For the special case of hard elastic spheres one m a y write for exp--flg(r12), containing the molecular interaction 9(r12) between the molecules 1 and 2, the unit step function: e(rl2 -- a) = 0 for r12 < a exp -- flg(r12) = ~(r12 -- a); (I0) #(r12 -- a) = 1 forrl2 >_ a The two configurational integrals which occur in (9) can then be written as: Q'= = f { I I r e(rl r - a)} dr1 = f * ~ e *eU d r l Oefl = ~.t
(1 la)
YI~ {~(rlr -- a) e ( r ~ -- a)} e(rl2 - a) d r l dr2 = 1 I'[ -~, d d double cell
~ ' ( r 1 2 - a)drl des
(llb)
410 z. w. SALSBURG, E. G. D. COHEN, B. C. RETHMEIER AND J. DE BOER I n the integral (! l a) t h e p r o d u c t is carried out o v e r all cells w h i c h are n e i g h b o u r i n g t h e cell ~ t h a t is considered a n d rx r is t h e distance b e t w e e n t h e c e n t r e of t h e molecule 1, s o m e w h e r e in cell a a n d a n o t h e r molecule p l a c e d in the centre of cell 7. I n (1 l a) one should t h u s s i m p l y i n t e g r a t e o v e r t h e v o l u m e left for t h e molecule 1 b e t w e e n t h e collision spheres (with radius a) o f the z s u r r o u n d i n g molecules, these being placed in the centers of t h e i r
Fig. 1. Cross-sections of the exact and approximate single- and double cell for a simple cubic lattice of hard spheres. The upper figure is a. cross-section along a (001) plane. The lower figure is a cross-section parallel to a (100) plane. The two sections cut each other along the line PQ. a is the lattice constant, a is the diameter of a sphere; dx = 2 ( a - - a),ds = 3 a - -
2a.
A FLUID OF HARD SPHERES
".
411
cells. Similarly in (1 I) the product is carried out over the cells y neighbouring the cell=cluster ~fl and thus the integration over the centers of 1 and 2 should be car~ed out over the total volume left between the collision spheres of t h e s u r r o u n d i n g molecules placed in the centers of the cells neighbouring the cell-cluster ~/5. Calculations have been carried out for a simple cubic as well as for a face centered cubic structure chosen as a basis for the cell configurations. The molecular volume v is related to the distance a between neighbouring cell centers b y the expression a 3 = y V / N = yv, where y = 1 for the simple cubic and y = ~/2 for the face centered cubic structure. The relevant quantities are, of course, the reduced quantities: v* = v/a 8 and a* = a/a, related b y the same relation: a .8 = yv*. It is of advantage to use in the present calculation as an auxiliary variable the quantity y = a* -- 1. As the highest density is reached for a value a* = 1 (giving v* = y-1 and y = 0), y is a measure for the amount of freedom which is left for the molecules. The relation between v* and y is: v*a = y -1 (y -}- 1)a
(12)
In this publication we shall make an approximate evaluation of the quantity w~, which characterizes the deviation from the free volume theory, b y replacing the actual space over which the integrations in the partition functions Qa~ and ~)~ should be carried out b y a rectangular parallelopiped and the corresponding cube respectively. (a) The siml~le cubic lattice: (z = 6, y = 1) In figure 1 the geometry of the exact single- and double cell and the approximation used, is illustrate'd for the case of the plane square lattice. In the approximation mentioned above the single cell integral (10) becomes simply I
I
=
8(a
-
=
8y3.3
(13)
and the double cell integral ( I I) can be written as:
'
2! J0 dZl
dyl
fo fo ; dxl
dz~ 0 dy2
fo
dx2 8(r12 -- a)
(14)
where dl=2(a--a) d3=3a--2a=
=2ya
dl
(is)
or d * - - - - - - = 2 y G
d8
( 3 y + 1)a or d ~ ' = - - = 3 y + G
I
(16)
The result of the integration of the sixfold integral (14) can be obtained from the general expression for Q ~ as given in the appendix b y equating dl = dg. and substituting the values of dl = d2 and d3 given in (15) and (16). t
412
Z. W. SALSBURG, E. G. D. COHEN, B, C. R E T H M E I E R AND J . DE BOER
Using the definition of w2 given b y (9) and the relations (15) a n d (16), the value of w2 for the simple cubic lattice can be evaluated as a function of y. !
w,
;
i
i
i
i
i
i
i"
'l~g.
0.7 o.6
0.5 0.4 0.3 0.2 0.1
01.0 111 1~2 ~ 11~ 1~5. 1.16 1:1 1JB 1'3 Fig. 2. w~ =
2.0
Q'o/Q~" - - 1 f o r a s i m p l e c u b i c l a t t i c e (s.c.) a n d a f a c e c e n t e r e d c u b i c l a t t i c e (f.c.c.) as a f u n c t i o n of v*l13 = (V/NaS)ll3.
In table I are given v* = (y + 1) 8, w~*) and dw~8)/dy as a function of y. A plot of w~") as a function of y is shown in fig. 2. I t is clear t h a t for small values of y = a* -- 1, when the available volume for the two molecules TABLE I Values associated with the double cell correction in a simple cubic lattice (s). y
v*
w,(s)
du,'s(8)/dy
0 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33
I
0.125
1.405
O. 187
1.521 1.643 1.772 1.907 2.048 2.197 2,353 2.482 2.744 3.049 3.375 4.096 4.913 5.832 6.859 8.000
0.204 0.220 0.237 0.255 0.274 0.294 0.315 0.333 0.372 0.417 0.485 0.604 0.673 0.700 0.702 0.690
0.5 0.518 0.539 0.569 0.590 0.612 0.642 0.678 O.726 0.773 0.932 1.170 1.297
*12V2 0.40 0.45 0.50 0.60 0.70 0.80 0.90, 1.00
0.962 0.451 0.119 --0.068 --0.166
A'cs=ZJpj, V/RT 0.229 0.214 0.218 0.225 0.229 0.232 0.239 0.247 0.258 0.270 0.323 0.375 0.386 0.259 0.119 0.032 --0.019 --0.051
AS,/R 0.244 0.320 0.338 0.354 0.372 0.389 0.405 0.422 0.440 0.454 0.483 0.514 0.556 0.622 0.655 0.668 0.669 0.663
to move in is small, the q u a n t i t y w2 is also small, as m a y be expected. The limiting value of w¢~8) for y = 0, is the same as t h a t for a one dimensional system of h a r d spheres and equals 1/8. I t should be noted t h a t for y = 0.5 when the molecules can just slip past each other in the double cell a point
A FLUID OF HARD SPHERES
413
of inflexion in the w~81 curve occurs. For values of y near 1, w2 shows a m a x i m u m of approximately 0.7. This is a value which is somewhat too large to allow, strictly speaking, the application of the equations (4)-(8). Thus in this region of w2-values poorer results might be expected. Values of y > 1, for which the cells are no longer closed by the collision spheres of the neighbouring molecules are not considered.
i I I I I
I
.....
I i
I
Fig. 3. Cross-sections of the exact and the a p p r o x i m a t e singlea face c e n t e r e d c u b i c l a t t i c e of hard speres. T h e upper figure is a (001) plane, t h e l o w e r figure a l o n g a (110) plane. T h e t w o sections t h e .......... line. a is t h e lattice c o n s t a n t , o the d i a m e t e r dl = { a ~ / 6 - - -
a n d double cell for cross-section along a c u t each other along of a sphere;
2a, d2 ---- a x / 3 - - - 20, ds --~ 3a -- 2e.
b) T h e / a c e centered cubic lattice" (z = 12, 7 = V 2 ) . In figure 3 we have
414
Z. W. S A L S B U R G , E. G. D. C O H E N , B. C. R E T H M E I E R A N D J. D E B O E R
illustrated how the inscribed parallelopipeda for the single cell and for the double cell were chosen. This leads to a single cell integral: !
(17)
Q~ = dodld2
and to a double cell integral:
' ' J;'
Q0C]~= ~.[
dz1
dyl
Jo" f;' fo" fo" dxl
dzz
dy2
dxz £ (r12 -- ,)
(18)
where: do =
2a -- 2~ =
dl
§aV'6-
=
dz =
2y0.
20" =
a 3 , / 3 " - - 20. =
(I 9a) (1.633y
(1.732y
(19b)
-- 0.367)0. -- 0.268)~
(19c)
d = 3a - - 20. ---- ( 3 y -'F" 1) 0.
(19d)
The details of the integration of (18) are given in the Appendix. The results come down to the following. The calculations were continued up to y = 1, although the maximum value of y for which the collision spheres of the nearest neighbours still touch is y = %/2-- 1 = 0.414; but due to the presence of the next nearest neighbours the cell can still be considered to be "closed" till y = 1, when the collision spheres of these next nearest neighbours just touch. The condition, that the dimensions of the single- and double cell, in our parallelopipedum 'approximation, are non-negative leads to a second condition for y, because of the fact that dl > 0 only for y ~ 0,224. Then d2 and d3 are also non-negative because dl < d2 < d3. T A B E L II Values associated with the double cell correction in a face centered cubic lattice (/).
y 0.225 0.34 0.38 0.42 0.46 0.50 0,54 0.58 0.62
0.66 0.70 0.74, 0.78 0.82 0.86 0.90 0.94 0.98 1.00
]
] 1.300 1.702 1.856 2.024 2.202 2.385 2.583 2.79! 3.005 3.235 3.477 3.724 3.989 4.259 4.550 4.853 5.160 5.489 5.659
.,(I} 0.129 0.151 0.162 O. 174 O. 187 0.200 0.215 0.231 0.249 0.269 0.292 0.318 0.346 0.375 0.402 0.427 0.449 0.468 0.476
]
dws(1)ldy 0.278 0.310 0.334 0.359 0.388 0.423 0.469 0.535 0.614 0.682 0.712
0.699 0.656 0.586
0.502 0.418
I zI'cI='dPlVIRT] "4sz/R
0.218 0.229 0.240 0.254 0.269 0.289 0.320 0.356 0.384 0.384 0.359 0.328 0.285 ' 0.239 0.196
0.424 0.441 0.459 0.478 0.497 0.516 0.536 0.557 0.579 0.603" 0.629
0.656 0.681 O.703 0.723 0.739 0.752
:0.758
i
A F L U I D OF .HARD S P H E R E S
.
415
T h u s we have only to consider the following range of values of y:
0,2247 ~ y ~ 1
(20)
Using (9) and(19), w2 for the face centered cubic lattice, indicated as w~1), can be calculated for all y-values in the range (20). In table II are given v* = (y + 1)8V'2, w~11 and dw~ll/dy as a function of y. A plot of w~11in fig. 2 shows that the behaviour of w~n for 0.3 < y < 1.0 is very similar to that of w~sl between 0.1 < y < 0.8. The hmiting value of w,2(n for y = 0.2247 is 0.129. This can be calculated using the formulae for a two-dimensional lattice 5). In the neighbourhood of y = x/3- -- -I = 0.732, where t h e two molecules in the double cell can just slip past one another, the w~fl .curve shows a point of inflexion. With the values given in table i and table II the entropy correction AS = ( S - sIr"J'))/R and the correction on the .equation of state. A~ = ( p - - pIL'J'I)V/RT due to double cells can be calculated both for the simple cubic (indicated b y a subscript s) and for the face centered cubic !attice (indicated b y a subscript/). They are given in the last two columns of table I and II. c). The total pressure. To get the total .pressure p we still have to add the single cell contribution p(S.J.I. This we did not calculate from the Q~ as given b y (13) or b y (17), which were only used to evaluate the corrections on the pressure, due to the cell-cluster treatment. For the zeroth approximation pILJ.I i.e. the single cell theory, one should t
TABLE Ilia
T A B L E IIIb
Values of Ks = pVIRT for a simple cubic lattice according to the single cell theory and to the ceU-cluster theory in first approximation.
Values of K! = pV/RT for a face centered cubic lattice according to the single cell theory and the cell-cluster theory in first approximation. v*l/s K$(L.J.) K!
Y 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33
112v'2 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00
I
v'1/3 1.09 1.12 1.15 1.18 1.21 1.24 1.27 1.30 1.33 1.35 1.40 1.45 1.50 1.60 1.70 1.80 1.90 2.00
[ K/L'J') 12,111 9.333 7.667 6.556 5.762 5.167 4.704 4.333 4.030 3.828 3.500 3.222 3.000 2.667
2.429 2.250 2.111 2.000
I
K, I 1.072 9.547 7.884 6.781 5.991 5.399 4.942 4.580 4.288 4.098 3.823 3.597 3.386 2.926 2.548 2,282 2.092 1.949
0.42 0.46 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.82 0.86 0.90 0.94 0.98
1.265 1.301 1.336 !.372 1.408 1.443 1.479 1.515 1.550 1.586 1.621 1.657 1.693 1.728 1.764
3.38 I 3.174 3.000 2.852 2.724 2.613 2.515 2.429 2.351 2.282 2.220 2.163 2. l I l 2.064 2.020
3.599 3.403 3.240 3.106 2.993 2.902 2.835 2.785 2.735 2.666 2.579 2.491 2.396 2.303 2.216
416
z. w . SALSBURG, E. G. D. COHEN, B. C. RETHMEIER AND J. DE BOER
use of course the best possible single cell approximation.We used the sphericalized single cell configurational partition function: 4~r 4st • {)~ = ~ ( a - - a ) 3 = 3 y3a3 (21) which leads to the following expressions for p(•'J')V/RT: p(pJ.
v _
_
RT
y +
RT
1
(22)
y
(22) is identical with the expression that would have been derived using (13), because all Q~ of the form Q= ~ y a will lead to the same pressure. Numerical values for K(sL'J')-~ p~L'J')V/RT and for K~r"J')= p~L"r')V/RT are given in table I I I a and IIIb respectively, together with values for K, = i
= paV/RT and Kf = plV/RT. § 4. Discussion o/the results. In a system of hard spheres the effect on the entropy of taking into account cell-clusters of two cells is given by: S = S (L'dl) + R(in (1 -- a~)z/9')/(1 -- z~)
(23)
where ~ is given in terms of w9 by equation (5). The second term is just equal to the extra factor which occurs in the partition function because of the taking into account of cell-clusters of two cells, as m a y be seen from equ. (4). The increase in entropy due to the double cells is plotted in figure 4 for 0.8
i
.
i
i
i
!
w
!
i
0.6 Z-1
.
0.~
0.2
°1.0
'
1'.2
'
1;4
'
118
'
!
t
Fig. 4. ,Entropy correction due to the introduction of double cells for~the simple cubic lattice (s.c.) and the face centered cubic lattice (f.c.c.). the simple cubic lattice and for the face centered cubic lattice respectively as a function of the reduced volume. Physically this increase in entropy is caused by the increase in the number of possible configurations for the system which is determined by the values of w~. There are no data available
A FLUID OF HARD SPHERES
417
for the entropy from the Monte Carlo results to which reference was made in the introduction, these being related only to the equation of state.'Still it is of interest that the taking into account of the correlated motion of two molecules in a double cell gives a considerable increase in the entropy. pv
T 0.8
0.6
0.2
1.0
12~
1£
1.6
-0.;
-0.~
Fig. 5. E q u a t i o n of s t a t e c o r r e c t i o n s o n t h e free v o l u m e t h e o r y d u e t o d o u b l e cells for t h e s i m p l e c u b i c l a t t i c e (s.c.) a n d t h e face c e n t e r e d c u b i c l a t t i c e (f.c.c.) as a f u n c t i o n of v'x/3. T h e c u r v e M.C. was d e r i v e d f r o m t h e M o n t e Carlo c a l c u l a t i o n s b y s u b t r a c t i n g t h e free v o l u m e v a l u e s for a face c e n t e r e d c u b i c lattice.
The influence of the introduction of double cells on the equation o/state is given in figure 5, where we have plotted the corrections on the free volume ~(L.J.) ) V / R T and A KI= (pf_p~L.J.I) V / R T for a simple cubic theory A Ks= 14~_ ~r~--r8 and a face centered cubic lattice respectively. The maximum of both curves corresponds to the inflexion point in the w2 curve, where the double cell is large enough that the two molecules can just pass each other. We have also plotted in this figure the corrections on the free volume theory that can be derived from the Monte Carlo calcuiations, b y subtracting from the KMV = pMoV/RT, as given b y R o s e n b l u t h and R o s e n b l u t h 2) the K~1"'J'~ given in table IIIb. It is seen that the qualitative behaviour of the two curves is very similar and that the order of magnitude of the corrections Physica X X l I I
27
4 1 8 Z. W. SALSBURG, E. G. D. COHEN, B. C. RETHMEIER AND J. DE BOER
are roughly the same, though the two curves are shifted with respect to each other in the direction of the positive v*l/3--axis. This, however, might be due to our restriction in considering only corrections due to double cells. For, the positive correction to the free volume pressure can be Associated with the tendency of the molecules to slip past one another. The rigid structure of our double cell permits this to occur at a much lower density than would be expected in an exact treatment. PV
!
T
i
b"
"'"- ~ ...." ~.. ~..~:..
"-. "--..~ s.clc.ct.)
" 'x..,
I f.c.c.
0
i|
0.9
~S.C.
1.0
" ~ ~ ' ~ ' ~ ' ~ ' ~ "
I
1.1
i
1.2
I
1.3
I
1.4
I
1.5
I
1.6
I
1.7
I
1.8
I
1.9
t
2.0
.__.~ V ~ Fig. 6. Calculation of p V/RT as a function of v*Zla according to various theories compared w i t h the Monte Carlo results: K • from t h e K i r k w o o d integral e q u a t i o n 4) ; BG : from the Born and Green integral e q u a t i o n 4) ; MC : from the Monte Carlo calculations 1) ; s.c. : cell-cluster t h e o r y based on a simple cubic lattice; f.c.c. : cell-cluster t h e o r y based on a face centered cubic lattice.
Finally in figure 6 we have plotted the results of various calculations for = p V / R T as a function of the reduced volume. The calculations based on a simple cubic lattice are seen to be in very poor
agreement with the Monte Carlo results. This means t h a t a simple cubic lattice is not a good lattice structure to serve as a basis for a quasi-crystalline theory of a liquid of hard spheres. It should be noted the discrepancy is mainly due to the single cell approximation which differs widely from the Monte Carlo curve as is illustrated by the fact that it approaches to the asymptote v .118 = 1, which line is cut by the Monte Carlo curve.
A FLUID OF HARD SPHERES
419
The agreement of the Monte Carlo results with the calculations based upon a /ace centered cubic lattice is much better. The corrections, though in absolute value of the right order of magnitude, have the wrong sign. Because of various approximations made, the Kycurve could not be continued below v*~/~ = 1.265 that is below y = 0.42. We have also plotted in figure 6 the results obtained by K i r k w o o d , M a u n and A l d e r 4) for x = p V / R T from a numerical integration of the Born and Green (KBa) and Kirkwood (KK) integral equations for the radial distribution function. They agree very well with Monte Carlo results at low densities, b u t show considerable deviations at high densities. A first step to improve our calculations would be to calculate w~, using the true form and size of the single- and double cells. This will be done in a forthcoming publication 5) for the two-dimensional trigonal lattice. Another important improvement of the theory would be the introduction of vacant sites into the basic lattice. This cell-cluster theory with holes would start with a number of lattice sites larger than the number of molecules. Then in every configuration there are a number of vacant lattice sites and a number of cell-clusters containing less molecules than cells. Such a theory would not only be an extension of the cell-cluster theory as developed in I and II, but also of the existing hole theories, which allow only for occupied and unoccupied single cells 6). Also this we reserve for a later publication 7).
¢
A p p e n d i x . Calculation o/ Q~# i n a lace-centered cubic lattice. We consider
the evaluation of the sixfold integral
:2.1l
f ~;1Fa* dZlJo
d y l J 0t'a3 dxl
f o;~dZ2JoFa*dy2Jot'a~d x 2 ~ ' ( r 1 2 - - a )
(A l)
where r~, = (xz - - Xl) ~ + (y2 -- Yl) z + (z2 -- Zl) 2.
We introduce a set of relative coordinates X, Y, Z:
X = x2 -- Xl Y = Yz - - Yl
(A 2)
Z = 22 -- Zl and if we write x, y, z for Xl, Yl, Zl respectively, then (A 1) can be written as: ' = 1o~jo ra~ jr_a~~ - ~ L ( Z ) d Z dz Q~t~
(A 3a)
L ( Z ) -ra~rd.,-y K ( Z , Y ) d Y dy -.10 J - - y
(A 3b)
where and
K(Z,
Y) = f # f _ ~ - x
e(R -- ,,) dX dx
t (A 3c)
420
z. w . SALSBURG, E. G. D. COHEN, B. C. R E T H M E I E R A N D J . DE B O E R
with R 2 = X ~.+ y 2 + Z L In each of the above double integrals we interchange the order of integration and this leads to the expression: Q'a~ = 4 f g l foa' foa3 N ( X , Y , Z)
e(R -
~) d X d Y d Z
(A 4)
with N ( X , Y , Z) = (d3 -- X)(d2 -- Y ) ( d l -- Z). Between the three lengths dl, d2 and ds hold the following inequalities: dl < d2 < d~ The integral (A 4) can best be considered in four separate cases. Case I" a K dl, or 0.8371 < y K 1. In this case the integral can be computed in the following manner: Q ~ I = 4[ f o~lf o~ f od3N d X d Y d Z -- f f f R~. ~st ootant N d X d Y dZ] = 222
=½ dld~d3 -- 4 f f f R ~ . 1st octant N d X d Y d Z
(A5)
This is illustrated in figure 7a. The integration can easily be carried out using spherical polar coordinates leading to:
Q~
a6 rqn,~*2a*~a*2
a*~*a*
+ 1 5 ~ (dl*d~ * + d 1*d*3 +
d~*d3) * -- 16(d* + d* + d*) + 5].
(A 6)
where d* = di/a(i = 1, 2, 3). Case II: dl _< a _< dz or 0.7321 _< y ~ 0.8371. The integral can be obtained from the calculation for case I if we correct for the spherical cap, as is illustrated b y figure 7b. Thus t
= '~0 + 4fffoao i N dX dY dZ = ,q,lli + ~ .amc,m dgf00°d0sin 0 fdlcosO * NR 2 dR =w*t~
(A 7)
after the introduction of spherical polar coordinates. 0o is given b y the equation: 0o = arc cos d*. The computation of the integral in (A 7) is lengthy b u t straightforward and leads to" an Q'(II) * 3* - - H I + ~ = v tl'(i) ~ + - ~ [5Glz¢ d~d
30d~'(d~' + d'~)(d*
-- arc cos d*) + + 16 (d* + d~) (I + ¼d~'2) (1 -- d l*~,&]/. +
V1--~I'
(A 8)
with (71 = d~'4 -- 6d~'2 + 8d~' -- 3 H1 ---- d *° -- 5d1.4 + lSd~ 2 -- lSd~' + 5
(A 9a) (A 9b)
A FLUID OF HARD SPHERES
421
Case I I I : d2 ~ ~ ~ ~/d~ + d~ or 0.6063 ~ y £ 0.7321. The calculation is similar to case II. W e m u s t add an additional correction-term due to a second cap outside the parallelopiped, as illustrated b y figure 7c. T h u s one has: '(III) ~'(II) NdXdVdZ (A 10)
Z
d
d2
Y
V
X~ 3
(a)
(b)
z
~0 (c)
/~X3
(d)
Fig. 7. Regions of integration in the four different cases for which Q ~ for a face centered cubic lattice has to be evaluated:
(a) case (b)
,,
I" ~ ~ al II:dl~
a ~ d~
(c) case m : a~ ~ o ~ ~/d~ + d~ (d)
,,
IV: ~/dl~ + d ~ K
a K d8
422
A FLUID OF HARD SPHERES
The evaluation of the integral proceeds in exactly the same w a y as in case II. Thus we get: Q'(III) t~'(II) U6 ~f~
= ,¢~
+ -ff~ [5Ee~d* d* - - F e +
+ 30d*(d* + d~)(d'~ V 1 + 16(d~' + d~')(1 +
arccos d*) +
d.2-
¼d.2) (1
--
d ~ ) 310-
(A 11)
with E2 = d~ 4 -- 6d~'2 + 8d* -- 3 v 2 = a *° -
4 +
15d 2 - - 16d
(A 12a) + S
(A 12b)
Case IV: x/d~ + d~ ~ a ~ d8 or 0 ~ y ~ 0.6063. The integral that we want to compute is the integral of N over t h a t part of the parallelepiped that does not overlap the sphere R < a. This is illustrated in figure 7 d. The integral can be computed directly from:
Q'(IV) ap = 4f0a' dZ(dl -- Z ) f o ~ d Y (de -- V ) f ~ , a _ v , _ z ,
d X ( d 3 - - X ) (A 13)
The lengthy computation is straightforward and gives:
O'6 Q'(IV) a/~ ~___60 E _[A3 + BI(1 -- d*2) 1/2 + Be(1 -- d*2) ~/~ +
4' + C~ ( 1 - d .2 - - d'~2) '/2 + D1 arcsin +
(1 - -
d'e) ~/' -t-
De arcsin (I -- d*d~2)',/, + 30d~ (d~ arcsin d* +
+ d~' arcsin d~') +
d'd*
80
d r d~ d* arcsin (l -- d'2)112(1 -- d ~ ) '/~
]
(A 14)
Here are A3
B1 Be C3 D1 De
= 3 0 d *2 I d~*2 d 3*2 -- 16d~ + 5 d *2 1 d 2*2 (6 -- dl*e--d2*S~j (A15a) = d~(16 + 18d* -- 4d*') (A lSb) = d~(16 + 18d~'2 -- 4d~ 4) (h 15c) = d~ {4(d*' + d~ 4) -- 18 (d~ ~ + d~ e) -- 12d~ 2 d~ a -- 16} (A 15d) * * "4 = lOded3(d t - - 6d~ 2 - - 3) (A 15e) = lOd~d~(d~ 4 - - 6d .2 - - 3) (A 15/)
Received 19-12-56. RE FERENCES 1) 2) 3) 4) 5) 6) 7)
R o s e n b l u t h , M. N. and R o s e n b l u t h , A. W., J. chem. Phys. 28 (1954) 881. De Boer, J., Prom Conf. theor. Phys. Japan, 1953, 507; Physica 20 (1954) 655. Cohen, E. G. D., De Boer, J. and S a l s b u r g , Z.W., P&ysiea -°! (1955) 137. K i r k w o o d , J. G., Maun, E. K. and Alder, B. J., J. chem. Phys. 18 (1950) 1040. Cohen, E. G. D. and R e t h m e i e r , B. C., Physica (to be published). R o w l i n s o n , J. S. and C u r t i s s , C. F., J. chem. Phys. 18 (1951) 1519. Cohen, E. G. D. and D a h l e r , J. S., Physiea (to be published).
Phys~ca X X l I I 407--422
C E L L - C L U S T E R T H E O R Y O F T H E L I Q U I D S T A T E IV A F L U I D OF H A R D S P H E R E S
by Z. W. SALSBURG*), E. G. D. COHEN, B. C. RETHMEIER and J. DE BOER (Instituut voor theoretische Physica, Universiteit van Amsterdam, Nederland) Synopsis • The cell-cluster theory of the liquid state, developed in two previous papers 1) 2), is applied here to a fluid of hard elastic spheres. The corrections introduced by considering cell-clusters of two cells are investigated. The double cell partition function is calculated in an approximate way and the influence on the entropy and the equation of state is studied. The results of the cell-cluster theory for the equation of state are compared with the recent Monte-Carlo calculations for a hard sphere fluid by R o s e n b l u t h and R o s e n b l u t h s).
§ 1. Introduction. Aside from the intrinsic theoretical interest, involved in the study of a fluid composed of hard elastic spheres, this hypothetical model is often considered to be a good starting point for the study of the thermodynamic properties of the liquid state. In particular also the relative ease of the calculations makes it an appropriate test case for proposed improvements of the theory of liquids. For that reason, we will apply the cell-cluster theory to a system of hard spheres in this publication. However, in discussing a fluid of elastic spheres, one is hampered by the fact that a fluid of elastic spheres does not actually exist, which makes it impossible to obtain direct experimental data for comparison. Theories could therefore only be compared amongst each other without knowing the true behaviour of a system of hard spheres at higher densities. This drawback of the present model is somewhat overcome by the recent Monte Carlo calculations, made by R o s e n b l u t h and R o s e n b l u t h 1), on the Maniac in Los Alamos. The results of these calculations which represent a direct numerical evaluation of the desired canonical average for a system of a limited number of molecules, takes the place in a certain sense of the ,experimental" data of a fluid of elastic spheres. These data are therefore the most suited for comparison with theoretical calculations made for a fluid of elastic spheres. *) Present address: Rice Institute, Texas, U.S.A.
-- 407 --
408
Z. W. SALSBURG, E. G. D. COHEN, B. C. R E T H M E I E R AND J. DE BOER
§ 2. The cell-cluster theory. The cell-cluster theory of the liquid state developed in two previous publications (12) and 118) of this series) is an extension of the cell theory of the liquid state proposed originally b y Lennard Jones. In Lennard-Jones' theory each molecule is supposed .to move, independent of the motions of the other molecules, in a cell, of volume V / N and in a potential field due to the neighbouring molecules. For the purpose of calculating the potential field in such a cell, all the other molecules are placed in the centers of their cells, which are taken to be the sites of a close packed cubic structure. In the cell-cluster theory a systematic extension of this theory is given, in which not only the motion of a single molecule in a single cell, but also the motion of two or more molecules in cell-clusters of two or more neighbouring cells is taken into account in a systematic way. The configurational partition function of l molecules in a cell-cluster of l neighbouring cells is written as: Q~..~ -- ( l ! ) - z f . . . f W ~ . . . ~ ( r l ... rz) d r l ... drz
(1)
Here W~...~(rl ... rz) is the configurational probability density for finding the molecules 1 ... l in a definite configuration rl ... rz in the cell-cluster ... ;t. They move in a potential field composed of two parts: (1) the mutual interaction of the l molecules and (2) the interaction of these l molecules with the remaining N - - l molecules, all being placed in the centers of their cells. This probability density is normalized b y making it equal to one for a configuration in which the l molecules inside the cell-cluster are also placed in the l cell centers. In terms of these configurational integrals Q~, Q~#, etc. of 1, 2 etc. molecules one now defines functions of the type: t
t
t
Vl = v~ = Q~ t
(2a) t
t
... etc .... as explained in detail in I and II'. The partition function m a y now be expressed in terms of these vz functions (comp. I and II) as:
ZN =
/(Z.J.)
,v
' [Z{m.}
IIz,k
(3)
where we have written wl~ ---- vzk/v11. Here the summation is carried out over all possible divisions of the system of N cells in cell-clusters. If mzk is the number of cell-clusters of l cells and type k in a particular division, g({m~k}) is the number of different divisions, which are geometrically possible corresponding to a given set {m~k}. As Z(~ "J') is the partition function, one obtains with Lennard-Jones' theory, based on. single cells only, the factor between [ ] represents the correction factor, which results from the cellcluster theory. The combinatorial problem of calculating g({mz~}) could be solved ex-
409
A F L U I D OF H A R D S P H E R E S
actly only in the one-dimensional case (See II % b u t for the case of more dimensions an approximate expression has been given, based on "the assumption, t h a t as a first approximation one could only consider single cells and cell-clusters of two neighbouring cells. U n d e r this assumption the following result was obtained with the m e t h o d of the m a x i m u m term: ZN = Z ~ "J'l [(1 -- ~)z/2/(1 -- z~)] 2v (4) where k, the fraction of the m a x i m u m possible n u m b e r of double ceils (equal to {zN), which contribute to the m a x i m u m term in the partition function, is given b y : = (2zw2 + 1 -- ~/4(z -- 1)92 + l)/2(z2w2 + 1) (5) F r o m this partition function one obtains for the Helmholtz' free energy, the pressure and the e n t r o p y the following expressions:
F
= F (L'J') - - N k T
i n (1
" x),/
-
2
(6)
z{l + ¢z ( 09, p = p~L.J,I+ kT 2(1 -- a$)(1 -- z$) dw2 \---~--v/r da~ ( a w z ~ (1--~)z/2 z{1 "F ( z - - 2 ) ~ } S = S IL'J') + N k T 2(1 -- k)(1 -- z$) dwz \-~-V/T + N k in 1 -- z$
i7) (8)
which will form the basis of the calculations of the next section.
§ 3. Equation o/state o / a fluid o/hard elastic spheres. According to the principles of the cell-cluster t h e o r y given in the previous section the corrections on the free volume, or single cell t h e o r y are determined b y the q u a n t i t y : l
- -
l
l
92 = O•B Q QbQ=Qfl
t
t 2
1 = Q=fl/Q= -
1
(9)
from which, according to the equations (5)-(8) the corrections on the free energy, the pressure and the e n t r o p y can be evaluated in first approximation. For the special case of hard elastic spheres one m a y write for exp--flg(r12), containing the molecular interaction 9(r12) between the molecules 1 and 2, the unit step function: e(rl2 -- a) = 0 for r12 < a exp -- flg(r12) = ~(r12 -- a); (I0) #(r12 -- a) = 1 forrl2 >_ a The two configurational integrals which occur in (9) can then be written as: Q'= = f { I I r e(rl r - a)} dr1 = f * ~ e *eU d r l Oefl = ~.t
(1 la)
YI~ {~(rlr -- a) e ( r ~ -- a)} e(rl2 - a) d r l dr2 = 1 I'[ -~, d d double cell
~ ' ( r 1 2 - a)drl des
(llb)
410 z. w. SALSBURG, E. G. D. COHEN, B. C. RETHMEIER AND J. DE BOER I n the integral (! l a) t h e p r o d u c t is carried out o v e r all cells w h i c h are n e i g h b o u r i n g t h e cell ~ t h a t is considered a n d rx r is t h e distance b e t w e e n t h e c e n t r e of t h e molecule 1, s o m e w h e r e in cell a a n d a n o t h e r molecule p l a c e d in the centre of cell 7. I n (1 l a) one should t h u s s i m p l y i n t e g r a t e o v e r t h e v o l u m e left for t h e molecule 1 b e t w e e n t h e collision spheres (with radius a) o f the z s u r r o u n d i n g molecules, these being placed in the centers of t h e i r
Fig. 1. Cross-sections of the exact and approximate single- and double cell for a simple cubic lattice of hard spheres. The upper figure is a. cross-section along a (001) plane. The lower figure is a cross-section parallel to a (100) plane. The two sections cut each other along the line PQ. a is the lattice constant, a is the diameter of a sphere; dx = 2 ( a - - a),ds = 3 a - -
2a.
A FLUID OF HARD SPHERES
".
411
cells. Similarly in (1 I) the product is carried out over the cells y neighbouring the cell=cluster ~fl and thus the integration over the centers of 1 and 2 should be car~ed out over the total volume left between the collision spheres of t h e s u r r o u n d i n g molecules placed in the centers of the cells neighbouring the cell-cluster ~/5. Calculations have been carried out for a simple cubic as well as for a face centered cubic structure chosen as a basis for the cell configurations. The molecular volume v is related to the distance a between neighbouring cell centers b y the expression a 3 = y V / N = yv, where y = 1 for the simple cubic and y = ~/2 for the face centered cubic structure. The relevant quantities are, of course, the reduced quantities: v* = v/a 8 and a* = a/a, related b y the same relation: a .8 = yv*. It is of advantage to use in the present calculation as an auxiliary variable the quantity y = a* -- 1. As the highest density is reached for a value a* = 1 (giving v* = y-1 and y = 0), y is a measure for the amount of freedom which is left for the molecules. The relation between v* and y is: v*a = y -1 (y -}- 1)a
(12)
In this publication we shall make an approximate evaluation of the quantity w~, which characterizes the deviation from the free volume theory, b y replacing the actual space over which the integrations in the partition functions Qa~ and ~)~ should be carried out b y a rectangular parallelopiped and the corresponding cube respectively. (a) The siml~le cubic lattice: (z = 6, y = 1) In figure 1 the geometry of the exact single- and double cell and the approximation used, is illustrate'd for the case of the plane square lattice. In the approximation mentioned above the single cell integral (10) becomes simply I
I
=
8(a
-
=
8y3.3
(13)
and the double cell integral ( I I) can be written as:
'
2! J0 dZl
dyl
fo fo ; dxl
dz~ 0 dy2
fo
dx2 8(r12 -- a)
(14)
where dl=2(a--a) d3=3a--2a=
=2ya
dl
(is)
or d * - - - - - - = 2 y G
d8
( 3 y + 1)a or d ~ ' = - - = 3 y + G
I
(16)
The result of the integration of the sixfold integral (14) can be obtained from the general expression for Q ~ as given in the appendix b y equating dl = dg. and substituting the values of dl = d2 and d3 given in (15) and (16). t
412
Z. W. SALSBURG, E. G. D. COHEN, B, C. R E T H M E I E R AND J . DE BOER
Using the definition of w2 given b y (9) and the relations (15) a n d (16), the value of w2 for the simple cubic lattice can be evaluated as a function of y. !
w,
;
i
i
i
i
i
i
i"
'l~g.
0.7 o.6
0.5 0.4 0.3 0.2 0.1
01.0 111 1~2 ~ 11~ 1~5. 1.16 1:1 1JB 1'3 Fig. 2. w~ =
2.0
Q'o/Q~" - - 1 f o r a s i m p l e c u b i c l a t t i c e (s.c.) a n d a f a c e c e n t e r e d c u b i c l a t t i c e (f.c.c.) as a f u n c t i o n of v*l13 = (V/NaS)ll3.
In table I are given v* = (y + 1) 8, w~*) and dw~8)/dy as a function of y. A plot of w~") as a function of y is shown in fig. 2. I t is clear t h a t for small values of y = a* -- 1, when the available volume for the two molecules TABLE I Values associated with the double cell correction in a simple cubic lattice (s). y
v*
w,(s)
du,'s(8)/dy
0 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33
I
0.125
1.405
O. 187
1.521 1.643 1.772 1.907 2.048 2.197 2,353 2.482 2.744 3.049 3.375 4.096 4.913 5.832 6.859 8.000
0.204 0.220 0.237 0.255 0.274 0.294 0.315 0.333 0.372 0.417 0.485 0.604 0.673 0.700 0.702 0.690
0.5 0.518 0.539 0.569 0.590 0.612 0.642 0.678 O.726 0.773 0.932 1.170 1.297
*12V2 0.40 0.45 0.50 0.60 0.70 0.80 0.90, 1.00
0.962 0.451 0.119 --0.068 --0.166
A'cs=ZJpj, V/RT 0.229 0.214 0.218 0.225 0.229 0.232 0.239 0.247 0.258 0.270 0.323 0.375 0.386 0.259 0.119 0.032 --0.019 --0.051
AS,/R 0.244 0.320 0.338 0.354 0.372 0.389 0.405 0.422 0.440 0.454 0.483 0.514 0.556 0.622 0.655 0.668 0.669 0.663
to move in is small, the q u a n t i t y w2 is also small, as m a y be expected. The limiting value of w¢~8) for y = 0, is the same as t h a t for a one dimensional system of h a r d spheres and equals 1/8. I t should be noted t h a t for y = 0.5 when the molecules can just slip past each other in the double cell a point
A FLUID OF HARD SPHERES
413
of inflexion in the w~81 curve occurs. For values of y near 1, w2 shows a m a x i m u m of approximately 0.7. This is a value which is somewhat too large to allow, strictly speaking, the application of the equations (4)-(8). Thus in this region of w2-values poorer results might be expected. Values of y > 1, for which the cells are no longer closed by the collision spheres of the neighbouring molecules are not considered.
i I I I I
I
.....
I i
I
Fig. 3. Cross-sections of the exact and the a p p r o x i m a t e singlea face c e n t e r e d c u b i c l a t t i c e of hard speres. T h e upper figure is a (001) plane, t h e l o w e r figure a l o n g a (110) plane. T h e t w o sections t h e .......... line. a is t h e lattice c o n s t a n t , o the d i a m e t e r dl = { a ~ / 6 - - -
a n d double cell for cross-section along a c u t each other along of a sphere;
2a, d2 ---- a x / 3 - - - 20, ds --~ 3a -- 2e.
b) T h e / a c e centered cubic lattice" (z = 12, 7 = V 2 ) . In figure 3 we have
414
Z. W. S A L S B U R G , E. G. D. C O H E N , B. C. R E T H M E I E R A N D J. D E B O E R
illustrated how the inscribed parallelopipeda for the single cell and for the double cell were chosen. This leads to a single cell integral: !
(17)
Q~ = dodld2
and to a double cell integral:
' ' J;'
Q0C]~= ~.[
dz1
dyl
Jo" f;' fo" fo" dxl
dzz
dy2
dxz £ (r12 -- ,)
(18)
where: do =
2a -- 2~ =
dl
§aV'6-
=
dz =
2y0.
20" =
a 3 , / 3 " - - 20. =
(I 9a) (1.633y
(1.732y
(19b)
-- 0.367)0. -- 0.268)~
(19c)
d = 3a - - 20. ---- ( 3 y -'F" 1) 0.
(19d)
The details of the integration of (18) are given in the Appendix. The results come down to the following. The calculations were continued up to y = 1, although the maximum value of y for which the collision spheres of the nearest neighbours still touch is y = %/2-- 1 = 0.414; but due to the presence of the next nearest neighbours the cell can still be considered to be "closed" till y = 1, when the collision spheres of these next nearest neighbours just touch. The condition, that the dimensions of the single- and double cell, in our parallelopipedum 'approximation, are non-negative leads to a second condition for y, because of the fact that dl > 0 only for y ~ 0,224. Then d2 and d3 are also non-negative because dl < d2 < d3. T A B E L II Values associated with the double cell correction in a face centered cubic lattice (/).
y 0.225 0.34 0.38 0.42 0.46 0.50 0,54 0.58 0.62
0.66 0.70 0.74, 0.78 0.82 0.86 0.90 0.94 0.98 1.00
]
] 1.300 1.702 1.856 2.024 2.202 2.385 2.583 2.79! 3.005 3.235 3.477 3.724 3.989 4.259 4.550 4.853 5.160 5.489 5.659
.,(I} 0.129 0.151 0.162 O. 174 O. 187 0.200 0.215 0.231 0.249 0.269 0.292 0.318 0.346 0.375 0.402 0.427 0.449 0.468 0.476
]
dws(1)ldy 0.278 0.310 0.334 0.359 0.388 0.423 0.469 0.535 0.614 0.682 0.712
0.699 0.656 0.586
0.502 0.418
I zI'cI='dPlVIRT] "4sz/R
0.218 0.229 0.240 0.254 0.269 0.289 0.320 0.356 0.384 0.384 0.359 0.328 0.285 ' 0.239 0.196
0.424 0.441 0.459 0.478 0.497 0.516 0.536 0.557 0.579 0.603" 0.629
0.656 0.681 O.703 0.723 0.739 0.752
:0.758
i
A F L U I D OF .HARD S P H E R E S
.
415
T h u s we have only to consider the following range of values of y:
0,2247 ~ y ~ 1
(20)
Using (9) and(19), w2 for the face centered cubic lattice, indicated as w~1), can be calculated for all y-values in the range (20). In table II are given v* = (y + 1)8V'2, w~11 and dw~ll/dy as a function of y. A plot of w~11in fig. 2 shows that the behaviour of w~n for 0.3 < y < 1.0 is very similar to that of w~sl between 0.1 < y < 0.8. The hmiting value of w,2(n for y = 0.2247 is 0.129. This can be calculated using the formulae for a two-dimensional lattice 5). In the neighbourhood of y = x/3- -- -I = 0.732, where t h e two molecules in the double cell can just slip past one another, the w~fl .curve shows a point of inflexion. With the values given in table i and table II the entropy correction AS = ( S - sIr"J'))/R and the correction on the .equation of state. A~ = ( p - - pIL'J'I)V/RT due to double cells can be calculated both for the simple cubic (indicated b y a subscript s) and for the face centered cubic !attice (indicated b y a subscript/). They are given in the last two columns of table I and II. c). The total pressure. To get the total .pressure p we still have to add the single cell contribution p(S.J.I. This we did not calculate from the Q~ as given b y (13) or b y (17), which were only used to evaluate the corrections on the pressure, due to the cell-cluster treatment. For the zeroth approximation pILJ.I i.e. the single cell theory, one should t
TABLE Ilia
T A B L E IIIb
Values of Ks = pVIRT for a simple cubic lattice according to the single cell theory and to the ceU-cluster theory in first approximation.
Values of K! = pV/RT for a face centered cubic lattice according to the single cell theory and the cell-cluster theory in first approximation. v*l/s K$(L.J.) K!
Y 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33
112v'2 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00
I
v'1/3 1.09 1.12 1.15 1.18 1.21 1.24 1.27 1.30 1.33 1.35 1.40 1.45 1.50 1.60 1.70 1.80 1.90 2.00
[ K/L'J') 12,111 9.333 7.667 6.556 5.762 5.167 4.704 4.333 4.030 3.828 3.500 3.222 3.000 2.667
2.429 2.250 2.111 2.000
I
K, I 1.072 9.547 7.884 6.781 5.991 5.399 4.942 4.580 4.288 4.098 3.823 3.597 3.386 2.926 2.548 2,282 2.092 1.949
0.42 0.46 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.82 0.86 0.90 0.94 0.98
1.265 1.301 1.336 !.372 1.408 1.443 1.479 1.515 1.550 1.586 1.621 1.657 1.693 1.728 1.764
3.38 I 3.174 3.000 2.852 2.724 2.613 2.515 2.429 2.351 2.282 2.220 2.163 2. l I l 2.064 2.020
3.599 3.403 3.240 3.106 2.993 2.902 2.835 2.785 2.735 2.666 2.579 2.491 2.396 2.303 2.216
416
z. w . SALSBURG, E. G. D. COHEN, B. C. RETHMEIER AND J. DE BOER
use of course the best possible single cell approximation.We used the sphericalized single cell configurational partition function: 4~r 4st • {)~ = ~ ( a - - a ) 3 = 3 y3a3 (21) which leads to the following expressions for p(•'J')V/RT: p(pJ.
v _
_
RT
y +
RT
1
(22)
y
(22) is identical with the expression that would have been derived using (13), because all Q~ of the form Q= ~ y a will lead to the same pressure. Numerical values for K(sL'J')-~ p~L'J')V/RT and for K~r"J')= p~L"r')V/RT are given in table I I I a and IIIb respectively, together with values for K, = i
= paV/RT and Kf = plV/RT. § 4. Discussion o/the results. In a system of hard spheres the effect on the entropy of taking into account cell-clusters of two cells is given by: S = S (L'dl) + R(in (1 -- a~)z/9')/(1 -- z~)
(23)
where ~ is given in terms of w9 by equation (5). The second term is just equal to the extra factor which occurs in the partition function because of the taking into account of cell-clusters of two cells, as m a y be seen from equ. (4). The increase in entropy due to the double cells is plotted in figure 4 for 0.8
i
.
i
i
i
!
w
!
i
0.6 Z-1
.
0.~
0.2
°1.0
'
1'.2
'
1;4
'
118
'
!
t
Fig. 4. ,Entropy correction due to the introduction of double cells for~the simple cubic lattice (s.c.) and the face centered cubic lattice (f.c.c.). the simple cubic lattice and for the face centered cubic lattice respectively as a function of the reduced volume. Physically this increase in entropy is caused by the increase in the number of possible configurations for the system which is determined by the values of w~. There are no data available
A FLUID OF HARD SPHERES
417
for the entropy from the Monte Carlo results to which reference was made in the introduction, these being related only to the equation of state.'Still it is of interest that the taking into account of the correlated motion of two molecules in a double cell gives a considerable increase in the entropy. pv
T 0.8
0.6
0.2
1.0
12~
1£
1.6
-0.;
-0.~
Fig. 5. E q u a t i o n of s t a t e c o r r e c t i o n s o n t h e free v o l u m e t h e o r y d u e t o d o u b l e cells for t h e s i m p l e c u b i c l a t t i c e (s.c.) a n d t h e face c e n t e r e d c u b i c l a t t i c e (f.c.c.) as a f u n c t i o n of v'x/3. T h e c u r v e M.C. was d e r i v e d f r o m t h e M o n t e Carlo c a l c u l a t i o n s b y s u b t r a c t i n g t h e free v o l u m e v a l u e s for a face c e n t e r e d c u b i c lattice.
The influence of the introduction of double cells on the equation o/state is given in figure 5, where we have plotted the corrections on the free volume ~(L.J.) ) V / R T and A KI= (pf_p~L.J.I) V / R T for a simple cubic theory A Ks= 14~_ ~r~--r8 and a face centered cubic lattice respectively. The maximum of both curves corresponds to the inflexion point in the w2 curve, where the double cell is large enough that the two molecules can just pass each other. We have also plotted in this figure the corrections on the free volume theory that can be derived from the Monte Carlo calcuiations, b y subtracting from the KMV = pMoV/RT, as given b y R o s e n b l u t h and R o s e n b l u t h 2) the K~1"'J'~ given in table IIIb. It is seen that the qualitative behaviour of the two curves is very similar and that the order of magnitude of the corrections Physica X X l I I
27
4 1 8 Z. W. SALSBURG, E. G. D. COHEN, B. C. RETHMEIER AND J. DE BOER
are roughly the same, though the two curves are shifted with respect to each other in the direction of the positive v*l/3--axis. This, however, might be due to our restriction in considering only corrections due to double cells. For, the positive correction to the free volume pressure can be Associated with the tendency of the molecules to slip past one another. The rigid structure of our double cell permits this to occur at a much lower density than would be expected in an exact treatment. PV
!
T
i
b"
"'"- ~ ...." ~.. ~..~:..
"-. "--..~ s.clc.ct.)
" 'x..,
I f.c.c.
0
i|
0.9
~S.C.
1.0
" ~ ~ ' ~ ' ~ ' ~ ' ~ "
I
1.1
i
1.2
I
1.3
I
1.4
I
1.5
I
1.6
I
1.7
I
1.8
I
1.9
t
2.0
.__.~ V ~ Fig. 6. Calculation of p V/RT as a function of v*Zla according to various theories compared w i t h the Monte Carlo results: K • from t h e K i r k w o o d integral e q u a t i o n 4) ; BG : from the Born and Green integral e q u a t i o n 4) ; MC : from the Monte Carlo calculations 1) ; s.c. : cell-cluster t h e o r y based on a simple cubic lattice; f.c.c. : cell-cluster t h e o r y based on a face centered cubic lattice.
Finally in figure 6 we have plotted the results of various calculations for = p V / R T as a function of the reduced volume. The calculations based on a simple cubic lattice are seen to be in very poor
agreement with the Monte Carlo results. This means t h a t a simple cubic lattice is not a good lattice structure to serve as a basis for a quasi-crystalline theory of a liquid of hard spheres. It should be noted the discrepancy is mainly due to the single cell approximation which differs widely from the Monte Carlo curve as is illustrated by the fact that it approaches to the asymptote v .118 = 1, which line is cut by the Monte Carlo curve.
A FLUID OF HARD SPHERES
419
The agreement of the Monte Carlo results with the calculations based upon a /ace centered cubic lattice is much better. The corrections, though in absolute value of the right order of magnitude, have the wrong sign. Because of various approximations made, the Kycurve could not be continued below v*~/~ = 1.265 that is below y = 0.42. We have also plotted in figure 6 the results obtained by K i r k w o o d , M a u n and A l d e r 4) for x = p V / R T from a numerical integration of the Born and Green (KBa) and Kirkwood (KK) integral equations for the radial distribution function. They agree very well with Monte Carlo results at low densities, b u t show considerable deviations at high densities. A first step to improve our calculations would be to calculate w~, using the true form and size of the single- and double cells. This will be done in a forthcoming publication 5) for the two-dimensional trigonal lattice. Another important improvement of the theory would be the introduction of vacant sites into the basic lattice. This cell-cluster theory with holes would start with a number of lattice sites larger than the number of molecules. Then in every configuration there are a number of vacant lattice sites and a number of cell-clusters containing less molecules than cells. Such a theory would not only be an extension of the cell-cluster theory as developed in I and II, but also of the existing hole theories, which allow only for occupied and unoccupied single cells 6). Also this we reserve for a later publication 7).
¢
A p p e n d i x . Calculation o/ Q~# i n a lace-centered cubic lattice. We consider
the evaluation of the sixfold integral
:2.1l
f ~;1Fa* dZlJo
d y l J 0t'a3 dxl
f o;~dZ2JoFa*dy2Jot'a~d x 2 ~ ' ( r 1 2 - - a )
(A l)
where r~, = (xz - - Xl) ~ + (y2 -- Yl) z + (z2 -- Zl) 2.
We introduce a set of relative coordinates X, Y, Z:
X = x2 -- Xl Y = Yz - - Yl
(A 2)
Z = 22 -- Zl and if we write x, y, z for Xl, Yl, Zl respectively, then (A 1) can be written as: ' = 1o~jo ra~ jr_a~~ - ~ L ( Z ) d Z dz Q~t~
(A 3a)
L ( Z ) -ra~rd.,-y K ( Z , Y ) d Y dy -.10 J - - y
(A 3b)
where and
K(Z,
Y) = f # f _ ~ - x
e(R -- ,,) dX dx
t (A 3c)
420
z. w . SALSBURG, E. G. D. COHEN, B. C. R E T H M E I E R A N D J . DE B O E R
with R 2 = X ~.+ y 2 + Z L In each of the above double integrals we interchange the order of integration and this leads to the expression: Q'a~ = 4 f g l foa' foa3 N ( X , Y , Z)
e(R -
~) d X d Y d Z
(A 4)
with N ( X , Y , Z) = (d3 -- X)(d2 -- Y ) ( d l -- Z). Between the three lengths dl, d2 and ds hold the following inequalities: dl < d2 < d~ The integral (A 4) can best be considered in four separate cases. Case I" a K dl, or 0.8371 < y K 1. In this case the integral can be computed in the following manner: Q ~ I = 4[ f o~lf o~ f od3N d X d Y d Z -- f f f R~. ~st ootant N d X d Y dZ] = 222
=½ dld~d3 -- 4 f f f R ~ . 1st octant N d X d Y d Z
(A5)
This is illustrated in figure 7a. The integration can easily be carried out using spherical polar coordinates leading to:
Q~
a6 rqn,~*2a*~a*2
a*~*a*
+ 1 5 ~ (dl*d~ * + d 1*d*3 +
d~*d3) * -- 16(d* + d* + d*) + 5].
(A 6)
where d* = di/a(i = 1, 2, 3). Case II: dl _< a _< dz or 0.7321 _< y ~ 0.8371. The integral can be obtained from the calculation for case I if we correct for the spherical cap, as is illustrated b y figure 7b. Thus t
= '~0 + 4fffoao i N dX dY dZ = ,q,lli + ~ .amc,m dgf00°d0sin 0 fdlcosO * NR 2 dR =w*t~
(A 7)
after the introduction of spherical polar coordinates. 0o is given b y the equation: 0o = arc cos d*. The computation of the integral in (A 7) is lengthy b u t straightforward and leads to" an Q'(II) * 3* - - H I + ~ = v tl'(i) ~ + - ~ [5Glz¢ d~d
30d~'(d~' + d'~)(d*
-- arc cos d*) + + 16 (d* + d~) (I + ¼d~'2) (1 -- d l*~,&]/. +
V1--~I'
(A 8)
with (71 = d~'4 -- 6d~'2 + 8d~' -- 3 H1 ---- d *° -- 5d1.4 + lSd~ 2 -- lSd~' + 5
(A 9a) (A 9b)
A FLUID OF HARD SPHERES
421
Case I I I : d2 ~ ~ ~ ~/d~ + d~ or 0.6063 ~ y £ 0.7321. The calculation is similar to case II. W e m u s t add an additional correction-term due to a second cap outside the parallelopiped, as illustrated b y figure 7c. T h u s one has: '(III) ~'(II) NdXdVdZ (A 10)
Z
d
d2
Y
V
X~ 3
(a)
(b)
z
~0 (c)
/~X3
(d)
Fig. 7. Regions of integration in the four different cases for which Q ~ for a face centered cubic lattice has to be evaluated:
(a) case (b)
,,
I" ~ ~ al II:dl~
a ~ d~
(c) case m : a~ ~ o ~ ~/d~ + d~ (d)
,,
IV: ~/dl~ + d ~ K
a K d8
422
A FLUID OF HARD SPHERES
The evaluation of the integral proceeds in exactly the same w a y as in case II. Thus we get: Q'(III) t~'(II) U6 ~f~
= ,¢~
+ -ff~ [5Ee~d* d* - - F e +
+ 30d*(d* + d~)(d'~ V 1 + 16(d~' + d~')(1 +
arccos d*) +
d.2-
¼d.2) (1
--
d ~ ) 310-
(A 11)
with E2 = d~ 4 -- 6d~'2 + 8d* -- 3 v 2 = a *° -
4 +
15d 2 - - 16d
(A 12a) + S
(A 12b)
Case IV: x/d~ + d~ ~ a ~ d8 or 0 ~ y ~ 0.6063. The integral that we want to compute is the integral of N over t h a t part of the parallelepiped that does not overlap the sphere R < a. This is illustrated in figure 7 d. The integral can be computed directly from:
Q'(IV) ap = 4f0a' dZ(dl -- Z ) f o ~ d Y (de -- V ) f ~ , a _ v , _ z ,
d X ( d 3 - - X ) (A 13)
The lengthy computation is straightforward and gives:
O'6 Q'(IV) a/~ ~___60 E _[A3 + BI(1 -- d*2) 1/2 + Be(1 -- d*2) ~/~ +
4' + C~ ( 1 - d .2 - - d'~2) '/2 + D1 arcsin +
(1 - -
d'e) ~/' -t-
De arcsin (I -- d*d~2)',/, + 30d~ (d~ arcsin d* +
+ d~' arcsin d~') +
d'd*
80
d r d~ d* arcsin (l -- d'2)112(1 -- d ~ ) '/~
]
(A 14)
Here are A3
B1 Be C3 D1 De
= 3 0 d *2 I d~*2 d 3*2 -- 16d~ + 5 d *2 1 d 2*2 (6 -- dl*e--d2*S~j (A15a) = d~(16 + 18d* -- 4d*') (A lSb) = d~(16 + 18d~'2 -- 4d~ 4) (h 15c) = d~ {4(d*' + d~ 4) -- 18 (d~ ~ + d~ e) -- 12d~ 2 d~ a -- 16} (A 15d) * * "4 = lOded3(d t - - 6d~ 2 - - 3) (A 15e) = lOd~d~(d~ 4 - - 6d .2 - - 3) (A 15/)
Received 19-12-56. RE FERENCES 1) 2) 3) 4) 5) 6) 7)
R o s e n b l u t h , M. N. and R o s e n b l u t h , A. W., J. chem. Phys. 28 (1954) 881. De Boer, J., Prom Conf. theor. Phys. Japan, 1953, 507; Physica 20 (1954) 655. Cohen, E. G. D., De Boer, J. and S a l s b u r g , Z.W., P&ysiea -°! (1955) 137. K i r k w o o d , J. G., Maun, E. K. and Alder, B. J., J. chem. Phys. 18 (1950) 1040. Cohen, E. G. D. and R e t h m e i e r , B. C., Physica (to be published). R o w l i n s o n , J. S. and C u r t i s s , C. F., J. chem. Phys. 18 (1951) 1519. Cohen, E. G. D. and D a h l e r , J. S., Physiea (to be published).