6916
J . Phys. Chem. 1989, 93, 6916-6919
Why Does the Camahan-Starling Equation Work So Well?+ Yuhua Song, E. A. Mason,* and Richard M. Stratt**l Department of Chemistry, Brown University, Providence, Rhode Island 0291 2 (Received: February 7 , 1989; In Final Form: April 21, 1989)
It is possible to view the Carnahan-Starling equation of state for hard spheres as arising from a kind of mean-field theory. We use this perspective to arrive at a generalized equation of state for hard spheres in any number of dimensions that still retains the extraordinary accuracy of the original Carnahan-Starling formula.
Introduction In the years since its derivation in 1969, the Carnahan-Starling equation of state for hard-sphere fluids1 has achieved the status of being one of the premier practical working tools for studying liquids. It was originally derived from the observation that the numerical values of the known hard-sphere virial coefficients came remarkably close to fitting a simple algebraic expression. When this expression was substituted into the virial expansion for the equation of state and summed to infinite order, the resulting formula for the pressure turned out to be virtually indistinguishable from that obtained via computer simulations over the entire fluid range.' Since its inception this formula has taken on a life of its own. It has been used to construct a similarly accurate (and similarly useful) expression for the hard-sphere radial distribution function.2 It has been generalized to mixtures3 and even to inhomogeneous fluid^.^ The statistical-mechanical origins of this result being what they were, though, one cannot help but wonder if there is really some fundamental reason for all of this success. What one would like to know is if there is a systematic approach to liquids, or to many-body problems more generally, lurking behind this serendipity. This question is made somewhat more intriguing by two other observations. It is possible to "derive" the Carnahan-Starling equation of state by a suitable (and apparently arbitrary) linear combination of the compressibility and virial equations of state resulting from Percus-Yevick t h e ~ r y . Whether ~ this finding has any real significance is completely unknown. More interesting, perhaps, is that the Carnahan-Starling equation does as well as it does in spite of the fact that it has a totally unphysical high density behavior. Recent studies have suggested that the pressure should diverge as the density approaches that of random closest packing.6 The Carnahan-Starling equation, however, does not diverge until a packing fraction 7 = (?r/6)pu3 = 1, the point at which there is no open space in the system if p is the number density and u the hard-sphere diameter. Spherical objects, of course, can never exceed the closest packing value of 7 = ( ~ / 6 ) 2 ' / ~ = 0.1405.
We suggest in this paper that there is, in fact, a systematic way of arriving at the Carnahan-Starling equation, a way which regards the equation as the outcome of an unusual kind of mean-field theory. The thesis we present here is that the success of this equation is largely due to lack of correlation between the motions of a particle in different directions. This very feature is none other than the one exploited by Maxwell in his original derivation of the equilibrium velocity distribution function.' Of course, any position distribution derived in this fashion will not be exact, but the approximation is quite different from the assumption that the motions of different particles are totally uncorrelated, an idea which is reasonable only at low densities. Furthermore, the separation into independent one-dimensional problems leads quite naturally to a multidimensional generalization. This paper is dedicated to that multidimensional character, Robert Zwanzig, on the occasion of his sixtieth birthday. 'Alfred P. Sloan Foundation Fellow.
0022-3654/89/2093-6916$01 S O / O
The general result we arrive at turns out to be surprisingly simple. Since the equation of state for hard spheres in any number of dimensions can be written in terms of the value of the radial distribution function at contact, g(u+),6 P P / P = 1 + bpg(c+)
(1)
with Pp the pressure divided by kBT and b the second virial coefficient, it suffices to write an expression for this contact value. What we find is that 1-an
where 7 is the packing fraction in d dimensions 9
and CY
= bp/2"'
(3)
b = 8[71-''/~/I'(d/2)4 (4) is chosen to reproduce the exact third virial coefficient, BS, a = d - 2"'(B3/b2) (5)
The third virial coefficient has been evaluated for all values of d,8*9
where 2Fl is the usual hypergeometric function.1° The above result for g(u+) is exact in one dimension, and in three dimensions reduces to precisely the Carnahan-Starling formula. Nor is its success limited to one and three dimensions; in other dimensions it is superior to any other current theoretical result, as demonstrated in what follows.
Derivation We first give a heuristic derivation of eq 2 based on physical arguments, and then present a Carnahan-Starling-style derivation based on known virial coefficients. A . Physical Arguments. The derivation of the nature of the divergence near 9 = 1 is the first order of business. The physical (1) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (2) Verlet, L.; Weis, J. J. Phys. Reu. A 1972, 5, 939. (3) Mansoori, G . A,; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J . Chem. Phys. 1971, 54, 1523. (4) Stoessel, J. P.; Wolynes, P. G., preprint. (5) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: New York, 1986; pp 121-123. (6) Song,Y.; Stratt, R. M.; Mason, E. A. J. Chem. Phys. 1988,88, 1126. Tobochnik, J.; Chapin, P. M. J . Chem. Phys. 1988, 88, 5824. (7) Maxwell, J. C. Philos. Mag. 1860, 19, 19. Reprinted in: The Scientic Papers of James Clerk Maxwell; Niven, W. D., Ed.; Cambridge University Press: London, 1890 (also Dover Publications, New York, 1965), Vol. I, p 377. Also reprinted in: Kinetic Theory I; Brush, S. G., Ed.; Pergamon Press: London, 1965; p 148. (8) Luban, M.; Baram, A. J. Chem. Phys. 1982, 76, 3233. (9) Joslin, C. G. J . Chem. Phys. 1982, 77, 2701. (10) Luke, Y. L. The Special Functions and Their Approximations; Academic Press: New York, 1969; Vol. I.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6917
The Carnahan-Starling Equation TABLE I: Values of BJb' and a for Hard (Hyper)Spheres of DhlWlSiOllP~tyd d Bilb2 a 1 0 (4/3) - (3'/2/?r) 0.435 99 0.500 00 518 -0.050 72 (4/3) - (3'l2/2r) 53/2' -1.625 00 (413) - (35/2/5*) -4.910 12 -1 1.062 50 289/21° -22.030 54 (4/3) - (31 X 35/2/140r) 6413/215 -41.101 56
TABLE II: Expansion Coefficients c, of Eq 13 for g(o') n d = l d = 2' d = 3a d = 4b 2.000 3.000 4.000 1 1 (4) (2) (3) 9.519 3 .OO 1 2 1 6.091 (3) (6) (10) 3 1 3.977 10.10 (10) (4) 15.01 4.916 4 1 (15) (5) 21.53 5.817 5 1 (21) (6)
meaning of g(u+) is that it is the ratio of the number density of particles in contact to the average number density. It is greater than unity because two particles close to each other tend to be pushed still closer by the impact of the other particles in the fluid." Thus g(u+) will be proportional to the probability that a second particle will be found within a region of thickness dr surrounding a reference particle. Our basic assumption is that this probability is the product of independent probabilities for each dimension; in three dimensions this would be
Virial coefficients summarized in ref 25. ref 8 and 9.
which has an obvious generalization to an arbitrary number of dimensions. The probability of a particle being located between x and x d x is proportional to the ratio of two volumes: the volume between x and x d x (which is Ld-'dx if the total volume is V = Ld) and that part of the total volume that is not excluded by the other particles (which, at the mean field level, is V( 1 - 7)if 7 is the packing fraction). Hence,
+
+
p(x)
-
fld-')ldd x = - - 1 V(1
- 7))
1-7)
Virial coefficients from
1.o
0.8
0.6
do+)-' 0.4
0.2
dx v'ld
The overall probability for d dimensions is thus
0.0
0
0.2
0.4
0.6
08
1
7)
Figure 1. Reciprocal of the pair distribution function at contact for hard (hyper) spheres (d = 1-6) as a function of packing fraction (density). The arrows mark the density at which freezing occurs for d = 2 and 3. Numerical simulations: 0,ref 16; 0 ,ref 17; 0,ref 18; A, ref 19. -, present work, eq 2; scaled-particle theory, ---, Percus-Yevick theory.
from which we conclude that
e..,
This proportionality merely gives the most divergent contribution to g(u+) near 7) = 1, because we did not specify that x had to be located at a distance u from the reference particle. The correct form would presumably be g(a+) =
Ad -
(1 - d d whereflq) is some weak function of density. The simplest guess forflq) is a linear function giving the correct low-density behavior; that is, one that gives the correct third virial coefficient f(0) = 1 - a7)
(12)
with a given by eq 5 . As already noted, this gives a result that is exact for d = 1 and that is identical with the Camahan-Starling result for d = 3. Since third virial coefficients for all d a r e known, this result completely defines the equation of state. Values of B3/bz and a for d = 1-9 are collected in Table I. Some further physical insight into the foregoing result can be obtained by noticing that the product behavior of P(xl,...,xd), and the divergence at 1 = 1 instead of a t random closest packing (7 = 0.69 in 3-4:is mimicked by a model of hard parallel (hyper) cubes first popularized by Z w a n ~ i g . ~Thus ~ - ~ the ~ expression of (11) Cutchis, P.; van Beijeren, H.; Dorfman, J. R.; Mason, E. A. Am. J . Phys. 1977, 45, 970.
(12) Zwanzig, R. W. J . Chem. Phys. 1956, 24, 855. (13) Mason, E. A.; Dorfman, J . R.; Zwanzig. R. Am. J . Phys. 1970, 38, 435.
(14) Kirkpatrick, T. R. J . Chem. Phys. 1986, 85, 3515.
eq 2 for g(u+) is probably an accurate result for this somewhat artificial model, which also happens to behave much like a hard-sphere model at lower densities. This latter point is shown by the fact that the third virial coefficient for hard parallel cubes (d = 3) is B3/b2= 9/16, as first calculated by Z w a n ~ i g , ' ~not J~ far from the hard-sphere value of B3/b2= 518. B . Summation of a Virial Expression. The same result can be obtained by some modest numerology with known virial coefficients. We write a virial expansion for do+) with the linear factor explicitly removed g(u+) = (1
-q)(l
+ Cl7) + C27)Z + ...)
(13) with a given by eq 5 . Then the c,, insofar as they are known, are nearly integers, as shown by the summary of Table 11. If we adopt the integer values shown in parentheses in the table, then the c, are reproduced by the binomial coefficient formula, n+d-1 cn=(n
1-
-(n+d-l)! n!(d-l)!
Assuming that this result holds generally, we can sum the series and obtain
which is the same result as before. (15) Mason, E. A,; Spurling, T. H. The Virial Equation ofState; Pergamon Press: London, 1969; p 187.
6918
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Song et al.
TABLE 111: Virial Coefficients for d = 2 and 3“ d = 2
Bdb2 B4fb3 4ib4 8 6 1 bS B7/b6
d=3
accurate6
present
SP
accurate6
present
PYIV)
SP PY (c)
0.7 8 200 0.53223 0.33356 0.1989 0.1148
0.78200 0.53200 0.33650 0.20350 0.11938
0.75000 0.50000 0.3 1250 0.18750 0.10938
0.62500 0.28695 0.1 1025 0.0389 0.01 37
0.62500 0.28125 0.10938 0.03906 0.013 18
0.62500 0.25000 0.08594 0.02734 0.00830
0.62500 0.29688 0.12109 0.04492 0.01562
OSP = scaled-particle theory. PY(v) = Percus-Yevick virial. PY(c) = Percus-Yevick compressibility. 6Reference25. TABLE I V Virial Coefficients for d = 4, 5, and 6’ d=5
d = 4
B3fb2 B4/b3
d=6
accurateb
present
accurateb
present
PY(V)
PY(C)
accurateb
present
0.50634 0.1519
0.50634 0.15942
0.41406 0.0761
0.41406 0.09033
0.41406 0.04785
0.41406 0.08905
0.34094 0.0336
0.34094 0.04928
‘Same key as for Table 111. bReferences8 and 9.
Comparisons A . Radial Distribution Function at Contact. There are several sources of g(a+) for direct comparison with the present eq 2 , including computer s i m ~ l a t i o n s , l ~scaled-particle -~~ theory,20,21 and solutions of the Percus-Yevick integral e q ~ a t i o n . ~All ~-~~ agree with the exact result for d = 1 (16) Scaled-particle theory gives 1-,
9
g(a+) =
2 and 3,25and through B4 for higher dimension^.*^^ There are two sets of virial coefficients from the Percus-Yevick equation, depending on whether one proceeds through a virial calculation of the pressure or from the compressibility result. Comparison of the known values with the various approximations is given in Table 111 for d = 2 and 3, and in Table IV for d = 4, 5 , and 6 . It can be seen that eq 2 is superior in almost all cases shown. It does start to fail at still higher dimensions, however, since it predicts positive B4 for all d , whereas B4 actually becomes negative for d > I.’.’ C. Excess Entropy. The excess entropy of a dense fluid (relative to the ideal gas at the same density) is of interest in a number of connections. It is given by
d=2 (1 - 0)’’ 9
v2
L
4
(1 -
d3
I-=+:
=
The integral is easily calculated to yield Sex(9) --
, d=3
Nk
The Percus-Yevick equation can be solved exactly in all odd dimensions; the results for d = 3 and 5 arez3
- In (1 - q ) ,
d = I
-Sex(9) - - 2 a In ( 1 - 7) - 2(11 -- Va17 , Nk
d=2
(23)
9
1+, g(a+) =
(1 g(.+)
=
(1
+ 189 + 67’)’/’
- VIZ’
--
d=3
- 1 + 339 + 877’ 607(1 - s ) ~
Nk
+ 67’ ,
For d = 2 numerical calculations are available.z2 Ail of these results are compared in Figure 1 ( d = 1-6). The agreement of eq 2 with the numerical simulations is excellent up to at least the density of freezing, and eq 2 is always better than any of the other approximations. The Percus-Yevick results become worse as d increases, possibly because the divergence at 9 = 1 a p p e a r ~ ’ ~ to , * ~behave as (1 - ~ ) - ( ~ + l ) instead / ~ of as (1 9rd.
B. Virial Coefficients. Another useful comparison is through the virial coefficients, since they are known through B7 for d = ~~
~
~~
(1 - 9)“’
d=5
(20)
~~
d-1
~
(16) Erpenbeck, J. J.; Wood, W. W.J . Star. Phys. 1984, 35, 321. (17) Erpenbeck, J. J.; Luban, M. Phys. Rev. A 1985, 32, 2920. (18) Tobochnik, J.; Chapin, P. M. J . Chem. Phys. 1988, 88, 5824. (19) Mfchels, J. P. J.; Trappeniers, N. J. Phys. Lett. 1984, 104, 425. (20) Fnsch, H. L. Adu. Chem. Phys. 1964, 6, 229. (21) Reiss, H. Adu. Chem. Phys. 1965, 9, 1. (22) Lado, F. J. Chem. Phys. 1968, 49, 3092. (23) Leutheusser, E. Physica 1984, 127A, 667. (24) Leutheusser, E. J . Chem. Phys. 1986, 84, 1050.
All of these expressions diverge at 9 = 1, as expected, since our model for the system becomes completely ordered in this limit.
Discussion There may be alternatives to our picture of why the Carnahan-Starling equation of state for hard spheres does as well as it does, but the analysis presented here has the advantage of generalizing naturally to an equivalent result in any number of dimensions. That this generalization leads to extremely accurate results is still another point in its favor-though perhaps the fact that it fails to correct the high-density behavior of the Carnahan-Starling formula is not. One observation that this development does not address is the point mentioned in the Introduction that one can derive the original Carnahan-Starling equation from a weighted average of Percus-Yevick compressibility and virial equations of state. We shall therefore close with a remark about this feature. In dimensions (25) Summarized by: Devore, J. A . J . Chem. Phys. 1984, 80, 1304.
J . Phys. Chem. 1989, 93, 6919-6926 other than three, the two Percus-Yevick results continue to bracket the true results. In five dimensions, for example, the exact fourth virial coefficient is very close to being the same two-thirds compressibility, one-third virial, combination that generates the Carnahan-Starling equation in three dimensions. Our five-dimensional formula, however, does not predict a fourth virial coefficient between the two Percus-Yevick coefficients. The two-thirds/one-third “rule” may thus be more of a statement about
6919
the Percus-Yevick approximation than about the CarnahanStarling equation. Acknowledgment. We thank Jan Tobochnik for helpful comments and for providing us with his computer simulation data. This work was supported by the National Science Foundation under Grants CHE-8420214, CHE-8815163, CHE-8509416, and CHE-88 19370.
Kinetics of Crystallization and Morphology of Poly(piva1oiactone): Regime I I Transition and Nucleation Constantst
-
III
Daniel B. Roitman,*Hew6 Marand,# Robert L. Miller, and John D. Hoffman* Michigan Molecular Institute, Midland, Michigan 48640 (Received: February 7, 1989; I n Final Form: April 24, 1989) Growth rates of the crystalline objects formed from the subcooled melt in poly(pivalolactone),PPVL, were measured optically at various temperatures T, between 180 and 218 OC. Spherulites were formed over most of this range, but axialite-like objects were formed at the highest T,; the growth data refer to the dominant a-crystal form. The results provided clear evidence of a regime I1 I11 transition near 203 OC. The regime behavior was in good agreement with expectation; K,(IIi$Kg(II) was 1.99 & 0.05. From the Kg values, and a determination of the layer thickness bo based on microbeam X-ray studies and related data, it was found that uue = 1748 erg2 cm-“. (The growth front corresponds to 120.) With a u, value of 58 erg cm-2 obtained independently from a T,’ vs 1/1 plot, it was determined that u = 30 erg cm-*, leading to an a of 0.25 in the modified Thomas-Staveley expression. These values of u and a are discussed briefly. The work of chain folding derived from uc is 7.5 kcal mol-’ which is within the anticipated range. WAXD studies confirmed the antiparallel arrangement of the chains in both regimes I1 and 111. Evidence based in part on the presence of cracks in the crystals corresponding to 120 cleavage planes indicates that (i) the chain folding which occurs in melt-crystallized PPVL is rather regular and (ii) in accord with nucleation theory the folding is more regular in regime I1 than regime 111. Finding (i) is supported by a recent “gambler’s ruin” type calculation by Mansfield which shows that an enhanced degree of “tight” folding is demanded when antiparallel packing is present.
-
I. Introduction Interest in poly-/3-lactones has increased in recent years. Substitutions in the 2- and 3-positions relative to the ester group lead to series of materials spanning a wide range of physicochemical pr0perties.I Research in the areas of blending and copolymerization of polylactones in general, and poly(piva1olactone), PPVL, in particular, has been active for some time.24 The chemical structure of PPVL is H CHsO
[ IiH:
-c-c-c-0-
1”
One major aim of this work is to check certain aspects of the kinetic theory of crystallization with chain folding by using spherulite growth rate, melting point, and X-ray data on PPVL. This theory of chain folding, initiated in 19605 and often termed “LH”, has been extended and refined over the years and accounts for a number of aspects of the crystallization behavior and morphology of highly flexible m a c r ~ m o l e c u l e s . ~According ~~ to this approach, the initial lamellar thickness, fg*, is kinetically determined. An important feature of this model is that the growth rate G varies as exp[-K,/T(AT)], where the nucleation constant K,,which can be determined experimentally with considerable precision, contains the factor bouu,. Here bo is the layer thickness, Dedicated with deep respect to Professor R. W. Zwanzig in recognition of his outstanding contributions to our understanding of the nature of condensed systems. * To whom correspondence should be addressed. *Present address: Dow Chemical Company USA, 2800 Mitchell Drive, Walnut Creek, CA 94598. *Present address: Virginia Polytechnic Institute and State University, Chemistry Department, Blacksburg, VA 2406 1.
0022-3654189 12093-6919$01.50/0 , I
I
-
u the lateral surface free energy, u, the fold surface free energy, AT = Tm0- T, the undercooling, T, = T the isothermal crystallization temperature, and Tm0the melting temperature of a very large extended-chain crystal of the molecular weight under consideration. The quantities u and a, are of fundamental interest. A specific aim of the present study is to determine these quantities for PPVL, beginning with the known value of bouue. In the present work we determine c and u, for PPVL by the following approach. The quantity K , is determined for meltcrystallized PPVL from the crystal growth rates by appropriate plots, and from this one finds bouue. The relationship between bouu, and Kg is certain in the case of PPVL because the crystallization regimes are known (see below). By microbeam X-ray and other studies we determine bo. This permits an accurate value of uue to be obtained. The value of a, was determined in a parallel study using a T,’ vs 1/ I plot,I5 which also gives a reliable estimate
(1) Koleski, J. V.; Lundberg, R. D. J . Polym. Sei., Part A-2 1972, 10, 323. (2) Allegrezza, A. E.; Lenz, R. W.; Cornibert, J.; Marchessault, R.H. J . Polym. Sci., Polym. Chem. Ed. 1978, 16, 2617. (3) Bluhm, T. L.; Hamer, G. K.; Marchessault, R. H.; Fyfe, C. A.; Veregin, R. P. Macromolecules 1986, 19, 2871. (4) Borri, C.; Briickner, S.; Crescenzi, C.; Della Fortuna, G.; Mariano, A.; Scarazzato, P. Eur. Polym. J . 1971, 7, 1515. ( 5 ) Lauritzen, J. I., Jr.; Hoffman, J. D. J . Res. Natl. Bur. Stand., Sect. A 1960, 64, 73. (6) Hoffman, J. D. SPE Trans. 1974, 4 , 315. (7) Lauritzen, J. I., Jr.; Hoffman, J. D. J. Appl. Phys. 1973, 44, 4340. (8) Hoffman, J. D.; Davis, G. T.; Lauritzen, J. I., Jr. In Treatise on Solid Stare Chemistry; Hannay, N. B., Ed.; Plenum Press: New York, 1976; Vol. 3, Chapter 7. (91 Hoffman, J. D. Polymer 1982, 23, 656. (10) Hoffman, J. D. Polymer 1983, 24, 3. (11) Clark, E. J.; Hoffman, J. D. Macromolecules 1984, 17, 878. (12) Hoffman, J. D.; Miller, R. L. Macromolecules 1988, 21, 3038. (13) Hoffman, J. D.; Miller, R. L. Macromolecules, in press. (14) Hoffman, J. D. Macromolecules 1986, 19, 1124. (15) Marand, H.; Hoffman, J. D.; Briber, R. M.; Barnes, J. D., to be submitted to Polymer for publication.
0 1989 American Chemical Society
