Density fluctuations and the specific heat near the critical point

Physica 74 (1974) 266-276 0 North-Holland Publishing Co.

DENSITY

FLUCTUATIONS

AND THE SPECIFIC HEAT

NEAR THE CRITICAL

POINT

0. K. RICE and D.R. CHANG Department of Chemistry, University of North Carolina, Chapel HilI, North Carolina 27514, USA

Received 27 November

1973

Synopsis A physically intuitive relation is developed between the correlation length for density fluctuations and the specific heat near the critical point of a liquid-vapor system. First, the average spatial extension of a fluctuation (the fluctuation volume) is inferred by considering at what point the volume, in which the fluctuation is supposed to occur, becomes so small that the standard formula relating the compressibility to the density fluctuation breaks down, and it is shown that the length of this extension obeys the same scaling law as the correlation length. The fluid is then divided into cells equal to the fluctuation volume, and the number of possible states of the system arising from the fluctuations in these cells is determined and assumed to be a factor in the partition function. The divergent part of the specific heat is then calculated as a function of temperature, and the result is compared with experimental data and shown to fit known scaling relations. The size of the fluctuation volume required to get the correct specific heat is reasonable. Application is also made to a two-dimensional Ising lattice. Finally, the relation between long-range and short-range fluctuations is considered.

1. General ideas. There have been a number of suggestions that the correlation length for density fluctuations in a fluid may play a fundamental role in the determination of the thermodynamic properties. An early attempt was made by Fixman’) who actually arrived at a proper scaling relation. More recently this idea has been applied by Widom and Fisk2) to a discussion of surface tension. The present paper is an attempt to relate the correlation length for a one-component fluid to the specific heat along the critical isochore near the critical point. In a certain sense this has already been done for lattice gases, inasmuch as correlations and specific heats have been calculated, but we wish to examine a very direct relationship, which we believe adds physical insight. The correlation length is only one aspect of the density fluctuations. The other is what may be called the intensity of the fluctuations, or the root-mean-square of the fluctuation in a given volume. 266

DENSITY

AND SPECIFIC HEAT NEAR CRITICAL POINT

267

The mean square (LI~)~ of the fluctuations of the number of particles in a volume v is given by (~In/n)~ = kTx,/v,

(1)

where xT is the isothermal compressibility, provided v is large enough. For ordinary compressibilities far from the critical point u may be almost as small as molecular dimensions, but near the critical point where xT is very large, such a small value of u would give impossibly large values of (An)‘, and the formula breaks down. The fluctuations near the critical point in such a small volume will remain much the same as far from the critical point. Rut there will be fluctuations correlated over large distances which can affect the thermodynamic properties. The correlation length, though usually defined through the exponential fall-off of fluctuations around a given point, can for our purposes, be considered to be the average distance between a region of fluctuation in one direction and a region in which the fluctuation occurs in the opposite direction. This will define the smallest volume in which eq. (1) may be expected to give a fair approximation. It is therefore suggested2) that it will be reasonable to think of the fluid as divided up into volumes equal to L?, where v = 13. If the fluctuation has a mean absolute value equal to dn, this means that in each volume l3 there will be about x’dn sets of states (we multiply by the gaussian integral, equal to x*), all resembling each other fairly closely, which are possible in the volume t3. In one mole, with a volume V, there will be (~*dn)“‘~’ such sets of states, and so the fluctuations will contribute a factor equal to this to the partition function, and there will be a “fluctuation free energy”, AA, given by dA = -(kTV/S’)

In (78 dn).

(21

It may at first seem strange that a factor which multiplies the original partition function, should not be reflected directly in the entropy. However, it is to be noted that this factor changes with the temperature. It, therefore, reflects a difference in the density of levels at different energies, so it appears in the entropy only after differentiation with respect to temperature. To express this mathematically, we write the canonical partition function in the form (P.F.) = j g(E,) eeELjkT dE,_,

(3)

where g(E,) dE, is the acluul number of energy levels of the whole system (say, 1 mole) in the range EL to EL + dE,. At any given temperature we may write (P.F.) = g(T) j emELIkTdEL, where g(T) is the average value of g(E,) for this particular temperature.

It is this

268

O.K. RICE

AND

D.R.

CHANG

quantity, g(F,C’) which we approximate by (T?L~PZ)~“~,since we assume that without this factor the energy-level density would be roughly uniform over a limited range of conditions near the critical point. 2. Local$uctuations and the signiJicance of E. As noted in section 1, local fluctuations are not strongly affected on passing through the critical point (but see section 6). If the local fluctuation is considered to be the fluctuation in a small volume, this might be expected, if only because such a small volume would be perturbed by the neighboring volumes. Over and above any such effect, it will be shown that the fluctuation in a volume, U, of the order of molecular dimensions does not depend only on the compressibility at the critical density, but is influenced by the compressibility at other densities, which does not become infinite. Let us fix attention upon a small volume ZI,and let us suppose that thermodynamic functions can be defined for it, even though it be near molecular dimensions. Suppose that there has been a fluctuation in density such that this small volume contains the centers of 6n more molecules than the number n it contains on the average. The chemical potential will then be ,u + Z,U,and the average chemical potential of the region entered by the 6n molecules will be closely equal to ,u + + 8~. The 6n molecules must have come from the rest of the system, but since there are also regions of low density it seems best to suppose that these Sn molecules have come from one of these regions, also of volume v; then the freeenergy change of the two volumes will be

(P + t Q4 an-(p

- *@4)&z = 6p6n.

(5)

We may write

where Q = n/v and (a&%),,, = v- ‘. To get a rough average value of the total change in free energy in this process we may set it equal to kT, and, if we further make use of the thermodynamic relation (a,@& = v/n, eqs. (5) and (6) yield (6n)2 w nkT (S@P)

= n’kTR/v.

(7)

It will be noted that eq. (7) is equivalent to eq. (l), except for the substitution of ji for its, and will approach eq. (1) for X~ Q v/kT. Eq. (7) will hold approximately in the regions where v is of the order of molecular dimensions even where v < x,kT if R is defined, as indicated, in terms of the finite differences SP and 6~ and so does not become infinite. Since 2 must thus itself depend upon an, this is an implicit equation for 6n. At values of v less than molecular dimensions, it must

DENSITY

AND SPECIFIC HEAT NEAR CRITICAL POINT

269

eventually give way, to3) (8n)z = n2kTxo/v,

(8)

where x0 is the isothermal compressibility of an ideal gas. The fact that v is not completely independent of the rest of the system, and the use of ordinary thermodynamic functions for a part of the system containing only a few molecules, mean that eq. (7) is not exact, but it should be a reasonable approximation and, because R cannot become infinite, eq. (7) suggests no possibility of an infinity from local fluctuations. We may inquire how small v may be before eq. (1) breaks down. We know that along the critical isotherm P - P, = a (e - ec)‘, where a is a constant and 6 is a critical exponent. The slope vanishes at e = PC, but for any isotherm above T, the slope remains nonvanishing. If we start from the critical isochore we may write P - P, = 6P and c, - PC = Se, and we assume 6P = (8P/i?e)),,c 6e + a (Se)‘.

(9)

Now the breakdown in eq. (1) may be expected to occur for a 6e such that the two parts of eq. (9) become of the same order of magnitude, for this is where SP/Se ceases to be linear. Also, because of the rapid increase of the second term of eq. (9) (the exponent 6 in general being large), this value of Fe may be considered to be a practical limit to the density of fluctuation, so the fluid can be considered to be composed of regions of the size of the v determined by this 6e E Q,, with fluctuations also determined by Q,,,. From eq. (9)

Se, = [a-l (8P/8e)T,c]1’ca-1).

(10)

We may find the corresponding value of v, which we shall call v,, by substituting into eq. (l), written in the form [since n is proportional to e and xT = e- ’ (Se/SP),] : (se)” = ekT(&/aP),

I.

(11)

V

If we solve for v and use eq. (10) we obtain v,

=

63 =

ekT!2/‘d-1’ (&$ap)(,q+c1)‘(6-1).

(12)

In eq. (12) we have assumed that v, may be set equal to E3. If we set (T - TJT, = t, then (S@P),,. behaves as twY,and if we suppose that t behaves as t-‘, then we see that 3v = y (6 + 1)/(6 -

1).

(13)

O.K. RICE AND D.R. CHANG

270

This is a known scaling relation4) for the exponent, Y,which describes the behavior of the correlation length, so this gives us some confidence that our l is, at the least, proportional to the correlation length. In two dimensions, eq. (13) becomes 2Y = y (6 + l)/(S - l),

(14)

which, with the known values Y = 1, y = 1.75 and 6 = 15, is satisfied exactly for the two-dimensional Ising lattice. There is one aspect of eq. (10) which is rather strange. It is seen that a~,,, approaches zero, although rather slowly, as t --f 0. However, if we should select some value of o less than v, , we see that 6~ would not build up as rapidly as called for by eq. (1) because ji does not remain constant. Thus within 0, we have a relatively less varying e than would be expected from (I), so it seems reasonable to suppose v, roughly proportional to P, although this might break down very close to T,. We are not attempting to get precise results so much as to get a simple physical picture. 3. Calculation of the specijic heat. The contribution AC, of the fluctuations to the specific heat in the neighborhood of the critical point may be obtained by twice differentiating eq. (2) : AC,

= -T(a’

AA/dT2),r.

(15)

We assume that dn may be calculated by extracting the square root of eq. (l), using v = C3, and in the light of the discussion of section 2 we write, for T > T, along the critical isochore, 5 = let-’ and xT = Bt-)‘. We also note that n = ~5~. We then find AA = -$kTVt3’&3

In (x~~kTBt-~-~‘&.

(16)

In differentiating this expression we may consider T, where it occurs as a direct factor, to be constant and equal to T,, and we take V = V, . We then find (where No is Avogadro’s number) A&/R

= 3~ (3~ -

x

1) (V,/&) t3Y-2&3

In (~&CT&~)

- (y + 3~) In t -

(y + 3~) (6~ 3Y(3Y-1)

I) >.

(17)

From eq. (17) we note that, if, in the asymptotic limit, C, N tma we have 3v - 2 = -a,

(18)

DENSITY AND SPECIFIC HEAT NEAR CRITICAL POINT

271

a relation which was derived as an inequality (3~ 1 2 - LX)by Josephsons), and is a consequence of scaling6). Actually it is equivalent to (13) when combined with other scaling relations. One rather remarkable feature of these considerations can be seen from eq. (16) It will be noted that the contribution to -4A vanishes at the critical point and increases away from the critical point. In other words, the effect of the fluctuations is self-limited, and the increase in the correlation length results in a smaller combinatory factor. Eq. (17) also offers the possibility of evaluating the magnitude of C,, and we shall attempt to do this in section 4. Our considerations thus far have dealt with the situation above the critical point. Below the critical point we have to take into account the fact that the fluctuations occur simultaneously in two different phases. Actually, as we shall see, this does not affect the result to first order. Below the critical point we may write the molal free energy A = X,A, + &A,,

(19)

where AI and A, are molal free energies in the liquid and gaseous phases, respectively, while X, and X, are the mole fractions of any given sample in the liquid and gaseous phases, respectively. We may write for the contribution to the molal free energy in the liquid phase, in analogy to eq. (16), ‘4A, = -+kTV,t3’&,-3

In (xe:kTB’t-Y’-3Y’~b3),

(20)

where V, is the molal volume of the liquid and the primes denote quantities evaluated below T,; there is a similar expression for AA,. Here t = (T, - T)/T,, i.e., it is taken as the absolute value. We may write V, = V, + V, - V,, but v, - V, goes to zero at the critical point, so the term in dAl containing it will go to zero more rapidly than the term containing V, and may be neglected. In other words, we may replace V by V, near the critical point. Similarly we may replace el bye,, V,by V,,ande,bye,. Thus AA, and AA, become equal to first approximation, and recalling that X, + X, = 1, we see that, dA = X, dAl + X, AA, is given along the critical isochore by the same expression below T, (except for the primes) as above T,. It is noted, however, that because of the definition of t as an absolute value, dt/dT = 1 above T, and - 1 below T,. This results in the entropy arising from fluctuations, LIS, having the opposite signs in the two cases. 4s is zero at the critical point and is positive above and negative below. Thus dC, is always positive as it must be, and it also will be given by the same expression (except for the primes) above and below T,.

272

O.K. RICE AND D. R. CHANG

This may also be seen directly (but not independently) from Widom’s suggestion2) that the mean fluctuation in a phase has, on the coexistence curve, the value necessary to give a region of the density of the other coexistent phase. Thus the relative change of density will vary with temperature as indicated by (dn)2/n2 = ct2fl where c is a constant and p is the critical exponent for the coexistence curve. Consider now the scaling relation6) 3v = y + 2j3; when substituted into the last equation together with n2 = e216 = p2&-6Y this shows (An)2 to be proportional to t-y-3v as in eq. (16). Since the dependence under the In sign in the expression for AA will be the same as in eq. (IQ it is clear that differentiating twice will again give eq. (17). 4. Application to Xe, C02, 4He and 3He. Sufficient data are available to evaluate eq. (17) for Xe7), C02”), 4Heg) and 3He10). Wherever possible we have used the values derived from light-scattering intensities. In table I, the calculated values of AC,/R (not in parentheses), obtained from the data listed, have been compared with C,*/R, the experimental value, from which has been subtracted the ideal-gas value. For Xe, CO2 and 4He the value of Cy*/R was estimated roughly from fig. 2 of Lipa, Edwards and Buckinghaml’), while for 3He we estimated C,IR from fig. 3 of Brown and Meyer12) and calculated C,*/R from it. It is noticed at once that the calculated value of A&/R is much greater than C:/R, but this may well have to do with the appropriate choice of to. The latter quantity in table I is a measure of the range of correlations and occurs in an exponential fall-off factor. This may be only indirectly related to the size of regions in the fluid of high and low density. Such a region has a fall-off around a maximum, so the appropriate value of e. to use in eq. (17) may be appreciably larger than the exponent in an exponential fall-off expression. The values in table I for Xe and CO, which are shown in parentheses were obtained by multiplying to by 6, while those for 4He were obtained by multiplying E. by 3, a reasonable factor, which reduces A&/R to a reasonable magnitude. In examining the trends with temperature we first note that, in the range of temperatures for which data are available, the logarithmic term in eq. (17) (which does not occur in some theories based on scaling or the renormalization group) has an important influence. The actual temperature range for which unequivocal values of the singular part of C, may be obtained is quite limited. It may thus be that the quoted valuesll) of 01(which have not taken into consideration the possibility of a logarithmic term) are not too meaningful. On the other hand, the temperature coefficients of dC, or the values of OLthat we will obtain from our calculations, depend strongly upon the observed values of v, and the indications seem to be that these may not be highly accurate. In the cases of Xe and CO,, subtracting the ideal-gas value from C, to get C,* does not seem adequate. It appears certain that ‘the ideal-gas value is an underestimate of the nonsingular part of C, since the molecules are vibrating in their

213

DENSITY AND SPECIFIC HEAT NEAR CRITICAL POINT TABLE I

Calculation of AC, for Xe, COz , 4He, and 3He (unprimed symbols T > T,; primed symbols T i To) Xe T, (in K) ec (in g cm-j) V

LO

(in

Al

Bx 10s (in cm3 erg-‘) AC,/R (talc.) (t = 10-J) (t = 10-b) C$R (expt.) (t = 10-3) (t = 10-d) V’ Y’ lb

(in

A)

B’ x 10” (in cm3 erg-l) AC,IR (&c.) (t’ = 10-S) (t’ = 10-b)

C$R (expt.) (t’ = 10-S) (t’ = 10-b)

4He

3He

289.7 1.105 0.58 f 0.05 1.21 + 0.03 3.0 * 0.10

304.1 0.468 0.633 f 0.01 1.219 + 0.01 1.50 + 0.09

5.20 0.070 0.66 f 0.02 1.31 + 0.02 4.2 f 0.6

3.31 0.0415 0.59 * 0.04 1.13 * 0.05 4.6 f 1.9

0.117

0.0775

2.94

18.6

351 (2.3) 991 (5.8)

970 (6.5) 1950 (11.5)

24.7 (1.09) 38:6 (1.61)

62.5 153

4 6.5 0.57 f 0.05 1.21 + 0.02 1.8 f 0.2

6 10

1.1 2.0 0.68 + 0.02 1.32 + 0.02 2.6 + 0.7

0.42 1.2 0.67 f 0.1 1.15 + 0.02 4.0 f 3

0.0283

0.82

5.18

2280 (16.8) 4730 (29.3)

65.5 (2.96) 90.5 (3.83)

28.5 40.5

12.5 18.5

4.5 6

2.6 3.4

mutual potential-energy fields. If the vibrations were truly harmonic twice the ideal-gas value should be subtracted, but this would no doubt be an overestimate. Since the amount to be subtracted is constant, the effect of this correction will be to increase the relative change of the singular part of C, with temperature in Xe and COz. Just what correction should be made for 4He and 3He is not clear, on account of the large effects of quantization. The theory seems to overestimate the difference in AC, below and above T, in the case of Xe, and to underestimate it slightly for 4He. Because of the extreme sensitivity of the theory to to it is not impossible that this could be caused in part by experimental errors in this quantity, and the large discrepancy in the 3He seems probably to arise from this cause. It may also be noted that because of the lack of complete symmetry of the coexistence curve in actual systems, the apparent value of C, may be affected by the heat of evaporation. The asymmetry

274

O.K. RICE AND D.R. CHANG

is in the direction to cause evaporation on heating, but this would become less as T, is approached, thus causing a raising and flattening of the C, vs. t curve. This is in the right direction for Xe but not for 4He. 5. The two-dimensional lattice gas. Since many of the properties of the twodimensional lattice gas (Ising lattice) are known precisely, it seems desirable to see how our theory works out in this case. Here t”’ must be substituted for t3” and 6: for tg in eq. (16); and of course V is an area and Q is a surface density. We consider a square lattice and take the distance between adjacent lattice points as the unit distance. Then e is equal to the fraction of lattice points occupied; ee = 4. Ywill b e equal to N,&, where No is Avogadro’s number, so that V, = 2N,, . For the two-dimensional lattice Y = 1 and y = 5. Thus, in place of eq. (17) we will have A&/R

= 25;’ [ - (15/4) In t + In (7&T&/4)

- (45/g)].

(21)

For the square two-dimensional lattice to1 = 1.76 above T,.Sufficiently close to the critical point only the first term in the bracket will be important, and it gives the expected logarithmic dependence on t. The other terms will presumably not be of importance until t is large enough so that L?is sufficiently small so that the approximations will not be expected to hold. Eq. (21) gives A&/R

w -23.2

In t,

(22)

while the exact result of 0nsager13) is -2 x 0.49 In t/mol.Thus, as in the threedimensional case, E needs to be taken larger than the value which is commonly designated as the correlation distance; in this case the factor is about 5. In the light of the symmetry of the specific-heat curve above and below the critical temperature, it appears that the value of to to be used in eq. (21) should be the same above and below T,. However, it is known that the correlation length is only half as large below T, as above14). It should be noted, however, that the correlation length is calculated for the magnetic Ising lattice by considering the correlation between spins, and no restraint is placed upon the number of spins in either direction. Thus, below T, the net magnetization can be in either direction. On the other hand, we are interested in the propagation of a fluctuation from a state of fixed average density (equivalent to a state of fixed total magnetization) at the coexistence curve. It is difficult to decide whether this can make the difference of a factor of two in the correlation length. 6. The interdependence between short-range and long-range order. We may end this discussion with some remarks about short-range order, and how it is controlled by the long-range effects which occur at the critical point. In considering this

DENSITY

AND SPECIFIC HEAT NEAR CRITICAL POINT

275

question we shall confine ourselves to classical systems in which the interatomic potentials are described adequately by pair potentials, but the extension to a more general case will be evident. In such a system the mutual potential energy per molecule is given by the expression GIN=

$eJelZglZdrz,

(23)

where &I2is the pair potential, g,, is the pair distribution function and the integral is taken over all possible positions of one of the molecules 1 and 2. The singular part of the specific heat will be contained in the derivative of this expression with respect to T. Since al2 is short-ranged, it is seen that only the parts of g,, for small rl; (where r12 is the distance between the two molecules) contribute in any important way to the integral. Yet the calculations of the proceeding sections of this paper depend upon the behavior of the long-range parts of g,,. How is this apparent contradiction to be reconciled? The reconciliation depends upon the general thermodynamic equation c, =

(aElan, = T (as/az-jv.

(24)

It will be noted that the calculation of the specific heat depended upon the entropy, which in turn depended upon the number of available states in the system, which was determined near the critical point by the long-range fluctuation. But eq. (24) represents a condition of equilibrium. The short-range part of the distribution function has no choice but to adjust itself in such a way that eq. (24) is fullilled. Just how it will do this is impossible to see in detail, so any rational attempt to calculate the thermodynamic quantities near the critical point will consider long-range effects and ignore short-range effects. This is not merely a surmise, but is based on thermodynamic principles, and is the basis for treatments using the renormalization group. The role of a minimum length has been discussed in detailr5), yet the thermodynamic grounds for such a role, arising from the necessity of the energy to conform to the entropy, appears not to have been stressed. This discussion may also be phrased in a slightly different way. The Helmholtz free energy A = E - TS must be a minimum under all possible variations of the state of the system which take place at constant volume and constant temperature. If the entropy is principally determined by the long-range fluctuations we may consider only variations which leave the long-range situation essentially unchanged. Then the singular part of the entropy is also fixed. The variations in the shortrange part of g12 must be such as to make A a minimum. Furthermore there must be some restrictions upon the possible short-range variations Sg12, inasmuch as they must be consistent with the condition that the density remains tixed. The determination of 6g, ; and hence E becomes, therefore, a problem in the calculus of variations.

276

0. K. RICE AND D. R. CHANG

Acknowledgements. We thank Professor H. Meyer for assistance in collecting the data, and Professor B. Widom for some helpful correspondence. We are grateful to the National Science foundation, the Army Research Office (Durham), and the Material Research Center of the University of North Carolina (under a grant from the National Science Foundation) for financial assistance.

REFERENCES 1) Fixman, M., J. them. Phys. 36 (1962) 1957. 2) Widom, B., J. them. Phys. 43 (1965) 3892. Fisk, S. and Widom, B., J. them. Phys. 50 (1969) 3219. Widom, B., Lecture given at the Van der Waals Centennial Conference on Statistical Mechanics, Amsterdam, 1973. 3) Derived by Omstein, L. S. and Zemike, F., See Rice, 0. K., Statistical Mechanics, Thermodynamics, and Kinetics, W.H.Freeman and Co. (San Francisco, 1967) p. 349f.; regarding the breakdown of eq. (1) see pp. 226,232, 349, 354f. 4) Gunton, J.D. and Buckingham, M. J., Phys. Rev. Letters 20 (1967) 143. 5) Josephson, B.D., Proc. Phys. Sot. 92 (1967) 269, 276. 6) Kadanoff, L.P., Physics 2 (1966) 263. 7) Giglio, M. and Benedek, G.B., Phys. Rev. Letters 23 (1969) 1145. Smith, I. W., Giglio, M. and Benedek, G.B., Phys. Rev. Letters 27 (1971) 1556. 8) Lunacek, J.H. and Cannell, D.S., Phys. Rev. Letters 27 (1971) 841. 9) Kagoshima, S., Ohbayashi, K. and Ikushima, A., J. low Temp. Phys. 11 (1973) 765. Roach, P.R., Phys. Rev. 170 (1968) 213. 10) Ohbayashi, K. and Ikushima, A., to be published. Wallace, Jr. B. and Meyer, H., Phys. Rev. A2 (1970) 1563. 11) Lipa, J.A., Edwards, C. and Buckingham, M. J., Phys. Rev. Letters 25 (1970) 1086. 12) Brown, G.R. and Meyer, H., Phys. Rev. A6 (1972) 364. 13) Onsager, L., Phys. Rev. 65 (1944) 117. Fisher, M.E., Rep. Prog. Phys. 30 (1967) 615, table II. 14) Kaufmann, B., Phys. Rev. 76 (1949) 1232. Fisher, M.E., ref. 13, section 6.2. 15) Ma, S., Rev. mod. Phys. 45 (1973) 589.