The Journal of Physical Chemistry, Vol. 83, No. 14, 1979
Thickness of an Interface Near the Critical Point
2(26 - 1)/(6
+ 1) = (y + v ) / v
From eq 8a 2(26 - 1)/(6
+ 1) = 3 - 7
(34) (35)
which turns out to be equivalent to eq 10. If, instead of (31), we had followed (a P / ~ P along ) ~ the coexistence curve we would have obtained (y’+ v ’ ) / v ras the exponent in eq 33, and can thus, from this point of view, also write y ’ / v r = y/v. We may go one step further and make a comparison of the behavior of (a 3P)a p3)T and a 3P/aTa p2, assuming (a 2P/ap2)T depends only on 5. On the basis of the approximate equation of state holding along the critical isochore, with subscript c denoting values a t the given temperature but at p = pc: IP - PcI= I(aP/ap)T,,(p - PJI
+ a l p - pel* (36)
upon which eq 30 and 31 are based, we conclude that a 3P/aT a p 2 is independent of T - T, to first order, while, of course, ( a 3 P / a p 3 ) T Ip - pf3. Then setting up equations analogous to eq 30 and 31 we eventually find the relation v = (6 + 1)/(36 - 6) (37) It will be seen that if 6 = 5 then v = 2/3, and with 6 slightly less v will be slightly greater. Thus this gives slightly too large a value of v. We must conclude either that eq 36 needs slight modification, or else that other factors than
1863
5 have an effect on the higher derivatives. It seem likely that the fact that the values of v calculated in this way are as close as they are is accidental, for if the same calculation is made for two dimensions [the only difference being that in this case X = (6 + 1)/2], the value obtained for v, using the known value of 6 = 15, is 8/13, whereas v should be 1. On the other hand, if the calculation based on the assumption that (aP/ap), depends only on 4 is worked out for two dimensions (or for four dimensions) the correct value v = 1 (v = 1/2 for four dimensions) is obtained from the analogue of eq 34. References and Notes (1) 6.Widom in “Phase Transitions and Critical Phenomena”, Voi. 2, C. Domb and M. S. Green, Ed., Academic Press, New York, 1972, Chapter 3. (2) 8. Widom, Pbysica, 73, 107 (1974). (3) 0. K. Rice and D. R. Chang, Pbysica, 74, 266 (1974); 78, 490 (1974); 81A, 161 (1975). (4) 0. K. Rice and D. R. Chang, Physica, 78, 500 (1974). (5) M. E. Fisher, J . Math. Pbys., 5, 944 (1964). (6) 0. K. Rice, J . Pbys. Chem., 81, 1388 (1977). (7) J. D. van der Waals, Z. Pbys. Cbem., 13, 657 (1894); J. W. Cahn and J. E. Hiiiiard, J. Cbem. Pbys., 28, 258 (1958). (8) M. E. Fisher, Pbys. Rev., 180, 594 (1969). (9) C. Warren and W. W. Webb, J . Cbem. Pbys., 50, 3694 (1969). (10) S. Fisk and B. Wdom, J . Cbem. Pbys., 50, 3219 (1969); 0. K. Rice, ibid., 64, 4362 (1976). (11) R. B. Griffiths, J. Cbem. Pbys., 43, 1958 (1965). (12) K. E. Wilson, Pbysica, 73, 119 (1974). (13) 0. K. Rice and D. R. Chang, Pbysica, 83A, 609 (1976). (14) 0. K. Rice, J. Low Temp. Pbys., in press. (15) 0. K. Rice and D. R. Chang, Pbysica, 83A, 18 (1976).
Existence of Two Characteristic Lengths in Determining the Thickness of an Interface Near the Critical Point, and the Interface Profile 0. K. Rice Department of Chemistry, University of North Carolina, Chapel Hi//,North Carolina 27514 (Received December 15, 1977)
In the van der Waals, Cahn-Hilliard theory of interfacial tension there are two equal contributions to the interfacial tension, ul,which may be calculated from local thermodynamic functions, and u2, which involves density gradients. Expressions for u1 and u2 involve the thickness Az of the interface, but there appears to be a slight difference in the definition in the two cases, since the respective thicknesses (Az, for u1 and Azz for u2) behave slightly differently near the critical point. Examination with the aid of an approximate model profile indicates that Az, is related to the exponential drop off at the edge of the profile, whereas Az2 is related to the slope of the central part of the interface, with Azl always greater than Azz. The critical exponents are discussed.
In the van der Waals, Cahn-Hilliard theory of interfacial tension1 the assumption is made that the Helmholtz free energy per molecule 4 may be divided into two parts 4 = 41 + 42 (1) Where, in a one-component (liquid-vapor) system, to which we shall confine our discussion, 41is the local free energy, a function of the molecular (number) density p at a particular point in the system, whereas & depends upon such quantities as dp/dz and d2p/dz2, where z is the distance coordinate in the direction normal to the surface. z is measured from a geometrical surface defined for a plane interface by the equation
where pv and
p1 are
the densities of the bulk vapor and 0022-3654/79/2083-1863$01 .OO/O
liquid phases, respectively. In order for the free energy of the surface to be a minimum it is necessary that 41 and qb2 be equal. Thus we can analyze the interfacial tension in terms of either one. It is possible to evaluate u1 and u2 by means of integrals involving the surface quantities.2
Here the subscripts 1 and v always refer to the liquid and vapor states, respectively. The quantity 6z is an average distance defined to take care of the effect of the density gradient on a molecule situated in the gradient. The latter molecule is supposed to be affected by the molecules on 0 1979 American
Chemical Society
1864
0. K. Rice
The Journal of Physical Chemistty, Vol. 83, No. 14, 1979
I,
I
I I
I
- 3AZ/8
3AzIi3
z+
Figure 2. Modified interface profile. In the second modlfication the quantity here shown as Az becomes Az2.
portion will not be an exact straight line, and to connect smoothly on the exponential decay part we will need a different coefficient in the exponential of eq 9 and 10. Let us still approximate the central part of the profile by a straight line, calling the Az defined in this way (Le., as in Figure 2 and in eq 7 and 8) Azz, and we replace eq 12 by bo = A21/8 (13)
The actual value for u2 is probably a compromise between eq 4 and 5. At this point we made a simplification in the density profile: we assumed that dp/dz is constant over a distance Az between p1 and pv (see Figure 1) and wrote dp/dz = (PI - pV)/Az (6) When we did this we found that u1 was proportional to Az and u2 to (Az)-l. Furthermore, we found that to conform to the usual scaling laws it was necessary for Az to behave slightly differently in the two cases as the critical point was approached. This could not, of course, happen if eq 6 were exactly correct. In order to analyze this situation better we shall attempt to see how modifications of eq 6 will affect the evaluation of the integrals. A first suggestion is that we retain the straight line for p vs. z for a certain range, say for three-fourths the distance pv to p1, Le., from pv + Ap/8 to p1 - Ap/8, where Ap = p1 - pv, and add an exponential drop off, as in Figure 2. This means that we write p - pv = Ap/2 z Ap/Az (7) z = - 3 A ~ / 8 to z = 0 pi - p = Ap/2 - z Ap/Az (8) z = 0 to z = 3 A 2 / 8
+
p - pv = aoez/bo z = -m to z = - 3 8 ~ / 8
p1 - p = aoe-z/bo z = 3 ~ 2 / ato z =
(9)
(10)
where a. and bo are constants to be determined. If we assume that the portions join on smoothly and with uniform slope at z = f3&/8, a. and bo may be determined, and it is found that a. = e3Ap/8 (11) bo = A z / ~ (12) As a matter of fact, this modification of eq 6 will not suffice to bring out any difference in the behavior of the Az’s as they appear in u1 and u2. However, the modified profile is still a very rough approximation. The central
As we shall see, in order to have a consistent picture, Az, and Az2 will have to behave slightly differently as the critical point is approached. In general, we may expect the actual curvature of the nearly straight-line midsection of the profile to be such that the slope with which it connects to the exponential drop offs is less than it is a t z = 0. If the exponential part connects on smoothly, this would then make Azl > Az2 under all circumstances, an important point for the understanding of the phenomena. Of course, this second modification of the profile is still an approximation, but it seems probable that it is sufficient to give the essential features, in particular, to show that there is a characteristic distance associated with the central part of the profile and a different one associated with the exponential drop off at the ends. In order to evaluate the integrals in eq 3 and 5 we need to know 41 and tl as functions of p. We shall start with an assumed equation-of-state, which gives the pressure P consistently with the observations about the critical point. We have
P = P o + Pp,o(P - Pc)
f
alp - PCT
(14)
In these equations Pp = (aP/apIT; Pt = (aP/aT),; P,, = a2P/apaT; pc is the critical density; a is a constant; the subscript 0 indicates a value at some arbitrary temperature but along the critical isochore, i.e., p = pc; and 6 is a critical e ~ p o n e n t .In ~ eq 14 the upper sign is for p > pc and the lower sign for p pc; it also is to be noted that P,,oa It!’, where y is a critical e ~ p o n e n tand , ~ that the sign is positive for t > 1,but negative for t C 1,the region which interests us. We can write
41 - 41,l =
Spppp-2 dP
We will now divide up the intervals in eq 3 to reflect our model of the interface profile. Thus the limits on the (now) four integrals are ---a, to -3Az2/8, -3k2/8 to 0,O to 3Az2/8, and 3Az2/8 to m. Also we shall change the variable of integration from z to p, using eq 7-10 to obtain dz/dp. Thus
The Journal of Physical Chemistty, Vol, 83, No. 14, 1979
Thickness of an Interface Near the Critical Point
Looking again at eq 7-10 and 13, we may write this as
“s
dP Ap/2
dP +
+
p,+AP/8
SP1
“1
Pmbr-P;41’l dp (19)
PYAPI8
The integrands could in principle be evaluated from eq 14 and 16, but the algebra is involved and we will not go through it. Suffice it to remark that the integrals will be functions of q51,v, q51,1, pv, pl, and pc only, and that the exponential part and the central part are multiplied respectively by Az, and Az,. Though the integral for the central part has the greater range and the greater coefficient, it is seen that the denominator p - pv, in the region near pv, will be much smaller than the denominator Ap/2 which occurs in the integral for the central part. Thus, this would enhance the relative importance of the exponential part of the integral. In any case, we shall assume that Az, is more important than Az2 in determining the thickness of the interface, in view of the fact that we have noted that we expect Azl to be always greater than Az,. This means, of course, that if they do not have the same exponents Az2 will become negligible a t T,. We shall assume that Az,behaves like the correlation length 5 (i.e., has the same critical exponent), whereas when we discuss u2 we shall find (and, indeed, have found in previous work4) that Az, has to behave as tl-7, where q is a small positive quantity, one of the critical exponent^.^ We shall now turn to eq 5. Starting with the general thermodynamic equation (a Cl/d
P)T
It is seen that the situation is exactly the reverse of what was found with eq 19. The factor Ap/2 as compared with p - pv now favors the central contribution compared to the exponential. More important, it is noted that (Azl)-l and (Az2)-l appear separately and respectively in the exponential and central parts, and if Azl behaves as [ and Az2 behaves as (l-7, then the central part becomes definitely predominant as the critical point is approached. The central, straight-line part can be handled in the same way as the complete straight-line profile was in our earlier paper. If Az2 behaves4 as [l-v or as (tl-Y(lT) then we obtain from u2 the scaling law p = 2j3 v‘ - v‘q (22) where the quantities are the usual critical exponents ( p being for the interfacial tension), and this is consistent with the scaling law5 obtained from u1 p = y’ 2p - Y‘ (23) and the scaling law3 y’= 2v’ - v’q. We are now able to see a physical picture for the idea that there are two characteristic lengths connected with the thickness of an interface, which behave differently from each other, though they are still closely related. That there are two divergencies involved is not new, for it is well known that u2 can be expressed in terms of the second moment of the direct correlation function, and it is possible for the latter to diverge.2 Our model profile shows how two lengths can be involved, associates u1 and Az, which comes from the exponential fall off part of the profile, and uz with Az2 from the central part of the profile. Of course, as we have already noted, our profile is itself an approximation. We cannot expect that in reality Az, and Az2 will be as clearly separated as we have depicted them, but we do believe that the essence of the situation is brought out by our approximate profile. Since 11 is so small in three dimensions, it is seen that the behavior of Az, and Az2 will differ very slightly, and the experimental data are not yet good enough to detect any difference, and we may expect eq 6 or Figure 1to be in many respects an adequate approximation for the profile. In two dimensions the difference will be somewhat greater, since q = 114. In four dimensions q = 0, so Az, Az~. In the region near the tricritical point of 3He-4He mixtures a rather different situation occurs,6 since in this case q = 1. Applied directly, this would mean that Azz is independent of 5, which is different from our previous conclusion,’ although the calculation of Az2 (called there Az) goes through as before. Az2 i= 80 A, the difference being that now it is temperature independent, and of course is the length only of the central, almost straight-line part of the interface. We are not sure that this is the proper procedure, especially in view of the extreme asymmetry of the thermodynamic functions near the tricritical point. It would appear that it would be of considerable interest to investigate the interface profile near the tricritical point by optical means.8
+
+
P41 - Pv41,v
Pc
1865
= P-2[P - T(aP/aT)I
(20)
we could use eq 14 and 15, along with eq 7-10 and 13 to evaluate eq 5. Again this is a very tedious process algebraically, and we shall merely divide the integral into the appropriate intervals and use eq 7-10 and 13 to find dp/dz, which, it will be noted, occurs instead of dzldp in eq 5. We have
References and Notes (1) J. D. van der Waals, 2.fhys. Chem., 13, 657 (1894);J. W. Cahn and J. E. Hilllard, J. Chem. fhys., 28, 258 (1958). (2) 0.K. Rice, J . fhys. Chem., 81, 1368 (1977).Errata: Above eq 1.13,6 x should be 6z; in eq 1.17,d$ ,/dp should be a$ Jap. (3) E. H. Stanley, “Introduction to Phase Transitions and Critical Phenomena”, Oxford University Press, New York, 1971,Chapters 3 and 4. (4) 0. K. Rice, J . fhys. Chem., preceding article In this issue. (5) B. Widom, J . Chem. fhys., 43,3892 (1965). (6) P. Leiderer, H. Poisel, and M. Wanner, J . Low Temp. fhys., 28, 167 (1977). (7) 0.K. Rice, J . Low Temp. fhys., 29,269 (1977). (8) J. S.Huang and W. W. Webb, J . Chem. fhys., 47,3694 (1969).
