Fluctuations Near the Critical Point of One-Component Fluids (7) (a) E. J. R. Sudholter and J. B. F. N. Engberts, Red. Trav. Cbim. Pays-Bas, 96, 86 (1977); (b) work to be published. (8) (a) J. Oakes, J. Cbem. Soc., Faraday Trans. 2, 68, 1464 (1972); (b) J. H. Fendler and L. J. Liu, J. Am. Cbem. SOC.,97, 999 (1975); (c) H. Okabayashi, K. Kitrama, and M. Okuyama, 2. Naturforsch. A , 32, 1571 (1977); (d) K. Kalyanasundaram and J. K. Thomas, J . Am. Chem. Sm.,99,2039 (1977); J. Pbys. Chem., 81,2176 (1977); (e) B. B. Craig, J. Kirk, and M. A. J. Rodgers, Cbem. Pbys. Lett., 49, 437 (1977); (f) J. Llor and M. Cortijo, J . Chem. SOC.,Perkin Trans. 2, 73, 1111 (1977); (9) M. S. Fernlndez and P. Fromherz, J. Pbys. Cbem., 81, 1755 (1977); (h) P. Stllbs, J. Jermer, and B. Llndman, J . Colloid Interface Sci., 60, 232 (1977). (9) M. A. J. Rodgers, M. F. da Silva, and E. Wheeler, Cbem. Pbys. Lett., 43, 587 (1976); 53, 165 (1978). (10) P. Mukerjee, Adv. Collold Interface Sci., 1, 241 (1967). (11) C. Tanford in ref 6, Vol. 1, p 119. (12) Y. Moroi, T. Oyama, and R. Matuura, J. ColloidInterface Sci., 60, 103 (1977). (13) K. Shlnoda, Bull. Cbem. SOC.Jpn, 26, 101 (1953). (14) M. Corrin, J. ColloldInterface Sci., 3, 333 (1948). (15) Compare (a) E. M. Kosower and P. E. Klinedinst, Jr., J . Am. Chem. Soc., 78, 3493 (1956); (b) E. M. Kosower, ibid., 80, 3253 (1958); (c) E. M. Kosower, “An Introduction to Physical Organic Chemistry”, Wiley, New York, 1968. ( 16) Employing probe molecules, several workers have previously estimated Zvaiues for_microenvironments in micelles and polymers. See ref 8 b and P. Strop, F. Mike& and J. Kllal. J . Pbys. Cbem., 80, 694 (1976). (17) E. M. Kosower and J. A. Skorcz, J . Am. Cbem. Soc., 82, 2195 (1960). (18) P. Mukerjee and A. Ray, J . Phys. Cbem., 70, 2138, 2144, 2150 (1966).
The Journal of Physical Chemlstty, Vol. 83, No. 14, 1979
1859
(19) P. Mukerjee, J. R. Cardinal, and N. R. Desai in ref 6, Vol. 1, p 241. (20) Precipitation of 1 occurred above [NaI] > -0.1 M. It is of interest to note that for the CT complex of 1-dodecylpyrldlnium iodlde e is also about 1600, see ref 18. (21) One could imagine that the c values are partly affected by 1:2 donor-acceptor interactions because of the excess of positively charged ions at the CT sites in the Stern layer. However, the fact that the line shapes of the CT bands of ion pairs of 1 in chloroform and in micelles are nicely matchable does not support this possibility. (22)’ L. S. Romsted in ref 6, Vol. 2, p 509. (23) It is assumed that cmc values are negligible as compared wlth stoichiometric concentrations of 1. (24) P. Hemmes, J. N. Constanzo, and F. Jordan, J . Pbys. Cbem., 82, 387 (1978). (25) (a) I. Cohen and T. Vassiliades, J . Pbys. Cbem., 65, 1774, 1781 (1961); (b) E. W. Anacker and H. M. Ghose, ibkf., 67, 1713 (1963); (c) P. Mukerjee, K. J. Mysels, and P. Kapauan, ibu., 71, 4166 (1967); (d) A. Ray and G. NBmethy, J . Am. Cbem. Soc., 93, 6787 (1971); (e) J. W. Larsen and L. J. Magid, ibid., 98, 5774 (1974). (26) F. Hofmeister, Arch. Exp. Patho/. Pbarmacol., 24, 247 (1888). (27) H. L. Friedman and C. V. Krishnan In “Water, A Comprehensive Treatise”, Vol. 3, F. Franks, Ed., Plenum Press, New York, 1973, Chapter 1. (28) W. F. McDevit and F. A. Long, J. Am. Cbem. Soc., 74, 1773 (1952). (29) (a) C. A. Bunton, M. J. Minch, J. Hidalgo, and L. Sepulveda, J. Am. Chem. Soc., 95, 3262 (1973); (b) C. A. Bunton and M. J. Minch, J . Pbys. Cbem., 78, 1490 (1974). (30) J. Steigman, I. Cohen, and F. Spingola, J. Colloid Interface Sci., 20, 732 (1965). (31) M. J. Blandamer, Q . Rev. Cbem. SOC.,24, 169 (1970). (32) T. S. Sarma and J. C. Ahluwalia, Cbem. SOC. Rev., 2, 203 (1973). (33) N. Nishikido and R. Matuura, Bull. Cbem. Sac. Jpn., 50, 1690 (1977).
Fluctuations, Density Gradients, and Interfaces Near the Critical Point of One-Component Fluids 0. K. Rlce Department of Chemistry, University of Nortb Carolina, Chapel Hi//, Nortb Carolina 27514 (Received November 14, 1977)
In this paper we make use of the analogy between boundaries of fluctuations and interfaces between phases. Fluctuations involve an increase in local free energy from the changes in density involved; there is also a contribution to the free energy from the gradient in density and proportional to the square of the gradient. Evidence is now given that these parallel each other, which means that the gradient term contains a factor Eq, where [ is the correlation length and a critical exponent. Interfaces also involve free energies arising from the local density changes and from the gradient; these must be equal to each other. These two terms involve the thickness Az of the interface, but in each case the Az is defined slightly differently. If Az, is the value for the local density part and Az2 for the gradient part, then Azl a: [ and Azz 0: E’”. This factor which occurs in the expression for the gradient free energy in the fluctuations can also be attributed to a similar behavior of the thickness of the boundary layer between fluctuations. It is also shown that various scaling laws for the exponents can be derived from the relation between fluctuations and interfaces. It is indicated that the compressibility depends only on 5, but this is not true for other thermodynamic functions.
1. Introduction Widom1i2 has pointed out the similarity between the boundaries between fluctuations in density of different direction and the interfaces between phases in equilibrium, and has shown how a number of scaling relations can be obtained from the use of this analogy. We wish to consider this resemblance in somewhat more detail. This will enable us to see some of the scaling relations in a slightly different light, and examine some scaling relations which have not been examined from this point of view. In recent papers we have considered the relationship of fluctuations and thermodynamic properties near the critical point of fluid^.^ Fluctuations are self-limiting, because a fluctuation near the critical point takes the
portion of the fluid in the fluctuation away from the critical region, thus driving it to a condition with much smaller compressibility. The standard formula for the fluctuation 6p in the density p over a volume u where KT is the thermal compressibility, breaks down where the compressibility changes, i.e., where the linearity of the equation-of-state breaks down. The 6p where this occurs can then be substituted back into eq 1and the latter solved for u , giving a value, say, u, = t3, If E, then, is identified with the correlation length, values are found which are very close to optically observed values, and the exponent Y in [ = tot-”obeys standard scaling laws. Here tois a constant
0022-3654/79/2083-1859$01.00/0 0 1979 American Chemical Society
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The Journal of Physical Chemistty, Vol. 83, No. 14, 1979
and t = IT - T,I/T,, where T, is the critical temperature. It should, however, be noted that values of [ several times larger are needed to give a reasonable description of the specific heat. The fact that the fluctuation in density cannot rise to the value given by eq 1if u is very much smaller than urn makes the fluctuation more nearly uniform over the volume u, and is what gives rise to the correlation. The fluctuation occurs against an increase in free energy in the volume u, which will not in general greatly exceed kT (though a decrease in free energy is related to the number of states of the system crossed over in the course of the fluctuation). The rise in free energy is associated with local changes in density, but also with local gradients of density, there being a contribution to the free energy density which to the lowest order is equal to Q(VP)~,where Q may depend on temperature and pressure. Recently we have investigated whether such a term parallels the density fluctuation terms4 For such parallelism we concluded that Q 4" where 17 is equal to the critical exponent defined by F i ~ h e r which ,~ is related to the spatial decay of a fluctuation near the critical point. One of our objectives is to discuss in somewhat more detail reasons for believing that the parallelism should exist. We shall attempt to show that it closely related to what may be called a principle of geometrical similarity of the fluctuations. More recently we have considered the interfacial tension between two coexisting phases: which is a related problem, since the boundaries between fluctuations of different sign must resemble interfaces. The interfacial tension according to the van der Waals, Cahn-Hilliard theory' can be considered to consist of two parts. The first part is obtained by integrating across the surface the difference of the local free energy of the system, considered in a one-component system, as a function only of the density at any point, and the free energy of the bulk phases. The second part is obtained by integrating an expression proportional to the square of the density gradient, ( V P ) ~ . These two parts have to be equal to each other, to make the interfacial tension a minimum. We defined,7somewhat arbitrarily, the thickness of the interface Az The local free energy part of the interfacial tension is proportional to AZ, while the gradient part is proportional to l / A z . It was found that actually slightly different definitions of Az are implied in the two parts; in fact, they behave in slightly different ways as the critical point is approached. We will wish to explore the relationship of this fact to the behavior of Q. 2. Fluctuations and Scaling Laws Let us return now to a consideration of the properties of the fluctuations. If Q ( V P ) ~is a positive term, the possibility of gradients also works to reinforce the tendency toward uniformity of density over a range approximating 4. Lack of uniformity would mean stronger gradients and more of them, thus increasing the free energy. Indeed, if we should arbitrarily choose to consider a volume u > u,, this volume splits up into more or less independent fluctuations because of the high free energy required to maintain a volume u >> urn at a reasonably uniform density. Se we may conclude that just around urn the gradient free energy becomes of the same order of magnitude as the density-fluctuation free energy, and thus we understand why they will parallel each other. We may
0.K. Rice
look upon the gradient free energy in two different ways. In the first way we may say that Q( ( 0 ~ represents ) ~ ) the average free energy density due to gradients in the volume urn. Thus, the gradient free energy per fluctuation is given by f m = Q((VP)')Urn (2) On the other hand we may consider that within the fluctuations there are interfaces somewhat resembling the interfaces between two phases, and that these interfaces have an average effective interfacial tension Q. Furthermore, near the critical temperature where the fluctuations are large these interfaces will be nearly planar. Under such conditions it is known that, at equilibrium, half of the interface free energy is gradient free energy. Thus we may write fm
am
=
(3)
where a, is the average area of these surfaces within a fluctuation. Widom2 has made an equivalent conjecture. Now, as we have seen, in all fluctuations whose onedimensional extension is equal to or near the correlation length the interface or density free energy is about equal to the density-fluctuation free energy. Furthermore, the total free energy of fluctuation is always about kT per fluctuation. We also recall from the theory of interfacial tension that the profile of the interface is determined by the density difference of the two phases which have the same free energy.lr7 In this case we have a situation which is somewhat similar. A fluctuation of given size in either direction (toward greater or smaller density) involves essentially the same change in free energy. Therefore, we are dealing with an equilibrium between denser and less dense regions which is something like an equilibrium between two phases. In the case of two phases the surface area, which involves an overall increase in free energy, tends to be as small as possible. In the case of fluctuations there is a tendency for the fluctuations to be as small in volume as possible since this involves an increase in disorder. However small fluctuations mean a relatively large total surface, so these tendencies balance in the actual state of the system. Because of the various influences which tend to make the fluctuations more or less uniform over a certain region, we may expect the thickness of the boundary regions to be less than the linear dimensions of the fluctuations (thus enhancing their resemblance to interfaces), but still more or less of the same order of magnitude, and presumably varying directly with the linear dimensions of the fluctuations. (This statement will need some modification later, but can be accepted for the present.) These expectations can be expressed mathematically as follows: u, = [3
(4)
am
( 5)
VP
- t2
-
6P/E
(6)
-
where a, is the area of the surface of the fluctuation and where the sign implies both proportionality and an approximate concordance in order of magnitude. We assume that the geometrical character of the fluctuations is similar throughout the critical region. From this we infer that their effects on the free energy can a t most vary with t , and we write, as already noted
Q
= QoE"
(7)
By definition of the exponents, t = tot-"along the critical isochore (T > T,) and E' = along the coexistence curve
The Journal of Physical Chemistry, Vol. 83, No. 14, 1979
Fluctuations Near the Critical Point of One-Component Fluids
(T < T,), and from eq 1 ( 6 ~ a): tW3v ~ or t-y’+3v’ above or below T,. Since f, x hT is essentially independent of t , eq 2 gives, using (6) for T > T, -qv
for T
+ 2v - y = 0
< T, (along coexistence
curve)
+ 2v’ - y‘ = 0
-qv’
(84 (8b)
Equation 8b is a known scaling relation for the exponent q which governs the average decay with distance of a fluctuation at the critical point,3 and so can be used to identify the latter q with one we defined in eq 7. It is of interest that the assumption that q has the same value in both these equations requires that v’/y‘ = v/u. To apply eq 2 along the critical isotherm we let [ = tprA, where r = Ip - p,l/p,, and then ( 6 ~ a) r-(S-1)+3A. ~ Again Q a 47. This gives
+ 2x - (6 - 1) = 0 (9) Earlier3 it was found that A = (6 + 1)/3, which combined -qx
with eq 9 gives 6 = (5 - d / ( 1
+ 17)
(10)
also a known relation,8 involving Fisher’s exponent 7. From eq 3 we can get a further relationship, which, however, has already been derived in somewhat similar fashion by Widom.2 In general, surface tension u is stated to be proportional to t h ’ , where p’ is a critical exponent. Usually this is written as p , but since it holds along the coexistence curve below T,, we prefer to add the prime, in particular as we must also consider interfacial tension within the fluctuations above T,. Equation 3 then gives, along with eq 5
(W
p--2v=O I*‘ - 2v’ = 0
(Ilb)
Equation 1la cannot, of course, be checked experimentally, but numerical values for p’ and v’ (or p’ and v, assuming v = v’) are in rather good agreement with ( l l b ) , although this has been questioned in one caseag
3. Fluctuations and Interfaces The above relationships are based upon the assumptions that boundaries between fluctuations are similar to interfaces. If this is true the relationships should be able to tell us something about interfaces. We have remarked that the local-free-energy term in the interfacial tension (called ul) is proportional to Az, the thickness of the interface. From the relationships involved it was possible to infer thatlo p’ = y’
+ 2p
-
u‘
(12)
It is possible to obtain eq 12 in an entirely different way by consideration of the fluctuations, again by assuming that the boundaries between fluctuations resemble interfaces. We start with the Griffiths inequality’l y’Ip(6 - 1) (13) This is believed to be an equality, which in effect states that the behavior of the compressibility is, in the limit, the same along the coexistence curve as along the critical isotherm. From eq 8b the Griffiths inequality may be written v’(2 - q ) Ip(s - 1)
which from eq 10 gives
3v’ Ip(6
+ 1) = 3px
1861
(14)
If (14) is an equality it means that in the limit the behavior of the correlation length is the same along the coexistence curve and the critical isotherm. Let us now set Po = ?’/(a - 1) = 3v‘/(6 1). By eq 1, ( a P ) 2 0: t-y’+3v’ , and we see that ( 8 ~ ) ~P o , so that 6 p parallels the coexistence curve if p = Po (Le., if eq 13 and 14 are equalities) as it should according to Widom.2 Now suppose that, actually, p > Po. Along the coexistence curve it is still true that ( 8 ~ a) t2@o, ~ by the definition of Po, but now p - pc a tfl along the coexistence curve. Thus as we approach the critical point the ratio I d p l / ( p p,) becomes very large. It does not seem reasonable to suppose that fluctuations in density much greater than the difference in density of two phases which can be in equilibrium could be sustained, so we assume that the equality is correct. Equation 10 can be combined with (13) as an equality to give
+
7’= 2P(2 - d/(l + 7)
(15)
We can then eliminate 7 between eq 8b and 15, and by using eq l l b we again find eq 12. The density-gradient part of the interfacial tension (called u2) has been shown6 to be ~2
+
= l/2(6~)’(P bT dP/dT)(Ap)’/p;
AZ
(16)
Here 6z is a distance of the order of magnitude of molecular distances which measures the distance one must go in the direction (or opposite) of the gradient to get a good average of the effects on a specified molecule of the molecules on that side of the gradient. P is the total pressure, pc the critical density, and Ap the difference in density of the two phases. According to eq 16, since from its physical meaning 6z should remain finite at the critical point, we would have p’ = 2p + v’ (17) which, if Az is exactly the same quantity in the two equations, would be consistent with eq 12 only if y’ = 2v’. From eq 8b this would mean that q would have to be equal to zero. However, the thickness of the interface comes into and g2 as an average in certain integrals. Thus u1 comes from a simple integral of the excess free energy over the distance z from the Gibbs surface, and this results in an expression which, as we have seen, puts Az in the numerator (we shall call this Azl). On the other hand, the integral for uz is more involved. We have
where the integral goes from the body of one phase, across the interface, into the body of the other phase. 4 is the contribution to the partial molecular free energy arising from the gradient. It was found possible to express6 u2 in terms of $’, the local partial molecular free energy and the distance 6z defined above. The following expression was obtained:
Here p1 and pv are the molecular densities of the liquid and vapor phases, respectively. If we assume a simplified straight-line density for the interface, Le., assume dp/dz to vanish except at the interface and to be constant through the interface of thickness Az, we obtain
1862
The Journal of Physical Chemistry, Vol. 83, No. 14, 1979 U'
=
- M ~ 4 2 ( h , l- 41,v)(P1 - PV)/AZ
(20)
Because of the approximations involved, it is presumably better to approximate o2 with a Az that behaves slightly differently from the aZ used for 6,. We shall call this aZz. If Az,
a
t-"'
(21)
as is generally assumed, then if Azz a t-V'(lh)
(22)
+ 'v
(23)
eq 17 becomes pf =
2p
- 7vf
which is consistent with eq 12. Since C; a t-"',we may say from eq 22 that Az2 0: 5 l - V (24) Assuming that the C; (which we will call F2) which occurs in the gradient term, eq 2, differs from the C; which occurs in eq 3, we would write from modified eq 2 and 6
f,
=
QC;~-Y&J)'
(25)
We see that this equation is quite analogous to eq 16, with 6p taking the place of Ap and E2 taking the place of Az. This suggests that we set
C;,
a
PV
(26)
Then we see that the 4'7 factor comes in automatically, and we may consider the coefficient Q to be nonsingular. Thus we have obtained a consistent view of the gradient term in interfaces and fluctuations. Both the apparent factor C;q in Q and the apparent discrepancy between eq 12 and 17 can be ascribed to a common geometrical factor. In some cases it is advantageous to consider how a disturbance in the density falls off with distance, and to this end minimization of the Landau-Ginzburg form for the order-disorder part of the free-energy density expressed in terms of r = Ip - pcl/pc may be used q5 =
&(or)'+ Rr' + Ur'
(27)
It has been shown by Wilson'' that R and U may be treated as functions of 4, and Rice and Chang13 showed that better results were obtained if Q were taken as proportional to C;q. It is not obvious how this connects up with our other conclusions above, but it is to be recalled that eq 27 refers to average values at a distance from the original disturbance, and these averages will be affected by the structure of the fluctuations, which are more directly dealt with above. This structure may be the reason for introducing the factor 4" in eq 27, though, as noted, this is difficult to see directly. Since 7 is so small in ordinary three-dimensional systems, the difference between Az, and Az, is negligible in ordinary liquid-vapor systems, and indeed also in ordinary binary-liquid systems.6 Near the tricritical point of the 3He-4He, however, the situation is very different, since p = v = 7 = 1, and CL' = 2. This might point to considerable difference in the behavior of Az, and Az,. However, the sharp drop off in the spatial extension of a fluctuation implied by such a large value of ql might have some effect on both Az, and Az2, and an analysis of the situation, assuming they behave similarly, was attempted in a recent paper.14 It is seen that eq 21 states that, if 7 = 1,then Az, is independent of 5, whereas Az, is still proportional to 5. It may be that the true situation is best described by something between these two rather different points of view.
0.K. Rice
4. Relation between C; and Thermodynamic Quantities We shall start this section with some remarks concerning long- and short-range order near the critical point. The fluctuations which occur on account of the high compressibility near the critical point carry the system over a range of energy levels. The resulting contribution to the partition function is used to calculate the anomaly in the specific heat. It also contributes negatively to the inverse compressibility, and the resulting correction is larger than the inverse compressibility itself, meaning that the measured value of the latter must be the difference between two large quantities. In calculating this contribution to the partition function we used3 the resultant or measured value of the compressibility. We questioned whether this was the correct thing to do, or whether we should have used the uncorrected value to calculate the contribution of the single fluctuation to the partition function. While showing that it did not make too much difference in the actual results, we rather favored the use of the uncorrected compres~ibi1ity.l~ On further consideration, however, it would appear that it is correct to use the measured value of the compressibility. For it is seen that the corrections to the thermodynamic quantities involve the size and intensity of the fluctuations, which are indeed determined by the real values of the thermodynamic quantities. The final real value is, on the other hand, itself determined by the fluctuations. They adjust mutually to give a system in thermodynamic equilibrium. The compressibility is the important quantity in these considerations, and the average fluctuation in density, 6p,, depends upon the value of the compressibility. We found the following formula for 6pm along the critical isochore (as later modified, see second citation in ref 3): 6pm = [(a6)-1(aP/ap)T]1/(*-1)
(28)
where a is a constant. Along the critical isotherm we found 6pm =
IP
- ~ c 1 / ( 6 - 1)
(29)
However, when we note that ( a P / a p ) T = a6lp - pClh1eq 29 becomes the same as eq 28, except for the trivial factor (6 - l)-I, which may arise in part from slight inconsistencies between the ways (28) and (29) were derived, and in part from the difficulty in comparing the changes with respect to t with those with respect to p - pc. In any case, when either of these equations is used with eq 1 we see that C; depends in the same way upon (aP/ap),. This leads us to expect that, reciprocally, the singular part of (a P/a p)T depends only on 5, and to express this we write
eq 30 holding along the critical isotherm and eq 31 along the isochore. Since under these conditions C; a Ip - pJX and 5 a (7'- T J U ,respectively, we may write from eq 30 d(aP/ap)T/dC; m E ( S - Z ) / A - ( X + l ) / X - ~2(2S-l)/(S+l) (32) because 3X = 6 + 1. From eq 31 d(aP/aP)T/df
{-(?-1)/p(u+1)/u
= [-(?+u)/v
(33)
We, therefore, have the following relation between the exponents
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
