The Journal of
Physical Chemistry
0 Copyright, 1983, by the American Chemical Society
VOLUME 87, NUMBER 21
OCTOBER 13, 1983
Incorporation of the High-Frequency Dielectric Constant into the Kirkwood Dielectric Equation Applied to I c e John F. Nagle Depadment of Physics, Carnegie-Mellon University. Plttsburgh, Pennsylvania 152 13 (Received: August 23, 1982; In Final Form: November 18, 1982)
A derivation is given of the Kirkwood-Frohlich dielectric equation which, like the derivation of Harris and Alder, incorporatesthe effect of induced polarization from the beginning. This derivation yields the same equation as was derived by Frohlich, Buckingham, and many others and is different from the equation first derived by Harris and Alder and recently advocated by Stillinger for ice. For ice the two equations yield substantially different results for the dipole moment.
Introduction Because ice appears to be a simple example of a highly cooperative, disordered system, it is a valuable testing ground for theories and statistical mechanical calculations. Despite this apparent simplicity, the theory of the dielectric constant of ice has been rich in unforeseen problems. One problem that was unforeseen is that sums of correlation functions for ideal ice models are only conditionally convergent, so that one must distinguish the Kirkwood correlation function g from a similar polarization factor G.1*2 Another unforeseen problem is that the Kirkwood-Frohlich theory has recently been shown to be inconsistent if only short-range interactions are included in the calculation of g.la This finding raises serious doubts that the Kirkwood-Frohlich theory can be of practical value for ice.4 In contrast, the Onsager-Slater theory involves a renormalization of the electrostatic interactions in ice into a dipole-free Hamiltoniana4l5 This renormalization utilizes the special properties of the ice system and is not generalizable to an arbitrary dielectric system. However, for ice it results in a well-defined theory which (1)F. H.Stillinger in “Studies in Statistical Mechanics”, Vol. 8, J. L. Lebowitz and E. W. Montroll, Ed., North-Holland, New York, 1982, Chapter 6, pp 341-431. (2)A. Yanagawa and J. F. Nagle, Chem. Phys., 43,317 (1979). (3)D.J. Adams, Nature (London), 293,447 (1981). (4)J. F.Nagle, Nature (London), 298,401 (1982). (5) J. F. Nagle, Chem. Phys., 43, 317 (1979). 0022-3654/83/2087-4015$01.50/0
yields a value of N = 2.93 D which is in agreement with the value 2.91 D which follows from Hubmann’s experimentally determined value of the Bjerrum charge.25 These problems have recently been discussed at length elsewhere.14 The reason for mentioning them again here is that they are independent of another problem that has arisen and that will be discussed in this paper. The problem to be discussed arises in connection with the Kirkwood-Frohlich theory only. It involves the way in which the high-frequency dielectric constant, th, appears in the dielectric equation. There is agreement that, if t h = 1, then, in cgs units, the Kirkwood dielectric equation is
where to is the static dielectric constant and N / V is the density of dipoles with permanent moment p. However, in ice (as well as other materials) there is an additional molecular polarizability which also contributes to the static dielectric constant and which is not included in the contribution from the permanent moment p. As the frequency is raised, the dielectric response, t(u),goes through a Debye dispersion at about lo5 Hz due to the inability of the permanent dipoles to follow the electric field by reorien(6) J. F. Nagle, J. Glaciol., 21, 73 (1978).
0 1983 American Chemical Society
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The Journal of Physical Chemistty, Vol. 87,No. 21, 1983
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tation via bonding (Bjerrum) defects. At somewhat higher frequencies 40)plateaus to q, = 3.1 which corresponds to both electronic polarization and atomic movements not involving reorientation of hydrogen bonds.' The problem of how to account for the molecular polarizability was not seriously addressed by Kirkwood' and there is a consensus that his passing suggesting was inc o r r e ~ t . ' , ~FrohlichlO ~~ made two rather simple modifications of Kirkwood's derivation to give
appear to be redundant even to discuss the question again, much less to offer yet another counterderivation. However, in the context of ice Stillingerl has recently reopened the question by advocating eq 3. His derivation and the original one of Harris and Alder have the distinct advantages of appearing both rigorous and simple when compared to many of the derivations of eq 2 and eq 4. While the newer derivations of eq 4 may be the most rigorous, they are not so accessible to readers schooled in the ways that Onsager and Kirkwood treated these topics. Accordingly, a derivation of eq 2 along the lines of the derivations of Harris and Alder and of Stillinger may have some value.
Harris and Alders derived a different formula
Derivation of Eq 2 This derivation begins by following the general outline of the derivations of ref 1 and 8. The sample will be a sphere placed in an infinite region which, when originally empty, possessed a uniform electric field
More recently, another equation has appeared"
Eo = Eou,
For ice, one wishes to use the correct dielectric formula to calculate the dipole moment p in the solid phase from measurements of eo and f h and calculations of g. (As noted by Onsager,I2 it is unreasonable that formulae such as (2%+ + 2) Pvapor (5) P = 3(260 + Ch) can adequately account for the hydrogen bonding which enhances the dipole moment p in the solid or liquid as compared to the vapor and one should not use such formulae to judge the goodness of eq 1-3.) The problem is that much different results for p emerge from eq 2 and eq 3. For ice at 0 "C, eo = 91.2 and Eh = 3.1.13 Using g = 3 (for ice with bonding defects1s2but not including dipolar interactions in the calculation of g) one obtains p = 2.42 D from eq 2 and p = 1.84 D from eq 3. (Although the value of g with dipolar interactions is not known, comparable differences will emerge from eq 2 and 3 in any case.) It is of interest to know how much p changes in ice as compared to the vapor for which p = 1.85 D. The answers given by eq 2 and eq 3 are clearly quite different. For ice eq 4 gives p = 2.40 D which is very close to the result from eq 2. One possibility for the differences between eq 2 and 4 is that different dipole moments p2, p3, and p4 are being considered in each equation. Indeed, it has been suggested14that this accounts for the difference between eq 2 and eq 4. Since for ice this difference is so small, we will not be concerned with it and will follow Felderhof in classifying eq 4 as being essentially equivalent to eq 2. In contrast, different dipole moments do not appear to be the source of the difference between eq 2 and eq 3. Although the formula of Harris and Alder8 has been defended,15J6 it has been labeled erroneous so many times9,11,14,17-23 and Alder24now supports eq 4 that it may (7) J. G. Kirkwood, J . Chem. Phys., 7, 911 (1939). (8) F. E. Harris and B. J. Alder, J. Chem. Phys., 21, 1031 (1953). (9) A. D. Buckingham, Proc. R. So?. London, Ser. A, 238,235 (1956). (IO) H. Frohlich, "Theory of Dielectrics", Oxford University Press, London, 1949. (11) M. W. Wertheim, Mol. Phys., 36, 1217 (1978). (12) L. Onsager, J. Am. Chem. Soc., 58, 1486 (1936). (13) P. V. Hobbs, "Ice Physics", Clarendon Press, Oxford, 1974. (14) B. U. Felderhof, J. Phys. C, 12, 2423-38 (1979). (15) F. E. Harris and B. J. Alder, J. Chem. Phys., 22, 1806 (1954). (16) F. E. Harris, J. Chem. Phys., 23, 1663 (1955). (17) H. Frohlich, J . Chem. Phys., 22, 1804 (1954). (18)H. Frohlich, Physica, 22, 89&904 (1956). (19) B. K. P. Scaife, Proc. Phys. Soc., 70B,314 (1957).
(6)
along the z axis. Macroscopic electrostatics yields a field E inside the spherical sample related to Eo by
From the constitutive relation 47rP =
(to -
l ) E one has
where the field E and also the polarization P are uniform within the sphere. The right-hand side of eq 8 is now evaluated by use of statistical mechanics. Defining M(x,Eo) to be the total electrical moment of the sphere with N molecules in positions and orientations x = (xl, x 2 , ..., x N ) and in an external field Eo, one has a polarization P given by
P = (M(x,Eo)) / V
(9)
where the brackets indicate the canonical distribution average over x at temperature T. Combining eq 8 and writing eq 9 in detail yields
-t o-Eo
1
+2
--
47r
3vEo
Jdx (M(x,Eo)-u,) exp[-PU~(x,E~)l I d " exp[-PU~(x,Eo)I (10)
where UN(x,Eo)is the total energy of the system, including all interactions between molecules and the interactions of the molecules with the electric field Eo. At this point our derivation departs from those of ref 1, 8, and 9. To motivate this departure recall that, following Kirkwood,' the last term in eq 1-3 involves a formal separation of the spherical sample into a macroscopic inner sphere of volume V and a much larger outer spherical shell composed of the same material with the same dielectric constant. This formal division is essential to completion of the derivation of ref 1 and 8, but it is not used in the derivation of the (q, - I)(% + 2)/(q, + 2) term in eq 3. This (20) M. Mandel and P. Mazur, Physica, 24, 116 (1958). (21) R. H. Cole, Annu. Reu. Phys. Chem., 11, 149 (1960). (22) A. D. Buckingham, MTP Int. Reu. Sci. Phys. Chem. Ser. One, 2 (1972). (23) G. Stell, G. N. Patey, and J. S. Hoye, Adu. Chem. Phys., 48, 183-328 (1981). (24) E. L. Pollock, B. J. Alder, and G. N. Patey, Physica, 108A, 14-26 (1981). (25) M. Hubmann, Z. Phys. B, 32, 127 (1979).
Kirkwood Dielectric Equation Applied to Ice
The Journal of Physical Chemistry, Vol. 87, No. 21, 1983
uneven treatment of the two terms would appear to be more likely to lead to error than the consistent use of the concentric sphere geometry in the following derivation. Therefore, in eq 9 and 10 the configuration x = (xl, ...,x N ) designates only the molecules in the inner sphere. The supporting argument for this truncation of the averages to the inner macroscopic sphere is that the mean polarization is uniform throughout the sample so the polarization of any macroscopic part of the system is the same as the polarization of the entire system. Of course, it must not be forgotten that UN(x,Eo)now includes interactions between molecules in the inner sphere and those in the outer shell. In accord with the basic approximation in the Kirkwood theory such interactions will be taken into account by assuming that the outer shell can be treated as a macroscopic dielectric with dielectric constant e,,. As usual, one is interested in behavior in small fields Eo, so expansions of M(x,Eo)and UN(x,E0)are performed. We begin with M(x,O)in zero field. In first approximation this is just the sum of the “permanent” dipole moments poi of the individual i molecules N
Mo(x) = CWoi
(11)
i=l
which is independent of the field Eo. The identification of a permanent moment poi is not a trivial matter, as evidenced by Frohlich’s discussions10~17 (see especially pp 166-8 of ref 10). Before embarking upon this discussion, it may be helpful to the reader to know in advance that this is not a crucial point in this derivation. What will emerge in eq 15 is the distinction between Kirkwood’s pK and Frohlich’s pF,which for ice differ by only 1%. Let us identify Mo(x)and poi as the moments before long-range polarization interactions with the surrounding spherical shell are taken into account. (Short-range interactions such as hydrogen bonding are already included.) A net moment Mo(x)of a fixed configuration x then interacts with the outer shell which in turn reacts back on the inner sphere to induce a moment Ml(x,O)such that the true moment
M(x,O) = Mo(x) + M,(x,O)
(12)
Since interactions between the inner sphere and the outer shell are to be treated using macroscopic electrostatics, one needs only to know the total E, field inside the inner sphere with polarization P, with Eo = 0. The solution to this electrostatic problem is given by Frohlichlo (Appendix A.2, case c) 4aP, E, = -2to ti
+
where ci is the effective dielectric constant in the inner sphere, which is clearly not to because the configuration x is held fixed. The dielectric constant in the outer sphere is taken to be eo. The field E, induces a moment M I = Ec(th- 1)V/4a = -P,v(Eh - 1)/(2to + e;) (14) Before inserting Ml into 12 a joint decision must be made concerning the moment P, and the dielectric constant ti. One possibility is that P, = M(x,O)/V. A second is that P, = M(x,O)/V. In the latter case the effect of molecular and electronic induced polarization is already accounted for in M(x,O),so ti = 1. In the former case induced polarization is not accounted for, so q = q,. Both possibilities yield the same result for M(x,O),namely N
M(x,O)2
x& = hfo(x)(2tO+ 1)/(2to +
i=l
th)
(15)
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which is obtained from eq 12-14. We identify pi as the actual dipole moment in the medium with dielectric constant E,,. Frohlich calls this pi (ref 10) and pK (ref 17) after Kirkwood and it is Onsager’s12p. With this identification, then poi becomes Frohlich’s p (ref 10) or pF (ref 17). It is clear that pi has the more fundamental meaning and it, rathern than pF which is the p appearing in eq 2, should be used in the numerical comparisons of dipole moments in the Introduction. Next, the effects of an applied field Eoon the electrical moment are considered. For a fixed configuration x in the inner sphere the sample has an effective dielectric constant th in the inner sphere and a dielectric constant to in the outer shell. From classical electrostatics one then has a uniform field Ei in the inner sphere given by
The first factor in eq 16 can be thought of as the reduction of the field Eo inside the outer shell and the second factor then yields the “cavity” field inside the inner sphere, although the “cavity”, consisting of the inner sphere with fixed x , here has dielectric constant t h since the induced polarization is free to change with the field. Indeed, the field Ei induces additional polarization in the inner sphere according to M(X,Eo) = M(x,O) + E;V(th - 1 ) / 4 ~ (17a)
where eq 17c is obtained by utilizing eq 15. We now consider those configurations, designated as x’, that have the most probable values of the electric moment M(x’,Eo)in an electric field E,,. These states x’ will be the dominant configurations in all thermal averages for field Eo and temperature T. Thus
Substituting eq 18 into eq 17c yields
where E is the actual electric field inside the sample. This simple result states that the part of the mean moment due only to the permanent dipole moment Mo(x) = Cpoiis determined by an effective dielectric constant to- th. The remaining induced moment Ml(x’,Eo)is just the difference between M(x’,Eo)and Mo(x’),namely M,(x’,Eo) = EV(th - 1 ) / 4 ~ (20) This decomposition of the mean permanent and induced dipole moments is identical with that postulated by Frohlichlo (p 46) in his brief derivation of eq 2. However, it should be emphasized that it is not postulated in the present derivation but emerges from it. It may also be noted that nothing in this entire paragraph will be used in the subsequent derivation. The last preparatory step obtains the first-order term in the field expansion of UN(x,Eo).The most direct way uses the interaction of the actual moment M(xJ0) of the inner sphere from eq 17b with the electric field in the inner sphere with both the configuration x and the induced
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The Journal of Physical Chemistry, Vol. 87, No. 21, 1983
polarization frozen so that the effective dielectric constant in the inner sphere is 1. This yields
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and and grouping terms to yield
where eq 17b allows the replacement of M(x,Eo)by M(x,O) to order Eo and eq 15 is also used to obtain the final expression. Using eq 17b and eq 21 we perform the thermal averages in eq 10 to yield
Discussion The critical difference between this derivation and those of Harris and Alder8 and Stillinger’ is the formal division of the system into concentric spheres from the beginning. In this regard this derivation is closest in spirit to the original one of Frohlich’O although one relation postulated by him is derived here. Not using this division of the system into concentric spheres leads one down the same paths trodden by Harris and Alder,8 Frohlich,”J8 Buckingham: Scaife,lgand Stillinger,’ to the difficult problem of treating ( [M(X,.E~)]~) consistently. In contrast, the path followed here seems more straightforward and would appear to be more consistent in that both the induced polarization term and the permanent dipole term are treated in the same way. In contrast to the difficult problem of proving the proper extension of the Kirkwood dielectric equation to accomodate th # 1,the extension of the Onsagel-Slater dielectric equation is very easy and yields the conventional result5 €0
where the distinction between p and pK is essentially that between Frohlich’s and Kirkwood’s dipole moments, whereby eq 23 becomes t0-1= 4rNgp2 3Eo (2c0 + 1) ~ E O (24) (Eh - 1) (2Eo + Eh) 3VkT (263 + th) (2to + th)
+-
Equation 24 can now be shown to be equivalent to eq 2 by writing
- 1 = (‘h - 1)
+
4aNGp2 3kTV
~
(26)
This follows because the effective field inside a slab sample is just the applied field Eowhich is the same for all states x (constant voltage ensemble). Furthermore, the “dipoles” do not interact, so inducing extra dipole moments does not affect the internal energies. Therefore, the effect of polarization is simply to add &(th - l)V/4r to the moment M(x,E0)which gives rise to the (Eh - 1) term in eq 26. Acknowledgment. Discussions with Dr. F. H. Stillinger, helpful correspondence with Professor A. D. Buckingham and Dr. E. L. Pollock, and the support of this research by National Science Foundation Grant DMR-8115979 are gratefully acknowledged. Registry No. Ice, 7732-18-5.
