ALLENP. MINTON
886 presence of potential drops due to dipole orientation in the droplet film and a t the oil-water interface.
Acknowledgments. K. R. is indebted to the Science Research Council of the United Kingdom for a post-
doctoral research asaociateship. The authors are greatly indebted to Professor A. W. Adamson of the University of Southern California for having brought this problem to their attention and for invaluable discussions.
Relations between Crystal Structure, Molecular Electronic Polarizability, and Refractive Properties of Ice I by Allen P. Minton National Institute of Arthritis and Metabolic Diseases, National Institutes of Health, Public Health Service, U. S.Department of Health, Educatwn and Welfare, Bethesda, Maryland 9001.$ (Received June SO, 1971) Publication costs assisted by the National Institute of Arthritis and Metabolic Diseases
A relation is derived which expresses the refractive index of a crystalline array of isotropically polarizable molecules as a function of the density, the scalar molecular electronic polarizability, and a structure factor calculated from the geometry of the array. The structure factor for the hexagonal ice I lattice is evaluated utilizing a theoretical lattice model and from the existing experimental data by means of the relation previously derived. The two values so obtained are found to be in fair agreement,and possible reasons for the discrepancy between them are discussed.
I. Introduction It is now well known that the permanent dipole moment p of a water molecule in condensed phases is larger than in the vapor phase, due to the polarizing effect of its neighb0rs.l The magnitude of p in ice I can be fairly reliably estimated at 2.4 debyes compared with the vapor phase value of 1.8debyes.2 I n contrast, the effect of local structure upon the molecular electronic polarizability and refractive properties of water does not appear to have received systematic study, although the results of Eisenberg,a studying liquid water, and B a t ~ a n o v ,studying ~ crystalline hydrates, have indicated that changes in the local environment of a water molecule are coupled to significant variations in refractive properties. The birefringence of ice is of intrinsic interest because, unlike that of ordinary molecular crystals, it cannot be accounted for on the basis of an anisotropic molecular electronic polarizability. Dielectric measurements indicate that at temperatures slightly below freezing, the relaxation time for random reorientation of HpO molecules in ice is of the order of to During the course of a birefringence measurement each molecule in the bulk of the crystal therefore assumes all six possible alternate hydrogen-bonded positions in the ice lattice. Since the birefringence of The Journal of Physical Chemistry, Vo‘ot.76, No. 6,1979
ice is independent of field strength (in the linear region) there can be no correlation between the average molecular orientation and the direction of the applied optical frequency field. Thus even though a single water molecule appears to be slightly anisotropically polarizable with respect to molecular coordinates,6 the timeaverage polarizability LY must be isotropic with respect to crystal or laboratory coordinates, and is given by a = -
+
amm
3
+
ann
(1)
where the a I Zare the diagonal elements of the molecular polarizability tensor defined with respect to molecular Cartesian coordinates Z,m,n. Since the polarizability is effectively isotropic, the birefringence must therefore be due to anisotropy of the local field at the site of a water molecule in ice. We present here a simple extension of the Lorentz(1) D. Eisenberg and W. Kauzmann, “The Structure and Properties of Water,” Oxford Press, New York, N. Y., 1969,p 105 ff. (2) A. P. Minton, Chem. Phys. Lett., 7 , 606 (1970).
(3) H.Eisenberg, J. Chem. Phys., 43,3887 (1965). (4) S.8.Batsanov, Russ. J. Phys. Chem., 34,32 (1960). (5) D.Eisenberg and W. Kauzmann, ref 1,p 112 ff. (0) W. H. Orttung and J. A. Meyers, J . Phys. Chem., 67, 1906 (1963).
REFRACTIVE ]?ROPERTIES
OF
887
ICE1
Lorenz theory’ which relates the structure of a crystal to anisotropy of the local field. The results obtained are equivalent to those obtained earlier by Braggs in a rather less straightforward manner. We next apply these relations to the hexagonal lattice of ice I and show that the simple model is consistent with the existing experimental data. We shall assume for the sake of ease in calculaticn that the water molecules in ice are electronically discrete. Possible consequences of this assumption will be discussed subsequently. Having made the assumption, it is a simple matter to show that the dipoles induced in these discrete molecules by an external field are of sufficiently small extension so that the electrostatic interactions between them are well described by the classical formulae for point dipoles. Finally, we neglect the permanent dipoles present in ice as not contributing to the induced polarization at optical frequencies.
11. The Refractive Index Equation for a Crystalline Array of Isotropically Polarizable Molecules Lorentzs showed that the effective field at the center of a spherical cavity of radius r, in an isotropic dielectric continuum of refractive index n was given by
where Eois the externally applied polarizing field. The effective field at the site of a reference molecule in a crystal is equal to the sum of the Lorentz field and that due to the molecules in the immediate vicinity of the designated site (lying inside the boundary of the Lorentz cavity), which we shall designate as El,,, the local field (3)
The local field Eloc at the site of a reference molecule (placed at the origin) is equal to the sum of the fields of the dipoles induced in neighbor molecules by the externally applied field. (The externally applied field itself is included in the Lorentz field.) The electric field of induced dipole i of moment il.oriented in the field direction (designated here as x ) and located at xt, vi,xt has a z component at the origin given by
(4)
+ +
where ri2 = xi2 yi2 xt2. If the field direction z is either parallel or perpendicular to the crystallographic axes, the internal field components perpendicular to the applied field will average to zero. In this case, because all molecules are taken ae identical, p z = p (oriented in the a direction) and we therefore obtain -
.
Eloc =
9E,“
i=l
=
g[ 3x62 r~-
i=l
r1.2]
A
where N , is the number of particles inside the Lorentz cavity of radius r,. We denote the sum on the righthand side of (5) by fz (for structure factor in the x direction) and remark that it must converge; it is not a function of N , (or r,) provided that they are sufficiently large. The converse would lead to the absurd result that the refractive properties are a function of the radius of an imaginary spherical boundary constructed for the purpose of calculation. Classical electrostatics yield the following relations
and (7) where all quantities are defined for electric field directed along the x coordinate. We can thus replace each term in eq 3 with an expression which is a function of n, and Canceling out go,and replacing N is linear in by a function of density p , molecular weight M , and Avogadro’s number Na,we may rearrange (3) to obtain
z0.
nz2- 1 - 4rNap a nz2 2 3M 1 - afz
+
The refractive index is thus expressed as a function of a molecular property (a)and a property of the crystal lattice (fz). It should be noted that a, even though a molecular property, would not be expected to be entirely independent of the molecular environment. The optical spectrum of an isolated molecule is altered upon its incorporation into a molecular crystal and this would presumably be reflected in its electronic polarizability as well. However, if the crystal structure is known, fz can be calculated in principle and, in conjunction with refractive index and density data, may be used to determine a in the crystal.
111. The Structure Factor Lorentz showed long ago9 that in a cubic crystal fz is identically zero due to the symmetry of the lattice. A similar argument can be used to show that this is also the case for face-centered cubic crystals. For fz = 0, equation 8 reduces to the familiar ClausiusMossotti relation. However, if the crystal is not isotropic, thenf, will not in general be equal to zero (although it may be infinitesimal) and there may be as many as three nonidentical structure factors. For a uniaxial crystal such as ice I there will be two such, corresponding to external electric fields applied parallel and perpendicular to the optic axis, which coincides with the crystallographic axis. A further simplifica-
d
(5)
(7) C. J. Bottcher, “Theory of Electric Polarization,” Elsevier, Amsterdam, 1952, Chapter 8. (8) W. L. Bragg, Proc. Roy. SOC.,Ser. A , 106, 346 (1924). (9) H. A. Lorentz, “Theory of Electrons,” Teubner Verlagsgesellschaft, Leipzig, 1909.
The Journal of Physical Chemistry, VoL 76, No. 6, 1972
ALLENP. MINTON
888
tion follows from the uniaxial symmetry requirement that cxt2/r16 = x y r 2 / r , Swhere the sum is taken over i
i
all lattice sites enclosed in a sphere about the reference molecule as origin. Upon writing out the expressions for f x , f,, and fz as indicated in eq 5, replacing ri2 by xt2 yi2 x f 2 and eliminating either x or y, it becomes apparent that f x = fu = - f , / 2 . Henceforth fz shall be denoted by f l i and fi (=f,) by fl to indicate applied field direction with respect to the crystallographic axis. Depending upon the magnitude off I I (and hence the difference between f l i andfL), the crystal will exhibit greater or lesser birefringence. The ordinary ray will correspond to light propagated such that its electric field vector is perpendicular to the optic (crystallographic, x ) axis, and the extraordinary ray will correspond to light propagated such that its electric field vector is parallel to this axis. Thus calculation of f i I (and thereby fL) in conjunction with a scalar molecular electronic polarizability will account for both ordinary and extraordinary refractive indices. The structure factor f r l was evaluated for the ice I lattice by performing the indicated sum in the following way. A computer program was written to generate the coordinates of the oxygen atoms on a hexagonal ice I lattice with = 2.77 8. (fll is proportional to the crystal density so that direct calculation at one intermolecular spacing-or crystal density-permits evaluation at any.) The sites were then ranked in order of their distance from a reference molecule set at the origin, and f l I summed over all molecules out to a given radius. I n Figure 1 the sum accumulated out to radius r is plotted as a function of r. The calculation was carried out to a radius of 32.32 8 from the reference site, which includes the 4340 nearest neighbors of the reference molecule. It can be clearly seen that even at the largest radius to which the calculation was carried, the sum fluctuates with successive contributions of small numbers of molecular induced dipoles equidistant from the reference molecule. These fluctuations cannot be attributed to round-off error in the double-precision
+ +
calculation and therefore indicate (surprisingly) that even at a radius of >30 A, the hexagonal ice lattice does not appear as a dielectric continuum to the reference molecule. Nevertheless, we may obtain from Figure 1 a fairly reliable estimate of f i 1. A statistical analysis of points beyond 22 8 indicates that they are distributed randomly about a mean value of 3.2 X A-a with a standard deviation of 0.7 X lowa
IV. Discussion Analogs of equation 8 may be written for both extraordinary (E) and ordinary (co) rays in terms of the parallel and perpendicular structure factors, respectively ne - 1 n,2
+2
- 4nN,p
Q!
3M 1 - afll
(9)
The refractive indices and birefringence of crystalline ice were measured by Ehringhaus'O who reported that at the sodium D-line (5894 8) and --3", ne = 1.3105 and n, = 1.3090. lnserting these values into equations 9 and 10, recalling that f l I = - l/zfl, and setting p = 0.9164 g/cm3l1 and M = 180.016, one obtains a = 1.5001 A3 and f l l = 2 X 10-3 A-3, Data at other wave lengths and temperatures yield different values of a but the same value of the structure factor. I n Table I, experimental values of the refractive index at several wavelengths and two temperatures are compared with those calculated from eq 9 and 10 using a structure factor f l l = 2 X loda. Note that excellent agreement is obtained across the visible spectrum and at two widely varying temperatures. The polarizability varies with wavelength as expected and slightly with temperature. This latter effect is probably due to slight changes in the electronic structure of the Hbonded water molecules arising from temperature variation in the equilibrium 0-0 distance in ice, a variation which is too small to significantly affect the structure factor. Table I: Experimental and Calculated Values of the Ordinary (n,) and Extraordinary ( n e )Refractive Indices of Ice I Temp, OC
N-3
-65
5
10
15
20
25
R tfil
Figure 1. f l I accumulated to r as a function of
T.
The Journal of Physical Chemistry, Vol. 76, No. 6,197.9
30
A,
A
p,
g/omv
Y-Calcd---
nw
ne
,Experimentalnu ne
4047 0.9164 1.3184 1.3200 1.3184 1.3200 1.3090 1.3105 1.3090 1.3105 5894 1.3065 1.3080 1.3066 1.3079 6907 4047 0.9237 1.3206 1.3222 1.3206 1.3222 1.3116 1.3101 1.3117 1.3101 6239
a,
ba
1.5419 1.5001 1.4891 1.5392 1.4932
(10) A. Ehringhaus, Neues Jahrb. Mineral., Geol., Beilage Band, B , 41, 342 (1917). (11) N.E.Dorsey, "Properties of Ordinary Water-Substance," Reinhold, New York, N. Y.,1940,p 484.
REFRACTIVE PROPERTIES OF ICEI The value of f l I obtained from the experimental data is somewhat (20-50%) lower than that predicted by the ice lattice calculation described in the preceding section. One possible source of this discrepancy may be an oversimplification in the model due to the assumption of electronically discrete molecules. The 0-Ha * hydrogen bond in ice is known to be partially covalent,12 but by assuming that the molecules are discrete we are neglecting the covalent character of this bond. To the extent that electrons are shared between neighboring HzO molecules in ice, it is incorrect to utilize the classical point dipole model to calculate the contribution of a molecule to the local field at the lattice site of its neighbor. Another possible source of the discrepancy is the experimental sample itself, a prism cut from ice formed on the surface of a vessel of conductivity water exposed to subfreezing temperatures.la Although Ehringhaus describes his sample as homogeneous with an optical axis perpendicular to the water-air interface a t which the ice was formed, there is a distinct possibility that this sample was not a single crystal, but rather a conglomerate of preferentially oriented crystallites which are too small to be discerned using crossed polarizers under low magnification. Such a conglomerate would e
0
889 indeed exhibit birefringence due to the preferential orientation of the crystallites but this birefringence would be less than that of a'single crystal. I n support of this hypothesis let it be noted that reliable methods of forming large single crystals of icez4 were only reported much later than the study of Ehringhaus. These procedures appear to require considerably more apparatus, time, and effort than that described in the earlier study. I n conclusion, the extension of the Lorentz-Lorenz theory described here appears to account quite well for the birefringence of ice by considering the anisotropy of the local field in the crystal. The apparent experimental structural factor is somewhat less than that calculated from a perfect lattice model, but it is not known whether the discrepancy arises from oversimplification in the theory or from microheterogeneity in the experimental sample. The author wishes to acknowledge that stimulating discussions with Professor Henryk Eisenberg led to the initiation of this study. (12) See A. P.Minton, Trans. Faraday Soc., 67, 1226 (1971) and references therein. (13) A. Ehringhaus, ref 10, p 363. (14) F.Jona and P. Schemer, Helv. Phgs. Acta, 25,35 (1952).
The Journal of Physical Chemistry, Vol. 76, No. 6, 197%
