STATE OF MATTER: Fluids, Simple and Complex 340 D. Chandler THE LIQUID E. W. Mantra// and J.L. Lebowitz {editors) © North-Holland Publishing Company, 1982 Rebertus, D. W., Berne, B. J. and Chandler, D. (1979). J. Chem. Phys. 70, 3395. "A Molecular Dynamics and Monte Carlo Study of Sol vent Effects on the Conformational Equi 1 ibrium of n-Butane in cc1 ." 4 Low Frequency Dielectric Properties of Reiss H. (1965). Adv. Chem. Phys. 9, 1. "Scaled Particle Methods in the Liquid and Solid Water Sta~istical Thermodynamics oTFTuids." Reiss, H. (1977). In Statistical Mechanics and Statistical Methods in Frank H. Stillinger Theory and Application. (V. Landman~,ed), Plenum, New York. "Scaled Particle Theory of Hard Sphere Fluids to 1976." Bell Telephone Laboratories Murray Hill, New Jersey U.S.A. 07974 Sandler, S. I. and Narten, A. H. (1976). Mol. ~h~. 32, 1543. "X-Ray Diffraction Study of Liquid Carbon Disulphide." I. INTRODUCTION Stanton, G. W., Clarke, J. H. and Dore, J. C. (1977). Mol. Phys. 34,823. "The Structure Factor of Liquid Bromine by Neutron Dillraction ll. Varia­ tion with Neutron Wavelength. One of the primary reasons that water is important to us is its ability to serve as a Stell, G. (1964). In of Classical Fluids. (H. L. solvent. That ability stems in large part from its high dielectric constant. This chapter Frisch and J. New York. "Cluster Expansions is devoted to clarifying the logical connections between the molecular structure of Stillinger, F. H., Jr. (1963). J. Chem. Phys. 38,1486. "Rigorous Basis water, the manner in which intermolecular forces cause water molecules to aggregate of the Frenkel-Band Theory of Association Equilibrium." in condensed phases, and the response of those phases to external electrical fields. Stillinger, F. H., Jr. (1971). J. Comp. Phys. 7, 367. "Pair Distribution in the Classical Rigid Disk and Sphere Systems." The last of these defines the dielectric properties. Topol, R. and Claverie, P. (1978). Mol. Phys. 35, 1753. "The Thermodynamic Functions of Molecular Liquids inthe Interaction Site Model." This survey will be restricted to pure water in liquid and solid forms. With respect Valleau, J.P. and Torrie, G. M. (1977). In Statistical Mechanics, Part A: to the latter, only those polymorphs of ice which can be prepared under ordinary pres- Equi 1 ibrium Theory. (B. J. Berne, ed), Plenum, New York. "A Guide to Monte Carlo for Statistical Mechanics: 2. Byways." sures, namely hexagonal ice Ih and the cubic modification Ic, will be considered. Verlet, L. (1968). Phys. Rev. 165, 201. "Computer 'Experiments' on Furthermore only linear dielectric response will be studied. But even with these res- Classical Fluids. 'fi::-- Eq;Jil i'brTum Correlation Functions." trictions the subject is a rich one containing several theoretical subtleties and many Weeks, J.D., Chandler, D. and Andersen, H. C. (l97la). :!._. Chem. ~- .2!±_, 5237. "Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids." important phenomena. Weeks, J.D., Chandler, D. and Andersen, H. C. (l97lb). :!._. Chem. ~- .2_2_, We begin this review in Section II with a brief outline of the relevant measure- 5422. "Perturbation Theory of the Thermodynamic Properties of Simple Liquids." ments. A comprehensive understanding of these measurements for liquid and solid Wi dom, B. ( 1963) . :!._. Chem. ~· ]1, 2808. "Some Topics in the Theory of Fluids." water is an indispensable prerequisite to motivate and to guide development of the Yarnell, J. L., Katz, M. J., Wenzel, R. G. and Koenig, S. H. (1973). Phys. subsequent theory. Rev. A 7, 2130. "Structure Factor and Radial Distribution Function fc;r:-- Uquid Argon at 85°K." Section III initiates the molecular theory with a discussion of several general issues that are applicable to all phases of water. One of these issues is the identification of a formal procedure for assigning multipole moments to molecules, a nontrivial matter 341 ·~ ;,!'}'( L} 342 FH Stillinger Low Frequency Dielectric Properties of Liquid and Solid Water 343 when electron distributions from neighboring molecules overlap. Another matter of P = (t-l)E/471" (2.2) general importance covered in Section III is the definition of molecular distribution Liquid water and cubic ice as bulk phases are both dielectrically isotropic, so for functions and an exposition of their basic properties. them the scalar t description is appropriate. The same is probably true for the am or- After having explored these few generally applicable ideas it has seemed advisable phous solid deposits that have been formed by slow condensation of water vapor on to treat liquid water, and the ices, as separate topics. These two cases present rather very cold surfaces (Narten et al. 1976). However ordinary hexagonal ice Ih has lower distinct statistical problems, and the dielectric behaviors at the molecular level respec- symmetry, so that its dielectric response demands use of a dyadic tensor quantity E; tively are clearly different. Consequently the discussion of bulk liquid water is con- Eq. (2.1) then becomes tained in Section IV, while Section V then considers the interesting and complex prob- D = E·E (2.3) !ems presented by ices lh and !c. and Eq. (2.2) must be modified accordingly: Although most of the conceptual aspects of the theory now seem well in hand there still remain several important quantitative problems to be resolved. These are P = (E-l)·E/471". (2.4) highlighted in the following development. It is the author's hope that the present sur- Hexagonal ice requires two independently specified tensor components of E describing vey will stimulate activity that soon closes these gaps in our understanding. respectively the response to electrical fields oriented parallel to the hexagonal c axis, For those readers interested in a broader treatment of the subject of aqueous and perpendicular to that axis (i.e. in the basal plane). In a Cartesian coordinate sys- dielectrics, the recent book by Hasted ( 1973) is recommended as a suitable guide. tern x,y,z with the z axis coincident with the hexagonal c axis, the dielectric tensor is diagonal and has the form II. DIELECTRIC MEASUREMENTS Eb 0 0 A. Background and Definitions E = I 0 fb 0 (2.5) 0 0 fc The macroscopic electrical behavior of isotropic dielectric substances is character- ized by the scalar dielectric constant L This quantity serves as the proportionality con- Similar remarks apply to the case of water at interfaces. Liquid water at a planar stant in the linear relation between the electric field E and the dielectric displacement interface, such as the liquid-vapor surface or the region near a planar electrode, can be D. expected to manifest tensor dielectric behavior with separate normal ( tN) and tang en- tial ( tT) components. Boundaries with two distinct curvatures generally will produce D = t E. (2.1) three independent tensor components. In any case the dielectric tensor is symmetrical It is often useful to consider as well the polarization vector P which is related to t and and can be diagonalized by a suitable choice of local coordinates. E by the following expression: Dielectric constants have values that depend on the angular frequency w of the per- 344 FH. Stillinger Low Frequency Dielectric Properties of Liquid and Solid Water 345 turbing field, and it has long been realized that these w variations contain important and fo is the standard static dielectric constant. However water does not fit this ideal information about dynamics of molecular processes occurring in the materials of description. Even in pure form it is electrically conducting owi~to formation of ions interest. Generally dielectric response will lag the excitation, giving rise to energy dis- by the reversible dissociation reaction sipation. Within the linear response regime it is conventional to separate f(w) into H20 : H+ + OH- . (2.9) real ( €') and imaginary ( €") parts as follows: In liquid water at 20°C the concentration of these ions is only 8.3XI0-8 molesfl, but f(w) = <'(w)-if"(w). (2.6) they have exceptionally high mobilities and thus might tend to interfere with high pre- [Similar representations are appropriate for each component of a tensor d cision measurement of the static dielectric constant (Stillinger 1978). The principle of causality demands that electrical response never precede the exci- If <J represents the low-frequency conductivity then in the neighborhood of w=O tation. This obvious requirement translates into a mathematical connection between <' the imaginary part of f(w) diverges thus: and €", the Kramers-Kronig equations (Kittel 1958). <"(w) = (47r<J/w) + O(w) (2.10) <'(w)-,'(c:o) = _1_ Jm ~ d (2.7) 7r o u2-w2 u, In order for this form to emerge from the second of the Kramers-Kronig relations (2.7) it is necessary for <'to possess a delta-function divergence at zero frequency: <"(w) =- k f f'(u)-f'{c:o) du. 7r o u2-w2 2 2 E'{w) = 47r <Ji5(w) + fo + O(w ). (2.11) Cauchy principal values are to be taken in these integrals at the poles of the integrands The physical meaning of this leading term is simply that a conductor can completely along the positive real axis. Notice the "infinite frequency" subtraction for <'; for most shield its interior from external electric fields. For practical reasons f' is always meas- cases of practical importance "infinite frequency" can be construed to lie above the ured at nonzero frequency so that in principle the delta function is not encountered; in characteristic frequency range of molecular vibrations but below the frequency range practice of course the nonzero conductivity can create troublesome space-charge of molecular electronic transitions. The two Kramers-Kronig relations arc inverse buildup within a measurement apparatus and thus lead to spurious results unless suit- integral transforms to one another (specifically, modified Hilbert transforms) and so able precautions are observed.