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Hamaker Constants of Inorganic Materials

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Hamaker Constants of Inorganic Materials ADVANCESIN COLLOID AND INTERFACE Advancesin Colloid and Interface Science SCIENCE ELSEVIER 70(1997) 125-169 Hamaker constants of inorganic materials Lennart Bergstr6m Institute for Surface Chemistry, P.O. Box 5607, S-114 86 Stockholm, Sweden Abstract Calculations of Hamaker constants using Lifshitz theory require the availability of accurate dielectric data, especially in the ultraviolet spectral region, and the use of a convenient and appropriate mathematical representation. In this review, a multiple oscillator model -- the so-called Ninham-Parsegian (N-P) representation -- has been used and spectral parameters for 31 different inorganic materials (including diamond) have been generated from critically evaluated optical data or collected from the litera- ture. For most materials, a two-oscillator model (one UV and one IR term) was used but more detailed representations were included when available. The spectral parameters presented here can be combined with previous data, mainly focused on hydrocarbon and organic systems, to yield an extensive spectral data base for both solids and liquids enabling Lifshitz calculations of Hamaker constants for many materials combinations. Non-retarded Hamaker constants for symmetric material combinations across vac- uum (Alvl) and water (Alwl) have been calculated for the different materials; these calculations were performed using the full Lifshitz theory. Asymmetric combinations, A lv3 and A lw3, against four commonly used materials in atomic force microscopy studies: silica, amorphous silicon nitride, sapphire, and muscovite mica, have also been covered. The use of a new dielectric representation for water resulted in significantly lower values of Alwl compared to previous calculations. Analytical approximations to the full Lifshitz theory were evaluated and found to give surprisingly accurate results (the Tabor-Win- terton approximation) for Alvl when the IR contribution is of minor importance. An attempt to make the TW approximation more general by establishing some scaling relationship between no and couv was met with little success; only the UV spectral parameters of the covalent oxides, sulphides and nitrides may be fitted to a simple power law relation. The Lifshitz calculations in this study were compared with an alternative method where a more detailed dielectric representation in the visible-ultraviolet spectral range was obtained through Kramers-Kronig (K-K) transformation of reflectivity data over a broad frequency range. Despite the difference in dielectric information, the two 0001-8686/97/$32.00 © 1997 -- Elsevier Science B.V. All rights reserved. PII: S0001-8686(97)00003-1 126 L. BergstrSm /Adv. Colloid Interface Sci. 70 (1997) 125-169 methods generally yield non-retarded Hamaker constants which do not differ signifi- cantly. This is not true for all materials, e.g. water, where a more detailed representation using either an N-P representation with several oscillators or the K-K representation must be used. It was shown that the omission of the static and low frequency contribution in the latter method may result in a significant underestimation of the value for Alwl when the dispersive contribution becomes very small. Keywords: Hamaker constant, van der Waals interaction, inorganic materials, interpar- ticle forces, AFM Contents I. Introduction .................................... 126 II. Theory ....................................... 129 IIA. Dielectric response ............................. 129 IIB. Calculation of non-retarded Hamaker constants ............ 134 III. Dielectric representation ............................. 137 IIIA. Dielectric data of materials ........................ 137 IIIB. Determination of spectral parameters .................. 139 IV. Spectral parameters of materials ........................ 142 IVA. Amorphous materials ........................... 142 IVB. Cubic halides ................................ 143 IVC. Cubic covalent materials ......................... 145 IVD. Hexagonal, tetragonal and trigonal materials .............. 147 IVE. Birefringent materials ........................... 149 V. Calculated Hamaker constants ......................... 150 VA. Importance of the IR range ........................ 150 VB. Full Lifshitz calculations and analytical approximations ........ 154 VC. Hamaker constants for use in AFM studies ............... 155 VD. Spectral parameter scalings ........................ 159 VE. Comparison with a Kramers-Kronig dielectric representation ..... 162 VI. Summary and conclusions ............................ 165 References ....................................... 167 I. Introduction The surface forces between two bodies in close proximity play an important role in many scientific and technical areas, e.g. adhesion, wetting, adsorption, colloidal phase equilibrium, flocculation rates, rheologi- cal properties and other colloid and interface phenomena [1,2]. In phenom- ena involving inorganic materials, examples include the important role of interparticle forces in determining the rheological behaviour of the ceramic suspensions in the powder preparation and consolidation step [2-4], and L. Bergstr6m/Adv. Colloid Interface Sci. 70 (1997) 125-169 127 the suggestion that a surface force balance controls the thickness of thin intergranular glass films in polycrystalline ceramics [5]. Among the many contributions to the interaction between surfaces, e.g. double layer, structural, steric, depletion, hydration and hydropho- bic forces, there is one type of interaction which is always present, the van der Waals interaction. This ubiquitous interaction may be of varying importance depending on the system. The Hamaker constant represents a conventional and convenient way of assessing the magnitude of this interaction. Recent work has shown that many inorganic systems are characterised by relatively high Hamaker constants [6,7], hence making this interaction of significant importance. It is obvious that accurate estimations of the Hamaker constants is necessary for a quantitative understanding of the effect of interparticle forces on various phenomena. The van der Waals force has an electrodynamic origin as it arises from the interactions between atomic or molecular oscillating or rotating electrical dipoles within the interacting media. There are three types of interactions which contribute to the van der Waals force: (i) The Keesom force which is the interaction between two permanent dipoles. (ii) The Debye force which is the interaction between one permanent dipole and one induced dipole. (iii) The London or dispersion force which is the interaction between two induced dipoles. Between molecules, each of these contributions has an interaction free energy which varies with the inverse sixth power of the distance at small and intermediate distances of separation (from 1-2 .~ to several nm). Hamaker [8] calculated the distance dependence of the free energy of macroscopic bodies by performing a pair-wise summation over all the atoms in the bodies. For example, the van der Waals interaction free energy, VvdW, per unit area, between semi-infinite parallel plates at separation, L, can be expressed as A Vvd w - 12xL 2 (1) where A is the Hamaker constant. As seen in Eq. (1), there is a direct proportionality between the magnitude of the van der Waals interaction and the Hamaker constant. The Hamaker constant is a materials constant that depends on the properties of the two materials and the intervening media. The distance dependence of the van der Waals energy depends essentially on the geometry of the two interacting bodies 128 L. BergstrSm /Adv. Colloid Interface Sci. 70 (1997) 125-169 being proportional to L -2 for parallel plates and scale to L -1 for two spherical particles at short separation distances where retardation can be ignored. In the original treatment, also called the microscopic approach, the Hamaker constant was calculated from the polarizabilities and number densities of the atoms in the two interacting bodies [8]. Lifshitz pre- sented an alternative, more rigorous approach where each body is treated as a continuum with certain dielectric properties [9]. This approach automatically incorporates many-body effects which is ne- glected in the microscopic approach. In the Lifshitz treatment, the van der Waals interaction is the result of fluctuations in the electromagnetic field between two macroscopic bodies, modified by the separating media, where the interaction can be referred to the standing waves which only occur at certain frequencies. Hence, the van der Waals interaction, and thus the associated Hamaker constants, can be estimated from a knowl- edge of the frequency dependent dielectric properties of the interacting materials together with the intervening medium, and the geometry of the bodies. The accuracy of the estimated Hamaker constants are directly related to the precision and accuracy of the dielectric spectra and the mathematical representation of this data. The main focus of this review is to provide a collection of dielectric representations for a range of inorganic materials presented in a con- densed but appropriate manner, suitable for calculations of accurate Hamaker constants. The dielectric representation follows the form presented by Hough and White [10] in their collection of spectral parameters and Hamaker constants for a number of materials, focusing mainly on liquids and organic solids. Since their work was published, collections of accurate dielectric data for many inorganic materials have
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