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Full Spectral Calculation of Non-Retarded Hamaker Constants
for Ceramic Systems from Interband Transition Strengths
- R. H. French DuPont Co. Central Research, Wilmington DE, USA
- R. M. Cannon Lawrence Berkeley Laboratory, Univ. of
California, Berkeley CA, USA
- L. K. DeNoyer Spectrum Square Associates Inc., Ithaca NY, USA
- Y.-M. Chiang Materials Science Department, Mass. Inst. of
Tech., Cambridge MA, USA
Abstract
The van der Waals (vdW) interaction is one of the key terms
in the force balances dictating wetting behavior and intergranular film
thicknesses. The characteristics of thin intergranular or surficial glass films
are of increasing importance due to their role in determining the properties of
polycrystalline ceramics. The Hamaker constant scales the London dispersion
force part of the vdW interaction for a particular configuration of grains and
films and is a direct function of the interband optical properties of the
interatomic bonds of the materials. For ceramics, much previous work focused on
simplified models, such as the Tabor-Winterton approximation (TWA), to determine
Hamaker constants based on refractive indices. Herein we develop full spectral
calculations of the Hamaker constants for various ceramic systems using
experimentally determined interband transition strengths ( )
to directly derive the London dispersion spectra ( )
from which spectral difference functions lead to direct determination of the
Hamaker constants. The results affirm the expectation that transitions involving
valence electrons provide the predominant contribution to the dispersion forces
for the compounds examined.
Calculations have been done for the planar case of a gap
between two semi-infinite bodies containing either vacuum or an intervening
glassy layer. The results indicate that the TWA is useful for oxides with
relatively low refractive indices, i.e., n ~ 1.4 - 1.8. However, when any
of the materials have larger indices, this approximation becomes inexact, and no
obvious, simple correction to the TWA gives uniformly good results, as the
behavior differs for simple covalent materials and for oxides with partially
filled d-shells but having similar refractive indices. An important consequence
is that Hamaker constants are smaller for such high index materials, especially
oxides, with intervening glassy films than might be expected from
approximations. Calculations have also been done for two other geometries, i.e.,
for an intervening film with a layer of a third material at both interfaces and
for glass coated free surfaces. The former of these provides first insights
regarding the behavior with nonuniform films which often differs markedly from
that expected for homogeneous films of the same average composition.
London Dispersion Transform
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A Kramers Kronig Dispersion Relation
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Calculation of the Full Spectral Hamaker Constant
Figure 1. Schematic geometry for Hamaker constant
calculations. a) corresponds to asymmetrical 1-2-3 Hamaker constants, where if
Material #1 and #3 are the same, then this is the simpler case of a symmetrical
1-2-1 Hamaker constant b) represents the more complex symmetrical 5 layer case
where a layer of Material #2 forms on the surface of Material #1 while Material
#3 is the central phase in the interfacial film. Reprinted with permission from
Elsevier Science.
From the interaction energy, we can, using Equation 4,
determine the Hamaker constant  .
Equation 22. 
Now the configurations shown in Figure 1 for the LD forces
can be calculated. These are the non-retarded (denoted NR) Hamaker constants for
three layer geometries with a single film ( , )
and five layer geometries with a three layer intervening film ( ).
These three types of Hamaker constants can be formulated on a common basis by
defining three appropriate versions of the function G(x) as follows:
Equation 23. 
where a is the thickness of the central layer (layer 2
for , and layer 3
for ),
Equation 24. 
and
Equation 25. 
where b is the invariant thickness of the intervening
film between each particle and the central film in the case of ,
Figure 1. D is the difference of the LD spectra
defined in Equation 26.
Equation 26. 
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