Lifshitz macroscopic approach

The method considered here for determination of these forces is the modified Lifshitz macroscopic approach, considering the microscopic approach results. This approach uses the optical properties of interacting macroscopic bodies to calculate the van der Waals attraction from the imaginary part of the complex dielectric constants. The other possible approach—the microscopic theory—uses the interactions between individual atoms and molecules postulating their additive property. The microscopic approach, which is limited to only a few pairs of atoms or molecules, has problems with the condensation to solids and ignores the charge-carrier motion. The macroscopic approach has great mathematical difficulties, so the interaction between two half-spaces is the only one to be calculated (Krupp, 1967). 

An attractive interaction arises due to the van der Waals forces between molecules of colloidal particles. Depending on the nature of dispersed particles, the Keesom forces (or the dipole-dipole interaction), the Debye forces (or dipole-induced dipole interaction), and the London forces (or induced dipole-induced dipole interaction) may contribute to the van der Waals interaction. First, the van der Waals interaction was theoretically computed using a method of the pairwise summation of interactions between different pairs of molecules of the two macroscopic particles. This method called the microscopic approximation neglects collective effects, and, as a consequence, misrepresents the Hamaker constant. For many problems of a practical use, however, specific features of the total interaction are determined by a repulsive part, and such an effective, gross description of the van der Waals interaction may often be accepted [3]. The collective effects in the van der Waals interaction have been taken into account in the calculations of Lifshitz et al. [4], and their method is known in the literature as the macroscopic approach. 

The retarded dispersion energy between macroscopic particles was treated by Liftshitz [28]. He considered half-spaces. Going half the way from the microscopic to the macroscopic approach, Lifshitz expanded the local fluctuations within the half-spaces in terms of plane waves and coupled them to the outgoing (reflected) radiation field. Then, satisfying the boundary conditions for the radiation field across the surfaces of the half-spaces under consideration, he found their force of attraction from Maxwell s stress tensor in the interspace. 

An alternative approach to the calculation of the Hamaker constant, A , in condensed phases is provided by the Lifshitz macroscopic theory [292,293], which is not limited by the assumption for pairwise additivity of the van der Waals interaction see Refs. 2,4, and 294. For the symmetric case of two identical phases 1 interacting across medium 3, the macroscope theory provides the expressiop [4]... 

Lifshitz (1955) approached the problem of the van der Waals interaction by examining the macroscopic properties of materials, as opposed to the Hamaker treatment of summing individual atomic interactions. The derivation is based on Maxwell s equations modified to allow rapid temporal fluctuations (Rytov, 1959). This gives an approximate expression for the free energy of interaction between two different (1 and 2) semi-infinite surfaces separated by a third material (3) of thickness ... 

The expressions and calculated values for VDW interactions are given in Refs. 13 and 33. The calculations are based on a pair-summation procedure which includes retardation corrections and many-body orientation effects, and as shown Ref. 34 gives approximately the same results as the macroscopic Lifshitz approach (35,36) or other macroscopic... 

Furthermore, these van der Waals interactions are important only near the interface, where it is unlikely that either Lifshitz or Hamaker approaches are accurate for spheres of molecular sizes. For example, the magnitude of the interaction for Na+ ions at. 5 A from the interface is only approximately 0.02kT (the values of B used in the calculation, Z Na = — 1X10 50 J m3 was obtained from fit by Bostrom et al. [17] and ZJNa= +0.8X10 511 J m3 was calculated by Karraker and Radke [18]). Eq. (8) might provide a convenient way to account for the interfacial interactions, if suitable values for Bt (not related to the macroscopic Hamaker constants) would be selected. 

Dispersion. Dispersion or London-van der Waals forces are ubiquitous. The most rigorous calculations of such forces are based on an analysis of the macroscopic electrodynamic properties of the interacting media. However, such a full description is exceptionally demanding both computationally and in terms of the physical property data required. For engineering applications there is a need to adopt a procedure for calculation which accurately represents the results of modem theory yet has more modest computational and data needs. An efficient approach is to use an effective Lifshitz-Hamaker constant for flat plates with a Hamaker geometric factor [9]. Effective Lifshitz-Hamaker constants may be calculated from readily available optical and dielectric data [10]. 

The major disadvantage of this microscopic approach theory was the fact that Hamaker knowingly neglected the interaction between atoms within the same solid, which is not correct, since the motion of electrons in a solid can be influenced by other electrons in the same solid. So a modification to the Hamaker theory came from Lifshitz in 1956 and is known as the Lifshitz or macroscopic theory." Lifshitz ignored the atoms completely he assumed continuum bodies with specific dielectric properties. Since both van der Waals forces and the dielectric properties are related with the dipoles in the solids, he correlated those two quantities and derived expressions for the Hamaker constant based on the dielectric response of the material. The detailed derivations are beyond the scope of this book and readers are referred to other publications. The final expression that Lifshitz derived is... 

Qualitatively the same result may be obtained if one utilizes more strict treatment of molecular interactions in disperse systems. This approach is based on the so-called macroscopic theory of van der Waals forces developed by E.M. Lifshitz, I.E. Dzyaloshinski and L.P. Pitaevski [14], In contrast to Hamaker s microscopic theory, the macroscopic theory does not use a simplified assumption of additivity of interactions between molecules, on which their summation is based (see Chapter I, 2). Mutual influence of molecules in condensed phases on each other may alter polarizabilities and ionization energies, making them different from those established for isolated molecules, which results in molecular interactions being not fully additive. 

The dispersion energy between two half-spaces is proportional to the inverse square of their separation d. Expression (4.76) was first derived by Lifshitz in 1955 on the basis of the fluctuation approach [28]. The macroscopic investigations presented here were reported by van Kampen et al. in 1968 [35]. 

Needless to say, equation (21.8) is a bit cumbersome and its original derivation is rather lengthy. However, many subsequent treatments of the macroscopic theory are now available which provide both a more readily understandable approach and many useful approximate expressions. In fact, by using the method of Parsegian and Ninham to determine the dielectric response function from absorption data and reflectance measurements, it is now quite straightforward to calculate dispersion forces from Lifshitz s theory. 

According to the more accurate macroscopic (Lifshitz) approach, the Hamaker constant for particles 1 and 2 in a medium 3 is given as (Israelachvili, 2011) ... 

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