Lifshitz macroscopic theory

The van der Waals interaction energy, E 22, per unit area between two half-spaces 1, 2 immersed in a medium 3 as a function of the separation, h, is described by the Lifshitz macroscopic theory as follows " ... 

For two spheres separated by the (shortest) distance, h, the interaction energy is given by the Lifshitz macroscopic theory ... 

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]... 

In the microscopic calculation pairwise additivity was assumed. We ignored the influence of neighboring molecules on the interaction between any pair of molecules. In reality the van der Waals force between two molecules is changed by the presence of a third molecule. For example, the polarizability can change. This problem of additivity is completely avoided in the macroscopic theory developed by Lifshitz [118,119]. Lifshitz neglects the discrete atomic structure and the solids are treated as continuous materials with bulk properties such as the... 

Additional complications can arise when two bodies, i.e. the tip and the sample, interact in liquid (Fig. 3c). The interaction energy of two macroscopic phases across a dielectric medium can be calculated based on the Lifshitz continuum theory. In contact, when the distance between the phases corresponds to the nonretarded regime, the Hamaker constant in Eq. (2) is approximated by ... 

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... 

The first improvement is based on the discovery that Eq. (2) of the Hamaker microscopic theory for spheres agrees with Eq. (4) of the continuum macroscopic theory when the Hamaker constant. A, in Eq. (2) is determined from Eq. (3) of the Lifshitz theory for parallel flat plates (Fig. 1). The combined Hamaker-Lifshitz function, A h), can be obtained by comparing the right-hand sides of Eqs. (1) and (3), giving... 

Ajj in condensed phases, called the macroscopic theory. The latter is not limited by the assumption for pairwise additivity of the van der Waals interaction (see also References 34, 254, and 265). The Lifshitz theory treats each phase as a continuous medium characterized by a given uniform dielectric permittivity, which is dependent on the frequency, v, of the propagating electromagnetic waves. Eor the symmetric configuration of two identical phases i interacting across a medium j, the macroscopic theory provides the expression ... 

Another (more strict) way of calculating the energy of dispersion interactions between the two volumes is based on Lifshitz s macroscopic theory and is briefly summarized in Chapter VII,2. 

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. 

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. 

Lifshitz developed the macroscopic theory relating the Hamaker constant to dielectric constants of the materials. Accordingly,... 

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). 

A first attempt at this was accomplished in [16] where the Lifshitz-Slyozov theory was adopted to allow for spatial variations including the concept of the local particle size distribution. In that work, however, undue emphasis was placed on the diffusion of the particles and the concept of forming macroscopic length scale patterns via the svegliabile mechanism was not realized (see a discussion in the other paper of the author in these proceedings). 

Dispersion forces are universal because they attract all molecules together, regardless of their specific chemical nature. The potential energy of dispersion attraction between two isolated molecules decays with the sixth power of the separation distance. Based on the so-called Hamaker theory (i.e., the method of pair-wise summation of intermolecular forces) or the more modern Lifshitz macroscopic treatment of strictly additive London forces, it is possible to develop the so-called Lifshitz-Van der Waals expression for the macroscopic interactions between macroscopic-in-size objects (i.e., macrobodies) [19, 21], Such an expression strongly depends on the shapes of the interacting macrobodies as well as on the separation distance (non-retarded or retarded interaction). For two portions of the same phase of infinite extent bounded by parallel flat surfaces, at a distance h apart, the potential energy of macroscopic attraction is ... 

Lifshitz (1955-60) developed a complete quantum electrodynamic (continuum) theory for the van der Waals interaction between macroscopic bodies. 

Other continuous profiles in e produce similarly intriguing behaviors. The nondivergence of free energy and of pressure, qualitatively different from the power-law divergences in Lifshitz theory, occurs here when there is no discontinuity in s itself or its z derivative. Deeper consideration of such behaviors would require going beyond macroscopic-continuum language. 

See the seminal paper by B. W. Ninham and V. Yaminsky, "Ion binding and ion specificity The Hofmeister effect and Onsager and Lifshitz theories," Langmuir, 13, 2097-108 (1997), for the connection between solute interaction and van der Waals forces from the perspective of macroscopic continuum theory. 

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 role of the medium, in which contacting and pull-off are performed, has been mentioned but not considered so far. However, the surroundings obviously influence surface forces, e.g., via effective polarizability effects (essentially multibody interactions e.g., by the presence of a third atom and its influence via instantaneous polarizability effects). These effects can become noticeable in condensed media (liquids) when the pairwise additivity of forces can essentially break down. One solution to this problem is given by the quantum field theory of Lifshitz, which has been simplified by Israelachvili [6]. The interaction is expressed by the (frequency-dependent) dielectric constants and refractive indices of the contacting macroscopic bodies (labeled by 1 and 2) and the medium (labeled by 3). The value of the Hamaker constant Atota 1 is considered as the sum of a term at zero frequency (v =0, dipole-dipole and dipole-induced dipole forces) and London dispersion forces (at positive frequencies, v >0). 

The complex Hamaker coeflBcients are predicted from individual self-interacting Hamaker coefficients (for example. An) evaluated by Visser (30) from direct Lifshitz solutions or Ninham and Parsegian s (36) macroscopic approximations. We used combining rules derived from thermodynamics and the Lifshitz theory by Bargeman and Van Voorst Vader (30, 37)... 

A more satisfactory method for calculating the attraction between colloidal particles is the macroscopic continuum theory due to Lifshitz (Lifshitz, 1956 Dzyaloshinskii et al., 1961) and subsequently elaborated by Ninham and coworkers (Mahanty and Ninham, 1976). This expresses the interaction in terms of the bulk dielectric properties of the two colloidal particles. The power of the Lifshitz formalism lies in its ability to encompass all many-body interactions to deal properly with the effects of intermediate substances (here the microscopic method is quite vague) and to include contributions from all resonant electronic and molecular frequencies. Its disadvantage lies in the dramatic increase in the complexity of the calculations, although such computations are readily performed with the aid of a digital computer. 

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