Hamaker theory

Solid-adsorbate interaction energy is also required, as is known from the Hamaker theory. 

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... 

For the van der Waals energy per unit area, El, between two half-spaces separated by a distance h, the Hamaker theory gives... 

The objections to the Hamaker theory were overcome by Lifshitz and his coworkers [Lifshitz 1956, Dzyaloshinskii 1961] using the bulk optical properties of the interacting bodies. The approach employed by Lifshitz uses the so-called Lifshitz-van der Waals constant h that depends only on the materials involved provided the separation distance is relatively small. Under some conditions the constant h can be related to the Hamaker constant by... 

For jump distances between 5-15 nm, the results fitted the Hamaker theory which states that the attractive van der Waals force is proportional to cylinder diameter and inversely proportional to the square of the gap. The Hamaker constant was found to be 10 J. These experiments were the first to reveal the true van der Waals forces at close approach. Previous tests had only shown the weaker, retarded forces which apply for gaps larger than 20 nm. These theories will be considered in Chapter 5. 

The forces of interaction (i.e., prior to contact) which a single, gas-borne particle can be subject to are treated from the perspective of its chemical and physical structure. To provide the requisite perspective for understanding the importance of these compositionally dependent factors, the role of the gas is discussed. Classical electrostatic and multipolar forces and the thermodynamic setting for any interaction involving a particle are described briefly. Principle emphasis in the chapter is given to the van der Waals forces. The modern (Lifshitz) theory is introduced and its relation to the classical Hamaker theory is described. A qualitative discussion of the computational approaches commonly used and experimental evidence for the theory are given. Inclusion of the chemical and physical factors necessary for treatment of cases that arise in actual application of the general theory is discussed. 

Historically, van der Waals forces between condensed media have been calculated by summing over pairwise interactions between the molecular constituents of each body [e.g., the first term in (5.23)]. This method, called Hamaker theory, first showed that the sum of the intermolecular interactions, as calculated by elementary second-order perturbation theory in the Schrbdinger equation, between macroscopic bodies gives realistic attractive forces in many cases. Its description can be found in any reference on modern dispersion or van der Waals forces between condensed media including LANGBEIN [5.34] and MAHANTY and NINHAM [5.35] at the advanced level or the excellent elementary introduction by PARSE6IAN [5.36]. 

Hamaker theory is the classical theory of van der Waals forces and is what is generally invoked even today in many fields including aerosols, despite its long-recognized inadequacies. Briefly, these inadequacies are as follows [5.351 ... 

The traditional approach to van der Waals force calculations employing Hamaker theory is illustrated in a series of papers [5.87-91] discussing anisomeric and layered-particle interactions. These papers give numerous explicit results including both indication of the conditions for repulsive interaction and the effects of geometry on the interaction forces. Their formulas are useful for qualitative purposes, but the limitations of the method raise questions both as to their general validity and certainly as to their potential for quantitative predictions. 

Estimate the dispersion interaction between two parallel plates of thickness d separated by a distance h using the Hamaker theory. 

To accoimt for the effect of the particle velocity, the expression for VLIM was multiphed with a parameter of hnear function of particle velocity. For particles of diameter smaller than 20 pm, the particle—surface adhesion energy was calculated using the Bradley—Hamaker theory. ° In this theory, the particle—surface adhesion energy is determined through the concept of London—van der Waals forces (this assumption is only vahd for small particle sizes). For the case of a particle colliding with a cylindrical fibre, the adhesion energy is given by ... 

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 ... 

Usually, additive particles are of maximum size of about 100 qm and the cohesive forces cannot be neglected. The significance of the cohesive forces is in the disintegration of the cluster, which requires that the hydrodynamic forces exceed the cohesive forces. Assuming the aggregates are formed by nontouching spherical particles of like material, the Bradley-Hamaker theory (Elmendorp, 1991) allows one to calculate the attractive van der Waals force between two particles as... 

For fine particles, the van der Waals force as a typical cohesive force is usually considered by using the Hamaker theory (Hamaker, 1937). For spherical particles, the van der Waals force is given as ... 

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