Continuum Theory of van der Waals Forces

The theory for the van der Waals interactions presented in the previous section applies to macroscopic media only in a qualitative sense. This is because (i) the additivity of the interactions is assumed — i.e., the energies are written as sums of the separate interactions between every pair of molecules (ii) the relationship of the Hamaker constant to the dielectric constant is based on a very oversimplified quantum-mechanical model of a two-level system (iii) finite temperature effects on the interaction are not taken into account since it is a zero-temperature description. Here, we present a simplified derivation of the van der Waals interaction in continuous media, based upon arguments first presented by Ninham et al a more rigorous treatment can be found in Ref. 4. The van der Waals interactions arise from the free energy of the fluctuating electromagnetic field in the system. For bodies whose separations 

the factor of 1/2 in the exponent accounts for the zero-point, quantum fluctuations of the system the Bose statistics imply that any mode can have an occupation number that ranges from 0 to oo. Performing the sum, we find 

For a layered system, where the inhomogeneity in the media and the dielectric constant is in the z direction only, we distinguish between the in-plane degrees of freedom and the perpendicular direction. Focusing on a system described by a length scale, L e.g., the thickness of a film bounded by two 

The summation over j accounts for a multiplicity of normal modes, all having the same value of . To obtain the interaction energy, only those modes whose frequency explicitly depends on the film thickness, L, need be considered. 

In the following example, we show that Eq. (5.22) can be written in the form 

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Lifshitz avoided this problem of additivity by developing a continuum theory of van der Waals forces that used quantum field theory. Simple accounts of the theory are given by Israelachvili and Adamson. Being a continuum theory, it does not involve the distinctions associated with the names of London, Debye and Keesom, which follow from considerations of molecular structure. The expressions for interaction between macroscopic bodies (e.g. Eqns. 2-4) remain valid, except that the Hamaker constant has to be calculated in an entirely different way. 

Both Hamaker and Lifshitz theories of van der Waals interaction between particles are continuum theories in which the dispersion medium is considered to have uniform properties. At short distances (i.e. up to a few molecular diameters) the discrete molecular nature of the dispersion medium cannot be ignored. In the vicinity of a solid surface, the constraining effect of the solid and the attractive forces between the solid and the molecules of the dispersion medium will cause these molecules to pack, as depicted schematically in Figure 8.5. Moving away from the solid surface, the molecular density will show a damped oscillation about the bulk value. In the presence of a nearby second solid surface, this effect will be even more pronounced. The van der Waals interaction will, consequently, differ from that expected for a continuous dispersion medium. This effect will not be significant at liquid-liquid interfaces where the surface molecules can overlap, and its significance will be difficult to estimate for a rough solid surface. 

It is unfortunate that this macroscopic-continuum limitation is sometimes forgotten in overzealous application. The same limitation also holds in the theory of the electrostatic double layers for which we often make believe that the medium is a featureless continuum. Neglect of structure in double layers is equally risky, though, and even more common than in the computation of van der Waals forces. 

The theories of van der Waals and double-layer forces are both continuum theories wherein the intervening solvent is characterized solely by its bulk properties such as refractive index, dielectric constant, and density. When a liquid is confined within a restricted space, it ceases to behave as a structureless continuum. At small surface separations, the van der Waals force between two surfaces is no longer a smoothly varying attraction instead, there arises an additional solvation force that generally oscillates between attraction and repulsion with distance, with a periodicity equal to some mean dimension of the liquid molecules. 

Van der Waals forces result from charge and electromagnetic-field fluctuations at all possible rates. We can frequency analyze these fluctuations over the entire frequency spectrum and integrate their force consequences over the frequency continuum. Alternatively, the modern theory shows a practical way to reduce integration over all frequencies into summation by the gathering of spectral information into a set of discrete sampling frequencies or eigenfrequencies. The nature and choice of the frequencies at which dielectric functions are evaluated reveal how the modern theory combines material properties with quantum mechanics and thermodynamics. 

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. 

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... 

Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of so-called retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. 

In the case of a gas, when the particle arrives at a point of the order of a mean free path from the surface, the continuum theory on which the calculation of resistance is based no longer applies, van der Waals forces contribute to adhesion, provided that the rebound effects discussed in the next section do not intervene. There are also thin layers of adsorbed liquids... 

The derivation of (7.16) is based on the assumption (hat the diffusion coefficients of the colliding particles do not change as the particles approach each other. That this is not correct can be seen qualitatively from the discussion in Chapter4 of the increased resistance experienced by a particle as it approaches a surface. The result is that the term (AH- A ) to decrease as the particles approach each other. This effect is countered in the neighborhood of the surface because the continuum theory on which it is based breaks down about one mean free path ( 0.1 /i m at NTP) from the particle surface in addition, van der Waals forces tend to enhance the collision rate as discussed later in this chapter. For further discussion of the theory, the reader is referred to Batchelor (1976) and Alam (1987). Experimental support for (7.16) is discussed in the next section. 

The Hamaker constant can be evaluated accmately using the continuum theory, developed by Lifshitz and coworkers [40]. A key property in this theory is the frequency dependence of the dielectric permittivity, e( ). If this spectrum were the same for particles and solvent, then A=0. Since the refractive index n is also related to t ( ), the van der Waals forces tend to be very weak when the particles and solvent have similar refractive indices. A few examples of values for for interactions across vacuum and across water, obtained using the continuum theory, are given in table C2.6.3. 

Computational techniques have extensively been used to study the interfacial mechanics and nature of bonding in CNT-polymer composites. The computational studies can be broadly classified as atomistic simulations and continuum methods. The atomistic simulations are primarily based on molecular dynamic simulations (MD) and density functional theory (DFT) [105], [106-110] (some references). The main focus of these techrriques was to understand and study the effect of bonding between the polymer and nanotube (covalent, electrostatic or van der Waals forces) and the effect of fiiction on the interface. The continuum methods extend the continuum theories of micromechanics modeling and fiber-reirrforced composites (elaborated in the next section) to CNT/polymer composites [111-114] and explain the behavior of the composite from a mechanics point of view. 

A more rigorous estimation of the effective Hamaker constant for mixtures avoids simplified combining rules and uses the Lifshitz theory (based on relative permittivities and refractive indices see Israelachvili (2011) and Chapter 2, Equation 2.8). The Lifshitz theory is particularly useful for calculating the van der Waals force for any surface and in any medium, also because it relates the Hamaker constant with the material properties (relative permittivity and refractive index). Thus, the theory shows how the van der Waals forces can be changed via changing the Hamaker constant. The Lifshitz theory is a continuum theory, i.e. the dispersion medium, typically water, is... 

So-called solvation/structural forces, or (in water) hydration forces, arise in the gap between a pair of particles or surfaces when solvent (water) molecules become ordered by the proximity of the surfaces. When such ordering occurs, there is a breakdown in the classical continuum theories of the van der Waals and electrostatic double-layer forces, with the consequence that the monotonic forces they conventionally predict are replaced (or accompanied) by exponentially decaying oscillatory forces with a periodicity roughly equal to the size of the confined species (Min et al, 2008). In practice, these confined species may be of widely variable structural and chemical types — ranging in size from small solvent molecules (like water) up to macromolecules and nanoparticles. 

For large separations, the force between two solid surfaces in a fluid medium can usually be described by continuum theories such as the van der Waals and the electrostatic doublelayer theory. The individual nature of the molecules involved, their discrete size, shape, and chemical nature was neglected. At surface separations approaching molecular dimensions continuum theory breaks down and the discrete molecular nature of the liquid molecules has to be taken into account. 

The concept behind this theory is illustrated in Fig. 17. The vibrating molecule is approximated as a spherical cavity within a continuum solvent, and the vibrational motion is approximated as a spherical breathing of the cavity. The radius of the cavity is determined by a balancing of forces the tendency of the solvent to collapse an empty cavity, the intermolecular van der Waals attraction of the vibrator for the solvent molecules, and the intermolecular repulsion between the solvent molecules and the core of the vibrator. When the vibrator is in v = 1, the mean bond length of the vibrating bonds is longer due to anharmonicity. The increased bond length... 

For large separations, the force between two solid surfaces in a liquid medium can usually be described by continuum theories such as the van der Waals and the electrostatic double-layer theories. The individual nature of the molecules involved, their discrete size, shape, and chemical nature can be neglected. At surface separations approaching molecular dimensions, continuum theory breaks down and the discrete molecular nature of the liquid molecules has to be taken into account. For this reason, it is not surprising that some phenomena cannot be explained by DLVO theory. For example, the swelling of clays in water and nonaqueous liquids and the swelling of lipid bilayers in water cannot be understood on the basis of DLVO theory alone. In this chapter, we consider surface forces that are caused by the discrete nature of the liquid molecules and their specific interactions. 

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