London-Hamaker forces

See R. H. French, "Origins and applications of London dispersion forces and Hamaker constants in ceramics," J. Am. Ceram. Soc., 83, 2117-46 (2000) K. van Benthem, R. H. French, W. Sigle, C. Elsasser, and M. Rtihle, "Valence electron energy loss study of Fe-doped SrTiOs and a E13 boundary Electronic structure and dispersion forces," Ultramicroscopy, 86, 303-18 (2001), and the extensive literature cited therein. Energy E" in those papers is written as hco" here. 

The combined effect of attraction and repulsion forces has been treated by many investigators in terms borrowed from theories of colloidal stability (Weiss, 1972). These theories treat the adhesion of colloidal particles by taking into account three types of forces (a) electrostatic repulsion force (Hogg, Healy Fuerstenau, 1966) (b) London-Van der Waals molecular attraction force (Hamaker, 1937) (c) gravity force. The electrostatic repulsion force is due to the negative charges that exist on the cell membrane and on the deposition surface because of the development of electrostatic double layers when they are in contact with a solution. The London attraction force is due to the time distribution of the movement of electrons in each molecule and, therefore, it exists between each pair of molecules and consequently between each pair of particles. For example, this force is responsible, among other phenomena, for the condensation of vapors to liquids. 

In the study of the texture of catalysts use is made of the physical adsorption of gases. Physical adsorption is an increase of concentration at the gas-solid or gas-liquid interface under the influence of integrated van der Waals-London forces, also known as de Boer-Hamaker forces. 

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). 

Particle-Collector Interactions. Dispersion forces and double-layer forces were the two interaction forces considered between the particle and collector. The London dispersion forces can be expressed with the Hamaker constant H, the distance between the two particles 5, and a retardation factor... 

The Derjaguin and Landau and Verwey and Overbeek, DLVO, theory, the most widely accepted for colloidal stability (13,14), is based on a model in which the rate of coagulation is determined by the diffusion of particles toward each other in the presence of a potential field. This field is the result of molecular attractive forces of the Van der Waals type and repulsive forces due to the interaction of the electric double layer around the particles. The attraction between particles immersed in a fluid is considered in this theory to result from London dispersion forces. Hamaker (15) has shown that the magnitude of the potential due to these forces increases rapidly as the particles are brought closer together. 

French R.H. Origins and applications of London dispersion forces and Hamaker constants in ceramics. J. Am. Ceram. Soc. 2000 83(9) 2117-2146 Frink L.J.D., Lupkowski M., Dodd T.L., Van Swol F. On the role of solvation forces in colloidal phase transitions. Langmuir 1993 9 1442-1445... 

The universal van der Waals attraction which occurs in all disperse systems is described in Vol. 1. The dipole-dipole, dipole-induced dipole and London dispersion forces for atoms and molecules are described. This is followed by the microscopic theory of Hamaker for colloidal particles and definition of the Hamaker constant. This microscopic theory is based on the assumption of additivity of all atom or molecular attractions in each particle or droplet. The variation of van der Waals attraction with separation distance h between the particles is schematically represented. This shows a sharp increase in attraction at small separation distances (of the order of a few nanometers). In the absence of any repulsion, this strong attraction causes particle or droplet coagulation which is irreversible. The effect of the medium on the overall van der Waals attraction is described in terms of the effective Hamaker constant which is now determined by the difference in Hamaker constant between the particles and the medium. The macroscopic theory of van der Waals attraction is briefly described, with reference to the retardation effect at long sepeiration distances. The methods that can be applied for determination of the van der Waals attraction between macroscopic bodies are briefly described. 

The Hamaker approach of pairwise addition of London dispersion forces is approximate because multi-body intermolecular interactions are neglected. In addition, it is implicitly assumed in the London equation that induced dipole-induced dipole interactions are not retarded by the finite time taken for one dipole to reorient in response to instantaneous fluctuations in the other. Because of these approximations an alternative approach was introduced by Lifshitz. This method assumes that the interacting particles and the dispersion medium are all continuous i.e. it is not a molecular theory. The theory involves quantum mechanical calculations of the dielectric permittivity of the continuous media. These calculations are complex, and are not detailed further here. 

Among dispersed magnetic particles, four different interparticle interactions exist— these are the van der Waals forces, magnetic attraction, steric repulsive forces, and electric repulsive forces. Van der Waals forces or London dispersion forces originate fi-om die interaction between orbital electrons or induced vibrating dipoles. For the equivalent two spherical particles, Hamaker s equation holds [95]. This force is strong only within short distances. 

Hamaker [32] first proposed that surface forces could be attributed to London forces, or the dispersion contribution to van der Waals interactions. According to his model, P is proportional to the density of atoms np and s in the particle and substrate, respectively. He then defined a parameter A, subsequently becoming known as the Hamaker constant, such that... 

The same logic that we used to obtain the Girifalco-Good-Fowkes equation in Section 6.10 suggests that the dispersion component of the surface tension yd may be better to use than 7 itself when additional interactions besides London forces operate between the molecules. Also, it has been suggested that intermolecular spacing should be explicitly considered within the bulk phases, especially when the interaction at d = d0 is evaluated. The Hamaker approach, after all, treats matter as continuous, and at small separations the graininess of matter can make a difference in the attraction. The latter has been incorporated into one model, which results in the expression... 

For a typical experimental hydrosol critical coagulation concentration at 25°C of 0.1 mol dm-3 for z = 1, and, again, taking if/d = 75 mV, the effective Hamaker constant, A, is calculated to be equal to 8 X 10 20 J. This is consistent with the order of magnitude of A which is predicted from the theory of London-van der Waals forces (see Table 8.3). 

The capture efficiency or Sherwood number was shown to be a function of three dimensionless groups—the Peclet number, the aspect ratio (collector radius divided by par-dele radius), and the ratio of Hamaker s constant (indicating the intensity of London forces) to the thermal energy of the particles. Calculated values for the rate of deposition, expressed as Ihe Sherwood number, are plotted in Figure 6 as a function of the three dimensionless groups. 

Hamakers (1937) expression for the unretarded London force was employed, whereas the expression for the force exerted by fluid motion was taken from the results of... 

Figure 1 suggests that as Hamaker s constant becomes very small compared to kT, vanishes. For this condition equation (16) is not valid. Instead, if the contribution to double-layer forces is much smaller than kT for x > 3, then only the gravitational contribution needs to be considered ... 

Van der Waals interactions are noncovalent and nonelectrostatic forces that result from three separate phenomena permanent dipole-dipole (orientation) interactions, dipole-induced dipole (induction) interactions, and induced dipole-induced dipole (dispersion) interactions [46]. The dispersive interactions are universal, occurring between individual atoms and predominant in clay-water systems [23]. The dispersive van der Waals interactions between individual molecules were extended to macroscopic bodies by Hamaker [46]. Hamaker s work showed that the dispersive (or London) van der Waals forces were significant over larger separation distances for macroscopic bodies than they were for singled molecules. Through a pairwise summation of interacting molecules it can be shown that the potential energy of interaction between flat plates is [7, 23]... 

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. 

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]. 

It is of importance for a knowledge of the forces acting between colloidal particles that the greatest distance at which the London forces are still important is not the radius of the atom but in fact of the order of magnitude of the radius of the particle itself, since the interaction between all the atoms in each of the colloidal particles must be summed, and this interaction, therefore, will increase with increasing size of the particles (Hamaker)1. This is quite different from, for example, the interaction between particles with a crystal lattice in which only purely electrostatic forces would act in this case the radius of action remains, even for large particles, of the order of the lattice constant and there is only a question of a surface action. The effect of the more deeply situated parts of the lattice does not appear outside on account of the mutual compensation of the action of the oppositely charged ions. 

For practical purposes it is expedient to drop pcirt of the formalism described so far, by substituting an explicit function for u(r). In particular, [2.5.3] can be used for the attractive part. One of the arguments which we cire now considering is that the r power law applies generally for all types of attraction between isolated pairs of molecules (sec. 1.4.4) in the unretarded range. As far as the London forces are concerned, these are reasonably additive, where reasonably means that attractive forces between macrobodies computed on this basis by Hamaker and de Boer differ by less than 10-20% from the more exact, but not so readily accessible, Lifshlts results. So one would anticipate that for simple liquids surface tensions. 

A variety of Interaction forces contribute to G(h) and 17(h), each with its own sign, magnitude and typical decay as a function of h. One of them is the London-Van der Wools force, also known as dispersion force, and already described extensively in chapter 1.4. For wetting films G(h) may be positive or negative, depending on the sign of the Hamaker constant examples of which can be found in... 

Figure 5.18 gives an impression of the agreement between theory and experiment. choosing representative parameter values. The London-Van der Waals component was computed from the Lifshits theory, as described in sec. 1.4.7. For water and silica the Hamaker constants are similar, as a result these forces do not contribute substcuitially, except at very low h, where 77... 

Depending on the geometrical model (Figure 17) used and on the theoretical approach taken, different relationships exist for the approximation of adhesion by van der Waals forces. The best known equations are those developed by Hamaker based on the microscopic theory of London-Heitler. For the model sphere/plane. Figure 17(a), a distance a< 100 nm, and the particle diameter x, the adhesion force A, i is... 

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