Flat plates, interaction

The interaction energy per sphere is linear in the sphere radius. If the flat plate interaction W has a characteristic range A, the interaction energy per sphere will scale like kRW. This allows predictions as a function of sphere size. 

Microscopic analyses of the van der Waals interaction have been made for many geometries, including, a spherical colloid in a cylindrical pore [14] and in a spherical cavity [15] and for flat plates with conical or spherical asperities [16,17]. 

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

Surface forces measurement directly determines interaction forces between two surfaces as a function of the surface separation (D) using a simple spring balance. Instruments employed are a surface forces apparatus (SFA), developed by Israelachivili and Tabor [17], and a colloidal probe atomic force microscope introduced by Ducker et al. [18] (Fig. 1). The former utilizes crossed cylinder geometry, and the latter uses the sphere-plate geometry. For both geometries, the measured force (F) normalized by the mean radius (R) of cylinders or a sphere, F/R, is known to be proportional to the interaction energy, Gf, between flat plates (Derjaguin approximation). 

For h < 26, the situation is much more complex. One not only needs to know 4>(z) for each layer, but how 4>(z) changes as the two particles approach, i.e. 4>(z,h) this may well depend on the time-scale of the approach, i.e. the equilibrium path may not be followed. Scheutjens and Fleer (25) in an extension of their model for polymer adsorption have analysed the situation for two interacting uncharged parallel, flat plates carrying adsorbed, neutral homopolymer, interacting under equilibrium conditions. Only a semi-quantitative picture will be presented here. 

Repulsive interaction per unit area between flat plates ... 

When heat and mass are transferred simultaneously, the two processes interact through the Gr and Gq terms in Eq. (10-12) and the energy and diffusion equations. Although solutions to the governing equations are not available for spheres, results should be qualitatively similar to those for flat plates (T4), where for aiding flows (Gr /Gq > 0) the transfer rate and surface shear stress are increased, and for opposing flows (Gr Gq < 0) the surface shear stress is predicted to drop to zero yielding an unstable flow. 

FIG. 11.8 Schematic illustration that shows how the repulsion between spheres may be calculated from the interaction between flat plates. 

Also, in the 1930 s London (9) indicated the quantum mechanical origin of dispersion forces between apolar molecules and in subsequent work extended these ideas to interaction between particles (10). It was shown that whereas the force between molecules varied inversely as the seventh power of the separation distance, that between thick flat plates varied inversely as the third power of the distance of surface separation. These ideas lead directly to the concept of a "long range van der Waals attractive force. A similar relationship was found for interaction between spheres (10). 

Fig. 1. Effect of surface potential fa) and Hamaker s constant (b) upon the total interaction energy profile for a sphere of radius 6-7 /an approaching a flat plate in a solution having s = 74-3 and tc 1 = 8A.

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]... 

Van der Waals interactions between identical solid particles are always attractive [7]. However, if the Hamaker constant of the suspending fluid is intermediate between the Hamaker constants of two different particles, the van der Waals interactions will be repulsive [9]. Moreover, in view of the finite speed of propagation of electromagnetic radiation, the response of a molecule to perturbations in the electric field deriving from another nearby molecule is not instantaneous. Retardation effects are observable at separation distances as small as 1 or 2 nm, and they become prominent at larger distances (>10 nm) [50]. Gregory [50] has proposed a simple expression for describing retarded van der Waals interactions between flat plates ... 

We shall further consider the interaction between two flat plates immersed into an aqueous electrolyte solution with an emphasis on the dielectric colloid material. These dielectric plates bear the surface charge of a constant density. All these assumptions simplify the boundary conditions (33)-(36) sufficiently, yielding... 

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]. 

These potentials for interactions between flat plates provide the basis for addressing interactions between spheres of radius a L through the Derjaguin approximation... 

How was the theoretical DLVO curve in Figure 1.12 obtained The DLVO model [18, 19] postulates that the appropriate thermodynamic potential energy of interaction between two parallel flat plates can be described in terms of two components a repulsive term VR, resulting from the overlap of electrical double layers, and an attractive van der Waals interaction, VA. It also assumes that these interactions are additive, so that the total potential energy can be written as... 

If you like, FR(DLVO) = U m (SI) for the Helmholtz free energy of the interaction for two flat plates is our baseline, a point at which everyone can agree. It was the next step in the SI formalism, the calculation of the Gibbs free energy, that sundered the colloid world. It may seem incredible to scientists outside the held that the basic thermodynamic relation... 

It is only recently that methods for calculating the interaction between dissimilar flat plates in a form convenient for AFM data analysis have been developed for both constant charge and constant potential interactions 46). Previously, the approximation due to Hogg et al. 147 was widely used because of the ease of calculation. This approximation, though, is valid for constant low-potential interactions only. 

Hamaker s linear superposition theory provides a single parameter (the Hamaker constant) to describe the interaction between macroscopic bodies. The force per unit area, /132, between infinite flat plates of media 1 and 2, separated by medium 3, is given... 

FIGURE lOE Geometrical coDstruction used in the calculation of the interaction between two dissimilar spherical particles from the interaciian of two infinite flat plates. 

The Deijaquin appitmmation [9] can be used to compute the interaction energy for two spheres of radius Oj and 02> (h if the flat plate... 

In principle G(h) is obtainable from /7(h) by integration. Alternatively, if G(h) is known, /7(h) Ccin be found by differentiation. Figure 5.12, which is an elaboration of fig. 1.4.2 gives an illustration. Figure 5.12a is typical for a Deryagin-Landau-Verwey-Overbeek (DLVO) type of interaction. For the case at hand (flat plates) G is expressed per unit area SI units are J m. The upper curve consists of an attractive Van der Waals part, a repulsive electrostatic component and a very short range steep repulsion. Such behaviour may be representative for various wetting films. 

Example A14.1. Calculating Interaction Energy Calculate the interaction energy of two flat plates with Gouy layers of 25-mV surface potential (assumed to be constant), at 25°C in a 10 M NaCl solution (1.0 mol m ), for a separation distance of 10 nm. A Hamaker constant /ln(2) = 10 J will be used. 

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