Forces between particles retarded

The earliest quantitative theory to describe van der Waals forces between two colloidal particles, each containing a statistically large number of atoms, was developed by Hamaker, who used pairwise summation of the atom-atom interactions. This approach neglects the multi-body interactions inherent in the interaction of condensed phases. The modem theory for predicting van der Waals forces for continua was developed by Lifshitz who used quantum electrodynamics [19,20] to account for the many-body molecular interactions and retardation within and between materials. Retardation is a reduction of the interaction because of a phase lag in the induced dipole response that increases with distance. 

Casimir and Polder also showed that retardation effects weaken the dispersion force at separations of the order of the wavelength of the electronic absorption bands of the interacting molecules, which is typically 10 m. The retarded dispersion energy varies as R at large R and is determined by the static polarizabilities of the interacting molecules. At very large separations the forces between molecules are weak but for colloidal particles and macroscopic objects they may add and their effects are measurable. Fluctuations in particle position occur more slowly for nuclei than for electrons, so the intermolecular forces that are due to nuclear motion are effectively unretarded. A general theory of the interaction of macroscopic bodies in terms of the bulk static and dynamic dielectric properties... 

The attractive-force component between particles in a colloidal system is developed by summation of the London dispersion forces between all atom pairs in the particles. Neglecting retardation effects, the expression for the attractive energy Va between two particles of radius a at a distance of separation Hq, for a > Ho, is... 

For the simplest case, the net attractive force between two particles at small separation distance after neglecting the retardation correction is given as... 

Hydrodynamic Forces Fluid mechanical interactions between particles arise because a particle in motion in a fluid induces velocity gradients in the fluid that influence the motion of other particles when they approach its vicinity. Because the fluid resists being squeezed out from between the approaching particles, the effect of so-called viscous forces is to retard the coagulation rate from that in their absence. 

Consider the coalescence of drops with fiilly retarded (delayed) surfaces (which means they behave as rigid particles) in a developed turbulent flow of a lowconcentrated emulsion. We make the assumption that the size of drops is much smaller than the inner scale of turbulence R Ao), and that drops are non-deformed, and thus incapable of breakage. Under these conditions, and taking into account the hydrodynamic interaction of drops, the factor of mutual diffusion of drops is given by the expression (11.70). To determine the collision frequency of drops with radii Ri and Ri (Ri < Ri), it is necessary to solve the diffusion equation (11.36) with boundary conditions (11.39). Place the origin of a spherical system of coordinates (r, 0,0) into the center of the larger particle of radius i i. If interaction forces between drops are spherically symmetrical, Eq. (11.36) with boundary conditions (11.39) assumes the form... 

We start with the case when the drop surface is completely retarded, in other words, drops can be considered as undeformed particles. We also assume that coalescence occurs only due to the joint action of turbulent pulsations and molecular attractive forces. The force of molecular attraction between two spherical par-tides is given by the formula (11.100), which implies that this force is determined by the distance between particle surfaces and does not depend on their mutual orientation, i.e. is spherically-symmetrical with respect to the center of the particle of radius. Since the force of molecular attraction manifests itself only at small clearances A between particles, we shall take its asymptotic expression at A->0... 

Coagulation has been considered to be the result of van der Waals attraction which draws two particles together at the moment of collision, unless opposed by a hydration barrier layer or by the electrostatic repulsion forces between the similarly charged particles, or both. There are therefore two factors that retard coagulation-of silica ... 

Figure 2. Retardation correction factor (f) for dispersion force attractions between spherical particles of radius (a) at separation distance (H), with dispersion force wavelength Xj. (10)...

Upon discharge from the nozzle, the droplets are simultaneously accelerated by gravity and retarded by fluid-fricuonal drag The drag force acts in both the honzontal (x directional) and vertical (j directional) dimensions. Thus, two-dimensional particle dvnamics must be considered to properly size the vertical dimension of the vapor space and the horizontal distance between inlet and outlet nozzles (Fig. 4). 

Suppose that the interaction forces establish an energy barrier that retards the motion of particles both toward and away from the collector. If this barrier reduces the adsorption and desorption rates significantly, particles near the primary minimum will have time to achieve a balance between the interaction forces and Brownian motion, before their population changes. Integration of Equation (6) with j 0 and D = mkT leads Lo a Boltzmann distribution... 

Casimir, H.B.G. Polder, D. The influence of retardation on the London-van der Waals forces. Physical Review 1948, 73, 360-372 Overbeek, J.T.G. The interaction between colloidal particles. 

---