Interparticle force calculations

Boundary layer formulation. Many membrane processes are operated in cross-flow mode, in which the pressurised process feed is circulated at high velocity parallel to the surface of the membrane, thus limiting the accumulation of solutes (or particles) on the membrane surface to a layer which is thin compared to the height of the filtration module [2]. The decline in permeate flux due to the hydraulic resistance of this concentrated layer can thus be limited. A boundary layer formulation of the convective diffusion equation can give predictions for concentration polarisation in cross-flow filtration and, therefore, predict the flux for different operating conditions. Interparticle force calculations are used in two ways in this approach. Firstly, they allow the direct calculation of the osmotic pressure at the membrane. This removes the need for difficult and extensive experi-... 

Equations 3 to 7 indicate the method by which terminal velocity may be calculated. Erom a hydrodynamic force balance that considers gravity, buoyancy, and drag, but neglects interparticle forces, the single particle terminal velocity is... 

The simple class of models just discussed is of interest because it is possible to characterize the decay of correlations rather completely. However, these models are rather far from reality since they take no account of interparticle forces. A next step in our examination of the decay of initial correlations is to find an interacting system of comparable simplicity whose dynamics permit us to calculate at least some of the quantities that were calculated for the noninteracting systems. One model for which reasonably complete results can be derived is that of an infinite chain of harmonic oscillators in which initial correlations in momentum are imposed. Since the dynamics of the system can be calculated exactly, one can, in principle, study the decay of correlations due solely to internal interactions (as opposed to interactions with an external heat bath). We will not discuss the most general form of initial correlations but restrict our attention to those in which the initial positions and momenta have a Gaussian distribution so that two-particle correlations characterize the initial distribution completely. Let the displacement of oscillator j from its equilibrium position be denoted by qj and let the momentum of oscillator j be pj. On the assumption that the mass of each oscillator is equal to 1, the momentum is related to displacement by pj =. We shall study... 

The DLVO-theory is named after Derjaguin, Landau, Verwey and Overbeek and predicts the stability of colloidal suspensions by calculating the sum of two interparticle forces, namely the Van der Waals force (usually attraction) and the electrostatic force (usually repulsion) [19],... 

From approximate calculations, Rumpf concluded that electrostatic forces have negligible influence on the strength of agglomerates. Goldstick [7] estimated that the maximum interparticle force due to magnetic attraction may be many times larger than the maximum strength due to electrostatic forces but is nevertheless small, even in comparison with van der Waals forces. 

Finally, the mean flux of particles of various sizes in the presence of bulk concentration gradients was calculated (Batchelor, 1976b, 1983), and the corresponding Fickian diffusion tensors systematically obtained. With no interparticle forces, the numerical results may be represented by the approx-... 

Repulsive interparticle forces cause the suspension to manifest non-Newtonian behavior. Detailed calculations reveal that the primary normal stress coefficient [cf. Eq. (8.7)] decreases like y 1. In contrast, the suspension viscosity displays shear-thickening behavior. This feature is again attributed to the enhanced formation of clusters at higher shear rates. 

Clearly, much work remains to be done in this subarea. Evolutionary operators need to be refined in order to deal efficiently with clusters and their ligand layers separately, as well as simultaneously. Suitable levels of theory for the interparticle forces have to be found and tested. And, of course, the size gap between application calculations and experiments needs to be closed. 

The dimerization is easily understood considering the optical potential created by the trapping laser. Figure 18.2b shows the calculated optical potential experienced by a silver nanoparticle that is fi ee to move in a Gaussian laser focus at a wavelength of 830 nm. The particle is also affected by the optical interparticle force from an immobilized silver particle located at different separations from the laser focus. It is clear that a deep potential minimum is induced when the trapped particle approaches the immobilized one, giving rise to spontaneous optical dimerization and a SERS hot spot in the optical trap. Note that the two particles are expected to ahgn parallel to the laser polarization in this case, as has been demonstrated experimentally recently [88]. 

On the other hand, solids are characterized by a very ordered structure in which each ion or molecule is surrounded by a fixed number of neighbors whose nature and orientation are determined by the interparticle forces in the crystal. These may be chiefly ion-ion interactions, as in an ionic crystal, or intermolecular forces, as in a molecular crystal. Because of the high state of order in crystals it is a reasonably straightforward problem to calculate their thermodynamic properties on the basis of quite simple statistical mechanical models. 

We can at this stage state that we are now in possession of a very accurate theoretical and numerical tool to calculate properties of any three-body system where the interparticle forces may be expressed as a function of the three interparticle distances. 

The classical DLVO theory of interparticle forces considers the interaction between two charged particles in terms of the overlap of their electric double layers leading to a repulsive force which is combined with the attractive London-van der Waals term to give the total potential energy as a function of distance for the system. To calculate the potential energy of attraction Va between solid spherical particles we may use the Hamaker expression ... 

For 6 > 6cr, a repulsive force is exerted between the two particles and / becomes larger than zero. For a more general case where 1 0, the interparticle force can also be calculated as a function of i/f and 9 using Eq. (14.1). A calculated result in Figure 14.4 for i/f = 30° shows that the equilibrium interparticle... 

Yeh and Berkowitz [9] proposed a simple and efficient method for treating systems with slab geometries. In their method, the nonperiodic dimension (e.g., the z direction in this case) was first extended to create some empty space, and then the periodic boundary conditions were applied in aU three directions to calculate interparticle forces. Finally, a correction force was added to each particle to remove the artifacts from the image charges due to periodic boundary conditions ... 

Similarly, the rate coefficient for a thermal reaction occurring with the influence of a spherically symmetric potential V(r) can be calculated from equation (63) by relating the cross-section to the potential. A useful relationship from classical scattering dynamics [16] is found in terms of the impact parameter, b. The impact parameter is the distance of closest approach between two particles in the absence of an interparticle force. At large separation, the collision trajectories of two particles will be parallel straight lines, and the impact parameter is the perpendicular distance between the trajectories. The cross-section is given by equation (64),... 

---