Dynamical interaction between particles

Father - My daughter, do you known that Solomon says If bad boys attract you, don t follow them. Daughter - But what should I do, if good boys attract me  

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S. R. Raghavan, J. Hou, G.L. Baker, and S.A. Kahn Colloidal Interactions between Particles with Tethered Nonpolar Chains Dispersed in Polar Media Direct Correlation Between Dynamic Rheology and Interaction Parameters. Langmuir 16, 1066 (2000). 

New difficulties arise when we try to take into account the dynamical interaction of particles caused by pair potentials U(r) mutual attraction (repulsion) leads to the preferential drift of particles towards (outwards) sinks. This kind of motion is described by the generalization of the Smoluchowski equation shown in Fig. 1.10. In terms of our illustrative model of the chemical reaction A + B —> B the drift in the potential could be associated with a search of a toper by his smell (Fig. 1.12). An analogy between Schrodinger and Smoluchowski equations is more than appropriate indeed, it was used as a basis for a new branch of the chemical kinetics operating with the mathematical formalism of quantum field theory (see Chapter 2). 

As it was mentioned above, up to now only the dynamic interaction of dissimilar particles was treated regularly in terms of the standard approach of the chemical kinetics. However, our generalized approach discussed above allow us for the first time to compare effects of dynamic interactions between similar and dissimilar particles. Let us assume that particles A and B attract each other according to the law U v(r) = — Ar-3, which is characterized by the elastic reaction radius re = (/3A)1/3. The attraction potential for BB pairs is the same at r > ro but as earlier it is cut-off, as r ro. Finally, pairs AA do not interact dynamically. Let us consider now again the symmetric and asymmetric cases. In the standard approach the relative diffusion coefficient D /D and the potential 1/bb (r) do not affect the reaction kinetics besides at long times the reaction rate tends to the steady-state value of K(oo) oc re. 

Beginning with pioneering works by Kuhn and Kuhn (1945), the relaxator attracted the attention of researchers (Bird et al. 1987b). Further, on, we shall consider the results concerning the dynamics of the dilute suspension of the dumbbell while the hydrodynamic interaction between particles inside each dumbbell is taken into account in correct form. 

In all but the most basic cases of very dilute systems, with microstructural elements such as rigid particles whose properties can be described simply, the development of a theory in a continuum context to describe the dynamical interactions between structure and flow must involve some degree of modeling. For some systems, such as polymeric solutions, we require modeling to describe both polymer-solvent and polymer-polymer interactions, whereas for suspensions or emulsions we may have an exact basis for describing particle-fluid interactions but require modeling via averaging to describe particle-particle interactions. In any case, the successful development of useful theories of microstructured fluids clearly requires experimental input and a comparison between experimental data and model... 

The force arising from the potential is F, while R is a gaussian random force. The net effect of the collisions , i.e. dynamical interactions between the particle and solvent molecules, is thus approximately accounted for by the frictional, or damping force, Fj. = —fiC,x, where is a friction constant related to the time correlation of the random force ... 

Molecular dynamics requires the description of the interactions between particles, the force field, the validity and quality of the results depend critically on the accuracy of this parameterization. The force field approach fixes molecular connectivity and it is not possible to create (or break) chemical bonds or study chemical reactions. Recently an approach that involves the combination of classical mechanics with electronic structure caleulations allows the intemuclear forces to be calculated on the fly from electronic structure calculation as the molecular dynamics develops [19],... 

Particles in a fluid in which a velocity gradient du/dy exists have a relative motion that may bring them into contact and cause coagulation (Figure 13.A.1). Smoluchowski in 1916 first studied this coagulation type assuming a uniform shear flow, no fluid dynamic interactions between the particles, and no Brownian motion. This is a simplification of the actual physics since the particles affect the shear flow and the streamlines have a curvature around the particles. 

In the preceding section our analysis for Brownian diffusion assumed the particles were diffusing points, whereas for interception the center of a particle of finite size was assumed to follow the undisturbed streamline near a large collector. In both cases, no other forces were considered to act on the particles, and when they struck the collector it was assumed that they adhered. In reality, however, even in the absence of inertia there may be other external forces acting on the particles, including London forces of attraction, gravitational hydro-dynamic interactions between the particle and collector, and double layer repulsive forces. 

The interaction between particles in combination with the dynamical equation determines how the system evolves in time. At the fundamental level, the only important force at the atomic level is the electromagnetic interaction. Depending on the choice of system description (particles), however, this may result in different effective forces. 

A spatially varying diffusivity can model not only inhomogeneities in the medium but also hydro-dynamic interactions between the Brownian particles and channel walls. The diffusivity is then replaced by a diffusion tensor D(r), and instead of Eq. 1, one obtains... 

Dukhin et al. [83-85] have performed the direct calculation of the CVI in the situation of concentrated systems. In fact, it must be mentioned here that one of the most promising potential applicabilities of these methods is their usefiilness with concentrated systems (high volume fractions of solids, 4>) because the effect to be measured is also in this case a collective one. The first generalizations of the dynamic mobility theory to concentrated suspensions made use of the Levine and Neale cell model [86,87] to account for particle-particle interactions. An alternative method estimated the first-order volume fraction corrections to the mobility by detailed consideration of pair interactions between particles at all possible different orientations [88-90]. A comparison between these approaches and calculations based on the cell model of Zharkikh and Shilov [91] has been carried out in Refs. [92,93],... 

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