Boltzmann model

The interaction potential model, originally proposed by Shan and Chen,37 and henceforth referred to as the S-C model, introduces k distribution functions for a fluid mixture comprising of k components. Each distribution function represents a fluid component and satisfies the evolution equation. The non-local interaction 

The free parameter, pk, relates to the speed of sound of a region of pure Mi component, as  

The conservation of momentum at each collision in the absence of interaction force (i.e. in the case of F = 0), requires u to satisfy the following relation, 

A simple long-range interaction force between particles of the fcth component at site x and the Mi component at site x is introduced and the total fluid/fluid interaction force on the Mi component at site x is given by  

The continuity and momentum equations can be obtained for the fluid mixture as a single fluid using Chapman-Enskog expansion procedure in the nearly incompressible limit 43 

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Theissen O, Gompper G and Kroll D M 1998 Lattice-Boltzmann model of amphiphilic systems Euro. Phys. Lett. 42 419... 

Prediction of pKaS of Titratable Residues in Proteins Using a Poisson-Boltzmann Model of the Solute-Solvent System... 

Langevin simulations of time-dependent Ginzburg-Landau models have also been performed to study other dynamical aspects of amphiphilic systems [223,224]. An attractive alternative approach is that of the Lattice-Boltzmann models, which take proper account of the hydrodynamics of the system. They have been used recently to study quenches from a disordered phase in a lamellar phase [225,226]. 

Ludwig Boltzmann, "Model," Encyclopedia Britannica (New York, 1911), XVH 638640. As discussed further, there is a problem with the distinction between "model" and "analogy." Georges Canguilhem has claimed to demonstrate that the distinction is perfectly impossible in Canguilhem. Etudes d histoire et de philosophie des sciences (Paris Vrin, 1979). 

The steady-state flow numerical experiment was primarily designed to evaluate the phasic relative permeability relations. The numerical experiment is devised within the two-phase lattice Boltzmann modeling framework for the reconstructed CL microstructure, generated using the stochastic reconstruction technique described earlier. Briefly, in the steady-state flow experiment two immiscible fluids are allowed to flow simultaneously until equilibrium is attained and the corresponding saturations, fluid flow rates and pressure gradients can be directly measured and correlated using Darcy s law, defined below. 

Wolf-Gladrow, D. A., "Lattice-Gas Cellular Automata and Lattice Boltzmann Models An Introduction". Springer-Verlag, Berlin (2000). 

Anderson, C. F., and Record, M. T. Jr. (1980). The relationship between the Poisson-Boltzmann model and the condensation hypothesis An analysis based on the low salt form of the Donnan coefficient. Biophys. Chem. 11, 353—360. 

Antosiewicz J, Blachut-Okrasinska E, Grycuk T, Lesyng BA (2000) Correlation Between Protonation Equilibria in Biomolecular Systems and their Shapes Studies Using a Poisson-Boltzmann Model, Math. Sciences Applications, GAKUTO International Series, 14, pp 11-17... 

Wagoner J, Baker N (2004) Solvation forces on biomolecular structures a comparison of explicit solvent and Poisson-Boltzmann models, J Comput Chem, 25 1623-1629... 

K, though for higher rotational levels the populations were higher than predicted by a Boltzmann model. The population ratio of the v = 1 level to the v = 0 represented a vibrational temperature of about 2200 K, which, though non-Boltzmann, is expected from the corona discharge source. Thus two vibrational levels of CN(B) could be studied relatively free from rotational congestion. 

One typical consequence is that we refrain from considering smeared out distributions, as In the Polsson-Boltzmann model. Another implication is that image charges have to be taken into account. 

In the Polsson-Boltzmann model the surface charge Is considered homogeneous and smeared-out. This assumption Is retained In many of the more... 

Charge rearrangement can occur in any circumstance where there are formally charged groups which have some degree of mobility. The most mobile, and easiest to treat are salt ions in the solvent. The commonest way of treating this is via the Boltzmann model, where the local concentration c(r) of an ion of type i, valence zj, and bulk concentration q° at a position r in the solvent is given by... 

The Poisson-Boltzmann model is the. other major model for protein electrostatics, which has been widely used, especially because of its ability to treat salt effects. In applications, where the protein is not highly charged the Boltzmann term in Eqn. (15) can be linearized since e< >/kT < 1, giving... 

Sharp, K. A. (1995). Polyelectrolyte electrostatics Salt dependence, entropic and enthalpic contributions to free energy in the nonlinear Poisson-Boltzmann model. Biopolymers. 36 227-243. 

Poisson-Boltzmann model The PB approximation is one of the more elaborate continuum solvation models that take into account not only the charge density of the solute but also the mobile charge density within the surrounding continuum. Continuum models of solvation are in general capable of calculating absolute free energies of solvation. 

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