Brownian hard particles

Fig.2.1. Eigenvalues of the energy scattering kernel for a hard-sphere Brownian test particle as a function of mass ratio, y = [2.16]...

The structure formation in an ER fluid was simulated [99]. The characteristic parameter is the ratio of the Brownian force to the dipolar force. Over a wide range of this ratio there is rapid chain formation followed by aggregation of chains into thick columns with a body-centered tetragonal structure observed. Above a threshold of the intensity of an external ahgn-ing field, condensation of the particles happens [100]. This effect has also been studied for MR fluids [101]. The rheological behavior of ER fluids [102] depends on the structure formed chainlike, shear-string, or liquid. Coexistence in dipolar fluids in a field [103], for a Stockmayer fluid in an applied field [104], and the structure of soft-sphere dipolar fluids were investigated [105], and ferroelectric phases were found [106]. An island of vapor-liquid coexistence was found for dipolar hard spherocylinders [107]. It exists between a phase where the particles form chains of dipoles in a nose-to-tail... 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

To evaluate the volume integrals in (84), the radial distribution function must be known. The pair distribution function affected by the Brownian motion and the relative electrophoretic velocity between a pair of particles is generally nonuniform and nonisotropic. When the particles are sufficiently small so that Brownian motion dominates, one can use a simple distribution function based on hard-sphere potential... 

For a binary mixture, if experimental diffusivities do not exist over the whole range of concentration, an interpolation of the diffusivities at infinite dilution D k] J is used. In calculating the diffusivities at infinite dilution by the Stokes-Einstein relation, we consider small isolated hard spheres, submerged in a liquid, that are subjected to Brownian motion The friction of the spheres in the liquid is given by the Stokes law Einstein used the Stokes law to calculate the mean-square displacement of a particle. The displacement increases linearly with time, and the proportionality constant is the Stokes-Einstein diffusivity... 

Figure 10. The potential F in the presence of lich the motion of the Brownian particle takes place. The right well is (xmsideted. The dashed curve expresses the harnuniic e q>anaon of the potential V around x a. The r ions 0 < x < a and a.< x

For monodisperse or unimodal dispersion systems (emulsions or suspensions), some literature (28-30) indicates that the relative viscosity is independent of the particle size. These results are applicable as long as the hydrodynamic forces are dominant. In other words, forces due to the presence of an electrical double layer or a steric barrier (due to the adsorption of macromolecules onto the surface of the particles) are negligible. In general the hydrodynamic forces are dominant (hard-sphere interaction) when the solid particles are relatively large (diameter >10 (xm). For particles with diameters less than 1 (xm, the colloidal surface forces and Brownian motion can be dominant, and the viscosity of a unimodal dispersion is no longer a unique function of the solids volume fraction (30). 

Figure 11 shows the relative-viscosity-concentration behavior for a variety of hard-sphere suspensions of uniform-size glass beads. Even though the particle size was varied substantially (0.1 to 440 xm), the relative viscosity is independent of the particle size. However, when the particle diameter was small ( 1 fJLm), the relative viscosity was calculated at high shear rates, so that the effect of Brownian motion was negligible. Figure 8 shows that becomes independent of the particle size at high shear stress (or shear rate). 

Since the particles forming a fractal gel are reactive for aggregation over their total surface, the free energy of the system would decrease when the coordination number of the particles increases, or in other words when more junctions between particles would be formed. However, since the particles are more or less immobilized in the network, this is hardly possible, especially if the gel is constrained in a vessel and sticks to the vessel wall. However, if

Brownian motion), so that a new junction between particles may occasionally form. This causes tension in the strand(s)... 

With surface forces absent, in the limit of Pe l, the distribution of particles is only slightly altered from the Einstein limit. To order cj> which takes into account two-particle interactions, Batchelor (1977) calculated the effect of Brownian motion on the stress field in a suspension of hard spheres and determined the low shear limit relative viscosity to be given by the Einstein relation with an added term equal to 6.14>. This result is found to agree satisfactorily with experiment for shear limit with interparticle surface forces, including questions as to the existence of a uniquely defined asymptotic limit, we choose not to discuss this case further, instead referring the reader to Russel et al. (1989) and van de Ven (1989). 

If particles are known to be spherical in shape and nondeformable in the relatively weak flow fields associated with Brownian motion (this may be expected in the case of synthetic latex particles, many proteins, and viruses and probably also holds for certain emulsion particles with rigid ordered interfaces, the Stokes radius will closely correspond to the hard sphere radius R, related to Rg through Rg = 3/5 R and may also be similar to that observed in the electron microscope Rem. The value of Rg should, however, on detailed inspection be greater than the radii measured by the latter methods because it includes bound solvent molecules. The discrepancy can be used to estimate the degree of solvation 81 grams solvent/gram of the particle through the relation ... 

Particles elevate the viscosity of the medium (water) through viscous interaction with the water. Thermal or Brownian motion of the particles contributes to this at low rates of shear, but this contribution diminishes with increasing shear rate. At very high rates of shear and with high particle volume fraction, instabilities in the tendency of particles to align in layers with the flow field can result in dilatency. The rheology of hard sphere dispersions has become quite well understood and quantified by theory and experiment, especially in the last decade. 

A way to relate particle size of hard spheres (solid or colloidal particles, or very viscous drops) to shear rate and themnal energy (Brownian motion) is by means of a dimensionless parameter, the Pedet number (Pe). which is... 

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