Inter-particle model

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

Small-angle neutron scattering (SANS) can be applied to food systems to obtain information on intra- and inter-particle structure, on a length scale of typically 10-1000 A. The systems studied are usually disordered, and so only a limited number of parameters can be determined. Some model systems (e.g., certain microemulsions) are characterized by only a limited number of parameters, and so SANS can describe them fully without complementary techniques. Food systems, however, are often disordered, polydisperse and complex. For these systems, SANS is rarely used alone. Instead, it is used to study systems that have already been well characterized by other methods, viz., light scattering, electron microscopy, NMR, fluorescence, etc. SANS data can then be used to test alternative models, or to derive quantitative parameters for an existing qualitative model. 

Nanocrystalline systems display a number of unusual features that are not fully understood at present. In particular, further work is needed to clarify the relationship between carrier transport, trapping, inter-particle tunnelling and electron-electrolyte interactions in three dimensional nan-oporous systems. The photocurrent response of nanocrystalline electrodes is nonlinear, and the measured properties such as electron lifetime and diffusion coefficient are intensity dependent quantities. Intensity dependent trap occupation may provide an explanation for this behaviour, and methods for distinguishing between trapped and mobile electrons, for example optically, are needed. Most models of electron transport make a priori assumptions that diffusion dominates because the internal electric fields are small. However, field assisted electron transport may also contribute to the measured photocurrent response, and this question needs to be addressed in future work. 

The results of this work show that even though the vapor pressure of vanadium is low, the transfer velocity of vanadium vapor is high and the rate of mass transfer in a fluidized bed is high. A high rate of vanadium transport to traps and a low rate of vanadium transport by transpiration are consistent with the vapor phase transport model. The vapor pressure of the vanadic acid follows a second order Freundlieh isotherm, which reflects a coverage dependent heat of adsorption. The rate of vanadium transfer from catalyst to trap is only weakly dependent on the number density of the catalyst or trap particles. This lack of dependence suggests that inter-particle collisions are not the dominant mechanism for vanadium transfer. Vanadium mobility in FCCU s is a complex issue dependent on many operating variables. 

The concentration of the electrolyte used in the mobile phase affects the value of the zeta potential and hence the flow. Knox et al. (13] investigated the effect of NaNOv concentration and pH on the zeta potential using ODS particles and found that it decreased at lower pHs and higher electrolyte concentration. From these results it would appear that a 0.001-0.01 M concentration range is the most acceptable for CEC. Very dilute solutions would give better zeta potentials, but would increase the thickness of the double layer and limit the particle size to a minimum of 1-2 pm 14(. Wan 14 extended Rice and Whitehead s theoretical model (15] of EOF in an open tube to predict the double-layer overlap effects in packed columns. The results published agreed with Knox s earlier work, the main conclusion being that electrolyte concentration has a major influence on EOF for low values of particle diameter and inter-particle porosity. 

Systems of randomly oriented magnetic nanoparticles randomly dispersed in a supporting medium or matrix and that interact via dipole-dipole forces (last subsection) are systems having several energetically equivalent supermoment orientational states, at given temperatures and applied fields. As such, it is relevant to compare their magnetic behaviors with both the observed behaviors of canonical SG systems (dilute magnetic alloys such as MnCu) and the theoretical predictions from overly simple SG models. This has lead to a productive examination of the effects of dipolar and other inter-particle interactions in synthetic nanoparticle model systems that is reviewed below. Hopefully, this will in turn motivate the development of more realistic theoretical models of disordered dipolar systems. 

As a result of magnetic inter-particle interactions, each nanoparticle will feel a net local interaction field that can be modelled by a time-dependent local applied field, Hint(t). If the time variation of the local interaction field is either very fast or very slow compared to the relevant characteristic times (e.g., i) of the particle, then Hi t(t) can in turn be modelled as a static local field, that will, of course, depend on temperature and macroscopic applied field. The distributions of particle positions, orientations, and supermoments will determine the distribution of local interaction fields. These interaction fields are present in zero applied field and dramatically affect the behaviors of the individual nanoparticles and, consequently, of the sample as a whole. They achieve this in two important ways. First, they change the equilibrium magnetic properties of the sample, giving rise, for example, to superferromagnetic ordering or interaction Curie-Weiss behaviors (see below). Second, and possibly more importantly, they affect dynamic response, via their influence on SP dwell times. 

Dpolar hteractbns. Inter-particle interactions are important and are often dipolar in nature. These interactions have, to date, not been modeled correctly. Realistic calculations must include both the spatial distributions of average dipolar interaction fields (Eqn. 19) and the temporal fluctuations of the local dipolar interaction field. The latter fluctuations must have characteristic times that are comparable to the SP fluctuation times of the particles since the interaction field is directly caused by the neighboring supermoments. Indeed, for this reason, the concept of an interaction field must be replaced by a proper handling of inter-particle spatial and temporal correlations. This will determine the dynamics in assemblies of interacting particles and has not yet been attempted. 

Mssbauer spectroscopy The inclusion of anisotropic fluctuations, modeled as T+ T in uniaxial symmetry, in the presence of applied magnetic fields, exchange anisotropy, or inter-particle interactions, must be used as a starting assumption unless the more restricted assumption that all relevant fluctuations are much faster than the measurement frequency (x+, t Xm, in uniaxial symmetry) is justified independently. All other relevant realistic features (distributions of characteristics and properties ) must also be included, by applying as many justified theoretical constraints as possible. 

Main recent developments in magnetic nanoparticle systems Measurements on single magnetic nanoparticles Synthetic model systems ofmagnetic nanoparticles Inter-particle interactions and collective behavior Noteworthy attempts at dealing with nanoparticle complexity Interpreting the Mossbauer spectra of nanoparticle systems Needed areas of development... 

Sommerfeld M (2001) Validation of a stochastic Lagrangian modelling approach for inter-particle collisions in homogeneons isotropic turbulence. Int J Multiphase Flow 27 1829-1858... 

Such relatively simple kinetic models also allowed one to analyze simultaneously the development of radical processes and mass-transfer on different levels (inside catalyst pores, in inter-particle space, in reactor bulk—see for instance Bristolfi et al, 1992 Couwenberg, 1994 Hoebink et al, 1994 Reyes et al, 1993). This can be considered as another important achievement. 

This brief summary of compatibilizer effects on the flow behavior of polymer blends substantiate that many aspects of the behavior remain unclear. For example, the effects of limited miscibihty of copolymer, the copolymer structure and molecular weight, micelle formations and inter-particle interactions are still beyond the capability of the proposed models. Similarly, little information has been published on copolymer/homopolymer blends. 

From the mechanical standpoint, the particles have a density and are interlinked by contacts which are defined by their normal and tangential stiffnesses, a friction coefficient and a microscopic cohesion. A cohesive behaviour model serves to define a fracture threshold that determines the normal and tangential forces that can be withstood by the inter particle links. Hence, a micro crack appear when the contact between two adjacent particles is broken. When a connection between several micro cracks takes place, it can generate a new fracture in the model. 

Additional initial and boundary conditions have to be specified for inter-particle transport according to the scenario to be modelled. 

Model calculations using smooth" geometric constructions. Wires 12 nm diameter, 21 nm centre-to-centre hexagonal standing rods cubic gyroid 50 nm unit cell, consisting of cylindrical rods of 12 nm diameter nanoparticles 20 nm diameter spheres, 60% porosity with point-like inter-particle contact. 

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