Suspension models hard sphere systems

This section on concentrated suspensions discusses the rheological behavior of sj tems which are colloidally stable and colloidally unstable suspensions. For stable sj tems, the rheology of sterically stabilized and electrostatically stabilized systems wiU be considered. For sterically stabilized suspensions, a hard sphere (or hard particle) model has been successfid. Concentrated suspensions in some cases behave rheologically like concentrated polymer solutions. For this reason, a discussion of the viscosity of concentrated polymer solutions is discussed next before a discussion of concentrated ceramic suspensions. 

Our main focus in this chapter has been on the applications of the replica Ornstein-Zernike equations designed by Given and Stell [17-19] for quenched-annealed systems. This theory has been shown to yield interesting results for adsorption of a hard sphere fluid mimicking colloidal suspension, for a system of multiple permeable membranes and for a hard sphere fluid in a matrix of chain molecules. Much room remains to explore even simple quenched-annealed models either in the framework of theoretical approaches or by computer simulation. 

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). 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

The objective of the polarization model is to relate the material parameters, such as the dielectric properties of both the liquid and solid particles, the particle volume fraction, the electric field strength, etc., to the rheological properties of the whole suspension, in combination with other micro structure features such as fibrillatcd chains. A idealized physical model ER system—an uniform, hard dielectric sphere dispersed in a Newtonian continuous medium, is usually assumed for simplification reason, and this model is thus also called the idealized electrostatic polarization model. The hard sphere means that the particle is uncharged and there are no electrostatic and dispersion interactions between the particles and the dispersing medium before the application of an external electric field. For the idealized electrostatic polarization model, there are roughly two ways to deal with the suspensions One is to consider the Brownian motion of particle, and another is to ignore the Brownian motion and particle inertia. For both cases the anisotropic structure of such a hard sphere suspension is assumed to be represented by the pair correlation function g(r,0), derived by... 

Figure 19-8. Measured second virial coefficients ofSTA (soUd squares) in dffferent background salt concentrations compared with data on a number of proteins (Lysotyme, BPTI open circles) in different buffer solutions. The second virial coefficients are nondimensionalized with the hard sphere value and plotted against the solubility (volume fraction 0at) of the respective species. The solid lines are calculations of the attractive Yukawa potential with two different ranges of attractions (2ak) of 7 and 15. The values of 7 and 15 indicate that attractions between the particles are short ranged. The experimental datafor STA (at high salt concentrations) and proteins collapse within the narrow range of attractions which are only a fraction of the particle diameter. The collapse also indicates thatproteins and STA are thermodynamically similar, iftwo suspensions have the same B2 then they have the same solubility. This plot also provides an opportunity to extract interaction potential parameters for a given experimental system in a model independent manner. For detailed discussions, please refer to (Ramakrishnan, 2000).

The mechanical properties of suspensions containing a narrow size distribution of particles have been studied extensively because they offer the best chances for testing models for flow behavior. The most detailed studies can be found for hard spheres where particles experience only volume exclusion, thermal and hydrodynamic interactions. Based on the models developed for these systems, a great deal can be learned about the behavior of suspensions experiencing longer range repulsions and attractions. 

Latex dispersions have attracted a great deal of interest as model colloid systems in addition to their industrial relevance in paints and adhesives. A latex dispersion is a colloidal sol formed by polymeric particles. They are easy to prepare by emulsion polymerization, and the result is a nearly monodisperse suspension of colloidal spheres. These particles usually comprise poly(methyl methacrylate) or poly(styrene) (Table 2.1). They can be modified in a controlled manner to produce charge-stabilized colloids or by grafting polymer chains on to the particles to create a sterically stabilized dispersion. Charge-stabiHzed latex particles obviously interact through Coulombic forces. However, sterically stabilized systems can effectively behave as hard spheres (Section 1.2). Despite its simpHcity, the hard sphere model is found to work surprisingly well for sterically stabilized latexes. 

A collection of hard, identical spheres is the simplest possible model system that undergoes a first order phase transition. For low packing fractions the particles are in a liquid state, but when the packing fractions exceeds a value of 49.4% a ordered solid state becomes more stable. This was first shown in computer simulations by Hoover and Ree [27] in 1968. The experimental realization of a colloidal suspension that closely mimics the phase behavior of hard spheres followed about 20 years later and was a milestone in soft matter physics [28, 29]. More recently the phase transition kinetics of hard sphere colloids has been studied extensively in experiments [5, 30, 31]. However as mentioned in the introduction the interpretation of the data with CNT was rather indirect. 

Further simplification may arise from the type of scientific question being addressed by the coarse-grained model. If the interest is not in specific material properties of a given chemical species, but rather in generic behavior and mechanisms, as is the case for the examples we will discuss in this article, then the details of the parameterization are less important than a model that is both conceptually simple and computationally efficient. The standard model for particulate suspensions is a system of hard spheres, while for polymer chains the Kremer-Grest model [12] has proved to be a valuable and versatile tool. Here, the beads interact via a purely repulsive Lermard-Jones (or WCA [13]) potential. 

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