Interaction between particles

The remainder of this contribution is organized as follows. In section C2.6.2, some well studied colloidal model systems are introduced. Methods for characterizing colloidal suspensions are presented in section C2.6.3. An essential starting point for understanding the behaviour of colloids is a description of the interactions between particles. Various factors contributing to these are discussed in section C2.6.4. Following on from this, theories of colloid stability and of the kinetics of aggregation are presented in section C2.6.5. Finally, section C2.6.6 is devoted to the phase behaviour of concentrated suspensions. 

In many colloidal systems, both in practice and in model studies, soluble polymers are used to control the particle interactions and the suspension stability. Here we distinguish tliree scenarios interactions between particles bearing a grafted polymer layer, forces due to the presence of non-adsorbing polymers in solution, and finally the interactions due to adsorbing polymer chains. Although these cases are discussed separately here, in practice more than one mechanism may be in operation for a given sample. 

Fundamental Limitations to Beers Law Beer s law is a limiting law that is valid only for low concentrations of analyte. There are two contributions to this fundamental limitation to Beer s law. At higher concentrations the individual particles of analyte no longer behave independently of one another. The resulting interaction between particles of analyte may change the value of 8. A second contribution is that the absorptivity, a, and molar absorptivity, 8, depend on the sample s refractive index. Since the refractive index varies with the analyte s concentration, the values of a and 8 will change. For sufficiently low concentrations of analyte, the refractive index remains essentially constant, and the calibration curve is linear. 

Dispersion of a soHd or Hquid in a Hquid affects the viscosity. In many cases Newtonian flow behavior is transformed into non-Newtonian flow behavior. Shear thinning results from the abiHty of the soHd particles or Hquid droplets to come together to form network stmctures when at rest or under low shear. With increasing shear the interlinked stmcture gradually breaks down, and the resistance to flow decreases. The viscosity of a dispersed system depends on hydrodynamic interactions between particles or droplets and the Hquid, particle—particle interactions (bumping), and interparticle attractions that promote the formation of aggregates, floes, and networks. 

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

The interaction between particles belonging to the same species is a hard sphere interaction... 

A simple model for interactions between particles in an associating bulk fluid consists of a particle-particle potential and the interactions between sites belonging to different molecules. Supposing that each molecule has M sites, the potential of interaction between molecules 1 and 2 is [14]... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

The charges of matrix ions are ez% = ez = ez and the density of the matrix subsystem is p (p+ = p- = p /2). We define the functions y (r) describing the interactions between particles. In particular, the interactions between matrix ions are given as... 

In the previous section we saw on an example the main steps of a standard statistical mechanical description of an interface. First, we introduce a Hamiltonian describing the interaction between particles. In principle this Hamiltonian is known from the model introduced at a microscopic level. Then we calculate the free energy and the interfacial structure via some approximations. In principle, this approach requires us to explore the overall phase space which is a manifold of dimension 6N equal to the number of degrees of freedom for the total number of particles, N, in the system. 

Equation (2) is valid only for very dilute suspensions of nondeformable, smooth, uniform spheres. It assumes a Newtonian liquid phase and neglects interaction between particles, a plausible condition when the volume of the solid phase is small compared with the liquid phase. 

In this model the gas particles are assumed to show no interactions between each other. This model can be realized or at least approached closely in a physical sense, since under conditions of low pressure and high temperatures interaction between particles becomes progressively weaker. Another example consists in the relationship between relativistic and classifical mechanics. The relativistic expression for momentum. 

Detailed consideration of the interaction between particles and fluids is given in Volume 2 to which reference should be made. Briefly, however, if a particle is introduced into a fluid stream flowing vertically upwards it will be transported by the fluid provided that the fluid velocity exceeds the terminal falling velocity m0 of the particle the relative or slip velocity will be approximately o- As the concentration of particles increases this slip velocity will become progressively less and, for a slug of fairly close packed particles, will approximate to the minimum fluidising velocity of the particles. (See Volume 2, Chapter 6.)... 

The interaction between particle and surface and the interaction among atoms in the particle are modeled by the Leimard-Jones potential [26]. The parameters of the Leimard-Jones potential are set as follows pp = 0.86 eV, o-pp =2.27 A, eps = 0.43 eV, o-ps=3.0 A. The Tersoff potential [27], a classical model capable of describing a wide range of silicon structure, is employed for the interaction between silicon atoms of the surface. The particle prepared by annealing simulation from 5,000 K to 50 K, is composed of 864 atoms with cohesive energy of 5.77 eV/atom and diameter of 24 A. The silicon surface consists of 45,760 silicon atoms. The crystal orientations of [ 100], [010], [001 ] are set asx,y,z coordinate axes, respectively. So there are 40 atom layers in the z direction with a thickness of 54.3 A. Before collision, the whole system undergoes a relaxation of 5,000 fsat300 K. 

From comparison of the optical properties of particles deposited on the same substrate and differing by their organization (Figs. 7 and 8) it can be concluded that the appearance of the resonance peak at 3.8 eV is due to the self-organization of the particles in a hexagonal network. This can be interpreted in terms of mutual dipolar interactions between particles. The local electric field results from dipolar interactions induced by particles at a given distance from each other. Near the nanocrystals, the field consists of the ap-... 

We introduced the technique for measuring the weak interaction forces acting between two particles using the photon force measurement method. Compared with the previous typically used methods, such as cross-correlation analysis, this technique makes it possible to evaluate the interaction forces without a priori information, such as media viscosity, particle mass and size. In this chapter, we focused especially on the hydrodynamic force as the interaction between particles and measured the interaction force by the potential analysis method when changing the distance between particles. As a result, when the particles were dose to each other, the two-dimensional plots of the kinetic potentials for each particle were distorted in the diagonal direction due to the increase in the interaction force. From the results, we evaluated the interaction coeffidents and confirmed that the dependence of the... 

Although the concept of non-interacting particles is an idealization, the model may be applied to real systems as an approximation when the interactions between particles are small. Such an approximation is often useful as a starting point for more extensive calculations, such as those discussed in Chapter 9. 

The forces of interaction between particles present barriers to their flow and dispersion. The major forces of interaction are van der Waals, electrostatic, and capillary forces [34],... 

As has already been mentioned, to carry out such a calculation is not a matter of thermodynamics, but requires adopting certain assumptions on the structure of the system and on interactions between particles. 

The velocity of motion of the particles depends on their dimensions and shape, on the interaction (e.g. association) between the solvent molecules and finally on the interaction between particles of the dissolved substance and solvent molecules. Consider the simplest case, where the molecule of the dissolved substance is much larger than the solvent molecule, is spherical and the interaction between the solute molecules and the solvent is negligible. Then the motion of the particles of the solute can be considered as the motion of spherical particles with radius rf through a viscous medium with viscosity coefficient rj. The velocity v is then described by the Stokes law ... 

All nt are determined, in principle, by the equations of statistical mechanics, since they are one-particle distributions.4 As such, each n, depends on the interactions between particles of species i and all other particles in the system, whether belonging to the metal or to the other phase. If there is a geographical separation of particles of species i from, say, particles of species k (as when i and k belong to different phases), the interaction between particles k and a particle of species i near the surface may be averaged over positions of particles /c, i.e., no correlation is assumed between the particles of the two species, so that the particles k become a source of external field for particles i. For a particle i far from the surface, the interaction is probably unimportant (unless it is a long-range electrostatic interaction). 

In this paper some of the current thinking in three closely-related areas is highlighted polymer adsorption the effect of polymer on the pairwise interaction between particles and the effect of polymers on dispersion stability. 

Pairwise Interaction Between Particles in the Presence of Polymer... 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

In addition, DNS of turbulent flow in a periodic box offer interesting opportunities for studying in a fully resolved mode the intimate details of the flow field, its interaction with particles and the mutual interaction between particles (including particle-particle collisions and coalescence). Such simulations may yield new insights see, e.g., Ten Cate et al. (2004) and Derksen (2006b). The same can be said about our understanding of particle-turbulence interactions in wall-bounded flows this has increased due to Portela and Oliemans (2003) exploiting both DNS and LES and due to Ten Cate et al. (2004). 

It is clear from the integration limits in Eq. (A.4) that the sequence of interactions between particles 1 and 2 must precede the interactions of the other types. Moreover, the expression (A.4) corresponds to the ensemble of contributions where the interactions between particles 1 and 3 are arbitrarily ordered with respect to interactions between particles 2 and 4 and it follows from the fact that the group (1,3) ignores the group (2,4) during the interval that extends from the initial instant to the instant rn. One can easily verify that this chronological ordering of the events is already contained in formula (A.3) (see the definition (61) of (2)2>(34))-... 

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