Interacting Particles

We may modify the above approaches to include interparticle forces. It is possible in this way to predict the non-Newtonian responses due to interparticle forces and yield values. These may be incorporated into cell models of suspensions. An early study of this was by Tanaka and White [36] using a power law fluid matrix. 

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Relaxations in the double layers between two interacting particles can retard aggregation rates and cause them to be independent of particle size [101-103]. Discrepancies between theoretical predictions and experimental observations of heterocoagulation between polymer latices, silica particles, and ceria particles [104] have promptetl Mati-jevic and co-workers to propose that the charge on these particles may not be uniformly distributed over the surface [105, 106]. Similar behavior has been seen in the heterocoagulation of cationic and anionic polymer latices [107]. 

A3.11.6.2 CLASSICAL SCATTERING THEORY FOR MANY INTERACTING PARTICLES... 

Vincent B, Edwards J, Emmett S and Greet R 1988 Phase separation in dispersions of weakly-interacting particles in solutions of non-adsorbing polymers Colloid Surf. 31 267-98... 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

Molecular dynamics (MD) studies the time evolution of N interacting particles via the solution of classical Newton s equations of motion. 

Simulations. In addition to analytical approaches to describe ion—soHd interactions two different types of computer simulations are used Monte Cado (MC) and molecular dynamics (MD). The Monte Cado method rehes on a binary coUision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Cado suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—soUd interactions has been provided (43). 

The kinetic theory of gases has been used so far, the assumption being that gas molecules are non-interacting particles in a state of random motion. This... 

The classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

Molecular simulation techniques, namely Monte Carlo and molecular dynamics methods, in which the liquid is regarded as an assembly of interacting particles, are the most popular... 

The study of the behavior of reactions involving a single species has attracted theoretical interest. In fact, the models are quite simple and often exhibit IPT. In contrast to standard reversible transitions, IPTs are also observed in one-dimensional systems. The study of models in ID is very attractive because, in some cases, one can obtain exact analytical results [100-104]. There are many single-component nonequilibrium stochastic lattice reaction processes of interacting particle systems [100,101]. The common feature of these stochastic models is that particles are created autocatalytically and annihilated spontaneously (eventually particle diffusion is also considered). Furthermore, since there is no spontaneous creation of particles, the zero-particle... 

T. M. Liggett. Interacting Particle Systems. New York Springer-Verlag, 1983, pp. 1-486. 

N. Konno. Phase transitions of interacting particle systems. Singapore World Scientific, 1994, pp. 1-228. 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

Suppose now that we have an ensemble of N non-interacting particles in a thermally insulated enclosure of constant volume. This statement means that the number of particles, the internal energy and the volume are constant and so we are dealing with a microcanonical ensemble. Suppose that each of the particles has quantum states with energies given by i, 2,... and that, at equilibrium there are Ni particles in quantum state Su particles in quantum state 2, and so on. 

Iti Chapter 1, we dealt at length with molecular mechanics. MM is a classical model where atoms are treated as composite but interacting particles. In the MM model, we assume a simple mutual potential energy for the particles making up a molecular system, and then look for stationary points on the potential energy surface. Minima correspond to equilibrium structures. 

Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quanmm mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

In Pauli s model, we still envisage a core of rigid cations (metal atoms that have lost electrons), surrounded by a sea of electrons. The electrons are treated as non-interacting particles just as in the Drude model, but the analysis is done according to the rules of quantum mechanics. 

Gelbart (1974) has reviewed a number of theories of the origins of the depolarized spectrum. One of the simplest models is the isolated binary collision (IBC) model of McTague and Bimbaum (1968). All effects due to the interaction of three or more particles are ignored, and the scattering is due only to diatomic collision processes. In the case that the interacting particles A and B are atoms or highly symmetrical molecules then there are only two unique components of the pair polarizability tensor, and attention focuses on the anisotropy and the incremental mean pair polarizability... 

The partition function Q here describes the whole system consisting of N interacting particles, and the energy states Ei are consequently for all the particles (in Section 12.2 we considered N non-interacting molecules, where the total partition function could be written in terms of the partition function for one molecule, Q — /N[). More correctly... 

A nuclear fission reaction will not occur unless the following occur (1) the total mass of the reaction products is less than the total mass of the interacting nuclei, and (2) the sum of the neutrons and the sum of the protons in the interacting particles ecjuals the sum of the neutrons and the sum of the protons in the products of the fission. 

The situation is more complicated for nonisolated systems consisting of strongly interacting particles and when the system is no longer in equilibrium with the environment. Kauffman [kauff93] notes that the second law really states that any system will tend to the maximum disorder possible, within the constraints due to the dynamics of the system. ... 

Though there was of course no way for Zuse to answer his second question (nor is there any way today), the fact that it is being asked at all underscores the essence of the second of the two paradigm shifts listed earlier in this chapter the notion that information is more fundamental than what have traditionally been used as fundamental variables (mass, energy, etc.). Zuse suggests that if only we could find an appropriate language or formalism with which to describe this primordial information, we would find, for example, that the information content of two or more interacting particles is conserved. 

Brueckner, K. A., and Levinson, C. A., Phys. Rev. 97, 1344, Approximate reduction of the many-body problem for strongly interacting particles to a problem of SCF fields/ ... 

If three-body collisions are neglected, which is permitted at sufficiently low densities, all the interactions take place between pairs of particles the two-particle distribution function will, therefore, satisfy Liouville s equation for two interacting particles. For /<2)(f + s) we may write Eq. (1-121) ... 

Molecular dynamics (MD) permits the nature of contact formation, indentation, and adhesion to be examined on the nanometer scale. These are computer experiments in which the equations of motion of each constituent particle are considered. The evolution of the system of interacting particles can thus be tracked with high spatial and temporal resolution. As computer speeds increase, so do the number of constituent particles that can be considered within realistic time frames. To enable experimental comparison, many MD simulations take the form of a tip-substrate geometry correspoudiug to scauniug probe methods of iuvestigatiug siugle-asperity coutacts (see Sectiou III.A). 

Fig. 6.16 Mossbauer spectra of 8 nm a-Fe203 particles at the indicated temperatures, (a) Data for phosphate coated (noninteracting) particles, and (b) data for the uncoated (interacting) particles. (Reprinted with permission from [77] copyright 2006 by the American Physical Society)...

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. 

Although we have explained Bose-Einstein condensation as a characteristic of an ideal or nearly ideal gas, i.e., a system of non-interacting or weakly interacting particles, systems of strongly interacting bosons also undergo similar transitions. Eiquid helium-4, as an example, has a phase transition at 2.18 K and below that temperature exhibits very unusual behavior. The properties of helium-4 at and near this phase transition correlate with those of an ideal Bose-Einstein gas at and near its condensation temperature. Although the actual behavior of helium-4 is due to a combination of the effects of quantum statistics and interparticle forces, its qualitative behavior is related to Bose-Einstein condensation. 

Consider two identical non-interacting particles, each of mass m, in a onedimensional box of length a. Suppose that they are in the same spin state so that spin may be ignored. 

In the most simple case of ideal, energy homogeneous surface the adsorption equilibrium of non-interacting particles is described by the Langmuir isotherm [33] ... 

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