Particles, description

Description Particle size (/tm) Minimum theoreticai piates/30 cm Exclusion limit (polystyrene molecular weight) Standard flow rate (ml/min) Maximum flow rate (ml/min) Maximum pressure drop/30 cm (bar)... 

Name Description Particle Size Bulk Density, G/ml... 

Filler Material Description Particle Size (pm) Particle Shape Density, g/cm ... 

The interaction between particles in combination with the dynamical equation determines how the system evolves in time. At the fundamental level, the only important force at the atomic level is the electromagnetic interaction. Depending on the choice of system description (particles), however, this may result in different effective forces. 

Particle velocity or flux distribution, density of particles, particle shape description, particle size distribution, particle composition... 

Below are brief descriptions of some of the particle-surface interactions important in surface science. The descriptions are intended to provide a basic understanding of how surfaces are probed, as most of the infonuation that we have about surfaces was obtained tluough the use of techniques that are based on such interactions. The section is divided into some general categories, and the important physics of the interactions used for analysis are emphasized. All of these teclmiques are described in greater detail in subsequent sections of the encyclopaedia. Also, note that there are many more teclmiques than just those discussed here. These particular teclmiques were chosen not to be comprehensive, but instead to illustrate the kind of infonuation that can be obtained from surfaces and interfaces. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Consider an ensemble of Brownian particles. The approach of P2 to as 00 represents a kmd of diflfiision process in velocity space. The description of Brownian movement in these temis is known as the Fo/c/cer-PIanc/c method [16]- For the present example, this equation can be shown to be... 

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling tln-ough barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in tliree dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

For homogeneous particles, it represents the number of distances within the particle. For inhomogeneous particles, it has to take into account the different electron density of the volume elements. Thus it represents the number of pairs of difference in electrons separated by the distance r. A qualitative description of shape and internal structure of the... 

This chapter deals with qnantal and semiclassical theory of heavy-particle and electron-atom collisions. Basic and nsefnl fonnnlae for cross sections, rates and associated quantities are presented. A consistent description of the mathematics and vocabnlary of scattering is provided. Topics covered inclnde collisions, rate coefficients, qnantal transition rates and cross sections. Bom cross sections, qnantal potential scattering, collisions between identical particles, qnantal inelastic heavy-particle collisions, electron-atom inelastic collisions, semiclassical inelastic scattering and long-range interactions. 

A fiirther theme is the development of teclmiques to bridge the length and time scales between truly molecular-scale simulations and more coarse-grained descriptions. Typical examples are dissipative particle dynamics [226] and the lattice-Boltzmaim method [227]. Part of the motivation for this is the recognition that... 

Colloidal particles can be seen as large, model atoms . In what follows we assume that particles with a typical radius <3 = lOO nm are studied, about lO times as large as atoms. Usually, the solvent is considered to be a homogeneous medium, characterized by bulk properties such as the density p and dielectric constant t. A full statistical mechanical description of the system would involve all colloid and solvent degrees of freedom, which tend to be intractable. Instead, the potential of mean force, V, is used, in which the interactions between colloidal particles are averaged over... 

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. 

The spin in quantum mechanics was introduced because experiments indicated that individual particles are not completely identified in terms of their three spatial coordinates [87]. Here we encounter, to some extent, a similar situation A system of items (i.e., distributions of electrons) in a given point in configuration space is usually described in terms of its set of eigenfunctions. This description is incomplete because the existence of conical intersections causes the electronic manifold to be multivalued. For example, in case of two (isolated) conical intersections we may encounter at a given point m configuration space four different sets of eigenfunctions (see Section Vni). 

In order to compute average properties from a microscopic description of a real system. one must evaluate in tegrals over phase space. For an A -particle system in an cn sem hie with distribution... 

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

As an exercise, let s adapt this model of the analytical approach to a real problem. For our example, we will use the determination of the sources of airborne pollutant particles. A description of the problem can be found in the following article ... 

Why are fhese beams, or jefs, distinguished from effusive beams by fheir description as supersonic In some ways fhis description is rafher misleading, firsf because particles in an effusive beam may well be fravelling af supersonic velocities and, second, because fhe name implies fhaf somefhing special happens when fhe particle velocities become supersonic whereas fhis is nof fhe case. Whaf supersonic is meanf fo imply is fhaf fhe particles may have very high Mach numbers (of fhe order of f 00). The Mach number M is defined as... 

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