Explicit particles

The measurement data in Figure 6 could be the indirect monitoring parameters (23) or the explicit particle size distribution (33), depending on the relative speed of the measurement system and process dynamics. 

Instead of the MC and MD methods using explicit particles, another method, that is, polymeric self-consistent field theory (SCFT) proposed by Edwards, is often used to study the phase separation of block copolymers. In SCFT, a polymer chain is treated as a Gaussian string, which is exposed to a set of effective chemical potentials ( ). The chemical potentials are used instead of the actual interactions between different components. Importantly, the relation between the external potentials and the concentration field ((/>) is bijective. 

Single-chain-in-mean-field (SCMF) simulation [40-42, 86] is an approximate, computational method that retains the computational advantage of self-consistent field theory but additionally includes fluctuation effects because, in contrast to self-consistent theory, SCMF simulations aim at preserving the instantaneous description of the fluctuating interactions of a segment with its environment. In this partide-based simulation technique, one studies an ensemble of molecules in fluctuating, real, external fields. The explicit particle coordinates are the degrees of freedom and not the collective variables, densities and fields. 

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

In extensively deionized suspensions, tliere are experimental indications for effective attractions between particles, such as long-lived void stmctures [89] and attractions between particles confined between charged walls [90]. Nevertlieless, under tliese conditions tire DLVO tlieory does seem to describe interactions of isolated particles at tire pair level correctly [90]. It may be possible to explain tire experimental observations by taking into account explicitly tire degrees of freedom of botli tire colloidal particles and tire small ions [91, 92]. 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Atoms not explicitly included in the trajectory must be generated. The position at which an atom may be placed is in some sense arbitrary, the approach being analogous to the insertion of a test particle. Chemically meaningful end states may be generated by placing atoms based on internal coordinates. It is required, however, that an atom be sampled in the same relative location in every configuration. An isolated molecule can, for example, be inserted into... 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

C- cutoff radius method - all non-bonded forces between particles within angstroms of each other are computed explicitly... 

For an initially fully saturated particle, the exchange rate is faster when the faster counterion is initially in the resin, with the difference in rate becoming more important as conversion from one form to the other progresses. Helfferich (gen. refs., pp. 270-271) gives explicit expressions for the exchange of ions of unequal valence. 

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

The explicit form of those equations that satisfy the preliminary data criteria, must then be tested against a series of data sets that have been obtained from different chromatographic systems. As an example, such systems might involve columns packed with different size particles, employed mobile phases or solutes having different but known physical properties such as diffusivity or capacity ratios (k"). 

It is seen that the optimum velocity is inversely proportional to the optimum particle diameter and it would be possible to insert the expression for the optimum particle diameter into equation (2) to provide an explicit expression for the optimum velocity. The result would, however, be algebraically cumbersome and it is easier to consider the effects separately. The optimum velocity is inversely... 

The singlet-level theory has also been used to describe the structure of associating fluids near crystalline surfaces [30,31,76,77]. The surface consists explicitly of atoms which are arranged on a lattice of a given symmetry. The fluid atom-surface atom potential can also involve an associative term, i.e., the chemical-type bonding of the adsorbate particles with the surface may be included into the model. However, we restrict ourselves to the case of a nonassociative crystalline surface first. 

The constants rc, u ic, etc. are specified in terms of microscopic parameters and the functions fc, f, f c tc. account for the various lateral interactions between the particles in the adsorbed and precursor states. We have factored out an explicit dependence on the coverages so that in the absence of any lateral interactions these functions are all equal to one. 

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