Particle connectivity model

The Monte Carlo-type electrode model is also called the particle connectivity model because its physics is straightforwardly based on KirchhofTs law for an electrical network, with particle resistance and interconnection resistances defined by a set of rules to mimic the current flow and electrochemical current generation within the microstructure. The electrochemical process is considered to take place with a constant resistance in agreement with intuitive notions about the mechanism. Variants of this concept attach correlated values to the resistances in the network to model polarisation more closely according to a percolation concept of active sites and passive connections [59]. Other specialised types of electrode models are mentioned briefly below,... 

In the simplest case, at N = 1, the considered subchain model of a macromolecule reduces to the dumbbell model consisting of two Brownian particles connected with an elastic force. It can be called relaxator as well. The re-laxator is the simplest model of a macromolecule. Moreover, the dynamics of a macromolecule in normal co-ordinates is equivalent to the dynamics of a set of independent relaxators with various coefficients of elasticity and internal viscosity. In this way, one can consider a dilute solution of polymer as a suspension of independent relaxators which can be considered here to be identical for simplicity. The latter model is especially convenient for the qualitative analysis of the effects in polymer solutions under motion. 

To ensure that only the finer fraction of the sediment slurry was processed, a shipboard centrifugal cone separator was connected to the slurry transfer hose to remove the coarse-sediment fraction. The cone separator was a 101.6-mm diameter, urethane-coated centrifugal cone (Demco 275). Under a normal operational pressure of 221 kPa the cone separator is capable of delivering 57.0 L/min of sediment slurry, whose sediment particle size ranged from 2 to 32 jitm as measured on the particle data Model 111 analyzer. 

The interaction in a two-body collision in a dilute suspension has been expanded to provide a useful and quantitative understanding of the aggregation and sedimentation of particulate matter in a lake. In this view, Brownian diffusion, fluid shear, and differential sedimentation provide contact opportunities that can change sedimentation processes in a lake, particularly when solution conditions are such that the particles attach readily as they do in Lake Zurich [high cc(i,j)exp]. Coagulation provides a conceptual framework that connects model predictions with field observations of particle concentrations and size distributions in lake waters and sediment traps, laboratory determinations of attachment probabilities, and measurements of the composition and fluxes of sedimenting materials (Weilenmann et al., 1989). 

The conductivity model estimates the critical volume fraction of paste that contains non-conducting or conducting additives which are not chemically active. In order to model the conductivity of the active material, the material is assumed to be made of spherical particles and modelled as nodes on a two-dimensional grid. Each node is connected to the surrounding eight nodes by a conductive pathway. The grid... 

As discussed in Chapter 1, a Gaussian chain is physically equivalent to a string of beads connected by harmonic springs with the elastic constant ikT/lP (Eq. (1.47) with 6 given by Eq. (1.44)). Here each bead is regarded as a Brownian particle in modeling the chain d3mamics. Such a model was first proposed by Rouse and has been the basis of molecular theories for the dynamics of polymeric liquids. 34... 

Descriptions of pore development and stracture require microscopic models of the particle. These models include intrinsic kinetics and pore stractural changes during bumoff. Three of the most popular mieroscopic models are a random eapillaiy pore model, one in which the pores are considered spherical vesicles connected by cyhndri-cal micropores, and one in which the pores have a treelike structure. These models allow for pore growth and coalescence in their respective fashions and provide estimates of reactive smface area. Parameters required for these models are obtained from experimental measurements of the various chars. 

Abstract Computer simulations are used to study the aggregation phenomena of volatile amphiphiles in a system displaying liquid/vapor coexistence. These molecular dynamics simulations are based on a simple, yet versatile, model used previously to study oil/water/amphiphile systems amphiphiles are nothing more than water and oil particles connected together by stiff springs. We observe a highly regulated self-assembly process wherein amphiphiles form bilayers within the liquid phase. The density of amphiphiles in a bilayer varies from a well-defined lower to upper limit as the overall concen-... 

The composition of the dense systems that have been simulated up to now is summarized in Table 1. The polymer chains are modeled as unbranched sequences of 100 isodieimetric units connected by links of length <7, while the filler particles are modeled as spherical entities with diameter a/. The simulated systems consist of three-dimensionally periodic arrays of cubic cells of edge 40 a containing Np polymer chains and Nf filler particles. The polymer units interact through a 12 — 6 Lennard-Jones potential Euu = — 2((r/ruu) ], where Tuu is the distance be-... 

In contrast to D, the prediction of other viscoelastic properties, such as the friction coefficient f or the zero-shear rate viscosity i/o, requires that the atomistic MD data be mapped upon a mesoscopic theoretical model. For unentangled polymer melts, such a model is the Rouse model, wherein a chain is envisioned as a set of Brownian particles connected by harmonic springs [25,28]. For entangled polymer melts, a better model that describes more accurately their dynamics is the tube or reptation model [26]. According to this model, the motion of an individual chain is restricted by the surrounding chains within a tube defined by the overall chain contour or primitive path. During the lifetime of this tube, any lateral motion of the chain is quenched. 

The value of x is determined by the geometry of the system, primarily by the particle size (radius, r, for spherical particles) and by the packing density of particles described by porosity, H. The porosity is a dimensionless characteristic defined as the ratio of the volume of pores, Vp to the total volume of the porous structure, V, that is, n = Vp/V. The x = Xir, H) dependence can be estimated from data on the degree of dispersion of the particles and the porosity of samples by employing the specific models for disperse structures. Eor example, in the case of loose monodisperse structures with spherical particles connected into crossing chains with n particles per chain between the nodes (Figure 3.17), the X function for the case when the porosity H does not exceed 48% can be described as... 

The harmonic oscillator is a model system in both quantum and classical mechanics. Classically, it is a particle moving back and forth at a frequency, - k/m, where k is the force constant and m is the particle mass. Its quantum mechanical description shows the system existing only with certain allowed energies, each separated by Planck s constant times the oscillator s frequency. A system of many particles connected by harmonic springs can exhibit normal mode vibration, whereby all the particles move with the same frequency. For a system of N particles, there are 3N - 6 normal modes, or 3N - 5 if the system is linear. 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

FIG. 6 Configuration snapshot of a spontaneously formed vesicle from doubletailed amphiphiles in the Larson model (a) entire vesicle (b) vesicle cut in half in order to show its inner side. Black circles represent head particles (+1), gray circles tail particles (—1), white circles the neutral connecting particles (0). (From Bernardes [126].)... 

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