Phase transitions, diffusion models

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

The bubble gas, with initial concentration c , leaves the dense phase at z = Lf with concentration Ci and enters the dilute phase. For the dilute phase the diffusion-model approach is more general, since a fair amount of gas mixing is observed in a large-diameter bed (H19). Here, however, the usual piston-flow assumption is made for simplicity, since in small-scale beds the particles in the dilute phase are suspended rather uniformly. Actually the observed flow properties of the phase, including the transition zone, seem to be more complicated. 

For the analysis heat and mass transfer in concrete samples at high temperatures, the numerical model has been developed. It describes concrete, as a porous multiphase system which at local level is in thermodynamic balance with body interstice, filled by liquid water and gas phase. The model allows researching the dynamic characteristics of diffusion in view of concrete matrix phase transitions, which was usually described by means of experiments. 

Hydrated bilayers containing one or more lipid components are commonly employed as models for biological membranes. These model systems exhibit a multiplicity of structural phases that are not observed in biological membranes. In the state that is analogous to fluid biological membranes, the liquid crystal or La bilayer phase present above the main bilayer phase transition temperature, Ta, the lipid hydrocarbon chains are conforma-tionally disordered and fluid ( melted ), and the lipids diffuse in the plane of the bilayer. At temperatures well below Ta, hydrated bilayers exist in the gel, or Lp, state in which the mostly all-trans chains are collectively tilted and pack in a regular two-dimensional... 

I. Jensen, H. C. Fogedby. Kinetic phase transitions in a surface-reaction model with diffusion Computer simulations and mean-field theory. Phys Rev A 2 1969-1975, 1990. 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

Then, there are model Hamiltonians. Effectively a model Hamiltonian includes only some effects, in order to focus on those effects. It is generally simpler than the true full Coulomb Hamiltonian, but is made that way to focus on a particular aspect, be it magnetization, Coulomb interaction, diffusion, phase transitions, etc. A good example is the set of model Hamiltonians used to describe the IETS experiment and (more generally) vibronic and vibrational effects in transport junctions. Special models are also used to deal with chirality in molecular transport junctions [42, 43], as well as optical excitation, Raman excitation [44], spin dynamics, and other aspects that go well beyond the simple transport phenomena associated with these systems. 

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

The model for phase transitions and chemical reactions takes into account thermal destruction of dust particles, vent of volatiles, chemical reactions in the gas phase, and heterogeneous oxidation of particles influenced by both diffusive and kinetic characteristics. 

The vast majority of biological membranes are in the liquid-crystalline phase. There are many experimental studies on model bilayer phase behavior [3]. Briefly, at low temperatures lipid bilayers form a gel phase, characterized by high order and rigidity and slow lateral diffusion. There is a main phase transition, as the temperature is increased, to the liquid-crystalline phase. The liquid-crystalline phase has more fluidity and fast lateral diffusion. 

Innocenti et al. have studied the kinetics [101] of two-dimensional phase transitions of sulfide and halide ions, as well as electrosorption valency [102] of these ions adsorbed on Ag(lll). The electrode potential was stepped up from the value negative enough to exclude anionic adsorption to the potential range providing stability of either the first or the second, more compressed, ordered overlayer of the anions. The kinetic behavior was interpreted in terms of a model that accounts for diffusion-controlled random adsorption of the anions, followed by the progressive polynucleation and growth. 

We have introduced in this Section a stochastic model for the A+ 2B2 —> 0 reaction which is equivalent to the ZGB-model [2] and thus remedies a deficiency of a previously presented model [13]. In this model we obtain for the case of no diffusion for the phase transition points y = 0.395 and y2 = 0.565, which are in good or fair agreement with the results of the ZGB-model (y = 0.395 and yi = 0.525). In the model [13] where the reaction occurs only if A particle jumps to active site occupied by a B particle, we obtain y — 0.27 and 7/2 = 0.65 (for D — 10). Because the reaction occurs only due to diffusion, we cannot directly compare this model with the ZGB-model in which no diffusion exists. But the value of t/2 is in agreement with computer simulations of the extended ZGB-model including diffusion (t/2 = 0.65 for a high diffusion rate) [3]. The value of y should not be influenced by the additional aspect of A-diffusion because too few A particles... 

For the case 5=1 and D = 1 the results of the stochastic model are in good agreement with the CA model y = 0.262). This is understandable because the different definition of the reaction which leads to a difference in the blocking of activated sites cannot play significant role because all sites are activated. The diffusion rate of D = 10 leads nearly to the same reactivity as if we define the reaction between the nearest-neighbour particles. If the diffusion rate is considerably lowered (D = 0.1), the behaviour of the system changes completely because of the decrease of the reaction probability. This leads to the disappearance of the kinetic phase transition at y because different types of particles may reside on the surface as the nearest neighbours without reaction, a case which does not occur at all in the CA approach. 

Let us study now a stochastic model for the particular a+ib2 -> 0 reaction with energetic interactions between the particles. The system includes adsorption, desorption, reaction and diffusion steps which depend on energetic interactions. The temporal evolution of the system is described by master equations using the Markovian behaviour of the system. We study the system behaviour at different values for the energetic parameters and at varying diffusion and desorption rates. The location and the character of the phase transition points will be discussed in detail. 

To demonstrate this, in Section 9.2.2 we have studied a stochastic model for an extended ZGB-model including diffusion, desorption and energetic interactions as additional steps. We have used different values of the diffusion and the desorption rates and different values for the energetic parameters. In the case of repulsive interactions the system s behaviour is strongly influenced by. Eaa for large values of Yqo and by for small values of kco-The former parameter leads to a smooth phase transition at yi and the latter to a sharp transition at 2/1 The sharpness and the location of the phase transitions depend also on the diffusion and desorption rate of the A particles. The A-diffusion leads to an increase of the value of 2/2 due to the higher reactivity of the A particles. At lower values of Yco the system behaviour is nearly not influenced by the diffusion. The A-desorption increases the values of the critical points and smoothes the phase transition at 2/2- This effect becomes very important if Ca is large. 

A correlation was predicted in terms of this model between the behavior of the concentration dependence of the temperature of the JT (or pseudo-JT) structural phase transition and the concentration dependence of the oxygen diffusion coefficient. 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

When modeling phenomena within porous catalyst particles, one has to describe a number of simultaneous processes (i) multicomponent diffusion of reactants into and out of the pores of the catalyst support, (ii) adsorption of reactants on and desorption of products from catalytic/support surfaces, and (iii) catalytic reaction. A fundamental understanding of catalytic reactions, i.e., cleavage and formation of chemical bonds, can only be achieved with the aid of quantum mechanics and statistical physics. An important subproblem is the description of the porous structure of the support and its optimization with respect to minimum diffusion resistances leading to a higher catalyst performance. Another important subproblem is the nanoscale description of the nature of surfaces, surface phase transitions, and change of the bonds of adsorbed species. 

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