System models

In the model calculations [13], the effective diffusion coefficients in the two-phase zone were used. It was established that diffusion coefficients were constant along the conode in case of a phase diagram with parallel conodes. Experimentally obtained zigzag diffusion paths in the two-phase zone of an Al-Cr-Ni system were also presented. The most detailed development of the theoretical approach [12, 13] for the investigation of diffusion interaction between the a phase of the solid solution and the a — f two-phase alloy was performed in [48]. The authors have obtained the dependencies of the contact zone microstructure on the initial conditions of diffusion annealing. The model is based on the following assumptions  

Another development of the approaches [12,13,48] is given in [35], as it describes diffusion interaction between two two-phase alloys with different volume fractions of phases, grounding upon the phase field model, which employs generahzed Cahn-HiUiard equations [34]. The model treats different components mobihties in different phases, and accounts for the surface energy between the particles of phases, which is connected with gradient terms from Cahn-HiUiard equations. 

This approach enables analysis of the KirkendaU effect in the contact zone. As a result of 2D computer simulations, different types of diffusion zone microstructure were obtained. The influence of KirkendaU effect on the behavior of the zigzag diffusion path is treated, and the possible type of the resulting diffusion zone between two-phase alloys without particles of the other phase is obtained. We concluded that the change of the rotation angle of the principle value vector of the effective diffusion coefHcients matrix and the concentration dependence of mass transfer coefficients lead to deviations of the averaged concentrations in the two-phase zone from the directions of the principle value vector obtained in the models developed earlier [12, 13, 48]. 

In this section, some recent fundamental investigations will be reviewed. Three catalytic model systems and the simplest possible electrochemical cell of the single-pellet type are chosen for illustration. Among the main techniques of investigation, special attention will be given to measurement of catalytic rate transients, cyclic voltammetry, and measurement of catalyst work function. 

We will focus on the investigation of the phenomenon of electrochemical promotion by using YSZ as the solid electrolyte. Two types of heterogeneous catalytic gas reactions will be discussed. One of them is the catalytic combustion of ethylene over RUO2 or I1O2 catalysts and the other is the reduction of NO by propylene in presence of oxygen over Rh catalysts. 

The catalytic oxidation of ethylene on metal oxides such as RUO2 and Ir02 is generally highly non-selective resulting in the complete oxidation of ethylene to CO2 and H2O (see Eq. 1). In such a case, the catalytic performance can be uniquely characterized by a single reaction rate, e.g., by that of CO2 production, rcoj (mol s ). The catalytic combustion of ethylene is a well-studied model reaction. In fact, ethylene is often used as a model compound for unsaturated hydrocarbon residues in automotive exhaust gases. 

Concerning the other model system, metallic rhodium is known to be an efficient catalyst for the reduction of NO. ° The propylene-NO- 

The reduction of NO with propylene is, however, not selective. In addition to the desired end-product (N2), a partially reduced by-product (N2O) is also formed  

Equations (7.57a) and (7.57b) provide two ways to calculate the pressure and chemical potential. The first one is to perform the appropriate derivative of the free energy, assuming that the latter can be evaluated for all states of interest. The second, more direct way is to employ the relations to the pressure and chemical potentials in the replicated system. This second strategy is particularly useful for the calculation of fit because lim o in closed form for a variety of model systems and closure relations, including the HNC approximation for molecular fluids [309, 310]. The pressure is more difficult because of the presence of the second, matrix-related term on the right side of Eq. (7.56). 

Finally, we note without proof that both pressure and chemical potential can also be obtained by integrating the compressibility given in Eq. (7.27). Explicitly, one has [308] 

The earliest applications of the replica integral equation approach date back to the beginning of the 1990s. They focused on quite simple QA systems such as hard-sphere (HS) and LJ (12,6) fluids in HS matrices (see, for example. Refs. 4, 286, 290, 298, 303, 312, and 313 for reviews). Fiom a technical point of view, these studies have shown that the replica integral equations yield accurate correlation functions compared with parallel computer simulation results [292, 303, 314, 315]. Moreover, concerning phase behavior, it turned out that the simple LJ (12,6) fluid in HS matrices already displays features also observed in experiments of fluids confined to aerogels [131, 132]. These features concern shifts of the vapor liquid critical temperature toward values 

Motivated by this success, a series of more recent replica integral equation studies has focused on the effects of more realistic features of both the adsorbed fluid and its interactions with matrix. Examples are studies of the influence of templated matrix materials [316], associating fluids [317], LJ mixtures [114, 311], and QA systems with ionic interactions [305, 306, 318-320]. However, until recently, only one study [309] has been available on QA systems with angle-dependent (specifically anisotropic steric) interactions. 

The hard core in Eq. (7.59) has been imposed for numerical convenience. As a consequence, it is mainly the van der Waals-like attractive (rather than the repulsive) part of the LJ (12,6) potential (oc r ) that contributes to the fluid fluid potential. The strength of the dipolar relative to the attractive LJ interactions is conveniently measured by the reduced (i.e., dimensionless) dipole moment m = fi/V a. Depending on this parameter, the Stockmayer fluid may serve as a simple model for polar molecular fluids [258, 259] (small m.) or for ferrofluids [227, 228] (large in). Here wc consider a system with dipole moment m = 2, whicli is a value typical for moderately polar molecular fluids [259] such as chloroform. For this value of m, GCEMC simulations have been presented in Section 6.4.1. 

Dependence of the fragility index, m, on the average coordination r for Ge-Se and Ge-As-Se glasses. The fragility index was derived from the activation energy for enthalpy relaxation measured by DSC using Moynihan s method. 

The same trends can be observed using other measures of fragility, including ACp and ATg/Tg. Note that ACp can be used 

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Face-centered cubic crystals of rare gases are a useful model system due to the simplicity of their interactions. Lattice sites are occupied by atoms interacting via a simple van der Waals potential with no orientation effects. The principal problem is to calculate the net energy of interaction across a plane, such as the one indicated by the dotted line in Fig. VII-4. In other words, as was the case with diamond, the surface energy at 0 K is essentially the excess potential energy of the molecules near the surface. 

There is quite a large body of literature on films of biological substances and related model compounds, much of it made possible by the sophisticated microscopic techniques discussed in Section IV-3E. There is considerable interest in biomembranes and how they can be modeled by lipid monolayers [35]. In this section we briefly discuss lipid monolayers, lipolytic enzyme reactions, and model systems for studies of biological recognition. The related subjects of membranes and vesicles are covered in the following section. 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

F( Pjy,) state but concludes that the adiabatic picture is largely correct. The issue of whether a reaction can be described by a single Bom-Oppenlieimer surface is of considerable interest in chemical dynamics [10], and it appears that the effect of multiple surfaces must be considered to gain a complete picture of a reaction even for as simple a model system as the F + H2 reaction. 

Miehelsen H A, Rettner C T and Auerbaeh D J 1993 The adsorption of hydrogen at eopper surfaees A model system for the study of aetivated adsorption Surface Reacf/onsed R J Madix (Berlin Springer) p 123... 

This solution can be obtained explicitly either by matrix diagonalization or by other techniques (see chapter A3.4 and [42, 43]). In many cases the discrete quantum level labels in equation (A3.13.24) can be replaced by a continuous energy variable and the populations by a population density p(E), with replacement of the sum by appropriate integrals [Hj. This approach can be made the starting point of usefiil analytical solutions for certain simple model systems [H, 19, 44, 45 and 46]. 

Wyatt R E, Hose G and Taylor H S 1983 Mode-selective multiphoton excitation in a model system Phys. Rev. A 28 815-28... 

Experimental investigations of the model system of dye molecules adsorbed onto surfaces of polystyrene spheres have finuly established the sensitivity and surface specificity of the SHG method even for particles of micrometre size [117]. The surface sensitivity of die SHG process has been exploited for probing molecular transport across the bilayer in liposomes [118], for measurement of electrostatic potentials at the surface of small particles [119] and for imaging... 

The electron-spm echo envelope modulation (ESEEM) phenomenon [37, 38] is of primary interest in pulsed EPR of solids, where anisotropic hyperfme and nuclear quadnipole interactions persist. The effect can be observed as modulations of the echo intensity in two-pulse and three-pulse experiments in which x or J is varied. In liquids the modulations are averaged to zero by rapid molecular tumbling. The physical origin of ESEEM can be understood in tenns of the four-level spin energy diagram for the S = I = model system... 

Levanon H and Mobius K 1997 Advanced EPR spectroscopy on electron transfer processes in photosynthesis and biomimetic model systems Ann. Rev. Biophys. Biomol. Struct. 26 495-540... 

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

Agrawal R and Kofke D A 1995 Thermodynamio and struotural properties of model systems at solid-fluid ooexistenoe. II. Melting and sublimation of the Lennard-Jones system Mol. Phys. 85 43-59... 

Hydrogen-bonded clusters are an important class of molecular clusters, among which small water clusters have received a considerable amount of attention [148, 149]. Solvated cluster ions have also been produced and studied [150, 151]. These solvated clusters provide ideal model systems to obtain microscopic infonnation about solvation effect and its influence on chemical reactions. 

Molecular dynamics tracks tire temporal evolution of a microscopic model system tlirough numerical integration of tire equations of motion for tire degrees of freedom considered. The main asset of molecular dynamics is tliat it provides directly a wealtli of detailed infonnation on dynamical processes. 

Prime K L and Whitesides G M 1991 Self-assembled organio monolayers—model systems for studying adsorption of proteins at surfaoes Science 252 1164-7... 

In practice, e.g., in nature or in fonnulated products, colloidal suspensions (also denoted sols or dispersions) tend to be complex systems, consisting of many components that are often not very well defined, in tenns of particle size for instance. Much progress has been made in the understanding of colloidal suspensions by studying well defined model systems, which allow for a quantitative modelling of their behaviour. Such systems will be discussed here. 

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. 

An important step in tire progress of colloid science was tire development of monodisperse polymer latex suspensions in tire 1950s. These are prepared by emulsion polymerization, which is nowadays also carried out industrially on a large scale for many different polymers. Perhaps tire best-studied colloidal model system is tliat of polystyrene (PS) latex [9]. This is prepared with a hydrophilic group (such as sulphate) at tire end of each molecule. In water tliis produces well defined spheres witli a number of end groups at tire surface, which (partly) ionize to... 

Anotlier model system consists of polymetliylmetliacrylate (PMMA) latex, stabilized in organic solvents by a comb polymer, consisting of a PMMA backbone witli poly-12-hydroxystearic acid (PHSA) chains attached to it [10]. The PHSA chains fonn a steric stabilization layer at tire surface (see section C2.6.4). Such particles can approach tire hard-sphere model very well [111. 

In addition to tire standard model systems described above, more exotic particles have been prepared witli certain unusual properties, of which we will mention a few. For instance, using seeded growtli teclmiques, particles have been developed witli a silica shell which surrounds a core of a different composition, such as particles witli magnetic [12], fluorescent [13] or gold cores [14]. Anotlier example is tliat of spheres of polytetrafluoroetliylene (PTFE), which are optically anisotropic because tire core is crystalline [15]. 

Even when well defined model systems are used, colloids are ratlier complex, when compared witli pure molecular compounds, for instance. As a result, one often has to resort to a wide range of characterization teclmiques to obtain a sufficiently comprehensive description of a sample being studied. This section lists some of tire most common teclmiques used for studying colloidal suspensions. Some of tliese teclmiques are discussed in detail elsewhere in tliis volume and will only be mentioned in passing. A few teclmiques tliat are relevant more specifically for colloids are introduced very briefly here, and a few advanced teclmiques are highlighted. 

Minimal END has also been applied to a model system for intramolecular electron transfer. The small triatomic system LiHLi is bent C2v structure. But the linear structure presents an unrestricted Haiti ee-Fock (TJHF) broken symmetry solution with the two charge localized stmctures... 

In our introductory remarks, we said that this section would be devoted to model systems. Nevertheless it is important to emphasize that although this case is treated within a group of model systems this model stands for the general case of a two-state sub-Hilbert space. Moreover, this is the only case for which we can show, analytically, for a nonmodel system, that the restrictions on the D matrix indeed lead to a quantization of the relevant non-adiabatic coupling term. 

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