Multi-component order parameter

Equation (5.21) assumes ternary interactions are small in comparison to those which arise from the binary terms. This may not always be the case and where evidence for higher-order interactions is evident these can be taken into account by a further term of the type Gijit = x< xj Xk Lijk, where Lijk is an excess ternary interaction parameter. There is little evidence for the need for interaction terms of any higher order than this and prediction of the thermodynamic properties of substitutional solution phases in multi-component alloys is usually based on an assessment of binary and ternary terms. Various other polynomial expressions for the excess term have been considered, see for example the reviews by Ansara (1979) and Hillert (1980). All are, however, based on predicting the properties of... 

It is possible in many cases to predict highly accurate phase equihbria in multi-component systems by extrapolation. Experience has shown extrapolation of assessed (n — 1) data into an nth order system works well for n < 4, at least with metallurgical systems. Thus, the assessment of unary and binary systems is especially critical in the CALPHAD method. A thermodynamic assessment involves the optimization of aU the parameters in the thermodynamic description of a system, so that it reproduces the most accurate experimental phase diagram available. Even with experimental determinations of phase diagrams, one has to sample compositions at sufficiently small intervals to ensure accurate reflection of the phase boundaries. 

The shape of the ellipsoid was on average non-cylindrical. In fact since the distribution of eigenvalues was fairly symmetric, it is possible to describe with simply a rhombic component. The TCF of the form of the EFG shows a rapid initial decay to a plateau with a order parameter of 0.8 of the time zero value. The reorientational motion is multi exponential and is the main cause of the decay of the EFG-TCF. 

In this endeavor synchrotron spectroscopy has played an important role in understanding the effect of fundamental parameters such as electronic density of states and short-range atomic order. The primary advantages of using the synchrotron are (1) the ability to probe these parameters in situ while the interface is under electrochemical control and (2) the fact that these can be measured with element specificity. The latter is particularly useful when investigating multi-component alloy clusters. In addition, this technique lends itself to systems with limited long-range order, which is typical for these nanoclusters used in fuel-cell electrode interface. This chapter describes some recent results with in-situ X-ray absorption spectroscopy, which has provided a direct probe into the variations of the Pt i/-band vacancy (normalized with respect to number of surface atoms) between... 

In this section the basic principles of membrane formation by phase inversion will be described in greater detail. All phase inversion processes are based on the same thermodynamic principles, since the starting point in all cases is a thermodynamically stable solution which is subjected to demixing. Special attention will be paid to the immersion precipitation process with the basic charaaeristic that at least three components are used a polymer, a solvent and a nonsolvent where the solvent and nonsolvent must be miscible with each other. In fact, most of the commercial phase inversion membranes are prepared from multi-component mixtures, but in order to understand the basic principles only three component systems will be considered. An introduction to the thermodynamics of. polymer solutions is first given, a qualitatively useful approach for describing polymer solubility or polymer-penetrant interaction is the solubility parameter theory. A more quantitative description is provided by the Flory-Huggins theory. Other more sophisticated theories have been developed but they will not be considered here. 

Thus at some points it is useful to think of the order parameter as an initiator (or terminator ) for a chain. (In our case with both ends equivalent we are not interested in distinguishing between the two.) However this viewpoint omits some important aspects related to the multi-component feature of M(r). We saw, for example, (in connection with certain correlations) that it helps to distinguish between components of M parallel to the applied field H and perpendicular to H. (Strangely enough, this distinction retains a meaning when we set n = 0.) These component aspects must be superimposed on the initiator aspect. 

It is now one of the fundamental problems of statistical physics and synergetics to derive the properties of physico-chemical multi-component systems on the macroscopic level from their constituent components on the elementary microscopic level. Part of the problem is to explain which macrovariables (including order parameters) may be relevant under given circumstances and to describe their dynamics by appropriate equations of motion [1.49, 50]. 

The type of the distribution of T-values is a much discussed topic. Experience shows that, in a mono-exponential case, the values should spread over an interval of more than 3 x Ti but not much over 4 x Ti, and a linear distribution appears to be slightly better than a logarithmic one. This is probably due to the fact that in a three-parameter exponential fit, the points with large x-values play as crucial a role in determining the relaxation rate as the slope at small x-values, and one needs both to determine R. On the other hand, it is evident that in multi-exponential cases, logarithmic distribution is often better suited for the task, especially when the relaxation rates of different sample components differ by an order of magnitude or more. 

The set of equations formed by fheseequations is then solved numerically. Such models have been used extensively to describe breakthrough curves onto activated carbon of mono-component solutions of metal ions, micro-organic compounds or dyes [20-22], Some studies have demonstrated that they could be used to model binary namic adsorption [23] but they may not be applied in the case of complex multi-solute solutions. In addition, they do not take into account the pore characteristics of activated carbon materials, which are known to influence strongly the adsorption of micro-orgaiucs. In these cases, statistical tools like neural networks may be used in order to introduce such parameters as explicative variables. 

Reduction of the number of independent parameters in the evolution of an indignant system proceeds continuously as a result of parameter synchronization and the determination of the correlative bonds between them. However, taking into account the difference between partial processes and their respective relaxation times, separate states of evolution can be noted, every one of which is described by its own number of independent parameters changing through transition from one state to another. For example, after the finishing time r(/V from the start of a single-component system, a so-called kinetic state of a process takes place [11,12] (ro and V are characteristic size of a particle and heat rate of their moving, respectively is an indication of particle interaction time under collision, in the order of 10" -10s). In this state a state of a system is fully determined by a partial distribution function tU] that rules by the temporary evolution of a system. Multi-partial distribution function and as a result a full one represent a function of (Oil... 

As usual, the SS-EOS scaling parameters of all conponents (HFC 134a, HFC 152a and PS) must be determined in order to develop the EOS for the multi-componmt system. The SS-EOS scaling parameters were determined fiom the thermodynamic properties for each component. Specifically, the scaling parameters for HFC 134a and HFC 152a were extracted fiom the gas-hquid saturation curves up to the critical point. The results of the optimization are shown in Fig. 5. 

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