Random phase volume model

Tlie ultimate test of new, theoretically motivated protocols for materials discovery is, of course, experimental. To motivate such experimentation, the effectiveness of these protocols is demonstrated by combinatorial chemistry experiments where the experimental screening step is replaced by hgures of merit returned by the random-phase volume model. The random phase volume model is not fundamental to the protocols it is introduced as a simple way to test, parameterize, and validate the various searching methods,... 

Tlie random phase volume model relates the figure of merit to the composition and noncomposition variables in a statistical way. The model is fast enough to allow for validation of the proposed searching methods on an enormous number of samples yet possesses the correct statistics for the hgure-of-merit landscape. 

The composition mole fractions are nonnegativc and sum to unity, and so the allowed compositions are constrained to lie within a simplex in d -1 dimensions. For the familiar ternary system, this simplex is an equilateral triangle, as shown in Fig. 3b. Typically, several phases will exist for different compositions of the material. The figures of merit will be dramatically different between each of these distinct phases. To mimic this expected behavior, the composition variables are grouped in the random phase volume model into phases centered around Vj points randomly placed within the allowed composition range. The phases form a Voronoi diagram (Sedgewick, 1988), as shown in Fig. 4. 

The random phase volume model is defined for any number of composition variables, and the number of phase points is defined by requiring the average spacing between phase points to be = 0.25. To avoid edge effects, additional points are added in a belt of width 2 around the simplex of allowed compositions. The number of phase points for different grid spacing is shown in Table 1. 

Fig. 4. The random phase volume model (from Falcioni and Deem, 2000). The model is shown for the case of three composition variables and one noncomposition variable. The boundaries of the X phases are evident by the sharp discontinuities in the figure of merit. To generate this figure, the z variable was held constant. The boundaries of the z phases are shown as thin dark lines.

The and are chosen so that the multinomial, crystallinity terms contribute 40% as much as the constant, phase terms on average. For both multinomials q = 6. As Fig. 4 shows, the random phase volume model describes a rugged figure-of-merit landscape, with subtle variations, local maxima, and discontinuous boundaries. 

Fig. 5. The maximum figure of merit found with different protocols on systems with different number of composition (x) and noncomposition (z) variables (from Falcioni and Deem, 2000). The results are scaled to the maximum found by the grid searching method. Each value is averaged over scaled results on 10 instances of the random phase volume model with different random phases. The Monte Carlo methods are especially effective on systems with a larger number of variables, where the maximal figures of merit are more difficult to locate.

How rough are real figures of merit, and can the random phase volume model be calibrated better ... 

Wang, Mukherjee, and Wang [124] investigated the effects of catalyst layer electrolyte and void phase fractions on fuel cell performance using a random microstructure. The model predicted volume fractions of 0.4 and 0.26 for void and electrolyte phases, respectively, as the optimal CL compositions. 

In addition to these pressure drop models, models to represent spreading of liquid in packed beds because of spatial variation in flow resistance are needed. In a randomly packed bed, the void fraction is not uniform. This implies that some flow channels formed within a packed bed offer less resistance to flow than other channels of equal cross-sectional area. Liquid will tend to move toward channels of lower resistance, leading to higher liquid hold-up in such channels. Thus, even if the initial liquid distribution is uniform, inherent random spatial variation of the bed leads to non-uniform liquid flow. Yin et al. (2000) assumed that the dispersion coefficient for liquid phase volume fraction is linearly proportional to the adverse gradient of... 

Reliable theories that accurately correlate or predict polyelectrolyte phase diagrams are lacking. Letting electrostatic excluded volume between chain segments be modeled at the level of the Debye-Hiickel approximation, a modified or effective Flory-Huggins parameter Xeff can be determined via the random phase approximation (141) ... 

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

Figure 4.26 shows a cell model of the three phases. Gas in the upper region has a very low density and the molecules are free to fly around. When the vapor condenses into a liquid (shown lower right), the density is greatly increased so that there is very little free volume space the molecules have limited ability to move around, and they have random orientation that is, they can rotate and point in random directions. When the liquid freezes into a solid (shown lower left), the density is slightly increased to eliminate the void space, the molecules have assigned positions and are not free to move around, and there is now an orientation order that is, they cannot rotate freely and they all point at the same direction. 

We now consider the random copolymer model in the presence of solvent— that is, for a copolymer volume fraction p0 < 1. We are not aware of previous work on this model in the literature, but will briefly discuss below the link to models of homopolymer/copolymer mixtures [57]. The excess free energy (86) then depends on two moment densities, rather than just one as in all previous examples. For simplicity, we restrict ourselves to the case of a neutral solvent that does not in itself induce phase separation this corresponds to X = 0, making the excess free energy... 

In this respect, Kuipers made an important point (as illustrated in Fig. 3.10c), namely that layers of thickness x which cover the support to a fraction 6, have the same dispersion as hemispheres of radius 2 x, or spheres with a diameter 3x. Even more interesting is the fact that these three particle shapes with the same surface-to-volume ratio give virtually the same fp/fs intensity ratio in XPS when they are randomly oriented in a supported catalyst The authors tentatively generalized the mathematically proven result to the following statement that we quote literally For truly random samples the XPS signal of a supported phase which is present as equally sized but arbitrarily shaped convex particles is determined by the surface/volume ratio. Thus, in Kuipers model the XPS intensity ratio fp/fs is a direct measure of the dispersion, independent of the particle shape. As the mathematics of the model is beyond the scope of this book, the interested reader... 

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