Bivariate surfaces

The substrate has been represented by a square lattice of M=LxL adsorbing sites with periodical boundaiy conditions. The heterogeneity has been introduced by considering the two referred kinds of sites. This is a so-called bivariate surface, in equal concentration, forming bd patches distributed in a chessboard-like ordered topography. 

Bulnes F., Ramirez-Pastor A.J. and Zgrablich G., Scaling Behavior in Adsorption on Bivariate Surfaces and the Determination of Energetic Topography. J. Chem. Phys. 115 (2001) 1513 Power Laws in Adsorption and the Characterization of Heterogeneous Substrates. Adsorption Sci. and Tech. 19 (2001) 229 Scaling Laws in Adsorption on Bivariate Surfaces Pf s. Rev. E, 65 (2002) 31603. 

Figure 10.7 Schematic representation of heterogeneous bivariate surfaces with chessboard, (a), random square patches, (b), ordered strips, (c) and random strips, (d), topography. The patch size in this figure is I = 4.

These findings provide for the first time a method to characterize the energetic topography (i.e., obtain the parameters from experimental measurements) of a class of heterogeneous surfaces that can be approximately represented as bivariate surfaces. 

Bulnes, F., Ramirez-Pastor, A.J., and Zgrablich, G. (2001). Scaling behavior in adsorption on bivariate surfaces and the determination of energetic topography. 

E Bulnes, A. J. Ramirez-Pastor, and G. Zgrablich, Scaling behavior of adsorption on patchwise bivariate surfaces revisited, Langmuir, vol. 23, no. 3, pp. 1264-1269, 2007. 

The CFD model described above is adequate for particle clusters with a constant fractal dimension. In most systems with fluid flow, clusters exposed to shear will restructure without changing their mass (or volume). Typically restructuring will reduce the surface area of the cluster and the fractal dimension will grow toward d — 3, corresponding to a sphere. To describe restructuring, the NDF must be extended to (at least) two internal coordinates (Selomulya et al., 2003 Zucca et al., 2006). For example, the joint surface, volume NDF can be denoted by n(s, u x, t) and obeys a bivariate PBE. 

The extension of DQMOM to bivariate systems is straightforward and, for the surface, volume NDF, simply adds another microscopic transport equation as follows ... 

TEMPERATURE EFFECTS ON THE SCALING PROPERTIES OF ADSORPTION ON BIVARIATE HETEROGENEOUS SURFACES... 

When only one sohd phase is present, the total concentration of the solution is indeterminate, and may be altered by addition of the other simple salt or of the double salt. The solubility of a salt is therefore not affected by the addition of a salt with a common ion. For the graphical representation of the equihbria in a three-component system, it is convenient to use a three-dimensional system of coordinates, of which the axes are the temperature and the concentrations of the two simple salts. Each point in the space corresponds to a definite vapour pressure. Monovariant equilibria are represented by lines, and bivariant equihbria by surfaces in the space model. (See van t Hoff, Bildung und Spaltung von Doppelsalzen, Leipzig 1897 also van t Hoff u. Meyerhoffer, Zeitschr. /. physikcd. Chemie, 30, 64 (1899), and others. Experimental methods of determining the transition point are also described there.)... 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

Ternary Systems.—Wq pass over the binary system FeClg—HgO, which has already been discussed (p. 187), and the similar system HCl—HgO (see Fig. 132), and turn to the discussion of some of the ternary systems represented by points on the surface of the model between the planes XOT and YOT. As in the case of carnallite, a plane represents the conditions of concentration of solution and temperature under which a ternary solution can be in equilibrium with a single solid phase (bivariant systems), a line represents the conditions for the co-existence of a solution with two solid phases (univariant systems), and a point the conditions for equilibrium with three solid phases (invariant systems). 

The representation of this equation for anything greater than two variables is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation 1.28 is of the form Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important here, one important property is that they have a well-known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic surface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

Two components present in a single phase constitute a tervariant system, characterized by three degrees of freedom. The equilibrium condition between two phases is a bivariant system, while three phases in equilibrium would be invariant. For a system of two components to be invariant, there must be four phases in equilibrium. From the phase rule, one immediately concludes that there cannot be more than four phases in equilibrium under any set of environmental conditions. The graphical expression of phase relationships on a two-dimensional surface will require the a priori specification of a number of conditions. Fortunately, for the two component systems of greatest interest to pharmaceutical scientists (hydrates and their anhydrates), the normal studies are conducted at atmospheric pressure, which immediately fixes one of the variables. For phase equilibria, this allows the construction of the usual planar diagrams. 

On a phase boundary, two phases are present (P = 2) and, from equation (S1.4), F = 2. The variance is seen to be equal to two. Thus, phase boundaries form two-dimensional surfaces in the -pressure-temperature-composition representation. A two-component system containing two phases is called a bivariant system. In a bivariant system, it is possible to change any two of the three variables temperature, pressure and composition independently but the third will be fixed by the values selected for the other two. On a phase diagram drawn for a system at atmospheric pressure, phase boundaries are drawn as lines. 

Figure 2.28 reports the Bivariate Distribution calculated for a 70/30 mixture of an AB random copolymer with an A homopolymer produced by anionic living polymerization. Figure 2.28 shows that the surface possesses two maxima, illustrating the usefulness of the bivariate plot. 

Fig. 18.5 Inter- and intramolecular branching distribution of tailored LLDPEs two side views and surface contours of the averaged 3D bivariate (MW-1-hexene content) distribution from SEC and TREE corresponding to M and T effects. The profile on the right corresponds to 180° rotation over the vertical axis of the teyi-hand-side profile (Vadiamudi et ai. 2009)...

Equilibria of two phases in two-component systems have a variance f = c-n + 2 = 2 (bivariant), and hence they are represented by two-dimensional surfaces in the phase space. 

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