Mixtures tangent construction

At an even lower temperature, T, a sample in equilibrium will consist of the crystalline phase h-Y2C>3(ss), MgO(ss) or a two-phase mixture of these (see Figure 4.8(f)). The compositions of the two phases in equilibrium are again given by the common tangent construction. 

Fig. 3.10 The dimensionless semi-grand potential 12 as a function of volume fraction (j>. Schematic view of the common tangent construction (straight lines) to determine the phase coexistence in mixtures of colloidal hard spheres and phs. (i) gas-4iquid coexistence, (ii) fluid-solid coexistence, (iii) gas-liquid-solid triple coexistence, and (iv) fluid-solid coexistence near a metastable (dashed lines represent the common tangent construction for this case) gas-liquid coexistence...

It follows from the construction in fig. 12.1(b) that for any composition between one end of the curve of fig. 12.1(a) and the nearest point B, partition of the mixture into two phases of different compositions would lead to an increase in the total value of AG ix. Compositions in the two corresponding regions are therefore stable against fluctuations of composition. In contrast, for any composition between the two points B where the common tangent... 

Each chaotic state C, consists primarily of what at first glance appears to be a stochastic mixture of the adjacent periodic states and thus, for example, consists of a mixture of Po and P, as Fig. 6 of ROUX et al. [23] illustrates. However, maps constructed from the time series clearly yield smooth curves, not a scatter of points. These maps indicate that behavior is deterministic, not stochastic. Moreover, it is difficult to imagine stochastic processes that would lead to period doubling, the universal sequence, and tangent bifurcations, yet all of these phenomena associated with chaos are found in one-dimensional maps and in experiments on nonequilibrium chemical reactions, as will now be described. 

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