Topology three-phase

The microstructure of the multiphase media is often the product of phase transitions, e.g. (i) capillary condensation in the porous media, (ii) phase separation in polymer/polymer and polymer/solvent systems, (iii) nucleation and growth of bubbles in the porous media, (iv) solidification of the melt with a temporal three-phase microstructure (solid, melt, gas), and (v) dissolution, crystallization or precipitation. The subject of our interest is not only the topology of the resulting microstructured media, but also the dynamics of its evolution involving the formation and/or growth of new phases. 

There are some important geometrical constraints for the phase equilibria topology in isobarothermal sections of ternary systems. For example, the Dutch physical chemist Franciscus Antonius Hubertus Schreinemakers (1864-1945) developed rules that determine the arrangement of stable and metastable univariant equilibria where they intersect at an invariant point (Schreinemakers, 1912, 1915). (Incidentally, Schreinemakers also authored the third volume of Die Heterogenen Gleichgewichte von Standpunkte des Phosenlehre.) Schreinemakers found that the extrapolations of one-phase field boundaries must either both fall inside a three-phase field or one inside each of the two two-phase fields. There are actually a number of Schreinemakers mles that are helpful for... 

A more exact procedure is to solve the Bom-von Karman equations of motions 38) to obtain frequencies as a function of the wave vector, q, for each branch or polarization. These will depend upon unit-cell symmetry and periodicity, force constants, and masses. Thus, for a simple Bravais lattice with identical atoms per unit cell, one obtains three phase-frequency relations for the three polarizations. For crystals having two atoms per unit cell, six frequencies are obtained for each value of the phase or wave vector. When these equations have been solved for a sufficient number of wave-vectors, g hco) can, in principle, be obtained by direct count . Thus, a recent calculation (13) of g to) based upon a normal-mode calculation that included intermolecular forces gave an improved fit to the specific heat data of Wunderlich, and showed additional peaks of 140, 90 and 60 cm in the frequency distribution. Even with this procedure, care must be exercised, since it has been shown that significant features of g k(o) may be rormded out. Topological considerations have shown that significant structure in g hco) vs. ho may arise from extreme or saddle points in the phase-frequency curves (38). 

Welchko, B. A., Lipo, T. A., Jahns, T. M., Schulz, S. E. (2004). Fault tolerant three-phase ac motor drive topologies A comparison of features, costs, and hmitations. IEEE Transactions on Power Electronics, 19(4), 1108-1116. 

Figure 17.22 shows some examples of how these two elements combine. For example, a binary minimum melting loop (top of Figure 17.22) can be considered to be produced by combining two simple melting loops. A simple peritectic can be considered to be what happens when a solvus intersects a simple melting loop, and a eutectic what happens when a solvus intersects a binary minimum melting loop. You can try to make these intersections with other topologies, but they will generally not obey the phase rule. (The difference between a peritectic and a eutectic is illustrated in Figure 17.23. In both, three phases exist...

In Fig. 4.3 we also plot the (equilibrium) binodals using FVT outlined in Chap. 3 for hard spheres plus penetrable hard spheres with diameters of 2Rg. Qualitatively, the phase diagram topology is quite well predicted. For q = 0.08, only equilibrium fluid, crystal and fluid + crystal regions are found and predicted. Both for q = 0.57 and 1 the phase diagram contains fluid, gas, liquid and crystalline (equilibrium) phases. In the different unmixing regions one now finds gas-liquid coexistence with a critical point, three-phase gas-liquid-crystal and... 

Without loss of generality, the failure management procedure can be conveniently abstracted as a phased-mission process [20], consisting of three phases. The first phase includes operations occurring before physical intervention on the network, e.g., organisation of work team, planning and transportation on site. This phase is considered to end when the failed component (e.g., pipe) is excluded from the network, which comprises the first variation of topology. Hence, load nodes are partitioned into three classes ... 

Anantharaju N, Panchagnula MV, Vedantam S, Neti S, Tatic-Lucic S (2007) Effect of three-phase contact tine topology on dynamic contact angles on heterogeneous surfaces. Langmuir 23 11673 11676... 

There are three kinds of diffusion (i) within the isotropic phase (ii) the interface (between the isotropic and the crystalline phases) and (iii) the crystalline phase. In the case of a polymer system, the topological nature of polymer chains assumes an important role in all three kinds of diffusion, which has been shown in the chain sliding diffusion theory proposed by Hikosaka [14,15]. It is obvious that any nucleus (a primary nucleus and a two-dimensional nucleus) and a crystal can not grow or thicken without chain sliding diffusion. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

---