Phase equilibrium, coherent

We note here that gel is a coherent solid because its structure is characterized by a polymer network, and hence, the above theoretical considerations on crystalline alloys should be applicable to gels without essential alteration. It is expected that the curious features of the first-order transition of NIPA gels will be explained within the concept of the coherent phase equilibrium if the proper calculation of the coherent energy and the elastic energy of the gel network is made. This may be one of the most interesting unsolved problems related to the phase transitions of gels. 

A VARIETY of good reference sources are available for those who wish to learn more about phase equilibrium calculations and the recent advances in the subject. A partial list of source books is given below. Some of them are recent and provide up-to-date developments, and some dated sources introduce the basic principles in a coherent and easy-to-understand fashion. 

A third definition of surface mobility is essentially a rheological one it represents the extension to films of the criteria we use for bulk phases and, of course, it is the basis for distinguishing states of films on liquid substrates. Thus as discussed in Chapter IV, solid films should be ordered and should show elastic and yield point behavior liquid films should be coherent and show viscous flow gaseous films should be in rapid equilibrium with all parts of the surface. 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

The results of a thermodynamic analysis of the interactions in Equations (127) and (128), as presented in [452], show that a coherent shell of tantalum and niobium hydroxides is formed on the surface of the columbite or tantalite during the interaction with sulfuric acid. The formation of the shell drives the process towards a forced thermodynamic equilibrium between the initial components and the products of the interaction, making any further interaction thermodynamically disadvantageous. It was also shown that, from a thermodynamic standpoint, the formation of a pseudomorphic structure on the surface of columbite or tantalite components is preferable to the formation of tantalum and niobium oxysulfates. Hence, the formation of the pseudomorphic phases catalyzes the interaction described by Equation (127) while halting that described by Equation (128). 

In a basic pulsed NMR experiment (for I = 1/2), when a sample is placed in the applied magnetic field (B0), the nuclear spins distribute themselves between parallel and antiparallel positions, according to Boltzmann distribution [Eq. (11)] (Figure 21 A). The number of spins in the parallel position is slightly greater than that in the antiparallel position. At equilibrium, the spins are processing randomly (i.e., lack phase coherence). The populations... 

In order to obtain estimates of quantum transport at the molecular scale [105], electronic structure calculations must be plugged into a formalism which would eventually lead to observables such as the linear conductance (equilibrium transport) or the current-voltage characteristics (nonequilibrium transport). The directly measurable transport quantities in mesoscopic (and a fortiori molecular) systems, such as the linear conductance, are characterized by a predominance of quantum effects—e.g., phase coherence and confinement in the measured sample. This was first realized by Landauer [81] for a so-called two-terminal configuration, where the sample is sandwiched between two metalhc electrodes energetically biased to have a measurable current. Landauer s great intuition was to relate the conductance to an elastic scattering problem and thus to quantum transmission probabilities. 

The flow of energy from the ground state can also be calculated when it is assumed that the rate of electronic dephasing is small (i.e., L%Pg = 0) and/or the rate of pure vibrational dephasing is small (i.e., L%Hg = 0). These conditions apply when the rate of relaxation to equilibrium is small relative to the rate of loss of phase coherence. Under these conditions... 

The equilibrium interfaces of fluid systems possess one variant chemical potential less than isolated bulk phases with the same number of components. This is due to the additional condition of heterogeneous equilibrium and follows from Gibbs phase rule. As a result, the equilibrium interface of a binary system is invariant at any given P and T, whereas the interface between the phases a and /3 of a ternary system is (mono-) variant. However, we will see later that for multiphase crystals with coherent boundaries, the situation is more complicated. 

---