First type stability boundary

In the first case, the boundary describes the correlation between pH and Eh values at equal concentrations of two competing migration forms of B, component in solution. Any compormd of B, component has the greatest concentration in solution within the boundaries of its stability fields only. Thus, the first boundary type is defined by a change of the dominant compounds of B. component only. 

First, let us examine the completed Eh-pH diagram for Mn-HjO-Oj in Figure 12.7. There are typically four different types of boundaries shown on these diagrams. The top line, labeled O2/H2O, represents conditions for water in equilibrium with Oj gas at 1 atm. Above this line, a Pq greater than 1 atm is required for water to exist, so that because the diagram is drawn for a pressure of 1 atm, water is not stable above this line. Similarly, the bottom line H2O/H2 represents conditions for water in equilibrium with H2 gas at 1 atm. Below this line, Phj values greater than 1 atm are required for water to exist, that is, at 1 atm water is not stable. Therefore, the water stability field is between these two lines. 

The second type of boundary separates the stability fields of minerals or solid phases such as hausmannite (Mn304) and pyrochroite [Mn(OH)2]- These are true phase boundaries hausmannite is thermodynamically tmstable below the hausmannite/pyrochroite boundary and pyrochroite is unstable above it. Thus these first two kinds of boundary represent thermodynamic stabUity fields for different substances. Notice that on this diagram they all have the same slope (equal to the Nernst slope). 

In this chapter, we will focus on the stability boundaries of the first type. Since the periodic trajectory persists in this case at the critical moment, we can construct a small cross-section and our problem reduces to the study of a Poincare map. In some suitable coordinates on the cross-section, the Poincare map can be written in the form... 

To answer the question raised in the preceding paragraph, we must first itemize the main types of boundaries of stability regions of equilibrium states and periodic orbits. To do this we must undertake a systematic classification of the information we have presented in all previous chapters. We will pay special attention to the features that distinguish each type of boundaries. 

In the final two sections, mechanisms of flame stabilization and processes of flame spread will be considered. In the first of these sections, we shall see that the onset of combustion can cause the boundary-layer approximation to fail. Physical aspects of various types of flame stabilization will be reviewed. The discussion of flame spread also will focus on the many different types of physical processes that may be involved. The presentation should serve to emphasize approximate unifying concepts as well as various currently outstanding unknowns. 

Fig. 51. Plol of — t)(iiZ)/i)Z z=u versus /j(0) for the cases of a second-order wetting transition (a) and a first-order wetting transition (b). The solution consistent with the boundary condition always is found by intersection of the curve /x2(0) - 1 with the straight line [h + gq.(0)]/y. In case (a) this solution is unique for all choices of k /y (keeping the order parameter g/y fixed). Critical wetting occurs for the case where the solution (denoted by a dot) occurs for y (0) = +1, where then ) i(Z)/dZ z u = I) and hence the interface is an infinite distance away from the surface. For ft] > ftlc the surface is non-wet while for fti > ft C the surface is wet. In case (b) the solution is unique for ft] < ft (only a non-wet state of the surface occurs) and for ft > ft (only a wet state of the surface occurs). For ft > ft] > ft three intersections (denoted by A, B, C in the figure) occur, B being always unstable, while A is stable and C metastable for ft]c > ft > ft and A is metastable and C stable for ft " > ft > ftic. At ftic where the exchange of stability between A and C occurs (i.e., the first-order wetting transition) the shaded areas in fig. 51b are equal. This construction is the surface counterpart of the Maxwell-type construction of the first-order transition in the bulk lsing model (cf. fig. 37). From Schmidt and Binder (1987).

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word hydrodynamic is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition. 

It was Sato and Toth who showed that when low-energy imperfections such as antiphase domain boundaries were introduced into an ordered structure without changing the near-neighbor coordination, the positions of (some of) the Brillouin zone boundaries were altered so that they followed an expanding Fermi surface and maintained structural stability despite the increase in electron concentration due to alloying. Consider, for example, the ordered AuCu I structure. The reduced Brillouin zone is made up of 100 planes and the second extended zone is made up of (002) and (110 -type planes. Mapped back in the reduced zone, the second zone has a square cross section normal to the axis. With two electrons per primitive cell, the Fermi surface overlaps the 001 planes of the first zone and touches the 110 planes of the second zone. 

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