Invariant phase system

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

Figure 4.1 shows these diffusion couples below (left) and above (right) the decomposition temperature of the , phases.10 It is obvious that the phase bands of the , phases are present between the fi and 6 phases the left column of the microstructures while it is absent in the right column showing the presence of an invariant phase reaction. The accordingly invariant temperatures are given in Table 4.1. For the Nb-C system the formation of a phase band was most difficult to observe because the peritectoid temperature is rather low and the carbon diffusivity is slow. 

The scalar product of two tensor operators is invariant of the phase system... 

As already mentioned, the sequential coupling of functionally terminated chains of different chemical structure can be used to make block copolymers,59,354,355 including those in which one or more of the blocks is a polysiloxane.74,115,356 If the blocks are relatively long, separation into a two-phase system almost invariably occurs. Frequently, one type of block will be in a continuous phase and the other will be dispersed in it in domains having an average size the order of a few hundred angstroms. Such materials can have unique mechanical properties not available from either species when present simply in homopolymeric form. Sometimes similar properties can be obtained by the simple blending of two or more polymers.357... 

Polymerization in third phase Commercially, the preparation of beads by polymerizing a suspension of a 2-phase emulsion in a third phase appears to be more viable The third phase should ideally be one in which both acrylic esters and allylamine hydrochlorides are insoluble. However, because of the opposite solubility properties of these two monomers, one of them is invariably soluble in a given third phase. It is believed that if one phase is dispersed in the continuous phase, then that should shield the first phase from the third. However, when the two-phase system is added to a third phase, the two-phase emulsion immediately breaks up. In most cases, the two-phase emulsions also disintegrate on heating and so adding the two-phase emulsion to a heated third phase usually proves disastrous. 

The integral in the left side of Equation (33) is called invariant since it does not depend on structural details of the specimen. In the case of a two-phase system... 

These two theorems are general and include as particular cases the theorems established in chap. XVIII, 6 and in chap. XXIII. They do not however apply to mono variant or invariant systems. Thus the eutectic point, which is certainly an indifferent point, does not represent, mathematically, an extreme value of or p for it is the point of intersection of two curves each of which refers to a two-phase system e.g. solution + ice or solution-f salt) under constant pressure. Only at the eutectic do three phases (solution + salt + ice) coexist. A mono variant three-phase system does not have an isobaric curve. 

Corresponding data for the propane/AOT/water system at 25 C are presented in Figure 8 for W - 1, 5, and 20. In a single phase at W 1 (a) hydrodynamic diameter is nearly invarient with pressure (3.8 0.3 nm) with a slight increase suggested at the very lowest pressures. In a single phase system at W 5 (b),... 

Our quantitative findings concern the phase behavior of the L and microemulsion systems are summarized in Figure 3. In the absence of alcohol, the maximum MMA content of the Lj phase is fixed by a limiting MMA/SLS mole ratio of approximately three. Higher mole ratios will invariably lead to a two-phase system. With alcohol... 

The main theoretical results derived from the ideal two-phase model, i.e., Equation (5.70) relating the invariant Q to the phase volumes and the Porod law (5.71), are no longer valid and need be modified when in the two-phase system the phase boundaries are diffuse. To see the modifications necessary to these theoretical results, we represent by p r) the scattering length density distribution in a two-phase material with diffuse boundaries and by p d(r) the density distribution in the (hypothetical) system in which all the diffuse boundaries in the above have been replaced by sharp boundaries. The two are then related to each other25 by a convolution product... 

If a two-phase system in the composition range between/and h is heated, then at t liquids/and g are present, and vapor h appears. The system at t is invariant. Since the vapor is richer in butanol than the original overall composition, the butanol-rich layer evaporates preferentially, leaving liquid/and vapor h. As the temperature rises, the liquid is depleted in butanol finally only vapor remains. 

The phase diagram for a typical one-component system is illustrated in Fig. 9-1. The solid, liquid, and vapor regions are one-phase systems for which / = 2. The curves, whose slopes are given by Eq. (9-14), represent states of coexistence of pairs of phases and are univariant. At the triple point, three phases coexist, and the system then is invariant. 

The process design and the reaction parameters in the Haifa Chemicals process were selected to reduce the limitations imposed by the properties of the extractant. Due to the low efficiency of the extractant, completion of acid extraction from the separated reaction mother liquor is impractical. To avoid losses of reagent and of by-product, the reaction is performed in the presence of the extractant, forming an invariant system. The acidity of the aqueous phase is maintained constant throughout the extraction at a level that provides suitable extraction. The four-phase system (solid KNO, solid KCl, an aqueous solution, and an extractant phase) is, however, difficult to control, a fact that interferes with production of large, easy to wash KNO j crystals. 

In a situation compatible with the lubrication approximation, perturbations due to the proximity of a solid surface are weak. In this case, the translational invariance of an unbounded two-phase system is weakly broken, and both the shift of the equilibrium chemical potential due to interactions with the solid surface and the deviation from the zero-order density profile are small. Since molecular interactions have a power decay with a nanoscopic characteristic length, this should be certainly true in layers exceeding several molecular diameters. A necessary condition for the perturbation to remain weak even as the liquid-vapor and liquid-solid interfaces are drawn together still closer, as it should happen in the vicinity of a contact line, is smallness of the dimensionless Hamaker constant % = asps/p — 1- Even under these conditions, the perturbation, however, ceases to be weak when the density in the layer adjacent to the solid deviates considerably from p+. This means that low densities near the solid surface are strongly discouraged thermodynamically, and a... 

A heterogeneous-catalyst is a solid composition which can effect or accelerate reaction by contact between its surface and either a liquid-phase reaction mixture (in which the catalytic material must be essentially insoluble) or gaseous reactants. In liquid-phase systems, one or more of the reactants may be introduced as a gas, but access of such reactants to the (fully wetted) surface of the catalyst is almost invariably by dissolution in the reaction medium and subsequent diffusion. 

For both two- and three-phase systems and for V g>0.08m/s (implying heterogeneous regime, particularly for r>0.2m), the quotient (k ale ) is invariant with respect to the column diameter. 

Figure 23.10 shows a Kratky plot [ ( AX/Ail Q) vs Q for a polyimide sample made from the condensatirm of pyromellitic-dianhydride and oxydianUine (PMDA-ODA). The integrated area under this curve is the invariant which for a 2-phase system is given by... 

This is the type of information that is presented in a Salinity Requirement Diagram. Figure 6 is the Salinity Requirement Diagram for the system under discussion. The vertical bars show, as a function of overall surfactant concentration, the range of brine salinity over which the system is in a Type III phase environment (although not necessarily three phases). The position of the circle on the bar indicates midpoint salinity at that overall surfactant concentration. Optimal salinity for oil-displacement efficiency should be close to that level of salinity. The number within the circle is the volume fraction of surfactant in the invariant phase at midpoint salinity. Healy and Reed (12) found lower microemulsion /ex cess brine and microemulsion/excess oil interfacial tensions for systems in which the volume fraction of surfactant in... 

In the case where one is not strictly dealing with a two-phase system because of, for instance, the polymer crystallizing in spherulites, a modification has to be made. The relative invariant is now given by... 

---