Phase rule interface

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Equation (6.42) introduces a new independent variable of the system the mean curvature c = (c1 +C2). This variable must be taken into account in the Gibbs phase rule, which now reads F + Ph = C + 2 + 1. The number of degrees of freedom (F) of a two-phase system (Ph = 2) with a curved interface is given by... 

A schematic illustration of the method, and of the correlation between binary phase diagram and the one-phase layers formed in a diffusion couple, is shown in Fig. 2.42 adapted from Rhines (1956). The one-phase layers are separated by parallel straight interfaces, with fixed composition gaps, in a sequence dictated by the phase diagram. The absence, in a binary diffusion couple, of two-phase layers follows directly from the phase rule. In a ternary system, on the other hand (preparing for instance a diffusion couple between a block of a binary alloy and a piece of a third... 

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. 

It should be further mentioned that the choice of the ensemble also depends on whether or not the solvent is explicitly modeled. In particular, care has to be exerted concerning the number of intensive variables of the ensemble. This can be understood considering the Gibbs phase rule for interfaces [93] f = 2 + c - p, with / being the number of independent intensive variables needed to describe the interface, p the number of different phases in the interface, and c denoting the number of different components. A lipid bilayer comprising one sort of lipid embedded into an implicit solvent corresponds to a one-component system c = 1, in one-phase state so that p = 1 hence / = 2. On the other hand, a model with explicit solvent yields c = 2, p = 1, and / = 3. Thus, implicit solvent models can be simulated within nXT ensemble while for explicit-solvent models an additional intensive quantity has to be controlled, e.g., the nPzzXT ensemble is appropriate. 

Degrees of freedom can also be described as the number of intensive variables that can be changed (within limits) without changing the number of phases in a system. This point of view is perhaps more useful to someone looking at a phase diagram thus divariant, univariant, and invariant systems correspond to areas, lines, and points in a P-T projection. We prefer however to emphasize the fact that coexisting phases reduee the number of independent variables, and that some systems have all their properties determined. This fact is very useful, as we will elaborate on below, and its explanation in terms of the Phase Rule is a very beautiful example of the interface between mathematics and physical reality. 

The number of components (C) will be taken to be 3 latex particles, free polymer and dispersion medium. The number of degrees of fre om (F) is 2, the latex particles concentration being taken as an extensive variable. The two intensive variables are the temperature and the concentration of free polymer, both of which can be varied independently. In accordance with the Phase Rule, the number of phases is given by P=C-F+2 = 3, one of which is the vapour phase. This means that two liquid phases are possible. These will be separated by a macroscopic liquid/liquid interface. Note that no information is obtained from this relationship as to the dispersed or flocculated nature of the latex particles in either liquid phase, nor to the respective concentrations of each component in the two phases. 

The phase rule is a very beautiful example of the interface between mathematics and physical reality. 

Systems with an Upper Critical Solution Temperature. In the case described in Fig. 2.1, which is typified by the system phenol-water, the solubilities of A in J5 and J5 in A increase with increase in temperature, so that at some elevated temperature the two conjugate solutions become identical and the interface between them consequently disappears. This temperature, termed the critical solution temperature (C.S.T.), or conso-lute temperature, occurs at the point M in the figure and represents the temperature above which mixtures of A and B in any proportions form but one liquid phase. Point M is the maximum on the continuous solubility curve but is not ordinarily at the midpoint of composition, nor are the solubility curves ordinarily symmetrical. The C.S.T. is the point where the two branches of the solubility curve merge, and the constant temperature ordinate is tangent to the curve at this temperature. The phase rule may be applied to this significant point ... 

Two phases which are in equilibrium must always have the same temperature ( 1 4 and 1 13). In addition, they must have the same pressure, provided that they are not separated by a rigid barrier or by an interface having appreciable curvature ( 2 9a). Finally, any substance which is able to pass freely between the two phas must have the same chemical potential in each of them ( 2 96). These important criteria of equilibrium, expressed in terms of the iniensive properties T, p and /, lead directly to the phase rule of Willard Gibbs. 

Example 1.9 Gibbs phase rule (flat interface) Derive the phase rule F = c + 2 — p, where Fis the number of degrees of freedom, c is the number of components, and p is the number of phases. Assume the interface between the... 

The phase rule presented above is valid if all the components are present in all phases. It should he modified when some of the components are absent in one or more phases. We will see in Chapter 2 that the above phase relationship should be modified when the interface between the phases is curved. A modification is also needed when there is influence of gravity on equilibrium. 

Example 2.9 Gibbs Phase Rule for curved interfaces Deri ve the phase ru 1 e for a composite system of p phases and c components with curved interfaces F = c + l where F is the number of degrees of freedom. If some of the interfaces are flat, then F = c + 1 — I, w here 7 is the number of flat interfaces between the bulk phases. 

Solution The Gibbs Phase Rule for a fiat interface between the phases, which W as established in Example 1.9 of Chapter 1, is based on the assumption that the PdV work is the only mode of work. As W c have seen in this chapter, the equilibrium conditions for systems with curved interfaces and under the influence of gravity are different from the equilibrium conditions of systems with flat interfaces and negligible gravity. For systems with curved interfaces and also with gravity effect, the Gibbs Phase Rule should be modified. In this example, we will only consider the effect of the curved interface. 

For systems in which there is a three-phase contact line between the phases as a result of a solid phase, the concept of contact angle is introduced. For such systems, the Phase Rule remains the same (Li et al, 1989). For highly curved interfaces where the thickness of the heterogeneous region between the phases is not small compared to r, there are other considerations in the derivation of the Phase Rule (Li et al., 1989 Li, 1994). 

Let us give two examples for the use of the Phase Rule for curved interfaces. First, consider a single component gas-liquid system with a curved interface between the gas and liquid phases for this system F = 2. Therefore, we can fix two intensive variables, say temperature and pressure of the vapor phase. Then the system is fully defined. We can also specify temperature and curvature. Note that for a single-component gas liquid system with a flat interface, F 1. If we fix the temperature, the vapor pressure is fixed. In the second example, wc consider a two-component two-phase system with a curved interface between the gas and liquid phases, F = 3. Unlike the system with a flat interface, specifying the temperature and pressure of the gas phase does not specify the system. Wc also need to specify the interface curvature. 

Example of multiphase flash and stability analysis. We will, in detail, discuss the stability analysis of a three-component system of Ci/CO /nCif at T = 294.0K and P — 67 bar with — 0.05. 2 co.> = 0.90, and = 0.05. At fixed temperature and pressure, from the phase rule F — c - -2 — p, there can be a maximum of three phases when the interface between the phases is flat. The first question is what types of phases may exist—gas, liquid, or solid. As we will see in Chapter 5, a solid phase does not exist for the above system. Therefore one might expect (1) a single gas phase or a single liquid phase, (2) gas and liquid phases, (3) liquid and liquid phases, or (4) gas-liquid-liquid phase separation. The difficulty in liquid-liquid (L-L) and vapor-liquid-liquid (V-Lr-L) and higher-phase equilibria (for more than three components) is how many phases should be considered for flash calculations. One approach is to determine whether one, two, or more phases are to be considered without prior knowledge of the true number of phases. In certain cases, as we will see in the next chapter, it is possible from thermodynamic stability analysis to determine the true number of phases a priori without performing a flash. However, in general, we do not know the true number of phases. One may, therefore, follow a sequential approaches outlined next for the Ci/C02/nCiQ example. 

The starting point of the multicomponent thermodynamic theory of capillarity is the classical work of Gibbs [2]. In the subsequent development, great attention has been paid to the phase interfaces, both from thermodynamic [4] and molecular [16] points of view. With regard to thermodynamic behavior of the bulk phases, few important problems have been studied, such as the phase rule [17] or generalization of the Kelvin equation to a multicomponent case [4,18]. Much more is known about capillary equihbrium in a single-component case [19-21]. Recently, the interest in multicomponent capillary equihbria has appeared in connection with the petroleum appUcations [22-25],... 

The thermodynamic equilibria of amphiphilic molecules in solution involve four fundamental processes (1) dissolution of amphiphiles into solution (2) aggregation of dissolved amphiphiles (3) adsorption of dissolved amphiphiles at an interface and (4) spreading of amphiphiles from their bulk phase directly to the interface (Fig. 1.1). All but the last of these processes are presented and discussed throughout this book from the thermodynamic standpoint (especially from that of Gibbs s phase rule), and the type of thermodynamic treatment that should be adopted for each is clarified. These discussions are conducted from a theoretical point of view centered on dilute aqueous solutions the solutions dealt with are mostly those of the ionic surfactants with which the author s studies have been concerned. The theoretical treatment of ionic surfactants can easily be adapted to nonionic surfactants. The author has also concentrated on recent applications of micelles, such as solubilization into micelles, mixed micelle formation, micellar catalysis, the protochemical mechanisms of the micellar systems, and the interaction between amphiphiles and polymers. Fortunately, almost all of these subjects have been his primary research interests, and therefore this book covers, in many respects, the fundamental treatment of colloidal systems. 

Although occasionally papers appear speaking of the inapplicability of Gibbs phase rule [Li, 1994, 13] or beyond the Gibbs phase rule [Mladek et al, 2007], this invariably means no more than that one of the ceteris paribus conditions Gibbs already mentioned is not fulfilled for example, the phase rule doesn t cover systems in which rigid semi-permeable walls allow the development of pressure differences in the system. Gibbs explicitly allows for the possible presence of other thermodynamic fields. An extended phase rule has been proposed for, inter alia, capillary systems (in which the number and curvature of interfaces/phases play a role), multicomponent multiphase systems for which relative phase sizes are relevant [Van Poolen, 1990], colloid systems (for which, even if in equilibrium, it is not always easy to say how many phases are present), unusual crystalline materials, and more. 

In all cases the electrode reaction secures continuity of current flow across the interface, a relay type of transfer of charges (current) from the carriers in one phase to the carriers in the other phase. In the reaction, the interface as a rule is crossed by species of one type electrons [e.g., in reaction (1.22)] or ions [e.g., in reaction (1.21)]. 

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