Constraints and Metastable States

We introduced the idea of constraints and metastable states in 4.4.3. Because of the importance of these concepts, we take a closer look at them here. 

An example, would be the cylinder containing a gas and having an internal piston that we considered in Chapter 4 (Figure 4.1). In this text I identify such states having an extra constraint as metastable states, but Callen does not use the term metastable in this connection. 

Stable A term describing a system in a state of equilibrium corresponding to a local minimum of the appropriate thermodynamic potential for the specified constraints on the system. Stability cannot be defined in an absolute sense, but if several states are in principle accessible to the system under given conditions, that with the lowest potential is called the stable state, while the other states are described as metastable. Unstable states are not at a local minimum. Transitions between metastable and stable states occur at rates that depend on the magnitude of the appropriate activation energy barriers that separate them. 

It is important to note that the cases of metal crystallites on a substrate and of a substrate of arbitrary thickness deposited upon a metal foil are not equivalent from a thermodynamic point of view because the constraints to which each of these systems are subjected are different. In the first case, a monolayer of TiOx will cover the metal, the amount being determined by the equilibrium with the surface of the substrate. For the second, the entire deposit of TiC>2 must be located on the surface. Since the coverage by a monolayer leads to the smallest free energy, the excess of TiOx should form in the latter case a tridimensional structure with the least possible surface area over the smallest possible part of the substrate surface, thus minimizing the free energy. There are, however, kinetic difficulties to achieve such a structure. For this reason, if coo<0, it is likely that a metastable state of extended patches... 

Although, as we have seen, each system has a minimum number of independent variables that must be specified or fixed for the system to be at stable equilibrium, systems frequently have more than this number of variables fixed, and they can then be said to be in metastable equilibrium states. In fact our definition of a metastable state can be restated as one that has more than the minimum number of constraints necessary to fix the equilibrium state. To illustrate what we mean by a metastable state, and the wide-ranging nature of the definition, we consider next three examples. 

To reiterate, systems at stable equilibrium must be described in general in terms of two fixed state variables. If two different equilibrium states having the same values of two state variables exist, then either both are metastable, or one is stable and one is metastable, and the metastable states in fact should be described in terms of three or more constraint variables. Usually, however, the third constraint is an activation energy barrier and is not thought of as a third variable (though in principle it is). For each choice of the two state variables that the two states have in common, there exists a function (another state variable) that is minimized or maximized at stable equilibrium therefore, by comparing values of this variable one can tell which of the two states is more stable. Note finally that although we have many potentials, the existence of the entropy parameter is the fundamental fact that allows us to define them all. It appears in one way or another in all thermodynamic potentials. 

If you write AGt,p < 0, the meaning is quite clear a macroscopic difference between two quantities, G initial and G final But the differential form dGp,p < 0 is less clear. It implies that there is a function G of which dG is the differential. If dG refers to an increment of a process leading from a metastable state (having three constraint variables) to a stable state (having two constraint variables), this in turn implies that we have a functional relationship between G and the three constraint variables. This should have the form... 

However, if more than the minimum two constraints apply to the system, then any equilibrium state achieved will be in our terms a metastable state, (14.25) does not apply, and the difference in chemical potential between products and reactants is not zero. In our example, a solution might be supersaturated with H4Si04 but prevented from precipitating quartz by a nucleation constraint, so that /iH4Si04 A si02 2//H2O > 0. 

As we have just noted, Arpb is not necessarily zero, and is not if the system is in a metastable state, but when the system achieves equilibrium with respect to the minimum two constraints (what we have called stable equilibrium), ArP becomes zero, the activities in the Q term take on their stable equilibrium values, and aj is called K instead of Q. Thus at stable equilibrium,... 

Lynch and Pettitt (36) employed a pure Nose thermostat and RATTLE-like algorithm for bond constraint, as well as a single fraction particle. Our approach (40) used quaternions (41) to allow separate Nose-Hoover thermostats for translational and rotational modes. We found that the system s tendency to freeze in metastable states could be overcome by introduction of multiple fractional particles (four were used). Curve (b) of Fig. 3 is an extreme example of runaway insertions at high chemical potential as compared to curve (a) where the actual chemical potential of water resulted in a stable density virtually undistin-guishable from the experimental one. 

In the equilibrium position, the piston is not locked in place, i.e., there is no constraint other than U and V, no third constraint. In any other position the piston must be locked in place, because the pressure on one side is greater than the pressure on the other side. Nevertheless, such locked positions are unchanging, and are equilibrium states. We want to distinguish between these states having an extra constraint, and those equilibrium states that have only two constraints, so we call the three-constraint states metastable equilibrium states. 

The USV surface defined by this equation has an energy U defined for given values of the two constraint variables S and V. It refers to changes in energy between stable equilibrium states of a system (which may take place reversibly or irreversibly). At given values of S and V, a metastable state will have greater values of U than those represented by this surface, and it will be held in that state by a third constraint of some kind. Examples of equations having a third constraint are shown in 4.7. 

Thus Reiss prefers to consider all states of equilibrium on an equal basis, as long as the number of constraints is specified. He attaches no special significance to the third constraint, and considers metastability in the conventional sense (very slow reaction rates). However, it seems to me that the concept of metastabiUty, so common in geochemistry, is clarified by defining it in terms of a third constraint, rather than in terms of reaction rates. 

He also says that metastable states have a good deal in common with constrained states, but the only example he gives is of a supercooled vapor phase which requires no physical constraints, but rather some good luck and/or a sense of humor to be treated as an equilibrium state. He does not distinguish between real states and thermodynamic states. In addition, his statements about the applicability of the fundamental equation do not take into account all possibilities. 

I identify metastable states with the third and higher constraints, and... 

T in Fig. 14.1) and truly amorphous material (middle bar B in Fig. 14.1) - as a model for semicrystalline polyethylene. The goal of studying the interface between crystal and melt also at temperatures different than the melting temperature requires that one imposes certain constraints on the system to keep the interlameUar domain in a metastable state [27]. Crystallization at low temperatures is prevented by keeping the volume constant, while in order to prevent melting at high temperatures the crystal sites are immobile. 

At every point in a phase diagram like Figure 11.14, if more than one phase is present and we have true thermodynamic equilibrium (systems like that in Figure 11.14 often display metastable equilibria), we can say that the system has taken up the state with the lowest Gibbs energy consistent with the external constraints and the initial state. That means that for... 

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