Gibbs phase rule violation

Abstract The Gibbs phase rule relating the number of degrees of freedom / of a system to the number of components c and the number of coexisting phases p is a central, universally used relation, expressed by what is probably the simplest formula in the natural sciences,/ = c — p + 2. Research into the behavior of small systems, notably atomic clusters, has shown in recent years that the phase rule is not as all-encompassing as is often assumed. Small systems can show coexistence of two or more phases in thermodynamic equilibrium over bands of temperature and pressure (with no other forces acting on them). The basis of this apparent violation of the phase rule, seeming almost like violation of a scientific law, is in reality entirely understandable, consistent with the laws of thermodynamics, and even allows one to estimate the upper size limit of any particular system for which such apparent violation could be observed. 

A pure substance can also have three phases present. According to the Gibbs phase rule, each phase in such a system has zero degrees of freedom. They do not have any independent properties therefore, all intensive properties in each of the three phases are specified. Consider a system in which a pure substance exists in the solid, liquid, and vapor phases. The properties of each phase can have only one value. Since the temperature and pressure are equal in all the phases, they are fixed for the entire system. For example, the values for P and T for water in a system with solid, liquid, and vapor are fixed at 611.3 [Pa] and 0.01[°C], respectively. This state is known as the triple point.In this case, we can specify neither T nor P, since neither property is independent. In other words, both properties we specify to constrain the state must be related to the fraction of matter in each of the three phases present. A pure substance cannot have more than three phases, as such a state would violate the Gibbs phase rule. 

The crystallization process of flexible long-chain molecules is rarely if ever complete. The transition from the entangled liquid-like state where individual chains adopt the random coil conformation, to the crystalline or ordered state, is mainly driven by kinetic rather than thermodynamic factors. During the course of this transition the molecules are unable to fully disentangle, and in the final state liquid-like regions coexist with well-ordered crystalline ones. The fact that solid- (crystalline) and liquid-like (amorphous) regions coexist at temperatures below equilibrium is a violation of Gibb s phase rule. Consequently, a metastable polycrystalline, partially ordered system is the one that actually develops. Semicrystalline polymers are crystalline systems well removed from equilibrium. 

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