# Phases and Phase Diagrams

## Introduction

In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties. A straightforward way to describe phase is "a state of matter which is chemically uniform, physically distinct, and (often) mechanically separable." Ice cubes floating on water are a clear example of two phases of water at equilibrium. In general, two different states of a system are in different phases if there is an abrupt change in their physical properties while transforming from one state to the other. Conversely, two states are in the same phase if they can be transformed into one another without any abrupt changes. There are, however, exceptions to this statement, such as the liquid-gas critical point. Moreover, a phase diagram is a type of graph used to show the equilibrium conditions between the thermodynamically-distinct phases. Common components of a phase diagram are lines of equilibrium or phase boundaries, which refer to the lines that demarcate where phase transitions occur. A triple point is, in a pressure-temperature phase diagram, the unique intersection of the lines of equilibrium between three states of matter, usually solid, liquid, and gas. The solidus is the temperature below which the substance is stable in the solid state. The liquidus is the temperature above which the substance is stable in a liquid state. There may be a gap between the solidus and liquidus; within the gap, the substance consists of a mixture of crystals and liquid. Phase diagrams are useful for material engineering and material applications. With their aid, scientists and engineers understand the behavior of a system which may contain more than one component.

## Definitions

Concept Definition
Bimodal curves In the phase diagram, the compositions which are in equilibrium with each other as a function of temperature.
Coexisting compositions Different compositions in phase diagram that are at local energy potentials and have the same free energy.
Equilibrium state Reversible
Glass Transition Below a transition temperature, individual polymers lose mobility and form a solid lacking a crystalline structure.
Gradient energy coefficient Has the dimension of length squared. The length in question must be related to the range of intermolecular forces involved. where k is the gradient energy coefficient (Jones p. 34)

Helmholtz free energy $F=U-TS$, also commonly denoted $A$. The Helmholtz free energy is a thermodynamic potential relating to the amount of useful work one may extract from a system that is isothermic (no temperature change) and isochoric (no volume change). We say "useful" work to mean work that can be extracted from the system, and not work done to change the system itself. An equivalent result is that the Helmholtz energy is minimized at equilibrium for an isothermic and isochoric system.

If a reversible process is performed on a system, the change in free energy is $\delta F=-P \delta V-S\delta T$.

Mean field approximation The interaction between molecules, even of different species is the same. The volumes of the molecules are the same.
Non-equilibrium state History dependent
Nucleation A large enough fluctuation of composition in the metastable regime to produce a region of lower chemical potential.
Nucleation – Homogeneous vs heterogeneous Nucleation normally occurs at nucleation sites on surfaces containing the liquid or vapor. Suspended particles or minute bubbles also provide nucleation sites. This is called heterogeneous nucleation. Nucleation without preferential nucleation sites is homogeneous nucleation. Homogeneous nucleation occurs spontaneously and randomly, but it requires superheating or supercooling of the medium. []
Order parameter Zero for disordered phase
Phase In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (density, crystal structure, index of refraction, etc.).[]
Phase change In thermodynamics, a phase transition is the transformation of a thermodynamic system from one phase to another.[]
Phase diagram A phase diagram In physical chemistry, mineralogy, and materials science is a type of graph used to show the equilibrium conditions between the thermodynamically-distinct phases []
Phase transition Phase change to non-equilibrium state
Phase transition, first order Order parameter changes discontinuously, e.g. melting of a crystal
Phase transition, second order Order parameter changes continuously, e.g. liquid-to-gas at critical point
Regular solution model A mean field theory to predict the free energy of mixing.\: The interaction between molecules, even of different species is the same. The volumes of the molecules are the same.
Self-assembly Spontaneous arrangement
Spinodal In the phase diagram, the locus of compositions where the second derive in the free energy is zero.
Spinodal decomposition Fluctuations in the unstable portion of the phase diagram grow continuously, and the
c d
c d