A summary of classical concepts from thermodynamics concerning phase transitions and a survey of phase transitions for a variety of thermotropic and lyotropic systems. The Landau theory for first- and second-order transitions applied to nematics and smectics is discussed in detail. Classic lattice models (Ising, Heisenberg, XY, Lebwohl–Lasher, …) and critical exponents, often associated with certain liquid crystals transitions, are introduced. Examples of experimental phase diagrams for thermotropic and lyotropic liquid crystals are shown.

Many different phases of matter can be characterized by the symmetries that they break.The Ising model for interacting spins illustrates this idea.In the absence of a magnetic field, there is a critical temperature, below which there is ferromagnetic ordering, and above which there is not.The magnetization is the order parameter for this transition: it is non-zero only when there is ferromagnetic ordering.The ferromagnetic phase transition in the Ising model is explored using the approximate method of mean field theory.Exact solutions are known for the Ising model in one and two dimensions and are discussed, along with numerical solutions using Monte Carlo simulations.Finally, the ideas of broken symmetry and their relationship to phase transitions are placed in the general framework of Landau theory and compared to results from mean field theory.

The quantum mechanical exchange interaction gives rise to magnetic moments and their interactions in materials, which give rise to patterns and structures in the orientations of magnetic moments at low temperatures. With increasing temperature, pressure, and magnetic field, magnetic structures are altered, and Chapter 21 describes several trends that can be understood by thermodynamics. The critical temperature of magnetic ordering, the Curie temperature TC, is calculated. Compared to chemical ordering, the strengths and alignments of magnetic moments have more degrees of freedom, allowing for diverse magnetic structures. These include ferrimagnetism, frustrated structures, and spin glasses. The vectorial character of spin interactions can give rise to localized spin structures such as skyrmions. An electromechanical phase transition can occur when the energy for a displacement of positive and negative ions in a unit cell is comparable to thermal energies. This ferroelectric transition has some similarities to the ferromagnetic transition, but is described by Landau theory. Domains in ferroelectric and ferromagnetic materials can reduce the energy in surrounding elastic and magnetic fields, andthe width of a boundary between two magnetic domains is estimated.

Diffusionless transformations occur when atoms in a crystal move cooperatively and nearly simultaneously, distorting the crystal into a new shape. The martensite transformation is the most famous diffusionless transformation, owing to its importance in steel metallurgy. In a martensitic transformation the change in crystal structure occurs by shears and dilatations, and the atom displacements accommodate the shape of the new crystal. The atoms do not move with independent degrees of freedom, so the change in configurational entropy is negligible or small. The entropy of a martensitic transformation is primarily vibrational (sometimes with electronic entropy, or magnetic entropy for many iron alloys). This chapter begins with a review of dislocations, and how their glide motions can give crystallographic shear. Some macroscopic and microscopic features of martensite are then described, followed by a two-dimensional analog for a crystallographic theory that predicts the martensite “habit plane” (the orientation of a martensite plate in its parent crystal). Displacive phase transitions are explained more formally with Landau theories having anharmonic potentials and vibrational entropy. Phonons are discussed from the viewpoint of soft modes and instabilities of bcc structures that may be relevant to diffusionless transformations.