Phasons,topological defects and dynamics in incommensurate matter

Abstract

In this thesis, we derive some physical consequences of the existence of a phason mode in incommensurate materials. We consider two main examples of such materials-quasicrystals and the incommensurate twist grain boundary (i-TGB) phase of liquid crystals. The three principal results on quasicrystals are: • an elucidation of the structure of small-angle grain boundaries, • a method for determining the full six-dimensional Burgers vector for quasicrystal dislocations from diffraction contrast, and • new predictions which allow a detailed test of the entropic model of quasicrystals. For the i-TGB phase, we present results on the nature of the broken symmetry, linear and nonlinear elasticity, and hydrodynamics, including the effects of thermal fluctuations. In the Introduction (Chapter I), we describe the structure of quasicrystals and the i-TGB phase. We identify the broken symmetry variables in quasicrystals using the Landau theory and briefly review earlier work on continuum elasticity theory and dislocations in quasicrystals. We also give a physical picture of the diffusive nature of phason dynamics. Chapter II contains a derivation of the “invisibility condition” for transmission electron micrographs of dislocations in quasicrystals using the density-wave framework. We show why this condition is essential for a complete analysis of quasicrystal dislocations. In Chapter III, we generalize Frank’s formula for the dislocation content of small-angle grain boundaries in crystals and the Read-Shockley treatment of small-angle grain boundaries in crystals to the case of quasicrystals. We find that even for a symmetric tilt boundary, dislocations with at least two types of Burgers vectors are required; these dislocations have to be arranged quasiperiodically along the boundary. We calculate the dependence of the grain boundary energy on the angle of mismatch and discuss the possible clumping of dislocations to form composites. In Chapter IV, we describe the entropic model of quasicrystals and show how it leads to a square-gradient elastic free energy in the phason variable. We give an intuitively appealing demonstration showing that the free energy of a dislocation line is logarithmic in the system size within this model. Assuming the validity of this model, we derive predictions for the temperature dependence of the grain boundary structure and free energy, the nature of the elastic instability, and the behavior of sound damping near this instability. We believe that these predictions will provide decisive tests for the entropic model of quasicrystals. A simple analysis of Griffith cracks in quasicrystals is presented in Chapter V. We obtain the expression for critical crack length and show how this could be used as a test for the entropic model, in addition to the tests proposed in Chapter IV. In Chapter VI, we consider a model of the i-TGB phase in which dislocation lines are not ordered within any grain boundary. We study the nature of the broken continuous symmetry and set up the nonlinear continuum elasticity theory of this phase from symmetry considerations. Using a perturbative, momentum-shell renormalization group treatment, we show how thermal fluctuations and nonlinearities affect the elastic properties. In the elastic Hamiltonian, we find elastic constants that depend logarithmically on the length scale. We then study the dynamics of this phase by setting up the generalized Langevin equations and obtain the linear mode structure. We also study the effects of out-of-plane fluctuations of the grain boundaries on the viscosity to show that certain components of the viscosity tensor diverge at small frequencies.