Defects in Continuous Media

Abstract

Whenever one deals with an extended system where longrange order is induced by short-range interactions one may find topological defects. These are imperfections in the ordering that cannot appear or disappear individually inside the material, like, for example, interstitials and vacancies in a crystal. Defects are important because they determine to a large extent the mechanical and other properties of real materials.""] The theory of defects in crystals is quite complicated, mainly due to the three-dimensional nature and the discreteness of the underlying lattice. As a result, one has a considerable number of different topological point-and linedefects. The discreteness also makes the detailed description very complicated and quantitative results about the energy and other properties are therefore very difficult to obtain. A simple continuum description['"] is only valid far from the defect core and often does not adequately explain important properties. From the experimental &de, the detailed structure of single defects cannot be observed, so there is considerable room for controversy.The situation is more promising for systems where the order can be described genuinely by a continuum theory since the spatial variations connected with the ordering are slow at the molecular level. Good candidates for a continuum description are systems undergoing a second-order phase transition near to the transition point. However, even more appropriate are fluids undergoing a transition to a macroscopically ordered state via a symmetry-breaking instability. Here, hydrodynamic equations can describe the phenomena to virtually arbitrary precision and moreover, the defects are usually easily visible due to their large size.We here consider extended fluid layers where the longrange order is two-dimensional. The most prominent example is the Rayleigh-BCnard instability, where a simple fluid, like water, is driven into a state with permanent convection due to buoyancy effects by heating from below (in the usual case of a positive thermal expansion coefficient).['b1 When the fluid layer is bounded by rigid plates and the experiment

Evolution of disorder in two-dimensional stripe patterns: ‘‘Smectic’’ instabilities and disclination unbinding

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Long-term vortex interaction in active media

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