Microstructure in crystalline solids
by Stefan Müller
The behavior of many materials is governed by microscopic structures invisible to the naked eye. Nature often uses a complex hierarchy of scales to obtain materials with high strength and small weight, e.g. finely structured surfaces as the skin of fishes can reduce the resistance to flow. Also man-made materials exhibit often a complex internal structure: light materials can be strengthened through a network of strong fibers providing high traction resistance with small weight; porous materials have also a very small weight, and their complex pore network gives good shock-absorption properties. Also the behavior of apparently homogeneous materials as steel is strongly modified by an internal structure of microscopically small grains. New optical properties can be obtained coating surfaces with structures of size comparable to the wavelength of light. Computer hard disks store information in form of tiny magnetic domains.
In the last years the study of such microstructures and of their effect on the global behavior of materials has become a very active area of mathematical research. The main aim is to find general principles that lie at the basis of the variety of microstructures, and concepts that capture the essentials of specific microstructures and therefore allow one to avoid the enormously complicated description of all details. This search lead to new, and often unexpected, connections to classical areas of mathematics, and in some cases to the solution of long-standing open problems. Experimentally observed structures inspired new mathematical constructions. Vice-versa, mathematical models delivered new criteria, which can be useful in the search of new materials with special properties.
Research is not restricted to the behavior of individual materials. The analysis of microstructure plays always a role, when effects of different length scales interact. For example, the behavior of flows is in large part determined by a hierarchy of vortices of different sizes. The structure of soil, with a fine network of inclusions and cavities, is of great relevance for the analysis of the flow of underground water, or for the diffusion of pollutants. Similar issues arise in the consideration of many different time scales, as for example is the case in simulations of the behavior of biomolecules.
Particularly interesting are materials whose internal structure is not prescribed, but can be altered under interactions with the environment. The study of such materials, which are often called smart materials due to their capacity of responding to changes in the environment, is a field of research in rapid evolution. One particular class of such materials are the so-called shape-memory alloys. At low temperature they are easily deformed into any shape, but a short heating is sufficient to bring them back to the original, memorized, shape. They have found many applications in everyday life (e.g. unbreakable glass frames), in technological applications (e.g. valves which close automatically at a certain temperature, robotic arms), in aerospatial technology (self-extracting antennas) and above all in medical technology (implants, flexible surgical instrumentation). One application which at the moment is in particularly rapid evolution are the so-called "stents", which are used to stabilize coronaric artherias, and which in some cases are able to avoid a bypass surgery.
Where does shape-memory come from? The origin lies in the fact that the material changes its internal crystalline structure with changing temperature. One example is shown in Figures 1 and 2. At high temperature only the cubic crystalline structure is possible, when the temperature is lowered under the critical one, one side of the cube is stretched, and the other two contracted. This way three different parallelepipedal structures are originated.
Figure 1 Cubic and tetragonal crystal lattices
Figure 2 From a single cubic phase three different tetragonal phases can arise
The existence of many different possible microscopic mixtures of these three basic structures is what allows one to easily deform the material at low temperature (Figure 3). However, rising again the temperature there is only a single cubic structure left, hence the material comes back to a fixed, prescribed shape under heating (Figure 4).
Figure 3 Depending on the different mixing of the various phases the crystal can be elongated or compressed
Figure 4 Idealized representation of the memory-shape effect
The complex microstructure which arises by mixing the different phases can be easily observed experimentally (Figure 5). Therefore the shape-memory alloys constitute an ideal model system to study the influence of external parameters on the formation of microstructure.
The classical crystallographic theory is essentially based on the local conditions which have to be satisfied at the boundary between two phases. Around ten years ago a new point of view was developed. The microstructure originates from the tendency of the material to minimize its free energy. Therefore the theory of microstructures is connected with the calculus of variations, one of the central areas of mathematics, which is devoted to the quest of optimal shapes.
Figure 5 Microstructure in a Cu-Al-Ni single crystal (courtesy of C. Chu and R. D. James, Department of Aerospace Engineering and Mechanics, Minneapolis). The different phases can form striped structures only in definite directions (see also Figure 9), which can then be combined to form complex patterns.