Crystal growth theory

Basics

The thermodynamic theory of crystal growth 　J.W. Gibbs's famous paper "On the Balance of Multiphase Matter" published in 1878 laid the foundation of the thermodynamic theory. He analyzed the conditions for the formation of new phases in fluids, and pointed out that the reduction of free energy of natural bodies facilitates the formation of new phases, but the surface energy hinders it. Only through thermal fluctuations to overcome the barrier required to form the critical size crystal nucleus can the nucleation of the crystal be achieved. By the 1920s, M. Folmer and others developed the classic nucleation theory and pointed out the promotion of nuclei (non-uniform nucleation) by the walls or impurity particles. Once the crystal nucleus has been formed (or a seed crystal has been prepared in advance), what follows is the problem of crystal growth. Gibbs considered that the surface energy coefficient of the crystal is anisotropic, and the minimum free energy in the equilibrium state is attributed to the minimum surface energy, so the equilibrium shape of the crystal can be derived from the pole figure of the surface energy. The theory of crystal equilibrium shape has been used by P. Curie et al. to explain the polyhedral shape of the growing crystal. However, the crystal growth is carried out under conditions that deviate from equilibrium, and the control effect of the surface energy on the crystal shape is limited to the crystals below the micron size. Once the crystal size is large, the ability of the surface energy to directly control the shape is lost. The decisive role is the anisotropy of the growth rate of each crystal plane. In this way, the problem of crystal face growth kinetics is highlighted.

Kinetic theory

The kinetic theory of crystal growth The kinetics of crystal growth refers to the deviating driving force (supercooling or supersaturation) and the rate of crystal growth The relationship between it is closely related to the microscopic morphology of the crystal surface. Research in this area began in the 1920s. The smoothness (atomic scale) of the crystal face plays a key role in the growth kinetics. On the rough crystal surface, atoms can be filled almost everywhere to become growth sites, which leads to the rapid linear growth law. As for the ortho-planes that deviate from the low-index planes, W. Kossel and F. Stransky proposed a crystal plane step-kink model, where the kinks of the steps on the crystal plane are the place for growth. From this, the corresponding growth law can be derived. As for the smooth dense planes (these are the slowest growth rates, and therefore the most common in crystal growth), when a layer of atoms is filled, there are no steps on the surface to provide a place to continue to fill the atoms, and thermal activation must be used to overcome the formation of two After the barrier of the dimensional crystal nucleus, it can continue to grow. In this way, the two-dimensional nucleation rate controls the growth rate of the crystal plane, leading to an exponential growth law. Growth can only be observed under very high driving forces (for example, 50% supersaturation). However, the actual measurement results are significantly contradictory to this inference. In order to explain the growth of smooth crystal planes under the action of low driving force, F.C. Frank proposed in 1949 that screw dislocations would form permanently unfilled steps at the outcropping of crystal planes and promote the growth of crystal planes. The spiral steps observed on the crystal growth surface confirmed Frank's idea. In this important paper entitled "Crystal Growth and Surface Equilibrium Structure" in 1951 by W. Burton, N. Cavreira and Frank, a comprehensive study of the crystal growth kinetics of ideal crystals and actual crystals was carried out. Elaboration has become an important milestone in the development of crystal growth theory.

Features

The smoothness of the surface is related to factors such as crystal structure, material characteristics, crystal plane orientation and temperature. The periodic bond theory proposed by P. Hartman is to determine the degree of smoothness according to the number of periodic bond chains in the crystal plane. More physical theories are based on the statistical mechanics of crystal planes. K.A. Jackson’s theory clarifies the relationship between phase transition entropy and surface smoothness; Burton and Cabrera’s theory points out that at a certain critical temperature, the surface may undergo a smooth-rough transition. In recent years, there have been more in-depth theoretical discussions on these issues, and the computer simulation of crystal planes intuitively reproduces the previous theoretical assumptions, and is extended to a non-equilibrium state.

Transport theory and morphological stability of crystal growth 　Crystal growth is a discontinuous process in space, and crystallization only occurs at the solid-fluid interface. There are heat and mass transport processes in fluids and solids. This type of transport problem can usually be dealt with by macro-physical methods, that is, it can be transformed into the solution of partial differential equations under boundary conditions. Of course, this boundary value problem has its particularity, that is, as the crystal grows, the boundary is moving. As early as 1891, J. Stefan first dealt with the problem of ice thickness in the polar regions, so this type of problem is called Stefan's problem. The external boundary conditions of the Stefan problem should simulate the actual conditions of the growth system. The analytical solution is limited to a few simple geometric shapes.

Heat and mass transfer in the fluid phase can be achieved by convection, so the heat transfer and solute diffusion in the fluid are often confined to the boundary layer at the solid-liquid interface. In this way, the boundary layer theory of fluid mechanics can be applied to the corresponding Stefan problem. But the fluid effect of crystal growth also has its complicated side, especially involving the flow instability and unsteady flow. It is extremely difficult to perform exact theoretical calculations, so it often resorts to simulation experiments or analysis of crystal growth layers.

Important issues

An important issue in crystal growth morphology is the stability of the morphology: specifically, whether the growth interface can be maintained continuously. Although some interfaces can satisfy the solution of Stefan’s problem, they do not actually appear because such interfaces are unstable to interference. Imagine that a certain flat interface is disturbed at a certain instant, causing the interface to locally protrude. There are two possibilities for its evolution over time: one is that the amplitude of the interference gradually decays, and finally the interface returns to its original state, indicating that the original interface is stable; the other is that the amplitude of the interference gradually increases, indicating that the original flat interface is Unstable, may be transformed into uneven cell interface, or even develop into dendrites (den-drites). For pure materials, a positive temperature gradient (melt temperature above the freezing point) stabilizes the interface, while a negative temperature gradient (melt temperature below the freezing point) causes interface instability. Usually crystal growth is always carried out under positive temperature gradient conditions, but the instability of the flat interface is often observed. In the 1950s, B. Chalmers proposed the effect of component undercooling caused by solutes to explain. By the early 1960s, W.W. Mullins and R.F. Sekka used self-consistent dynamics to deal with interface stability problems, derive more correct stability criteria, and can track interface instability and the initial evolution process. The theory of interface stability has also been extended to the solidification, dendrite growth and smooth interface instability of eutectic alloys, and it is still under development.

Bibliography

RLParker,Crystal Growth Mechanism: Energetics,Kinetics and Transport, Seitz, Turnbull and Ehrenreich,ed.,Solid State Phys., Vol.25, Academic Press, New York, 1970.

Min Naibin: "The Physical Basis of Crystal Growth", Shanghai Science and Technology Press, Shanghai, 1982.