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We have studied the interface between regions of opposite spin in the simple cubic Ising model. Low-temperature expansions of moments of the gradient of the density profile and of the slope at its midpoint suggest that the interface width diverges at a temperature ${T}_{R}$ about half the critical temperature. We discuss the physical significance of this transition and implications for general theories of interfacial properties.

The idea that there could be a “roughening” of the interface of a crystal in equilibrium with its vapor at a particular temperature TR was first suggested by Burton and Cabrera (1949) and further developed in a now classic article by Burton, Cabrera and Frank (BCF) (1951). Representing the crystal surface by a two-dimensional (2D) Ising model they suggested that there would be large fluctations in the surface structure at the Ising model’s critical temperature TC(2D) and a disappearance of the nucleation barrier to crystal growth. Jackson (1953, 1967) further developed and extended these ideas to the case of melt growth and showed that the morphology and growth mechanism of a wide class of crystals could be understood by assuming they were grown above or below the appropriate surface roughening temperature.

The van der Waals picture focuses on the differing roles of the strong short-ranged repulsive intermolecular forces and the longer ranged attractions in determining the structure and dynamics of dense fluids and solids. According to this physical picture, the attractive interactions help fix the volume of the system, but the arrangements and motions of molecules within that volume are determined primarily by the local packing and steric effects produced by the repulsive forces. This very useful approach, its limitations, and its successful application to a wide variety of static and dynamic phenomena in condensed matter systems are reviewed.

An understanding of the frictional sliding on faults that can lead to earthquakes requires a knowledge of both constitutive behavior of the sliding surfaces and its mechanical interaction with the loading system. We have determined the constitutive parameters for frictional sliding of initially bare surfaces of Westerly granite, using a recently developed high pressure rotary shear apparatus that allows long distances of sliding and therefore a greater assurance of attaining steady state behavior. From experiments conducted at room temperature and normal stresses of 27–84 MPa several important results have been found. (1) A gouge layer 100 to 200 μm thick was developed from the initially bare rock surfaces after 18 to 70 mm of sliding. (2) The steady state frictional resistance, attained after about 10 mm of sliding, is proportional to the negative of the logarithm of the sliding velocity. (3) Abrupt changes in the velocity of sliding result in initial changes in the frictional resistance, which have the same sign as the velocity change, and are followed by a gradual decay to a new steady state value over a characteristic distance of sliding. This velocity weakening behavior is essentially identical with that found by several previous workers on the same material at lower normal stress. (4) Our results are well described by a two state variable constitutive law. The values of the constitutive parameters are quite similar to those found previously at low normal stress, but the characteristic distance is about an order of magnitude smaller than that found at 10 MPa normal stress with thicker layers of coarser gouge. (5) We have approximated our results with a one state variable constitutive law and compared the results with the predictions of existing nonlinear stability analysis; in addition, we have extended the stability analysis to systems possessing two state variables. With such formulations good agreement is found between the experimentally observed and theoretically predicted transitions between stable and unstable sliding. These results allow a better understanding of the instabilities that lead to earthquakes.

We analyze the derivation of boundary-layer models for dendritic growth and investigate the extent to which they yield information about the existence of a continuous family of steady-state needle crystal solutions. Although recent work has established that there exists only a discrete set of solutions for viscous fingering in a Hele-Shaw cell, side walls play an important role in this system, and we argue on physical grounds that the same mechanism may not apply to free dendrites. After a discussion highlighting the physical differences in these two systems, we analyze the model equations for dendritic growth, which first suggested the breakup of the family. We develop a systematic, and in principle exact, boundary-layer formalism for diffusion-controlled dendritic growth starting from the full heat-conduction equation. A consistent application of the formalism generates an expansion of the smooth steady-state solutions in powers of (1-\ensuremath{\Delta}), where \ensuremath{\Delta} is the dimensionless undercooling, but gives no indication as to whether or not a family of such solutions exist. Different physically motivated approximations yield different model equations, including the boundary-layer model of Ben-Jacob and co-workers, with very different properties. Steady-state predictions of all such models are arbitrary. We show that a proper phase-space description requires an infinite-dimensional phase space, in which there are stable directions not found in the boundary-layer model.

It is argued that surface undulations observed after self-sustained rapid crystallization of amorphous films result from a thermal instability that induces a periodically varying crystallization rate. Its physical origin is discussed for a simple nonlinear heat-conduction model which yields good agreement with experimental observations. A numerical analysis of the nonlinear oscillations shows that these may, in turn, bifurcate via a series of period-doubling bifurcations.

The equilibrium structure and thermodynamics of a simple liquid is discussed. The particular system considered in our calculations is the Lennard- Jones liquid in which the intermolecular interaction is the pairwise additive 6- 12 potential. The potential is separated into two parts, a reference part containing all the repulsive forces and a perturbation containing all the attractive forces. (This separation is distinctly different from the often-used separation of the potential into positive and negative parts.) It was found that the effects of the two parts of the potential on the structure of the liquid are most easily discussed in terms of the structure factor (or equivalently, the Fourier transform of the pair-correlation function) rather than the pair-correlation function itself. On the basis of the results a very simple prescription is given for calculating the radial distribution function of a simple liquid at high densities which is more accurate than those obtained by any previously reported theory.

It is shown that the repulsive (not merely the positive) portion of the Lennard-Jones potential quantitatively dominates the equilibrium structure of the Lennard-Jones liquid. A simple and accurate approximation for the radial distribution function at high densities is presented.

In his recent Buckley Prize lecture, J. C. Phillips showed a slide (Fig. 1) in which he compared the actual usage of, as opposed to the purely theoretical interest in, various electronic structure methods in the band theory of solids. In this slide pseudopotentials and the APW method heavily outweighed all others, and in the upper right-hand corner was a tiny square representing LCAO. If Jim had, however, been interested in comparing the total number of electronic structure calculations without limiting himself to solids, all other methods would have paled in comparison to LCAO. At least half of all LCAO calculations are Huckel theory ones. In terms of the practical application of quantum theory to real electronic structures, then, one must submit that the Huckel theory and other rather crude LCAO concepts are by far the most successful. It is reasonable to suppose that most of this success must be accounted for by the fact that these methods really work.

Fault slip occurs via earthquakes or stable creep depending on the manner in which frictional resistance varies with sliding velocity and displacement. If the form of this dependence is a function of rock type, observed spatial variations in slip behavior along faults could be explained by lithologic variation. We have used a rotary shear apparatus to investigate the frictional response of dolomite marble to changes in sliding velocity. We have modeled the results using a state variable constitutive law for rock friction and a numerical method to solve for the interaction between the friction law and the compliance of our machine. Dolomite marble displays steady state velocity strengthening, as contrasted with granite and quartzite, which show velocity weakening at the same conditions. To model the dolomite results, it is necessary to use two state variables, and surprisingly, the parameters specifying the decay magnitudes associated with these state variables differ in sign from each other. The magnitude of the parameter associated with the longer decay is negative, leading to a downward decay of frictional strength associated with a decrease in velocity. If the shorter decay is of sufficient magnitude, as we observe at low velocity, it is possible for the machine-rock system to exhibit instability and oscillatory behavior even though the steady state response is velocity strengthening. Another way in which dolomite contrasts with granite and quartzite is that the friction parameters vary strongly with velocity. A transition from unstable to stable sliding with increasing velocity has been observed previously, and our observed variation of the constitutive parameters with velocity can explain this behavior. There is some indication that the evolution of the state variables for dolomite occurs as a function of time rather than displacement, in contrast with granite and quartzite. The frictional behavior of dolomite shows that the constitutive laws that have been developed based on results from granite and quartzite are inadequate to describe all rock types and that more general laws are required.

We present a simple derivation of the pair approximation equations for spin distribution functions in the Ising model. In a uniform system, the pair equations are equivalent to those found by the Bethe or the quasichemical methods. Our derivation requires self‐consistency between approximations for the singlet and pair distribution functions and is easily generalized to study inhomogeneous systems. A simple and efficient numerical method for solution of the equations is presented. We study in particular properties of the interface or free surface in an anisotropic Ising model with coupling between spins in adjacent planes given by J⊥, which may differ from the coupling between spins in the same plane. In the limit J⊥→∞, the model is equivalent to the ’’solid‐on‐solid’’ model used in theories of crystal growth. In this limit the mean field theory for the interface fails completely, while the pair approximation gives results at low temperatures in good agreement with previous calculations. Other possible ap...