Peter W. Voorhees

The theory of Ostwald ripening

Developments in the theory of Ostwald ripening since the classic work of I. M. Lifshitz and V. V. Slyozov (LS) are reviewed and directions for future work are suggested. Recent theoretical work on the role of a finite volume fraction of coarsening phase on the ripening behavior of two-phase systems is reformulated in terms of a consistent set of notation through which each of the theories can be compared and contrasted. Although more theoretical work is necessary, these theories are in general agreement on the effects of a finite volume fraction of coarsening phase on the coarsening behavior of two-phase systems. New work on transient Ostwald ripening is presented which illustrates the broad range of behavior which is possible in this regime. The conditions responsible for the presence of the asymptotic state first discovered by LS, as well as the manner in which this state is approached, are also discussed. The role of elastic fields during Ostwald ripening in solid-solid mixtures is reviewed, and it is shown that these fields can play a dominant role in determining the coarsening behavior of a solid-solid system.

Quantum dot formation on a strain-patterned epitaxial thin film

We model the effect of substrate strain patterning on the self-assembly of quantum dots (QDs). When the surface energy is isotropic, we demonstrate that strain patterning via embedded substrate inclusions may result in ordered, self-organized QD arrays. However, for systems with strong cubic surface energy anisotropy, the same patterning does not readily lead to an ordered array of pyramids at long times. We conclude that the form of the surface energy anisotropy strongly influences the manner in which QDs self-assemble into regular arrays.

Direct measurement of dopant distribution in an individual vapour–liquid–solid nanowire

Semiconductor nanowires show promise for many device applications, but controlled doping with electronic and magnetic impurities remains an important challenge. Limitations on dopant incorporation have been identified in nanocrystals, raising concerns about the prospects for doping nanostructures. Progress has been hindered by the lack of a method to quantify the dopant distribution in single nanostructures. Recently, we showed that atom probe tomography can be used to determine the composition of isolated nanowires. Here, we report the first direct measurements of dopant concentrations in arbitrary regions of individual nanowires. We find that differences in precursor decomposition rates between the liquid catalyst and solid nanowire surface give rise to a heavily doped shell surrounding an underdoped core. We also present a thermodynamic model that relates liquid and solid compositions to dopant fluxes.

Morphological instability in epitaxially strained dislocation‐free solid films: Linear stability theory

The morphological instability of a growing epitaxially strained dislocation‐free solid film is analyzed. An evolution equation for the film surface is derived in the dilute limit of vacancies based on surface diffusion driven by a stress‐dependent chemical potential. From the time‐dependent linear stability problem the conditions for which a growing film is unstable are determined. It is found that the instability is driven by the lattice mismatch between the film and the substrate; however, low temperatures as well as elastically stiff substrates are stabilizing influences. The results also reveal that the critical film thickness for instability depends on the growth rate of the film itself. Detailed comparison with experimental observations indicates that the instability described exhibits many of the observed features of the onset of the ‘‘island instability.’’

Quantitative serial sectioning analysis.

A method for serial sectioning is presented that allows one to take about 20 sections per hour with spacings in the range 1–20 µm between sections. The alignment of the cross‐sections is done with a linear variable differential transformer; it is thus independent of the microstructure of the sample and does not rely upon markers implanted in the sample. The alignment errors as well as tilts and rotation errors between sections associated with the new method are found to be negligible. Once all the sections are captured in a computer a three‐dimensional image can be constructed. This image can be viewed interactively and rotated, thus allowing the direct observation of three‐dimensional shapes. It can further be used to determine a vast array of microstructural parameters including those that cannot be determined from planar sections. The technique is illustrated through the reconstruction of the microstructure of a cast standard aluminium alloy specimen.

THE EQUILIBRIUM SHAPE OF A MISFITTING PRECIPITATE

Abstract We examine the equilibrium morphologies of precipitates with either a tetragonal or purely dilatational misfit in an elastically anisotropic medium with cubic symmetry under conditions of plane strain. We find that particles with a dilatational misfit are nearly spherical at small sizes, take on four-fold symmetric shapes at intermediate sizes and then undergo a supercritical symmetry-breaking bifurcation to two-fold symmetric shapes aligned along the elastically soft directions of the crystal. A tetragonal misfit breaks this four-fold to two-fold supercritical bifurcation when the direction of the tetragonality is coincident with one of the elastically soft directions of the crystal. Such a tetragonal misfit can lead to two-fold equilibrium particle shapes which are local energy minima, or metastable, and in some cases have large negative interfacial curvatures. When the tetragonality is not in an elastically soft direction, the supercritical bifurcation is not broken and the particles can take on unusual diamond-like or S-shaped morphologies.

Ostwald ripening during liquid phase sintering—Effect of volume fraction on coarsening kinetics

Phase coarsening, also termed Ostwald ripening, is generally thought to be a slow, diffusion-controlled process which occurs subsequent to phase separation under extremely small under- or over-saturation levels. The theory due to Lifshitz, Slyozov, and Wagner (LSW), which predicts the coarsening kinetics and the particle distribution function, are applicable todilute systems only, in which particle-particle interactions are unimportant. Most liquid phase sintered systems, however, have large enough volume fractions of the dispersed phase to violate the essential assumptions of LSW theory. Recent progress will be described on simulating Ostwald ripening in randomly dispersed, high volume fraction systems. A fast algorithm for solving the multiparticle diffusion problem (MDP) will be described, permitting simulation of coarsening dynamics by cyclic time-stepping and updating the diffusion solution for large random particle arrays. The rate constants, controlling the growth of the average particle, and the particle distribution functions were obtained by numerical simulations up to a volume fraction of 0.55. A new statistical mean field theory has now been developed which reproduces the MDP simulation data accurately, and finally makes clear how the linear mean-field approximations employed by LSW theory must be modified to describe real systems. The predictions of the mean field are found to compare favorably with experimental measurements made over a wide range of volume fraction solid of the kinetics of Ostwald ripening in liquid phase sintered Fe-Cu alloys. The new theory provides a comprehensive approach to understanding microstructural coarsening in liquid phase sintered systems.

Ostwald ripening in ternary alloys

A theory of coarsening in an isothermal, ternary alloy is developed in an effort to understand the effects of a third chemical component on the ripening behavior of a two-phase system. The analysis is valid for a general, nonideal, nondilute solution, but is limited to extremely small volume fractions of the coarsening phase and neglects off-diagonal terms in the diffusion matrix. The Gibbs-Thompson equation in a ternary system undergoing coarsening reveals that the concentrations at the particle/matrix interface are dependent on the far-field supersaturations as well as on the particle radius. In addition, the capillary length depends on the diffusivities of the two components. An asymptotic analysis shows that the exponents of the temporal power laws for the average particle radius, number of particles per unit volume, and the matrix supersaturations are the same as that found in the binary limit; however, the amplitudes of the power laws are modified. We find that the trajectory of the matrix supersaturation must lie along a tie-line, but the trajectory of the particle composition does not. An expression for the effect of dilute ternary additions to the coarsening rate of a binary alloy is also given.

Ostwald ripening in a system with a high volume fraction of coarsening phase

Experiments on the coarsening behavior of two-phase mixtures in a model Pb-Sn system are reported. This system fulfills most of the assumptions of theory and has the particular advantage that all the materials parameters necessary for a comparison between the experimentally measured and theoretically predicted coarsening kinetics are known. We have examined the coarsening of Sn-rich and Pb-rich solid phases in contact with eutectic liquid in the volume fraction solid range above approximately 0.6 where the development of a solid skeletal structure inhibits sedimentation. Particle intercept distributions are measured and found to be time independent when scaled by the average intercept. This invariance is interpreted as evidence that scale factor coarsening is present. The intercept distributions are in good agreement with the predictions of theory. Measurements of average intercept diameter as a function of time establish unambiguously that the coarsening follows the theoretically predictedt1/3 kinetics. The coarsening rate constants are measured as a function of volume fraction solid and are found to exceed the values calculated from theory using the known thermophysical properties of the Pb-Sn system by factors ranging from approximately 2 to 5.

Ostwald ripening in concentrated alloys

The temporal power laws describing the coarsening process in a concentrated nonideal solid solution are derived. It is shown that the interfacial energy [sigma] can be determined in such a system from measurement of the coarsening kinetics, if an isothermal stress-free two-phase mixture is assumed. In the derivation, a modified Gibbs-Thomson equation is used and the effects of nonideal solution thermodynamics and nonzero solubilities of solute in each phase on the flux conservation condition at the interface are taken into account. The resulting rate constant is then used to analyze the coarsening process of Ni[sub 2]Al precipitates in a homogeneous Ni-Al matrix. A model for the solution thermodynamics of the matrix phase is used to compute the various thermodynamic factors and phase compositions necessary to evaluate the rate constant. The resulting value of [sigma] is approximately an order of magnitude smaller than derived on the basis of the assumptions used in the classical theory by Lifshitz and Slyozov, and Wagner. However, it is possible to extract only an approximate value of [sigma] from the experimental data due to an unfortunately large uncertainty in the value of the interdiffusion coefficient present during the coarsening experiments.