Robert T. Dehoff

Mechanism of steady-state grain growth in aluminum

Grain growth, defined as the increase in volume of the average grain, is found, in its steadystate, to be directly proportional to the time of isothermal annealing. During steady-state grain growth the grain corners are found all to be quadruple points, the grain edges all triple lines and the ratio of corners to faces to edges to be 6:7:12. The rate constant for steady-state grain growth is shown to be calculable from first principles and from properties that can be measured independently of the growth observation. It is the product of four individual constants, namely: 1) a dimensionless topological constant ⊝ that is characteristic of steady-state grain growth in any material, 2) the mobility ώ of the average grain boundary in the specific material, 3) the surface tension y of the average grain boundary in the specific material and 4) a dimensionless structural constant σ which expresses the curvature of surface of the grain boundary in the array of grain forms obtaining in the specific specimen of the material and which can be determined metallographically. The topological changes that constitute steady-state growth are shown to exist as a logical sequence of simple events.

The analysis of the evolution of particle size distribution during microstructural change

The concept of the growth path envelope is introduced and combined with that of the rate of change of the number of particles to provide a detailed description of the dynamics of microstructural change in particulate systems. A graphical method is presented, which permits the determination of the nucleation rate and growth path envelope from quantitative measurements of the variation of the particle size distribution during the process. A straightforward mathematical relationship between the size distribution function at any time, nv(R,t), and the nucleation rate, N(Τ) and growth path envelope is derived: nv(R,t) = N(Τ)/G(R,Τ where Τ is the nucleation time for particles that grow to size R at time t, and G(R,Τ) is a properly evaluated growth rate for these particles. It is shown that this equation provides a basis for constructing models of the dynamics of microstructural change in geometrically complex systems.

Application of Volatility Diagrams for Low Temperature, Dry Etching, and Planarization of Copper

A thermodynamic analysis of the dry etching process for copper using halides is undertaken with the development of volatility diagrams. The construction of such diagrams for the Cu-Cl system is illustrated for temperatures ranging from 50 to 200°C. The copper condensed phase species and the partial pressures of the predominant copper vapor phase species are depicted on these diagrams as a function of the partial pressure of chlorine at various temperatures. From these diagrams, it is apparent that reactive ion etching (RIE) of copper based on a procedure previously utilized for aluminum, is inefficient at temperatures less than 200°C. However, closer examination of the metastable equilibrium line between condensed CuCl 2 and gaseous Cu 3 Cl 3 reveals the possibility of extremely high etch rates at low temperatures, if a reducing environment based on hydrogen is utilized for etching CuCl 2 . The design of a multistep, low temperature RIE process for copper, that first requires the formation of condensed CuCl 2 followed by the hydrogen based etching of condensed CuCl 2 land hence copper) due to the formation of Cu 3 Cl 3 gas, is discussed. Some important aspects for the successful implementation of this etching scheme are considered. The further utilization of this process for the purpose of reactive ion planarization of copper as a potential replacement for the problematic chemical mechanical planarization step in the damascene process, is also suggested. The role of capillarity effects in this dry planarization process is discussed briefly.

Estimation of dihedral angles from stereological counting measurements

Abstract The dihedral angle at the edge of a microstructural feature is defined as the angle between the vectors that are normal to the surfaces that meet to form the edge. This angle may vary with position along the edge. The development presented in this paper demonstrates that the average value of the dihedral angle may be determined by an appropriate combination of two simple stereological counting measurements. The result also provides a demonstration that the average dihedral angle measured on a section through the edge is equal to average of the true angle measured in three dimensions.

Effect of La2Zr2O7 on Interfacial Resistance in Solid Oxide Fuel Cells

The impact of La 2 Zr 2 O 7 (LZO) on interfacial resistance (R p ) at the La 0.78 Sr 0.20 MnO 3-δ /yttria-stabilized zirconia interface was studied upon isothermal sintering at 1200°C for 2-25 h. Quantification of triple phase boundary length was performed by applying focused ion beam/scanning electron microscopy (FIB/SEM) serial-sectioning techniques and classical stereology. Electrochemical impedance spectroscopy was used to characterize the R p . The effect of LZO formation on microstructural models for R p was analyzed with respect to previous works that did not include this effect. LZO formation modifies the TPB length, rapidly increases R p , and needs to be controlled in analytical microstructural R p models.

The Trouble with Diffusion

The phenomenological formalism, which yields Ficks Laws for diffusion in single phase multicomponent systems, is widely accepted as the basis for the mathematical description of diffusion. This paper focuses on problems associated with this formalism. This mode of description of the process is cumbersome, defining as it does matrices of interdiffusion coefficients (the central material properties) that require a large experimental investment for their evaluation in three component systems, and, indeed cannot be evaluated for systems with more than three components. It is also argued that the physical meaning of the numerical values of these properties with respect to the atom motions in the system remains unknown. The attempt to understand the physical content of the diffusion coefficients in the phenomenological formalism has been the central fundamental problem in the theory of diffusion in crystalline alloys. The observation by Kirkendall that the crystal lattice moves during diffusion led Darken to develop the concept of intrinsic diffusion, i.e., atom motion relative to the crystal lattice. Darken and his successors sought to relate the diffusion coefficients computed for intrinsic fluxes to those obtained from the motion of radioactive tracers in chemically homogeneous samples which directly report the jump frequencies of the atoms as a function of composition and temperature. This theoretical connection between tracer, intrinsic and interdiffusion behavior would provide the basis for understanding the physical content of interdiffusion coefficients. Definitive tests of the resulting theoretical connection have been carried out for a number of binary systems for which all three kinds of observations are available. In a number of systems predictions of intrinsic coefficients from tracer data do not agree with measured values although predictions of interdiffusion coefficients appear to give reasonable agreement. Thus, the complete connection has not been made, even for binary systems. The theory has never been tested in multicomponent systems. An alternative path to understanding diffusion behavior in multicomponent systems is presented which is based upon a kinetically derived version of the flux equations. While this approach has problems of its own, it has the potential for providing a new range of insights into the process, and for devising simple models for predicting composition evolution in multicomponent systems.

Scale factor coarsening

Abstract Theories of coarsening of precipitate particles predict a simple behavior at sufficiently long times in which successive microstructural states are statistically geometrically similar, and experience only a change in scale. The kinetics of particle coarsening in this asymptotic, scale-factor-growth condition thus can be described in terms of the time dependence of the scale of the system. This scale change is usually described in terms of the change in the average particle radius r. In fact, any geometric property that changes with time may be used to monitor the kinetics when the system follows scale factor growth. In particular, the simple counting measurements of stereology are convenient measures of scale in geometrically similar structures; their estimation requires significantly less effort than does the average particle size. This paper reviews the strategy for adapting stereology counting measurements to the description of scale factor coarsening.

Classical Stereological Measures

This chapter reviews the most commonly used stereological measurements. More detailed discussions and derivations may be found in the collection of traditional texts in the field (Saltykov, 1958; DeHoff and Rhines, 1968; Underwood, 1970; Weibel, 1978; Kurzydlowski and Ralph, 1995; Howard and Reed, 1998). Each section of the chapter focuses on a manual stereological measurement. In each case, the procedure is illustrated with a microstructure and a superimposed grid.

Metric and topological contributions to the rate of change of boundary length in two-dimensional grain growth

Abstract The boundary in a two-dimensional grain structure decreases its length during grain growth. Four distinct processes contribute to this length change: (1) the smooth motion of the grain edges toward their centers of curvature; (2) migration of grain corners resulting from the edge motion; (3) the annihilation of small grains (topological process I); (4) the switching process (topological process II). The contribution of each of these four processes to grain growth is examined. It is demonstrated that almost all of the length change in grain growth is directly due to length changes associated with the topological processes.

Topography and microstructure

Abstract Three levels of description of the geometric state are considered in discussing the “three-dimensional topography” of a microstructure. The first level is simply a list of the features that exist in the structure; the second associates with each member of this list, numbers that describe its extent or configuration; the third is a complete description in terms of the distribution of features in space. Experimentally, most of the metric properties may be estimated from counting measurements made upon a representative microsection; whereas the topological properties require a serial sectioning analysis. The counting measurements are then combined with some elementary statistical analysis in order to describe anisotropies, gradients, and spatial relationships between microstructural features.