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spring 1998 logo1998 MRS Spring Meeting & Exhibit

April 13 - 17, 1998 | San Francisco
Meeting Chairs: John A. Emerson, Ronald Gibala, Caroline A. Ross, Leo J. Schowalter

Symposium BB—Computational and Mathematical Models of Microstructural Evolution


Jeffrey Bullard 
Dept of MS&E
Univ of Illinois-Urbana
Room 201C Metal & Mining Bldg
Urbana, IL 61801

Long-Qing Chen
Dept of MS&E
Pennsylvania State Univ
118 Steidle Building
University Park, PA 16802/5006

Rajiv Kalia 
Dept of Physics & Astronomy & Comp Sci 
Louisiana State Univ 
CCLMS Nicholson Hall Rm#225B 
Baton Rouge, L A 70803-4001 

A. Marshall Stoneham
Dept of Physics & Astronomy
Univ College London
Ctr for Materials Research

Proceedings published as Volume 529 
of the Materials Research Society 
Symposium Proceedings Series.

* Invited paper

Chair: Jeffrey W. Bullard 
Tuesday Morning, April 14, 1998 
Golden Gate A3
8:30 AM BB1.1 LOCALIZED SURFACE INSTABILITIES OF STRESSED SOLIDS. Jerome Colin, Jean Grilhe, Nicole Junqua, Universite de Poitiers, Laboratoire Metallurgie Physique, Poitiers, FRANCE. 
Sinusoidal instabilities are well adapted to describe the beginning of the surface evolution under constant or homogeneous applied stress, but the study of a large class of problems such as linear defect nucleation, stressed plate striction or butterfly transformation of cubic precipitates involves non-homogeneous stress and so requires the use of a localized perturbation. A wavelet shaped deformation has been introduced onto the free surface of a homogeneously stressed solid submitted to an additional non-homogeneous stress. An energy variation calculation has been performed to determine the opening parameter of the wavelet, corresponding to the wavelenght of the sinusoidal instabilities, which is supposed to appear. Localized surface instabilities have been then used to explain the transformation to a butterfly shape of a cubic precipitate submitted to epitaxial stress. Calculation shows that under a critical size the precipitates are stable. Above this size, localized instabilities can appear on each of the precipitate-matrix interface and the butterfly transformation and the partitionning of the precipitates occur. 

8:45 AM BB1.2 
UNIVERSAL PINCH OFF OF RODS BY CAPILLARITY-DRIVEN SURFACE DIFFUSION. Harris Wong, Louisiana State Univ., Mech. Eng. Dept., Baton Rouge, LA; Michael J. Miksis, Northwestern Univ., ESAM Dept., Evanston, IL; P.W. Voorhees, Northwestern Univ., Dept. Mat. Sci. & Eng., Evanston, IL; Stephen H. Davis, Northwestern Univ., ESAM Dept., Evanston, IL. 

A circular rod with surface energy is unstable and eventually will breakup into spherical beads, an equilibrium configuration that minimizes the surface energy. This work studies the film profile near a pinch-off point. Surface diffusion is taken as the dominant mechanism for mass transport. It is found that during pinching the surface profile locally near the pinch-off point can be self-similar. This local solution asymptotes far from the neck to two opposing cones with a unique half-cone angle of 46.04 degree. This unique angle is independent of the material or initial perturbations of the rod. The presence of the unique cone angle during pinching implies that the topological singularity introduces a universality in the morphology of the rod both prior to and after pinch off. The existence of the unique cone angle is supported by a couple published numerical simulations for the pinching motion of axisymmetric cylinders. The results obtained here for a rod also apply to the pinch off of a cylindrical pore channel embedded in a solid matrix. 

9:00 AM BB1.3 

Solid inclusions are added to the melt during the solidification processing of metal matrix composites. The influence of these inclusions on the stability of the planar solid-liquid interface during the unidirectional solidification of a binary mixture is examined. The inclusions are modeled as a doubly periodic array of spherical particles at (2nL,2mL),z0), n,m varying from  to , . The insertion of these particles into the fluid will necessitate modifications to the concentration C from its ambient value C0 due to the requirement that C have zero normal derivative on each particle surface. It will be assumed that the particle volume fraction  is small enough for the O effects to suffice. The calculations show that, due to the presence of particles, the effective segregation coefficient is given by , where  is the volume fraction of particles ( and k is the segregation coefficient in the absence of particles. The decrease in the solutal diffusion coefficient due to the particles is parametrized by  ( is negative) as  where D is the solutal diffusion coefficient of the binary mixture without paticles. The analysis shows that  has no effect on the value of keff to leading order in . A linear stability analysis of the planar interface reveals that the presence of the particles is destabilizing and the threshold parameters for the onset of interfacial deformations altered. For instance, a  volume fraction decreases the critical values of the concentration and wavenumber by  and  respectively. 

9:15 AM BB1.4 
MICROSTRUCTURAL EVOLUTION AND METASTABILITY IN ACTIVE MATERIALS. Richard Jordan, Univ. of Michigan, Dept of Math, David Kinderlehrer, Dept Math Sciences and Center for Nonlinear Analysis, Carnegie Mellon Univ., Pittsburgh, PA; Felix Otto, Dept. Math, Univ of California, Santa Barbara, CA. 

Many shape-memory alloys with highly mobile, complex, and easily transformed microstructure display hysteresis during a loading cycle. This suggests that they are metastable systems in this environment. Issues related to this are the energetic origin of the metastability and the dynamical mechanism of evolution. We explore this with special attention given to a model problem consisting of the evolution of the microstructure in a mosaic of twinned and compound twinned lamellar structures in CuAlNi. We suggest an an approach towards modeling of the system interactions across a wide range of length scales, termed a heirarchy of scales approach. This can explain the presence of local energetic minima owing to partially coherent interfaces, for example. The coarse graining techniques associated with this heirarchy of scales also suggests a dynamical mechanism for microstructural evolution. We also discuss some developments in this direction with Shlomo Ta'asan.These results are consistent with the work of Abeyaratne, Chu, and James (*), who reported the experiments and offered the first theory. In addition, the coarse graining methods suggest a tactic for simulation and we present some results in this direction. 

9:30 AM BB1.5 
THEORETICAL ANALYSIS OF WEDGE-SHAPED TRANSGRANULAR VOID DYNAMICS IN METALLIC THIN FILMS UNDER ELECTROMIGRATION CONDITIONS. Henry Ho and Dimitrios Maroudas, Dept of Chemical Engineering University of California, Santa Barbara, Santa Barbara, CA. 

Electromigration-induced void propagation is among the most serious failure mechanisms in metallic thin films, which are used for device interconnections in integrated circuits. A common transgranular void morphology in metallic films with bamboo grain structure is the wedge-shaped void consisting of two planar facets. Such void surfaces nucleate at the metallic film's edges, undergo faceting under electromigration conditions due to the strongly anisotropic nature of surface adatom mobility, and may evolve unstably due to growth of shape fluctuations of the planar facets. This presentation focuses on the morphological stability of a two-dimensional model of a wedge void surface as a function of its orientation with respect to the direction of the applied electric field. In addition, nonplanar steady surface morphologies that bifurcate from the planar wedge facets are examined systematically based on asymptotic nonlinear bifurcation analysis. Our theoretical analysis follows a continuum formalism of surface mass transport under the action of an electric field including contributions to the surface atomic flux from curvature-driven surface diffusion and electromigration-induced drift. Both our morphological stability analysis and nonlinear bifurcation analysis are based on a local approximation for the electric field component tangent to the void surface for analytical tractability. Special emphasis is placed on the anisotropy of the surface diffusivity, which is approximated by a three-parameter function of the void surface orientation. The analytical results are compared with fully self-consistent dynamical simulations based on the boundary element method for the electric field computation coupled with new methods for free surface propagation that we have developed. Analytical and computational results are in agreement when current crowding effects around the void surface are not strong enough, thus validating the local approximation for the electric field. Our numerical simulations demonstrate a wealth of nonlinear dynamical phenomena associated with morphological instabilities of the flat faceted surfaces. These instabilities are consistent with experimental observations and are discussed in the context of recent experimental measurements. 

10:15 AM *BB1.6 
CRYSTALLINE ANISOTROPY AND THE DYNAMICS OF MICROSTRUCTURAL EVOLUTION. Jean E. Taylor, Mathematics Department, Rutgers University, Piscataway, NJ. 

Weighted mean curvature is the local rate of change of total surface energy with volume swept out under deformations of the surface. It is therefore both a thermodynamic potential to add to the specific bulk energy difference and (by definition) a variational quantity. Its computation, for both sharp and diffuse interfaces, will be reviewed. The correspondence between motion by weighted mean curvature and solutions of the anisotropic Allen-Cahn equation will be demonstrated; this correspondence holds even in the extreme cases where there are facets and/or corners in the surfaces. Examples of computations of motion by weighted mean curvature, for surfaces and for curves and including the cases where there are triple junctions, will be shown. Of particular interest is the difference between the limit of small surface energy and zero surface energy in phase transformations, and this too will be demonstrated and shown to be part of a comprehensive variational approach. 

10:45 AM *BB1.7 
VARIATIONAL METHODS AND GRADIENT FLOWS IN MICROSTRUCTURAL EVOLUTION. W. Craig Carter, Materials Science and Engineering Laboratory, NIST, Gaithersburg, MD; Jean E. Taylor, Mathematics Department, Rutgers University, Piscataway, NJ; John W. Cahn, Materials Science and Engineering Laboratory, NIST, Gaithersburg, MD. 

The application of variational methods and gradient flows to principles of microstructural evolution are reviewed and illustrated with calculated examples. Gradient flows indicate the variation for which a specified free energy functional decreases most rapidly. For various microstructural evolution problems the choice of inner product is of central importance. The L2 inner product gives microstructural evolution equations appropriate for non-conserved field parameters, such as e.g., the equations of motion by weighted mean curvature as in the growth of grains or domains, regardless of whether is for sharp interfaces or the Allen-Cahn equation for diffuse interfaces. The H-1 inner product gives microstructural evolution for conserved quantities, such as the mass in e.g. the diffusion equation, the Cahn-Hilliard equation, or the Mullins equation for surface diffusion. We show how these familiar microstructural evolution equations are derived from an equation for the free energy, choice of the appropriate inner product, and the algorithms of gradient flow. The variational approach is not limited to continuous functions or to free energies: we provide an example of L2 gradient flow by calculating the motion of a triple junction with completely faceted grain boundaries in two-dimensions. Use of gradient flows provides an apparatus for the derivation of microstructural evolution when coupled driving forces and fields are present and provides theoretical tools for singular and non-differentiable cases where traditional methods can fail. 

11:15 AM BB1.8 
THE INFLUENCE OF ANISOTROPIC GRAIN BOUNDARY ENERGY ON TRIPLE JUNCTION MORPHOLOGY AND GRAIN GROWTH. Alexander H. King, Dept. of Materials Science & Engineering, State University of New York, Stony Brook, NY. 

We study the conditions required of equilibrium configurations at triple junctions, under the influence of grain boundary energy that can vary with boundary inclination, through application of the Hoffman-Cahn capillarity vector. It is shown that relatively simple geometric constructions can be used, which are analogous to the force-triangle method used under isotropic conditions. Particular attention is paid to cases in which one, two or three of the boundaries are trapped in orientations that are associated with energy cusps, and it is shown that these are often able to accommodate wide variations of grain boundary energy and/or inclination. Some cases generate bi-stable or tri-stable configurations in which one or two of the boundaries can flip between different stable inclinations. The case of a twin boundary meeting an isotropic grain boundary is analyzed in detail, and shown to produce some unique solutions for the inclination of the grain boundary relative to the twins.Cases of triple junction pinning will be described, and the principles of their influence upon the equilibrium configurations will be explained semi-quantitatively. 

11:30 AM BB1.9 
ENVIRONMENTAL NOISE EFFECTS IN STATISTICAL COARSENING THEORY. S.P. Marsh, C.S. Pande, Naval Research Laboratory, Washington, DC; M.E. Glicksman, Rensselaer Polytechnic Institute, Troy, NY; D.I. Zwillinger, Aztec Corporation, Waltham, MA. 

Statistical mean-field theories of phase coarsening (Ostwald ripening) employ an effective diffusion distance that is a function of both the domain size and of the global volume fraction to formulate the characteristic growth rate of each size class. The corresponding single-valued growth rate function yields self-similar size distributions that are narrower than those generally observed experimentally and in numerical simulations. A stochastic growth function has been derived that permits an extension of the mean-field formalism to account for variations in the size-dependent growth rates ariosing from local interactions. The resulting continuity analysis yields a Fokker-Planck equation that tends to broaden the corresponding size distribution. Effects of the noise term on coarsening rate constants and on the shape of the size distribution at various volume fractions will be discussed. 

Chair: Jeffrey W. Bullard 
Tuesday Afternoon, April 14, 1998 
Golden Gate A3
1:30 PM *BB2.1 
PHASE FIELD COMPUTATIONS OF DENDRITIC SOLIDIFICATION. Robert F. Almgren, Univ of Chicago, Dept of Mathematics, Chicago, IL. 

I will discuss mathematical and practical aspects of using phase field models to simulate the development of material microstructure by solidification from the melt. Recent advances in the construction of the models and in the analysis of their sharp-interface asymptotic limit hold out the promise of soon being able to carry out realistic and accurate computations. In addition, the models have several features of independent mathematical interest. 

2:00 PM BB2.2 

During the last decade the non-linear Cahn-Hilliard equation (CHE) has become a powerful tool for description of phase transformations. While the linearized version of the CHE is valid only for the early stages of spinodal decomposition a numerical solution allows the modelling of the whole process of decomposition. An additional term that considers the local elastic strain energy, which is caused by the concentration dependence of the lattice parameter, has been introduced by J.W. Cahn and extensively applied by A. Khatchaturyan, L.-Q. Chen and Y. Wang. Together with the Allen-Cahn equation arises a set of non-linear differential equations, which can be used for simulation of a wide range of phase transitions. Hithero this simulation technique has been applied for more or less qualitative simulations with simulation parameters of model systems. Owing to the availability of the thermodynamics of alloys by means of the CALPHAD method and the numerical algorithms for solving non-linear differential equations, nowadays the CHE can be used for simulation of the kinetics of phase transformations of several ënear commercialí alloys. In this work, we use the formulation of the free energy of GP-zones in AlCu-alloys, given by I. Hurtado. We simulate the kinetics of formation of GP-zones for several temperatures using different spatial discretizations and interfacial energies. The results yield the concentration profiles and the temporal evolution of the length and the aspect ratios of the GP-zones. They will be discussed with respect to the influence of numerical errors. The limitations of the CH concept for the prediction of phase transformation kinetics will be explained and discussed. 

2:15 PM BB2.3 

The phase-field model, a continuum thermodynamic model, will be used to simulate the pearlite to austenite transformation in steels. The initial microstructure will be a lamellar, eutectic structure of ferrite and cementite. This structure will be heated to a temperature in the autenite region. At this temperature carbon diffuses from the cemetite to the ferrite phase with an accompanying phase transformation from pearlite to austenite. The incorporation of three phases, diffusion and coarsening due to capillarity into the phase-field model will be shouwn in this paper. 

2:30 PM BB2.4 
STRESS-INDUCED PRECIPITATE MORPHOLOGICAL EVOLUTION AND RAFTING IN Ni-BASED SUPERALLOYS. D.Y. Li, L.Q. Chen, Penn State University, Department of Materials Science and Engineering, University Park, PA. 

The morphological evolution and rafting kinetics of precipitate particles in Ni-Al superalloys under applied stresses were investigated. A computer simulation model based on the diffuse-interface field model was employed. The elastic inhomogeneity was taken into account using the effective medium approximation recently proposed by Khachaturyan. It is shown that an applied stress results in the shape change of a precipitate particle from four-fold to two-fold symmetry. At high applied stresses, a particle may split to two particles through diffusional redistribution of solute atoms, similar to coherency-strain-induced particle splitting. Similar splitting phenomenon was also observed during rafting of a multi-particle system at relatively low volume fractions. The relationships between the rafting orientation, and the magnitude and direction of the applied stress, the elastic inhomogeneity, and lattice misfit strain will be discussed. 

Chair: Jeffrey W. Bullard 
Tuesday Afternoon, April 14, 1998 
Golden Gate A3
3:15 PM *BB3.1 

Cement-based materials are usually composites, where the matrix consists of cement paste. Cement paste is a material formed from the hydration reaction of cement, usually a calcium silicate material, with water. The microstructure of cement paste changes drastically over a time period of about one week, with slower changes occurring over weeks to months. The effect of this hydration process on the changing microstructure can be clearly seen using computer simulation techniques applied to large three dimensional models. Percolation theory can be applied to understand the evolving microstructure in terms of the four percolation thresholds that are of importance in the cement paste microstructure. Several of these percolation thresholds have been clearly confirmed by experiments. 

3:45 PM BB3.2 
CELLULAR AUTOMATA FOR MICROSTRUCTURAL EVOLUTION OF NEAR-CRITICAL FLUIDS WITHIN NANOTUBES. Illam Park, Dept of Chemistry and Biochemistry, University of California, Los Angeles, CA.. 

A lattice-gas cellular automaton model is used to simulate microstructural evolution of near-critical fluids within cylindrical nanotubes. The goal of present paper is to elucidate mechanisms underlying the filling of nanotubes with substances in the vapor phase, understanding of which has been of a great concern in synthesis of metallic nanofibers within carbon nanotubes, characterization of mesoporous carbon molecular sieves by high-resolution adsorption, and microbial-growth in porous membranes. In the present model, the temporal rate of phase transition in a cellular automaton system is determined by temperature, vapor concentration (or chemical potential) and free energy of the substrate. Phase transition is permitted only on the interfacial areas exposed to the bulk phase while thermal interaction upon releasing and absorbing the latent heat of phase transition is simplified by a mesoscopic Newton's law of cooling. Simulations are performed using the 3-d lattice gases and quasi-crystalline nanotubes with the pore widths of 2 to 50 A and the surface free energies of 5 to 20 KJ/mol. The simulation results show that the microstructural evolution of the vapor in a nanotube involves subsequent changes in interfacial area and strength, and accessible pore volume. The process of the growth ceases as parts of the independently growing domains merge, leaving the bulk gas inaccessible to the effective interfacial area. It is important to note that the properties of the growth depend not only on the thermodynamic properties of the bulk fluid and the surface but also on non-thermodynamic parameters such as vapor dosage and the length of a nanotube. We will discuss how to externally control the microstructural evolution within nanotubes based on the simulation results. The limit of template-mediated growth due to the metastable nature of the filling processes and the crystallinity of nanotubes, is also discussed in conjuction with previous experimental observations available from the literature. 

4:00 PM BB3.3 
CELLULAR AUTOMATION MODELING OF ALLOY SOLIDIFICATION USING LOCAL KINETIC ANISOTROPY RULES. Ralph E. Napolitano Jr., National Institute of Standards and Technology, Materials Science and Engineering Laboratory, Metallurgy Division, Gaithersburg, MD; Thomas H. Sanders Jr., Georgia Institute of Technology, School of Materials Science and Engineering, Atlanta, GA. 

Morphological evolution of a dendrite growth fro nt in a binary alloy is simulated using a cellular automaton approach to establish the feasibility of modeling such growth with a local rule-based scheme. The motivation for this work is derived from the need to predict the development of solidification structures within real components of complex geometry, where significant constraint of the thermal and solutal fields may exist. Such cases present complex boundaries and large domain sizes, which may preclude the effective use of more conventional methods. In this work, a model is presented which couples a two-dimensional alternate-direction implicit finite-difference diffusion solution with a cellular automation algorithm to simulate morphological evolution in alloys solidifying under directional growth conditions. As the thermal and solutal fields evolve, they are incorporated into the automation, which governs the growth of the solid according to alloy thermodynamics, interface curvature, and kinetic anisotropy. Alloy solidification is simulated over a range of experimental conditions, producing various structures. Issues addressed include morphological instability, primary spacing selection, interface curvature, kinetic anisotropy, microsegregation, and tip conditions. 

4:15 PM BB3.4 

A 3D model has been developed to simulate both primary static recystallization and recovery of cold worked aluminium alloys. The model is based on a modified cellular automaton approach and incorporates the influence of crystallographic texture and microstructure in respect to both mechanisms mentioned above. The model takes into account oriented nucleation using an approach developed by Nes for aluminium alloys. The subsequent growth of the nuclei depends on the local stored energy of the deformed matrix (i.e. the driving pressure) and the misorientation between a growing nucleus and its surrounding matrix (i.e. the grain boundary mobility). This approach allows to model preferred growth of grains that exhibit a maximum growth rate orientation relation ship, e.g. for Al alloys a  relationship with the surrounding matrix. The model simulates kinetics, microstructure and texture development during heat treatment discrete in time and space. 

4:30 PM BB3.5 
GEOMETRIC ANALYSIS OF PARTICLE PACKINGS IN THE STUDY OF SUPERHEATED AND SUPERCOOLED LIQUIDS. Srikanth Sastry, Pablo G. Debenedetti, Princeton University, Dept of Chemical Engineering, Princeton, NJ; Frank H. Stillinger, Bell Laboratories, Lucent Technologies, Murray Hill, NJ and Princeton University, Princeton Materials Institute, Princeton, NJ. 

We have developed a new computational method for exact quantification of the void space in packings of uniform and polydisperse spherical particles, which permits determination of void space connectivity and quantitative measures such as volumes and surface areas of connected regions of the void, or cavities. Application of this method to study mechanically stable packings of particles obtained from simulations of model atomic liquids leads to insights relevant to processes defining limits to the metastable liquid state, namely nucleation and glass formation. Density fluctuations in superheated liquids are shown to be strongly correlated with the highly heterogeneous morphology observed for the mechanically stable packings. The density dependence of such heterogeneous morphology leads to the prediction of a lower density limit to the formation of a glass upon supercooling the liquid, which is borne out be simulation results. Further, the temperature dependent properties of mechanically stable particle packings leads to the identification and interpretation of distinct regimes of complex dynamics in glass forming liquids. 

4:45 PM BB3.6 
ORGANIZATION OF IMPURE MEDIUM THROUGH CYCLING. M. Ausloos, M. Kramer, N. Vandewalle, R. Cloots, Univ. of Liege, SUPRAS, Liege, BELGIUM. 

The dynamic epidemic Eden model has been used in order to describe the evolution of a medium containing mobile but chemically inert impurities.