The Morphological Evolution of Low Volume Fraction Tin Dendrites During Coarsening
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INTRODUCTION
COARSENING is a widespread phenomenon which occurs in nearly all two-phase systems, affecting metallic, ceramic and polymeric materials in solid, liquid and gaseous states. Driven to reduce the excess free energy associated with interfacial area, the system works to decrease the amount of interface, resulting in an increasing length scale within the system. Coarsening, also known as Ostwald ripening, is generally well understood for the simplest case of twophase systems with spherical particles. To minimize surface energy in such a system, small spheres shrink while large spheres grow, maintaining the second phase at a nearly constant volume while decreasing the surface area to volume ratio. Over time, the average particle radius increases, thus the number density of particles decreases. This process is due to the Gibbs–Thomson effect, the dependence of equilibrium concentration at an interface on the mean curvature, H, of the interface, CL ¼ C1 þ CH
½1
where CL is the composition in the liquid at a curved interface, C¥ is the equilibrium composition of the liquid at a flat interface, and C is the capillary length (determined by the material parameters including solid–liquid interface energy). The mean curvature of the interface is defined as AMBER L. GENAU, Assistant Professor, is with the Department of Materials Science and Engineering, University of Alabama at Birmingham, Birmingham, AL 35205. Contact e-mail: genau@uab. edu PETER W. VOORHEES, Professor, is with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. Manuscript submitted January 31, 2012. Article published online September 6, 2012 406—VOLUME 44A, JANUARY 2013
1 1 1 þ H¼ 2 R1 R2
½2
where R1 and R2 are the maximum and minimum radii of curvature, respectively. This curvature dependence of the concentration results in a concentration gradient, causing mass to diffuse from a small particle to a larger particle. Coarsening of spherical structures can therefore be described by only one variable, the radius of the spheres. A theory which predicts the growth rate of the average particle size for a polydisperse array of particles embedded in a matrix was developed by Lifshitz and Slyozov[1] and Wagner[2] in 1961. LSW theory also predicts the condition of self-similarity, meaning that the microstructure appears the same as it coarsens except for a scaling factor (i.e., the morphology is timeindependent when scaled by a time-dependent characteristic length). Most systems however do not present simple spherical particles. Metal alloys, for example, usually solidify by forming dendrites, complicated tree-like structures with secondary, tertiary and sometimes higher-order branches. Between the tips and the bases of the dendrites is a region of solid plus liquid, known as the mushy zone, where coarsening occurs under nearly isothermal conditions. Dendritic coarsening is an area of particular technological importance because the coarsening of these structures can significantly impact on the properties of
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