Exploring The Consequences of Negative Triple Junction Energy
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EXPLORING THE CONSEQUENCES OF NEGATIVE TRIPLE JUNCTION ENERGY Gaurav K. Gupta and Alexander H. King School of Materials Engineering, Purdue University, West Lafayette, IN 47907-1289
ABSTRACT We have investigated the consequences of negative triple junction energy in grain growth. Two and three-dimensional models for total system energy, incorporating varying triple junction energy are developed. These models show that there is a decrease in overall system energy with grain size corresponding to the driving force for grain growth. Although the free energy available to drive grain growth is reduced under some conditions, it is never removed for any reasonable values of the triple junction energy. INTRODUCTION Ultrafine-grained materials are characterized by a very high density of grain boundaries. Palumbo and Aust [1] have pointed out that in the nanometer grain size regime, the density of triple junctions also becomes a potentially important consideration, with the volume of triple junction material exceeding the volume of grain boundary material at a grain size of about 3 nanometers. If triple junctions have properties that differ from those of the grain boundaries [2] then they may exert a powerful influence upon materials behavior in nanostructured materials. Srinivasan, Cahn and Kalonji [3] have suggested that triple junctions may have “negative” energy and this would certainly be expected to result in unusual behaviors wherever there is a large concentration of triple junctions in a material. In fact, “negative” triple junction energy corresponds to an energy that is lower than that of the adjacent grain boundaries, but is still greater than the energy of the corresponding crystalline material. Even with this understanding, its existence remains somewhat controversial [4]. The purpose of this paper is to investigate the possible effects of reduced triple junction energy on the overall driving force for grain growth, to see if it can contribute to the stabilization of ultrafine microstructures. At its most basic level, the driving force for grain growth is the reduction of interfacial energy that accompanies increasing grain size, and we compute this overall interfacial energy taking account of grain boundaries, triple junctions and quadruple points as potentially different components of the microstructure. The computations are performed for model microstructures corresponding to columnar-grained thin films, and equiaxed bulk polycrystals. 2-D MODEL We consider a microstructure comprising uniformly sized regular, hexagonal grains with edge length ‘a’ and a grain boundary thickness ‘δ ’. The grain boundaries (GBs) are rectangular slabs and the triple junctions (TJ’s) are equilateral triangular prisms as shown in Fig.1. There are no quadruple points. The energy of the system depends on the amounts of the various features present in it, which vary with the grain size. Fig.2 shows a plot of the cumulative volume fractions of the various features with varying equivalent cylindrical grain diameter. Assigning reasonabl
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