Kinetics of thermal grain boundary grooving for changing dihedral angles
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P. Sachenko Department of Mechanical Engineering, Oakland University, Rochester, Michigan 48309
J.H. Schneibel Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6115 (Received 31 October 2001; accepted 22 March 2002)
In his classic paper on thermal grain boundary grooving Mullins [W.W. Mullins, J. Appl. Physics 28, 333 (1957)] assumes that the dihedral angle at the groove root remains constant and predicts that the groove width and depth grow ⬀t0.25. Here, we derive models describing groove growth while the dihedral angle changes. In our grooving experiments with tungsten at 1350 °C in which the dihedral angle decreased, the growth exponent for the groove depth reached values as high as 0.44 while the growth exponent for the width decreased slightly from Mullins’ value of 0.25. Hence groove width data alone are not sufficient for verifying Mullins’ growth law unless the dihedral angle is constant. The observed changes in the dihedral angle are used as an input for numerical simulations. With the simulations we are able to extract the surface diffusion constants. Atomic force microscope observations of groove widths and depths in tungsten are in excellent agreement with the simulations.
I. INTRODUCTION
In 1957 Mullins1 developed his classical model for thermal grain boundary grooving by surface diffusion. One of his assumptions was that the dihedral angle included by the surfaces adjacent to a grain boundary remains constant during growth. This assumption ensures self-similarity of the growing grooves. Because of this assumption, groove width and groove depth follow the same growth exponent. Therefore, and because of experimental ease, only the groove width has been determined in most experimental work on grain boundary grooving. The few papers (Refs. 2–4) providing information on the growth of the depth of grooves are listed in Table I. Table I indicates that the growth exponents derived from depth measurements show typically more scatter than those derived from width measurements, While the relatively poor precision of depth measurements is one factor responsible for this, there is also the possibility that the dihedral angle changes during annealing, thus invalidating one key assumption in Mullins’ model. The dihedral angle at a grain boundary groove is determined by the values of the grain boundary and surface free energies (an energy per unit interfacial area). If the surface free energy varies in time due to changes in the surface composition (segregation, adsorption) the dihedral angle will change. This considerably complicates the growth J. Mater. Res., Vol. 17, No. 6, Jun 2002
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kinetics of the grooves. In the present work we describe cases where the dihedral angle decreases during grain boundary grooving. Mathematical models for grain boundary grooving with changing dihedral angles are developed based on the surface diffusion model established by Herring.5 They are modifications of Mullins’s model.1 Simulations
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