A phase field model of the premelting of grain boundaries in pure materials
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A phase eld model of the premelting of grain boundaries in pure materials Alexander E. Lobkovsky and James A. Warren
Metallurgy Division National Institute for Standards and Technology 100 Bureau Drive, Stop 8555 Gaithersburg, MD 20899 ABSTRACT
We present a phase eld model of solidi cation which includes the e ects of the crystalline orientation in the solid phase. This model describes grain boundaries as well as solid-liquid boundaries within a uni ed framework. With an appropriate choice of coupling of the phase eld variable to the gradient of the crystalline orientation variable in the free energy, we nd that high angle boundaries undergo a premelting transition. As the melting temperature is approached from below, low angle grain boundaries remain narrow. The width of the liquid layer at high angle grain boundaries diverges logarithmically. INTRODUCTION
The energy of a boundary between two crystalline grains depends on the relative orientations of the grains and the inclination of the boundary plane. It is often the case that this energy is larger than twice the energy of the liquid-solid boundary. In this case one expects the grain boundary to become metastable (or unstable) with respect to the formation of a layer of liquid. The nature of this transition is a matter of continuing interest. Hsieh and BalluÆ [1] observed melting of grain boundaries in 99.9995% pure aluminum within a degree of the melting temperature Tm . The measured width of the liquid layer was consistent with a divergence at Tm. Indirect probes of the structure of the grain boundary, such as the measurement of its mobility [2], shear resistance [3] and di usion coeÆcients [4] all found discontinuous jumps in these quantities at some characteristic temperature below the melting point. A number of molecular dynamics simulations [5, 6, 7, 8] of grain boundaries also reported a 1 Y1.9.1
divergence of the width of the disordered layer for high angle grain boundaries as the temperature approached Tm . A thermodynamic approach [9, 10] to interfacial melting reveals that premelting may be complete (the width of the liquid layer diverges as T ! Tm ) or incomplete (it stays nite) depending on the functional dependence of the free energy on the thickness of the liquid layer. In addition, there may be a discontinuous jump in the width of the liquid layer as a function of temperature. An explicit calculation of the free energy of a liquid layer is diÆcult and has been performed only in a few cases. Kikuchi and Cahn calculated it within a lattice gas model [11]. They concluded that when the interaction of the solid-liquid interfaces is short range, the transition, if it exists, is continuous and the width of the liquid lm diverges as a logarithm of the undercooling. Elbaum and Schick calculated the energy of a water lm on ice due to the Van Der Waals dispersion forces [12] and concluded that premelting there is incomplete. Here we analyze the behavior of grain boundaries near the melting temperature using a simpli cation of a phase eld model introduce
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