doi:10.1016/S0167-2789(02)00377-9 
Copyright © 2002 Elsevier Science B.V. All rights reserved.
Phase field model of premelting of grain boundaries
a Physics Department, Northeastern University, Boston, MA 02115, USA
b National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
Received 5 November 2001;
revised 16 January 2002;
accepted 16 January 2002
Communicated by E. Bodenschatz
Available online 17 March 2002.
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Abstract
We present a phase field model of solidification which includes the effects of the crystalline orientation in the solid phase. This model describes grain boundaries as well as solid–liquid boundaries within a unified framework. With an appropriate choice of coupling of the phase field variable to the gradient of the crystalline orientation variable in the free energy, we find 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. In addition, for some choices of model coupling, there may be a discontinuous jump in the width of the fluid layer as function of temperature.
Author Keywords: Grain boundary; Phase field; Premelting
PACS classification codes: 61.72.Bb; 64.70.Nd; 81.10.Fq
Fig. 1. Order parameters in a flat stationary grain boundary as a function of the coordinate ζ normal to the boundary. The orientation θ varies smoothly between 0 and Δθ in the grain boundary ζ
[−ℓ,ℓ] and is constant in the interior of the grains. The phase field φ approaches 1 exponentially away from the boundary.
Fig. 2. Amount of liquid W in units of as a function of temperature L for a low-angle grain boundary. h=φ2, and .
Fig. 3. Energy γgb in units of of the low-angle grain boundary (same values of the parameters as in Fig. 2) and the saddle point solution (upper branch).
Fig. 4. The spinodal temperature Ls, above which the grain boundary cannot exist, as a function of the misorientation. Note that the spinodal temperature vanishes at a critical misorientation.
Fig. 5. Critical misorientation in units of as function of . h=φ2. Dashed line is the analytical result in the limit of large .
Fig. 6. Amount of liquid W for h=1 in units of as function of temperature L for and three different misorientations Δθu<Δθl, Δθl<Δθ2<Δθu and Δθ3<Δθl.
Fig. 7. Upper and lower critical misorientations as a function of the h=1 case.
Fig. 8. Grain boundary mobility as a function of misorientation both in units of for h=φ2 and three choices of P. The slopes of the two straight line segments are +1 and −1. The temperature is L=−0.001, . Note that above the critical angle, changes in P do not influence the grain boundary mobility.
Fig. 9. Mobilities of the wet and dry boundary solutions. Here , and h=φ2.
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