Phase-field modeling of diffusion controlled phase ... - KTH Mechanics

574 LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATIONfusion are taken into account. The paper is organizedas follows: Section 1 gives an overview of the modelderivation; numerical aspects are presented in Section2; in the next two sections results of truly non-isothermalcalculations of dendritic growth for two temperatureregimes are presented and discussed.J B and the heat flux J e are given by the linear lawsof irreversible thermodynamicsJ B L A BB δS Lδx A Be δSB δe(6)2. MATHEMATICAL MODELIn this section the phase-field model given in Ref.[1] is considered with modifications accounting fortemporal and spatial evolution of the temperaturefield. The formulation is based on an entropy functionalS s(f,x B ,e) 22 |f|2d (1)where the thermodynamic entropy density s is a functionof the phase-field variable f varying smoothlybetween 0 in the solid and 1 in the liquid, the molefraction x B of a solute B in solvent A and the internalenergy density e, and is a spatial region occupiedby a mixture.Anisotropy is included in the system because thephase change kinetics depends upon the orientationof the interface¯h ¯(1 g cos kb) (2)where ¯ is related to the surface energy s and interfacethickness, g is the magnitude of anisotropy in thesurface energy, k specifies the mode number and theexpression b arctan(f y /f x ) gives an approximationof the angle between the interface normal and theorientation of the crystal lattice.The evolution of the non-conserved phase-fieldvariable is governed byḟ M fδSδf(3)where M f is related to the interfacial mobility. Theevolution of x B and e is governed by the normal conservationlawsJ e L A eB δS Lδx A ee δSB δe(7)L A BB and L A ee are related to the inter-diffusionalmobility of B and A and the heat conduction, respectively.The coefficient L A Be L A eB describes the crosseffects between heat flow and diffusion and will beneglected. The second law requires M f , L A BB and L A eeto be positive. It should be emphasized that the gradients(δS/δx B ) and (δS/δe) are to be evaluated isothermallyand under fixed composition, respectively.The variational derivatives in equations (6) and (7)are given byδSδx B ∂s∂x B m Bm ATV m(8)δSδe ∂s∂e 1 T(9)where T is the temperature and the quantity on theright-hand side of equation (8) is known as the interdiffusionpotential in binary substitutional alloys. Thechemical potentials m A and m B under the assumptionof an ideal mixture have the following formm A °m A (f, T) RT ln(1x B ) (10)m B °m B (f, T) RT ln x B (11)The expressions of the molar Gibbs energy for purematerials are presented as in Ref. [7]°m AV m W A g(f)T [e s A(T A m)c A T A m (12) p(f)H A ]1 T T A mc A T ln T T A mẋ BV m·J B (4)°m BV m W B g(f)T [e s B(T B m)c B T B m (13) p(f)H B ]1 T T B mc B T ln T T B mė ·J e (5)where V m is the molar volume. The diffusional fluxwhere g(f) f 2 (1f) 2 and W A , W B are constants.e s A(T A m) and e s B(T B m) are the energy densities of pure