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

I. Loginova et al. / Acta Materialia 51 (2003) 1327–13391329substitutional atom and will be approximated asconstant, is the phase-field variable taken 0 ina and 1 in g. The u-fraction u C is defined from thenormal mole fraction of C, x C asu C x C(2)1x CAs already mentioned, the temperature T isassumed constant all over the whole system. However,it may of course vary in time but heat conductionis assumed so rapid that all temperaturegradients can be neglected. The molar Gibbsenergy G m is postulated as a function of the phasefieldvariable:G m (1p(f))G a m p(f)G g m g(f)W (3)whereg(f) f 2 (1f) 2 (4)p(f) f 3 (1015f 6f 2 ) (5)and the choice of the parameter W will be discussedlater. G a m and G g m denote the normal Gibbsenergy functions of the a and g phases and aretaken from the assessment of Gustafson [8]. Thecomplete expressions are given in Appendix A. Itshould be mentioned that g(f) and p(f) have beenchosen so that dp/df 30g(f).The evolution of the phase-field variable isgoverned by the Cahn–Allen equation [9](6)ḟ M fdGdf M f 1 V m∂G m∂f e2 2 fThe kinetic parameter M f is related to the interfacialmobility as will be shown later.The evolutionof the concentration field is governed by the normaldiffusion equation. When u-fractions are usedand the molar volume is approximated as constantthe normal diffusion equation can be rewritten asu˙ C·JVC (7)mThe diffusional flux of carbon J C is given by theOnsager linear law of irreversible thermodynamics:J C L d Gd u c (8)The quantity dG/du C is the normal chemicalpotential of C denoted by m C . If the so-called gradientterms are neglected we have∂ G m /∂ u C m Cand Eq. (8) may be expanded in terms of the concentrationand phase-field gradientsJ C 1 DV C u C L ∂2 G mf (9)m ∂u∂fThe first term corresponds to the normal Fick’s lawand we may thus identify the normal diffusioncoefficient of C asD C V m L ∂2 G m∂u 2 C(10)The second-order derivative corresponds to Darken’sthermodynamic factor and the parameter L is relatedto the diffusional mobility [10] M C by means ofL u CV my Va M C (11)where y Va denotes the fraction of vacant interstitials,i.e. 1-u C for g and y Va 1u C /3 for a. Fora given C content the fraction of vacancies wouldthus depend on the character of the phase, i.e. itwill depend on the phase-field variable. We havepostulatedu C y Va (1p(f))u C (1u C /3) p(f)u C (1u C )(12)The diffusional mobility in the two phases coulddiffer by several orders of magnitude, therefore wehave chosen the following combination:M C (M a C) 1p(f) (M g C) p(f) (13)For substitutional solutes the mobility in thecenter of the interface is most probably muchhigher than in the any of the crystalline phases. Forinterstitial solutes like carbon it may not be muchhigher and the approximation represented by Eq.(13) may not be too crude. The mobilities of carbonin a and g were taken from Ågren [11,12].The complete expressions are given in AppendixB.