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

1330 I. Loginova et al. / Acta Materialia 51 (2003) 1327–13393. Solute drag modeling of the g→atransformation in Fe–CIn the solute drag modeling the thickness (δ) ofthe interface is considered small enough comparedto the curvature of the interface to approximate thediffusion field inside the interface as planar, seee.g. in ref. 2. The steady-state solution of Eq. (7)will be a good approximation inside the interfacebecause it is so much thinner than the distance ittravels. Eq. (7) takes the formv(uVC u a C) J C (14)mwhere v is the interface migration rate. The flux isgiven by Eqs. (8) or (9) and is set to 0 in the growinga phase. For a given combination ofv and u a C the solution of Eq. (14) yields a concentrationprofile across the interface and thus also theC content on the g side of the interface, i.e. u g/aThe dissipation of Gibbs energy due to diffusioninside the interface, expressed per mole of Fe anddefined as positive, is given byG diffm V mvdC .dm CJ C dz (15)dzBy combining Eqs. (14) and (15) and makinguse of ∂ G m /∂ u C m C one obtainsG diffm (u C u a C) d ∂ G mdz (16)dz∂ u CdIn order to perform calculations the variation inthermodynamic properties and mobility inside thephase interface must be known. In the solute dragtheory G m is postulated as a function of both distanceand composition. Several choices are possible.One choice is to use Eqs. (3) and (13) insidethe interface but rather than Eqs. (4) and (5) onesimply postulates that g(f) 0 and p(f) z/d.This approach was taken by Hillert and Sundman[2] and will be used in the present report.The interface has a finite mobility due to interfacialfriction and often a linear relation betweeninterface migration rate and driving force isobserved experimentally, i.e.v M V mG i m (17)where G i m is the driving force needed to movethe interface and M is the interfacial mobility.G i m is defined positive for a spontaneous reactionand is the Gibbs energy dissipated by the interfacefriction. The total Gibbs energy dissipated in theinterface is thus given by the sum ofG diffm and G i m. The dissipation must be suppliedfrom the total driving force available over the interface.It is given per mole of atoms and expressedin terms of the individual chemical potentials onp. 152 in ref. [13]. By instead introducing theGibbs energy per mole of Fe, G m , and its firstderivative we obtainG totm G g mG a m(u g/aC u a C) ∂Gg m∂ u g C(18)where G totm is defined as positive for the consideredreaction to occur. For a given migrationrate v we may, by combining Eqs. (9), (14) and(16)–(18) find the u a C and u g/aC that makes the dissipatedGibbs energy exactly matchG totm . In thelimit of low migration rates u a C and u g/aC willapproach the local equilibrium values predicted bythe phase diagram but at high velocities they willapproach each other.The solute-drag modeling of the interfacial reactionsmay be combined with a treatment of C diffusionin g ahead of the interface. By such anapproach it is possible to describe the gradual deviationfrom local equilibrium as the interfacemigration rates increases. In principle such modelingcould be based on numerical methods as inthe DICTRA software [14] or semi-analyticalmethods as the Green function formalism usedrecently by Enomoto [15]. For the sake of simplicitywe will here take a simpler approach basedon the linear-gradient approximation. For the thickeningof a grain-boundary precipitate it yields thefollowing expressionv D (u g/aC u gC ) 2(19)2(u a CuC g/a )(u a CuC g )where is half the thickness of the grain-boundaryprecipitate and u gC is the carbon content in g faraway from the interface, i.e. the initial content of