CHEMICAL POTENTIAL APPROACH TO DIFFUSION-LIMITED MICROSTRUCTURE EVOLUTION

Transcription

1 CHEMICAL POTENTIAL APPROACH TO DIFFUSION-LIMITED MICROSTRUCTURE EVOLUTION M.A. Fradkin and J. Goldak Department of Mechanical & Aerospace Engineering Carleton University, Ottawa K1S 5B6 Canada R.C. Reed Department of Materials Science & Metallurgy Cambridge University, U.K. Abstract A general approach to solving the diffusion problem is developed for chemical potential flows in multicomponent system. A partial equilibrium type of interface boundary conditions of equal chemical potential is applied to the interface between contacting phases. The resulting non-linear diffusion equation is solved numerically. Thermodynamic data from a phase diagram optimization package have been used. The redistribution of carbon between austenite and ferrite (martensite) in low-carbon steel (Hillert s problem) has been studied. Presented at the Symposium Modern Methods for Modeling Microstructural Evolution AMS International Materials Week 95 Cleveland, OH

2 1 Introduction There are some established thermodynamical criteria to determine the state of equilibrium in a heterophase system containing many different chemical components. The general principle is that there is no mass transfer across the interface in the state of thermodynamical equilibrium for a heterophase system. This means that the chemical potentials in contacting phases are equal for all the chemical constituents. This conditions can be used to calculate the interface chemical composition for all phases. As mass transfer processes for different components often have time scales differing in the orders of magnitude, the partial equilibrium concept can be naturally introduced that corresponds to a zero mass flux for the particular component that has the fastest diffusion kinetics. The diffusion-limited evolution of the heterophase microstructure is determined by the growth/dissolution process for individual phases as well as change in their chemical composition through multi-component diffusion. The proper choice of boundary conditions for a mass-transfer equation is crucial in any attempt to simulate this evolution by means of analytical or numerical solution of the kinetic equation. The usual method is to solve a linear diffusion equations for individual phases subject to boundary conditions of prescribed compositions at the interfaces and continuity of the mass fluxes across the interfaces. In the present paper we suppose a different approach based on the kinetic equation for a chemical potential distribution rather then for compositional field. As the inhomogeneity of chemical potential fields is the driving force for diffusion and, generally speaking, for microstructure evolution toward equilibrium configuration, this method corresponds to the enthalpy formulation of the heat transfer problem. This algorithm can utilize a wide variety of thermodynamical and thermochemical data currently available for computer calculation of equilibrium phase diagram in various systems. The approach is particularly convenient when dealing with boundary conditions at the interface, because there is no boundary singularities in the equilibrium chemical potential fields. This is in a contrast with the compositional fields that have step-like behavior at the interface. We present in this paper the results of analysis of the kinetics of a carbon redistribution between ferrite (martensite) and supercooled austenite in a lowcarbon steel (Hillert problem) which has been carried out using the proposed chemical potential approach. 2 Kinetic Equation for Chemical Potential The diffusion flux of component A can be written in the form [1] j A = Γ A x A µ A = D A x A, (1) where x A is the content of component A, µ A is its chemical potential, Γ A is mobility of component A and D A is its tracer diffusion coefficient. Then we have a relationship between diffusion coefficient and mobility Γ A = D A x A ( ) 1 ( ) 1 = D A. (2) x A ln x A 2

3 chemical potential of C (J/mol) Ferrite 400 o C Austenite weight % of C Figure 1: Carbon chemical potential in ferrite and austenite at 400 o C vs carbon concentration Multiplying the continuity equation for the mass transfer of component A div(j A ) + x A = 0 (3) by the derivative of its chemical potential with respect to composition we get the kinetic equation for the distribution of µ A div(j A ) + µ A x A = 0 (4) Supposing Γ A being constant we can write and substituting this into Eq.(4) we can obtain div(j A ) = Γ A (x A µ A + µ A x A ) (5) µ A = Γ A µ A x A (x A µ A + µ A x A ). Finally we can write the kinetic equation for the chemical potential in the form of a nonlinear diffusion equation µ A ( ) = Γ A µ A + µ A 2 ln x A (6) 3

4 6 weight % of C in austenite o C weight % of C in ferrite Figure 2: Interface carbon concentration in ferrite and austenite at 400 o C calculated from the partial equilibrium condition of equal chemical potential 3 Boundary Conditions Partial equilibrium condition with regards to component A µ α A = µ β A (7) means no flux of component A across the interface between the phases α and β. This condition should be satisfied for all phases present in the system. In order to determine the (partial) equilibrium concentration of the component A on both sides of the α/β interface some thermodynamical model is needed that can provide the chemical potential of component A in all phases. There is the only general interface boundary condition for non-linear diffusion equation (6) which is the conservation of mass for component A at the α/β interface expressed through the chemical potential of component A D α A ( µ α A ln x α A ) 1 µ α A D β A ( µ β ) 1 A ln x β µ β A = M α/β A (8) A where superscript α, β stands for a particular phase and the right-hand side describes the effective sink of a component A due to the movement of α/β interface. In what follows we do not consider an interface movement, so, we have M α/β A = 0. We do not have to set up explicitly the boundary condition for the value of chemical potential of A on both sides of the interface. 4

5 4 Example: Carbon Redistribution Between Ferrite and Austenite As an example of this algorithm let us consider the carbon diffusion in a mixture of ferrite (martensite) and supercooled austenite. This problem, known as the Hillert problem, has been the subject of numerous analytical as well as computational studies in recent years (see [2] and references therein). This microstructure can appear as a result of fast transformation upon rapid cooling of austenite when the slow carbon diffusion process does not have enough time to change the carbon distribution during transformation. It is supposed that during bainite formation in a low-carbon steel at low temperatures the ferrite plates grow so fast into austenite that carbon does not diffuse from ferrite into austenite. The supersaturation of austenite with respect to carbon takes place and the diffusion process, which starts after growth is practically over, lead to an escape of carbon from the ferrite plates. Immediately after the transformation, carbon has a homogeneous distribution with composition equal to the average steel composition, however the carbon chemical potential has step-like distribution. Then diffusion leads to a homogeneous chemical potential and step-like distribution of carbon at infinite time. 4.1 Interface Carbon Concentration The carbon content on both sides of the ferrite/austenite interface can be found through the solution of Eq.(7) for carbon chemical potential in both phases. In order to calculate the dependence of carbon chemical potential on the composition some thermodynamic data are required and we have used a thermodynamical model for steel developed previously for phase diagram optimization and described elsewhere [3]. A compositional dependence of the carbon chemical potential for ferrite µ α C and austenite µ γ C at 400 o C is shown in the Fig.1. It can be approximated by empirical dependencies µ α C = log x C (9) µ γ C = x C (10) where chemical potential is measured in kj/mol and x C is an atomic fraction of carbon. From the chemical potential calculated in both phases one can find (partial) equilibrium interface composition of carbon shown in Fig.2. We suppose the problem geometry as a periodical arrangement of parallel ferrite plates in austenite and this transforms the diffusion equation into one-dimensional form with coordinate axis going along a normal plate direction. 4.2 Carbon Redistribution Kinetics For the carbon chemical potential in ferrite described by Eq.(9) the kinetic equation takes very simple form. If µ C = A + B log x C then ln x C = B 5

6 µ C, kj/mol 40.0 t 3 t 2 Ferrite t t=0 Austenite x Figure 3: Evolution of the carbon chemical potential in ferrite/austenite bilayer at 400 o C. Initial step-like distribution at t = 0 is shown by thin line. The distribution is shown for three subsequent time moments t 1 < t 2 < t 3 by thick dots. and Eq.(6) takes the form ( = D C µ C + µ C 2 ) B. (11) In this case Γ C = D C /B and assumption of constant Γ leads to the independence of carbon diffusion coefficient on composition. This is the case according to various studies, (e.g. see [4] and references therein). For linear dependence of the carbon chemical potential on composition that takes place in austenite (10) we have µ C = A + B x C ln x C = B x C and kinetic equation for chemical potential in austenite can be written in the form ( = D C µ C + µ C 2 ) B x C. (12) The mobility of carbon takes a form Γ C = D C /(B x C ) and carbon diffusion coefficient appears to be proportional to composition. The strong compositional dependence of the coefficient of carbon diffusion in austenite has been previously found in many experiments, see [5] and references therein. 6

7 x C, wt % 4.0 t 3 t 2 t 1 Ferrite 0.4 t=0 Austenite x Figure 4: Evolution of the carbon composition in ferrite/austenite bilayer at 400 o C. Notations are the same as at Fig.3 We have solved numerically the one-dimensional kinetic equation for carbon redistribution between ferrite and austenite at 400 o according to Eqs.(14) and (16). Initial carbon concentration of was 0.4 wt%. The equilibrium concentration of carbon in ferrite and austenite at 400 o C is 0.02 and 3.77 wt%, respectively for a steel with 0.4 wt% of carbon. The tracer diffusion coefficient for carbon in ferrite and austenite was and m 2 /s, respectively. The evolution of chemical potential and carbon concentration is shown in Figs.3 and 4 where arbitrary distance units are used because of the scale invariance of Eq.(6). A large difference in the carbon diffusion coefficient between ferrite and austenite appears to lead to different timescales in the kinetics of the chemical potential evolution. As one can see from Fig.3, the diffusion in austenite is a limiting stage for this process. The carbon chemical potential in ferrite is almost constant whereas main changes take place in austenite. The interface carbon concentration on the austenite side changes slightly with time after initial jump to almost equilibrium value. References [1] C.P. Flynn, Point Defects and Diffusion (Oxford: Clarendon Press, 1972) ch.8 [2] M. Hillert, L. Höglund and J. Ågren, Acta Metal. Mater. 41, 1951 (1993) [3] R. Reed, J. Goldak and E. Hughes, Phase Diagrams as a minimisation problem, unpublished [4] J. Ågren, Acta Metall. 30, 841 (1982) [5] S.A. Mujahid and H.K.D.H. Bhadeshia, Acta Metal. Mater. 40, 389 (1992) 7