Fundamentals of diffusion in phase transformations

3

Fundamentals of diffusion in phase transformations

M. H i l l e r t, Royal Institute of Technology, (KTH) Sweden

Abstract: An atomistic model in the lattice-fixed frame of reference is the basis for the present discussion of the fundamentals of diffusion. It is shown that cross terms appear when gradients of some composition variable are introduced. It is demonstrated that an equation describing the movement of Kirkendall markers must be included in the new set of flux equations when the frame of reference is changed. It is argued that one should store information of mobilities rather than diffusivities and one should make calculations of diffusion in the lattice-fixed frame. Key words: frame of reference, driving force for diffusion, evaluation of mobilities, Kirkendall shift, deviation from local equilibrium.

3.1

Introduction

This chapter discusses some fundamental aspects on diffusion that the author, having a basic training as a chemical engineer, has found important in his work in the field of materials science and engineering. Many sections of the chapter are based on previous publications by the author. A wider coverage of the field, but still with an emphasis on materials, can be found in Kirkaldy and Young (1987) and Borg and Dienes (1988). Diffusion in a crystalline solution phase occurs by atoms jumping into vacant lattice sites. The fundamentals of diffusion are thus based on a latticefixed frame of reference. However, in the laboratory one studies diffusion relative to the length dimension of the specimens and one takes measurements relative to a frame of reference that is regarded as the volume-fixed frame. As an alternative, one may evaluate the experimental results in the numberfixed frame. Furthermore, in practical applications one is interested in making predictions in any of these frames. When discussing the fundamentals of diffusion, it is thus necessary to consider all frames and, in particular, to transfer information from one frame to another. Fundamentally, one may regard diffusion in the lattice-fixed frame as driven by thermodynamic forces and the kinetic coefficient is regarded as the mobility of the diffusing atom. The thermodynamic force for diffusion 94 © Woodhead Publishing Limited, 2012

Fundamentals of diffusion in phase transformations in steels

95

and its application to basic kinetic equations for diffusion, referred to the lattice-fixed frame, will be discussed in the first few sections. The kinetic equation is usually transferred to one of the other frames and at the same time one usually introduces a composition gradient instead of the thermodynamic force. The influence of the thermodynamic properties of the system will thus be included in the kinetic coefficients which change character drastically. They are then regarded as diffusion coefficients but in the present chapter they are called diffusivities for conformity with mobilities and the relationship between diffusivities and mobilities is discussed in detail. Due to the thermodynamic properties, the diffusivities may vary with composition much more than the mobilities which do not depend on the thermodynamic properties. Experimental information on diffusion in the volume-fixed or number-fixed frame is thus difficult to rationalize without considering the ‘thermodynamic factor’ separately. It may seem convenient to interpret experimental information directly in terms of parameters defined for the lattice-fixed frame, i.e. mobilities, and to save them in databases instead of diffusivities. The stored mobilities can then be used for practical applications either by continuously evaluating the required diffusivities in the volume- or number-fixed frame or directly in the lattice-fixed frame. In both cases one must have access to stored information on the thermodynamic properties. When calculating the progress of diffusion in a phase during a phase transformation, one must take into account the boundary conditions at the phase interface. It is the movement of the phase interface that represents the phase transformation and it is often assumed that phase interfaces are very mobile. In that case one can apply the approximation of local equilibrium between two phases where they meet at an interface and the boundary conditions for the diffusion inside the phases are given by the thermodynamics properties of the adjoining phases. For a binary system they are obtained directly from the phase diagram or they can be calculated directly under the assumption of local equilibrium between the two phases. However, in higher order systems there is a multitude of possible tie-lines because there is one more degree of freedom for each additional component. The correct tie-line can be found only together with the solution of the diffusion equation for the phases. That is usually done by some iteration process applied for each step in time when the process of diffusion is treated numerically. This may be avoided by modelling the transfer of atoms across the interface. Such a model can be based on an absolute reaction rate approach and will be discussed. Several factors that affect the progress of a diffusional phase transformation are connected to the moving phase interface. In general they will displace the local equilibrium between the two phases or even cause a deviation from local equilibrium. Such factors will be sketched or just mentioned in the last

© Woodhead Publishing Limited, 2012

96

Phase transformations in steels

few sections. Already the transfer of atoms across the interface will dissipate Gibbs energy which may become important at very high velocities. There may be a local volume change at the moving interface, which will cause a local build-up of high pressures if the material does not yield. Furthermore, there are factors that depend upon what happens inside the moving interface. They may be described with models that are based on the distribution of the components inside the interface. However, in this chapter the phase interface has only been modelled without descriptions of local properties or composition as a function of the position in the interface. Such models are regarded as sharp interface models.

3.2

Driving forces of simultaneous processes

We shall start by reviewing the basis of Gibbs’ thermodynamics in a way that leads directly to the definition of thermodynamic driving force for diffusion and other processes. The discussion is based on Chapter 1 in Hillert (2008). The second law of thermodynamics states that the entropy S of an isolated system can never decrease but it can increase by spontaneous internal processes. For each process the extent will be denoted by xi and for an isolated system with a number of simultaneous processes one has

dS = dipS = ∑ Aidxi > 0

[3.1]

The subscript ip stands for ‘internal process’. For a system open to exchange of heat, Q, and matter, N, with the surroundings, one has

dS = dQ/T + SmdN + ∑ Aidxi

[3.2]

Sm is the molar entropy of the system and it is here assumed that any addition of matter is made without changing the composition, i.e., with balanced amounts of various components. The coefficient Ai is regarded as the affinity of the process i and is sometimes regarded as its thermodynamic force. The addition of heat, dQ, enters into the first law which defines the change of internal energy, U:

dU = dQ + dW + HmdN

[3.3]

At this stage we shall simply regard Hm as a coefficient related to the increase of the content of matter. dW is the work done on the system. By only considering compression work, dW = –PdV, we obtain by combination of the two laws,

TdS = dU – (Hm – TSm)dN + PdV + T ∑ Aidxi

[3.4]

Introducing the notation G m = Hm – TS m, without yet discussing its interpretation, we can express Eq. [3.4] as



dU = TdS – PdV + GmdN – T ∑Aidxi

97

[3.5]

Equations [3.4] and [3.5] are two different ways of expressing the combined law but they are of limited use because the variations of S and U are given as combinations of variables some of which are often difficult to control experimentally. The most practical form of the combined law is obtained as follows, because T and P are convenient experimental variables and they are often kept constant. It introduces a new quantity, G = U –TS + PV: dG = d(U – TS + PV) = – SdT + VdP + GmdN – T ∑Aidxi

[3.6]

G is called Gibbs energy and Gm is thus the molar Gibbs energy, obtained as a partial derivative under constant T, P and xi. By regarding the addition of individual amounts of the components we instead write dG = d(U – TS + PV) = – SdT + VdP + ∑GidNi – T ∑Aidxi

[3.7]

Gi is defined as the partial derivative ∂G/∂Ni when all the other variables are kept constant including the amounts of the other components. It is also regarded as the chemical potential of component i and is then denoted by mi. It is also convenient to define a quantity called enthalpy, H = U + PV, for which we find dH = d(U + PV) = TdS + VdP + ∑ midNi – T ∑Aidxi

[3.8]

The production of entropy by internal processes is given by Eq. [3.1] and for the rate of internal production of entropy we find

s =

∂ip S ∂x = SAi i > 0 ∂t ∂t

[3.9]

Under constant T and P it is more convenient to consider the rate of Gibbs energy dissipation d(–G)/dt = T ∑Ai∂xi/∂t = ∑Xi∂xi/∂t = Ts > 0

[3.10]

∂xi /∂t is regarded as the flux of process i, and is denoted Ji.. This term makes most sense for processes of transportation, e.g. diffusion, but is used more generally. For convenience we have changed the notation of TAi to Xi. this quantity is regarded as the driving force for the process i under constant T and P. It is the partial derivative of G, ∂G/∂xi, when the other variables are kept constant. Under constant T and P one may write the rate of dissipation of Gibbs energy as – dG/dt = Ts = ∑XiJi > 0

[3.11]

It should be emphasized that there is no rule requiring that XiJi > 0 for each of a number of simultaneous processes. For a system with several internal processes, it is possible to consider other aspects and to find that it is more convenient to consider a different set of © Woodhead Publishing Limited, 2012

98


processes, all of which are linear combinations of the initial ones. This new way of looking at the system does not change what actually happens in the system. One may thus assume that the overall dissipation of Gibbs energy is the same and the value of ∑XiJi must be the same. This criterion can be used to derive the driving forces for the new processes from the driving forces for the initial ones.

3.3

Atomistic model of diffusion

Diffusion can be treated mathematically by starting with Fick’s law for diffusion and expressing material properties through diffusion coefficients. However, in ternary and higher order systems, one may need a large number of such coefficients. It may be difficult both to determine them experimentally and to handle them in calculating the progress of diffusion. By accepting an atomistic model one may considerably reduce the number of independent coefficients and may relate the diffusion coefficients to the thermodynamic properties; this procedure considerably decreases their remaining dependence on composition. In this section we shall thus develop an atomistic model which will then be used throughout the chapter. The discussion is based on Section 17.1 in Hillert (2008). The rate of many internal processes depends on the possibility of crossing an energy barrier, e.g. in the narrow passage between other atoms when an atom jumps from one lattice site into the neighbouring one in diffusion. The extra energy, Q, is provided by thermal fluctuations and the probability is proportional to exp(–Q/RT) according to Boltzmann statistics. However, the requirement is decreased by the driving force, and the flux may thus be represented by J = K exp –

Q – X /2 RT

[3.12]

From this equation it is evident that Q as well as X are expressed as J/mol if the gas constant R is given as J/mol,K. It is here assumed that the reaction path can be regarded as a distance between the initial and final states and also that the driving force is caused by a continuous change of the Gibbs energy of the system along the path. If X is the total Gibbs energy change between the two states then only half will be available at the top of the barrier if it is situated in the middle between the two states. According to the philosophy of the absolute reaction rate approach, originating from Eyring (Glasstone et al., 1941, ch. 9), one should also consider the rate of the reverse process and for the net rate we obtain



99

Q – X /2 Q + X / 2ˆ Ê J = K Á exp – – exp – RT RT ˜¯ Ë = K exp –

–X X ˆ Q Ê · exp – exp RT ÁË 2RT 2RT ˜¯

= K exp –

Q Q · 2sinh X @ K exp – ·X RT 2RT RT RT

[3.13]

This is a good approximation when X n, i.e., for interstitial elements. Using the number-fixed frame The number of atoms crossing the plane of markers is always ∑J *i if the flux of vacancies is excluded and the fraction of j atoms in the same amount of the material is xj. the aji factor in the number-fixed frame is thus independent of the possible existence of sublattices. We can thus use aji = dji – xj in Jj = ∑ ajiJ*j in the following equations. Insertion of J *j from Eq. [3.71] into Eq. [3.25] yields n +m

n +m

n +m m+ +1

i =1

i =1

k =1

J j = S a ji J i* = – S a ji L*i S

∂mi dyk for j ≤ n + m for ∂yk dz ∂y

[3.75]

eliminating the nth component in the substitutional sublattice and the vacancies from the interstitial sublattice, we obtain



117

n Ê n –1Ê ∂m j ∂m ˆ dy S n + m Ê ∂m ∂m ˆ dy I ˆ J j = – S a ji L*i Á S Á S – Si ˜ k + S Á Ii – I i ˜ k ˜ i =1 Ë k =1Ë ∂yyk ∂yn ¯ dz k = n++1 Ë ∂∂yyk ∂yVa ¯ dz ¯

d njk,VVaa dykS ,I =– S k ≠ n Vmf dz

[3.76]

n +m Ê ∂m ∂m ˆ d njk /Vmf = S a ji L*i Á Si – Si ˜ for for k < n i =1 Ë ∂yyk ∂yn ¯

[3.77]

n +m Ê ∂m ∂m ˆ dVa / V = S a ji L*i Á Ii – I i ˜ for for k > n mf jk i =1 Ë ∂yyk ∂yVa ¯

[3.78]

Again it should be remembered that L*i is equal to bSySi Mi*/Vmf for i < n but I it is equal to bIyIiyVa Mi*/Vmf for i > n, i.e., for interstitial elements. Andersson and Ågren (1992) have already discussed this case but their final equations are slightly different because they used a composition variable that does not appear in the thermodynamic model of interstitial solutions. Applications Interstitials in steels, mainly carbon and nitrogen, are much more mobile than substitutional elements. The difference is so large that one can treat most cases as one of two extreme cases. For short times one may neglect the diffusion of substitutional elements completely but should of course take into account their effect on the chemical potential of the rapidly diffusing interstitial element. That would be a purely thermodynamic effect. After long times the rapid diffusion of the interstitial elements may have stopped but one can consider the diffusion of substitutional elements. There may be simultaneous diffusion of the interstitial elements but only due to the redistribution of the substitutional elements. At that stage it may be a good approximation to assume that there is uniform carbon or nitrogen activity and their effect on the substitutional elements can be evaluated from their activities if one has access to a thermodynamic databank. The situation will be more complicated if there is a phase transformation controlled by long range diffusion of carbon or nitrogen. The boundary conditions at the phase interface may then be affected by very local redistribution of substitutional elements (Hillert, 1953; Kirkaldy, 1958; Hillert and Ågren, 1988). This is an important and active field of research that involves deviation from local equilibrium at phase interfaces for the substitutional elements but good local equilibrium for the interstitials. Other factors may also interact with phase interfaces and may cause deviations from


118


local equilibrium. The remainder of this chapter will discuss the boundary conditions and such factors briefly.

3.5.2

Boundary conditions at phase interfaces

In order to predict the progress of one-dimensional diffusion in a phase of limited extent, it is necessary to know the boundary conditions at the two sides of the phase. In a phase transformation there is a moving interface between the two phases and the conditions at the interface will define boundary conditions of both phases. It is very common to assume that there is local equilibrium between the two phases even though they may be separated by a moving phase interface. With that assumption one could fix the boundary conditions for a binary system for which the two-phase equilibrium at given T and P has no degrees of freedom. The compositions at the interface will then be constant during the process of diffusion. However, for each additional component there is a new degree of freedom and already for a ternary system the operating tie-line is affected by the diffusion process itself and may change with time. In computerized calculations of diffusion, one usually finds the operating tie-line by an iteration process for each step in time. Such a procedure gets more complicated for each additional component. The situation gets even more complicated if the assumption of local equilibrium at the interface could not be applied. In such a case it is necessary to define the conditions at the interface in some other way, which means that one must apply a model for the physical factor causing the deviation from local equilibrium. If the interface is curved, as it is for a spherical inclusion, the pressure difference will affect the local equilibrium. That effect can be easily handled with a computerized program for thermodynamic calculations by assuming that the included phase is under an increased pressure, DP = 2s/r. If the interface experiences a resistance when moving, i.e. some kind of friction, then the effect on the included phase can be translated into a pressure increase and be treated as such. Then it is necessary to find a suitable physical model and combine it with diffusion into a kinetic model for the overall process. Simple methods that have been tried are to assume a constant friction coefficient, which results in a friction proportional to the velocity of the interface, and a constant friction that implies that there is an energy barrier that must be surmounted. A more detailed model is to assume that the atoms have to cross the interface individually and to give the atoms of each component a mobility. For that model one could apply the formalism for diffusion and assume that the interface has a fixed thickness Dz comparable to the thicknesses of the space elements into which the system is already divided for numerical treatment of diffusion inside the phases. It should be possible to apply the atomistic model derived in Section 3.3 but without using the approximation x B¢ x B¢¢¢ @ x B because the compositions of the two phases



119

may be too different. The following flux equation has thus been proposed for the flux across the interface: J B = – M B xaB / b x Bb /a Dm B /D Dzz = – M B xaB / b x Bb /a (maB /b – m Bb /a )/Dzz [3.79] With the approximation x B¢ x B¢¢¢ @ x B, this equation reduces to Eq. [3.21], which has often been used for numerical solutions of diffusion inside a phase, divided into thin slices. The quantity Dz is then the distance between the centres of two neighbouring slices (Larsson et al., 2006). The Kirkendall shift is evaluated as Jk = –∑Ji = ∑MixiDmi/Dz and according to the Gibbs–Duhem relation, ∑xidmi = 0, it would be very close to zero if Dz is small and the mobilities are equal. It is evident that the Kirkendall shift is caused only by differences between the mobilities. On the other hand, if Eq. [3.79] without the approximation is applied to the two slices adjoining the interface, then the Kirkendall shift may be considerable. In that case, the interface plays the role of Kirkendall markers and the Kirkendall shift is identical to the rate of the phase transformation. The composition at the interface and its rate of movement will thus be calculated for each time step when the progress of diffusion in the whole system is calculated. If the initial conditions at the interface, represented by the two adjoining slices, are not correct, then there may be an automatic iteration towards better values with time. The calculated conditions at the interface will approach those for local equilibrium if high values have been chosen for the mobilities of the atoms for crossing the interface. A deviation from local equilibrium will be obtained if lower values are chosen. A limitation of this method is that it can only describe such conditions at the interface when each component is transferred to the side with the lower chemical potential of the component. Cases have been observed where atoms are transferred to the side with the higher potential and will be mentioned in the next section.

3.6

Trapping and transition to diffusionless transformation

It is known that a diffusional phase transformation may become partitionless or even diffusionless if the rate can be increased enough by increasing the supersaturation. Partitionless means that the new phase forms with the same composition as the parent phase, which may occur behind a steep pile-up of the minority component in the parent phase. Diffusionless means that the result is the same but there is no diffusion of the components relative to each other. It is also known that under less drastic conditions there may be


120


trapping, which means that one component is transferred across the moving phase interface against its own driving force. Of course, the diffusionless transformation is the extreme of trapping. Certainly, such phenomena cannot be modelled with the above mechanism of individual jumps of atoms to the side with lower chemical potential. There is a need for a more general model that can also describe the transfer of atoms to a higher potential. An attempt has been made to improve the above treatment by adding a third mechanism, a cooperative transfer of the components (Larsson and Borgenstam, 2007). There have been attempts to predict trapping on a strictly formal basis using the fact that two simultaneous processes may affect each other by cross terms in their rate equations (Baker and Cahn, 1971; Caroli et al., 1986). Instead of Eq. [3.21] one should then write the flux equations as

Jj = LjjXj + LjkXk

[3.80]

Trapping may occur if the two terms have different signs and the second one is larger. As demonstrated in Section 3.4.1, one may define a new set of processes and derive the driving forces for them by the criterion that the dissipation of Gibbs energy must be accounted for. In an attempt to give the introduction of cross terms a physical interpretation, a new set of processes has been introduced as linear combinations of the initial ones hoping to find a case where the cross terms have vanished. For a binary system the following new fluxes have been chosen:

JA* = Jj + Jk

[3.81]

JB*

[3.82]

= x kJ j – x j J k

If applied to diffusion processes inside a phase, the first flux gives the net flux of atoms, i.e., the Kirkendall shift and the second flux describes the mixing of the two components. This set of new processes thus happens to be the same as the set obtained when changing from lattice-fixed to number-fixed frame, although in Section 3.4 we started with fluxes without cross terms in the lattice-fixed frame. The ambition has been to find two new processes that are independent of each other, i.e. without cross terms. The coefficients for the new processes can be obtained as demonstrated in Section 3.4.1 and there is no general way to make them zero if they are zero for the initial set of processes. However, by requiring that they must be zero in the new set, one can easily derive what values they must have in the initial set. By using the initial set of processes with those values, one may thus model a transformation occurring with two independent processes without directly representing them in the computer program. Since the new set of processes is identical to the set introduced when changing from lattice- to numberfixed frame, this simply means that one would apply the alternative method described in Section 3.4.2 but with cross terms. At the same time there would



121

be no cross terms in Eqs [3.38] and [3.39]. This approach does not seem to have found much practical application. Here we have only considered the so-called sharp interface situated between two thin space elements. Wide interface models are more powerful and can be applied to describe such phenomena as segregation and solute drag during grain growth (Cahn, 1962) and phase transformations (Hillert, 2004) and trapping (Hillert and Sundman 1977). Finally, the phase field method must be mentioned where the width is not fixed in advance (Moelans et al., 2008). Instead, the interface is diffuse and the variation of the local state and composition along the direction of diffusion are optimized continuously by minimization of the overall Gibbs energy and the movement of the phase interface is obtained. This seems to be the most powerful method but requires more computing time.

3.6.1 Local change of volume at phase interface The introduction of mobilities for diffusion in the lattice-fixed frame in Section 3.3 was straightforward, but the introduction of new frames of reference revealed a complication. Formally, the equations were correct as far as the Kirkendall shift was defined as the net amount of material that crossed the plane of the Kirkendall markers. The complication appeared when the Kirkendall shift was measured as a length in the specimen because then it had to be assumed that the cross section of the system had not changed. It may be measured either in m2 for the volume-fixed frame or in a measure related to the number of atoms in the cross section. Primarily, one should expect that an expansion of the volume due to a Kirkendall effect should be homogeneous in all three dimensions. It may be argued that for onedimensional diffusion, a solid piece of material should be able to change its volume only in the direction of diffusion and that requirement could only apply to the cross section measured in m2. It is thus necessary to accept plastic deformation simultaneous to the diffusion. The situation is more complicated if there is diffusion in two or three dimensions and even more interesting if there is a diffusional phase transformation. In Section 3.5.2 it was mentioned that the movement of the interface has the same origin as the Kirkendall shift. The following discussion is similar to the one given by Hillert (2008, Section 17.4) but the method of transforming fluxes between different frames of reference will now be used. In order to evaluate the need of local plastic deformation at the moving phase interface during a transformation, it is necessary to apply a unique lattice-fixed frame of reference for each phase and also to consider the flux of atoms across the phase interface, Jjtrans. That flux is defined relative to the interface, which is generally moving with different velocities relative to the two phases. Those velocities will be denoted ua/b and u b/a for a b Æ a


122


transformation. When comparing the three fluxes, it is convenient to describe them in the same frame of reference and we shall choose a frame fixed to the interface. Since the flux leaving one phase must cross the interface and be received by the other phase, the three fluxes for each element must be equal when expressed in the same frame. According to Eq. [3.24] we obtain

Jja – xjaua/b/Vma = Jjtrans = Jjb – xjbub/a/Vmb

[3.83]

ua/b and ub/a are generally different and there would then be a local change of volume. The lattice of the b phase will move away from the lattice of the a phase with a velocity of Du = ua/b – ub/a. To find this quantity we shall first eliminate the flux across the phase interface, Jjtrans, from Eq. [3.83] and then sum over all components:

ua/b/Vma – ub/a/Vmb = ∑Jia – ∑Jib

[3.84]

We may then eliminate any one of the velocities from Eq. [3.83], obtaining

ua/b/Vma = [Jja – Jjb – xjb (∑Jia – ∑Jjb)]/(xja – xjb)

[3.85]

and similarly for ub/a. We may then take the difference and for a binary system we finally obtain

Du = ua/b – ub/a = Î(Vma xBb – Vmb xBa) (JAa – JAb )

– (Vma xAb – Vmb xAa) – (JBa – JBb )˚/(xAa – xAb)

[3.86]

In two- or three-dimensional diffusion experiments, this may cause severe deformations of the material, which may occur by several deformation mechanisms. One of them is stress-induced diffusion caused by the build-up of a high positive or negative pressure in a precipitating phase. A drastic example is the growth of a spherical nodule of graphite in a supersaturated matrix of an Fe-C solution phase (Hillert, 1957). The diffusion of carbon to the graphite would primarily dominate completely and the pressure in the graphite should build up until the hole in the matrix is expanding at the same rate. Otherwise, the diffusion of carbon has to stop due to a pressure-induced increase of the carbon potential in the graphite. In reality, a balance will be established where the diffusion of carbon is slowed down and the hole is expanded by stress-induced diffusion of iron or mechanical deformation of the matrix.

3.6.2 Composition of flux across phase interface It is possible to derive the net composition of the material transferred across the phase interface without modelling the mechanism of transfer but



123

considering the fluxes in the two phases. Inserting Eq. [3.85] into the first part of Eq. [3.83] we obtain after some manipulations: Jjtrans = Jja – xjaua/b/Vm = [xja(Jjb – xjb ∑Jib) – xjb (Jja – xja∑Jia)]/(xja – xjb)

[3.87]

By first summing Eq. [3.85] over all the components we obtain after similar manipulations: ∑Jitrans = ∑Jia – ua/b/Vm = [(Jjb – xjb∑Jib) – (Jja – xja∑Jia)]/(xja – xjb)

[3.88]

The mole fraction of component j in the material crossing the interface will thus be rans x trj ans = J ttra /SJ itrans = j

xaj (J bj – x bj SJ ib ) – x bj (J aj – xaj SJ ia ) (J bj – x bj SJ ib ) – (J aj – xaj SJ ia )

[3.89]

From this composition one can evaluate the driving force for the transfer as ∑xitransDmi to be used when trying possible models for the transfer.

3.7

Future trends

Most measurements of the rate of diffusion and most practical applications of diffusion have concerned binary alloys. That activity has resulted in an impressive amount of data on diffusivities that is available in various compilations. However, most commercial alloys contain more than one alloying element and that is particularly true for steels. Then there will be cross terms in the kinetic flux equation, which define new diffusivities. To determine all of them may require too much experimental work and one should not expect to find many such values in compilations. It may thus be predicted that experimental data will in the future be analyzed in terms of mobilities that are defined in the lattice-fixed frame. It may reasonably be assumed that there should be no cross terms in the lattice-fixed frame and, thus, only one mobility for each element, although it should vary with composition. Compared to the diffusivities, the variation with composition is much smaller for the mobilities because they do not contain the composition dependence of the thermodynamic properties. The reason is that the gradient of the chemical potential of a component is used as the driving force in the lattice-fixed frame. In order to use that formalism, it is thus necessary to have access to the thermodynamic properties of the system. In that field one encounters the same kind of problem when going to ternary and higher order systems. Binary phase diagrams are manifestations of the thermodynamic properties and are available in compilations for a large


124


number of systems. By combination with various kinds of thermodynamic measurements it has been possible to obtain reasonable descriptions of the thermodynamic properties of binary systems. This is called the CALPHAD approach (Saunders and Miodownik, 1998; Lukas et al., 2007). The same holds for ternary systems, although additional parameters must be taken into account. There is an increasing complexity for each additional component in a system. The situation is similar to that in the field of diffusion, but efforts to handle that kind of situation have already started in the field of thermodynamics. There are already computerized databases available together with powerful software for the calculation of various thermodynamic quantities. The results achieved so far in the field of thermodynamics have been made possible by an informal world-wide collaboration. Although the results are impressive and are already of considerable use, it is important that the work continues. It is evident that the same kind of effort is necessary in the field of diffusion and the experience gained in the field of thermodynamics may be useful in the field of diffusion. It is necessary to reach international agreement on each step in the work, the choice of ‘model’, i.e., choice of frame of reference, flux equation and kind of parameter to be evaluated and stored, e.g. diffusivity or mobility, the values of those parameters in simple systems, primarily pure elements and binary systems, and the method of describing their dependence on composition and temperature. Since the values of diffusivities in the volume-fixed frame and the driving forces in the lattice-fixed frame depend on the thermodynamic properties, it is necessary that the future work on diffusion in alloy systems be closely connected to the development in the field of thermodynamic properties of alloy systems. Hopefully, that will come naturally because many research groups will work in both fields. To the present author it seems that one should agree on analyzing experimental information in terms of mobilities and also to make numerical calculations in the lattice-fixed frame. The database should only contain the mobilities but that information, in combination with a thermodynamic database, will be used for evaluating diffusion coefficients for binary systems. Lists of such quantities are necessary for filling the need of making hand calculations for simple cases, but for more complicated cases it is necessary to have a powerful package of databases and programs for thermodynamics and diffusion.

3.8

Acknowledgement

The author owes much of his understanding of the fundamentals of diffusion to many discussions with Professor John Ågren, and this chapter was inspired by his paper with Dr J.-O. Andersson in 1992.



3.9

125

References

Ågren J (1982), J. Phys. Chem. Solids, 43, 421. Andersson J-O and Ågren J (1992), J. Appl. Phys., 72, 1350. Baker J C and Cahn J W (1971), in Solidification, eds. J. Hughel and G F Bolling, Metals Park, OH, ASM, p. 23. Borg R J and Dienes G J (1988), Solid State Diffusion, Boston, MA, Academic Press. Cahn J W (1962), Acta Metall., 10, 1. Caroli B, Caroli C and Roulet B (1986), Acta Metall., 34, 1867. Darken L S, (1948), Trans. AIME, 175, 184. Glasstone S, Laidler K J and Eyring H (1941), The Theory of Rate Processes, New York, McGraw-Hill. Hillert M (1953), Paraequilibrium (Internal report), Stockholm, Swedish Inst. Metal Res. Reprinted in Thermodynamics and phase transformations – The collected works of Mats Hillert, (2006), Les Ulis Cedex A, France, EDP Science. Hillert M (1957), Jernkontorets Ann., 141, 67. Hillert M (2001), Scripta Mater., 44, 1095. Hillert M (2004), Acta Mater., 52, 5289. Hillert M (2008), Phase Equilibria, phase diagrams and Phase Transformations – their Thermodynamic basis, 2nd edn, Cambridge, Cambridge University Press. Hillert M and Ågren J (1988), Diffusional transformations under local equilibrium in Fe-C-M systems, in Advances in phase transitions, Oxford, Pergamon Press. Hillert M and Sundman B (1977), Acta Metall., 25, 11. Kirkaldy J S (1958), Can. J. Phys., 36, 907. Kirkaldy J S and Young D J (1987), Diffusion in the Condensed State, London, The Institute of Metals. Larsson H and Borgenstam A (2007), Scripta Mater., 56, 61. Larsson H, Strandlund H and Hillert M (2006), Acta Mater., 54, 945. Lukas H L, Fries S G and Sundman B (2007), Computational thermodynamics, Cambridge, Cambridge University Press. Moelans N, Blanpain B and Wollants P (2008), CALPHAD, 32, 268. Saunders N and Miodownik A P (1998), CALPHAD – Calculation of Phase Diagrams, Oxford, Pergamon Press.


Fundamentals of diffusion in phase transformations

Fundamentals of diffusion in phase transformations

Suggest Documents