On two-phase coherent equilibrium in binary alloys

ACla "'1'1011. maIer. Vol. 38, No.4, pp. 561-572, 1990 Printed in Great Britain. All rights reserved

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0956-7151/90 $3.00 + 0.00 © 1990 Pergamon Press pic

ON TWO-PHASE COHERENT EQUILIBRIUM IN BINARY ALLOYS ZI-KUI LIU and J. AGREN Division of Physical Metallurgy, Royal Institute of Technology, S-IOO 44 Stockholm, Sweden (Received 10 May 1989; in revised farm 27 September 1989)

Abstract-The effect of coherency stresses on two-phase equilibria is treated adopting Williams' approach that the Gibbs energy of the coherent state is obtained by simply adding the elastic energy to the Gibbs energy of the unstressed state. Under those circumstances the general equilibrium conditions are reasonably simple and may be applied in practical calculations for alloys where the thermodynamic data are available. The stability of the coherent state is discussed in terms of Gibbs energy diagrams. Some general features and the inapplicability of the phase rule are further discussed. In addition the coherent IX-y equilibrium in the binary Fe-Mn system is calculated assuming simple geometry and applying assessed thermodynamic functions. Resume--on traite I'effet des contraintes de coherence sur les equilibres biphases en adoptant l'approche de Williams dans laquelle on obtient I'energie de Gibbs de I'etat non contraint. Les conditions generales d'equilibre sont alors raisonnablement simples et peuvent etre appliquees a des calculs pratiques sur les alliages pour lesquels on dispose de donnees thermodynamiques. La stabilite de I'etat coherent est discutee en fonction des diagrammes d'energie de Gibbs. On discute en outre quelques caracteristiques generales et les conditions dans lesquelles la regie des phases ne peur s'appliquer. On calcule egalement l'equilibre coherent IX-y du systeme binaire Fe-Mn en supposant une geometrie simple, et en appliquant des fonctions thermodynamiques imposees. Zusammenfassung-Der EinfluB der Koharenzspannungen auf zweiphasige Gleichgewichte wird mit dem Ansatz von Williams behandelt. Danach ergibt sich die Enthalpie des koharenten Zustandes aus einer einfachen Addition der elastischen Energie zur Enthalpie des spannungsfreien Zustandes. Unter diesen Umstanden werden die allgemeinen Gleichgewichtsbedingungen hinreichend einfach und konnen fiir praktische Rechnungen auf Legierungen, deren thermodynamischen Daten verfiigbar sind, angewandt werden. Die Stabilitat des koharenten Zustandes wird anhand von Enthalpie-Diagrammen diskutiert. AuBerdem wird das koharente IX-y-Gleichgewicht im binaren System Fe-Mn unter der Annahme einfacher Geometrie und mit ageschatzten thermodynamischen Funktionen berechnet.

1. INTRODUCTION

follow. Consequently, their method has not been used very much and it seems as it was never applied to real materials. It was not until quite recently that some of the more general features of coherent equilibria were explored. The starting point is the work by Williams [6,7]. In his remarkable paper from 1984 [7] Williams applied a very simple approach and based on graphical constructions and a sound intuition he drew a number of important conclusions. Williams' very simple idea was that the Gibbs energy of a coherency stressed system is obtained by adding an extra term to the Gibbs energy of the unstressed system. The extra term is simply the elastic energy due to the coherency stresses. If the extra term depends on the composition or the amounts of the two phases the state that minimizes the total Gibbs energy of the system will be different from the one of the unstressed system. The mathematical problem is then to find the new equilibrium state. Williams found that a number of situations could occur depending on the characteristics of the elastic term.

It has been known for a long time that coherency stresses may play an important role in phase equilibria. In the early sixties Cahn [1] was able to explain the depression of the spinodal decomposition by taking into account the coherency stresses caused by the concentration dependent lattice parameter. A decade later the first general set of equilibrium conditions valid at a coherent phase interface were derived by Robin [2]. Larche and Cahn in a series of three papers [3-5] considered not only the interface but the whole material and derived the general thermomechanical equilibrium conditions. Their approach was to define an energy density field and obtain the total energy of a system by integrating the energy density over the volume of the system. The equilibrium field was subsequently obtained by searching for the minimum in the total energy under the constraint of a constant entropy and maintained coherency. Their treatment is elegant and they arrive at the conditions which assure thermal, diffusional and mechanical equilibrium. However, their papers contain a lot of heavy algebra and are difficult to 561

562

ZI-KUI LIU and AGREN:

TWO-PHASE COHERENT EQUILIBRIUM

We will not review his paper further but refer the reader to Refs [6] and [7]. We will mention, though, Williams' two most extraordinary conclusions. The first one is that if the elastic energy is sufficiently high then the two-phase field of the phase diagram will degenerate into a line although the transformation is still of first order. In that case a two-phase mixture is never stable. At equilibrium one will observe one phase or the other, never both at the same time. Cahn and Larche [8] termed this point, which is not a critical point in the usual sense because the transformation remains of first order, the Williams point. Another spectacular feature found by Williams is what he called "forbidden zones". In those zones of the phase diagram, located between the incoherent compositions of the two phases, the two-phase mixture is not stable because its Gibbs energy is higher than that of the one-phase state. The Williams point may then be regarded as the special case when the whole two-phase field is a forbidden zone. The remarkable conclusion is that there must be a discontinuous change in the fraction of phases as a forbidden zone is entered from the two-phase field. The fraction of one of the phases jumps discontinuously to zero. An important question which was not discussed in any detail by Williams is if the two-phase mixture inside a forbidden zone is metastable, i.e. corresponds to a local minimum in Gibbs energy, or if it is truly unstable, i.e. corresponds to a maximum or a plateau point. In an attempt to explore Williams' ideas on a more quantitative basis Cahn and Larche [8] assumed that the Gibbs energy of an unstressed phase could be approximated as a second order polynomial of its composition. They further assumed that the Gibbs energy curves of the two phases had the same curvature and that the elastic contribution is proportional to the product of the volume fraction of the two phases. It was then possible for them to find the analytical solutions and the existence of the Williams point was confirmed. However, due to the assumption that the two Gibbs energy functions have the same curvature they found no forbidden zones in their analysis. It is interesting to notice that Cahn and Larche did not draw those important conclusions from their previous general treatments but from the much more simplified treatment of Williams. It thus seems as Williams in his approach intuitively caught the essential physical aspects of coherent equilibria. Johnson and Alexander [9] applied the same procedure as Larche and Cahn [3-5] and derived a general set of thermomechanical equilibrium conditions that took into account also the effect of interfacial segregation. However, they did not apply their method to real materials in that report. Somewhat later Johnsson and Voorhees [10], after having made some approximations, applied the general equilibrium conditions in an extensive analysis of the general characteristics. Even though they used a simplifying

assumption concerning the shape of the Gibbs energy curves of the two phases their analysis gives some insight in the complex behaviour of real materials. In addition they made their analysis on a particular geometry. The two phases were assumed to occupy shells of concentric spheres. This choice of geometry allowed them to solve their elastic problem exactly. They could then take into account the influence of differences in elastic properties and an isotropic strain caused by different molar volumes. Johnson and Voorhees could now obtain an analytical solution. The whole behaviour of the system was obtained by solving a cubic equation. Their analysis confirmed the results of Cahn and Larche [8]. In addition they discussed a large number of different features and in particular they analyzed the stability applying bifurcation theory. They found regions, situated inside the two-phase field of the ordinary phase diagram, where no solution except the one corresponding to a one-phase structure existed. Such regions could be termed as forbidden although Williams defined his forbidden regions in a different way. As the result of numerous investigations applying the so called Calphad technique the thermodynamic behaviour is nowadays known in quite a detail for many alloy systems of practical interest and the technique of calculating phase equilibria (in absence of coherency stresses) in real alloys is well established. Since one would expect coherency effects to be of great practical importance in the solid state it should be quite valuable to have a reasonably straight forward technique to account for coherency stresses in solid-state equilibrium calculations. The purpose of the present study is to develop such a technique suitable for practical calculations on real systems where thermodynamic data may be available. The technique is not as general as the one by Larche and Cahn but is simple enough to be applied in the Calphad type of practical calculations. Some new aspects of general character will be discussed and finally a coherent austenite-ferrite equilibrium of the Fe-Mn system will be calculated. 2. MATHEMATICAL DERIVATIONS

2.1. Simplified equilibrium equations of a coherent system

The equilibrium state of a system is obtained by minimizing the appropriate state function. For a system under a given temperature T, external pressure P and over-all composition, i.e. a closed system, the appropriate state function is the Gibbs energy G. Let us now adopt Williams' hypothesis that the Gibbs energy of the coherent system is obtained by simply adding the elastic energy to the total Gibbs energy of the corresponding unstressed system and then look for the minimum of the total Gibbs energy. It should be pointed out that the more general thermomechanical equilibrium conditions derived

ZI-KUI LID and AGREN:

by Larche and Cahn [3-5] and by Johnson and Alexander [9] reduce to Williams' hypothesis if the composition within each phase is homogeneous. In many situations this should be a fairly reasonable approximation and it simplifies the calculations considerably. We thus have the following expression for the total Gibbs energy, expressed per mole of atoms in the system, Gm=fG~+(I-f)Gl;,+~G~

(I)

(2)

where x~, XB and x~ are the mole fractions of element B, in the system, IX and y phases, respectively. From now on we will restrict the discussion to substitutional binary systems and the subscript B will be dropped. However, general conditions are readily formulated for arbitrary numbers of substitutional and interstitial elements. This will be dealt with in a coming paper. The problem is now to find the values of f, x" and x Y that yield the lowest value of Gm • The minimization is conveniently performed by applying the Lagrange method. An extra unknown, a so called multiplier, is introduced for each constraint and a new function, the Lagrangian L, is defined. L = Gm + A[X O- fx" - (1- f)x Y ]

(3)

Gm is given by equation (I) and A is the Lagrange multiplier corresponding to equation (2). The extrema of Land Gm are now found by setting the partial derivatives of L with respect to the unknown variables equal to zero. In the present case this yields 3 equations. The constraint, equation (2), yields one more equation and we can thus uniquely determine the 4 unknowns f, x", Xl and A. For simplicity it is assumed that ~G~ only depends on the relative phase amounts. We thus have the following system of equations

oL dG" -=f_m_Af =0 ax" dx"

oL ox

dGl'

_=(I_f)_m-A(I-f)=O Y dx l

(6)

xO-fx"-(I-f)xl=O.

(7)

After eliminating the Lagrange multiplier A from the equations and rearranging we obtain the following equations in addition to equation (7) Gl - G"

where f is the mole fraction of IX phase and G~ and Gl;, are the molar Gibbs energies of the IX and y phases, respectively, in the unstressed state. ~G~ is the stored elastic energy caused by the coherency stresses. It should be noticed that the effect of surface tension is neglected in this equation. As mentioned the equilibrium state is now obtained by minimizing Gm under the constraint that the over-all composition is fixed. In a substitutional system, where the number of lattice sites are conserved, the composition can be changed only by exchange of atoms and we will thus have the following constraint x~ - fXB - (I - f) x~ = 0

563

TWO-PHASE COHERENT EQUILIBRIUM

(5)

m

m

+ (x" -

dGI d~GeI Xl) __m = _ _m dxl df dG~

dx"

dGl;, dx Y

(8)

(9)

It is interesting to notice that the above equations have a straight forward physical interpretation. The left hand side of equation (8) represents the driving force needed in order to form a small amount of IX phase from a large reservoir of y having the composition Xl. Equation (8) thus states that this driving force is used to overcome the resistance caused by the elastic stress. The effect is formally the same as if the IX phase was under the extra pressure p= I

d~Gel

m

Vm df

(10)

where Vm is the molar volume. Equation (9) states that the diffusion potential is the same in both phases, i.e. there is no driving force for changing the composition. Now from equations (7), (8) and (9) one can see that the only difference between the treatments by Williams [7], and Cahn and Larche [8] and Johnson and Voorhees [10] is the right hand side of equation (8). Williams' treatment was based on the following expression (I - 2f) + f2(1 - w"lw Y ) 2 - - -----"..--- w" V £ [f (w"lw l - I) + If m

whereas Cahn and Larche considered

Johnson and Voorhees applied a rather complicated expression [their equation (21) with a minus sign]. In the above expressions Wi is a parameter containing the elastic constants of the i phase, £ the mismatch between the phases, E the Young's modulus and v Poisson's ratio. As can be seen, the elastic energy expression used by Williams degenerates to the one used by Cahn and Larche when the two phases have the same elastic properties and are isotropic. 2.2. Stability of a coherent system

The equations derived in the previous section yield only the extrema of the Gibbs energy but yields no information about the type of extrema, i.e. if it is a minimum, a maximum or a plateau point. In order to investigate the stability we should look for the change in the total Gibbs energy when the

564

ZI-KUI LID and AGREN:

TWO-PHASE COHERENT EQUILIBRIUM

fraction of the two phases are changed by a small amount 11f. We consider only changes where the diffusional equilibrium is maintained i.e. all changes are made in such a way that equation (9) is always obeyed. The state is stable if the corresponding change in Gm is positive. A Taylor expansion around some state f yields dGm I1Gm = df I1f

1 d2Gm 2 df211f +.

+2

".

(11)

If the first derivative is non-zero the second-order term is usually not important and it is sufficient to see whether the quantity (dG m/df)l1fis positive or not. When the first-order term is small or vanishes the second order term becomes important and the stability is controlled by the sign of the second derivative d 2Gm/df2. From equation (1) we can derive dG dG" dx" m _ G" GY + f m df - m m dx" df

In order to obtain equation (12) the derivative must be taken in such a way that the over-all composition is fixed. When we further apply the condition that the diffusion potential is the same in both phases and combine with the derivative of equation (2) with respect to f we obtain dG dGY dl1G el = G" - GY - (x"_xy)_m + __ m. df m m dx Y df

~

(13)

As expected, the equilibrium condition, i.e. equation (8), is identical to the condition dG m =0. df

(14)

In a similar way we get the second-order derivative of the total Gibbs energy as d 2Gm (x" - x y)2F" FY df2 = FYf + F"(l- f)

d211G~

+

dj2

(15)

where F" and FY are the second derivatives of G~ and Gin with respect to x" and xY, respectively. We now have the following possibilities: (a) If the first derivative, given by equation (13), is positive for all physical values of f, then the y one phase is stable. If this is the case no other solutions, metastable or unstable, exist. Any small amount of rx precipitation will increase the total Gibbs energy. (b) On the other hand, if the first derivative is negative for all physical values of f, then the rx one phase is stable and this is the only possible solution. (c) When the first derivative is equal to zero the two-phase mixture corresponds to an extremum in

the Gibbs energy. The stability now depends on the second order derivative. If the second order derivative, expressed by equation (15), is positive the twophase mixture represents a local minimum of the total Gibbs energy and if it is negative a maximum. If it is equal to zero the solution corresponds to a so called turning point, discussed by Johnson and Voorhees [10]. At such a point a minimum and a maximum in the Gibbs energy merge to a plateau point, and beyond that point this solution does not exist. It is also possible to have another type of turning point. This may be the case if, at some composition, a one-phase structure is stable. Consider for example a stable one phase y, i.e. f = O. The equilibrium condition is then given by the inequality· dGm/df > O. This minimum is sharp because if it were not for the physical constraint 0:;:; f :;:; 1 it would have been possible, at least mathematically, to reduce the Gibbs energy further. If, upon changing the composition, one reaches a point where dGm/df = 0, the stability depends on the second derivative. If d2Gm/df2>0 the one phase y now represents a normal smooth minimum. As dGm/df becomes negative the minimum point will be gradually displaced, i.e. there will be a continuous change in the equilibrium fractionf as the over-all composition is changed. If, on the other hand, d 2Gmi f2 < 0 the one-phase y state would change from a sharp minimum to a smooth maximum and the y one-phase state would be unstable. The equilibrium state would then change discontinuously to some other state. This is thus a turning point because an equilibrium solution disappears. However, at this kind of turning point the condition d2Gm/df2 < 0 rather than d2Gm/df2 = 0 is valid. In the next section we will discuss the properties of the solutions in detail. From equation (15) we can now conclude that in the absence of coherency stresses the second-order derivative is always positive and the minimization does always yield a local minimum and a metastable equilibrium provided that the phases themselves are stable, i.e. provided that F" and FY are both positive. In order to find the global minimum of the total Gibbs energy one has further to compare the total Gibbs energy at all local minimum points as well as the two one-phase states in order to find the state with the lowest Gibbs energy. 3. GENERAL DISCUSSION ON COHERENT EQUILIBRIUM BY MEANS OF GIBBS ENERGY DIAGRAMS

In this section we will discuss some of the general properties which the coherent equilibrium state will exhibit regardless of the exact nature of the elastic contribution. We will first deal with the situation when there is no turning point in the system and thereafter when there are two or more turning points. It will be seen there cannot be one single turning point. Since the equilibrium and stability conditions

ZI-KUI LIU and AGREN:

TWO-PHASE COHERENT EQUILIBRIUM

under diffusional equilibrium are concerned with the Gibbs energy as functions of the fraction I it seems natural to base a general discussion on coherent equilibrium on graphical representations of these functions for the different cases. Figure I shows schematically how the total Gibbs energy as a function of the fraction I under diffusional equilibrium may look when there is no turning point. Each curve represents a fixed over-all composition. As mentioned, the equilibrium state corresponds to the derivative being zero and to a minimum, a maximum or a plateau point in Gibbs energy. In this case there is no turning point in the system because there is no composition where the first and second derivatives are both zero at the same I value. The stable state for the over-all composition corresponding to the curve a in Fig. I is the IX one phase. This can be seen because the first derivative is negative for all values off If/was not unity any increase in the amount of IX would result in a lowering of the Gibbs energy. This state also represents a global minimum. By a similar argument we can conclude that the composition represented by the curve e corresponds to a y one-phase state. For alloys with an intermediate over-all composition there is one minimum corresponding to a twophase mixture. There is no doubt that this is a global minimum of the Gibbs energy. From Fig. lone can also see that the curvature, i.e. the second derivative, at this point is positive. As the over-all content changes from the curve (a) to curve (e) the I value representing the minimum moves continuously from one to zero. It is thus clear that the equilibrium state changes continuously from single IX to single y phase as the over-all composition is changed. This is the normal behaviour observed in the absence of

o

1.0 Mole fraction of a phase

Fig. I. A schematic diagram showing the total Gibbs energy in a system without the turning point. The horizontal axis is the mole fraction of C( phase.

565

coherency. However, as pointed out by Cahn and Larche [8] and by Johnson and Voorhees [10], the phase compositions are not constant but vary with the over-all composition contrary to the situation at incoherent equilibrium. From equation (15) one can see that the second derivative may vanish if the quantity d2!!.G~/d/2 is sufficiently negative. If !!.G~ has a sufficiently strong variation with composition a turning point may emerge. At such a point both the first and second derivatives are equal to zero. i.e. we have a plateau point in the Gibbs energy. Figure 2 shows schematically the situation when there are turning points. It differs from Fig. I in that there is not only a smooth minimum but also a smooth maximum. For one particular composition those extrema merge to a plateau point where both the first and second derivatives are zero, see curve (e). Curves above the one with the plateau point are monotonous and have no extremum corresponding to a two-phase state. The solution corresponding to the smooth minimum and the two-phase state has thus disappeared. The composition having the plateau point represents a turning point. At the turning point the solution will jump from a two-phase mixture to a one-phase state. It is thus clear that compositions beyond the turning point fall inside a forbidden zone in the sense that no two-phase solution at all exists there. If we look at the left-hand side of the diagram, i.e. 1= 0, we find that at the composition corresponding to the curve (f) the one-phase y is stable and any other state is truly unstable. As the over-all content changes in such a way that we move downwards in the diagram the sharp minimum at 1= 0 will change to a maximum, which may be smooth or sharp, and the y one-phase state becomes unstable. At this composition the y one phase changes from a stable or

o

1.0 Mole fraction of a phase

Fig. 2. A schematic diagram showing the total Gibbs energy in a system with two turning points. The horizontal axis is the mole fraction of C( phase.

566

ZI-KUI LIU and AGREN: TWO-PHASE COHERENT EQUILIBRIUM

metastable to a truly unstable state. This composition thus represents a turning point of the second type, see Section 2.2(c). It is important to emphasize that Williams defined the forbidden zone as a region where the two-phase mixture has a higher Gibbs energy than the single phase. It is thus possible that there is a local minimum inside the forbidden zone. If this is the case there must also be a maximum and when moving deeper inside the forbidden region one would eventually reach the corresponding turning point. It is thus clear that Williams' forbidden regions are not necessarily regions of true instability. We now suggest that the term forbidden zone should not be used or should be used only for the region of true instability, i.e. the region beyond a turning point. In the present example, see Fig. 2, one of the turning points is atf = 0 and the other one represents a two-phase mixture. Let us now consider the metastable state and let us assume that there is a spontaneous change in the system if the conditions are changed in such a way that a stable or metastable state becomes unstable. That change will occur at different compositions depending on how the over-all composition is changed. This situation was termed as "path dependence" by Johnson and Voorhees [10]. It is possible that the two turning points both represent one-phase states or that they both represent two-phase states. It is also possible to have more than two turning points. 4. INAPPLICABILITY OF PHASE RULE

As noticed by Cahn and Larche [8] and discussed by Johnson [11] the introduction of coherency stresses will make the Gibbs phase rule invalid. Johnson assumed a special geometry where the different phases occupied shells of concentric spheres. We will now show that one obtains the same results as Johnson regardless of assumed geometry if the elastic contribution to the Gibbs energy is a function of the fraction of the phases and their compositions only, i.e. dG~ = dG~(f·, fY, . ..

,xL Xl, ... )

(16)

As emphasized by Hillert [12] the Gibbs phase rule expresses the number of potentials that can be varied independently in a system without changing the number of phases. There are n + 2 potentials in a system with n components, namely the pressure, the temperature and the n chemical potentials. An n + 2 dimensional space is thus spanned by those potentials. However, a point in this space will not in general correspond to a possible state of a system because the Gibbs-Duhem relation imposes a relation between all the potentials. Thus we can only vary n + 1 potentials independently. For a certain phase we can in principle fix n + 1 potentials and then calculate all size independent properties of the phase including the remaining dependent potential. A certain phase will thus

form an n + 1 dimensional "surface" in the n + 2 dimensional potential space. The equilibrium conditions further require that a potential must have the same value in all phases when equilibrium is established. A two-phase equilibrium is thus represented by the intersection of two n + 1 dimensional surfaces which is an n dimensional "line". By the same token a three-phase equilibrium is represented by the intersection of three n + 1 dimensional "surfaces" and is an n dimensional "point". It is thus clear that the dimensionality of a subspace representing p phases in equilibrium is n + 1- (p -1) = n + 2 - p. This dimension is the number of degrees of freedom and the ordinary Gibbs phase rule is written v =n

+2 -po

(17)

As can be seen from our general equilibrium conditions the equilibrium can be calculated by specifying n + 2 - p potentials if there are no coherency stresses. This calculation would give us the value of all n + 2 potentials in the system and the compositions of the phases. However, the amount of the phases does not enter into the equilibrium conditions. In order to calculate the amounts we have to add also n mass-balance equations, one for each component. Those equations will contain also n new variables, the over-all content of each component, which must be fixed in order to impose a well defined system. We thus have, for a given over-all content, p new unknowns and n new equations. We can now vary n + 2 - p + (p - n) = 2 potentials independently, namely pressure and temperature, i.e. v = 2, without changing the number of phases. This may be regarded as a trivial statement of the phase rule. It simply expresses the fact that for a given over-all composition we can always change the pressure and temperature a little without changing the number of phases in the system. Let us now consider how the situation is altered when coherency stresses are introduced. We have still the same equilibrium conditions. However, those will now in general contain the relative amount of the phases as new variables because the elastic contribution will in general be non-linear infand the right-hand side of equation (8) will then depend on f It is thus clear that in addition to the n + 2 - P potentials we must specify the p - 1 phase fractions in order to define the equilibrium state in the general case and there are thus p - I extra degrees of freedom, i.e. we have

v=n+2-p+(p-I)=n+1.

(18)

This equation is identical to the modified phase rule suggested by Johnson [11]. The situation is schematically shown in Fig. 3 for a binary system under fixed external pressure P and temperature T, i.e. v = n + 1 - 2 and n = 2 yields v = 1. The dashed curves represent the normal equilibrium for the unstressed system which is given by the intersection of the two one-phase lines, i.e. a point

567

ZI-KUI LIU and AGREN: TWO-PHASE COHERENT EQUILIBRIUM

1000 +----'------'---'----'----'----'----+ 900 800 700 600 500 a

f~,:~----~ Unstressed equilibrium " , ,

y

400

.'\

300 200 100

o-!L----r------r----,----,r--..,.---r>--_f_ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 xMn

Chemical Potential of B

Fig. 3. Potential diagram. The dashed lines represent the chemical potential of A as a function of the chemical potential of B for unstressed IX and I' respectively. The equilibrium between IX and I' is given by the intersection of the two lines. The solid lines represent the corresponding relations in the stressed state. with v = O. The situation for the stressed system is represented by the solid lines. Observe that there is a discontinuous change in the chemical potentials as the coherent two-phase state is entered.

5. CALCULATION OF THE F.C.C. AND B.C.C. COHERENT EQUILIBRIUM IN THE Fe-Mn SYSTEM We will now turn to a practical calculation of coherent equilibrium from thermodynamic functions. As an example we have chosen the IX and 1', Le. the f.c.c.-b.c.c., equilibrium in the Fe-Mn system which was recently assessed by Huang [13]. The ordinary phase equilibrium between the IX and y phases and its extension to lower temperatures, calculated from Huang's description, are shown in Fig. 4. For the elastic-energy contribution to the total Gibbs energy we will apply the following type of expression, which was also applied by Williams [7] and by Cahn and Larche [8] (19)

Fig. 4. The ordinary phase equilibrium of the IX and I' phases and its extension to lower temperatures in the Fe-Mn system calculated from Huang's [13) description.

dependence between the applied pressure and the phase compositions and the phase fraction f First we calculate the two phase region boundary at 600°C for all Gel where the two-phase region can exist. Then we choose one Gel value according to the lattice mismatch between the two phases and calculate the coherent equilibrium at different temperatures. The results will now be discussed in detail. 5.1. Coherent phase equilibrium at 600°C The molar Gibbs energy of the IX and y phases as functions of composition at 600°C are shown in Fig. 5. At this temperature the ordinary equilibrium = 0.0325 corresponds to the phase compositions and x~ = 0.1353, where the x's denote the mole

x:

-33

-34 -35 -36

I:l

-37

'E

where Gel is an elastic parameter representing the elastic properties of the phases. As mentioned earlier the effect of the elastic energy on the two-phase equilibrium can be taken into account formally by considering the IX phase as being under a pressure proportional to the first derivative of the elastic energy with respect to the amount of IX phase, i.e. equation (10). The PV m term is thus added to the Gibbs energy of the IX phase. In the present analysis the PARROT program developed by Jansson [14] was used for the equilibrium calculations. The flexibility of the program allows the user to introduce a

a

-38 -39 -40 -41

-42+--,---,r----r-......--,--.----y---+ o 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

xMn Fig. 5. The molar Gibbs energy of the IX and y phases as the functions of the Mn content at 600°C calculated from Huang's [13] description.

568

ZI-KUI LIU and AGREN: TWO-PHASE COHERENT EQUILIBRIUM 1000 900

,. ,. ,. ,. ,. ,.

\ \

800

\

700

\

1000

,. ,.

800

Numbers are over-all Mn content, Gel =200 llmol

600 400

\

600

!

\

500

~

!

\

~'l:a

t:l

400

a

300

y

200 0 -200 -400

200

-600

100

-800 0.05

0.10

0.15

0.20

0.25

f

XMn

Fig. 6. The calculated phase stability diagram. The solid line represents the global minimum of the total Gibbs energy, dashed and dash-dotted lines represent the turning points explained in the text. The horizontal axis is the over-all Mn content and the vertical axis is the Gel value. The upper part has been magnified in Fig. 13. fraction of Mn in the (J, and y phases, respectively. At ordinary equilibrium those compositions are independent of the over-all composition. As the coherency is introduced this is no longer so and the width of the two-phase field in the phase diagram varies with the elastic energy contribution. In Fig. 6 it is shown how the phase boundary varies with the Gel value plotted on the vertical axis. The solid lines represent the global minimum boundary of the two phase region. It should be emphasized that the phase compositions do not in general coincide with the phase boundary composition because they depend on the over-all composition. The dashed and dash-dotted lines in Fig. 6 represent the turning point and will soon be explained.

0.03

-1 000 -I--.,.--~-r-.---r---,r--r---,-""'-+ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0

0.044

Fig. 8. The first derivative of the total Gibbs energy in Fig. 7 with respect to the mole fraction of the IX phase. The numbers are the over-all Mn content. The horizontal axis is the mole fraction of the IX phase. At low Gel values there are no turning points or metastable solutions in the system. Figures 7 and 8 show the calculated Gibbs energy and its first derivative as a function of the mole fraction of (J, phase for Gel = 200 limo!. As the over-all composition XO increases the equilibrium state changes from (J, one phase to y one phase. The phase compositions and the mole fraction of (J, phase are shown as a function of the over-all composition in Figs 9 and 10. In addition Figs 9 and 10 contain curves calculated from some other Gel values which will be discussed later. As Gel reaches above a critical value, around 270 llmol in the present model, there will be a discontinuous change of the phase amount on passing the y (y + (J, boundary. As an example the Gibbs energy and its first derivative as a function of the mole fraction of (J, phase for Gel = 350 llmol is shown 1.0

-33

0.9 Numbers are over-all Mn content, Gel = 200 llmol

r--"S

-35

Numbers are Gel values

0.8

-34

f--

0.7

0.03

0.6

0.044

..e.

....

:>l

0.5

0.07

r} -36

0.09

-37

0.4

0.108

0.3

0.12

0.2 0.1

-38

o

O-t--,---,,----,--'j---,--+--.---+ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f

Fig. 7. The total Gibbs energy of the system for 200 limo!. The numbers are the over-all Mn content. The horizontal axis is the mole fraction of the IX phase.

Gel =

o

0.02.0.04 0.06 0.08 0.10 0.12 0.14 0.16 XMn

Fig. 9. The phase compositions of the IX and)' phases as the function of the over-all Mn content. The numbers are the Gel values.

ZI-KUI LIU and AGREN: TWO-PHASE COHERENT EQUILIBRIUM 0.25 +-_-'--_.L-_I...----JL...---'_----'-_-'-_-+

569

600 +----'---'----'---'----'----'---'---'---'-+ Numbers are over-all Mn content, Gel = 350 llmol

500 590

0.20 350

0.15

200

400

~

Iny

200 I--_ _~ ,/ ,/ ,/

o-F====:::::::::::::-O:O

,/

0.10

Numbers are Gel values Ina

-200

0.05

0.05

-400

o-t'--,------,.----.----r--,.---r--,...--+ o 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 xMn Fig. 10. The mole fraction of the IX phase as the function of the over-all Mn content. The numbers are the Gel values.

-600 +----.-.-----r-.-----.-.-----.-.-----.-+ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f

Fig. 12. The first derivative of the total Gibbs energy in Fig. II with respect to the mole fraction of the IX phase. The numbers are the over-all Mn content. The horizontal axis is the mole fraction of the IX phase.

point is on the solid line of the I' jy + ce boundary in Fig. 6. The I' one phase is still metastable because the formation of a small amount ce phase will increase the total Gibbs energy as can be seen from Figs II and 12. As the over-all concentration becomes even lower there is another turning point where the I' one phase will change from being metastable to truly unstable. This point is on the dash-dotted line which emerges on the left side of the I' jy + ce phase boundary of Fig. 6. As the over-all concentration continues to decrease, the amount of ce phase in the two phase mixture will increase until there is no I' phase left. In this case there is no discontinuous change of phase amount on the cell' + ce boundary. The phase compositions and the mole fraction of ce phase versus the over-all composition are shown in Figs 9 and 10. One can see the jumps of the phase composition and the phase amount discussed in the previous -34500 +---I.._.!--l-_.L--L_'---'----''----'--+ paragraph. Numbers are over-all Mn content. Gel = 350 llmol If we assume that a metastable state is maintained and that a spontaneous transition occurs first on -35000 passing into a region of instability we can see that the structure for a given over-all composition will depend on whether this composition is approached from the left or from the right. As we mentioned before this phenomenon was termed as "path dependent" by Johnson and Voorhees [10]. When the Gel increases above another critical value, which is around 550 J Imol, there will be a discontinuous change of phase amount on both the I'll' + ce and the cell' + ce boundaries. The upper region of Fig. 6 has been enlarged and is shown in Fig. 13. -37000 +--.--r--r-,-----r-r--r---,r--.--__+_ The Gibbs energy and its first derivative as a function o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 of the mole fraction of ce phase for Gel = 590 Jlmol is f Fig. II. The total Gibbs energy of the system for shown in Figs 14 and 15, respectively. As the over-all concentration changes downwards from the I' oneGel = 350 limo!. The numbers are the over-all Mn content. The horizontal axis is the mole fraction of the IX phase. phase region one first has the kind of behaviour as

in Figs 11 and 12, respectively. Now there are turning points in the system. As mentioned a turning point represents the transition from the metastable to the unstable state or vice versa. In Figs II and 12 one can see that, as the over-all concentration decreases from I' one phase, a turning point emerges in the two-phase region around XO = 0.09 which is on the dashed line of the right side of the I' jy + ce phase boundary in Fig. 6. At that over-all concentration the two phase mixture changes from being unstable to metastable. The I' one phase still represents the global minimum of the total Gibbs energy. As the over-all concentration decreases further one will reach a point where the two phase mixture has the lowest total Gibbs energy. The global equilibrium state will then jump from the I' one phase to the two phase mixture. This

570

ZI-KUI LIU and AGREN: TWO-PHASE COHERENT EQUILIBRIUM 720

400

/A

700

1

300

Number.; are over-an Mn content, Gel = 590 J/mol

'I

680

,

y(a)

a(y)

660

,

/11 I 1

,

'0

~

.

640

,I I

~

~

620

I

I

600

,

'./

580

"

"

;::;

~la

./1

I I

".'/

I" I

]

y(a,a+y)l/

I

a (y,a+y)

200

,I

y(a+y)

I I

ro n

00

~

~

~

~

00

X ,·10- 3 Mn

Fig. 13. The upper part of Fig. 6 magnified. The phase states inside parentheses represent metastable equilibria. described in the previous paragraph, i.e. a turning point emerges inside the one-phase field where y is globally stable and the two-phase mixture changes from being unstable to metastable. As the over-all composition is further decreased we cross the dashdotted curve and the rx -phase state now becomes metastable while the one phase y still represents the globally stable state. Upon crossing the solid line, i.e. the yIy + rx boundary, the y one phase becomes metastable and the coherent mixture becomes globally stable. The rx one-phase state is still a metastable state. The two phase field is very narrow at this high Gel value and only a small decrease in Mn content is needed in order to cross the next solid line, the rxly + rx boundary, where the rx one-phase state becomes globally stable and the two-phase structure becomes metastable. Upon decreasing the Mn content by a very small amount the two-phase structure becomes .34500 -f----"-_.l..----L_.l.-.---'--_L-....L.----l_...L.-+ Number.; are over-aU Mn content, Gel

=590 J/mol

-35000 -r---__

] ;::; -35500

1------0~.06~5

0.065

0.052

540 +-----f'----r--r-r-'-r-----,r--+--,--.--+ ~

0.0786

-400

I I

m n

0.084

-300

Y

I

0.09

-100

I

560

..;.

0

-200

I

"

100

J

0.0786

-36000 L_-----:::.:O.O:::.:84:..:......--------j 0.09

-36500 -r----r--r--r-,----.--r-.,-----,--,--+ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f

Fig. 14. The total Gibbs energy of the system for Gel = 590 J/mo\. The numbers are the over-all Mn content. The horizontal axis is the mole fraction of the Cl phase.

0.045

·500 -1--.-----.-...---.-----.-,=::::::;::::=:::::;=-...---+ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f

Fig. 15. The first derivative of the total Gibbs energy in Fig. 14 with respect to the mole fraction of the (X phase. The numbers are the over-all Mn content. The horizontal axis is the mole fraction of the (X phase. unstable, which is on the dashed line on the left side of the rx/y + rx boundary. We then enter a regime where the rx one-phase state is globally stable and the y one-phase state is metastable. From Fig. 6 we can see that this regime extends down to an Mn content around 0.052 where we cross the dash-dotted line denoting the limit of stability of the one phase y. Beyond that point the one phase rx represents the only solution. As the Gel increases the two phase region shrinks, see Fig. 6. When it reaches above a critical value the two-phase region representing the global minimum of the total Gibbs energy will disappear. This happens at Gel = 593 J/mol at this temperature. This critical point was called the Williams point by Cahn and Larche [8]. Let us now consider the situation for the composition corresponding to the Williams point in some more detail. It is easy to understand that this composition must be the same as the so-called To composition where the Gibbs energy curves of the two phases intersect. Above the Williams point the Gibbs energy of the coherent two-phase mixture always falls above the Gibbs energy of the one-phase states. Below the Williams point the Gibbs energy of the two-phase mixture has a global minimum for some over-all compositions. As the Williams point is approached by increasing the Gel value, the Gibbs energy level of this global minimum will approach that of the two one-phase states. The two one-phase states have the same Gibbs energy because this is at the To composition. Exactly at the Williams point the two one-phase states and the coherent two-phase mixture all have the same Gibbs energy. As we increase Gel further the minimum corresponding to two-phase mixture will be higher than the one-phase states and represents a metastable equilibrium. As we continue to increase the Gel value the uppermost dashed line is crossed and

571

ZI-KUI LID and AGREN: TWO-PHASE COHERENT EQUILIBRIUM the local minimum disappears and the two-phase mixture becomes truly unstable. It is interesting to notice, see Fig. 13, that the two-phase mixture can be metastable above the Williams point. In fact, there is a metastable Williams point, see point A in Fig. 13, that marks the end of metastability. 5.2. Coherent phase equilibrium for Gel = 1 kllmol

It remains to be discussed how large Gel is in the present case. If we adopt the expression given by Cahn and Larche [8]. 2 EVf GeI = _ _

(20)

I-v we can calculate

VI - V" m

V~

m

3.643 - 2· 2.86 3 2. 2.86 3 = 0.022

v =0.3.

Those parameters yield Gel = 1 kJ/mol which will be used in present calculation where we study the effect of temperature. The same type of calculation as in the previous section was now repeated keeping Gel fixed and varying the temperature. The result is shown in Figs 16 and 17. The two-phase region boundary for the global minimum of the total Gibbs energy at different temperatures is shown in Fig. 16 with the ordinary phase diagram superimposed. The dashdotted and dashed lines in Fig. 16 represent the turning point and the dotted line is the so called To line where the two phases have the same Gibbs energy. The corresponding phase compositions are 1000 900 800 ~

If

700

2

600

~

500

e ~

y

400 300 200

u+y(coherenl)

100 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

xMn Fig. 16. The two-phase region boundary for the global minimum of the total Gibbs energy at coherent equilibrium for Gel = 1kJ/mol with the ordinary phase diagram in Fig. 4 superimposed. The dash-dotted and dashed lines are the turning points explained in the text. The dotted line is so-called To line. AMM 38/4-C

700

u

0

If

y

600

2

[

E

~

400 300 200 100

Fig. 17. The phase compositions corresponding to the coherent equilibrium phase boundary in Fig. 16 with the ordinary phase equilibrium superimposed. The dotted line is the so called To line.

V = 7.l·1O- 6 m 3/mol =

800

xMn

E = 2.2' 1011 N/m 2

f

900

0-f'L--,--.--......---.----.-"----r"---~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7

from following parameters

Gel

1000

shown in Fig. 17. Above the Williams point, which is at 525°C, the two phase region does not exist. The metastable Williams point is at about 550°C. As can be seen this diagram has the same topology as Figs 6 and 13. Outside the coherent two-phase field only incoherent two-phase equilibria or one-phase states are possible. The dash-dotted line denotes the limit of stability of the one phase y state. The dashed lines denote the limit of stability of the two phase mixture. From a practical point of view it is interesting to discuss the behaviour of an alloy which is initially in its y one-phase state and then quenched to a lower temperature where IX may form. Outside the coherent two-phase field and above the dash-dotted line only incoherent IX can form. This would correspond to the grain-boundary allotriomorphs which has been discussed extensively by Aaronson and co-workers [IS]. It is then tempting to identify the coherent two-phase field with the region where Widmanstatten ferrite can form because the Widmanstatten plates form coherently on the austenite. It must be emphasized though that the present calculation considers only equilibrium states and that kinetic factors are certainly important when considering the growth of Widmanstatten plates. If the y can be quenched below the dash-dotted line it becomes unstable and it is possible to form onephase IX. Such a transformation may then be diffusionless. However, it must be emphasized that all the present calculations are performed under diffusional equilibrium, i.e. equation (9) is always obeyed. In principle the diffusional equilibrium constraint can be replaced by some other constraint, e.g. one can force both the phases to have the same composition, and the Gibbs energy minimization will then yield a different result. When the diffusional equilibrium

572

ZI-KUI LIU and AGREN:

TWO-PHASE COHERENT EQUILIBRIUM

constraint is replaced it is found that the y phase can be maintained in a metastable state to still lower temperatures. This type of problem will be treated in a separate report. 6. SUMMARY

The effect of coherency stresses on two-phase equilibrium in binary alloys have been treated by a simplified method. It is assumed that the composition inside the individual phases is homogeneous and the Gibbs energy of the alloy will then be obtained by adding the strain energy due to the stresses to the Gibbs energy of the unstressed system. The equilibrium conditions are derived by minimizing the total Gibbs energy with respect to phase compositions and phase fractions. The resulting equations have been discussed and compared with the more general approach by Larche and Cahn and Johnson and Alexander. The equilibrium conditions and the stability of the coherent equilibrium state have been discussed in detail by the use of Gibbs energy diagrams. The method is applied to the b.c.c.-f.c.c. equilibrium in the Fe-Mn system where the thermo-

dynamic functions have been recently assessed. A coherent phase diagram has been calculated and discussed on a tentative basis. Acknowledgements-The authors are indebted to Professor Mats Hillert for many valuable suggestions. This is a part of a project which is sponsored by the Swedish board for technical development.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. II. 12. 13. 14. 15.

J. W. Cahn, Acta metall. 10 907 (1962). P.-Y. Robin, Am. Mineralog. 59, 1286 (1974). F. Larche and J. W. Cahn, Acta metall. 21,1051 (1973). F. Larche and J. W. Cahn, Acta metall. 26,53 (1978). F. Larche and J. W. Cahn, Acta metall. 26, 1579 (1978). R. O. Williams, Metall. Trans. llA, 247 (1980). R. O. Williams, CALPHAD 8, I (1984). J. W. Cahn and F. Larche, Acta metall. 32,1915 (1984). W. C. Johnson and J. I. D. Alexander, J. appl. Phys. 50, 2735 (1986). W. C. Johnson and P. W. Voorhees, Metal. Trans. 18A, 1213 (1987). W. C. Johnson, Metal. Trans. 18A, 1093 (1987). M. Hillert, Int. Metal Rev. 30,45 (1985). W. Huang, CALPHAD 13,234 (1989). B. Jansson, Trita-Mac-0234. Royal Inst. Tech., Stockholm, Sweden (1984). J. R. Bradley and H. I. Aaronson, Metall. Trans. 12A, 1729 (1981).