Thermodynamics of phase transformations in steels

2

Thermodynamics of phase transformations in steels J. Å g r e n, Royal Institute of Technology (KTH), Sweden

Abstract: The basics of thermodynamics are reviewed with special attention to phase transformations. The distinction between internal and external variables is emphasized and the general equilibrium conditions are derived from the combined first and second law. The concepts of entropy production and driving force as well as stability are discussed. The calculation of thermodynamic properties and phase equilibria is considered and the Calphad method is briefly reviewed, including modeling of substitutional and interstitial disorder. The thermodynamic bases of phase diagrams are examined, and finally, the effect of interfaces, fluctuations and thermodynamics of nucleation are reviewed. Key words: internal and external variables, entropy production, Gibbs energy, driving force, Le Chatelier’s Principle, phase diagrams, interfacial energy, fluctuations, nucleation.

2.1

Introduction: the use of thermodynamics in phase transformations

It is often taken for granted in textbooks that thermodynamics only applies at equilibrium, i.e. the condition when nothing more can change. From the viewpoint of analyzing phase transformations, where the change itself is of primary interest, thermodynamics would then seem rather useless. However, it is easy to demonstrate that such a conclusion is incorrect. At equilibrium, various properties of the system can be measured experimentally, e.g. volume, temperature, composition, amount and composition of the different phases present, etc. A liquid may be supercooled substantially below its solidification point and kept there long enough to allow various thermodynamic measurements. The properties of cementite in plain carbon steel may be evaluated although it is not an equilibrium phase. The decomposition into iron and graphite is simply too sluggish and occurs only for very high carbon contents and long heat treatment times. These examples hold for metastable equilibria where the stable conditions have not yet been reached due to the finite rate of kinetics. The fact that one can perform exactly the same thermodynamic measurements a little bit outside equilibrium and extrapolate these measurements rather far into the non-equilibrium regime indicates that 56 © Woodhead Publishing Limited, 2012

Thermodynamics of phase transformations in steels

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thermodynamics would also apply considerably outside equilibrium. But how far outside would it apply? In this chapter we will demonstrate how thermodynamics may be applied as far as needed to solve certain problems. We shall also state the rules which must be obeyed for a sound analysis. Once we have accepted that thermodynamics may be applied regardless of whether there is equilibrium or not, we have to consider the concept of local behavior, i.e. thermodynamics may be applied locally in a small volume element. This has as a consequence that, even though it is fruitless to apply thermodynamics to a complex body as a whole, it may still be helpful to apply thermodynamics locally to its parts and thus attach certain properties to each point in the body. Sometimes it is necessary to also include some influence of non-local effects, i.e. the thermodynamic properties may depend on neighboring regions and gradients have to be taken into account. The concepts of non-equilibrium states and local behavior (possibly including some non-local behavior) serve as the basis for thermodynamics of phase transformations. As we shall see, it allows the calculation of driving forces for various changes of a non-equilibrium system. A modern and thorough treatise of the thermodynamic basis of phase transformations is found in the recent book by Hillert (2008). The present chapter is largely inspired by that book and personal experience in teaching phase transformations and thermodynamics to graduate research students.

2.2

External and internal variables

2.2.1 Action and reaction A given system may be influenced from the outside in several different ways. These interactions involve • • •

heat work composition change.

A system with constant composition is called a closed system. If the composition may be changed it is called an open system. A closed system that is also thermally isolated, i.e. no heat is exchanged with the outside, is left completely alone and is called adiabatic or isolated. We may change the composition by adding (or removing) certain components. The definition of components is somewhat arbitrary but should reflect all possible composition changes we are interested in. In physical metallurgy it is usually most convenient to use the chemical elements as components but it may sometimes be more convenient to choose stoichiometric compounds, like oxides. The system Fe-O could thus be fully represented by the components Fe and O but FeO and Fe2O3 are also possible, although we would then have to accept the

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Phase transformations in steels

possibility of negative concentrations outside the interval between FeO and Fe2O3. Although this is perfectly thermodynamically consistent, it may lead to some difficulties when plotting phase diagrams using potentials as axes. In a system with c components we may thus perform the external actions in c + 2 different ways. These external actions can generally be characterized by well-defined variables, for example the volume and the amount of a certain component. As these variables are controlled from the outside by the experimentalist or the processing conditions they are called external variables. Thus, generally there will be c + 2 external variables. Once the external action is over and the external conditions are kept constant, the reactions inside the system will proceed until a new state of equilibrium is established and there will be no more changes unless there are new external actions on the system. The equilibrium state of the system is thus a function of c + 2 external variables which can be chosen in many different ways. We may thus call these variables external state variables. For example, the equilibrium state of a piece of iron would be a function of three external state variables, i.e. the number of moles of iron, pressure and temperature. As far as the properties of a material are concerned the absolute amount of a substance is not of interest and without loss of generality we may consider 1 mol of iron and only two variables, e.g. pressure and temperature, are then needed to uniquely define the equilibrium state. This fact is expressed by Gibbs phase rule, discussed in Section 2.4.5, and it is reflected in the phase diagram of pure iron which is shown in Fig. 2.1. All phase diagrams in the present chapter have been calculated using the Thermo-Calc code (Andersson et al., 2002). 3000 L 2500

2000 T (K)

a g

1500

1000 a

500

0

0 2 109

4

6

8

e

10 12 14 16 18 P (Pa)

2.1 Phase diagram of pure iron.


20


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In a system that is not yet in equilibrium, more variables are usually needed to characterize it completely. These variables cannot be controlled directly from the outside but their values evolve as a consequence of internal reactions in the system. Consequently they are called internal variables. Familiar examples in an alloy are the fraction and composition of different phases, the number or fraction of vacancies, the state of chemical and magnetic order, etc. At equilibrium the internal variables attain certain values which depend on the given external conditions. As these variables characterize the state of the system at a given instant, they are called internal state variables. The distinction between external and internal variables is essential but is unfortunately not always fully appreciated. As a simple example, one may consider the thermal vacancies of a pure metal at elevated temperatures. Suppose a piece of pure iron is taken from room temperature up to close to the melting point at atmospheric pressure and kept there for some time. In this case it is natural to consider the temperature T, pressure P and number of moles of iron NFe of the piece as the external variables. At the beginning there are few vacancies but their number will gradually increase and level out at a much higher value corresponding to equilibrium at the high temperature. The number NVa of vacancies is an internal variable. In principle it can have any value depending on the history of the material but for given P, T and eq eq NFe it has a well-defined value at equilibrium, i.e. N Va = N Va (P, T, NFe)

2.2.2 Extensive and intensive variables In the example of thermal vacancies, the total number of vacancies was considered. However, provided that the system is sufficiently large to have reasonable statistics, the total number of vacancies will be proportional to the size of the system. It is thus rather inconvenient to use NVa as an internal variable. It is more convenient to consider the fraction of lattice sites that will be vacant, i.e. the so-called site fraction yVa which is given by yVa = NVa/(NVa + NFe) and, for a given value of the site fraction, NVa is thus proportional to, NFe, i.e. NVa = NFeyVa/(1 – yVa) Thus yVa is an example of an intensive variable because its value will not change if a larger or smaller part of an equilibrium system is considered. On the other hand NVa is an extensive variable and so also is the system size defined as NFe. An extensive variable can thus be external or internal and the same holds for an intensive variable. In the above case T and P are both intensive and external variables. There are two kinds of intensive variables: potentials and densities. The first type will be discussed in more detail in Section 2.4. The second type is obtained by normalizing an extensive variable to the size of the system. The customary term density stems from the important example of mass density which is given by the mass of the system divided by its volume. Some examples are given in Table 2.1. Molar quantities are particularly useful


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Table 2.1 Examples of intensive variables obtained by normalizing extensive quantities to the system size System size as Mass m Volume V

Number of moles of j, Nj

Internal energy U

Volume V

Density 1 r = m/V

Concentration cj = Nj / V

Energy density UV = U/V

Number of moles N

Molar mass m/N

Mole fraction xj = Nj / N

Molar energy Um = U/N

Molar volume V/N = Vm

examples of densities and are denoted by a lower index or subscript m, e.g. the molar volume Vm. Traditionally the molar content is called mole fraction and denoted xj rather than Njm which would be more logical. There are many more possibilities and Table 2.1 lists only a few common examples. It is inconvenient to use extensive variables when we consider the local thermodynamic behavior because we would then have to specify the size of each volume element considered. On the other hand, we will not have that difficulty if we use intensive variables because we may imagine that these generally would vary with location in a system, i.e. they would be field variables. Integration over the whole system of a density-type of variable would then give the corresponding quantity for the whole system, e.g. its total energy or mass.

2.3

The state of equilibrium

When the external conditions are kept constant, systems usually evolve towards stationary states where the internal variables no longer change. Such a state is an equilibrium state provided that the external conditions do not impose a constant flow of matter or energy. The latter would be the case, for example, if different sides of the system are kept at constant but different temperatures. In that case equilibrium can never be attained and the stationary state rather corresponds to a steady-state solution of the appropriate transport problem. It should be mentioned, though, that in systems which are far from equilibrium sometimes not even stationary solutions are possible, but there will be solutions with oscillatory time dependence. As we shall explain in Section 2.4, the equilibrium state fulfills welldefined thermodynamic criteria. For given external conditions equilibrium is obtained for the combination of internal variables that gives an extremum in the appropriate thermodynamic function. For simplicity we shall here first consider external conditions where P, T and composition are kept constant. In that case we shall see in Section 2.4 that the appropriate thermodynamic function is Gibbs energy G. For an internal variable x, equilibrium is then given by the minimum in G (see Fig. 2.2).



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G

x

2.2 Equilibrium for the value of the internal variable x that gives the lowest Gibbs energy. G

x

2.3 Change in Gibbs energy during a phase transformation. G

G

x (a)

x (b)

2.4 (a) Metastable equilibrium, (b) unstable equilibrium (critical state).

According to the second law of thermodynamics, see Section 2.4, an ongoing spontaneous process is characterized by a steady decrease in Gibbs energy (see Fig. 2.3). On the other hand the situation may be as in Fig. 2.4(a). The equilibrium state is only a local minimum and there is a deeper minimum at


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another value of x. However, in order to reach that value, the system would first have to pass a maximum, Fig. 2.4(b), before it could steadily decrease its Gibbs energy towards the more stable equilibrium. The required increase in Gibbs energy violates the second law of thermodynamics and could only happen by means of a sufficiently large fluctuation (see Section 2.8). The system thus has some stability and the equilibrium in Fig. 2.4(a) is said to be metastable. The maximum in Fig. 2.4(b) is an unstable equilibrium because even an infinitely small fluctuation would lead to a decrease in Gibbs energy. An unstable equilibrium is often called a critical state.

2.4

The combined first and second law – its application

2.4.1 First and second laws of thermodynamics At a given instant the internal energy of a system U depends only on its internal state. Such a quantity is called a state quantity. Of course the internal state may depend on the history unless the system is at equilibrium, and thus U may indirectly depend on the history of a non-equilibrium system. For example, a highly deformed piece of steel will have a higher concentration of defects and, before these have been annealed out, U indirectly depends on the thermo-mechanical history. The first law states that U is a conserved quantity and thus obeys a conservation equation. In a closed system its value may be changed by adding or removing heat and work to or from the system. This is usually written

dU = dQ + dW

[2.1]

Here dQ and dW stand for amount of heat and work added. If we consider pressure-volume work only, the work addition is expressed as dW = –PdV. This is positive for P > 0 and dV < 0, i.e. when work is added to the system. In the second law another state function, the entropy S, is introduced. However, the entropy is not a conserved quantity and may thus be produced inside the system, but can never be consumed according to the second law. For a closed system the change in entropy S is thus given by

dS = dQ/T + diS

[2.2a]

diS ≥ 0

[2.2b]

The first term dQ/T is the entropy exchanged with the outside and may be positive or negative depending on whether heat is added to or removed from the system. It vanishes for an adiabatic system. The second term diS is called the internal entropy production and is always positive for spontaneous processes inside the system. As mentioned, this is the essence of the second law. Such processes are usually called irreversible processes because they



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always take the system towards equilibrium. They never occur in the reverse way which would take the system further away from equilibrium. If diS = 0 there can be no internal reactions. This may be the case if the system is at equilibrium or if the kinetics are so sluggish that no reactions occur during the observation. In the latter case the system is said to be frozen-in. It should be observed that the entropy change could have any sign because it also depends on the sign and magnitude of dQ. S has to increase by an internal process only if the system is adiabatic.

2.4.2

The combined law

The combined law is obtained by combination of eqs [2.1] and [2.2]. i.e. dU = TdS – PdV – TdiS

[2.3]

If the system is open, i.e. there is exchange of matter with the outside, eq. [2.3] is modified dU = TdS – PdV + ∑ mkdNk – TdiS

[2.4]

where Nk denotes the number of moles of component k inside the system. The new quantity mk is called the chemical potential for component k and will be discussed in more detail shortly. equation [2.4] is the combined law in its general form. Suppose that the internal reactions can be described with internal variables xj. We can then express the internal entropy production by di S = 1 S D j dx j T

[2.5]

The quantity Dj is called the driving force for the jth internal process. It is defined as positive for spontaneous increase in xj. The introduction of the driving force allows the distinction between equilibrium, Dj = 0, and frozenin state dxj = 0. Thus we can write the combined law as dU = TdS – PdV + ∑ mkdNk – ∑ Djdxj

2.4.3

[2.6]

Natural variables

equation [2.6] implies that the internal energy is a function U(S, V, Nk, x1, x2…). However, the internal energy is also a physical quantity which in principle can be measured under various conditions, e.g. at different T and P, and we could thus express it as another function having the external variables T and P rather than S and V. nevertheless, in thermodynamics it is generally acceptable to use the same symbol to denote a given quantity although it may refer to completely different mathematical functions. We can thus write U(T,P) and U(S,V) but always keep in mind that they are completely different functions. The variables S, V, Nk are called the natural


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variables of the internal energy because they appear as differentials in the combined law. Using the natural external variables we immediately find the following partial derivatives: ∂U = T ∂S

[2.7a]

∂U = –P P ∂V

[2.7b]

∂U = m k ∂∂N Nk

[2.7c]

∂U = – D j ∂x j

[2.7d]

each derivative tells how much the internal energy changes per unit amount of an extensive quantity, i.e. entropy, volume or number of moles of k, added to the system. In physics such derivatives are called potentials. The potential mk is called the chemical potential. It should be emphasized that the potential corresponding to volume is –P. We also note that the driving force Dj should be regarded as a potential. If the natural variables are not used, one has to state what variables were kept constant during the differentiation. For example, the heat capacity at constant volume is defined as Ê ∂U ˆ = cV ÁË ∂T ˜¯ VN

[2.8]

k

As T is not a natural variable for the internal energy, one cannot be certain what the other variables are and thus it is necessary in this case to include the suffices V and Nk to denote that U is considered as a function U(T,V, Nk). Observe that the same derivative taken under constant pressure, i.e. using a function U(T, P, Nk), is a completely different quantity.

2.4.4

Different forms of the combined law

We can generalize the combined law to include more types of interactions thus: dU = ∑ Yj dXj – ∑ Dj dxj

[2.9]

Here Yj denotes the potential which corresponds to the external extensive variable Xj. For example, we may consider Xj as the electric charge and Yj would then be the electric potential. The variables Xj and Yj are conjugated to each other and are called a conjugated pair. For a system where all Xj are kept constant the energy will be constant at equilibrium because the last



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summation in eq. [2.9] will vanish. If we consider the internal process of moving Xj from one part of the system to another we find by applying Eq. [2.9] to each part that the potential Yj must be the same in both parts. This leads to the important conclusion that a system in equilibrium must have the same value of a given potential over the whole system. For a system at equilibrium with all Xj kept constant, all imaginable internal changes would lead to an increase in internal energy, i.e. the equilibrium state is a minimum in internal energy. For example, a system with fixed entropy, volume and composition has its lowest possible internal energy. All non-equilibrium states would have a higher energy. We may rearrange eq. [2.4] to obtain dS = 1 dU + P dV – 1 ∑ µk dN dN k + di S T T T

[2.10]

We can thus conclude that a system at fixed energy, volume and composition has maximum entropy at equilibrium. Usually it is difficult to arrange an experiment in order to keep the entropy constant. It is somewhat easier to arrange for a fixed volume or energy. We may rearrange eq. [2.6]: d(U – TS) = – SdT – PdV + ∑ mkdNk – ∑ Djdxj

[2.11]

The function U – TS is called the Helmholtz energy and is usually denoted with F. Again we immediately obtain the partial derivatives ∂F = –S ∂T

[2.12a]

∂F = –P P ∂V

[2.12b]

∂F = m k ∂∂N Nk

[2.12c]

∂F = –D Dj ∂x j

[2.12d]

For the Helmholtz energy the natural variables are obviously T, V and Nk, and we may conclude that at equilibrium under fixed temperature, volume and composition we will have a minimum in Helmholtz energy. In experiments it is usually more convenient to control pressure and temperature and we may rearrange eq. [2.11] to obtain d(U – TS + PV) = – SdT + VdP + ∑ mkdNk – ∑ Djdxj

[2.13]

The function G = U – TS + PV is called the gibbs energy and has T, P and Nk as natural variables. Again we obtain the partial derivatives


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∂G = – S ∂T

[2.14a]

∂G = V ∂P

[2.14b]

∂G = m k ∂∂N Nk

[2.14c]

∂G = – D j ∂x j

[2.14d]

We also conclude that at equilibrium under fixed temperature, pressure and composition there will be a minimum in gibbs energy. Consider now an infinitely small subsystem of a larger equilibrium system and then let that subsystem grow by moving its imaginary boundary. During that imaginary growth there is no entropy production because nothing is really happening inside the system and the internal energy change becomes dU = (TSm – PVm + ∑ mkxk) dN

[2.15]

where N is the size of the subsystem. At equilibrium all potentials are constant and so are the molar quantities and we may directly perform an integration to obtain U. U = TS – PV + ∑ mk Nk

[2.16]

We thus find that G = U – TS + PV = ∑ mk Nk

[2.17]

In a similar way we may introduce other thermodynamic functions having other natural variables that may be convenient for other experimental conditions. One may mention, for example, the grand potential –PV, sometimes called the Landau free energy after the russian physicist L. Landau (see Landau and Lifshitz, 1970): d(U – TS – ∑ mkNk) = d(– PV) = – SdT – PdV – ∑ Nkdmk – ∑Djdxj

[2.18]

This function is useful when one considers equilibrium at fixed temperature, volume and chemical potentials.

2.4.5

Gibbs–Duhem relation and the phase rule

If one tries to replace also the volume with pressure in eq. [2.18] one obtains for a system at equilibrium



d(– PV + PV) = 0 = – SdT + VdP – ∑ Nkdmk

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[2.19]

This equation is called the Gibbs–Duhem relation and reveals the fact that there are only c + 1 independent potentials in a system with c components. Moreover, it is clear that Eq. [2.19] can be applied to each phase and it has its own values of S, V and Nk. We have already concluded that each potential is constant at equilibrium. If there are several phases present, they must thus all have the same value of T and P, etc., if equilibrium is to be maintained. In a one-phase system there are c + 1 independent potentials. For each additional phase there is an additional Gibbs–Duhem relation and for p phases there are thus c + 1 – (p – 1) = c – p + 2 independent potentials. The number of independent potentials is called the number of degrees of freedom f and the relation f = c – p + 2 is called the Gibbs phase rule.

2.4.6 Equilibrium conditions From the different forms of the first law we see directly which function should be minimized under given external conditions. The most common examples are summarized in Table 2.2. At equilibrium nothing is happening, i.e. everything is constant. One may then question how one could know what was kept constant before equilibrium was established and what function was minimized. For example, at equilibrium there is some pressure and volume which are both constant. Was the Gibbs energy or the Helmholtz energy minimized? The answer is that it does not matter which variables were kept constant as long as the same equilibrium state is approached. Provided that all calculations are performed correctly the minimization of the different functions would all give the same result. If we know the volume and the temperature of the system it is certainly most convenient to use the Helmholtz energy but we could equally well use any other function, e.g. Gibbs energy. However, if we use Gibbs energy we have to perform the calculation iteratively and find the pressure for which the derivative ∂G/∂P = V agrees with the fixed volume. It should be emphasized that Table 2.2 lists only the most commonly used Table 2.2 Functions to be minimized at equilibrium for various external conditions Fixed variables

Function to be minimized

Name

U, V, Nk S, V, Nk T, V, Nk T, P, Nk S, P, Nk U, V, µk T, P, Nk, µj

–S U F = U – TS G = U – TS + PV = F + PV H = U + PV –PV = U – ∑ Nk µk G – ∑ N jµ j

Negative entropy Internal energy Helmholtz energy Gibbs energy Enthalpy Grand potential


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conditions and corresponding functions. There are many more possibilities that are all derived from the various forms of the combined law but most of them have limited use. However, the last possibility in Table 2.2, which does not have a name, is actually quite useful. It can be used to calculate the equilibrium state of a system under fixed temperature and pressure where the potentials of some of the elements, but not all, are fixed. For the remaining elements their content is fixed. This may be the situation when a certain steel is equilibrated in an atmosphere of given nitrogen, oxygen or carbon activity.

2.4.7

Evaluation of the driving force

In Section 2.4.4 different expressions for the driving force Dj were derived from the various forms of the combined law. Its use will now be demonstrated for the case of constant T, P and composition, which is most important from the practical point of view. For a non-equilibrium system, eq. [2.13] then reads: dG = – ∑ Djdxj

[2.20]

Consider now a process where the x variables change from an initial state to a final state. Even though the driving forces may vary in a complicated way during the integration, the integral –S

final

Ústar start

ta tart D j dx j = G finall – G sstart

[2.21]

is easily evaluated as the difference in gibbs energy between the two states. We call this quantity the integrated driving force. As another example of the use of eq. [2.20], we now consider the formation of a b particle in an a matrix. We consider dNk moles of element k taken from a and transferred to b, i.e. dNk = –dNka = dNkb. In principle, one or several elements may be transferred. The gibbs energy of the whole system which has a fixed overall composition is given by G = Ga + Gb

[2.22]

We chose as our single internal variable the number of moles of b that has formed and the driving force from eq. [2.14d] thus yields Ê a ∂N N ka ∂G b ∂N N kb ˆ D = – ∂G = – ∂Gb = – S Á ∂Ga + k Ë ∂N ∂x ∂N N N k ∂N N b ∂N N b ˜¯ N kb ∂N

[2.23]

From eq. [2.14c] we introduce the chemical potentials and applying dNk = –dNka = dNkb we can write eq. [2.23] as



D = S (mak – m kb ) k

∂N Nk ∂N Nb

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[2.24]

It remains to be discussed what the composition of the material transferred from a to b is. In the special case where the composition of b cannot change, the transferred composition must be the same as in b and Nk = xkb Nb, i.e. D = S (mak – m kb ) xkb

[2.25]

k

Applying eq. [2.17], eq. [2.25] may also be written D = S xkb mak – Gmb . This k

well-known relation or eq. [2.25] can be found in many textbooks, e.g. the very recent and extensive treatise by Aaronson et al. (2010).

2.4.8

Stability conditions

The different types of equilibrium were discussed in Section 2.3. They can be stated in a more precise mathematical form by investigating the combined law. For example, at given T, P and composition we use the gibbs energy and apply eq. [2.20]. However, at equilibrium all driving forces vanish and we have to perturb the system in order to check the stability. Consider first a case with a single internal variable which is only perturbed a small amount Dx from equilibrium. We find 2 DG = 1 ∂ G (Dx )2 2 ∂x 2

[2.26]

For stable equilibrium the perturbation must lead to an increase in gibbs energy, i.e. the second derivative must be positive. We can apply eq. [2.26] to a case where we consider as an internal process the transfer of component k from one part of the system to another. In Section 2.4.4 we used this ‘thought experiment’ to show that a potential must have the same value everywhere in an equilibrium system. now we also ask under what conditions such an equilibrium is stable. We identify x with Nk and find immediately that for a stable system the second derivative of G with respect to Nk must be positive. From the expression for the chemical potential we find that this condition may be written Ê ∂m k ˆ >0 ÁË ∂N ∂N k ˜¯ P,T ,N

[2.27]

j

The chemical potential has no natural variables and thus we have added what variables are fixed as suffices. In a stable system at fixed temperature and pressure the chemical potential of a component must always increase


70


when it is added to the system. By looking at eq. [2.9], which concerns U as a function of all extensive variables Xi, we may by similar considerations deduce that the following more general condition Ê ∂Yk ˆ >0 ÁË ∂X ∂X k ˜¯ X

[2.28]

j

must be obeyed in a stable system. Starting with eq. [2.9] we can now exchange an extensive variable with the corresponding potential and we find stability conditions of the type Ê ∂Yk ˆ >0 ÁË ∂X ∂X k ˜¯ Y X i

[2.29]

j

In the above derivative Yi denotes the potentials and Xj the extensive quantities, which are kept constant during the differentiation. We thus find that, regardless of whether potentials or extensive quantities are constant, a potential has to increase when the conjugated extensive quantity is added to a stable system. For example, we may immediately write the following stability criterion Ê ∂T ˆ >0 ÁË ∂S ˜¯ P, X

[2.30]

j

where Xj represents all the components. This may be written in a more transparent form in terms of cP, the heat capacity at constant pressure and defined as (∂H/∂T)P. From Table 2.2 we see that the enthalpy H may be written as a function of T and P, although these are not its natural variables, if we use the relation H = G + TS and the fact that S = – ∂G/∂T (see eq. [2.14a]), then Ê ∂H ˆ cP = Á ˜ = T ∂S ∂T Ë ∂T ¯ P Ê ∂T ˆ 1 T ÁË ∂S ˜¯ = ∂S = c > 0 Ê ˆ P P ÁË ∂T ˜¯ P

[2.31]

[2.32]

In a closed stable system the heat capacity is positive. Another condition is Ê ∂ – Pˆ ÁË ∂V ˜¯ > 0 T


[2.33]


71

By introducing the isothermal compressibility defined as Ê ∂V ˆ kT = – 1 Á ˜ V Ë ∂P¯ T

[2.34]

eq. [2.33] may be rephrased into its more familiar form 1 >0 Vk T

[2.35]

Thus the compressibility must be positive in a stable material. It should be observed that the stability criteria only involve conjugated variables. Consequently there is no requirement on, for example, the thermal expansion which is defined as a = (∂V/∂T)/V because V and T are not conjugated. In fact, although the thermal expansion is usually positive, there are important examples of substances with negative values, e.g. rubber (see, for example, Pellicer et al., 2001). The following general relation may be shown Ê ∂Yk ˆ ÁË ∂X ∂X k ˜¯ Y Y ºY

k –1 Xk +1…

1 2

Ê ∂Y ˆ ≤ Á k˜ Ë ∂X ∂X k ¯ Y Y º ºX 1 2

k –1 Xk +1…

[2.36]

If a potential is fixed rather than the conjugated extensive variable, the system is closer to instability. In a system with c components the most sensitive stability condition is thus given by, for example Ê ∂mc ˆ ÁË ∂N ∂N c ˜¯ TPm

≥0 2 …mc –1 N1

[2.37]

If also m1 is kept constant, the derivative will vanish due to the gibbs–Duhem relation. For alloys, eq. [2.37] may be given in a more convenient form. Introducing Gm as a function of the independent mole fractions x2, x3, etc., taking x1 as a dependent quantity and defining ∂G fi = mi – m1 = ÊÁ m ˆ˜ Ë ∂xxi ¯ PTx PT j

[2.38]

the limit of stability is written using the determinant

f22 … f2C    =0 fC 2 º fCC

[2.39]

where Ê ∂f ˆ fij = Á i ˜ Ë ∂xx j ¯

[2.40]


72


For a binary system this gives the well-known expression defining the socalled spinodal (see, for example, Cahn, 1961): d 2 Gm =0 dx B2

[2.41]

Inside the composition and temperature regime where this second derivative is negative, an initially homogeneous alloy may decompose by means of a spinodal mechanism.

2.4.9

Le Chatelier’s Principle

equation [2.36] has some quite interesting consequences. It was already pointed out that the driving force should be regarded as a potential conjugated to some internal variable. We could then write eq. [2.36] as: Ê ∂Yk ˆ Ê ∂Y ˆ ≤ Á k˜ ÁË ∂X Ë ∂X ∂X k ˜¯ D ∂X k ¯ x j

[2.42] j

We have considered a stable system and so both derivatives are thus positive. The derivative on the left-hand side is taken under a constant driving force, e.g. Dj = 0, which corresponds to equilibrium. The derivative on the righthand side is taken under constant xj, i.e. under frozen-in conditions. The important interpretation of eq. [2.42] is as follows. If an equilibrium system is exposed to an infinitesimal change dXk of an external condition such that there is no time for internal reactions, i.e. frozen-in conditions, the conjugated potential Yk will increase accordingly. When equilibrium is established the initial increase has diminished. For example, if the volume is suddenly compressed pressure will increase, but as equilibrium is approached pressure will decrease and approach a constant value which is larger than the original value. equation [2.41] may be regarded as a mathematical formulation of Le Chatelier’s Principle. A graphical illustration is given in Fig. 2.5. For a more thorough discussion the reader is referred to Hillert (1995).

2.5

The calculation of thermodynamic properties and equilibrium under fixed T, P and composition

In order to perform thermodynamic calculations for real materials it is necessary to know how their thermodynamic properties vary with internal and external conditions. In the previous section it was demonstrated that both equilibrium and non-equilibrium properties, driving forces and stability may be derived from a thermodynamic state function, such as gibbs energy. In most cases it does not matter what function one chooses but since the



73

D=0

Yb

x = const

Le Chatelier modification

dX b

Xb

2.5 Graphical illustration of Le Chateliers principle. The solid curve marks the relation between Yb and Xb under equilibrium.

majority of practical problems involve temperature, pressure and composition as external conditions, gibbs energy is most useful. nevertheless, it should be emphasized that some special problems cannot be analyzed with all state functions. For example, the critical point in the temperature-pressure diagram of a fluid cannot be very well handled using the Gibbs energy but needs a function that has volume as a natural variable, e.g. internal energy, entropy or Helmholtz energy. In the following we will limit the discussion to problems where gibbs energy is sufficient. We thus need a function: Gm = Gm(T, P, x1, x2 … x1, x2…), where the xs denote the mole fractions and the xs are various internal variables. From such a function we may derive all thermodynamic quantities of interest (see eq. [2.14]). For example, Vm = ∂Gm/∂P, Sm = ∂Gm/∂T a = (∂2Gm/∂P∂T)/∂Gm/∂P, cP = – T∂2Gm/∂T2, etc. The chemical potential is obtained as

m k = Gm +

∂Gm ∂Gm – S xj ∂xxk ∂xx j

[2.43]

and the driving forces Dj = –

∂Gm ∂x j

[2.44]

Here we have assumed that the xs are given as intensive variables. The above equations hold for a one-phase as well as a multi-phase system. In a database it is most flexible to store a function Gma = Gma (T, P, T, P, x1a, x2a… x1a, x2a…) for each separate phase a. The gibbs energy of a multiphase system is then obtained as © Woodhead Publishing Limited, 2012

74


G = S N a Gma (T , P, x1a , x2a … x1a , x2a …) a

[2.45]

where the summation is performed over all phases in the system and Na is the number of moles of atoms in each phase. The mole fractions x, the xs and the N as are now all internal variables. However, they are not independent because the overall composition of the system is fixed as expressed by the mass balances N k = xk N = S N a xak a

[2.46]

The mole fractions are also related because for each phase a c

S xa = 1 k =1 k

[2.47]

The equilibrium is now calculated by finding the minimum in Eq. [2.45] under the constraints given by eqs [2.46] and [2.47]. It should be noted that often there are constraints on some or all of the x values. The minimization can be performed with many different methods and we shall not discuss these further in this chapter.

2.6

Gibbs energy of phases in steel – the Calphad method

In order to find expressions for the Gibbs energy of the individual phases the so-called Calphad method (Saunders and Miodownik, 1998; Lukas et al., 2007) has been found most valuable. In this method one formulates models for the individual phases. Such models can have varying degree of sophistication, ranging from simple regular solutions to cluster methods. So-called Monte Carlo methods should also be mentioned although they are somewhat different in character as they are not based on expressions of suitable state functions. nevertheless, they are very powerful for some problems but due to laborious computations they are typically applied to very simple alloy systems and they will not be discussed further here. As a rule, the simpler a model is, the more complex alloys, in terms of number of elements and phases, may be analyzed.

2.6.1

Modeling of disorder and entropy

All methods based on mathematical expressions for a state function have Boltzmann’s classical expression for the entropy as a starting point: S = kB ln W

[2.48]



75

Here W is the number of different ways a system can arrange itself under given external conditions and kB is Boltzmann’s constant. For example, in a crystal with N atoms and NVa vacant lattice sites, there is N + NVa lattice sites and the vacancies may be distributed on those sites in W =

(N + NVa )! N ! NVa!

[2.49]

different ways if they are randomly distributed. Inserting eq. [2.49] into eq. [2.48] and applying Stirling’s approximation ln(x!) @ x ln x – x, which is justified because N and NVa are typically very large, we obtain S@–

NkB ((1 – yVa ) ln (1 – yVa ) + yVa Va ln yV Va a) (1 – yVaa )

[2.50]

where yVa is the fraction of vacant lattice sites. The quantity N/(1 – yVa) is thus the number of lattice sites including the vacant ones. The entropy per mole of lattice sites is thus Sm = – R((1 – yVa) ln (1 – yVa) + yVa ln yVa)

[2.51]

where R, the gas constant, is simply kB multiplied by the Avogadro number. The gibbs energy for a system with vacancies should also contain the extra energy associated with formation of a vacancy as well as the pressure volume work to expand the lattice and some changes in vibrational modes around the vacancy. We may lump all these contributions in a parameter °GVa which we could formally define as the change in Gibbs energy to form the first vacancy counted per mole of vacancies. If we assume that it is constant we obtain for a system with 1 mole of lattice sites Gm = yVa °GVa + RT((1 – yVa) ln(1 – yVa) + yVa ln yVa)

[2.52]

In order to use the Gibbs energy to find the equilibrium state we have to consider a fixed number of atoms rather than lattice sites and we should thus look at the gibbs energy per mole of atoms which is Gm/(1 – yVa). The equilibrium fraction of vacancies is thus obtained as ∂[Gm /( //(1 (1 – yVVaa )] (1 ∂Gm Gm 1 = + =0 ∂∂yyVa (1 – yVa ) ∂y (1 ∂yVa (1 – yVa )2

[2.53]

equation [2.53] may be rewritten Gm +

∂Gm ∂Gm – yVa =0 ∂yyVa ∂yVa ∂y

[2.54]

By comparison with Eq. [2.42] we find that the expression on the left-hand side is actually the chemical potential of vacancies, i.e. mVa = (∂G/∂NVa)


76


T,P,N.

At equilibrium the chemical potential of vacancies thus vanishes. By applying Eq. [2.54] to Eq. [2.52] we find mVa = °GVa + RT ln yVa

[2.55]

and the equilibrium fraction of vacancies is thus eq

yVa = exp (–°GVa/RT)

[2.56]

In fact, eq. [2.51] may be applied to any system where there is a random mixture of constituents. The only problem is to define the constituents, i.e. to find those things that mix randomly. From Eqs [2.44] and [2.53] it is clear that mVa = (∂G/∂NVa)T,P,N = – DVa serves as the driving force for creation or annihilation of vacancies and we may express it in terms of the equilibrium vacancy content by combining eqs [2.55] and [2.56] as DVa = –RT ln(yVa/ eq yVa). The entropy production DVadyVa is thus always positive as the system approaches equilibrium.

2.6.2

Regular solution type models

Consider, for example, a substitutional alloy, where one may assume a random mixture of its c components. expressed per mole of atoms we would then have c

c

k =1

k =1

Gm = S xk °Gk + R RT T S xk ln xk + E Gm

[2.57]

If we consider only two components and replace x with y and set the last term EGm = 0, we recover as a special case the expression for thermal vacancies. We have an expression like eq. [2.56] for each phase in the system and indicate the phase with an upper index, e.g. for the phase a we write Gma, °Gma , EG ma . The last quantity EG ma is called the excess energy and takes into account contributions which are not included in the first two sums and, like the mixing entropy, it vanishes for a pure component. The quantity °G ka thus denotes the gibbs energy of pure k in the a phase. The simplest expression for the excess energy is called the regular solution and is given by E

Gma = S S xk x j Lakj k j >k

[2.58]

Of course, the regular solution parameters Lkja are zero when k = j and we also obviously have Lkja = Ljka . When experimental data are analyzed they are usually found to be functions of temperature and it is common to apply expansions of the type a + bT + cT ln T + … where the coefficients a, b and c are fitted to experimental data. However, often experimental data cannot be fitted satisfactorily unless Lkja is allowed to vary also with composition. The



77

most commonly used method is to expand Lkja in a so-called redlich–Kister polynomial m

Lakj = 0 Lakj + 1 Lakj (xk – x j ) + 2 Lakj (x (xk – x j )2 … = S  Lakj (xk – x j )  =0

[2.59]  a Lkj

 a Ljk

 a Lkj

 a Ljk.

=– and otherwise = Such a When  is uneven, we have model is usually referred to as being of the regular solution type. In the special case where only the first two terms are non-zero, it is referred to as a subregular solution. If the experimental information is such that many terms in the expansion are needed to have a good fit to experimental data, it is a strong indication that the model, i.e. eq. [2.57], is not realistic. even though higher-order expansions may represent the experimental information satisfactorily, they will lead to difficulties when extrapolating the data and usually one only uses terms up to the second order. For a binary regular solution phase a with components A and B, eq. [2.43] yields the following expressions for the chemical potentials of A and B, respectively.

2.6.3

a mA = °GAa + RT ln(1 – xB) + xB2 LAB

[2.60a]

a mB = °GBa + RT ln xB + (1 –xB)2LAB

[2.60b]

Phases with substitutional and interstitial components

Interstitially dissolved atoms do not occupy normal lattice sites and consequently they do not mix with substitutional atoms. It is more reasonable to assume that they mix randomly with vacant interstices or with other types of interstitial atoms. We should thus regard such a phase as consisting of two sublattices, one where the substitutional atoms mix and one where the interstitials and vacant sites mix. It should be emphasized, as discussed in the previous section, that the substitutional atoms also mix with vacancies but usually their fraction is so low that it can be neglected except in those cases when one is particularly interested in the vacancy content. We write a formula unit of such a phase (M1M2…)a(C, D … Va)c and Eq. [2.57] is thus modified into Gm = S S yM yC °GMC + RT [a S yM ln yM + cS yC ln yC ] + E Gm MC

M

C

[2.61]


78


The ys stand for the fraction of lattice sites, substitutional or interstitial, that is occupied by a certain component M or C, and a and c are the number of moles of each type of lattice site. In FCC there is one interstitial site per substitutional site and a = c = 1. In BCC there are three interstitial sites per substitutional and a = 1, c = 3. The suffix m in eq. [2.61] stands for 1 mole of formula units and MC stands for the real or hypothetical compound corresponding to the formula unit (M)a(C)c, often called an end member. In particular, the formula unit (M)a(Va)c means that all interstitials are vacant and it thus corresponds to pure M. It is straightforward to show that eq. [2.43] is then modified into

m M a Cc = G m +

∂Gm ∂Gm C ∂Gm + – S yj j =1 ∂yyM ∂yyc ∂yy j

[2.62]

The summation is taken over all components, i.e. over both sublattices including the vacancies. The expression for the chemical potential of M thus becomes C È ∂G ∂G ∂Gm ˘ m M = 1 ÍGm + m + m – S y j aÎ ∂yyM ∂yyVa j =1 ∂yy j ˙˚

[2.63]

The chemical potential for an interstitial component C is obtained from mMaCc = amM + cmC

[2.64]

and becomes ∂G ∂G mC = 1 ÈÍ m – m ˘˙ c Î ∂yyC ∂yyVa ˚

[2.65]

The excess energy term is usually expressed as a regular solution type on each sublattice leading to expressions of the type E

Gma = S S S yk yi y j Lakij ki

[2.66]

j k i>k i> k

where k and i refer to components on the same sublattice and j is on the other sublattice. As an example, we give the expression for Fe-Mn-C in austenite (FCC). g

g

g

g

Gmg = yFeyC °GFeC + yMnyC °GMnC + yFeyVa °GFe + yMnyVa °GMn + RT [yFe ln yFe + yMn ln yMn + yC ln yC + yVa ln yVa] g g + yFe yMn yVa °LFe,Mn:Va + yFe yMnyC°LFe,Mn:C g g + yMnyC yVa °LMn:C,Va + yFeyC yVa °LFe:C,Va g

g

[2.67] g

g

In eq. [2.67] we have used the facts that °GFeVa = °GFe and °GMnVa = °GMn. It


Thermodynamics of phase transformations in steels g

g

g

79 g

should also be observed that the parameters LFe,Mn:Va, °GFeC, °GMnC, LFe:C,Va g and LMn:C,Va refer to the binary Fe-Mn, Fe-C and Mn-C systems, respectively. Moreover, two y fractions may be eliminated because yFe + yMn = 1 and g yC + yVa = 1. Only the parameter LFe,Mn:C represents a ternary effect. By means of Eq. [2.63] it is straightforward to derive the expression for the chemical potentials. The leading terms are

g

g

mFe = °GFe + yMnyC °DGMnC + RT [ln(1 – yMn) + ln(1 – yC)] +…

[2.68a] g

g

mMn = °GMn – (1 – yMn)yC °DGMnC + RT [ln yMn + ln(1 – yC)] +…

[2.68b]

mC =

g °GC

+

g yMn °DGMnC

+ RT ln[yC/(1 – yC)] +…

[2.68c]

where we have introduced g

g

g

g

+

g LFe:C,Va

g

°DGMnC = °GFe + °GMnC – °GFeC – °GMn

[2.69a]

g °GC

[2.69b]

=

g °GFeC

–

g °GFe

g °DGMnC

The parameter represents the effect of Mn content on the carbon chemical potential and the interaction between carbon and manganese in iron. g The parameter °GC represents the reference state for carbon in austenite.

2.6.4 Activity and reference states The activity aref j of a component j is defined relative to a chosen reference state from the chemical potential mj by means of

ref mj = mref j + RT ln aj

[2.70]

Thus the activity is unity in the chosen reference state. For a binary regular solution, with components A and B, combination of Eqs [2.60] and [2.70] yields

a aBref = xB exp((°GBa – mBref + (1 – xB)2LAB )/RT )

[2.71]

In metallurgy several choices of reference states are common. For example, one may choose pure B in the phase and at the temperature under consideration. This is called the Raoultian reference and in a binary regular solution the Raoultian activity is given by

a aB = xB exp((1 – xB)2LAB /RT )

[2.72]

Evidently the B activity approaches the mole fraction xB in the neighborhood of pure B. This fact is referred to as Raoult’s law. On the other hand, for sufficiently low B contents the activity will be proportional to xB, with a


80


a proportionality factor exp(LAB /RT ) which may be larger or smaller than unity depending on the sign of the regular solution parameter. The proportionality factor is called the activity coefficient and the behavior at low B contents is referred to as Henry’s law. Another possibility is to take as reference the most stable state of the component under the temperature of interest, for example, b. In that case a the proportionality factor becomes exp((°G Ba –°G bB + (1 – xB)2LAB )/RT). This has the consequence that the activity of pure B in a will be larger than unity if it is not the most stable state at the temperature of interest. Still another possibility, which is sometimes used in steel refining, is to prescribe that the activity of a certain alloy element is unity when there is 1 mass% of the element in the molten steel. Its activity, close to the composition of 1 mass%, will then simply be equal to the content expressed in mass%. Actually, the issue of reference state must be considered even when the activity has not yet been introduced. This is because there is no unique zero level for the energy and consequently not for the thermodynamic functions that depend on the energy either. According to the third law of thermodynamics, all substances have the same value of entropy at 0 K and by convention it is set to 0. There is no such natural choice of reference for the energy, and for each component one may choose any reference that is convenient provided that all subsequent calculations are performed consistently and based on the chosen references. In the above discussion we considered various choices of reference for expressing the activity. They all had in common that we considered references at the same temperature and pressure as the conditions of interest, i.e. the reference varies with temperature. If such a reference is chosen, it is not possible to represent the temperature dependencies of individual phases but only the temperature dependence relative to the reference. In order to represent temperature dependencies, e.g. entropy and heat capacity, one needs to use temperature independent references. The most common choice is the so-called stable element reference (SER). In this reference, the reference for entropy is taken as 0 at 0 K and for enthalpy it is SER the enthalpy of the most stable phase at 25°C (298.15 K), i.e. mref k = Hk .

2.7

Various kinds of phase diagrams

Phase diagrams play a very important role when interpreting phase transformations in steel and, in fact, many conclusions can be drawn directly from the phase diagram. A phase diagram indicates which phase or phases are thermodynamically stable in the different areas of the diagram. A phase diagram may be plotted with different quantities on the axes and for each particular problem a certain set of axes is usually most convenient. The simplest phase diagram is one with potential quantities on all axes. However, the Gibbs phase rule (see Section 2.4.5), reveals that the number of potentials



81

that can be varied independently is f = c – p + 2 which has its largest value f = c + 1 when p = 1, i.e. when there is only one phase. This would thus be the dimensionality of the full phase diagram if all phase regions should be represented. Usually one wants to plot two-dimensional diagrams and thus, except for the case of a pure element where c = 1, it is necessary to reduce the dimensionality by sectioning or projections. Some important aspects of phase diagrams will be discussed in this section.

2.7.1 Potential diagrams For a pure element like Fe the Gibbs phase rule yields f = 3 – p and the full phase diagram is two-dimensional (see the phase diagram for pure Fe depicted in Fig. 2.1). Pressure and temperature have been chosen as potential axes but one could also have chosen pressure and chemical potential or temperature and chemical potential. The topology of such a diagram is simple and given by the phase rule. Regions with two coexisting phases have f = 1 and are lines, and regions with three coexisting phases f = 0 are represented by points. From the meaning of the diagram it is self-evident that such a three-phase point appears where two two-phase lines meet and a third two-phase line will then leave the point. According to the phase rule, four phases cannot coexist because they would require f = –1. For a binary system the phase rule gives f = 4 – p and the full potential phase diagram will be three dimensional. We may lower the dimensionality by sectioning the full diagram at a given potential. For that case we will write the phase rule

f = c – p + 2 – n s

[2.73]

where ns is the number of potentials that have been given a fixed value. Usually one considers the phase diagram at atmospheric pressure. From Eq. [2.73] it is obvious that from a topological point of view sectioning by keeping one of the potentials fixed is equivalent with having one component less. A binary potential phase diagram at a fixed pressure would thus have the same topology as the diagram for a pure element. This is demonstrated by Fig. 2.6 which shows the metastable Fe-C phase diagram plotted at P = 1 atm with temperature and C activity as axes. The reference state for C is graphite at the temperature under consideration. Equilibrium with graphite is thus represented by a vertical line at ACR(C) = 1. In many practical situations the potential diagram in Fig. 2.6 may be more instructive than the conventional diagram. One example is when iron is in contact with a carbon-rich atmosphere having a certain carbon activity and one would like to know what phase will form in contact with the gas. From Eq. [2.73] it also follows that the topology of the Fe-C-N potential diagram would look the same if sectioned at P = 1 atm and a certain N activity.


82

Phase transformations in steels 1600 1500

Temperature (°C)

1400 Liquid

1300 1200 1100 1000

g

900

Cementite

800 700 a

600 500

0

0.5

1.0

1.5

2.0 2.5 ACR(C)

3.0

3.5

4.0

2.6 The metastable Fe-C phase diagram plotted at P = 1 atm with temperature and C activity as axes. The reference state for C is graphite at the temperature under consideration.

2.7.2 Molar diagrams and mixed diagrams Although two phases in equilibrium have the same values of the potentials, they usually have different values of the molar quantities, e.g. the mole fractions. In a stable system a potential must increase when the amount of the corresponding component increases (see Section 2.4.8). Along a twophase line in a potential phase diagram the phase corresponding to the higher chemical potential thus must have a larger value on the corresponding molar quantity. If a potential axis is replaced with the corresponding mole fraction, the two-phase line in the potential diagram opens up into a two-phase field. The two points representing two phases in equilibrium may be tied together with a straight line, a tie-line. An example is shown in Fig. 2.7, where the binary Fe-C system is plotted with molar enthalpy instead of temperature on one axis. A number of tie-lines have been included in the figure. This type of diagram is useful to show how much heat needs to be added in order to reach a certain part of the phase diagram. The reference temperature for enthalpy is 25°C (298 K). For example, a steel with eutectoid composition, i.e. xC = 0.034, needs at least 30 kJ mol–1 to be austenitized and c. 70 kJ mol–1 to be fully molten. If we want to exchange carbon content in Fig. 2.7 to carbon activity,



83

8 7 Liquid

Hm

6 5 g

4 3

104

2 1 0

0.05

0.10 0.15 Mole_fraction C

0.20

0.25

2.7 The metastable Fe-C phase diagram plotted at P = 1 atm with molar enthalpy and C mole fraction as axes.

certain measures have to be taken in order to obtain a true phase diagram, i.e. a diagram where each point corresponds to a well-defined set of phases. This is discussed in detail by Hillert (2008) and in the present case it turns out that we have two choices. Either we may keep the molar enthalpy on the y-axis and instead plot aC/aFe on the x-axis, or we may plot the enthalpy per mole of iron atom on the y-axis and the carbon activity on the x-axis. The latter diagram is shown in Fig. 2.8. As can be seen it has the same topology as a usual binary phase diagram but we have exchanged the potential and molar quantities. From Eq. [2.73] we may calculate the dimensionality of a phase-field also when we have exchanged some potentials with molar quantities. If there are no molar quantities, the dimensionality is directly given by Eq. [2.73] but with each potential that is exchanged with a molar quantity increased by one until it is the same as the dimensionality d of the diagram, usually two, i.e.

f = c – p + 2 – n s + n m

[2.74]

where nm is the number of molar quantities introduced. If the calculated f is larger than d, the dimensionality of a phase field will simply be d. For the binary system sectioned once at P = 1 atm, we find f = 3 – p + nm. Thus a three-phase region will be a point in the diagram with only potentials, a line if one potential is exchanged with a molar variable, and an area if both potentials are exchanged with molar quantities.


84

Phase transformations in steels 12 10

Liquid

Hm/Fe

8 6 g

4 2 10

a

4

0

Cementite

0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 Carbon activity (relative graphite)

4.0

2.8 The metastable Fe-C phase diagram plotted at P = 1 atm with enthalpy per mole of Fe and carbon activity relative graphite as axes.

2.7.3 Sections For steels with two alloy elements the complete potential phase diagram will be three dimensional even if it is sectioned at P = 1 atm. In order to have a two-dimensional diagram we may then section it at constant value of another potential, usually the temperature. As discussed earlier, it would then have the same topological properties as a unary system. The most common way to plot ternary systems is the isobaric, isothermal sections with molar quantities on the axes. They then have the same topology as Fig. 2.7. All phase fields are then areas. At equilibrium a potential has the same value in all phases. If only potentials have been sectioned, all tie-lines will then lie in the plane of the phase diagram. When the tie-lines are in the plane, one may apply the well-known lever rule to evaluate the fraction of the different phases from the average value of the molar quantity. This useful property is lost when the tie-lines are not in the plane. The latter situation is the case if a ternary system is sectioned in a different way which will now be discussed. Consider a ternary system sectioned at atmospheric pressure and at a given mole fraction of one of the components. If one plots temperature on the y-axis and the molar content of one of the other elements on the x-axis, one obtains a so-called isoplethal section, sometimes called vertical section. Such isopleths are common in the literature. In this case we have ns = 2, nm = 2, and f = 5 – p and consequently all one-, two- and three-phase regions will be areas in a two-dimensional diagram. However, in this case it is evident that the tie-lines are generally out of plane because the diagram is sectioned at a constant molar quantity and the lever rule cannot be applied.



85

An example is shown in Fig. 2.9 showing an isopleth for the Fe-Cr-C system sectioned at xCr = 0.05 and P = 1 atm. According to Eq. [2.74] a four-phase equilibrium is given by a line, see the horizontal line representing the fourphase equilibrium g + a + M7C3 + Cementite at 757°C in Fig. 2.9.

2.8

Effect of interfaces

2.8.1 Surface energy and surface stress The surface energy is related to the physical process of changing the interfacial area in a system, for example during growth of a particle. In the simplest case only the interfacial area is changed and all other quantities stay constant. This is generally not the case during a precipitation process where both volume fraction of the precipitated phase and the interfacial area increase. It is approximately the case during coarsening where small particles dissolve and large particles grow leaving the volume fraction approximately constant but decreasing the total interfacial area. It is also the case during grain growth in a one-phase material where the total grain boundary area decreases leaving everything else unchanged. Already Gibbs (1875) and more recently Cahn (1980) pointed out that the surface of a solid can have its physical area changed by either of two different processes when an external force is applied: creation or destruction of surface area without changing surface structure and properties per unit area; 1600

xCr = 0.05

1400 Liquid

g 1200

g + M7C3 + Cementite 7C 3

1000

g+ M

Temperature (°C)

∑

800 g + a + M 7C 3 + Cementite 757°C

600

g + Cementite

a + M7C3 a + Cementite 0

0.05

0.10 0.15 Mole_fraction C

0.20

0.25

2.9 Isopleth at xCr = 0.05 of Fe-Cr-C phase diagram plotted at P = 1 atm. The tie-lines are not in plane.


86

∑


elastic deformation eij of the surface layer, keeping the number of surface lattice sites constant while changing the form, physical area and properties.

They defined the surface energy s as the work of creating a new unit area of the surface and the surface stress fij as related to the work of elastically deforming the surface. If dA0 denotes the area change due to the creation of new surface we have for the total area change dA: dA = A0(de11 + de22) + dA0

[2.75]

The change in internal energy caused by the area change may then be written dU = A0(f11 de11 + f22 de22 + f12de12) + s dA0

[2.76]

and the surface stresses and surface energy are thus defined as Ê ˆ fij = 1 Á ∂U ˜ A0 Ë ∂e ij ¯ A

[2.77a]

s = ÊÁ ∂U ˆ˜ Ë ∂A ∂A0 ¯ e ij

[2.77b]

0

and

The surface stress thus is a tensorial quantity and it follows that Ê ∂s ˆ ÁË ∂e ij ˜¯ = fij

[2.78]

For the sake of simplicity the following discussion will be limited to cases with isotropic surface properties. Thus the surface energy is constant and does not depend on the orientation of the surface and the only non-zero surface stresses are f11 = f22 = f. even though gibbs called f surface tension, this term should be avoided. The reason is that the term surface tension is often misused for surface energy and it only has a meaning for the isotropic case, whereas both surface stress (a 2*2 tensor) and surface energy (a scalar) are generally applicable. Consider a b-particle in a matrix a assumed to have a constant hydrostatic pressure, equal to the external pressure. Strictly speaking, we thus exclude cases where the matrix is solid and has been deformed elastically. This assumption will be strictly valid when the matrix is liquid or gaseous. Suppose the particle and the matrix are at equilibrium and the total volume is kept constant. Allow the volume of the particle to change by elastic compression or expansion without any transfer of atoms between the two phases under adiabatic conditions. We then have dU = 0, i.e.



dU = dU a + dUb + dU s = 0

87

[2.79]

The first two terms represent the change in internal energy of the a matrix and the b particle respectively, whereas the last term represents the change caused by the change in interfacial area. Under the present assumptions the result becomes particularly simple because the only contribution to the change in internal energy of a and b stems from the pressure volume work and deformation of the interface. We thus have

dU = PadVb – PbdVb + f dAel = 0

[2.80]

where dAel = A0(de11 + de22) and dV = dV a + dV b = 0 have been used. It thus follows that at equilibrium the particle will be under a compressive pressure given by

P b – P a = f dAel/dV b

[2.81]

For a spherical particle with radius r we have dAel/dV b = 2/r and the familiar condition for mechanical equilibrium is obtained

P b – P a = 2f/r

[2.82]

We may thus note the compressive internal stress of the b particle is goverened by the surface stress rather than the surface energy. The relation between surface stress and internal pressure is quite general and may be used to calculate, for example, the critical size during nucleation if the internal pressure is known. If a thin film of a liquid in contact with a large reservoir of the same liquid is slowly expanded a distance dx by pulling the bar in Fig. 2.10, the new interfacial area, upper and lower side, will be formed by transferring atoms from the reservoir to the film. The elastic deformation of the interface will not change and thus dU = 2s dx. This quantity must equal the work performed on the system by opposing the surface stress Fdx = 2f dx because these are the only tensile forces that a liquid can support. If b is a liquid, the surface stress thus has the same value as the surface energy. If the experiment is performed by pulling the bar very quickly, or if the liquid is very viscous,

F



2.10 The thin-film experiment to measure surface stress between liquid and atmosphere.


88


there will not be time enough for the liquid to relax and tensile stresses will build up both at the interface and in the bulk between the two interfaces. As equilibrium is approached the stresses in the bulk will vanish and the surface stress will relax to the equilibrium value being equal to the surface energy. The important conclusion is thus that, if the b phase is a liquid droplet, we could as well calculate the internal pressure from the surface energy. In general, however, the surface energy and the surface stress will be different and the internal pressure must be calculated from the surface stress. However, this does not mean that the surface energy is less important. As we shall see, the interfacial energy plays the major role in the effect of interfaces on phase equilibria whereas the internal pressure caused by the surface stress is of secondary importance.

2.8.2 Effect on phase equilibria – isotropic case We shall consider the simple case of a b phase inclusion in an a matrix. The b phase is assumed to be gaseous or liquid and we shall thus assume isotropic properties. The effect of capillarity can then be treated with any one of the characteristic state functions but it is most convenient to choose the function that has as its natural variables those state variables that will be kept constant for the system under consideration. From a practical point of view it thus seems convenient to keep temperature, the external pressure and the content of matter in the system constant and one should then use the Gibbs energy. The total Gibbs energy of a two-phase system consisting of the b inclusion in the a matrix is given by:

G = Gb + G a + G s

[2.83]

Gs is the contribution from surface free energy of the a/b interface. Due to capillarity, the internal pressure in the particle may be higher than in the a matrix. It is thus convenient to write Gb as

Gb = Nb [Um(P b ) – TSm(P b ) + PVm(P b )]

= G(P b ) + (P – P b )V (P b )

[2.84]

The values of T and P in Eq. [2.84] are those in the surroundings, according to the definition of Gibbs energy. In the present case the same value of P holds in the a matrix that is supposed to be liquid or gaseous but not to the b phase if the a/b interface is curved. Consider the transfer of dN jb atoms of component j from a to b in a multicomponent system, and the total change of Gibbs energy is obtained after first inserting Eq. [2.84] in Eq. [2.83]. The transfer may cause a change of Pb and we thus obtain



89

dG = G bj (Pb )dN bj + (∂G b /∂P)dPb – V b dPb – (Pb – P)dV b

– Gaj (P)dN bj + dGs = G bj (Pb )dN bj – (Pb – P)dV b

– maj (P) dN bj + dGs

[2.85]

We have here inserted the symbol for chemical potential, maj (P), instead of the partial Gibbs energy for the a matrix, G aj (P). We have not done so for the partial Gibbs energy of b because we hesitate to consider the chemical potential as identical to the partial Gibbs energy for a phase not under the same pressure as the surroundings. Moreover, we see no need to do so. For a spherical particle of radius r and isotropic interfacial energy s, the last term in Eq. [2.85] is dGs = (2s/r)V jbdN bj, where V jb is the partial molar volume of b. Of course s may contain an entropy contribution and is thus a surface Gibbs energy. From Eq. [2.85] the equilibrium conditions thus become

maj = G bj (Pb ) + (2s/r – (Pb – P))V bj

[2.86]

For the case when b is liquid and the surface energy and surface stress are identical we simply obtain,

maj = G bj (Pb )

[2.87]

This condition may be interpreted in terms of a normal common tangent construction with a Gibbs energy curve for the b phase corresponding to the pressure Pb = P + 2s/r. On the other hand, if b is a solid we may not assume that the surface stress and surface energy are identical but it may be a reasonable approximation that the b phase is incompressible. If this is the case then

G bj (Pb ) = G bj (P) + (Pb – P)V jb

[2.88]

and the equilibrium conditions are obtained as

maj = G bj (P) + 2s/rV jb

[2.89]

This condition may be interpreted in terms of a normal common tangent construction with a Gibbs energy curve for the b phase displaced upwards by a distance 2s/rV jb. Under both those simplifying but still reasonable approximations we find that the surface energy is the important quantity and not the surface stress. An expression for the critical size may be obtained by considering the growth of a b-phase particle without changing its composition, i.e. transfer of balanced amounts of all the components. Then we have dN jb = x jbdN b and obtain

dG = [G mb (Pb) – (Pb – P)Vmb – ∑ x jbmja(P)] dNb + dGs

[2.90]

For a spherical particle and balanced transfer we have dGs = (2s/r)V mb dNb and the critical size may be obtained from dG = 0 yielding © Woodhead Publishing Limited, 2012

90


– [G mb (Pb) – (Pb – P)Vmb – ∑ x jbmja(P)] = (2s/r)Vmb

[2.91]

It is instructive to consider two extremes, either the b phase may be approximated as incompressible or compressible as an ideal gas. Both extremes may serve as reasonably good approximations for many cases of practical importance. For the case when b is incompressible, eq. [2.91] becomes – [G mb (P) – ∑ x jbmja(P)] = (2s/r)Vmb

[2.92]

The quantity inside the square brackets is the driving force for initial precipitation (see eq. [2.25]), and is often denoted DGm. We thus arrive at the familiar textbook expression rc = 2s/(–DGm/Vmb )

[2.93]

This is a reasonable approximation if b is a condensed phase. When b is an ideal gas we have b

Gmb (P b ) = Gmb (P) P + RT ln P P

[2.94]

Using (Pb – P) = 2f/r, f = s and eq. [2.94], we may rewrite eq. [2.91] as b

[Gmb (P ) – S x bj maj (P )] = – RT ln P P

[2.95]

The quantity D°m = [G mb (P) – ∑ x jbmja (P)] is obviously a characteristic thermodynamic function for the gas and the a phase under consideration. Then, again using Pb – P = 2s/r, one can write Ê b ˆ RT PÁ P – 1˜ = P(e– D °m /RT – 1) = 2s P r Ë ¯

[2.96]

For the critical radius of a bubble it then follows that rc =

2s /RT RT P(e– Dm °°/RT – 1)

[2.97]

A pore may nucleate even if there is no gas at all provided that there is an external hydrostatic tension, i.e. P < 0. For a pore without gas Pb = 0 and the mechanical equilibrium gives directly rc = –

2ff P

[2.98]

Since the surface energy is equal to the surface stress when a is a liquid, we may also write rc = – 2s P

[2.99]



2.9

91

Thermodynamics of fluctuations in equilibrium systems

The second law states that all spontaneous processes have a positive entropy production. When equilibrium has been established all imaginable processes would cause a negative entropy production and the second law will then reverse them and drive the system back to the equilibrium state by spontaneous processes. However, the second law does not prohibit processes with negative entropy production. In fact, there is always a probability for such processes by means of fluctuations in a system at equilibrium. In order to understand this, we may first note that the internal entropy production TdiS may be given a straightforward meaning by the following argument. Consider our system, which we assume to be at equilibrium, together with its surroundings which we assume have the same temperature T as the system. We further consider the new system formed by our initial system and its surroundings as a closed isolated system, i.e. its energy, volume and composition are fixed. The entropy change of this new system is then TdStot = dS – dQ/T = TdiS. As already mentioned, the equilibrium state is then the state that maximizes the entropy of the new system comprised of the surroundings and our initial system. According to Boltzmann’s relation, the entropy of that system is Stot = kB ln W, where W is the number of ways in which the system can arrange its state. 1/W may thus be regarded as the probability that the system is arranged in one particular way. The probability of a fluctuation may be estimated as w/W, where w is the number of ways in which the system can arrange the particular fluctuation. The entropy change in the new system caused by the fluctuation would thus be DStot = kB ln w/W, i.e.

p = w/W = exp(DStot/kB)

[2.100]

But DStot = DiS and we have thus shown that Boltzmann’s relation permits violations of the second law. It should be noted that if DiS is positive, the system is not in equilibrium and the above reasoning does not apply. In order to see how likely a fluctuation is, we shall thus calculate the entropy production caused by the fluctuation, i.e. DiS, which will actually be a ‘consumption’ as it is negative for a system at equilibrium. Consider a fluctuation in some internal variable x. We take the average of x as zero, i.e. x = 0 which must also be the equilibrium value if the system is at equilibrium. DiS may be expanded around x = 0. The first derivative is zero due to the equilibrium condition ∂DiS/∂x|x = x = 0 and we thus have DiS = – ax2/2 + …, where a = ∂2DiS/∂x2 > 0 due to stability requirements. The probability that x will take a value in the range x to x + dx thus is

p(x)dx = A exp(–ax2/2kB)dx

A is determined from the normalization condition and we thus have


[2.101]

92


p(x )dx =

a exp(– ax 2 /2k )dx B 2p

[2.102]

This is a Gaussian distribution and the mean square fluctuation x2 = 1/a. In a system with constant T and P we find from the fundamental equation that TDiS = – DG and the constant a thus is 2 a = 1 ∂ D2G T ∂x

[2.103]

For example, as one approaches the limit of stability, where a = 1/T∂2DG/∂x2 Æ 0 and the mean square fluctuation 1/a Æ •, the fluctuations become much more likely.

2.10

Thermodynamics of nucleation

The entropy production TdiS is very important in order to understand fluctuations of a system at equilibrium. The probability p of having a negative TDiS by a fluctuation is p = exp(DiS/kB), i.e. violations of the second law are possible but there is a penalty which makes them very unlikely as soon as DiS becomes much bigger than Boltzmann’s constant kB. It is thus evident that the quantity –TDiS is precisely the one we usually call the activation barrier for nucleation. We shall now evaluate DiS for the formation of a critical spherical nucleus under fixed temperature, pressure and composition and one should thus consider gibbs energy. We obtain the activation barrier by integrating gibbs energy from the size 0 to the critical size, which is now easy because G is a state function and we obtain directly final fi nal nal initial nitiall –T D i S = G fina – G iinitia = n b Gmb + As – S nib mia

= n b [Gmb (P b ) – (P b – P )Vmb – S xib mia ] + 4p r 2s

[2.104]

where nb = V b/Vmb. As already mentioned, we usually call the quantity inside the square bracket the driving force for initial precipitation. It is interesting to notice that this quantity also appeared in our derivation of the equilibrium equations for the unstable nucleus. We may combine eq. [2.104] with the equilibrium condition, eq. [2.91], and obtain –TD i S = 4 p rc2s 3

[2.105]

It should be noticed that this is a quite general result and it holds regardless of whether or not b is incompressible. We can thus use eq. [2.104] to calculate the probability of reaching a critical size rc by means of a fluctuation if we apply eq. [2.100]. The result is



Ê 4 ˆ p = expÁ – p rc2s /kBT ˜ Ë 3 ¯

93

[2.106]

For an interfacial energy of 1 Jm–2 and a temperature of 1000 K, we find that the probability is negligible until the size comes down to the size of one or two atoms. The result depends very much on the value of the interfacial energy and a different interfacial energy would give a quite different result. Using the expression rc = 2s/(–DGm/Vmb ) for incompressible b, we obtain the classic textbook result: –T D i S = 16 p s 3 /(DGm /Vmb )2 3

2.11

[2.107]

References

Aaronson H. I., enomoto M. and Lee J. K., 2010. Mechanism of Diffusional Phase Transformations in Metals and Alloys, Boca raton, FL: CrC Press. Andersson J.-O., Helander T., Höglund L., Shi P. and Sundman B., 2002. Thermo-Calc & DICTrA, computational tools for materials science. CALPHAD 26, 273–312. Cahn J. W., 1961. On spinodal decomposition. Acta Metallurgica 9, 795–801. Cahn J. W. 1980. Surface stress and the chemical equilibrium of small crystals – I. the case of the isotropic surface. Acta Metallurgica 28, 1333–1338. gibbs J. W., 1875 [1948]. In Collected Works, 1. new Haven, CT: Yale University Press, pp. 215–331. Hillert M., 1995. Le Chatelier’s Principle – restated and illustrated with phase diagrams. Journal of Phase Equilibria, 16, 403–410. Hillert M., 2008. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis, 2nd edn, Cambridge: Cambridge University Press. Landau L. D. and Lifshitz e. M., 1970. Statistical Physics, Oxford: Pergamon Press. Lukas H. L., Fries S. g. and Sundman B., 2007. Computational Thermodynamics – the Calphad Method, Cambridge: Cambridge University Press. Pellicer J., Manzanares J. A., Zúñiga J., Utrillas P. and Fernández J., 2001. Thermodynamics of rubber elasticity. Journal of Chemical Education 78, 263–267. Saunders n. and Miodownik A. P., 1998. CALPHAD, Oxford: Pergamon Press.
