1 Thermodynamics and Phase Diagrams This chapter deals with some of the basic thermodynamic concepts that are required for a more fundamental appreciation of phase diagrams and phase transformations. It is assumed that the student is already acquainted with elementary thermodynamics and only a summary of the most important results as regards phase transformations will be given here. Fuller treatment can be found in the books listed in the bibliography at the end of this chapter. The main use of thermodynamics in physical metallurgy is to allow the prediction of whether an alloy is in equilibrium. In considering phase transformations we are always concerned with changes towards equilibrium, and thermodynamics is therefore a very powerful tool. It should be noted, however, that the rate at which equilibrium is reached cannot be determined by thermodynamics alone, as will become apparent in later chapters. 1.1 Equilibrium It is useful to begin this chapter on thermodynamics by defining a few of the terms that will be frequently used. In the study of phase transformations we will be dealing with the changes that can occur within a given system, e.g. an alloy that can exist as a mixture of one or more phases. A phase can be defined as a portion of the system whose properties and composition are homogeneous and which is physically distinct from other parts of the system. The components of a given system are the different elements or chemical compounds which make up the system, and the composition of a phase or the system can be described by giving the relative amounts of each component. The study of phase transformations, as the name suggests, is concerned with how one or more phases in an alloy (the system) change into a new phase or mixture of phases. The reason why a transformation occurs at all is because the initial state of the alloy is unstable relative to the final state. But how is phase stability measured? The answer to this question is provided by thermodynamics. For transformations that occur at constant temperature and pressure the relative stability of a system is determined by its Gibbs free energy (G). The Gibbs free energy of a system is defined by the equation G = H - TS (1.1) where H is the enthalpy, T the absolute temperature, and S the entropy of the system. Enthalpy is a measure of the heat content of the system and is given 2 Thermodynamics and phase diagrams by H = E + PV (1.2) where E is the internal energy of the system, P the pressure, and V the volume. The internal energy arises from the total kinetic and potential energies of the atoms within the system. Kinetic energy can arise from atomic vibration in solids or liquids and from translatkmal and rotational energies for the atoms and molecules within a liquid or gas; whereas potential energy arises from the interactions, or bonds, between the atoms within the system. If a transformation or reaction occurs the heat that is absorbed or evolved will depend on the change in the internal energy of the system. However it will also depend on changes in the volume of the system and the term PV takes this into account, so that at constant pressure the heat absorbed or evolved is given by the change in H. When dealing with condensed phases (solids and liquids) the PV term is usually very small in comparison to E, that is H — E, This approximation will be made frequently in the treatments given in this book. The other function that appears in the expression for G is entropy (5) which is a measure of the randomness of the system. A system is said to be in equilibrium when it is in the most stable state, i.e. shows no desire to change ad infinitum. An important consequence of the laws of classical thermodynamics is that at constant temperature and pressure a closed system (i.e. one of fixed mass and composition) will be in stable equilibrium if it has the lowest possible value of the Gibbs free energy, or in mathematical terms dG = 0 (1.3) It can be seen from the definition of G, Equation 1.1, that the state with the highest stability will be that with the best compromise between low enthalpy and high entropy. Thus at low temperatures solid phases are most stable since they have the strongest atomic binding and therefore the lowest internal energy (enthalpy). At high temperatures however the —TS term dominates and phases with more freedom of atom movement, liquids and gases, become most stable. If pressure changes are considered it can be seen from Equation 1.2 that phases with small volumes are favoured by high pressures. The definition of equilibrium given by Equation 1.3 can be illustrated graphically as follows. If it were possible to evaluate the free energy of a given system for all conceivable configurations the stable equilibrium configuration would be found to have the lowest free energy. This is illustrated in Fig. 1.1 where it is imagined that the various atomic configurations can be represented by points along the abscissa. Configuration A would be the stable equilibrium state. At this point small changes in the arrangement of atoms to a first approximation produce no change in G, i.e. Equation 1.3 applies. However there will always be other configurations, e.g. B, which lie at a local minimum in free energy and therefore also satisfy Equation 1.3, but which do not have the lowest possible value of G. Such configurations are called metastable Gibbs free energy G Equilibrium dG=0 Arrangement of atoms Fig. 1.1 A schematic variation of Gibbs free energy with the arrangement of atoms. Configuration 'A' has the lowest free energy and is therefore the arrangement when the system is at stable equilibrium. Configuration 'B' is a metastable equilibrium. equilibrium states to distinguish them from the stable equilibrium state. The intermediate states for which dG + 0 are unstable and are only ever realized momentarily in practice. If, as the result of thermal fluctuations, the atoms become arranged in an intermediate state they will rapidly rearrange into one of the free energy minima. If by a change of temperature or pressure, for example, a system is moved from a stable to a metastable state it will, given time, transform to the new stable equilibrium state. Graphite and diamond at room temperature and pressure are examples of stable and metastable equilibrium states. Given time, therefore, all diamond under these conditions will transform to graphite. Any transformation that results in a decrease in Gibbs free energy is possible. Therefore a necessary criterion for any phase transformation is AG = G2 - Gx < 0 (1.4) where Gj and G2 are the free energies of the initial and final states respectively. The transformation need not go directly to the stable equilibrium state but can pass through a whole series of intermediate metastable states. The answer to the question "How fast does a phase transformation occur?" is not provided by classical thermodynamics. Sometimes metastable states can be very short-lived; at other times they can exist almost indefinitely as in the case of diamond at room temperature and pressure. The reason for these differences is the presence of the free energy hump between the metastable and stable states in Fig. 1.1. The study of transformation rates in physical chemistry belongs to the realm of kinetics. In general, higher humps or energy barriers lead to slower transformation rates. Kinetics obviously plays a central 4 Thermodynamics and phase diagrams role in the study of phase transformations and many examples of kinetic orocesses will be found throughout this book. ' The different thermodynamic functions that have been mentioned m this section can be divided into two types called intensive and extensive properties The intensive properties are those which are independent of the size ot the system such as T and P, whereas the extensive properties are directly proportional to the quantity of material in the system, e.g. V, £, H, S and O. The usual way of measuring the size of the system is by the number of moles of material it contains. The extensive properties are then molar quantities, i e. expressed in units per mole. The number of moles of a given component in the system is given by the mass of the component in grams divided by its atomic or molecular weight. . , ■ • u The number of atoms or molecules within 1 mol of material is given by Avogadro's number (Na) and is 6.023 x 102 . 1.2 Single Component Systems Let us begin by dealing with the phase changes that can be induced in a single component by changes in temperature at a fixed pressure, say 1 atm. A single component system could be one containing a pure element or one type of molecule that does not dissociate over the range of temperature of interest. In order to predict the phases that are stable or mixtures that are in equilibrium at different temperatures it is necessary to be able to calculate the variation of G with T. 1,2.1 Gibbs Free Energy as a Function of Temperature The specific heat of most substances is easily measured and readily available. In general it varies with temperature as shown in Fig. 1.2a. The specific heat is the quantity of heat (in joules) required to raise the temperature of the substance by one degree Kelvin. At constant pressure this is denoted by Cp and is given by Therefore the variation of H with T can be obtained from a knowledge of the variation of Cp with T. In considering phase transformations or chemical reactions it is only changes in thermodynamic functions that are of interest. Consequently H can be measured relative to any reference level which is usually done by defining H = 0 for a pure element in its most stable state at 298 K (25 °C) The variation of H with T can then be calculated by integrating Single component systems Cf (Q) 0 -7-(K) Enthalpy H 0 (b) r (K) r(K) Fig. 1.2 (a) Variation of Cp with temperature, Cp tends to a limit of —3Í?. (b) Variation of enthalpy (H) with absolute temperature for a pure metal, (c) Variation of entropy (S) with absolute temperature. 6 Thermodynamics and phase diagrams Equation 1.5, i.e. H= f2" CpáT (1.6) J 298 The variation is shown schematically in Fig. 1.2b. The slope of thei/-T curve is G p. The variation of entropy with temperature can also be derived from the specific heat Cp. From classical thermodynamics T - (11 ™ Taking entropy at zero degrees Kelvin as zero, Equation 1.7 can be integrated to give 5= ľ^dr (1.8) as shown in Fig. 1.2c. Finally the variation of G with temperature shown in Fig. 1.3 is obtained by combining Fig. 1.2b and c using Equation 1.1. When temperature and pressure vary the change in Gibbs free energy can be obtained from the following result of classical thermodynamics: for a system of fixed mass and composition dG= SÚT+ VdP (1.9) At constant pressure dP = 0 and $,~* ™ This means that G decreases with increasing T at a rate given by —S. The relative positions of the free energy curves of solid and liquid phases are illustrated in Fig. 1.4. At all temperatures the liquid has a higher enthalpy (internal energy) than the solid. Therefore at low temperatures GL > Gs. However, the liquid phase has a higher entropy than the solid phase and the Gibbs free energy of the liquid therefore decreases more rapidly with increasing temperature than that of the solid. For temperatures up to Tm the solid phase has the lowest free energy and is therefore the stable equilibrium phase, whereas above Tm the liquid phase is the equilibrium state of the system. At Tm both phases have the same value of G and both solid and liquid can exist in equilibrium. Tm is therefore the equilibrium melting temperature at the pressure concerned. If a pure component is heated from absolute zero the heat supplied will raise the enthalpy at a rate determined by Cp (solid) along the line ab in Fig. 1.4. Meanwhile the free energy will decrease along ae. At Tm the heat Single component systems 1 Fig. 1.3 Variation of Gibbs free energy with temperature. supplied to the system will not raise its temperature but will be used in supplying the latent heat of melting (L) that is required to convert solid into liquid (be in Fig. 1.4). Note that at Tm the specific heat appears to be infinite since the addition of heat does not appear as an increase in temperature. When all solid has transformed into liquid the enthalpy of the system will follow the line cd while the Gibbs free energy decreases along ef. At still higher temperatures than shown in Fig. 1.4 the free energy of the gas phase (at atmospheric pressure) becomes lower than that of the liquid and the liquid transforms to a gas. If the solid phase can exist in different crystal structures (allotropes or polymorphs) free energy curves can be constructed for each of these phases and the temperature at which they intersect will give the equilibrium temperature for the polymorphic transformation. For example at atmospheric pressure iron can exist as either bcc ferrite below 910 °C or fee austenite above 910 °C, and at 910 °C both phases can exist in equilibrium. 1.2.2 Pressure Effects The equilibrium temperatures discussed so far only apply at a specific pressure (1 atm, say). At other pressures the equilibrium temperatures will differ. 8 Thermodynamics and phase diagrams Fig. 1,4 Variation of enthalpy (H) and free energy (G) with temperature for the solid and liquid phases of a pure metal. L is the latent heat of melting, 7"m the equilibrium melting temperature. For example Fig. 1.5 shows the effect of pressure on the equilibrium temperatures for pure iron. Increasing pressure has the effect of depressing the a/7 equilibrium temperature and raising the equilibrium melting temperature. At very high pressures hep E-Fe becomes stable. The reason for these changes derives from Equation 1.9. At constant temperature the free energy of a phase increases with pressure such that If the two phases in equilibrium have different molar volumes their respective free energies will not increase by the same amount at a given temperature and equilibrium will, therefore, be disturbed by changes in Single component systems O o Q) 1— O d» Q. £ (b Liquid iron -6 -iron Y-iron 2000-1600- 1200-800-400- 0 25 50 75 100 125 150 175 Pressure, kbar Fig. 1.5 Effect of pressure on the equilibrium phase diagram for pure iron. pressure. The only way to maintain.equilibrium at different pressures is by varying the temperature. If the two phases in equilibrium are a and ß, application of Equation 1.9 to 1 mol of both gives dGa = V^dP - 5adr dG& = V^dP - S^dT If a and ß are in equilibrium Ga = Gř therefore dGa = dGß and S& - Sa AS (1.12) (1.13) V&m - V"m AV This equation gives the change in temperature dT required to maintain equilibrium between a and ß if pressure is increased by dP. The equation can be simplified as follows. From Equation 1.1 G" = Ha - TSa Gß = H& - TS& Therefore, putting AG = G13 - Ga etc. gives AG = AH ~ TAS But since at equilibrium Gß = Gtt, AG = 0, and AH - TAS = 0 Consequently Equation 1.13 becomes dP \ AH dT eq TeqAV (1.14) 10 Thermodynamics and phase diagrams which is known as the Clausius-Clapeyron equation. Since close-packed •y-Fe has a smaller molar volume than a-Fe, AV = V* — V£, < 0 whereas AH = W - Ha > 0 (for the same reason that a liquid has a higher enthalpy than a solid), so that (dP/áT) is negative, i.e. an increase in pressure lowers the equilibrium transition temperature. On the other hand the hl L equilibrium temperature is raised with increasing pressure due to the larger molar volume of the liquid phase. It can be seen that the effect of increasing pressure is to increase the area of the phase diagram over which the phase with the smallest molar volume is stable (y-Fe in Fig. 1.5). It should also be noted that s-Fe has the highest density of the three ätiotropes, consistent with the slopes of the phase boundaries in the Fe phase diagram. 1.2.3 The Driving Force for Solidification In dealing with phase transformations we are often concerned with the difference in free energy between two phases at temperatures away from the equilibrium temperature. For example, if a liquid metal is undercooled by AT below Tm before it solidifies, solidification will be accompanied by a decrease in free energy AG (J mol-1) as shown in Fig. 1.6. This free energy decrease provides the driving force for solidification. The magnitude of this change can be obtained as follows. The free energies of the liquid and solid at a temperature T are given by GL - HL - TSL G* = H" TSS Therefore at a temperature T AG = AH - TAS (1.15) Temperature Fig. 1.6 Difference in free energy between liquid and solid close to the melting point The curvature of the Gs and GL lines has been ignored. where AH = HL - Hs and AS = 5L - 5s At the equilibrium melting temperature Tm the free energies of solid and liquid are equal, i.e. AG = 0. Consequently AG = AH - rmA5 = 0 and therefore at Tm AH L AS = — - — (1.16) 'm * m This is known as the entropy of fusion. It is observed experimentally that the entropy of fusion is a constant — R(8.3 J mol~x K"1) for most metals (Richard's rule). This is not unreasonable as metals with high bond strengths can be expected to have high values for both L and Tm. For small undercoolings (AT) the difference in the specific heats of the liquid and solid (C]; — C|) can be ignored. AH and AS are therefore approximately independent of temperature. Combining Equations 1.15 and 1.16 thus gives AG - L - T^r Tm i.e. for small AT LAT AG = —- (1.17) This is a very useful result which will frequently recur in subsequent chapters. 1.3 Binary Solutions In single component systems all phases have the same composition, and equilibrium simply involves pressure and temperature as variables. In alloys, however, composition is also variable and to understand phase changes in alloys requires an appreciation of how the Gibbs free energy of a given phase depends on composition as well as temperature and pressure. Since the phase transformations described in this book mainly occur at a fixed pressure of 1 atm most attention will be given to changes in composition and temperature. In order to introduce some of the basic concepts of the thermodynamics of alloys a simple physical model for binary solid solutions will be described. 1.3.1 The Gibbs Free Energy of Binary Solutions The Gibbs free energy of a binary solution of A and B atoms can be calculated from the free energies of pure A and pure B in the following way. 12 Thermodynamics and phase diagrams „ is assumed that A and B have the "-^.S^Ä^ and can be mixed in any proport,»n t £^^Äl»«ton is made ÄÄTľ-Ä - «BSin-there is a total of 1 mol of solution XA + XB = 1 (1-18) and XA and XB are the mole fractions of A and B respectively in the alloy. In order to calculate the free energy of the alloy, the mixing can be made in two steps (see Fig. 1.7). These are: 1. bring together XA mol of pure A and Xß mol of pure B; 2. allow the A and B atoms to mix together to make a homogeneous solid solution. After step 1 the free energy of the system is given by Gl = XAGA + XBG* J mol"1 (1.19) where GA and Gq are the molar free energies of pure A and pure B at the temperature and pressure of the above experiment, Gi can be most conveniently represented on a molar free energy diagram (Fig. 1.8) in which molar free energy is plotted as a function of XB or XA. For all alloy compositions Gx lies on the straight line between GA and GB. The free energy of the system will not remain constant during the mixing of the A and B atoms and after step 2 the free energy of the solid solution G2 can Before mixing After mixing MIX XA moi A FE.XÄGŕ XBmolB *a^a HE.XBGB v--------------Y-----------------/ Total free energy = ft =XAGA*XBGB 1mol solid solution ^. Total free energy q 2 "G\ + AG™* Fig. 1.7 Free energy of mixing. Free energy per mole before mixing 2 *■- í Fig. 1.8 Variation of G, (the free energy before mixing) with alloy composition (*A or XB). be expressed as G2 = Gl + AGmix (1.20) where AGmix is the change in Gibbs free energy caused by the mixing. Since d = Hi - TSX and G2 ~ H2 — TS2 putting AHmix - H2~ H, and A^mix = S2 ~ S\ gives AGmix = AHmix - TAS, (1.21) AHmix is the heat absorbed or evolved during step 2, i.e. it is the heat of solution, and ignoring volume changes during the process, it represents only the difference in internal energy (E) before and after mixing. ASmix is the difference in entropy between the mixed and unmixed states. 1.3.2 Ideal Solutions The simplest type of mixing to treat first is when AHmix - 0, in which case the resultant solution is said to be ideal and the free energy change on mixing is only due to the change in entropy: AGmix = -TbS^ (1.22) In statistical thermodynamics, entropy is quantitatively related to randomness by the Boltzmann equation, i.e. 5 = k In o> (1.23) where k is Boltzmann's constant and w is a measure of randomness. There are two contributions to the entropy of a solid solution—a thermal contribution 5th and a cbnfigurational contribution ^config • In the case of thermal entropy, w is the number of ways in which the thermal energy of the solid can be divided among the atoms, that is, the total number of ways in which vibrations can be set up in the solid. In solutions, additional randomness exists due to the different ways in which the atoms can be arranged. This gives extra entropy 5config for which w is the number of distinguishable ways of arranging the atoms in the solution. If there is no volume change or heat change during mixing then the only contribution to ASmix is the change in configurational entropy. Before mixing, the A and B atoms are held separately in the system and there is only one distinguishable way in which the atoms can be arranged. Consequently Si = kin 1 = 0 and therefore ASmix = S2- Assuming that A and B mix to form a substitutional solid solution and that all configurations of A and B atoms are equally probable, the number of distinguishable ways of arranging the atoms on the atom sites is Wconfi* - NA!NB! (L24) where NA is the number of A atoms and NB the number of B atoms. Since we are dealing with 1 mol of solution, i.e. Na atoms (Avogadro's number), NA = XANa and NB = XBNa By substituting into Equations 1.23 and 1.24, using Stirling's approximation (In N\ =* N InN - N) and the relationship1 Nak = R (the universal gas constant) gives ASmix = -R{XA In XA + XB In XB) (1.25) Note that, since XA and XB are less than unity, A5mix is positive, i.e. there is an increase in entropy on.mixing, as expected. The free energy of mixing, AGmix, is obtained from Equation 1.22 as AGmix = RT(XA In XA + XB In XB) (1.26) Figure 1.9 shows AGmix as a function of composition and temperature. The actual free energy of the solution G will also depend on GA and GB • From Equations 1.19, 1.20 and 1.26 AG mix A B Fig. 1.9 Free energy of mixing for an ideal solution. G = G2 = XAGA + XBGB + RT(XA In XA + XB In XB) (1.27) This is shown schematically in Fig. 1.10. Note that, as the temperature increases, GA and GB decrease and the free energy curves assume a greater curvature. The decrease in GA and GB is due to the thermal entropy of both components and is given by Equation 1.10. It should be noted that all of the free energy-composition diagrams in this book are essentially schematic; if properly plotted the free energy curves must end asymptotically at the vertical axes of the pure components, i.e. tangential to the vertical axes of the diagrams. This can be shown by differentiating Equation 1.26 or 1.27. Molar free energy Fig- 1.10 The molar free energy (free energy per mole of solution) for an ideal solid solution. A combination of Figs. 1.8 and 1.9. lO l řic/f/((/u^/řUřim.o u/m ynujc LimK' uriu 1.3.3 Chemical Potential In alloys it is of interest to know how the free energy of a given phase will change when atoms are added or removed. If a small quantity of A, ánA mol, is added to a large amount of a phase at constant temperature and pressure, the size of the system will increase by dnA and therefore the total free energy of the system will also increase by a small amount dG'. If dnA is small enough dG' will be proportional to the amount of A added. Thus we can write dG' = |xAdnA (ľ, P, nB constant) (1.28) The proportionality constant u,A is called the partial molar free energy of A or alternatively the chemical potential of A in the phase. u.A depends on the composition of the phase, and therefore dnA must be so small that the composition is not significantly altered. If Equation 1.58 is rewritten it can be seen that a definition of the chemical potential of A is ^ ■ SF) (L29) The symbol G' has been used for the Gibbs free energy to emphasize the fact that it refers to the whole system. The usual symbol G will be used to denote the molar free energy and is therefore independent of the size of the system. Equations similar to 1.28 and 1.29 can be written for the other components in the solution. For a binary solution at constant temperature and pressure the separate contributions can be summed: dG' = u.AdnA + m-b^b (1.30) This equation can be extended by adding further terms for solutions containing more than two components. If T and P changes are also allowed Equation 1.9 must be added giving the general equation dG' - -SdT + VdP + \xAdnA + u,BdrtB + |Xcd«c + ■ ■ ■ If 1 mol of the original phase contained XA mol A and XB mol B, the size of the system can be increased without altering its composition if A and B are added in the correct proportions, i.e. such that dnA:dnB = XA:XB. For example if the phase contains twice as many A as B atoms (XA = 2/3, XB = 1/3) the composition can be maintained constant by adding two A atoms for every one B atom (dnA:dnB = 2). In this way the size of the system can be increased by 1 mol without changing u.A and |xB. To do this XA mol A and XB mol B must be added and the free energy of the system will increase by the molar free energy G. Therefore from Equation 1.30 G = ílaXa + M-B^B J mor1 (1.31) When G is known as a function of XA and XB, as in Fig. 1.10 for example, u.A and (xB can be obtained by extrapolating the tangent to the G curve to the Molar free energy Fig. 1.11 The relationship between the free energy curve for a solution and the chemical potentials of the components. sides of the molar free energy diagram as shown in Fig. 1.11. This can be obtained from Equations 1.30 and 1.31, remembering that XA + XB = 1, i.e. dA"A = -dXB, and this is left as an exercise for the reader. It is clear from Fig. 1.11 that |iA and [iB vary systematically with the composition of the phase. Comparison of Equations 1.27 and 1.31 gives |xA and p,B for an ideal solution as u-A = GA + RT In XA |xB = GB + RT\nXB (1.32) which is a much simpler way of presenting Equation 1.27. These relationships are shown in Fig. 1.12. The distances ac and bd are simply —RT In XA and -RT\nXB. RTlnX^i >~RT\nXB Fig. 1.12 The relationship between the free energy curve and chem an ideal solution. 1.3.4 Regular Solutions Returning to the model of a solid solution, so far it has been assumed that AHmix = 0; however, this type of behaviour is exceptional in practice and usually mixing is endothermic (heat absorbed) or exothermic (heat evolved). The simple model used for an ideal solution can, however, be extended to include the A//mix term by using the so-called quasi-chemical approach. In the quasi-chemical model it is assumed that the heat of mixing, A//mix, is onlyxlue to the bond energies between adjacent atoms. For this assumption to be valid it is necessary that the volumes of pure A and B are equal and do not change during mixing so that the interatomic distances and bond energies are independent of corr ^si :>n. The structure of a . ..dry solid solution is shown schematically in Fig. 1.13. Three types of interatomic bonds are present: 1. A—A bonds each with an energy eAA, 2. B—B bonds each with an energy eBB, 3. A—B bonds each with an energy eAB ■ By considering zero energy to be the state where the atoms are separated to infinity eAA, eBB and eAB are negative quantities, and become increasingly more negative as the bonds become stronger. The internal energy of the solution E will depend on the number of bonds of each type PAA, PBB and PAB such that E = í*AAeAA + ■^BBeBB + ^ABeAB Before mixing pure A and B contain only A—A and B—B bonds respectively and by considering the relationships between PAA, PBB anc* ^ab m the solution it can be shown1 that the change in internal energy on mixing is given Fig. 1.13 The different types of interatomic bond in a solid solution. by AHmix = FABe (1-33) where e = £ab - Ksaa + eBB) (1.34) that is, e is the difference between the A—B bond energy and the average of the A—A and B—B bond energies. If e = 0, Arřmix = 0 and the solution is ideal, as considered in Section 1.3.2. In this case the atoms are completely randomly arranged and the entropy of mixing is given by Equation 1.25. In such a solution it can also be shown1 that Far = NazXAXB bonds mol -i AB (1.35) where Na is Avogadro's number, and z is the number of bonds per atom. If g < 0 the atoms in the solution will prefer to be surrounded by atoms of the opposite type and this will increase PAB, whereas, if e > 0, PAB will tend to be less than in a random solution. However, provided e is not too different from zero, Equation 1.35 is still a good approximation in which case AH^ = OXaX- a-^-b where ft = /Vazs (1.36) (1.37) Real solutions that closely obey Equation 1.36 are known as regular solutions. The variation of A#mix with composition is parabolic and is shown in Fig. 1.14 for ft > 0. Note that the tangents at XA = 0 and 1 are related to ft as shown. A/Ymtx per mol Fig. 1.14 The variation of Ai/mjx with composition for a regular solution. 20 Thermodynamics and phase diagrams The free energy change on mixing a regular solution is given by Equations 1.21, 1.25 and 1.36 as AGmix = axAXB + RT(XA In XA + XB In XB) (1.38) Ai/„ -TAS, This is shown in Fig. 1.15 for different values of ti, and temperature. For exothermic solutions A//mix < 0 and mixing results in a free energy decrease at all temperatures (Fig. 1.15a and b). When AHmix > 0, however, the situation is more complicated. At high temperatures 7ASmix is greater than A/rmix for all compositions and the free energy curve has a positive curvature at all points (Fig. 1.15c). At low temperatures, on the other hand, TA5mix is smaller and AGmix develops a negative curvature in the middle (Fig. 1.15d). Differentiating Equation 1.25 shows that, as XA or XB -* 0, the -rA5mix curve becomes vertical whereas the slope of the A//mix curve tends to a finite A ~^mix B (q) fi< 0, high T (b) fi< 0,low T AH mix A ^-------^ B A (c) Q>0, high T (d), Q > 0 low T Fig. 1.15 The effect of &Hmix and T on AGmi)t. B Binary solutions 21 value O (Fig. 1.14). This means that, except at absolute zero, AGmix always decreases on addition of a small amount of solute. The actual free energy of the alloy depends on the values chosen for GA and GB and is given by Equations 1.19, 1.20 and 1.38 as G = XAGA + XBGB + OX^XB + RT(XA In XA + XB In XB) (1.39) This is shown in Fig. 1.16 along with the chemical potentials of A and B in the solution. Using the relationship XAXß ~ XAXB + XßXA and comparing Equations 1.31 and 1.39 shows that for a regular solution U.A = GA + fl(l - XAf + RT\n XA (1.40) and M-B = GB + fi(l - XBf + RT\n XB 1.3.5 Activity Expression 1.32 for the chemical potential of an ideal alloy was simple and it is convenient to retain a similar expression for any solution. This can be done by defining the activity of a component, a, such that the distances ac and bd in Fig. 1.16 are —RT In aA and —RT In aB. In this case I^a = Ga + ^^ In aA and u.B = GB + RT In aB (1.41) ► -RTir\ aB 0 *B—*- 1 Pig. 1-16 The relationship between molar free energy and activity. *f"«i>wvj mím* fnuäc uiugritrrtA In general aA and aß will be different from -3fA and XB and the relationship between them will vary with the composition of the solution. For a regular solution, comparison of Equations 1.40 and 1.41 gives «A XA In ^ = a =^(i-*A)2 and (1.42) In XB RT (1 - XBf Assuming pure A and pure B have the same crystal structure, the relationship between a and X for any solution can be represented graphically as illustrated in Fig. 1.17. Line 1 represents an ideal solution for which aA = XA and aB - XB. If &Hmix < 0 the activity of the components in solution will be less in an ideal solution (line 2) and vice versa when AHmix > 0 (line 3). The ratio (äA/XA) is usually referred to as -yA, the activity coefficient of A, that is 7a = aA/XA (1.43) For a dilute solution of B in A, Equation 1.42 can be simplified by letting XB —> 0 in which case aB 7b Xv =» constant (Henry's law) and -Ya = ~ = 1 (Raoult's law) Xa (1.44) (1.45) a, / / A2 l\_0t0*y0^ __Henry's (aw 0 Xq —*- _ ///■______nenrys>iuw - A B a ! A 2^ ^s. 3 \1 \\ n ^s^M Xa 0 A B (a) (b) Fig. 1.17 The variation of activity with composition (a) aß (b) aA. Line 1: ideal solution (Raoult's law). Line 2: áHmill < 0. Line 3: AHmi;( > 0. tSinary solutions 15 ton 1 44 is known as Henry's law and 1.45 as Raoult's law; they apply to solutions when sufficiently dilute. S nee activity is simply related to chemical potential via Equation 1.41 the ť*tv of a component is just another means of describing the state of the Donent in a solution. No extra information is supplied and its use is simply ľmatter of convenience as it often leads to simpler mathematics. Activity and chemical potential are simply a measure of the tendency of an atom to leave a solution. If the activity or chemical potential is low the atoms are reluctant to leave the solution which means, for example, that the vapour sure of the component in equilibrium with the solution will be relatively tow It will also be apparent later that the activity or chemical potential of a component is important when several condensed phases are in equilibrium. 1.3.6 Real Solutions While the previous model provides a useful description of the effects of configurational entropy and interatomic bonding on the free energy of binary solutions its practical use is rather limited. For many systems the model is an oversimplification of reality and does not predict the correct dependence of AGmix on composition and temperature. As already indicated, in alloys where the enthalpy of mixing is not zero (e and ft f 0) the assumption that a random arrangement of atoms is the equilibrium, or most stable arrangement is not true, and the calculated value for AGmix will not give the minimum free energy. The actual arrangement of atoms will be a compromise that gives the lowest internal energy consistent with sufficient entropy, or randomness, to achieve the minimum free energy. In systems with e < 0 the internal energy of the system is reduced by increasing the number of A—B bonds, i.e. by ordering the atoms as shown in Fig. 1.18a. If e > 0 the internal energy can be reduced by increasing the number of A—A and B—B bonds, i.e. by the clustering of the atoms into A-rich and B-rich groups, Fig. 1.18b. However, the degree of ordering or (a) (b) (c) Fig. 1.18 Schematic representation of solid solutions: (a) ordered substitutional, (b) clustering, (c) random interstitial. 24 Thermodynamics and phase diagrams clustering will decrease as temperature increases due to the increasing importance of entropy. In systems where there is a size difference between the atoms the quasi-chemical model will underestimate the change in internal energy on mixing since no account is taken of the elastic strain fields which introduce a strain energy term into A#mix. When the size difference is large this effect can dominate over the chemical term. When the size difference between the atoms is very large then interstitial solid solutions are energetically most favourable, Fig. 1.18c New mathematical models are needed to describe these solutions. In systems where there is strong chemical bonding between the atoms there is a tendency for the formation of intermetallic phases. These are distinct from solutions based on the pure components since they have a different crystal structure and may also be highly ordered. Intermediate phases and ordered phases are discussed further in the next two sections. 1.3.7 Ordered Phases If the atoms in a substitutional solid solution are completely randomly arranged each atom position is equivalent and the probability that any given site in the lattice will contain an A atom will be equal to the fraction of A atoms in the solution XA, similarly Xß for the B atoms. In such solutions PAB, the number of A—B bonds, is given by Equation 1.35. If Í1 < 0 and the number of A—B bonds is greater than this, the solution is said to contain short-range order (SRO). The degree of ordering can be quantified by defining a SRO parameter s such that *ab - /^(random) řAB(max) - PAB(random) where PAB(max) and PAB(random) refer to the maximum number of bonds possible and the number of bonds for a random solution, respectively. Figure 1.19 illustrates the difference between random and short-range ordered solutions. In solutions with compositions that are close to a simple ratio of A: B atoms another type of order can be found as shown schematically in Fig. 1.18a. This is known as long-range order. Now the atom sites are no longer equivalent but can be labelled as A-sites and B-sites. Such a solution can be considered to be a different (ordered) phase separate from the random or nearly random solution. Consider Cu-Au alloys as a specific example. Cu and Au are both fee and totally miscible. At high temperatures Cu or Au atoms can occupy any site and the lattice can be considered as fee with a 'random' atom at each lattice Binary solutions 25 (a) (b) r i.i I .'' (a) Random A~B solution with a total of 100 atoms and XA = XB = 0.5, ,,- "j , 5 = 0. (b) Same alloy with short-range order PAB = 132, PAB(max) -200, S = (132 - 100)/(200 - 100) = 0.32. point as shown in Fig. 1.20a. At low temperatures, however, solutions with X = XAvi = 0.5, i.e. a 50/50 Cu/Au mixture, form an ordered structure in which the Cu and Au atoms are arranged in alternate layers, Fig. 1.20b. Each atom position is no longer equivalent and the lattice is described as a CuAu superlattice. In alloys with the composition Cu3Au another superlattice is found, Fig. 1.20c. The entropy of mixing of structures with long-range order is extremely small and with increasing temperature the degree of order decreases until above some critical temperature there is no long-range order at all. This temperature is a maximum when the composition is the ideal required for the superlattice. However, long-range order can still be obtained when the composition deviates from the ideal if some of the atom sites are left vacant or if some atoms sit on wrong sites. In such cases it can be easier to disrupt the order with increasing temperature and the critical temperature is lower, see Fig. 1.21. The most common ordered lattices in other systems are summarized in Fig. 1.22 along with their Structurbericht notation and examples of alloys in which they are found. Finally, note that the critical temperature for loss of long-range order increases with increasing Í1, or A//mix, and in many systems the ordered phase is stable up to the melting point. 1.20 Ordered substitutional structures in the Cu-Au system: (a) high-tempera-disordered structure, (b) CuAu superlattice, (c) Cu?Au superlattice. 26 Thermodynamics and phase diagrams D *-> O 0; Qc 500r 400 3001- Q.200 6 a» 1001- _i_________i_ 0 0-1 0-2 0-3 0-4 05 06 07 0-8 09 1-0 Cu X Au Au 3 Fig. 1.21 Part of the Cu-Au phase diagram showing the regions where the Cu3Au % and CuAu superlattices are stable. ?, «*st (o) 'Cu OZn 'Cu OAu iCuqAu Cd) iff (e) ►AL OFe 'Cd OMg Fig. 1.22 The five common ordered lattices, examples of which are: (a) L20: CuZn, FeCo, NiAl, FeAl, AgMg; (b) Ll2:Cu3Au, Au3Cu, Ni3Mn, Ni3Fe, Ni3Al, Pt3Fe: (c) Ll0:CuAu, CoPt, FePt; (d) D03:Fe3Al, Fe3Si, Fe3Be, Cu3Al; (e) D019:Mg3Cd, Cd.iMg, Ti3Al, Ni3Sn. (After R.E. Smallman, Modern Physical Metallurgy, 3rd edition, Butterworths, London, 1970.) 1.3.8 Intermediate Phases Often the configuration of atoms that has the minimum free energy after mixing does not have the same crystal structure as either of the pure components. In such cases the new structure is known as an intermediate phase. Binary solutions 27 Intermediate phases are often based on an ideal atom ratio that results in a minimum Gibbs free energy. For compositions that deviate from the ideal, the free energy is higher giving a characteristic 'U' shape to the G curve, as in pig. 1.23. The range of compositions over which the free energy curve has a meaningful existence depends on the structure of the phase and the type of interatomic bonding—metallic, covalent or ionic. When small composition deviations cause a rapid rise in G the phase is referred to as an intermetallic compound and is usually stoichiometric, i.e. has a formula AWB„ where m and n are integers, Fig. 1.23a. In other structures fluctuations in composition can be tolerated by some atoms occupying 'wrong' positions or by atom sites being left vacant, and in these cases the curvature of the G curve is much less, Fig. 1.23b. Some intermediate phases can undergo order-disorder transformations in which an almost random arrangement of the atoms is stable at high temperatures and an ordered structure is stable below some critical temperature. Such a transformation occurs in the ß phase in the Cu-Zn system for example (see Section 5.10). The structure of intermediate phases is determined by three main factors: relative atomic size, valency and electronegativity. When the component atoms differ in size by a factor of about 1.1-1.6 it is possible for the atoms to fill space most efficiently if the atoms order themselves into one of the so-called Laves phases based on MgCu2, MgZn2 and MgNi2, Fig 1.24. Another example where atomic size determines the structure is in the formation of the interstitial compounds MX, M2X, MX2 and M6X where M can be Zr, Ti, V, Cr, etc. and X can be H, B, C and N. In this case the M atoms form a cubic or hexagonal close-packed arrangement and the X atoms are small enough to fit into the interstices between them. Idea composition IT" *g- 1.23 Free energy curves for intermediate phases: (a) for an intermetallic com-°*"rá with a very narrow stability range, (b) for an intermediate phase with a wide stability range. Thermodynamics and phase diagrams Fig. 1.24 The structure of MgCu2 (A Laves phase). (From J.H. Wernick, chapter 5 in Physical Metallurgy, 2nd edn., R.W. Cahn (Ed.) North Holland, 1974.) The relative valency of the atoms becomes important in the so-called electron phases, e.g. a and ß brasses. The free energy of these phases depends on the number of valency electrons per unit cell, and this varies with composition due to the valency difference. The electronegativity of an atom is a measure of how strongly it attracts electrons and in systems where the two components have very different electronegativities ionic bonds can be formed producing normal valency compounds, e.g. Mg2+ and Sn4" are ionically bonded in Mg2Sn.2 1.4 Equilibrium in Heterogeneous Systems It is usually the case that A and B do not have the same crystal structure in their pure states at a given temperature. In such cases two free energy curves Equilibrium in heterogeneous systems 29 must be drawn, one for each structure. The stable forms of pure A and B at a given temperature (and pressure) can be denoted as a and ß respectively. For the sake of illustration let a be fee and ß bcc. The molar free energies of fee A and bcc B are shown in Fig. 1.25a as points a and b. The first step in drawing the free energy curve of the fee a phase is, therefore, to convert the stable bcc arrangement of B atoms into an unstable fee arrangement. This requires an increase in free energy, be. The free energy curve for the a phase can now be constructed as before by mixing fee A and fee B as shown in the figure. -AG • for a of composition X is given by the distance de as usual. A similar procedure produces the molar free energy curve for the ß phase, Fig. 1.25b. The distance af is now the difference in free energy between bcc A and fee A. It is clear from Fig. 1.25b that A-rich alloys will have the lowest free energy as a homogeneous a phase and B-rich alloys as ß phase. For alloys with (a) A X B (b) A Xb B Hg. 1.25 (a) The molar free energy curve for the a phase, (b) Molar free energy curves for a and ß phases. ™. y..*»*, »«*6'""W compositions, near the cross-over in the G curves the situation is not so straightforward. In this case it can be shown that the total free energy can be minimized by the atoms separating into two phases. It is first necessary to consider a general property of molar free energy diagrams when phase mixtures are present. Suppose an alloy consists of two phases a and ß each of which has a molar free energy given by Ga and Gß, Fig. 1.26. If the overall composition of the phase mixture is X% the lever rule gives the relative number of moles of a and ß that must be present, and the molar free energy of the phase mixture G is given by the point on the straight line between a and ß as shown in the figure. This result can be proven most readily using the geometry of Fig. 1.26. The lengths ad and cf respectively represent the molar free energies of the a and ß phases present in the alloy. Point g is obtained by the intersection of be and dc so that beg and acd, as well as deg and dfc, form similar triangles. Therefore bg/ad = bc/ac and ge/cf = ab/ac. According to the lever rule 1 mol of alloy will contain bc/ac mol of a and ab/ac mol of ß. It follows that bg and ge represent the separate contributions from the a and ß phases to the total free energy of 1 mol of alloy. Therefore the length 'be' represents the molar free energy of the phase mixture. Consider now alloy X° in Fig. 1.27a. If the atoms are arranged as a homogeneous phase, the free energy will be lowest as a, i.e. Gg per mole. However, from the above it is clear that the system can lower its free energy if the atoms separate into two phases with compositions c^ and ßi for example. The free energy of the system will then be reduced to Gx. Further reductions in free energy can be achieved if the A and B atoms interchange between the a and ß phases until the compositions ae and ßc are reached, Fig. 1.27b. The free energy of the system Ge is now a minimum and there is no desire for further change. Consequently the system is in equilibrium and ae and ße are the equilibrium compositions of the a and ß phases. This result is quite general and applies to any alloy with an overall composition between ae and ße: only the relative amounts of the two phases change, Molar free energy Gia+ß) Fig. 1.26 The molar free energy of a two-phase mixture (a + ß). LJUM-IUUI ItA-llt III /(CtCK/gCřřCWlW dydl *- n in the lever rule. When the alloy composition lies outside this range, Í*..". ..i -i the minimum free energy lies on the Ga or Gp curves and the "ouíľibriúm state of the alloy is a homogeneous single phase. '"From Fig. 1-27 it can be seen that equilibrium between two phases requires that the tangents to each G curve at the equilibrium compositions lie on a ommon line. In other words each component must have the same chemical potential in the two phases, i.e. for heterogeneous equilibrium: The condition for equilibrium in a heterogeneous system containing (a) Fig. 1.27 (a) Alloy X° has a free energy Gx as a mixture of otj + ß,. (b) At -equilibrium, alloy X° has a minimum free energy Ge when it is a mixture of ae + ße ■ 32 Thermodynamics and phase diagrams two phases can also be expressed using the activity concept defined for homogeneous systems in Fig. 1.16. In heterogeneous systems containing more than one phase the pure components can, at least theoretically, exist in different crystal structures. The most stable state, with the lowest free energy, is usually defined as the state in which the pure component has unit activity. In the present example this would correspond to defining the .activity of A in pure a - A as unity, i.e. when XA = 1, a% = 1. Similarly when XB = 1, aB = 1- This definition of activity is shown graphically in Fig. 1.28a; Fig. 1.28b and c show how the activities of B and A vary with the composition of the a and ß phases. Between A and ae, and ßc and B, where single phases are stable, the activities (or chemical potentials) vary and for simplicity ideal solutions have been assumed in which case there is a straight line relationship between a and X. Between ae and ßc the phase compositions in equilibrium do not change and the activities are equal and given by points q and r. In other words, when two phases exist in equilibrium, the activities of the components in the system must be equal in the two phases, i.e. (1.47) a% = ah aB - flB -RTinctj (b) aB I (c) °*l 0 t^Lf I. A ae ße B ' 1 t */ lť'\ i \ p ľ-RTina, 0 X, B 1 Fíg. 1.28 The variation of úa and aB with composition for a binary system containing two ideal solutions, a arid ß. Binary phase diagrams 33 ,.5 ií„iji>IW Diagrams i -i * iPii- ^ection it has been shown how the equilibrium state of an alloy h' nht 'i '«• ' fr°m tne ^ree ener§y curves at a given temperature. The next '. r. iii -^ I >w equilibrium is affected by temperature. I $ j \ \ <■■ > i Phase Diagram Th' šmink vi - -»e to start with is when A and B are completely miscible in both t Ik. "ll|ul '1C* hqurô states and b°tn are ideal solutions. The free energy ot puu' \ "i'll pure B will vary with temperature as shown schematically in f [n | i ! k * uilibrium melting temperatures of the pure components occur * hen ii* - ('' i-e- at rm(A) and Tm(B). The free energy of both phases dtCTvM-«." * r- nperature increases. These variations are important for A-B Ulov* -il^" *»k* they determine the relative positions of G%, G&, G% and G& fin il><- UK»! i liee energy diagrams of the alloy at different temperatures, Fig. 1.2y. At a high temperature Tx > Tm(A) > Tm(B) the liquid will be the stable phase for pure A and pure B, and for the simple case we are considering the liquid also has a lower free energy than the solid at all the intermediate compositions as shown in Fig. 1.29a. Decreasing the temperature will have two effects: firstly G\ and Gb will increase more rapidly than GSA and G%, secondly the curvature of the G curves will be reduced due to the smaller contribution of -7"A5mix to the free energy. At Tm(A), Fig. 1.29b, G% = G^, and this corresponds to point a on the A-B phase diagram, Fig. 1.29f. At a lower temperature T2 the free energy curves cross, Fig. 1.29c, and the common tangent construction indicates that alloys between A and b are solid at equilibrium, between c and B they are liquid, and between b and c equilibrium consists of a two-phase mixture (S + L) with compositions b and c. These points are plotted on the equilibrium phase diagram at T2. Between T2 and Tm(B) GL continues to rise faster than Gs so that points b and c in Fig. 1.29c will both move to the right tracing out the solidus and liquidus lines-in the phase diagram. Eventually at Tm(B) b and c will meet at a single point, d in Fig. 1.29f. Below rm(B) the free energy of the solid phase is everywhere below that of the liquid and all alloys are stable as a single phase solid. 1.5.2 Systems with a Miscibility Gap Figure 1.30 shows the free energy curves for a system in which the liquid phase is approximately ideal, but for the solid phase Atfmix > 0, i.e. the A and B atoms 'dislike' each other. Therefore at low temperatures (T3) the free energy curve for the solid assumes a negative curvature in the middle,