Chapter 18: Solutions - Berkeley · PDF fileNotes on the Thermodynamics of Solids J.W. Morris, Jr.; Fall, 2008 Page 396 Chapter 18: Solutions Chapter 18: Solutions

Notes on the Thermodynamics of Solids J.W. Morris, Jr.; Fall, 2008

Page 396

C h a p t e r 1C h a p t e r 1 88 : S o l u t i o n s: S o l u t i o n s Chapter 18: Solutions.......................................................................................................396

18.1 Introduction .................................................................................................396 18.2 Ideal Solutions .............................................................................................397

18.2.1 A solution of ideal gases...............................................................397 18.2.2 The ideal solution..........................................................................400 18.3.3 The vapor pressure of a component in an ideal solution ...............401

18.3 Thermodynamics of Real Solutions.............................................................401 18.3.1 Excess thermodynamic quantities .................................................401 18.3.2 The activity and the activity coefficient .........................................403

18.4 Dilute Solutions ...........................................................................................404 18.4.1 The chemical potential in a dilute solution.....................................404 18.4.2 The fundamental equation of a dilute solution...............................407 18.4.3 Generalization to "semi-dilute" solutions ......................................408

18.5 Dilute Solutions: Applications.....................................................................409 18.5.1 Equilibrium of two solutions with the same solvent .....................409 18.5.2 Influence of a dilute solute on two-phase equilibrium...................410 18.5.3 The vapor pressure of a dilute solution: Raoult's Law ..................411 18.5.4 Osmotic pressure..........................................................................412 18.5.5 Dilute solutions with the same solute: Henry's Law .....................413 18.5.6 Solute-solvent equilibria between dilute solutions ........................414 18.5.7 The interference of solutes in a dilute ternary solution...................415

18.6 Behavior near a Critical Point ......................................................................416 18.7 The Simple Solution ....................................................................................419

18.7.1 The thermodynamics of the simple solution..................................419 18.7.2 Decomposition and ordering in the simple solution......................421

18.8 The Phase Diagram of a Binary Solution.....................................................422 18.8.1 Special points in the T-x diagram..................................................422 18.8.2 Binary phase diagrams..................................................................425 18.8.3 Solid solution diagrams ................................................................425 18.8.4 Low-temperature behavior of a solid solution...............................427 18.8.5 Phase diagrams with eutectic or peritectic reactions ......................430 18.8.6 Structural transformations in the solid state ..................................433 18.8.7 Systems that form compounds......................................................435 18.8.8 Mutation lines in binary phase diagrams.......................................441 18.8.9 Miscibility gap in the liquid ..........................................................442

18.1 INTRODUCTION This chapter treats multicomponent fluid phases (solutions) that are sufficiently simple that their thermodynamic behavior can be studied in some detail. We are, of course, particularly interested in solid solutions, which obey the thermodynamics of fluids if the stress is hydrostatic. The theory developed in this chapter includes that part of the


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thermodynamics of solid solutions that is independent of the details of the atom arrangements. We begin with a discussion of the one completely solvable case: a solution of ideal gases. We shall then use classical thermodynamics to generalize this result to the simplest model of a real solution, the ideal solution. While almost no real solutions (except the ideal gases) are ideal solutions over the whole range of composition, we shall discover that every solution behaves very nearly like an ideal solution in the limit of small solute concentration (the dilute solution). This result, and its logical extension to higher solute concentrations through an expansion of the free energy in powers of the concentration, is the basis for most of the useful theory of solutions, and we shall explore its applications in some detail. We shall then consider models that attempt a global representation of the fundamental equa-tion of a solution. Finally, we shall discuss the phase diagrams that represent the equilib-rium of binary solutions and offer a simple classification of the more important of these. 18.2 IDEAL SOLUTIONS 18.2.1 A solution of ideal gases The fundamental equation of a mixture of ideal gases was developed as an exercise earlier in the course. The classical derivation is based on Dalton's Law,

Pk = NkTV 18.1

where Pk is the partial pressure of the kth specie, and the caloric equation of state

E = NCT = 32 NT 18.2

where N is the total mole number. From eq. 18.2 the energy can be written as the sum of the energies of the individual species:

E = ∑k

Ek = ∑k

32 NkT 18.3

Given the additivity of the energy and the pressure,

S = ET +

PVT - ∑

i µiNi = ∑

k Sk 18.4

where Sk is the entropy the kth specie would have if it filled the volume V, at temperature, T, alone. It follows that a component in a solution of ideal gases has the entropy it would have if it were present alone.


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The entropy of a one-component ideal gas can be found by integrating the thermal and caloric equations of state, and is

⁄S(E,V,N) = 32 Nln

E

N + NlnV

N + Nsº 18.5

The entropic form of the fundamental equation of a mixture of ideal gases is, therefore, ⁄S(E,V,{N}) = N⁄s(e,v,{x})

= ∑k

3

2 Nkln

Ek

Nk + Nkln

V

Nk + Nksº

k

= 32 Nln

E

N + NlnV

N - N∑k

xkln(xk) + N∑k

xksºk 18.6

Comparing eqs. 18.5 and 18.6, the major difference between the entropy functions of the one-component and n-component ideal gas is the inclusion of the extra term Smix = - N∑

k xkln(xk) 18.7

on the right-hand side of 18.6. This term is called the entropy of mixing, and plays a cen-tral role in the thermodynamics of solutions. Note that the entropy of mixing is a straight-forward consequence of Dalton's Law; it does not enter because the species interact, but because they do not interact. Its appearance is a consequence of writing the entropy as a function of the global content of the mixture (E,N) rather than the contents of the individual species (Ek, Nk). Essentially, it results from the fact that the various species in the mixture fill the same space at the same time. The Helmholtz free energy can be found by expressing S as a function of T,V and {N}, and can be written ⁄F(T,V,{N}) = N⁄f(T,v,{x}) = V⁄Fv(T,{n})

= - VT

3n

2 ln(T) + n - ∑k

nkln(nk) + ∑k

nkfºk

= - NT

3

2ln(T) + ln(v) + 1 - ∑k

xkln(xk) + ∑k

xkfºk 18.8


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The advantage of using the Helmholtz free energy is that it is easily computed from statistical thermodynamics, which lets us evaluate the unknown constants, fºk . The result is

fºk =

32 ln

mk

2πN0Ó2 18.9

where mk is the atomic mass of the kth specie, N0 is Avogadro's number, Ó = h/2π, h is Planck's constant, and T is measured as RK, where R is the molar gas constant and K is the temperature in degrees Kelvin. [I leave it as an exercise to confirm that, with this result, the quantity in braces on the right-hand side of 18.8 is dimensionless.] The form of the fundamental equation that is most commonly used in the study of solutions is the Gibbs free energy. To find it we compute the chemical potentials of the components as a function of T, P, and {x}. From equation 18.8,

µk =

∆ ¡F

∆Nk = -

3T2 ln(T) + T ln

Nk

V - Tfºk

= T ln(Pk) - 5T2 ln(T) - Tfº

k 18.10

Equation 18.10 shows that the chemical potential of a component in a solution of ideal gases is the chemical potential it would have if it were present by itself. Since Pk = xkP, where xk is the atom fraction of the kth component,

µk = T ln(P) - 5T2 ln(T) - Tfº

k + T ln(xk)

= µ0

k(T,P) + T ln(xk) 18.11 where µ0

k(T,P) is the chemical potential of component k at temperature T and pressure P. The Gibbs free energy per mole is, then, ⁄g(T,P,{x}) = ∑

k xkµ0

k(T,P) + T∑k

xk ln(xk) 18.12

where the first term is the Gibbs free energy per mole when the components are separated at constant P and T, and the second term (which is negative) is the free energy of mixing. Note that the free energy of the mixing comes from the entropy of mixing; it does not rep-resent any physical interaction between the species, but is due to the fact that they do not interact, and can, hence, fill the same space at the same time.


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18.2.2 The ideal solution The mixture of ideal gases is one example of an ideal solution, which is a solution of atoms of species that have no preferential interaction with one another. The ideal solu-tion is the simplest solvable model of a real solution. It is useful to define an ideal solution by its Gibbs free energy:

an ideal solution is a solution that obeys a fundamental equation of the form 18.12, where µ0

k(T,P) is the chemical potential of a pure sample of the kth component at temperature T and pressure P.

Equation 18.12 shows that the change in free energy on forming an ideal solution from the pure components at (T,P) is always negative. Hence species that form ideal solutions are miscible in any proportions. The entropy of mixing of an ideal solution is the same as that of a mixture of ideal gases (eq. 18.6). The chemical potential of the kth component of an ideal solution is given by equation 18.11. The molar enthalpy of an ideal solution is h = g + Ts = ∑

k xk

µ0k(T,P) + Ts0

k(T,P)

= ∑

k xkh0

k(T,P) 18.13

where s0

k(T,P) is the molar entropy of the kth component in its pure state at (T,P). Equation 18.13 shows that the heat of mixing, Îh, is zero; the enthalpy of an ideal solution is just the weighted average of the enthalpies of its pure components, so no heat evolves when the solution forms. It follows as a corollary that the miscibility of the species that form an ideal solution is due to the entropy of mixing. The relative chemical potential of the kth component in an ideal solution is

–µk(T,P,{x}) =

∆ ¡g

∆xk = µ0

k - µ0n + T ln

xk

xn

= –µ0k(T,P) + T ln

xk

xn 18.14

where n designates the reference component, or solvent, and –µ0

k(T,P) is the relative chemi-cal potential of the pure solute. Note that –µk depends on the concentrations of the other solutes only through their effect on xn, the mole fraction of the solvent. For a binary solu-tion,


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–µk(T,P,x) = –µ0k(T,P) + T ln

x

1 - x 18.15

where x is the mole fraction of solute. 18.3.3 The vapor pressure of a component in an ideal solution One of the properties of an ideal solution that can be easily calculated is the equilib-rium vapor pressures of its components. Let the ideal solution be a condensed phase in equilibrium with a vapor that is a mixture of ideal gases (this is almost always a good ap-proximation since the total vapor pressure of a condensed phase is usually very small). The condition of chemical equilibrium requires that the chemical potential of every component be the same in the solution and the vapor. From equation 18.10 the chemical potential of the kth component in the vapor phase is µk = µk(T) + T ln(Pk) 18.16 where µk(T) is a function of the temperature only and Pk is the partial pressure of the kth component. The chemical potential in the solution is, from equation 18.11 µk = µ0

k(T,P) + T ln(xk) 18.17 where µ0

k(T,P) is the chemical potential of the pure component in the condensed state at (T,P), and is a weak function of P when P is small. Equating the chemical potentials yields

Pk = xk exp

µ0

k - µk

T ~ xkP0k(T) 18.18

where P0

k(T) is the vapor pressure of the kth component in its pure form. It follows that the vapor pressure of a component in an ideal solution depends only on its own atom frac-tion. Conversely, the concentration of a component in an ideal solution in equilibrium with a dilute vapor depends only on its partial pressure in the gas. 18.3 THERMODYNAMICS OF REAL SOLUTIONS 18.3.1 Excess thermodynamic quantities There are two model systems that provide good reference states for the thermody-namics of a real solution in the sense that their properties are well-defined and relatively simple to measure experimentally. These are the mixture of pure components and the ideal solution. If the solution were a simple mixture of the pure components at temperature, T, and pressure, P, the molar density of the Gibbs free energy would be


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gº(T,P,{x}) = ∑

k xkµ0

k(T,P) 18.19

while if the solution were ideal its molar free energy would be gi(T,P,{x}) = ∑

k xkµ0

k(T,P) + T∑k

xk ln(xk) 18.20

It is often useful to measure the thermodynamic content of the solution in terms of its de-viation from one of these two reference values. The excess of the Gibbs free energy of a multicomponent system over that of a simple mixture is called the excess free energy of mixing, Îgº(T,P,{x}) = g - gº

= ∑k

xk µk - µ0

k = ∑k

xkÎµ0k 18.21

and is a measure of the tendency of a mixture of components to form a solution. The more negative this quantity, the more reactive the mixture. The excess of the Gibbs free energy of a solution over the value it would have if the solution were ideal is the excess free energy of solution, Îgi(T,P,{x}) = g - gi = ∑

k xkÎµ0

k - T∑k

xk ln(xk)

= ∑

k xkµi

k 18.22

where µi

k is the non-ideal part of the chemical potential, defined by the relation µi

k = µk - µ0k + T ln(xk) 18.23

µi

k measures the deviation of the kth component from ideal behavior. It is sometimes informative to divide the excess free energy of solution into its en-thalpic and entropic parts. Since the excess enthalpy of an ideal solution is zero, the excess enthalpies of mixing and solution are the same, the heat of mixing,

Îhº = h - hº = - T2 ∆

∆T

ÎgºT


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= - T2 ∑k

xk ∆

∆T

Îµ0

kT 18.24

The heat of solution is zero when the solution is ideal, that is, when the species in solution do not interact preferentially with one another. The heat of solution is negative when the species in the solution bond preferentially to one another. In this case they prefer one an-other as neighbors; solid solutions with Îhº < 0 tend to form ordered compounds at low temperature. The heat of solution is positive when the species prefer neighbors of their own kind. Solutions with Îhº > 0 tend to decompose into pure components at low tem-perature. The excess entropy of solution is

Îsi = -

∆Îgi

∆T = - ∑k

xk

∆µi

k∆T 18.25

The excess entropy of solution is zero when the solution is ideal, and also vanishes when the deviation from ideality is independent of the temperature. The latter condition is as-sumed in many of the more useful models of solid solutions. When the excess entropy of solution is not zero it is generally negative and is due to a preferential order or clustering that reduces the degeneracy of the configuration. 18.3.2 The activity and the activity coefficient The chemical potential of a real solution can always be written µk(T,P,{x}) = µ0

k(T,P) + µik(T,P,{x}) + T ln(xk) 18.26

where µi

k is a measure of the non-ideality of the kth component. To preserve the functional form of the chemical potential of the ideal solution it is customary to define the activity, ak, and the activity coefficient, ©k, by the relations µk = µ0

k(T,P) + T ln(ak) = µ0

k(T,P) + T ln(©kxk) 18.27 where the activity of the kth specie is

ak(T,P,{x}) = exp

µk - µ0

kT 18.28


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and the activity coefficient is

©k(T,P,{x}) = (xk)-1exp

Îµk

T 18.29

The activity is a convenient measure of the contribution of the kth component to the excess free energy of mixing, which can be written Îgº = T∑

k xk ln(ak) 18.30

In this notation the heat of mixing becomes

Îhº = - T2∑k

xk

∆[ln(ak)]

∆T 18.31

which shows that Îhº vanishes when the activity is independent of the temperature. The activity coefficient is a convenient measure of the contribution of the kth com-ponent to the excess free energy of solution, which can be written Îgi = T ∑

k xk ln(©k) 18.32

Given the emphasis that is placed on the activity and activity coefficient in many undergraduate texts that treat the thermodynamics of solutions, it is worth noting that these quantities are not independent thermodynamic parameters in any sense. They are simply ways of writing a part of the chemical potential in a form that is sometimes mathematically convenient. 18.4 DILUTE SOLUTIONS 18.4.1 The chemical potential in a dilute solution One of the most important results in the thermodynamics of solutions was the demonstration, by Gibbs, that dilute solutions have a quasi-ideal behavior. This result es-tablished two older "laws" in the chemistry of liquid solutions: Henry's Law, which states that the vapor pressure of a solute above a dilute solution is proportional to its concentra-tion, and Raoult's Law, which states that the vapor pressure of the solvent decreases in proportion to the total concentration of solute. It also provided a very useful framework for the analysis of phase equilibrium and phase diagrams in solid solutions, since most solid phases accept only a small addition of solute before the solubility limit is exceeded.


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Consider a dilute binary solution, and let the molar density of the solute, n1, be changed at constant values of T, P and n2. The Gibbs-Duhem equation requires that the chemical potentials of the solute and solvent change in the relation

n1

dµ1

dn1 + n2

dµ2

dn1 = 0 18.33

where the total differential has been used since only the single variable, n1, is changed. Equation 18.33 continues to hold in the limit n1 “ 0. But since n2 is finite there, we must have either

dµ2

dn1 = 0 18.34

or

dµ1

dn1 =

An1

18.35

where A is a constant. Since equation 18.33 holds for arbitrary components and arbitrary types of solutions, either 18.34 or 18.35 must hold in general. However, equation 18.34 cannot hold in general. It is not true for solutions of ideal gases, for which

dµ2

dn1 = -

Tn2

18.36

in the limit of small values of n1. It is also untrue for common liquid and solid solutions. We are, therefore, forced to choose equation 18.35. Equation 18.33 then shows that

n2

dµ2

dn1 = - A 18.37

where A is independent of n1 and n2. In the limit n2 “ 0 the system behaves as an ideal gas. Hence A = T 18.38 We can now evaluate the chemical potential of the dilute solute from equation 18.35. Setting A = T and dividing the denominator of both sides of the equation by n2 ~ n2 + n1 = n, we have

dµ1dx =

Tx 18.39


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where x is the mole fraction of component 1. Equation 18.39 has the integral µ1(T,P,x) = µd

1(T,P) + T ln(x) 18.40 Equation 18.40 is formally identical to the expression for the chemical potential for the ideal solution with the difference that µd

1(T,P) is simply a constant of integration. There is no a priori reason to believe that µd

1 has any simple relation to the chemical potential, µ01 ,

of the pure component. Rewriting equation 18.40 in terms of the activity coefficient of the solute gives the result µ1 = µ0

1 + T ln[©1x1] 18.41 where ©1 is a constant. Equation 18.41 confirms one form of Henry's Law: a dilute solute has a constant activity coefficient. The chemical potential of the solvent also has a simple behavior. Given equation 18.38, equation 18.37 can be written

dµ2dx = - T 18.42

which has the integral µ2 = µ0

2(T,P) - Tx ~ µ0

2(T,P) + T ln(1-x) 18.43 where the constant of integration is fixed by the fact that µ2 reduces to its value in the pure component when x “ 0. Equation 18.43 confirms one form of Raoult's Law: the solvent of a dilute solution behaves as if the solution were ideal. Equations 18.40 and 18.43 are readily generalized to solutions of an arbitrary num-ber of components, provided that all except one, the solvent, are present in very dilute con-centration. A dilute solution behaves as if it were ideal with the exception that the constant potential of a dilute solute is µd

k rather than µ0k . We call these solutions quasi-ideal.

Two important features of the behavior of real solutions can be inferred from the behavior of the relative chemical potential of the solute in the dilute limit. From equations 18.40 and 18.43, when x is small

–µ1(P,T,x) = –µd1(P,T) + T ln

x

1-x 18.44


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where –µd

1 = µd 1 (P,T) - µ0

1(P,T) 18.45 In the limit x “ 0 the relative chemical potential increases without bound. It follows that:

1. any chemical specie is at least slightly soluble in any other; 2. the function ¡g(x) at constant (T,P) has infinite negative slope in the limits x = 0 and x = 1.

More generally, since most real solid solutions are dilute, the theory of dilute solu-tions is often sufficient to treat the equilibrium behavior of multicomponent solids. The application of the theory involves only one unknown, the function µd

1 (P,T) , which can be obtained experimentally. Where the dilute theory is inadequate it can be developed into a series in powers of x, as we shall show below. 18.4.2 The fundamental equation of a dilute solution The fundamental equation of a dilute solution is formally identical to that of an ideal solution with the difference that the chemical potential µ0

k(T,P) is replaced by the asymp-totic value, µd

k(T,P) , for each of the solute species. It is, however, useful to write the fun-damental equation in a form that makes its functional dependence on the concentration vari-ables more explicit. For a multicomponent fluid, equation 18.40 gives the chemical potential of the kth dilute solute µk(T,P,{x}) = µd

k(T,P) + T ln(xk) 18.46 while equation 18.43 can be generalized to give the chemical potential of the solvent in the form µn(T,P,{x}) = µ0

n(T,P) - T∑k

xk 18.47

Hence the molar density of the Gibbs free energy is

g = ∑k=1

n xkµk


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= xnµ0n(T,P) - xnT∑

k xk + ∑

k

xkµd

k(T,P) + Txkln(xk)

~ µ0n(T,P) + ∑

k

xk–µdk(T,P) + Txkln

xk

e 18.48

where the unlabeled sum is taken over the (n-1) solute species and we have neglected terms of order (xk)2. Equation 18.48 has the form g = µn + ∑

k –µkxk 18.49

where the relative chemical potential,

–µk(T,P,{x}) =

∆ ¡g

∆xk = –µd

k(T,P) + Tln(xk) 18.50

is accurate to within terms of order xk. 18.4.3 Generalization to "semi-dilute" solutions To generalize the fundamental equation to slightly higher concentrations note that the function,

¡g'(T,P,{x}) = ¡g(T,P,{x}) - ∑k

Txkln

xk

e 18.51

is well-behaved in the limit xk “ 0 and has well-behaved first derivatives there (unlike g, whose partial derivatives with respect to the xk are singular when xk “ 0). Assuming that its higher derivatives are well-behaved as well we can obtain the free energy at finite {x} by expanding ¡g'({x}) about {x} = {0}. To second order,

¡g'({x}) ~ ¡g'({0}) + ∑i

∆ ¡g

∆xi {0} xi +

12∑

ij

∆2¡g

∆xi∆xj {0} xixj

= µ0n + ∑

i –µd

i xi + 12∑

ij µijxixj 18.52

where the second-order coefficient

µij =

∆2¡g

∆xi∆xj {0} 18.53


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Hence, to second order in the concentration,

¡g({x}) ~ µ0n + ∑

i –µd

i xi + 12∑

ij µijxixj + ∑

i Txiln

xi

e 18.54

where the coefficients are functions of T and P. The relative chemical potentials of the sol-utes in the semi-dilute solution are –µi ~ –µd

i + ∑j

µijxj + Tln(xi) 18.55

The expansion can, of course, be continued to higher orders of {x}. However, the second order expansion is sufficient to treat most of the solid solutions of interest in mate-rials science, since most solid phases have a limited range of solubility. Note that the same expansion applies to non-stochiometric compounds if the solvent is taken to be the sto-chiometric compound and the solutes are the species that are present in excess. Equation 18.54 can be evaluated experimentally by measuring the values of the constant parameters as functions of temperature (their pressure dependence is rarely of interest). Several ther-modynamic data bases have been compiled that contain the coefficients (or others that are equivalent to them) and can be used to compute the thermodynamic properties of solid solutions. 18.5 DILUTE SOLUTIONS: APPLICATIONS We include a series of examples here to illustrate the applications of the theory of dilute solutions. All except the last assume that the first-order expansion is sufficient. 18.5.1 Equilibrium of two solutions with the same solvent Let two phases (å and ∫) be dilute solutions with the same solvent, and let them be in equilibrium with one another. Let xå and x∫ be the solute concentrations in the two so-solutions. The solutes need not be the same so long as they are dilute; in fact, either "solute" may be a mixture of several solutes whose total content is small. The two phases have the same temperature, but do not necessarily have the same pressure since they may be separated by a membrane that restricts their mechanical interaction. The condition of chemical equilibrium of the solvent specie is µå

n(T,På,xå) = µ∫n(T,P∫,x∫) 18.56

Since the mole fraction of solute is small, µå

n(T,På,xå) = µ0ån (T,På) - Txå 18.57


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and µ∫

n obeys a similar equation. Equations 18.56 and 18.57 are sufficient to solve three important problems: the change in the temperature or pressure of two-phase equilibrium between å and ∫ when a solute is added, the influence of a solute on the vapor pressure of a dilute solution, and the osmotic pressure across a semi-permeable membrane that is restrictive to the solute. We solve these in turn. 18.5.2 Influence of a dilute solute on two-phase equilibrium Let two phases of a pure material be in equilibrium at (T0, P0). Then µ0å

n (P0,T0) = µ0∫n (P0,T0) 18.58

If a dilute solute is added to the two phases then the equilibrium temperature and pressure change slightly to satisfy equation 18.56. Since the changes, ∂T and ∂P are small when xå is small, the chemical potential of å at T0 + ∂T, P0 + ∂P can be found in terms of its value at T0, P0:

µ0ån (P0+∂P,T0+∂T) = µ0å

n (P0,T0) +

∆µ0å

n∆T ∂T +

∆µ0å

n∆P ∂P

= µ0å

n (P0,T0) - så∂T + vå∂P 18.59 where så and vå are the molar entropy and volume of the solvent in the å phase. Writing the same equation for the ∫ phase and equating the two chemical potentials to preserve equilibrium yields the equation (xå - x∫)T0 ~ (vå - v∫)∂P - (så - s∫)∂T

~ (vå - v∫)∂P - Qå∫T0

∂T 18.60

where Qå∫ is the latent heat of the transformation å“∫, and we have neglected terms of order xå∂T and (∂T)2. Equation 18.60 determines the change in the temperature and pres-sure of the transformation in terms of the properties of the pure phases and the solute con-centrations. First consider the effect on the å“∫ transformation temperature at constant pres-sure. Since ∂P = 0,

∂T = - T02

Qå∫ (xå - x∫) 18.61


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The sign of the temperature change is fixed by the product of the signs of the latent heat and the composition difference. If å is the high-temperature phase then the latent heat is posi-tive; heat is released. The transformation temperature then decreases if å is relatively rich in solute, as it usually is since the high-temperature phase has higher entropy and can nor-mally accept a greater concentration of the solute. If å is relatively poor in solute the trans-formation temperature rises. A familiar example of this effect is the influence of dissolved salt on the boiling and freezing points of water. Assuming that the solute is relatively insoluble in ice, as most salts are, then x∫, the concentration in the low-temperature phase, is negligible. The freez-ing point is depressed by an amount that is proportional to the concentration of the salt in water:

∂T = - xåT02

Q 18.62

where Q is the (positive) latent heat of freezing. One therefore spreads salt on highways in wintertime to melt the ice. Now consider boiling a solution of salt in water. If the solute is relatively non-volatile, as salt is, then equation 18.62 still holds, but å is now the low-tem-perature phase. The latent heat is negative and the boiling point is raised. One therefore cooks with salty water to raise the boiling point and speed the process. If the temperature is fixed then the transformation pressure is altered by the amount

∂P = T0(xå - x∫)

vå - v∫ 18.63

If ∫ is the high pressure phase then it has the lower molar volume and normally has a lower solubility limit. Hence the transformation pressure is usually raised by adding a solute. An exception is the freezing of water at temperatures near T0. Water has a lower molar volume than ice, but a higher affinity for salt. Hence ∂P is negative, and ice melts at lower pressure when salt is added. 18.5.3 The vapor pressure of a dilute solution: Raoult's Law The change in the vapor pressure of a condensed phase when a dilute solute is added can be computed from equation 18.63. The pressure, P0, at which a condensed phase is in equilibrium with its vapor at temperature T is the vapor pressure at that tempera-ture. When a solute is added the total pressure of the vapor changes to P0 + ∂P, where ∂P is given by equation 18.63. Neglecting the molar volume of the condensed phase (∫) with respect to that of the vapor (å),

∂P = T(xå - x∫)

vå = På(xå - x∫) 18.64


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where we have assumed that the vapor is sufficiently dilute to behave as an ideal gas. Since ∂P = (På

n + Pås ) - P0 = ∂På

n + xåPå 18.65 where ∂På

n is the change in the vapor pressure of the solvent and Pås = xåPå is the vapor

pressure of the solute, then, to within a term of order xå∂P,

∂På

nP0

= - x∫ 18.66

Equation 18.66 is Raoult's Law: the relative decrease in the vapor pressure of a condensed phase on adding a dilute solute is equal to the molar concentration of the solute. 18.5.4 Osmotic pressure Let two dilute solutions with the same solvent be separated by a membrane that is permeable to the solvent but impermeable to the solute. The pressure differs on the two sides of the membrane. The condition for chemical equilibrium of the solvent requires that its chemical potential be the same on both sides. Since the solvent is in the same phase on both sides, µ0

n(På,T) - xåT = µ0n(P∫,T) - x∫T 18.67

where å and ∫ now refer to the two different sides of the membrane. To within a term of the order ˚T(P∫ - På)2, µ0

n(P∫,T) = µ0n(På,T) + v(P∫ - På) 18.68

where v is the molar volume of the solvent at På, T. It follows that the pressure difference across the membrane is

ÎP = (P∫ - På) = Tv(x∫ - xå) 18.69

When the solute is present only on one side,

ÎP = Txv 18.70

where x is the solute concentration and the pressure is greater on the solute side. Equation 18.70 is known as van't Hoff's Law.


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When the solvent is a condensed phase, v is small and equations 18.69-70 hold for large ÎP since the compressibility is small. It follows from equation 18.70 that a dilute solute concentration can generate a substantial pressure. The most familiar example occurs in the supply of water to plants. By absorbing water through an organic membrane that is impermeable to the salts that are added to the water on the inside, a plant can create a pres-sure differential that helps to raise the water to branches and leaves far above the ground. 18.5.5 Dilute solutions with the same solute: Henry's Law Let two dilute solutions with the same solute be in equilibrium with one another. The condition of chemical equilibrium for the solute is µå

1(T,P,xå) = µ∫1(T,P,x∫) 18.71

where å and ∫ are the two phases, the subscript 1 designates the solute, and x is the con-centration of solute. Since both solutions are dilute µdå

1 (T,P) + T ln[xå] = µd∫1 (T,P) + T ln[x∫] 18.72

Hence the ratio of the solute concentrations in the two solutions is

xå

x∫ = exp

µd∫

1 - µdå1

T = ©∫

©å 18.73

where © is the value of the activity coefficient of the solute in the dilute solution limit. A common example of the equilibrium of two solutions with the same solute is the equilibrium of a dilute solution in a condensed phase with its vapor. If phase (å) is the vapor and På

1 is the vapor pressure of the solute, then, approximating the vapor as an ideal gas, equation 18.72 can be written µd∫

1 (T,P) + T ln[x∫] = T ln På1 + ç(T) 18.74

where ç(T) is a function of the temperature only. Equation 18.74 has the solution På

1 = K(P,T)x∫ 18.75 where K(P,T) is independent of the composition. Equation 18.74 is Henry's Law:

the partial pressure of a solute above a dilute solution is linearly proportional to the concentration of the solute.


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A second example, which is more important in materials science, is the equilibrium of liquid and solid or two solid phases in a binary system near x = 0. Let T0 be the tem-perature at which phases å and ∫ are in equilibrium in the pure solvent at reference pres-sure, P. Equation 18.73 then determines the equilibrium concentrations in the two phases as a function of x when a dilute solute is added. This equation hence determines the shape of the two-phase region in the T-x diagram of the binary solution near the x = 0 axis. The ratio xå/x∫ is called the separation coefficient, and is often approximated as the constant, kå∫. 18.5.6 Solute-solvent equilibria between dilute solutions Let two phases, å and ∫, be in equilibrium. Let component (1) be a dilute solute in phase å and the solvent in phase ∫. Hence the activity coefficient of species (1) is a con-stant, ©å

1 , in phase å and is unity in phase ∫. The condition of equilibrium is T ln ©å

1xå = T ln[x∫] 18.76

which shows that the ratio of the concentrations in the two phases is

xå

x∫ = ©å1

-1 18.77

In order for equation 18.77 to be consistent with the problem as it was set we must have ©å

1 >> 1 18.78 so that xå << x∫. If this is not the case the dilute solution of component 1 cannot be in equilibrium with a phase in which it is the solvent. From the definition of the activity coefficient, equation 18.29, as it applies to the dilute solution,

©å1 = exp

µdå

1 - µ01

T 18.79

Equation 18.78 is always satisfied at sufficiently low temperature if µdå

1 > µ01 18.80

Hence the equilibrium is always possible if the chemical potential of component 1 in phase å in the limit of zero concentration is greater than its chemical potential in the pure state. Since the few atoms of species 1 that are present in phase å in the limit of zero concentra-tion are surrounded by atoms of the ∫ solute, equation 18.80 is the formal expression of the


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familiar statement that if species 1 bonds to itself in preference to the ∫ solvent the solution will decompose at low temperature. Note that, given equation 18.80, the activity coefficient of species 1 in the dilute å solution increases without bound as the temperature approaches zero. It follows that

the concentration of a component in a dilute solution that is in equilibrium with a phase in which it is the solvent vanishes as the temperature ap-proaches zero.

It follows that the dilute solution behaves according to the Third Law in the sense its equi-librium concentration approaches zero with the temperature. As a simple extrapolation, consider the behavior of a binary system that decom-poses into two dilute solutions at low temperature. Each component is then present in a solution in which it is dilute in equilibrium with a second in which it is the solvent. Both obey equations of the form 18.77. It follows that the equilibrium state of the binary system approaches a mixture of two pure phases as T “ 0. Moreover, since the chemical poten-tials become independent of temperature in the limit of zero temperature, the solute concen-tration in each phase approaches zero exponentially, according to the relation

xå = exp

- Kå

T 18.81

where xå is the solute concentration in the åth phase and Kå is a positive constant. 18.5.7 The interference of solutes in a dilute ternary solution The governing equations of the "semi-dilute" solution, equations 18.54 and 18.55, are useful to treat the mutual interaction of multiple solutes in a dilute solution. One simple example is the influence of a ternary addition on the solubility of one specie in another. The solubility limit of a binary solution is defined in practice as the maximum amount of the solute specie that can be added, and is, hence, determined by the two-phase equilibrium line in the binary phase diagram. The two-phase equilibrium depends on the properties of both phases, which are ordinarily solutions themselves. When a ternary species is added in dilute concentration its effect on the solubility limit depends on its influ-ence on both of the phases in two-phase equilibrium. To simplify the problem somewhat, let a binary system contain two phases, å and ∫, that are in equilibrium, where å is a dilute solution of component (1) in component (n). Now let a small quantity of a second component, (2), be introduced into (å), but not into (∫). This condition is satisfied to a good approximation when the activity coefficient of (2) in ∫ is much larger than it is in å. It is exact when (2) is introduced into å and å and ∫ are separated by a membrane that is impermeable to (2); a situation that is encountered in prac-tice when the diffusivity of (2) in å is much larger than its diffusivity in ∫. If ∫ is large


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enough to act as a reservoir the relative chemical potential of (1) in å is unchanged by the addition of component (2). Hence –µå

1(xå1) = –µå

1(xå1 + ∂xå

1, xå2) 18.82

where ∂xå

1 is the change in the solubility of (1) due to the addition of (2). The right-hand side of equation 18.82 can be evaluated by expanding about xå

2 = 0. To first order in the concentration variation

0 =

∆ –µå

1∆x1 0

∂xå1 +

∆ –µå

1∆x2 0

xå2

=

T

xå1 + µ11 ∂xå

1 + µ12xå2 18.83

where the µij are the expansion coefficients that appear in equation 18.55. Since xå

1 is small we neglect µ11 with respect to the first term in the bracket. The fractional change in the solubility of (1) is, hence,

∂xå

1

xå1

= -

µ12

T xå2 18.84

Since µ12 = µ21, to the accuracy of equation 18.55 the influence of component (1) on the solubility of (2) is the same as the influence of (2) on the solubility of (1). It follows that if components (1) and (2) have limited solubilities, xå

1 and xå2 , in phase å, the maxi-

mum solubilities in a dilute ternary solution that includes both (1) and (2) differ from these by the amounts

∂xå1 ~ -

µ12

T xå2 xå

1 ~ ∂x2å 18.85

Since the coefficient, µ12, can have either sign the addition of a second element may either increase or decrease the solubility. 18.6 BEHAVIOR NEAR A CRITICAL POINT The preceding sections have described the evaluation of the fundamental equation of a solution by expanding it in powers of the concentration, x, of the solute added to a pure component. The pure component may be a monatomic material, a stoichiometric com-pound; in either case the addition of a second atomic species or compound creates a dilute solution whose fundamental equation can be estimated. In the present section we consider a second case in which the fundamental equation can be evaluated by expansion, when the


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state of the solution is very close to the critical state at the top of a miscibility gap. The method of expansion in this case is very nearly the same as that used near the critical point of a one-component fluid in Chapter 17. A critical point in the concentration of a binary solution occurs where two distinct solutions that are in equilibrium with one another become indistinguishable. The critical point lies at the top of a miscibility gap in the T-x phase diagram, and is located at the tem-perature, Tc, and composition, xc. Above Tc a system with the critical concentration, xc, is a homogeneous solid solution. Below Tc the homogeneous solution decomposes into two solutions with different compositions. In order for a miscibility gap to appear in the phase diagram of a binary solution the two phases must differ only in composition; they must, therefore, have the same symmetry. Miscibility gaps can divide liquid solutions or solid solutions that are based on the same crystal structure, but evidently cannot separate a liquid solution into a solid and a liquid, or a solid solution into two solutions with different crystal structures. It follows from the phase rule that a critical state in a binary solution has one degree of freedom. The critical state is a point in a T-x diagram, but a line in the three-dimensional T-x-P diagram. The general thermodynamic criteria for a critical state in a binary solution follow from the general theory given in Chapter 16, and can be written

∆ –µ

∆x TP = 0 18.86

∆2–µ

∆x2 TP = 0 18.87

∆3–µ

∆x3 TP > 0 18.88

Since a stable solution must satisfy the condition

∆2¡g

∆x2 =

∆ –µ

∆x TP ≥ 0 18.89

the critical point is a stability limit for the homogeneous solution. The fundamental equation for the critical point can be developed just as in the case of the critical point in the one-component fluid. Let ∂x = x - xc 18.90 ∂T = T - Tc 18.91


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Then the compositional derivative of the relative chemical potential can be written, to lowest order,

∆ –µ

∆x TP = ∑1(∂T) + ∑2(∂x)2 18.92

where ∑1 and ∑2 are constant coefficients. It follows from equations 18.88-89 that both ∑1 and ∑2 are positive. Integrating equation 18.92 gives the relative chemical potential near the critical point

–µ(T,x) = –µ(Tc,xc) + ∑1(∂T∂x) + 13 ∑2(∂x)3 18.93

The stability of the solution near the critical point can be found from equation 18.92. Since the equation is quadratic in ∂x, there are two solutions to the homogeneous equation

∆ –µ

∆x TP = 0 18.94

When ∂T > 0 both solutions are imaginary; the binary solution is stable for all concentra-tions near xc. When ∂T < 0 instabilities occur at

∂xi = ± - ∑1∂T

∑2 18.95

which shows that the system must separate into two distinct phases. The compositions of the two phases that are in equilibrium at ∂T < 0 (call them å' and å") are fixed by the con-dition –µ(xå') = –µ(xå") 18.96 which has the solution

∂xå' = - ∂xå" = - 3∑1∂T

∑2 18.97

It follows that the miscibility gap has a parabolic shape near the critical point, and that a parabolic instability or spinodal curve lies within the miscibility gap and converges at the critical point. If a solution is varied in composition at a fixed temperature, ∂T < 0, it is stable in phase å' until the composition reaches the value ∂xå' given by equation 18.97. The solution then enters a region of two-phase equilibrium in which å' and å" coexist. If the nucleation of å" is prevented then å' is preserved until the composition reaches the in-stability at ∂x = - ∂xi. At this point the solution is unstable with respect to transformation into a mixture of å' and å".


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The fundamental equation of the binary solution near the critical point can be found by integrating equation 18.93 with respect to ∂x at constant T. The result is

¡g(T,x) = µn(T,xc) + –µ(Tc,xc)∂x + ∑12 ∂T∂x2 +

∑212 ∂x4 18.98

where µn is the chemical potential of the solvent. This function has a positive curvature when T > Tc. When T < Tc the function has positive curvature to the left of - ∂xi and to the right of ∂xi, but has negative curvature between the two spinodal limits. 18.7 THE SIMPLE SOLUTION It is useful to have a model of a binary solution that yields a simple fundamental equation at all compositions, but can generate binary phase diagrams that have the geomet-ric features most commonly observed in real systems. Such a solution must, of course, have the same symmetry at all compositions, otherwise its fundamental equation could not be continuous for all x. The simplest possible model solution is the ideal solution. However, the ideal so-lution does not represent the behavior of typical real solutions in several respects. It has zero heat of mixing, does not exhibit a miscibility gap, and has non-vanishing entropy in the limit T “ 0. We therefore seek the simplest model of a real solution by perturbing the ideal solution. We require that, like the ideal solution, the Gibbs free energy of a binary solution reduce to the free energies of the pure components when x is zero or one, and yield chemical potentials that have the correct form when either component is dilute. We also require that the enthalpy of mixing be non-zero. 18.7.1 The thermodynamics of the simple solution The simplest realistic model of the binary solution that has these properties has a fundamental equation of the form ¡g(T,P,x) = xµ0

1 + (1-x)µ02 + h0[x(1-x)] + Tx ln(x) + T(1-x) ln(1-x) 18.99

where x is the mole fraction of component 1,

µ0k = µi

k + 12 vkk 18.100

is the chemical potential of the pure component, k, in the mean field approximation, and

h0 = v12 - 12 (v11 + v22) 18.101


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is a constant if the mean interaction potentials, vij , are. We shall call a binary solution that obeys equation 18.99 a simple solution. It is a special case of the regular solution defined by Fowler and Guggenheim (Statistical Thermodynamics). The relative chemical potential of the simple solution is

–µ = µ01 - µ0

2 + h0(1-2x) + T ln

x

1-x 18.102

. The chemical potential of the solvent can be found from the relation µ2 = g - –µx = µ0

2 + h0x2 + T ln(1-x) 18.103 while the chemical potential of the solute is µ1 = –µ + µ2 = µ0

1 + h0(1-x)2 + T ln(x) 18.104 The chemical potentials have the correct behavior in the dilute solution limit. When x is small, µ1 = (µ0

1 + h0 ) + T ln(x) = µd1 + T ln(x)

µ2 = µ0

2 + T ln(1-x) = µ02 - Tx 18.105

The chemical potentials in the limit x “ 1 are obtained from these by interchanging the in-dices 1 and 2. Since the first correction term is quadratic in x (or (1-x)) the simple solution remains dilute in the thermodynamic sense for a reasonable range of concentrations near x = 0 or 1. The heat of mixing is equal to the change in the enthalpy on making the solution

Îh0 = - T2

∆

∆T

Îg0

T Px = x(1-x)h0 18.106

which vanishes only if h0 does, that is, if the solution is ideal. The enthalpy of mixing has the sign of h0. We discussed previously that a negative heat of mixing suggests that the species in solution prefer to bond to one another and hence promotes the formation of solu-tions at high temperature and ordered compounds at low temperature. This qualitative result is consistent with equation 18.101, which shows that h0 is negative when the mean potential of the interaction between the different atoms is less than the average of the inter-action between atoms of the same kind. When h0 is positive the species have an energetic preference to separate, and we should expect the phase diagram of the solution to exhibit a miscibility gap at low temperature.


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18.7.2 Decomposition and ordering in the simple solution To determine the existence of a miscibility gap in the simple solution we compute the second compositional derivative of the free energy, which is

gxx = T

x(1-x) - 2h0 18.107

The second derivative vanishes (that is, the solution reaches a stability limit) when

2h0 = T

x(1-x) 18.108

When h0 < 0 equation 18.108 has no solutions for positive values of x; the solution is sta-ble for all x and T. When h0 > 0, however, equation 18.106 has the two solutions x = x0 and x = (1-x0), where

x0(1-x0) = T

2h0 18.109

These solutions define a spinodal gap that is symmetric about x = 0.5. However, the value of the left-hand side of 18.109 has a maximum at x0 = 0.5, at which point the two branches of the instability curve converge. When T > h0/2 the solutions to 18.109 are imaginary. It follows that the top of the miscibility gap falls at the critical point located at

Tc = h0

2

xc = 0.5 18.110 As T “ 0 the instability points approach x0 = 0 and 1. Since the equilibrium concentra-tions of the two phases lie outside the spinodal lines it follows that the solution decomposes into its two components as T “ 0, in agreement with the Third Law. The simple solution model has two serious limitations in its ability to represent the behavior of real solutions. First, it predicts a symmetric phase diagram with a miscibility gap (if one exists) that is centered at x = 0.5, and predicts that the two species have the same limiting potential in the dilute limit. This problem can be overcome in the model by developing h0 as a power series in (x): h0 = h0 + h1x + h2x2 + ... 18.111


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The second limitation concerns the low-temperature behavior of the solution when h0 < 0. In this case there is no miscibility gap; the solution is retained to arbitrarily low temperatures. But the entropy

s = -

∆ ¡g

∆T = xs01 + (1-x)s0

2 - x ln(x) - (1-x) ln(1-x) 18.112

contains configurational terms that do not vanish in the T “ 0 limit for finite x. Hence the simple solution does not obey the Third Law when h0 < 0. However, equation 18.101 shows that a negative value of h0 occurs in solutions whose species prefer to bond to one another. It follows that a solution with h0 < 0 forms ordered compounds at low tempera-ture. The free energy curves for the ordered phases fall below the free energy of the solu-tion at low temperature, and the phase diagram near T = 0 contains only the pure compo-nents and ordered compounds as stable states. It is straightforward to construct solution models that predict ordering at low temperature, but it is simpler to write these for crystal lattices, so we shall defer a discussion of them until we treat the solid state. 18.8 THE PHASE DIAGRAM OF A BINARY SOLUTION The phase equilibria within a solution are given by the phase diagram, which is just a map of the phase fields as a function of the values of the constitutive coordinates. While most solutions of interest in materials science are, in fact, multicomponent solutions, we shall restrict the discussion here to the phase diagrams of binary solutions, which are rela-tively easy to visualize and interpret in terms of the simple solution model. The value of the phase diagram is two-fold. First, it maps the equilibrium behavior of the solution. Second, while it is extremely difficult to compute the fundamental equation of a solution, the fundamental equation can always be found by expansion about a point where its value is known. The fundamental equation is most easily found at special points in the phase diagram where the fundamental equation must have a simple form or where the fundamental equations of several phases must have equal value. We have already used this principle to discuss the fundamental equation of a dilute solution and a solution near a criti-cal point, and to study phase equilibrium near T = 0 and near a phase transformation point of the pure solvent. In the following we first identify the special points, and then discuss the form of the various possible phase diagrams of binary solutions. 18.8.1 Special points in the T-x diagram A simple first- or second-order expansion is valid for at least some finite range of values of the coordinates near the known point. The positions in the T-x phase diagram of a binary solution where the thermodynamic potential is known, or can be found with rela-tive ease, include the pure component lines at the boundary of the diagram, and particularly the phase transformation points on the pure-component lines, the T = 0 line, where only pure components and ordered compounds can be at equilibrium, and the isolated points that represent critical phases, congruent phases and three-phase equilibria. As we have seen,


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the fundamental equation also has a relatively simple behavior near a limit of stability of the solution which terminates a metastable region, but we shall defer a further discussion of these limits until we treat the thermodynamics of order reactions. The pure component lines lie at the boundaries, x = 0 and x = 1. The thermody-namics of the solution near a pure-component line can be found from the dilute solution model. If the thermodynamics of the pure component are known the fundamental equation of the dilute solution contains only one unknown term: the activity coefficient of the solute. Even at slightly higher concentrations the fundamental equation can be developed in a power series so that only a few terms are required. The phase transformation points on the pure-component lines are points from which two-phase regions emanate that represent di-lute solutions in the two phases. The shape of the two-phase region near the transformation point on the one-component line can be found from dilute solution theory as discussed in Section 18.6.5. The zero temperature line lies at the base of the phase diagram. According to the Third Law the stable phases that fall on this line must be pure components or ordered com-pounds. As we found in Section 18.6.6, both the pure-component phases and the ordered compounds become dilute solutions at finite temperature with solute concentrations that in-crease according to the Arrhenius equation that governs the activity. They are hence dilute solutions over the whole equilibrium range for at least some range of finite temperatures. Since many of the solutions that are important in materials science are effectively at low temperature over the range of practical interest, they can be treated as dilute solutions over the whole range of interest. The critical points in the T-x diagram are the peaks of miscibility gaps, and are as-sociated with the reaction å “ å' + å" 18.113 where å' and å" differ from å (and from one another) only through their compositions. The fundamental equation of the solution near the critical point can be found by mathemati-cal expansion as sown in Section 18.8. The three-phase equilibrium points in the T-x diagram are isolated points in the T-–µ diagram of the solution where three two-phase equilibrium lines meet. In the T-x diagram each of these equilibrium lines splits into a two-phase region with a finite extent. Since the temperature of the three-phase equilibrium is fixed, the three two-phase equilibrium regions meet at an isothermal line along which three phases are in equilibrium. There are three three-phase reaction points on the three-phase line in the T-x diagram. Two of these are the points at which the isothermal three-phase line terminates at a single-phase region, and are the points where the reaction å “ å + ∫ + © 18.114


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occurs when the composition is varied at constant temperature. The third point lies at an interior point of the three-phase line, and determines the reaction å “ ∫ + © 18.115 when the temperature is changed at constant composition. Primarily for historical reasons four different terms are used to describe the three-phase equilibrium lines in the T-x diagram. If å in equation 18.115 is the high-temperature phase, that is, if the arrow is in the direction of decreasing temperature, the special point is called a eutectic point if å is a liquid, and is called a eutectoid point if å is a solid. If å is the low-temperature phase then the special point is called a peritectic point if either ∫ or © is a liquid, and is called a peritectoid point if both ∫ and © are solid. Since the phase diagram is morphologically identical at a eutectic and a eutectoid, and can be made identical at a peritectic or peritectoid by simply reversing the sign of the temperature, the terminology can be simplified substantially by referring to them as a eutectic and inverse eutectic, respec-tively. The three-phase equilibrium determines the compositions of the three phases. In the T-x diagram, the compositions of the terminal solid solutions that lie at the ends of the three-phase line are given by their intersection points, while the composition of the third phase is given by its intersection with the three-phase line at the eutectic point. Since the three two-phase regions are lines in the T-–µ plot that split into bands in the T-x plot, all three one-phase regions are cusped at their intersections with the three-phase line. It is, of course, also possible to find a four-phase equilibrium in a T-x or T-–µ dia-gram. However, it is monumentally unlikely unless the pressure is very carefully selected, since four-phase equilibria are isolated points in the three-dimensional P-T-–µ diagram. Since the chemical potentials of the two components are the same in all three phases at a eutectic, it is possible to evaluate the fundamental equations and interpret the two-phase equilibria near the eutectic by mathematical expansion about the three-phase equilibrium point. A congruent point in the T-x diagram of a binary system is a point where two solu-tions transform into one another without changing composition. The reaction is å “ ∫ 18.116 which occurs at constant composition. Hence the single-phase regions of å and ∫ touch at the congruent point. The phase transformations of the pure components are always con-gruent. In order for å and ∫ to have a congruent point at intermediate x there must be a temperature at which the fundamental curves of the two phases touch without crossing (a point at which two fundamental curves cross is always within a two-phase region by the common tangent rule. To touch without crossing the two curves must be equal and tangent


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at the congruent point, but must have different curvature there. This leads to three mathe-matical conditions: gå(T,P,x) = g∫(T,P,x) 18.117

–µå(T,P,x) =

∆ ¡g

∆x = –µ∫(T,P,x) 18.118

∆ –µå

∆x =

∆2¡gå

∆x2 >

∆ –µ∫

∆x 18.119

that ensure that the reaction å “ ∫ occurs congruently on cooling at the composition, x. If å is the high-temperature phase the congruent point falls at a local minimum in the tempera-ture at which it is stable. If å is the low-temperature phase the congruent point falls at a lo-cal maximum in the temperature of its one-phase field. A mutation, such as the Curie point in iron, is a line in the T-x diagram. Mutations of solutions are always congruent. 18.8.2 Binary phase diagrams Many binary systems contain several solid phases, and hence have phase diagrams with rather complicated forms. However, most of these diagrams can be simplified and understood by breaking them into parts that involve the equilibrium of only a few phases. In this section we consider possible binary phase diagrams for systems that contain one, two or three solid phases, and also describe one common example of a phase diagram with two liquid phases. Almost all binary phase diagrams can be divided into segments whose behavior is like that of one of the diagrams listed below. 18.8.3 Solid solution diagrams

L

å

A B

T

x

Fig. 18.1: The simplest phase diagram for the solid solution. The systems that form solid solutions at all compositions (at least at intermediate temperature) have one of three phase diagrams: the simple solution diagram discussed in


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Section 18.5, or a slight modification of it that has a congruent point either at the top or the bottom of the two-phase (å+L) region. Of course, solid solutions are only possible when the two components have the same crystal structure in the solid state. The simplest phase diagram for the solid solution is re-drawn in Fig. 18.1. This diagram appears when the free energy curve of the solid solution first cuts the liquid free energy curve at the higher of the two melting points of the pure components, and cuts it last at the lower melting point, so there is no congruent point. Many binary systems have this simple phase diagram, including Ag-Au, Ag-Pd, Au-Pd, Bi-Sb, Nb-Ti, Nb-W, Cd-Mg, Cr-W, Cu-Ni, Cu-Pt, Cu-Pd, Hf-Zr, Mo-Ta, Mo-Ti, Mo-V, Mo-W, Ge-Si, Pd-Rh, Ta-Ti, Ta-V, Ta-Zr, U-Zr, and V-W.

L

å

A B

T

x

TA

TB

...

Fig. 18.2: Solid solution with a high-temperature congruent point. However, almost as many binary solutions have congruent points in their phase di-agrams, which shows that the free energy curves of the liquid and the solid solution touch before they cross at x = 0 or x = 1. If the first contact that happens between the liquid and solid free energy curves on cooling falls at an intermediate composition then the system has a elevated congruent point, as in Fig. 18.2. If the last contact on cooling falls at an inter-mediate composition the system has a depressed congruent point, as in Fig. 18.3.

L

å

A B

T

x

TA

TB

Fig. 18.3: Solid solution with a low-temperature congruent point.


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There are very few binary systems with elevated congruent points; the Pb-rich so-lution in the Pb-Tl system is one of the few examples. On the other hand, depressed con-gruent points are common. Au-Cu, Au-Ni, Nb-Mo, Nb-Ni, Nb-V, Co-Pd, Cs-K, Fe-Ni, Fe-Pd, Hf-Ta, Mn-Ni, Mn-Fe, Pu-U, Ti-V and Ti-Zr, show this behavior, among others. Perhaps the strangest example is the behavior of the Ti-Zr system. The high temperature solid structures of both components are BCC, and both transform to HCP on cooling. The phase diagram contains two solid solutions, ∫(BCC) at high temperature and å(HCP) at lower temperature. Both the liquid-å equilibrium and the å-∫ equilibrium are separated by two-phase regions with depressed congruent points like that shown in Fig. 18.3. There is a simple thermodynamic reason for the preference for a low-temperature congruent point. The high-temperature phase is the more disordered one, and generally has a higher entropy of mixing. As a consequence its free energy curve tends to have a deeper trough at intermediate composition, so that the liquid and solid free energy curves contact at intermediate composition at a temperature below the melting points of the pure components. 18.8.4 Low-temperature behavior of a solid solution One of the fundamental laws of thermodynamics is the Third Law, which asserts that the entropy of an equilibrium phase vanishes in the limit T “ 0. The Third Law has the consequence that the equilibrium state cannot be a solid solution in the limit of zero temperature. If a solid solution is cooled sufficiently its equilibrium state must be a per-fectly ordered phase or a simple mixture of perfectly ordered phases. This criterion can be satisfied in two simple ways: the solid solution can decompose into two terminal solutions at low temperature or the system can rearrange itself into an ordered compound or mixture of ordered compounds. In many real systems this low-temperature behavior intrudes at temperatures so low that it is never observed; such systems are solid solutions for all practical purposes. However, in other cases complete solubility is lost at moderate temperature through the formation of either a miscibility gap or an ordered phase. We consider the two possibilities in turn. Phase diagrams containing a miscibility gap A possible binary phase diagram that contains a miscibility gap is shown in Fig. 18.4. The system freezes into a solid solution (å) at all compositions. However, at lower temperature the solid solution spontaneously decomposes into two solid solutions, å' and å'', that have the same structure but different compositions. The two-phase, å' + å'' region within the miscibility gap contains the same information as any other two-phase region in a binary phase diagram. The compositions of the two phases, å' and å'', are determined as a function of temperature by the isothermal tie-lines. The phase fractions are determined from the tie-line by the lever rule. As T “ 0 the compositions of the terminal solid solutions approach x = 0 and x = 1 to satisfy the Third Law.


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As suggested by the simple solution model, a miscibility gap is due to a preference for bonds between atoms of like kind, with the consequence that the energy is lowered when the system segregates into A-rich and B-rich solutions. At higher temperature the en-ergetic preference for decomposition is outweighed by the entropic preference for the solid solution.

L

å

A B

T

x

å' å''

...

Fig. 18.4: The phase diagram of a binary system that contains a miscibility gap in a homogeneous solid solution. The two-phase regions are shown shaded; the horizontal lines are the tie-lines.

The binary systems whose components are mutually soluble at intermediate tem-perature, but become immiscible at lower temperature include Au-Ni, Au-Pt, Cr-Mo, Cr-W, Cu-Ni, Cu-Rh, Ir-Pa, Ir-Pt and Ta-Zr. Ceramic systems such as NiO-CaO also form solid solutions with low-temperature miscibility gaps. According to the Third Law, the miscibility gap must extend to the pure component lines at T = 0, as drawn in Fig. 18.4. Not all of the diagrams that appear in compilations of binary phase diagrams are drawn this way since decomposition is kinetically slow and difficult to observe at low temperature. Phase diagrams with low-temperature ordered phases The phase diagram of a binary system that forms an ordered phase at low tempera-ture is shown in Fig. 18.5. The single-phase region of the ordered phase is closed at a congruent point at its top (T0) and asymptotes to a point in the limit T “ 0, to satisfy the Third Law. The conditions at the two limiting temperatures have the consequence that the ordered phase field has a shape something like that of an inverted teardrop. At finite tem-peratures the ordered phase has at least a slight solubility for the species A and B, and is in equilibrium over a range of compositions about its stoichiometric value. The single-phase © field is bounded by two-phase (å+©) fields that separate it from the single-phase å field on either side. If only one ordered phase is present, then the two-phase regions that bound it must spread across the phase diagram in the limit T “ 0 so that the equilibrium phases at T = 0 are the stoichiometric © ordered phase and the å phase at x = 0 or x = 1, in agreement with the Third Law. In many systems that order, several ordered phases are present.


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L

å

A B

T

x

©

...

Fig. 18.5: Phase diagram of a binary system that has an ordered phase (©) at low temperature. The two-phase regions are shaded with horizontal tie-lines.

The free energy relations that give rise to a phase diagram like that in Fig. 18.5 are illustrated in Fig. 18.6. The free energy curve of the ordered compound lies above that of the å solid solution when T > T0 and passes through it to create a congruent point at T = T0. The free energy curve of the ordered © phase has a strong minimum at its stoichiomet-ric composition. When T < T0 the free energy curve of the © phase lies below that of the å solid solution only at compositions near the stoichiometric value. Hence there are common tangents between the © and å free energy curves on both sides of the © curve. The © phase is the equilibrium phase at compositions near the stoichiometric value; the å phase is at equilibrium at compositions that deviate significantly from the stoichiometric value to either side.

g

xA B

å

©

g

xA B

å

©

åå+©

©å+©

å

14A

...

Fig. 18.6: Free energy relations leading to the appearance of an ordered compound: (a) T > Tc; (b) T < Tc.

Many binary systems whose components have complete or extensive solid solubil-ity at intermediate temperature are known to order into one or more stoichiometric com-


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pounds at lower temperature. Examples include Au-Cu, Cu-Pt, Cd-Mg, Co-Pt, Cu-Pd, Fe-Ni, Fe-Pt, Fe-V, Mn-Ni, Ni-Pt, and Ta-V. 18.8.5 Phase diagrams with eutectic or peritectic reactions Binary systems that have two distinct phases in the solid state often have phase dia-grams of the simple eutectic or peritectic (inverted eutectic) form. The eutectic diagram The simple eutectic diagram is drawn in Fig. 18.7. It takes its name from the eutectic reaction which occurs at the minimum point of the liquid phase field.

T

A Bx

å ∫

L

...

Fig. 18.7: A binary system with a simple eutectic diagram. The two-phase regions are shown shaded with horizontal tie-lines.

A system that has a eutectic phase diagram is usually one whose components have different crystal structures in the pure form. Since components with different structures cannot form a continuous range of solid solutions, there are always at least two phases in the solid state and the eutectic diagram is one of the simplest the system can have. Among the systems with simple eutectic diagrams are Ag-Bi, Al-Ge, Al-Si, Al-Sn, Au-Co, Au-Si, Bi-Cu, Bi-Cd, Bi-Sn, Cd-Pb, Cu-Li, In-Zn, Pb-Sb, Pb-Sn, Si-Zn and Sn-Zn. Ceramic systems with simple eutectic diagrams include MgO-CaO, among others. All of these sys-tems have components with different crystal structures, and are sufficiently different chemically that it is plausible that they form no stable compounds. However, there are also systems that have simple eutectic phase diagrams even though their components have the same crystal structure. Examples include Ag-Cu, in which both components are FCC, Cd-Zn, both components HCP, and Na-Rb, both com-ponents BCC. The most plausible interpretation of the eutectic behavior in this case is that the two terminal solutions have a miscibility gap at a temperature so high that the liquid phase is re-tained to temperatures well below Tc. The relations between the free energy curves that


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lead to a eutectic diagram in a system whose solid phases have a miscibility gap is dia-grammed in Fig. 18.8. Fig. 18.8a diagrams a situation in which a solid phase has decom-posed into two solutions with the same structure, å and ∫, at a temperature at which the liquid is still stable. As the temperature decreases the solid free energy curve drops with respect to the liquid, and leads to a eutectic diagram. Fig. 18.8b shows the situation just above the eutectic point where all three phases appear. Fig. 18.8c shows the situation just below the eutectic point where only å and ∫ solid solutions appear at equilibrium.

g

xA

å ∫

B

L

xA

å ∫

B

Lå ∫å+L ∫+L

L

xA

å ∫

B

å ∫

L

å+∫

... Fig. 18.8: Free energy relations leading to a eutectic diagram for a system

whose components have a miscibility gap at high temperature. (a) Liquid phase stable; (b) three phases appear at lower T; (c) two solid phase appear below the eutectic point.

The peritectic diagram The classic peritectic phase diagram is drawn in Fig. 18.9. It is characterized by the appearance of a peritectic reaction of the form ∫ + L “ å 18.150 that appears at the top of the å field.

A B

x

å

∫

L

T

...

Fig. 18.9: A simple peritectic phase diagram in a binary system.


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The peritectic reaction is, essentially, an inverse eutectic. The classic eutectic reac-tion occurs when the free energy curve of the liquid cuts through a common tangent to the curves of two solid phases on heating. The peritectic occurs when the free energy curve of a solid phase cuts through a common tangent to the curves of liquid and solid phases on cooling. The relations between the free energy curves just above and just below the peri-tectic point are illustrated in Fig. 18.10. Just above the peritectic the common tangent to the L and ∫ phases lies below the å free energy curve, as in Fig. 18.10a. At the peritectic the å free energy curve contacts that common tangent, and drops below it as the system is cooled further to create the configuration shown in Fig. 18.10b.

g

xA

å

∫ L

L L+∫ ∫

B

g

xA

å ∫

L

LL+å

å å+∫ ∫

B

... Fig. 18.10: Free energy relations leading to a peritectic phase diagram. (a)

T just above the peritectic temperature; (b) T just below.

g

xA

å

∫

B

L

xA

å

∫

B

Lå å+L

L

xA

å

∫

B

å ∫

L

å+∫

... Fig. 18.11: A peritectic reaction in a system with a high-temperature misci-

bility gap. (a) The liquid phase is stable at a high temperature below the miscibility gap in the solid; (b) two-phase equilib-rium just above the peritectic; (c) two-phase equilibrium below the peritectic temperature.

Simple peritectic diagrams are much less common that simple eutectic ones. The thermodynamic reason is apparent from Fig. 18.10. To create a peritectic point the free en-ergy curve of the solid must cut that of the liquid at finite x, that is, at a composition away from the axis. For that to happen the free energy of the solid phase must decrease more quickly than that of the liquid at small x. Since the liquid has higher entropy, this is only


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likely to happen when the enthalpy of the solid phase drops rapidly with the solute content, that is, when there is a strong preferential bonding between the two components in the å phase. But this is precisely the situation that leads to the formation of stable compounds between the two components. A system that has a simple peritectic diagram is, therefore, likely to be one that also forms stable compounds. There is at least one example, Ag-Pt (FCC), in which two components with the same crystal structure have a peritectic phase diagram. As in the case of a eutectic diagram between elements with the same structure, this suggests that the components have a high-temperature miscibility gap, leading to free energy relations like those shown in Fig. 18.11. The condition is that the second solid phase that intersects the liquid curve (∫ in the case shown) makes its first appearance as a stable phase by cutting the tie-line between the liquid and the å solid solution. 18.8.6 Structural transformations in the solid state When one of the components of a binary system undergoes a structural transforma-tion on cooling, not only is a new structure introduced into the binary phase diagram, but new two-phase equilibria appear. Since the phases that are connected by the structural transformation are different, they respond differently to the introduction of the solute. The result is a two-phase equilibrium field between them. Figs. 18.12 and 18.13 show the com-mon forms of the binary phase diagram of a system in which one component (A) under-goes a structural transformation (© “ å) as the temperature is lowered. The configurations at the solid-solid transformation are geometrically identical to those at the eutectic and peri-tectic points of the liquid-solid transformation. If å is the low-temperature phase of a component (A) that also has a high-tempera-ture phase, ©, then the free energy of å falls below that of © as the temperature is lowered. On the x=0 axis (where the system has only one component) the two free energies cross at the transition, which happens at a particular temperature. However, since the two phases are distinct they respond differently to the solute, and hence have different free energy curves at finite x. As these curves pass through one another they generate a two-phase re-gion, just as in the liquid-solid case. The shape of the two-phase (å+©) region depends on where the å free energy curve first contacts the © curve as the temperature is lowered. If the first contact is between the å and © curves rather than between the å curve and the ©-∫ common tangent then the behavior is just like that near a eutectic point in the liquid-solid case; the high-temperature phase field (©) extends to a temperature minimum at finite x, as shown in Fig. 18.42. The reaction at the bottom of the © field is © “ å + ∫ 18.151 This reaction is called a eutectoid reaction since it is eutectic-like, but involves only solid phases.


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T

A Bx

å∫

L

∫+L

å + ∫

©©+L

©+∫

...

Fig. 18.12: A binary system with a eutectic reaction at the bottom of the liquid phase field and a eutectoid reaction at the bottom of the phase field of the high-temperature (©) phase. The two-phase fields are shaded with isothermal tie-lines.

In a system that has a eutectoid reaction the first contact between å and © is ordinar-ily at x = 0, in which case the phase diagram near the reaction looks like that shown in Fig. 18.18. However, it is also possible that the first contact occurs slightly off the x-axis at fi-nite composition. In this case the å field has a maximum at a congruent point between © and å, while the © field has a minimum at a eutectic point at slightly higher composition. We shall not illustrate this case. Eutectoid reactions are common in binary systems that include a component that transforms in the solid state. The most familiar example is the ©“å transformation in Fe-Fe3C phase diagram that is the basic phase diagram of carbon steel. Other examples in-clude the ∫“å transformation of Ti in Ti-Cr and Ti-W, the å“∫ transformation in Mn in Ni-Mn, and the å“∫ transformation in Th in Th-U and Th-Zr. Eutectoid reactions also occur in the transformations of many intermetallic and oxide compounds. The second possibility is that the å free energy curve cuts the ©-∫ common tangent as the system is cooled before it contacts the © free energy curve outside the common tan-gent. Then the situation near the transition temperature is like that illustrated for the peri-tectic transition in the liquid-solid case. The å phase field extends to a temperature maxi-mum at finite x, and the configuration near the transition has a shape like that drawn in Fig. 18.18. The reaction at the maximum point of the å field is © + ∫ “ å 18.152 and is called a peritectoid reaction. Peritectoid reactions are reasonably common. Examples include the ©“å transition of Fe in Fe-Nb and Fe-Ta, the å“‰ transition of Co in Co-Cr and Co-W, and the å“∫


Page 435

transition of Mn in Mn-Cr. Peritectic reactions are also found in a number of intermetallic and oxide compounds.

T

A Bx

å

∫

L

©

...

Fig. 18.13: A binary system with a eutectic reaction at the bottom of the liquid phase field and a peritectoid reaction at the top of the phase field of the low-temperature (å) phase. The two-phase fields are shaded with isothermal tie-lines.

18.8.7 Systems that form compounds A substantial fraction of all binary systems form ordered compounds in the solid state. In fact, it is common for several compounds to appear in the phase diagram. To ex-plore the influence of ordered compounds on the shape of the phase diagram we consider systems that contain a single one. Fig. 18.15 illustrates the phase field of a compound that emerges directly from a solid solution. We now consider compounds in systems that con-tain two terminal solid solutions. Four cases are reasonably common: (1) a compound first appears at a congruent point in the liquid; (2) a compound first appears at a peritectic point in a two-phase region (å+L); (3) a high-temperature compound disappears at a eutectoid; (4) a low-temperature first appears at a peritectoid. Finally, we consider the equilibrium phase fields near a structural transformation of an ordered compound. Compounds that form directly from the liquid Many binary systems have stable compounds that can be formed directly from the liquid at a congruent point. The simplest phase diagram for a system of this type is shown in Fig. 18.14. The compound essentially divides the phase diagram into two eutectic dia-grams between the compound and the terminal solid solutions. Phase diagrams like that shown in Fig. 18.14 govern a large number of binary sys-tems, including Al-Sb, Al-Ca, Al-Au, As-In, As-Pb, Ca-Mg, Nb-Cr, Cd-Sb, Cd-Te, Cr-Ta, Cr-Zr, Ga-Sb, Hf-V, In-Sb, Mg-Pb, Mg-Si, Mg-Sn, Mo-Pt, Pb-Te, Sn-Te, and Zn-Te, among others. Note that phase diagrams of this form are particularly common in the III-V and II-VI systems. In these cases the stable compounds are the semiconducting III-V and II-VI compounds.


Page 436

T

A Bx

å∫

L

©

...

Fig. 18.14: The phase diagram of a system that forms a stable compound at an intermediate composition.

The reaction at the congruent point in Fig. 18.14 is L “ ©, where © is the com-pound. Compounds of this type are particularly easy to make since they can be gotten by direct solidification (casting or crystal growth) from a liquid of appropriate composition. Phase diagrams of this type are basic to a number of technologically important processes. Perhaps the most important is the growth of large crystals of III-V and II-VI semiconduct-ing compounds from the melt, which is only possible when the compound has a congruent point with the liquid. Compounds that form through a peritectic reaction If a binary system contains a single compound (©) whose free energy curve is such that its first appearance breaks a solid-liquid (å+L) tie-line then the compound is derived from a peritectic reaction (å + L “ ©) and the simplest phase diagram is like that shown in Fig. 18.15.

T

A Bx

∫

L

å

©

...

Fig. 18.15: The phase diagram of a binary system with an intermediate compound that forms by a peritectic reaction.


Page 437

Several binary systems that have phase diagrams that closely resemble Fig. 18.15, including Bi-Pb, Cd-Sn, Hg-Pb, In-Pb, Hf-W, Mo-Zr, Ru-W and Sn-Tl. The phase dia-grams of Mo-Hf and Sb-Sn differ from Fig. 18.15 only in that the ∫ phase at the far end of the phase diagram has a peritectic rather than a eutectic relation to ©. The phase diagrams of many binary systems that form multiple compounds are such that some of these com-pounds form through peritectic reactions and have local phase relationships like those in the left-hand side of Fig. 18.15. Because the © compound in Fig. 18.15 is the product of a peritectic reaction it can-not be cast or grown directly from the melt. Moreover, many of the more useful com-pounds of this type include elements that diffuse slowly in the solid state so that it is diffi-cult to make the compound by holding the system at a point within equilibrium phase field. Technologically important compounds that have this behavior include the A15 supercon-ducting compounds such as Nb3Sn and Nb3Al, high-temperature intermetallic structural materials like Ni3Al, and low-density intermetallics with potential high-temperature struc-tural applications such as the Al-Ti intermetallics. Complex processing techniques such as reaction from a ternary solution, vapor deposition, or powder processing are required to synthesize these compounds. Finally, note that a phase diagram like that shown in Fig. 18.15 has a eutectic reac-tion, but the phases that border the eutectic include intermediate compounds (© in the fig-ure). Nonetheless, a system that has the eutectic composition will solidify into a eutectic microstructure. One or both of the interleaved phases are intermetallics rather than terminal solid solutions. Compounds that disappear at a eutectoid Many binary phase diagrams contain ordered compounds that only appear at inter-mediate temperature. They are stable at high temperature, but eventually disappear if the system is cooled. For this to happen in a simple system that contains only one ordered compound the common tangent to the free energy curves of the terminal solid solutions must fall beneath the free energy curve of the compound at sufficiently low temperature. This is more likely to happen if the terminal solution is more stable than the compound, and is hence most often observed in systems whose compounds result from a peritectic reaction like that shown in Fig. 18.15. Fig. 18.16 contains a sketch of a simple phase diagram containing a single ordered compound that is confined to intermediate temperature. The top of the phase field of the compound is a peritectic point, å+L “ ©. The phase field terminates in a eutectoid reaction, © “ å+∫. Several binary systems have phase diagrams that resemble Fig. 18.16 very closely, including Bi-Pb, Cd-Sn and Ru-W. Many other systems contain compounds whose ther-mal stability is limited by the intrusion of other ordered compounds.


Page 438

T

A Bx

∫

L

å

©

...

Fig. 18.16: Phase diagram of a simple binary system that forms a com-pound at intermediate temperature.

Compounds that form at a peritectoid If an intermetallic compound first appears in the solid state then it intrudes either into a single-phase region or a two-phase region. In the former case the maximum tempera-ture of the equilibrium field of the compound is a congruent point, as in Fig. 18.5, where the compound forms by a reaction of the type å “ ©. In the latter case the maximum tem-perature is the temperature at which the free energy curve of the compound cuts a two-phase tangent line. The top of the field is a peritectoid point, and the compound forms by a reaction of the type å+∫ “ ©. A simple phase diagram for a binary system with a compound that forms by a peri-tectoid reaction is shown in Fig. 18.18. Several binary systems have phase diagrams that resemble this one, including Ru-Mo, Ru-Nb and Pd-V. In binary systems that contain several compounds it is common that one or more appear at low temperature through peri-tectic reactions.

L

å

A B

T

x

©

∫

...

Fig. 18.17: Phase diagram of a simple system in which a compound ap-pears through a low-temperature peritectoid reaction.


Page 439

Structural transformation of a compound Compounds may undergo structural transformations just as pure phases do. In fact, there is a richer set of possibilities for transformations in compounds, since compounds can change in chemical order as well as in basic lattice structure. A transformation occurs in a compound when there are two separate phases of es-sentially the same compound (nearly the same stoichiometric composition) whose free en-ergies become equal at some temperature and composition. If the high-temperature phase of the compound were cooled to that temperature and composition it would transform ho-mogeneously to the low-temperature phase. If the two phases are related by a first-order transformation, that is, if they are distinct phases at the transformation point, then they are represented by different free energy curves and their first contact on lowering the tempera-ture is at an isolated point. A compound differs from a pure component in that its composition can deviate from stoichiometry in either the positive or the negative sense. At finite temperature the free energy curve of a compound is continuous through its stoichiometric composition and its chemical potential is not infinite there. This has the consequence that the free energy curves of two phases of essentially the same compound (that is, compounds that have the same stoichiometric composition in the limit T “ 0) may first touch one another on cooling at a composition that is off-stoichiometric and possibly outside the equilibrium phase field of the high-temperature phase.

g

å∫

xA Bå ∫

©

©

∂

å+© ∫+©

å∫

xA Bå ∫

©

©

∂

å+© ∫+©

...

Fig. 18.18: Possible shapes of the free energy curves near the transforma-tion ©“∂. (a) The ∂ free energy curve contacts within the © stability range. (b) The ∂ free energy curve contacts the ©-∫ common tangent.

The two possibilities are illustrated in Fig. 18.18, which shows free energy curves at a temperature just above that at which a compound, ©, transforms to a second compound, ∂, in a system whose phase diagram is like that in Fig. 18.14 at temperatures above that shown. In the left-hand figure the free energy curve of the ∂ phase contacts that of the ©


Page 440

phase within its region of stability. In the right-hand figure the ∂ free energy curve contacts the common tangent between the © phase and the primary ∫ solution.

T

A Bx

å∫

L

©

∂

...

Fig. 18.19: Phase diagram of a binary system in which a high-temperature compound, ©, transforms to a low-temperature compound, ∂, at a congruent point, as in Fig. 18.48a.

The situation shown in Fig. 18.18a leads to a phase diagram like that shown in Fig. 18.19. The contact of the © and ∂ free energy curves gives rise to a congruent point in the © phase field at which © “ ∂ without change of composition. The congruent point is en-closed by two-phase (©+∂) fields that terminate at eutectic points for the reactions © “ å+∂ and © “ ∫+∂. Structural transformations of compounds that lead to a phase relationship like that drawn in Fig. 18.19 occur in a number of binary systems, including W-C, Ag-Ga, Ag-Li, Au-Zn, Cu-In, Cu-Sn, Mo-C, Mn-Zn, Ni-S, Ni-Sn and Ni-Sb.

T

A Bx

å∫

L

©

∂

... Fig. 18.20: Phase diagram for a system in which a compound transforms

through a peritectic reaction, as in Fig. 18.48b. When the configuration of the free energy curves near the structural transformation resembles that in Fig. 18.18b, the low-temperature phase, ∂, first appears as the product of


Page 441

a peritectic reaction, ∫+© “ ∂. The given fact that ∂ becomes stable means that the ∂ curve is displaced downward relative to the © free energy curve as the temperature decreases. The form of the phase diagram near the peritectic can be approximated by translating the ∂ free energy curve in 18.18b downwards as T decreases and constructing the successive common tangents. The resulting phase diagram is drawn in Fig. 18.20. The © and ∂ sta-bility fields never touch; they are separated by a narrow two-phase region that terminates at a eutectic point where the reaction is © “ å + ∂. Compound structural transformations of the type that appears in Fig. 18.20 are found in many binary phase diagrams. Among the systems that have reactions of this type are Ag-Cd, Ag-In, Bi-Mg, Co-Cr, Ge-Cu, Cu-Sn, Hf-Ir, Mn-Ni, Mn-Pt, Mn-Zn, Mo-Pt, Ni-V, and Zn-Sb. As this extensive list suggests, the geometry of the transformation in Fig. 18.20 is, in fact, more common than that the congruent geometry shown in Fig. 18.19. Its prevalence reflects the narrow width of the equilibrium phase fields of most solid com-pounds; a small difference in the relative composition dependence of the free energies of the two phases can then shift the first intersection of the two curves out of the stability field of the high-temperature phase. 18.8.8 Mutation lines in binary phase diagrams In a mutation, one phase simply becomes another. There is no two-phase equilib-rium and, hence, there are no two-phase regions associated with mutations.

T

A Bx

å ∫

L

å'

...

Fig. 18.21: A eutectic system with a mutation in the å terminal solid solu-tion, indicated by the dashed line.

However, in a binary system the critical temperature for a mutation can be a func-tion of composition, and almost always is. Hence the mutation appears as a simple curve in a pseudo-single phase region that contains both of the phases that are related by the mu-tation. There is also no discontinuity in the boundary of the pseudo-single phase region where the mutation line contacts it. The composition of the phase in equilibrium in a two-phase region is fixed by temperature. Hence a mutation line is a horizontal isotherm through a two-phase region that gives the temperature at which one phase mutates.


Page 442

T

A Bx

å∫

L

©

©'

...

Fig. 18.22: A binary system with an intermediate compound that under-goes a mutation, indicated by the dashed line.

Fig. 18.21 illustrates the appearance of a eutectic phase diagram with a mutation in the å-rich solid solution. The ferromagnetic transition in Fe and Ni and the rare earths leads to phase relationships like those shown in Fig. 18.21. Fig. 18.22 illustrates the phase relationships in a simple system with an intermediate ordered phase that mutates. Many intermediate compounds undergo ordering reactions that are mutations. The classic exam-ple is the ∫ “ ∫' transition in Cu-Zn. 18.8.9 Miscibility gap in the liquid As a final example we consider a binary system in which a miscibility gap intrudes in the liquid, as it does in many real systems. The simplest system of this type has only the two terminal solid solutions in the solid state. The phase diagram is drawn in Fig. 18.23.

T

A Bx

∫

å

L1 L2

...

Fig. 18.23: A possible phase diagram for a binary system with a miscibil-ity gap in the liquid. The shaded region is an equilibrium be-tween two liquid phases.


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A sequence of free energy relations that lead to a phase diagram like that shown in Fig. 18.23 is drawn in Fig. 18.24. Fig. 18.24a pertains to a temperature just below the melting point of the å phase. The miscibility gap in the liquid is due to the inflection in its free energy curve, which divides it into two stable phases with a common tangent. The three phases å, L1 and L2 appear in the section. Fig. 18.24b is drawn at a lower tempera-ture at which the liquid phase L1 no longer appears. As shown in the diagram the free en-ergy curve of the å phase has dropped with respect to that of the liquid, with the conse-quence that the lowest common tangent connects the å and L2 free energy curves directly. The ∫ free energy curve is everywhere above that of L2. As a consequence there are two phases in the section, å and L2. Fig. 18.24c illustrates behavior at a still lower temperature where both the å and ∫ free energy curves are well below that of the liquid. The lowest common tangent in this case connects å and ∫ directly; only these phases appear in an isothermal section through the phase diagram.

g

xA

å

L' L"

∫

åå+L'L' L'+L" L"

B xA

åL' L"

∫

å

B

å+L" L"

xA

å L' L" ∫

å

B

å+∫ ∫

... Fig. 18.24: Free energy relations at three temperatures in a system with the

phase diagram shown in Fig. 18.53: (a) just below the melting point of å; (b) at a T where only å and L2 appear; (c) at a T where only å and ∫ appear.

The binary systems that have phase diagrams that resemble Fig. 18.24 include Al-Bi, Al-In, Bi-Zn, Cu-Cr, Cu-Pb, Cu-Tl, Ni-Pb, Pb-Zn and Th-U. These systems contain species that are very different in their chemical behavior, which leads to the miscibility gap in the liquid. A like behavior is seen on the silica-rich side of the SiO2-MgO diagram. Similar phase relations are found at low temperature in the solid state in a number of systems that form extensive solid solutions, including Al-Zn, Nb-Zr and Hf-Ta. Phase fields like those in Fig. 18.23 result from a miscibility gap in the solid solution. In the case of Al-Zn the miscibility gap has a bottom because of its interaction with the terminal Zn solid solution. In Nb-Zr and Hf-Ta the bottom of the miscibility gap is due to interference by the low-temperature phase of one of the components; both Hf and Zr have structural transformations at low temperature.

Chapter 18: Solutions - Berkeley · PDF fileNotes on the Thermodynamics of Solids J.W. Morris, Jr.; Fall, 2008 Page 396 Chapter 18: Solutions Chapter 18: Solutions

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