Immiscibility–Two Mechanisms of Phase

There may be partial miscibility between two components in the solid state, which is usually proportional to T , unless the individual components undergo a phase change. To examine this phenomenon, a regular solid solution of two components is considered, which can be used to predict the pattern of the phase separation of the two partially miscible components. In light of Eq. (4.28), ΔmixGm = RT (xAln xA+ xBln xB) + ωxAxB. The first term of RT (xAln xA+ xBln xB) is the one that applies to ΔmixGm for an ideal solution. The second term ωxAxB represents the nonideality of the mixture.

If ω has a small positive value, ΔmixSm ensures that the two components of A and B mix in more or less all proportions. As ω increases, ΔmixSm is less able to dominate the positive ΔmixHmand ΔmixGm, although remaining negative overall, acquires a shape with two minima and one maximum, i.e., becoming partially immiscible as illustrated in Fig. 4.8(a).

The compositions of the two immiscible phases, x_B,1 and x_B,2, are eas-ily determined using the common tangent method described in the preced-ing section, which are those correspondpreced-ing to the two minima in the curve of ΔmixGm, where the first derivative of the curve is zero and the second is positive. This method has been used to calculate the phase diagram as illustrated in Fig. 4.8(b). Differentiating with respect to xB,∂ΔmixG_m

∂xB

= RT ln

 xB

1− xB



+ ω(1− 2xB) = 0. To satisfy the equation needs two xB val-ues denoting the minima corresponding to the two immiscible phases in equi-librium. The equation has a maximum at xB= 1/2. The solution of this equa-tion, when plotted against T , yields a line of miscibility gap in Fig. 4.8(b).

4.5 Equilibrium Conditions of Phases and Phase Diagram of Binary Systems 141

Further differentiation gives

∂²ΔmixGm

∂x²_B = RT

 1 xB

+ 1

(1− xB)



− 2ω = 0. (4.38)

Fig. 4.8 Plots of ΔmixGmversus xBfor a regular solution (a) and the correspond-ing phase diagram havcorrespond-ing the miscibility gap, showcorrespond-ing spinodal line (b).

This can be solved to give the upper consolute temperature (the temper-ature at which the two solutions just become miscible) as T_a = ω/2R, i.e., the plot has no curvature. In this phase diagram, the single-phase region, T > T_a, indicates that A and B are completely miscible. At T < T_a, the solution is separated into two phases of α1and α2. At points under the mis-cibility gap, the phase compositions vary with T . For example, at 800 K the phase compositions x_B,1 and x_B,2 are in equilibrium.

Another feature of the phase behavior is related to the points of inflection in the curves of ΔmixGm with respect to xB when their second derivative equals zero (Eq. (4.38)). These inflection points, called spinodal points, have a special significance in the study of phase transitions. The locus of spinodal points can be indicated with a dash line in phase diagrams as in Fig. 4.8(b).

To appreciate the importance of the spinodal curve, we consider the region to the right of the spinodal point in Fig. 4.9, where ΔmixGmcurve is concave downward and has regions of a negative curvature (∂²ΔmixGm/∂x²_B < 0).

In this region the solution may begin the process of decomposition into the equilibrium phases by incremental changes in composition without increasing Gmof the system, called the spinodal decomposition mechanism. A different situation exists in the region to the left of the spinodal point (the inflection point), in which the curvature is positive (∂²ΔmixGm/∂x²_B > 0). Hence, as

142 Chapter 4 Phase Diagrams

the material is separated into two phases, Gm of the system must increase before it can finally decrease, called the nucleation and growth mechanism.

This difference in path for Gmduring decomposition results in a difference in phase transition behavior, which is important for kinetics. It will be useful to discuss the two different phase transition mechanisms in the context of Gm

curves.

Fig. 4.9 The relation between ΔmixGm and composition on either side of the spinodal point.

The dynamics of phase separations is driven by fluctuations. In gen-eral, if a fluctuation leads to a decrease in ΔmixGm, it will happen spon-taneously. Consider a part of ΔmixGm curve where the curvature is nega-tive, ∂²ΔmixGm/∂x²_B < 0. Suppose that a very small fluctuation occurs and consider what happens to ΔmixGm for the small fluctuation. Apparently, ΔmixGm change is negative for an arbitrarily small fluctuation in a compo-sition that one part of the system gets more concentrated at the expense of another. The system is inherently unstable and this process is called spinodal decomposition under a condition,

∂²ΔmixGm

∂x²_B < 0. (4.39)

Now we consider the part of the curve where the curvature is positive but inside the miscibility gap (two-phase region). Apparently, ΔmixGm here in-creases. Therefore, small fluctuations in this region do not lead to phase separation and the system is “stable”. In other words, it is metastable with respect to infinitesimal composition fluctuations. Such a system is clearly unstable to the separation into the limiting compositions given by the com-mon tangent construction. How does the system phase separate? Clearly an average composition within the two-phase region, but outside of the spin-odal curves, requires large composition fluctuations to decrease the energy. A process requiring a large composition fluctuation is called “nucleation”. Nu-cleation is a phase transition that is large in degree (composition change) but small in extent (size). After a nucleus forms, the new phase grows. Together,

4.5 Equilibrium Conditions of Phases and Phase Diagram of Binary Systems 143

the process is called nucleation and growth and the transition is discontin-uous, whereas spinodal decomposition is small in degree but large in extent and it is continuous.

Nucleation is a topic of wide interest in many scientific studies and techno-logical processes, which is the starting of a phase transition in a small region.

It is used heavily in industry. The most widely used examples are directional solidification of vanes in different engines and the growth of monocrystalline silicon in semiconductor industry. Recently, these ideas are also utilized in the growth of quantum dots, nanowires, nanobelts, single crystalline thin films, etc. in nanotechnology.

In solidification, nucleation is the formation of a crystal phase from liq-uid phase. Nucleation without preferential nucleation sites is a homogeneous one, which occurs spontaneously and randomly, but it requires supercool-ing or superheatsupercool-ing of the medium. The creation of a nucleus implies the formation of an interface at the boundaries of a new phase. The change in Gibbs free energy is balanced by the energy gain of creating a new vol-ume, and the energy cost due to appearance of a new interface, expressed as ΔG = −(4π/3)r³ΔGv+ 4πr²γsL where ΔGv = GL − G^s being a change in Gibbs free energy per unit volume between the two phases, which relate to the ΔT as described in Eq. (4.32). The greater the supercooling or ΔT , the larger the ΔGv, which favors phase transition. When the solid nucleus is spherical in a radius r, we have

ΔG(r, T ) =−(4π/3)r³ΔGv(T ) + 4πr²γsL. (4.40) The first and second terms of the right hand of Eq. (4.40) and their addition (ΔG(r, T )) are given in Fig. 4.10. ΔG achieves maximum ΔG^∗ at r = r^∗, which means that the nucleus will certainly grow when r > r^∗. The nucleus with r^∗ is termed the critical nucleus. Letting dΔG/dr = 0, it reads

r^∗= 2γsL

V ΔGv

= 2γsLTmV

Δ^L_sHmΔT. (4.41)

Equation (4.41) shows that r^∗ ∝ 1/ΔT . Substituting Eq. (4.41) into Eq.

(4.40) gives the nucleation barrier energy or the driving force for nucleation ΔG^∗,

ΔG^∗= 16 3

πγsL³ T_m²V²

ΔH_m²ΔT². (4.42)

ΔG^∗ is an amount to overcome the additional energy associated with the interface or the interfacial energy, which will be discussed in Chapter 6 in detail.

Heterogeneous nucleation occurs much more often than homogeneous nu-cleation. It forms at preferential sites such as phase boundaries or impurities like dust and requires less energy than homogeneous nucleation. At such pref-erential sites, the effective interface energy is lower, thus diminishing ΔG^∗and facilitating nucleation. An interface promotes nucleation because γss < γsL

144 Chapter 4 Phase Diagrams

Fig. 4.10 ΔG(r, T ) function during solidification.

where subscript ss denotes solid-solid interface, which encourages particles to nucleate. The above inequation can be quantitatively described by another amount – the wetting contact angle θ between two phases, which is always greater than zero. The Gibbs free energy needed for heterogeneous nucleation is equal to the product of homogeneous nucleation and a function of θ,

ΔGhetero= ΔGhomo× f(θ) (4.43)

where f (θ) = 1/2 + 3 cos θ/4− cos³θ/4 < 1.ΔGhetero < ΔGhomo leads to less ΔT needed. θ determines the ease of nucleation by reducing the energy needed. It is important to note that r^∗remains unchanged while volume of the critical nucleus can be significantly small for heterogeneous nucleation where θ affects the shape of the nucleus. During the heterogeneous nucleation, some energy is released by the partial destruction of the previous interface. For ex-ample, precipitate particles can be formed at grain boundaries of a solid. This can interfere with precipitation strengthening, which relies on homogeneous nucleation to produce a uniform distribution of precipitate particles.

4.6 On Approximation of Gibbs Free Energy Change of