g(c) miscibility gap spinodal stable nonlinearly unstable linearly unstable 0 c-cs-cs+ c+ 1 VIII. Phase Transformations Lecture 38: Nucleation and Spinodal Decomposition MIT Student In this lecture we will study the onset of phase transformation for phases that differ only in their equilibrium composition, keeping in mind that it is possible for phases to differ in other physical quantities such as density, crystal structure, magnetization, etc. Consider a regular solution model for free energy, illustrated in figure 1. Suppose a homogeneous system is in region of its phase diagram (P,T,c)Regular solution model Figure 1 where the homogeneous phase is unstable thermodynamically. The system can lower its free energy by separating into two phases with an interface in between. In figure 1, c and c+ are common tangent points that mark the −boundaries of the miscibility gap. For any composition inside the miscibility gap (between c and c+), a phase separated system is energetically favorable. −1Lecture 38: Nucleation and spinodal decomposition 10.626 (2011) Bazant Outside the miscibility gap, the system remains homogeneous at equilibrium no matter what the composition. In his classical treatment of phase transformations, Gibbs distinguished between two types of transformations: those that are small in degree and large in extent (spinodal decomposition, linear instability), and those that are large in degree and small in extent (nucleation, nonlinear instability). Let’s examine both in a bit more detail. 1 Spinodal decomposition Compositions between cs and cs+ lie within the chemical spinodal, a re-−gion of linear instability where g""(c) < 0 and small fluctuations will grow spontaneously through a process called spinodal decomposition. Previously we showed that g¯""(c) < 0 is the condition for linear instability by graphical construction (ignoring variations of µ with ∇c). Spontaneous phase sepa-ration is favored when the chord to a free energy curve is lower everywhere than the curve itself: (a) g!!(c) < 0 promotes spinodal decom-position. Sp ontaneous separation into c0 + ∆c and c0 − ∆c lowers the total free energy (∆c << c0). (b) g!!(c) > 0 does not promote spon-taneous phase separation. Small com-position fluctuations raise the total free energy. 2 Classical nucleation Nucleation is a nonlinear instability that requires the formation of a large enough nucleus of the nucleating phase. The creation of a nucleus of the low-energy phase with concentration c+ form a matrix of concentration c∗ (where c− <c∗ <cs), is illustrated in figure 2a. There is a decrease in free −energy ∆¯g associated with the conversion of c to c+, and an increase in free ∗energy due to the interfacial energy γ. There is a critical radius where these 2Lecture 38: Nucleation and spinodal decomposition 10.626 (2011) Bazant (a) A simple spherical nucleus with a sharp interface.rcradius4πr4/3πr ∆g32γ 0 (b) A free energy barrier must be over-come for the nucleus to grow. Figure 2 ∆G two opposing effects exactly balance each other, as illustrated in figure 2b. Recall that the surface area of a sphere is proportional to r2, but its volume is proportional to r3. A nucleus larger than the critical radius will grow due to the volume free energy decrease, and a nucleus smaller than the critical radius will shrink due to its proportionally high surface area. The growth rate of a nucleus is proportional to ∆Ge− kT . The free energy change for the creation of a nucleus with radius r is: ∆G = V ∆g¯ + Aγ 4 (1) = πr 3∆g¯ +4πr 2γ3where ∆¯g =¯g(c+) − g¯(c ) is the volume free energy savings (thus ∆g<¯ 0) ∗and γ is the surface energy. The particle will grow when d∆G > 0 and shrink dr when d∆G < 0. At the critical radius: drd∆G =4πr 2∆g¯ +8πrγ = 0 (2)dr The critical radius is: 2γ r = rc = (3)|∆g¯| Unfortunately, the classical nucleation model generally does not agree well with experiment. Treating interfacial energy as a constant and modeling the interface as a sharp discontinuity are too simplistic. The discrepancy 3Lecture 38: Nucleation and spinodal decomposition 10.626 (2011) Bazant remained until Cahn and Hilliard introduced a model that treated the in-terface as a smoothly varying, diffuse quantity. 3 The Cahn-Hilliard model The Cahn-Hilliard model adds a correction to the homogeneous free energy function to account for spatial inhomogeneity. This correction comes from a Taylor expansion of g¯ in powers of ∇c combined with symmetry consid-erations. Similar models were used by Van der Walls to model liquid-vapor interfaces, and by Landau to study superconductors. While composition in a homogeneous system is a scalar, composition becomes a field for an inhomogeneous system. Thus the Cahn-Hilliard free energy is a functional of the composition field: 1 G[c(x)] = N2V !g¯(c)+ κ(∇c) dV (4)V 2g¯(c) is the homogeneous free energy, and NV is the number of sites per volume. κ∇2c, the so-called gradient energy, is the first-order correction for inhomogeneity which introduces a penalty for sharp gradients, allowing interfacial energy to be modeled. 3.1 Chemical potential Chemical potential is a scalar quantity that is only defined at equilibrium. Here we will introduce a chemical potential that is defined for an inhomo-geneous system, away from equilibrium. If we assume local equilibrium, we can define this potential field by employing the calculus of variations: δG G[c(x)+%δ !(x − y)] − G[c(x)] µ = = lim (5)δc !→0 %See 2009 notes for mathematical details. The variational derivative may be found by applying the Euler-Lagrange equation: δG ∂I d ∂I = (6) δc ∂c −dx ∂∇c where I is the integrand of G[c(x)]. The functional derivative of Eq. 4 is: µ =¯g"(c) − κ∇2c (7) 4Lecture 38: Nucleation and spinodal decomposition 10.626 (2011) Bazant 3.2 Anisotropy For a crystal, κ is anisotropic and must be represented as a tensor K. The free energy functional and corresponding chemical potential are: ! 1 G = NV g¯(c)+ c K c dV (8a)V 2∇ · ∇δG µ = =¯g"(c) cδ−∇ Kc · ∇ (8b) 3.3 Evolution equation for c In previous lectures we have arrived at a generalized expression for the flux of concentration: F'= −Mc∇µ (9) Since c is a