University:
Massachusetts Institute of Technology
Course:
3.205 | Thermodynamics and Kinetics of Materials
Academic year:

2024
Views:

152

Pages:

1
Author:

Rylee Roach

Massachusetts Institute of Technology Department of Materials Science and Engineering 77 Massachusetts Avenue, Cambridge MA 02139-4307 3.205 Thermodynamics and Kinetics of Materials Kinetics Lecture 10: Spinodal Decomposition Lecture References 1. Porter and Easterling, Phase Transformations in Metals and Alloys, Van Nostrand Reinhold (Interna tional), Wokingham, 1988. Section 5.5.5, pp. 308–314. 2. Ballufﬁ, Allen, and Carter, Kinetics of Materials, Chapters 17 and 18 (advanced treatment; also in cludes discussion of continuous order-disorder transformations—not covered in 3.205). Key Concepts • Spinodal decomposition is the name of the mechanism by which an initially uniform but composition ally unstable solution can transform to the equilibrium two-phase mixture. A homogeneous solution is unstable inside the “spinodal” region deﬁned by ∂ 2 f /∂c2 < 0, where f is an appropriate free energy per unit volume for the homogeneous solution. In the absence of elastic strain energy accompanying compositional variations, f is the solution’s chemical free energy per unit volume and ∂ 2 f /∂c2 = 0 deﬁnes the “chemical spinodal.” • As a system inside the spinodal region develops composition variations, the regions where there are composition gradients make an extra contribution to the system’s free energy, such that the total free energy of a system with an arbitrary one-dimensional composition proﬁle c(x) is given by � ∞� � F [c(x)] = A f (c) + K (dx/dx)2 dx (1) −∞ where K is a positive constant known as the gradient-energy coefﬁcient. • When gradient energy makes a signiﬁcant contribution to the system’s free energy (i.e., when the density of regions with composition gradients is high), gradient energy consumes some of the available driving force for diffusion, and the appropriate diffusion equation becomes � 2 � � 4 � ∂c ∂ c 2K ∂ c � =D − (2) 2 2 2 ∂t ∂x ∂ f /∂c ∂x4 • Inside the spinodal region where ∂ 2 f /∂c2 < 0, (??) will have solutions obtainable by separation of variables, with the form c = c◦ + eR(β)t cos βx (3) and substitution into the diffusion equation shows that the ampliﬁcation factor R(β) must obey � � � � 2K 2 � R(β) = −Dβ 1 + β2 (4) ∂ 2 f /∂c2 At a given temperature within the spinodal region, the maximum rate of spinodal decomposition will occur for waves with β = βmax at which dR(β)/dβ = 0.

3.205 Lecture 10 Review

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