The Cahn-Hilliard equation

Diffusion and microstructural evolution

The Cahn-Hilliard (CH) equation describes microstructural evolution in cases where the microstructure is completely described by the composition field variable:

\frac{\partial c}{\partial t} = \nabla \cdot M \nabla \mu \;\;\;\;\; \dots (1),

where the composition c (\mathbf{r}, t) is a function of position \mathbf{r} and the time t, \mu is the chemical potential and M is the mobility; the operator \nabla is the vector differential operator.

The classical equation that describes microstructural evolution through diffusion is the Fick’s second law, which has the following form:

\frac{\partial c}{\partial t} = \nabla \cdot D \nabla c \;\;\;\;\; \dots (2),

where, D is the diffusion coefficient.

This strong resemblance between Fick’s second law (Eq. 2) and CH equation (Eq. 1) is not accidental; CH can indeed be considered as a generalisation of Fick’s second law; in fact, CH is the result of a study of spinodal decomposition — a phenomenon which also brought out the limitations of Fick’s second law. In this post, I will discuss CH equation in the context of spinodal decomposition.

Spinodal decomposition

Let us consider the phase diagram of a binary alloy of the type shown below; let us consider an alloy of composition c_0 that is quenched from a high temperature T_1 in the single phase region to a temperature T_2 deep in the two phase region.

This phase diagram indicates that the correpsonding free energy as a function of composition, at the temperature T_2, looks, schematically, as shown below.


I have marked five points on the free energy curve; the points A, B and C correspond to the extrema — A and C correspond to minima while B corresponds to a maxima; in other words, the second derivative of the free energy G with respect to composition, \partial^{2} G/\partial c^{2} is positive at A and C and is negative at B; and, at the points marked D and E, the second derivative is zero; i.e., \partial^{2} G/\partial c^{2} = 0.

Suppose the chosen alloy composition c_0 is such that it falls in between the points D and E. In such a system, even a small composition fluctuation will grow spontaneously. What is more, if Fick’s second law is used to describe the system, the movement of atoms will be such that the diffusion is against the concentration gradient — making it necessary to assume that the diffusion coefficient D is negative in this region. This apparent discrepancy is overcome by realising that the diffusion takes place in such a way as to decrease inhomogeneities in chemical potential \mu (and to decrease the total free energy of the system — the chemical potential \mu = \partial G/\partial c (upto a constant)). Thus, the first modification that needs to be done to the Fick’s law is to replace composition gradients by gradients in chemical potential (and to calculate the chemical potential from the free energy) — which results in the CH equation.

Diffuse interface approach

Replacing \nabla c by \nabla \mu, and eliminating the necessity of assuming negative diffusion coefficients in the study of spinodal decomposition is but only one aspect of the CH equation. The second and the more fundamental aspect of CH is related to the fact that the widths of the interfaces in the system are not, a priori, fixed to some value — usually zero. Thus, in contrast to the usual models (like Fick’s law) which arbitrarily fix the interface width to zero (and hence are called sharp interface models), CH equation, by virtue of its allowing the system to choose its own interface width, belongs to another class of models called the diffuse interface models.

In addition to this fundamental correctness (in that this equation does not impose arbitrary restrictions on the system) and consistency (in that this equation is posed in terms of a continuously decreasing free energy), CH equation is also ideal for a numerical  study of microstructural evolution since it does not require that the interfaces be tracked at all times and what would be topological singularities (like the formation of new interfaces or the disappearance of the existing ones in the microstructure) in the corresponding sharp interface model are naturally accounted for in this model.

The CH series of papers

In a series of papers published in The Journal of Chemical Physics, Cahn and Hilliard (and Cahn, by himself) provide the context as well as formulation of the CH equation; the first of these papers, published by Cahn and Hilliard in 1958 [1] discusses the formulation of the free energy which takes into account the interfacial energies that result from composition gradients; the second, published by Cahn in 1959 [2] discusses the thermodynamic basis behind the free eenergy formulation; the third, published by Cahn and Hilliard in 1959 [3], the formulated free energy is used to study phase separation in immiscible liquid mixtures. In a paper published in Acta Metallurgica in 1961 [4], Cahn discusses the study of spinodal decomposition in solids (including the elastic stress effects due to the lattice parameter differences between the two phases). These four papers (sometimes along with another by Cahn and Allen [5] — which is a classic by itself and deserves a separate post for one of the future Giants’ shoulders carnival) forms the theoretical basis for almost all the diffuse interface studies on microstructural evolution in the metallurgical and materials science literature. A more pedagogical exposition of the CH equation can be found in the extremely readable accounts by Cahn [6] and Hilliard [7].

Numerical simulations and online resources

Cahn, in his 1961 Acta Metallurgica paper [4] traces some of the ideas of spinodal decomposition in solids to Gibbs, and in his Institute of Metals lecture in 1968 [6] traces the name and the concept of spinodal to van der Waals, a link to whose paper is available here. It is also well known that the work of Cahn and Hilliard is but a continuum version of that of Hillert (whose Ph D thesis, discussing spinodal decomposition was apparently given to Cahn by Hilliard [8]). However what makes CH as formulated by Cahn and Hilliard very different from the preceding works is (a) the use of diffuse interface approach, (b) the exposition of the formulation in variational terms using thermodynamic principles, and (c) the ease with which CH can be extended and numerically simulated. Thus, it is no wonder that with the extensive use of numerical simulations in the metallurgical and materials literature CH has become the necessary tool in the arsenal of every theoretical materials scientist.

For those of you who might be interested in solving the CH equation numerically, a C code that uses spectral methods (and FFTW) to solve CH spinodal decomposition is available for download (under GPL license) from here. For some more material on CH equation, diffuse interface modelling and references to recent reviews, go here. Here is a page where you can see simulation of spinodal decomposition using CH equation.

References

[1] J W Cahn and J E Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28, 2, pp. 258-267, 1958.

[2] J  W Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, The Journal of Chemical Physics, 30, 5, pp. 1121-1124, 1959.

[3] J W Cahn and J E Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, The Journal of Chemical Physics, 31, 3, pp. 688-699, 1959.

[4] J W Cahn, On spinodal decomposition, Acta Metallurgica, 9, pp. 795-801, 1965.

[5] J W Cahn and S M Allen, A microscopic theory of domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics, Journal de Physique, 38, pp. C7-51, 1977.

[6] J W Cahn, Spinodal decomposition, The 1967 Institute of Metals Lecture, TMS AIME242, pp. 166-180, 1968.

[7] J E Hilliard, Spinodal decomposition, in Phase transformations, American Society for Metals publication, pp. 497-560, 1970.

[8] J W Cahn, Reflections on diffuse interfaces and spinodal decompositons, in The selected works of John W. Cahn, edited by W. Craig Carter and William C. Johnson, TMS, Warrendale, PA , pp. 1-8, 1998.

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8 Responses to “The Cahn-Hilliard equation”

  1. The Giant’s Shoulders: third edition « Entertaining Research Says:

    […] write about a series of papers that laid the foundation for the Cahn-Hilliard equation for the study of microstr…: In a series of papers published in The Journal of Chemical Physics, Cahn and Hilliard (and Cahn, […]

  2. tegar Says:

    Hi there, I was looking for Ref. 6, J.Cahn, Spinodal Decomposition, TMS AIME, but unable to find it. Do you by any chance have a copy of it?

    Thanks.

  3. Vioan Says:

    did you tried CH-muse? did you find the parameters for which you cam obtain spinodal and also nucleation process? thanks.

  4. Katharina Wagner Says:

    Is it possible to get the original papers?
    Especially:
    [7] J E Hilliard, Spinodal decomposition, in Phase transformations, American Society for Metals publication, pp. 497-560, 1970.

  5. Roman Says:

    The link above (http://gururajan.mp.googlepages.com/phase-field) does not work. Could you please fix it?
    Thanks!

  6. RT Says:

    What about BC applied in your C-H code, Neumann or periodic?

  7. How was my blogging last year: 2010 in review « Entertaining Research Says:

    […] The Cahn-Hilliard equation September 2008 7 comments 5 […]

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