## Posts Tagged ‘Cahn Hilliard equation’

### The Giant’s Shoulders: third edition

September 16, 2008

This is, by far, the smallest Giant’s Shoulders blog carnival; and, I hope it will remain so in future too. The next edition of the carnival will be hosted by Doctor Silence at second order approximation on October 15, 2008. So, have fun with this one and be ready with your entries for the next one!

[1] 350 BC Aristotle on mayfly

At any rate, the more I read Aristotle, and the more I understand both where things stood at his time and what he actually said, I find him to be an amazing natural historian, a good observer, and generally not a bad theoretician. Sure, his theories are wrong, and his overall philosophy of teleology in biological cases (not, I hasten to add, in his physics) is unnecessary now we have teleosemantic explanations (i.e., natural selection), but he is not the moron of popular history of biology; far from it.

Funnily enough, the EMBOR author of the canard, Katrin Weigmann, is trying to make the case that science is not infallible, while ignoring the very real actual achievements of the people she denigrates. Nobody thought science was infallible anyway, but trying to make out that errors were made where they weren’t doesn’t give one much confidence in any subsequent argument. And of course this is another case of scientists doing bad history for [scientific] political reasons.

[2] 1817 Defining Parkison’s disease

I definitely recommend Parkinson’s monograph, partially because it’s always interesting to read the old lit, and also because his case descriptions are incredibly vivid and empathetic. Although his methods of treatment probably brought little real cure, he was a compassionate physician and a brilliant man of his time, who put together all the dots to define what we now call Parkinson’s Disease.

[3] 1958/59 Modelling spinodal decomposition

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.

[4] 1963 Interference between different photons

One of the most famous statements concerning quantum mechanics, as it relates to the light particles known as photons, was made by the brilliant scientist Paul Dirac in his Quantum Mechanics book:

“each photon then interferes only with itself.  Interference between different photons never occurs.”

This statement is bold and unambiguous: in Dirac’s view, a photon only creates interference patterns by virtue of its own wave function, and wave functions of different photons do not interact.

The statement is bold, unambiguous, often quoted — and wrong!  In 1963, Leonard Mandel and G. Magyar of Imperial College disproved this statement with a clever and simple experiment and a two-page paper in Nature.

[5] 1973 Beginnings of Genetic Engineering

Cohen had previously determined how to make E. coli take in foreign DNA (a citation classic worthy feat in itself) when he transformed E. coli with a plasmid known as pSC101, that conferred resistance to the antibiotic tetracycline.

Boyer on the other hand had discovered EcoRI, a restriction enzyme that could snip open pSC101 while leaving “sticky ends“.

Like chocolate and peanut butter, the combination was unbeatable. Cohen and Boyer realized they could combine their techniques to create a new plasmid containing foreign DNA.

[6] 1977 Categorizing fundamental types of living beings

The “plant” and “animal” distinction is pretty classic – until comparatively recently, bacteria were assumed to be “plants”, just as fungi (”plants” that lacked chlorophyll) were. Non-photosynthetic bacteria were referred to as “schizomycetes” (literally “fission” [splitting in two] fungi, because they reproduce by splitting from one cell into two rather than forming spores), while bacteria with chlorophyll (cyanobacteria or “blue-green” algae, and possibly the “green sulfur bacteria”) were designated “schizophyta” (”fission plants”).

Within the last fifty years or so, though, it’s become obvious that bacteria were a different type of life from fungi, chlorophyll-containing plants, or animals. The latter critters have cells that in turn contain “organelles”, which are more or less very specialized “mini-cells” within themselves. The nucleus, for example, is a compartment within the cell where the cell’s DNA is kept and processed. Bacteria, it turned out, don’t have any of these organelles (in fact there’s good evidence that at least some if not all organelles used to be bacteria, but this post’s long enough already so I won’t go into that), and life was re-organized into the bacterial “prokaryotes” (”before nucleus”) and the “eukaryotes” (having a “true nucleus” – i.e. everything that isn’t bacteria).

From this paper we get the the modern fundamental three groups we still use today: Eukaryotes, Eubacteria [“True” bacteria], and the Archaea (or “Archaebacteria” as this paper names it). The name comes from the idea that the environment in which methanogens thrives resembles what has often been assumed to be the atmosphere of the very early Earth.

### The Cahn-Hilliard equation

September 13, 2008

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.