Morphological instabilities during growth: linear stability analyses

Morphological instabilities

Typically, when a liquid alloy solidifies, as heat is extracted and the solid nucleates and grows, even if the initial solid-liquid interface is planar, pretty soon it breaks up — resulting in cellular and dendritic structures; see this page for some samples of such structures and videos (both experimental and simulated). Similar break-up of planar interfaces can also happen when a solid grows in another that is supersaturated, purely by diffusion, at isothermal conditions.

A rigorous mathematical study of these kinds of instabilities of interfaces during growth were pioneered by Mullins and Sekerka in a couple of classic papers [1,2]; as the quote below shows, this work is considered as one of the key steps in the general area of study known as pattern formation:

A determination of the stability of simple solutions to moving-boundary equations with respect to shape perturbations is an important step in the investigation of a wide range of pattern-formation processes. The pioneering work of Mullins and Sekerka on the stability of the growth of solidification fronts and of Saffman and Taylor on moving fluid-fluid interfaces were major advances. The basic approach is to analyze the initial, short time, growth and/or decay of an infinitesimally small perturbation as a function of the characteristic length scale or wavelength of the perturbation. Although the linear stability approach, exemplified by this work, is not always sufficient, it is a basic tool in theoretical morphogenesis.

— Paul Meakin in Appendix A of his Fractals, scaling and growth far from equilibrium

In this blog post, I will talk about the papers of Mullins and Sekerka; the work of Saffman and Taylor as well as the insufficiency of linear stability analyses mentioned in the quote above deserve their own posts; and, may be someday I will write them.

Point effect of diffusion

Here is a schematic showing growth during a phase transformation regulated by (a) flow of heat and (b) diffusion. In case (a), the heat is getting extracted from the left hand side, resulting in the growth of the solid into the liquid. In case (b), it is the diffusion of solute from the supersaturated solid on the right hand side that results in the growth of the solid on the left hand side. In both cases, the interface is shown by the dotted lines in the schematic; though the interface is shown to be planar, as we see below, in most of the cases, it would not remain so.

In the second case, wherein one solid (say, 1) is growing into a supersaturated solid (say, 2), the schematic composition profile will look as shown here:

With the above composition profile, it is easier to see as to why one can expect the interface, shown to be planar in Fig. 1 (b) above can be expected not to remain planar: suppose there is a small protrubation on the planar interface; the sharper the disturbance, the larger the area (or volume) of material ahead from which, by diffusion, the material can be ferried to the interface, resulting in faster growth. This is known as the point effect of diffusion.

The figure below explains the point effect of diffusion:

as opposed to a case where a planar interface would result in a half-circle of radius r_D, where, r_D is the diffusion distance, if there is a protrubation with a sharp end, that sharp end can ferry material from a(n almost) circular region of radius r_D. Thus, it is favourable for the interface to break into a large number of such jagged edges purely from a point of view of growth; however, such jagged interfaces lead to higher interfacial areas and hence higher interfacial energies. Thus, the actual shape of the interface is determined by these two opposing factors – namely, interfacial energy considerations and the point effect of diffusion.

Even though the above explanation was in terms of diffusion, a similar effect can be shown to operate in the case of heat extraction also. In fact the generic way of looking at both is to consider the gradients in these fields– be it composition or temperature. In Chapter 9 of the book Introduction to nonlinear physics (edited by Lui Lam), L M Sander explains the mechanism behind Mullins-Sekerka instability using a schematic of equipotential lines ahead of a bump — since they are bunched up ahead of a protuberance, it grows (p. 200 — Fig. 9.4). Of course, this explanation is a visual version of what Mullins and Sekerka have to say in their paper [1]:

The isoconcerntrates are then bunched together above the protuberances and are rarified above the depressions of the perturbation. The corresponding focussing of diffusion flux away from the depressions onto the protuberances increases the amplitude of the perturbation; we may view the process as an incipience of the so-called point effect of diffusion.

The analysis of Mullins and Sekerka

The mathematical analysis of Mullins and Sekerka [1] is aimed at understanding the morphology of the interface; as they themselves explain:

The purpose of this paper is to study the stability of the shape of a phase boundary enclosing a particle whose growth during a phase transformation is regulated by the diffusion of the material or the flow of heat. … The question of stability is studied by introducing a perturbation in the original shape and determining whether this perturbation will grow or decay.

Of course, in the case of a dilute alloy, during solidification, the solid-liquid interface is known to break-up and this break-up is more complicated since it involves simultaneous heat flow and diffusion; and, in another paper published shortly afterwards [2], Mullins and Sekerka analyse the stability of such an interface:

The purpose of this paper is to develop a rigorous theory of the stability of the planar interface by calculating the time dependence of the amplitude of a sinusoidal perturbation of infinitesimal initial amplitude introduced into the shape of the plane; … the interface is unstable if any sinusoidal wave grows and is stable if none grow.

Both these papers are models of clarity in exposition; Mullins and Sekerka are very careful to discuss the assumptions they make and the validity of the same; they also show how these assumptions are physical in most cases of interest.

As noted above, the actual break-up of an interface is determined by two competing forces — the capillary forces which oppose the break-up and the point effect which promotes break-up; what Mullins and Sekerka achieve through their analysis is to get the exact mathematical expressions (albeit under the given assumptions and approximations) for these two competing terms.

The continuing relevance

There are several limitations associated with the Mullins-Sekerka analysis; it is a linear stability analysis; it assumes isotropic interfacial energies; it neglects elastic stresses, if there be any.

Of course, there are many studies which try to rectify some of these limitations; for example, we ourselves have carried out Mullins-Sekerka type instability analysis for stressed thin films. Numerical studies and nonlinear analyses which look at morphological stability overcome the problems associated with the assumption of linearity that forms the basis of Mullins-Sekerka analysis.

But what is more important is that in addition to being the basis for these other studies, Mullins-Sekerka analysis, by itself, also continues to be of relevance — both from a point of view of our fundamental understanding of some of these natural processes and from a point of view of practical applications of industrial importance. I can do no better than to quote from this (albeit a bit old) news report:

Scientists in the 1940s and ’50s were well aware of instabilities and knew they played a role in formation of dendrites. But until Mullins and Sekerka published their first paper in 1963, no one had ever been able to explain the mechanisms that accounted for instabilities.

The Mullins-Sekerka theory provided a method that scientists and engineers could use to quantify all sorts of instabilities, said Jorge Vinals, an associate professor of computational science and information technology at Florida State University and a former post-doctoral fellow who studied under Sekerka.

Understanding instabilities is the first step in controlling them, so this methodology is important for engineers who need to make industrial processes as stable as possible, Vinals said. Physicists, on the other hand, find that interesting things happen when systems become unstable and so have an entirely different sort of interest in the theory. Mathematicians, for their part, have launched entire fields, such as non-linear dynamics and bifurcation theory, that explore the underlying mathematical descriptions of instabilities.

One example of how the theory has been put to use is in the semiconductor field, where computer chips are made out of large, single crystals of silicon that are sliced into thin wafers. In the early years, these single crystals measured just an inch in diameter; today, 12-inch diameter crystals are produced, resulting in wafers that each can yield hundreds of fingernail-size computer chips.

“You don’t just walk into the lab and build a bigger [silicon crystal] machine because in a bigger machine these instabilities can eat you alive,” Sekerka said. But by understanding the instabilities that occur as liquid silicon crystallizes, engineers have found ways to greatly reduce the formation of dendrites.

Sekerka, a Wilkinsburg native who earned his doctorate in physics from Harvard University, said he and Mullins weren’t thinking about such applications 40 years ago. Though working in a metallurgy department during Pittsburgh’s steel and aluminum heyday, they weren’t especially inspired by the needs of the metals industry, either.

“We were driven by intellectual curiosity more than the need to solve any particular problem,” he said. “Some of the greatest discoveries come from following intellectual curiosity.”

I will end this post with a link to the obituary of W W Mullins (by R F Sekerka H Paxton) and that of his wife June Mullins — to give an idea of the person behind these works.

References:

[1] W W Mullins and R F Sekerka, Morphological stability of a particle growing by diffusion or heat flow, Journal of Applied Physics, 34, 323-329,1963.

[2] W W Mullins and R F Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, Journal of Applied Physics, 35, 444-451, 1964.

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