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Spinodal decomposition

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A term introduced in the physical description of unmixing of metallic alloys by J.W. Cahn [a1], cf. Fig.a1.

Figure: s110240a

Schematic plot of a "quenching experiment" of a mixture and the resulting build-up of concentration fluctuations in a mixture. Shown is the temperature () versus concentration () plane, while the third axis () is a spacial coordinate.

The thermodynamic state of such a mixture is described by the variables temperature and relative concentration , and one considers a situation where in the "phase diagram" of this system one finds a "miscibility gap" , i.e. there is a curve in the -plane in Fig.a1 (this curve is labelled by the two branches , in Fig.a1, which merge in a critical point , ) underneath which the system cannot exist in a state of homogeneous concentration in thermal equilibrium, while it does exist in such a state above this curve (e.g., in the initial state at a temperature , in Fig.a1).

One now considers a rapid cooling experiment (quenching) where the system is brought at time from this temperature above the coexistence curve to a temperature below the so-called spinodal curve , defined by the condition that the second derivative of the free energy density vanishes. In this regime, the homogeneous state is intrinsically unstable, as is found from a linear stability analysis of concentration fluctuations [a1]. According to such a linear stability analysis, all long wavelength concentration fluctuations with wavelengths exceeding a critical wavelength are unstable, and the maximum growth rate occurs at a wavelength . While, according to such a linear theory, one would expect that this wavelength dominates in the late stages of the phase separation process (Fig.a1), actually the process is highly non-linear [a2], and so-called "coarsening" occurs (there is a dominant wavelength , but it depends upon the time after the start of the process, as [a2]).

If the thermal fluctuations are neglected, spinodal decomposition is described by the following non-linear diffusion equation for the local concentration at a point of -dimensional infinitely extended space:

(a1)

where has the physical meaning of a mobility, comes from a gradient energy contribution, denotes the Laplace operator, and the free energy density can, e.g., be written in terms of as

with , constants. A parabolic spinodal curve results from

The linear stability analysis of (a1) yields, writing ,

(a2)

and is solved by introducing spacial Fourier transforms, , i.e., one finds an exponential time dependence (),

(a3)

One sees from (a3) that the rate is positive if

and

The "critical wavelength" mentioned above is .

Of course, it is clear that the linear analysis is relevant at best for the early stages of the process. Solving the full non-linear partial differential equations (a1) is a challenging problem of numerical mathematics. For applications in physics additional complications occur: in the initial stages of unmixing, the deterministic equation (a1) needs to be amended by a stochastic "random force" term to represent statistical fluctuations. In addition there occurs a coupling to further dynamical variables (in solids: the elastic displacement field ; in fluid mixtures: the velocity field ). Nevertheless, (a1) is a very useful starting point, and numerous experimental applications exist [a2].

References

[a1] J.W. Cahn, "On spinodal decomposition" Acta Metall. , 9 (1961) pp. 795–801
[a2] K. Binder, "Spinodal decomposition" P. Haasen (ed.) , Materials Science and Technology. Phase Transformations in Materials , 5 , VCH (1991) pp. 405–471
How to Cite This Entry:
Spinodal decomposition. K. Binder (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Spinodal_decomposition&oldid=15567
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098