Spinodal decomposition

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s110240a.gif" />

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 (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102401.png" />) versus concentration (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102402.png" />) plane, while the third axis (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102403.png" />) is a spacial coordinate.

The thermodynamic state of such a mixture is described by the variables temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102404.png" /> and relative concentration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102405.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102406.png" />-plane in Fig.a1 (this curve is labelled by the two branches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102408.png" /> in Fig.a1, which merge in a critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s1102409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024010.png" />) underneath which the system cannot exist in a state of homogeneous concentration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024011.png" /> in thermal equilibrium, while it does exist in such a state above this curve (e.g., in the initial state at a temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024013.png" /> in Fig.a1).

One now considers a rapid cooling experiment (quenching) where the system is brought at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024014.png" /> from this temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024015.png" /> above the coexistence curve to a temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024016.png" /> below the so-called spinodal curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024017.png" />, defined by the condition that the second derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024018.png" /> of the free energy density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024020.png" /> are unstable, and the maximum growth rate occurs at a wavelength <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024021.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024022.png" />, but it depends upon the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024023.png" /> after the start of the process, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024024.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024025.png" /> [a2]).

If the thermal fluctuations are neglected, spinodal decomposition is described by the following non-linear diffusion equation for the local concentration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024026.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024028.png" />-dimensional infinitely extended space:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024029.png" />

(a1)

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024030.png" /> has the physical meaning of a mobility, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024031.png" /> comes from a gradient energy contribution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024032.png" /> denotes the Laplace operator, and the free energy density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024033.png" /> can, e.g., be written in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024034.png" /> as

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024035.png" />

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024037.png" /> constants. A parabolic spinodal curve results from

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024038.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024039.png" />

The linear stability analysis of (a1) yields, writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024040.png" />,

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024041.png" />

(a2)

and is solved by introducing spacial Fourier transforms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024042.png" />, i.e., one finds an exponential time dependence (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024043.png" />),

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024044.png" />

(a3)

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024045.png" />

One sees from (a3) that the rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024046.png" /> is positive if

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024047.png" />

and

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024048.png" />

The "critical wavelength" mentioned above is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024049.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024050.png" />; in fluid mixtures: the velocity field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110240/s11024051.png" />). 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