Spinodal

Definition

This model calculates the SAS signal of a phase separating system undergoing spinodal decomposition. The scattering intensity $I(q)$ is calculated as

$$ I(q) = I_{max}\frac{(1+\gamma/2)x^2}{\gamma/2+x^{2+\gamma}}+B
$$
where $x=q/q_0$, $q_0$ is the peak position, $I_{max}$ is the intensity at $q_0$ (parameterised as the $scale$ parameter), and $B$ is a flat background. The spinodal wavelength, $\Lambda$, is given by $2\pi/q_0$.

The definition of $I_{max}$ in the literature varies. Hashimoto *et al* (1991) define it as

$$ I_{max} = \Lambda^3\Delta\rho^2
$$
whereas Meier & Strobl (1987) give

$$ I_{max} = V_z\Delta\rho^2
$$
where $V_z$ is the volume per monomer unit.

The exponent $\gamma$ is equal to $d+1$ for off-critical concentration mixtures (smooth interfaces) and $2d$ for critical concentration mixtures (entangled interfaces), where $d$ is the dimensionality (ie, 1, 2, 3) of the system. Thus 2 <= $\gamma$ <= 6. A transition from $\gamma=d+1$ to $\gamma=2d$ is expected near the percolation threshold.

As this function tends to zero as $q$ tends to zero, in practice it may be necessary to combine it with another function describing the low-angle scattering, or to simply omit the low-angle scattering from the fit.

References

H. Furukawa. Dynamics-scaling theory for phase-separating unmixing mixtures: Growth rates of droplets and scaling properties of autocorrelation functions. Physica A 123, 497 (1984).
H. Meier & G. Strobl. Small-Angle X-ray Scattering Study of Spinodal Decomposition in Polystyrene/Poly(styrene-co-bromostyrene) Blends. Macromolecules 20, 649-654 (1987).
T. Hashimoto, M. Takenaka & H. Jinnai. Scattering Studies of Self-Assembling Processes of Polymer Blends in Spinodal Decomposition. J. Appl. Cryst. 24, 457-466 (1991).

Authorship and Verification

**Author:** Dirk Honecker **Date:** Oct 7, 2016
**Last Modified by:** Steve King **Date:** Oct 25, 2018
**Last Reviewed by:** Steve King **Date:** Oct 25, 2018