# spinodal

Spinodal decomposition model

Parameter |
Description |
Units |
Default value |
---|---|---|---|

scale |
Scale factor or Volume fraction |
None |
1 |

background |
Source background |
cm |
0.001 |

gamma |
Exponent |
None |
3 |

q_0 |
Correlation peak position |
Å |
0.1 |

The returned value is scaled to units of cm^{-1} sr^{-1}, absolute scale.

**Definition**

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

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

whereas Meier & Strobl (1987) give

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.

**Source**

**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