spinodal

Spinodal decomposition model

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

gamma

Exponent

None

3

q_0

Correlation peak position

Å-1

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

\[I(q) = I_{max}\frac{(1+\gamma/2)x^2}{\gamma/2+x^{2+\gamma}}+B\]

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.

../../_images/spinodal_autogenfig.png

Fig. 120 1D plot corresponding to the default parameters of the model.

Source

spinodal.py

References

  1. 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).

  2. 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).

  3. 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