.. _spinodal:

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       |Ang^-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

.. math::
    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

.. math::
    I_{max} = \Lambda^3\Delta\rho^2

whereas Meier & Strobl (1987) give

.. math::
    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.


.. figure:: img/spinodal_autogenfig.png

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


**Source**

:download:`spinodal.py <src/spinodal.py>`

**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