teubner_strey

Teubner-Strey model of microemulsions

Parameter	Description	Units	Default value
scale	Scale factor or Volume fraction	None	1
background	Source background	cm-1	0.001
volfraction_a	Volume fraction of phase a	None	0.5
sld_a	SLD of phase a	10-6Å-2	0.3
sld_b	SLD of phase b	10-6Å-2	6.3
d	Domain size (periodicity)	Å	100
xi	Correlation length	Å	30

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

Definition

This model calculates the scattered intensity of a two-component system using the Teubner-Strey model. Unlike dab this function generates a peak. A two-phase material can be characterised by two length scales - a correlation length and a domain size (periodicity).

The original paper by Teubner and Strey defined the function as:

\[I(q) \propto \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background}\]

where the parameters \(a_2\), \(c_1\) and \(c_2\) are defined in terms of the periodicity, \(d\), and correlation length \(\xi\) as:

\[\begin{split}a_2 &= \biggl[1+\bigl(\frac{2\pi\xi}{d}\bigr)^2\biggr]^2\\ c_1 &= -2\xi^2\bigl(\frac{2\pi\xi}{d}\bigr)^2+2\xi^2\\ c_2 &= \xi^4\end{split}\]

and thus, the periodicity, \(d\) is given by

\[d = 2\pi\left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} - \frac14\frac{c_1}{c_2}\right]^{-1/2}\]

and the correlation length, \(\xi\), is given by

\[\xi = \left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} + \frac14\frac{c_1}{c_2}\right]^{-1/2}\]

Here the model is parameterised in terms of \(d\) and \(\xi\) and with an explicit volume fraction for one phase, \(\phi_a\), and contrast, \(\delta\rho^2 = (\rho_a - \rho_b)^2\) :

\[I(q) = \frac{8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi} {a_2 + c_1q^2 + c_2q^4}\]

where \(8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi\) is the constant of proportionality from the first equation above.

In the case of a microemulsion, \(a_2 > 0\), \(c_1 < 0\), and \(c_2 >0\).

For 2D data, scattering intensity is calculated in the same way as 1D, where the \(q\) vector is defined as

\[q = \sqrt{q_x^2 + q_y^2}\]

Fig. 123 1D plot corresponding to the default parameters of the model.

Source

teubner_strey.py

References

1. M Teubner, R Strey, J. Chem. Phys., 87 (1987) 3195

2. K V Schubert, R Strey, S R Kline and E W Kaler, J. Chem. Phys., 101 (1994) 5343

3. H Endo, M Mihailescu, M. Monkenbusch, J Allgaier, G Gompper, D Richter, B Jakobs, T Sottmann, R Strey, and I Grillo, J. Chem. Phys., 115 (2001), 580

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