lamellar_hg_stack_caille

Random lamellar head/tail/tail/head sheet with Caille structure factor

Parameter	Description	Units	Default value
scale	Scale factor or Volume fraction	None	1
background	Source background	cm-1	0.001
length_tail	Tail thickness	Å	10
length_head	head thickness	Å	2
Nlayers	Number of layers	None	30
d_spacing	lamellar d-spacing of Caille S(Q)	Å	40
Caille_parameter	Caille parameter	None	0.001
sld	Tail scattering length density	10-6Å-2	0.4
sld_head	Head scattering length density	10-6Å-2	2
sld_solvent	Solvent scattering length density	10-6Å-2	6

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

This model provides the scattering intensity, \(I(q) = P(q)S(q)\), for a lamellar phase where a random distribution in solution are assumed. Here a Caille \(S(q)\) is used for the lamellar stacks.

The scattering intensity \(I(q)\) is

\[I(q) = 2 \pi \frac{P(q)S(q)}{q^2\delta }\]

The form factor \(P(q)\) is

\[P(q) = \frac{4}{q^2}\big\{ \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right] + \Delta\rho_T\sin(q\delta_T)\big\}^2\]

and the structure factor \(S(q)\) is

\[S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right) \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right)\]

where

\begin{align*} \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right) && \\ \gamma_E &= 0.5772156649 && \text{Euler's constant} \\ \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} && \text{Caille constant} \end{align*}

\(\delta_T\) is the tail length (or length_tail), \(\delta_H\) is the head thickness (or length_head), \(\Delta\rho_H\) is SLD(headgroup) - SLD(solvent), and \(\Delta\rho_T\) is SLD(tail) - SLD(headgroup). Here \(d\) is (repeat) spacing, \(K\) is smectic bending elasticity, \(B\) is compression modulus, and \(N\) is the number of lamellar plates (Nlayers).

NB: When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect. And due to a complication of the model function, users are responsible for making sure that all the assumptions are handled accurately (see the original reference below for more details).

Non-integer numbers of stacks are calculated as a linear combination of results for the next lower and higher values.

Be aware that the computations may be very slow.

The 2D 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. 48 1D plot corresponding to the default parameters of the model.

Source

lamellar_hg_stack_caille.py \(\ \star\ \) lamellar_hg_stack_caille.c

References

1. F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502

2. J Berghausen, J Zipfel, P Lindner, W Richtering, J. Phys. Chem. B, 105, (2001) 11081-11088

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