Lamellar Stack Paracrystal

# Note: model title and parameter table are inserted automatically This model calculates the scattering from a stack of repeating lamellar structures. The stacks of lamellae (infinite in lateral dimension) are treated as a paracrystal to account for the repeating spacing. The repeat distance is further characterized by a Gaussian polydispersity. **This model can be used for large multilamellar vesicles.**

Definition

In the equations below,

- *scale* is used instead of the mass per area of the bilayer $\Gamma_m$ (this corresponds to the volume fraction of the material in the bilayer, *not* the total excluded volume of the paracrystal),

- *sld* $-$ *sld_solvent* is the contrast $\Delta \rho$,

- *thickness* is the layer thickness $t$,

- *Nlayers* is the number of layers $N$,

- *d_spacing* is the average distance between adjacent layers $\langle D \rangle$, and

- *sigma_d* is the relative standard deviation of the Gaussian layer distance distribution $\sigma_D / \langle D \rangle$.

The scattering intensity $I(q)$ is calculated as

$$ I(q) = 2\pi\Delta\rho^2\Gamma_m\frac{P_\text{bil}(q)}{q^2} Z_N(q)
$$
The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness $t$ (compare the equations for the lamellar model).

$$ P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2
$$
$Z_N(q)$ describes the interference effects for aggregates consisting of more than one bilayer. The equations used are (3-5) from the Bergstrom reference:

$$ Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)} + x_N S_N + (1 - x_N) S_{N+1}
$$
where

$$ S_N(q) = \frac{a_N}{N}[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2
$$
and

$$ a_N = 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) \\ quad - 4w^{N+2}\cos(Nq \langle D \rangle) + 2 w^{N+3}\cos[(N-1)q \langle D \rangle] + 2w^{N+1}\cos[(N+1)q \langle D \rangle]
$$
for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$.

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

$$ N_L = x_N N + (1 - x_N)(N+1)
$$
The 2D scattering intensity is the same as 1D, regardless of the orientation of the $q$ vector which is defined as

$$ q = \sqrt{q_x^2 + q_y^2}
$$

Reference

#. M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, *J. Phys. Chem. B*, 103 (1999) 9888-9897

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