lamellar_stack_paracrystal

Random lamellar sheet with paracrystal structure factor

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
thickness sheet thickness 33
Nlayers Number of layers None 20
d_spacing lamellar spacing of paracrystal stack 250
sigma_d Sigma (polydispersity) of the lamellar spacing 0
sld layer scattering length density 10-6-2 1
sld_solvent Solvent scattering length density 10-6-2 6.34

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

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 Γm (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 Δρ,
  • thickness is the layer thickness t,
  • Nlayers is the number of layers N,
  • d_spacing is the average distance between adjacent layers D, and
  • sigma_d is the relative standard deviation of the Gaussian layer distance distribution σD/D.

The scattering intensity I(q) is calculated as

I(q)=2πΔρ2ΓmPbil(q)q2ZN(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).

Pbil(q)=(sin(qt/2)qt/2)2

ZN(q) describes the interference effects for aggregates consisting of more than one bilayer. The equations used are (3-5) from the Bergstrom reference:

ZN(q)=1w21+w22wcos(qD)+xNSN+(1xN)SN+1

where

SN(q)=aNN[1+w22wcos(qD)]2

and

aN=4w22(w3+w)cos(qD)4wN+2cos(NqD)+2wN+3cos[(N1)qD]+2wN+1cos[(N+1)qD]

for the layer spacing distribution w=exp(σ2Dq2/2).

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

NL=xNN+(1xN)(N+1)

The 2D scattering intensity is the same as 1D, regardless of the orientation of the q vector which is defined as

q=q2x+q2y
../../_images/lamellar_stack_paracrystal_autogenfig.png

Fig. 50 1D plot corresponding to the default parameters of the model.

Reference

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