lamellar_Slab_APL_nW -

    # Note: model title and parameter table are inserted automatically
This model provides the scattering intensity, $I(q)$, for a lyotropic lamellar
phase where a random distribution in solution are assumed. The SLD of the outer,
solvent-exposed region is taken to be different from the SLD of the inner region.

This model is intended to be used with input parameters including the molecualr
volumes and  bound coherrent scattering length of the inner portion of the
amphiphile (solvent not exposed), the outer portion of the amphiphile (outer
solvent exposed), and solvent molecule. 

The variable parameters are intended to be the APL, the average area per
amphiphile (lipid) molecules at the inner/outer interface and n_W, the number
of solvent (water) molecules residing in the outer region of the bilayer.

This model is adapted from the lipid bilayer model of Nagle and Wiener(1988) and fitting function lamellar_hg.

Note that the model can be applied with other combinations of input parameters/
assumptions; and is ideally applied as a simultaneous fit to datasets with
multiple independent measurments; such as neutron contrast variation strategies.


The scattering intensity $I(q)$ is

.. math::

   I(q) = 2\pi\frac{\text{scale}}{2(\delta_H + \delta_T)}  P(q) \frac{1}{q^2}

The form factor $P(q)$ is

.. math::

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

where $\delta_T$ is *length_tail*, $\delta_H$ is *length_head*,
$\Delta\rho_H$ is the head contrast (*sld_head* $-$ *sld_solvent*),
and $\Delta\rho_T$ is tail contrast (*sld_tail* $-$ *sld_solvent*),
$\length_tail$ equals (*Volume_tail*$/$*APL*),
$\length_head$ equals ((*Volume_head*$+$*n_H*$*$*Volume_water*)$/$*APL*),
$\sld_head$ equals (*B_head*$+$*n_H*$*$*B_water*)$/$(*Volume_head*$+$*n_H*$*$*Volume_water*),
$\sld_tails equals (*B_tail*$/$*Volume_tail*).

The total thickness of the lamellar sheet is $\delta_H + \delta_T + \delta_T + \delta_H$.
Note that in a non aqueous solvent the chemical "head" group may be the
"Tail region" and vice-versa.

The 2D scattering intensity is calculated in the same way as 1D, where
the $q$ vector is defined as

.. math:: q = \sqrt{q_x^2 + q_y^2}


.. [#] F Nallet, R Laversanne, and D Roux, *J. Phys. II France*, 3, (1993) 487-502
.. [#] J Berghausen, J Zipfel, P Lindner, W Richtering, *J. Phys. Chem. B*, 105, (2001) 11081-11088
.. [#] Nagle, J., & Wiener, M. (1988). Structure of fully hydrated bilayer dispersions. Biochimica et Biophysica Acta (BBA)-Biomembranes, 942(1), 1-10
.. [#] Tan, L., Elkins J.G., Davison, B.H., Kelly, E.G., Nickels, J.D. Implementation of Slab Model of Bilayer Structure in SASview. Submitted (2020)

Authorship and Verification

* **Author:luoxi Tan, James G. Elkins, Brian H. Davison, Elizabeth G. Kelly and Jonathan D. Nickels

import numpy as np
from numpy import inf

name = "lamellar_Slab_APL_nW"
title = "Random lamellar phase with Head and Tail Groups"
description = """\
    [Random lamellar phase with Head and Tail Groups]
        I(q)= 2*pi*P(q)/(2(H+T)*q^(2)), where
        P(q)= see manual
        layer thickness =(H+T+T+H) = 2(Head+Tail)
        SLD_c = Tail scattering length density
        SLD_h = Head scattering length density
        SLD_w = solvent scattering length density
        background = incoherent background
        scale = scale factor
        B_h = Bound coherrent scattering length of lipid headgroup
        B_c = Bound coherrent scattering length of lipid tails
        B_w = Bound coherrent scattering length of water
        APL = Average area per lipid
        V_w = Molecualar volume of water
        V_h = Molecular volume of lipid headgroup
        V_c = Molecualr volume of lipid tails
        N_w = Number molecules of water molecule permeating into bilayer region
        Dc = Average length of lipid tail
        Dh = Average length of lipid headgroup
category = "shape:lamellae"

# pylint: disable=bad-whitespace, line-too-long
#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["B_h", "1e-6*Ang",       58,   [-inf, inf],  "volume",  "Bound coherrent scattering length of lipid headgroup"],
              ["B_c", "1e-6*Ang",       -9.05984,   [-inf, inf],  "volume",  "Bound coherrent scattering length of lipid tails"],
              ["B_w", "1e-6*Ang", 19.145, [-inf,inf], "volume",    "Bound coherrent scattering length of water"],
              ["APL", "Ang^2", 60,   [0,inf], "volume",    "Area"],
    ["V_w", "Ang^3", 30.4,   [0,inf], "volume",    "Molecualar volume of water"],
    ["V_h", "Ang^3", 211,   [0,inf], "volume",    "Molecular volume of lipid headgroup"],
    ["V_c", "Ang^3", 896.608,   [0,inf], "volume",    "Molecualr volume of lipid tails"],    
          ["N_w", "None", 20,   [0,inf], "sld",    "Number molecules of water molecule permeating into bilayer region"]
# pylint: enable=bad-whitespace, line-too-long

# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?

def Iq(q, 
    SLD_h = (B_h + N_w*B_w)/(V_h +N_w*V_w) #the calculation of sld of headgroup
    SLD_c = (B_c/V_c) #the calculation of sld of tail
    SLD_w = (B_w/V_w) #the calculation of sld of solvent
    Dc = (V_c)/(APL) #the calculation of length of lipid hydrocarbon chains
    Dh = (V_h + N_w*V_w)/APL #the calculation of length of lipid headgroup
    qsq = q*q # q square 
    drh = (SLD_h - SLD_w) #delta rho_H (the head contrast)
    drt = (SLD_c - SLD_w) #delat rho_T  (the tail contrast)
    qT = q*Dc
    Pq = drh*(np.sin(q*(Dh + Dc))-np.sin(qT)) + drt*np.sin(qT)
    Pq *= Pq 
    Pq *=4.0/(qsq)
    inten = 2*np.pi*Pq/qsq
    inten /=2.0*(Dc +Dh)
    return inten

def random(): #the random function which can generate random number for the vriable parameters
    """Return a random parameter set for the model."""
    APL = np.random.uniform(1, 500) 
    N_w = np.random.uniform(0, 100)
    pars = dict(
        APL = APL,
        N_w = N_w
    return pars  

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