be_polyelectrolyte

Polyelectrolyte with the RPA expression derived by Borue and Erukhimovich

Parameter	Description	Units	Default value
scale	Source intensity	None	1
background	Source background	cm-1	0.001
contrast_factor	Contrast factor of the polymer	barns	10
bjerrum_length	Bjerrum length	Å	7.1
virial_param	Virial parameter	Å3/mol	12
monomer_length	Monomer length	Å	10
salt_concentration	Concentration of monovalent salt	mol/L	0
ionization_degree	Degree of ionization	None	0.05
polymer_concentration	Polymer molar concentration	mol/L	0.7

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

Definition This model calculates the structure factor of a polyelectrolyte solution with the RPA expression derived by Borue and Erukhimovich[1]. Note however that the fitting procedure here does not follow the notation in that reference as ‘s’ and ‘t’ are not decoupled. Instead the scattering intensity \(I(q)\) is calculated as

\[ \begin{align}\begin{aligned}I(q) = K\frac{q^2+k^2}{4\pi L_b\alpha ^2} \frac{1}{1+r_{0}^2(q^2+k^2)(q^2-12hC_a/b^2)} + background\\k^2 = 4\pi L_b(2C_s + \alpha C_a)\\r_{0}^2 = \frac{1}{\alpha \sqrt{C_a} \left( b/\sqrt{48\pi L_b}\right)}\end{aligned}\end{align} \]

where

\(K\) is the contrast factor for the polymer which is defined differently than in other models and is given in barns where \(1 barn = 10^{-24} cm^2\). \(K\) is defined as:

\[ \begin{align}\begin{aligned}K = a^2\\a = b_p - (v_p/v_s) b_s\end{aligned}\end{align} \]

where \(b_p\) and \(b_s\) are sum of the scattering lengths of the atoms constituting the monomer of the polymer and the sum of the scattering lengths of the atoms constituting the solvent molecules respectively, and \(v_p\) and \(v_s\) are the partial molar volume of the polymer and the solvent respectively

\(L_b\) is the Bjerrum length(Å) - Note: This parameter needs to be kept constant for a given solvent and temperature!

\(h\) is the virial parameter (ų/mol) - Note: See [1] for the correct interpretation of this parameter. It incorporates second and third virial coefficients and can be Negative.

\(b\) is the monomer length(Å), \(C_s\) is the concentration of monovalent salt(mol/L), \(\alpha\) is the ionization degree (ionization degree : ratio of charged monomers to total number of monomers), \(C_a\) is the polymer molar concentration(mol/L), and \(background\) is the incoherent background.

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

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

Fig. 81 1D plot corresponding to the default parameters of the model.

References

[1]	(1, 2) V Y Borue, I Y Erukhimovich, Macromolecules, 21 (1988) 3240

[2]	J F Joanny, L Leibler, Journal de Physique, 51 (1990) 545

[3]	A Moussaid, F Schosseler, J P Munch, S Candau, J. Journal de Physique II France, 3 (1993) 573

[4]	E Raphael, J F Joanny, Europhysics Letters, 11 (1990) 179

Authorship and Verification

• Author: NIST IGOR/DANSE Date: pre 2010
• Last Modified by: Paul Kienzle Date: July 24, 2016
• Last Reviewed by: Paul Butler and Richard Heenan Date: October 07, 2016