gel_fit

Fitting using fine-scale polymer distribution in a gel.

Parameter	Description	Units	Default value
scale	Source intensity	None	1
background	Source background	cm-1	0.001
guinier_scale	Guinier length scale	cm^-1	1.7
lorentz_scale	Lorentzian length scale	cm^-1	3.5
rg	Radius of gyration	Å	104
fractal_dim	Fractal exponent	None	2
cor_length	Correlation length	Å	16

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

This model was implemented by an interested user!

Unlike a concentrated polymer solution, the fine-scale polymer distribution in a gel involves at least two characteristic length scales, a shorter correlation length ( \(a1\) ) to describe the rapid fluctuations in the position of the polymer chains that ensure thermodynamic equilibrium, and a longer distance (denoted here as \(a2\) ) needed to account for the static accumulations of polymer pinned down by junction points or clusters of such points. The latter is derived from a simple Guinier function. Compare also the gauss_lorentz_gel model.

Definition

The scattered intensity \(I(q)\) is calculated as

\[I(Q) = I(0)_L \frac{1}{\left( 1+\left[ ((D+1/3)Q^2a_{1}^2 \right]\right)^{D/2}} + I(0)_G exp\left( -Q^2a_{2}^2\right) + B\]

where

\[a_{2}^2 \approx \frac{R_{g}^2}{3}\]

Note that the first term reduces to the Ornstein-Zernicke equation when \(D = 2\); ie, when the Flory exponent is 0.5 (theta conditions). In gels with significant hydrogen bonding \(D\) has been reported to be ~2.6 to 2.8.

Fig. 89 1D plot corresponding to the default parameters of the model.

References

Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C Han, J. Chem. Phys. 1992, 97 (9), 6829-6841

Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R Rennie, Erik Geissler, Macromolecules 1991, 24, 543-548