# 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
fractal_dim Fractal exponent None 2
cor_length Correlation length 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.

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