Gel Fit

*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

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