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## gel_fit

Fitting using fine-scale polymer distribution in a gel.

Parameter | Description | Units | Default value
--- | --- | --- | ---
scale | Scale factor or Volume fraction | None | 1
background | Source background | cm-1 | 0.001
guinier_scale | Guinier term scale | cm^-1 | 1.7
lorentz_scale | Lorentz term 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 (\(\xi\)) to describe the rapid fluctuations in the position of the polymer chains that ensure thermodynamic equilibrium (based on an Ornstein-Zernicke, or Lorentz, model), and a longer distance (denoted here as \(R_g\)) needed to account for the static accumulations of polymer pinned down by junction points or clusters of such points (based on a simple Guinier model). The relative contributions of these two contributions, \(I_L(0)\) and \(I_G(0)\), are parameterised as lorentz_scale and guinier_scale, respectively.

See also the lorentz model and the gauss_lorentz_gel model.

Definition

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

\[I(Q) \approx \frac{I_L(0)}{\left(1+\left[(D+1)/3\right]Q^2\xi^2 \right)^{D/2}} + I_G(0) \cdot \exp\left( -Q^2R_{g}^2/3\right) + B\]

Note that the first term reduces to the Ornstein-Zernicke equation when the fractal dimension \(D = 2\); 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. 108 Figure 1: 1D plot corresponding to the default parameters of the model.

Source

`gel_fit.py` \(\ \star\ \) `gel_fit.c`

References

1. Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C Han, J. Chem. Phys. 1992, 97 (9), 6829-6841. DOI: 10.1063/1.463637

2. Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R Rennie, Erik Geissler, Macromolecules 1991, 24, 543-548. DOI: 10.1021/MA00002A031

Authorship and Verification

- Author:

- Last Modified by: Steve King Date: November 22, 2022

- Last Reviewed by: Paul Kienzle Date: November 21, 2022