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. 106 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