Polymer Excl Volume

This model describes the scattering from polymer chains subject to excluded volume effects and has been used as a template for describing mass fractals.

Definition

The form factor was originally presented in the following integral form (Benoit, 1957)

$$ P(Q)=2\int_0^{1}dx(1-x)exp\left[-\frac{Q^2a^2}{6}n^{2v}x^{2v}\right]
$$
where $\nu$ is the excluded volume parameter (which is related to the Porod exponent $m$ as $\nu=1/m$ ), $a$ is the statistical segment length of the polymer chain, and $n$ is the degree of polymerization.

This integral was put into an almost analytical form as follows (Hammouda, 1993)

$$ P(Q)=\frac{1}{\nu U^{1/2\nu}} \left\{ \gamma\left(\frac{1}{2\nu},U\right) - \frac{1}{U^{1/2\nu}}\gamma\left(\frac{1}{\nu},U\right) \right\}
$$
and later recast as (for example, Hore, 2013; Hammouda & Kim, 2017)

$$ P(Q)=\frac{1}{\nu U^{1/2\nu}}\gamma\left(\frac{1}{2\nu},U\right) - \frac{1}{\nu U^{1/\nu}}\gamma\left(\frac{1}{\nu},U\right)
$$
where $\gamma(x,U)$ is the incomplete gamma function

$$ \gamma(x,U)=\int_0^{U}dt\ \exp(-t)t^{x-1}
$$
and the variable $U$ is given in terms of the scattering vector $Q$ as

$$ U=\frac{Q^2a^2n^{2\nu}}{6} = \frac{Q^2R_{g}^2(2\nu+1)(2\nu+2)}{6}
$$
The two analytic forms are equivalent. In the 1993 paper

$$ \frac{1}{\nu U^{1/2\nu}}
$$
has been factored out.

**SasView implements the 1993 expression**.

The square of the radius-of-gyration is defined as

$$ R_{g}^2 = \frac{a^2n^{2\nu}}{(2\nu+1)(2\nu+2)}
$$
.. note:: This model applies only in the mass fractal range (ie, $5/3<=m<=3$ ) and **does not apply** to surface fractals ( $3<m<=4$ ). It also does not reproduce the rigid rod limit (m=1) because it assumes chain flexibility from the outset. It may cover a portion of the semi-flexible chain range ( $1<m<5/3$ ).

A low-Q expansion yields the Guinier form and a high-Q expansion yields the Porod form which is given by

$$ P(Q\rightarrow \infty) = \frac{1}{\nu U^{1/2\nu}}\Gamma\left( \frac{1}{2\nu}\right) - \frac{1}{\nu U^{1/\nu}}\Gamma\left( \frac{1}{\nu}\right)
$$
Here $\Gamma(x) = \gamma(x,\infty)$ is the gamma function.

The asymptotic limit is dominated by the first term

$$ P(Q\rightarrow \infty) \sim \frac{1}{\nu U^{1/2\nu}}\Gamma\left(\frac{1}{2\nu}\right) = \frac{m}{\left(QR_{g}\right)^m}\left[\frac{6}{(2\nu +1)(2\nu +2)} \right]^{m/2} \Gamma (m/2)
$$
The special case when $\nu=0.5$ (or $m=1/\nu=2$ ) corresponds to Gaussian chains for which the form factor is given by the familiar Debye function.

$$ P(Q) = \frac{2}{Q^4R_{g}^4} \left[\exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right]
$$
For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the $q$ vector is defined as

$$ q = \sqrt{q_x^2 + q_y^2}
$$

References

H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247
B Hammouda, *SANS from Homogeneous Polymer Mixtures - A Unified Overview, Advances in Polym. Sci.* 106 (1993) 87-133
M Hore et al, *Co-Nonsolvency of Poly(n-isopropylacrylamide) in Deuterated Water/Ethanol Mixtures* 46 (2013) 7894-7901
B Hammouda & M-H Kim, *The empirical core-chain model* 247 (2017) 434-440

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