Scatter mass radius

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When analyzing the structure of soft matter , the scattering mass radius (also called gyration radius or sometimes radius of gyration ) is an important parameter for characterizing the spatial expansion of irregularly shaped particles . The more compact an object, the smaller its scattering mass radius. It can be determined by scattering experiments .

The definition of the scattering mass radius is similar to that of the moment of inertia . If the particles consist of N similar building blocks with position vectors , the square of the scattering mass radius is defined as the mean square distance of the building blocks (for example the monomers of a polymer chain) to the center of gravity of the particle: ${\displaystyle {\vec {r}}_{1}\dots {\vec {r}}_{\text{N}}}$ ${\displaystyle R_{\text{G}}}$ ${\displaystyle {\vec {r}}_{\text{S}}}$

${\displaystyle R_{\text{G}}^{2}={\frac {1}{N}}\sum _{i=1}^{N}\left|{\vec {r}}_{i}-{\vec {r}}_{\text{S}}\right|^{2}}$.

If the mass distribution in the particle is given by a mass density , the following results for the scattering mass radius: ${\displaystyle \rho ({\vec {r}})}$

${\displaystyle R_{\text{G}}^{2}={\frac {1}{M}}\int {\mbox{d}}^{3}r\,\rho ({\vec {r}})\left|{\vec {r}}-{\vec {r}}_{\text{S}}\right|^{2}}$,

where M is the mass of the particle.

In the extreme case of a homogeneous sphere with radius  R :

${\displaystyle R_{\text{G, Kugel}}={\sqrt {\frac {3}{5}}}\cdot R}$

The scattering mass radius can be determined by scattering experiments on a dilute suspension or solution of the particles: for small scattering vectors , the structure function S can be approximated by ${\displaystyle {\vec {q}}}$ 

${\displaystyle S({\vec {q}})\approx N\left(1-{\frac {q^{2}\cdot R_{\text{G}}^{2}}{3}}\right),}$

it then only depends on the number  N of the building blocks and the scatter mass radius. This relationship is known as Guinier's Law .

Light scattering is often suitable for measuring the scattering mass radius of suspended colloids or dissolved polymers . In order to characterize the spatial extension of the tangled chain molecules in a polymer melt , small-angle neutron scattering is a good choice ; In order to obtain a scatter contrast, a small proportion of deuterated polymer is added to the melt .

Individual evidence

1. ^ Friedrich R. Schwarzl: Polymer mechanics: Structure and mechanical behavior of polymers . Springer, 1990, ISBN 3-540-51965-3 ( limited preview in Google book search).