Radius of Gyration - Molecular Applications

Molecular Applications

In polymer physics, the radius of gyration is used to describe the dimensions of a polymer chain. The radius of gyration of a particular molecule at a given time is defined as:

R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2},

where is the mean position of the monomers. As detailed below, the radius of gyration is also proportional to the root mean square distance between the monomers:

R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{2N^{2}} \sum_{i,j} \left( \mathbf{r}_{i} - \mathbf{r}_{j} \right)^{2}.

As a third method, the radius of gyration can also be computed by summing the principal moments of the gyration tensor.

Since the chain conformations of a polymer sample are quasi infinite in number and constantly change over time, the "radius of gyration" discussed in polymer physics must usually be understood as a mean over all polymer molecules of the sample and over time. That is, the radius of gyration which is measured is an average over time or ensemble:

R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N} \langle \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2} \rangle,

where the angular brackets denote the ensemble average.

An entropically governed polymer chain (i.e. in so called theta conditions) follows a random walk in three dimensions. The radius of gyration for this case is given by

Note that although represents the contour length of the polymer, is strongly dependent of polymer stiffness and can vary over orders of magnitude. is reduced accordingly.

One reason that the radius of gyration is an interesting property is that it can be determined experimentally with static light scattering as well as with small angle neutron- and x-ray scattering. This allows theoretical polymer physicists to check their models against reality. The hydrodynamic radius is numerically similar, and can be measured with Dynamic Light Scattering (DLS).

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