# softmatter:Radius of Gyration

The radius of gyration Rg is a measure of the distribution of a collection of particles around the center of mass of the particles. The radius of gyration can be represented as scalar value or as a vector quantity (termed the principle radii of gyration). The radius of gyration and principle radii of gyration are often applied to individual polymers or aggregates of particles/polymers (such as spherical micelles).

## End to End distance

A flexible, freely jointed polymer chain--a chain with no Angle Bending or Bond Rotation restrictions--can access a huge number of configurations, however, over long times, the distribution of configurations is equivalent to a random walk process. The distance between the ends of the polymer chain composed of n beads, average bond length l, and characteristic ratio (polymer) $C_\infty$ can be expressed as [1]:

$\langle R^2 \rangle_0 = C_\infty n l^2$ ,

where,

$C_\infty = \frac{b_n^2}{l^2}$

where, bn is the effective random-walk step of the polymer.

## Radius of Gyration (scalar)

If we consider a polymer chain composed of n beads, the radius of gyration is defined as the root-mean-square average distance between any bead on the chain and center of mass of the chain. This can be expressed as [1]:

$R_g^2 = \left \langle \frac{1}{2}n^{-2}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}(R_{ij})^2\right) \right \rangle$,

where (Rij)2 is the squared radial distance between beads.

There are various mathematical constructions of the radius of gyration that are equivalent, including:

$R_g^2 = \left \langle \frac{1}{n} \sum_{k=1}^{n} \left( \mathbf{q}_{k} - \mathbf{q}_{com} \right)^{2} \right \rangle$,

where qk corresponds to the coordinates of particle k, and qcom is the cooridnates of the location of the center of mass of the chain.

For a freely jointed polymer chain, the radius of gyration can be related to the end to end polymer length as follows [1]:

$R_g = \frac{\langle R^2 \rangle_0^{1/2}}{\sqrt{6}}$