# softmatter:Radius of Gyration

The radius of gyration R_{g} 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).

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## 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) can be expressed as [1]:

,

where,

where, *b*_{n} 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]:

,

where (*R*_{ij})^{2} is the squared radial distance between beads.

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

,

where *q*_{k} corresponds to the coordinates of particle *k*, and *q*_{com} 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]:

## References

## Additional Sources

- The radius of gyration of a polymer chain is addressed in depth in chapter 2.2 of the first edition of The Structure and Rheology of Complex Fluids starting on page 71.