softmatter:Radius of Gyration
From NSDL Materials Digital Library Wiki
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) can be expressed as :
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 :
where (Rij)2 is the squared radial distance between beads.
There are various mathematical constructions of the radius of gyration that are equivalent, including:
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 :
- 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.