Radius of gyration

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The radius of gyration [R_] describes the distribution of particles (or infinitesimal elements) in a D-dimensional space by relating it to an equivalent distribution in a D-dimensional sphere, usually a circular (D=2) or spherical (D=3) distribution.

Contents

1 Definition
2 Molecular applications
3 Applications in structural engineering
4 Applications in mechanics
5 Derivation of identity
6 References

Definition

The formal definition for [N] particles is

[R_^ \equiv \frac \sum_^ \left( \mathbf_ - \mathbf_ \right)^]

where [\mathbf_] is the mean position of the particles

[\mathbf_ \equiv \frac \sum_^ \mathbf_ ]

The radius of gyration is also proportional to the root mean square distance among the particles

[R_^ \equiv \frac} \sum_ \left( \mathbf_ - \mathbf_ \right)^]

(The identity of this second definition with the first is derived below.)

The squared radius of gyration can also be computed by summing the principal moments of the gyration tensor.

If the positions of the particles are not constant, the radius of gyration can be generalized by taking the ensemble average, e.g.,

[R_^ \equiv \frac \langle \sum_^ \left( \mathbf_ - \mathbf_ \right)^ \rangle]

similar to the hydrodynamic radius, where the angular brackets [\langle \ldots \rangle] denote the ensemble average.

Molecular applications

The radius of gyration (possibly multiplied by a constant factor) is a useful estimate of the size of a molecule. Size exclusion chromatography ideally separates molecules by their radius of gyration. Similarly, the hydrodynamic drag force on molecules may be estimated from its radius of gyration and Stokes' law [f = 6\pi \eta R], although the numerically similar hydrodynamic radius may be better for this purpose.

Applications in structural engineering

In structural engineering, the two-dimensional radius of gyration is used to describe the distribution of cross-sectional area in a beam around its centroidal axis. The radius of gyration is given by the following formula

[R_^ = \frac]

where I is the second moment of area and A is the total cross-sectional area. The gyration radius is useful in estimating the stiffness of a beam. However, if the principal moments of the two-dimensional gyration tensor are not equal, the beam will tend to buckle around the axis with the smaller principal moment. For example, a beam with an elliptical cross-section will tend to buckle around the axis with the smaller semiaxis.

Applications in mechanics

The radius of gyration can be computed in terms of the second moment of inertia I and the total mass M:

[R_^ = \frac]

It should be noted that I is a scalar, and is not the moment of inertia tensor.

Derivation of identity

To show that the two definitions of [R_^] are identical, we first multiply out the summand in the first definition

[R_^ \equiv \frac \sum_^ \left( \mathbf_ - \mathbf_ \right)^ = \frac \sum_^ \left[ mathbf_ cdot mathbf_ + mathbf_ cdot mathbf_ - 2 mathbf_ cdot mathbf_ right]]

Carrying out the summation over the last two terms and using the definition of [\mathbf_] gives the formula

[R_^ \equiv -\mathbf_ \cdot \mathbf_ + \frac \sum_^ \left( \mathbf_ \cdot \mathbf_ \right)]

Similarly, we may multiply out the summand of the second definition

[R_^ \equiv \frac} \sum_ \left( \mathbf_ - \mathbf_ \right)^ =\frac} \sum_ \left[ mathbf_ cdot mathbf_ + mathbf_ cdot mathbf_ - 2mathbf_ cdot mathbf_ right]]

which can be written

[R_^ \equiv- \left( \frac \sum_^ \mathbf_ \right) \cdot \left( \frac \sum_^ \mathbf_ \right) + \frac} \sum_ \left( \mathbf_ \cdot \mathbf_ + \mathbf_ \cdot \mathbf_ \right)]

Substituting the definition of [\mathbf_] and carrying out one of the summations in the final term (and renaming the remaining summation index to k) yields

[R_^ \equiv -\mathbf_ \cdot \mathbf_ + \frac \sum_^ \left( \mathbf_ \cdot \mathbf_ \right)]

proving the identity of the two definitions.

References

Grosberg AY and Khokhlov AR. (1994) Statistical Physics of Macromolecules (translated by Atanov YA), AIP Press. ISBN 156396710

Flory PJ. (1953) Principles of Polymer Chemistry, Cornell University, pp. 428-429 (Appendix C of Chapter X).

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