What are molecular application of radius of gyration?

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-

{displaystyle 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 {displaystyle mathbf {r} _{mathrm {mean} }} is the mean position of the monomers. As detailed below, the radius of gyration is also proportional to root mean square distance between the monomers-

{displaystyle 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 gyration tensor.

Since 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 as an average over time or ensemble:

{displaystyle R_{mathrm {g} }^{2} {stackrel {mathrm {def} }{=}} {frac {1}{N}}leftlangle sum _{k=1}^{N}left(mathbf {r} _{k}-mathbf {r} _{mathrm {mean} }right)^{2}rightrangle }

Where the angular brackets {displaystyle langle ldots rangle }<> denotes the ensemble average.

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

{displaystyle R_{mathrm {g} }={frac {1}{{sqrt {6}} }} {sqrt {N}} a}

Note that although {displaystyle aN}aN represents contour length of the polymer, {displaystyle a}a is strongly dependent of polymer stiffness and can vary over orders of magnitude. {displaystyle N}N is reduced accordingly.

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

Derivation of Identity

To show that the two definitions of {displaystyle R_{mathrm {g} }^{2}} are identical, we first multiply out the summand in the first definition:

{displaystyle R_{mathrm {g} }^{2} {stackrel {mathrm {def} }{=}} {frac {1}{N}}sum _{k=1}^{N}left(mathbf {r} _{k}-mathbf {r} _{mathrm {mean} }right)^{2}={frac {1}{N}}sum _{k=1}^{N}left[mathbf {r} _{k}cdot mathbf {r} _{k}+mathbf {r} _{mathrm {mean} }cdot mathbf {r} _{mathrm {mean} }-2mathbf {r} _{k}cdot mathbf {r} _{mathrm {mean} }right]}

Carrying out the summation over the last two terms and using the definition of {displaystyle mathbf {r} _{mathrm {mean} }} gives the formula, we get-

{displaystyle R_{mathrm {g} }^{2} {stackrel {mathrm {def} }{=}} -mathbf {r} _{mathrm {mean} }cdot mathbf {r} _{mathrm {mean} }+{frac {1}{N}}sum _{k=1}^{N}left(mathbf {r} _{k}cdot mathbf {r} _{k}right)}

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