PS674

r and s

The conformations of linear chains are often discussed in terms of averages of their end-to-end vector, r, squared end-to-end distance, r², and squared radius of gyration, s². These three properties are defined here for a fixed conformation.

The end-to-end vector, r

A chain of n bonds (n+1 atoms) can be represented as a sequence of n vectors, l_i, joined head-to-tail, as depicted in PC Fig. 7.13. r is the sum of all of the l_i, with the understanding that all vectors are expressed in the same coordinate system.

The squared end-to-end distance, r²

The squared end-to-end distance is the dot product of r with itself. The dot product yields a double sum with all possible terms of the type l_i dot l_j. When i = j, this dot product is just the square of the length, l_i². Therefore r² is the sum of one term that is independent of conformation,

and another term (a double sum with i < j) that depends on conformation, because we must know the relative orientation of the two different bond vectors.

The squared radius of gyration, s²

The squared radius of gyration depends on the coordinates of all of the atoms and on their masses. In most cases of interest in an introduction to polymer science, the masses can be taken to be identical, and we focus on the coordinates. The squared radius of gyration can be expressed in terms of the distances between all pairs of atoms,

^-2

_ij

or, equivalently, in terms of the vectors that point to each atom, r_i, from an arbitrarily chosen origin.

^-1

where g is the center-of-mass vector,

^-1

The radius of gyration can be defined for any collection of points, without the need to specify their connectivity. In contrast to the end-to-end distance, it is well defined for cyclic and branched chains.

Special cases

Two special cases of the squared radius of gyration are useful as benchmarks in polymer science:

A uniform sphere of radius R has s² = (3/5)R². Therefore the minimum radius of gyration for a chain of molecular weight M (molecular mass M/L, where L denotes Avogadro's number) and partial specific volume v is
s²_min = (3/5)(3vM/ 4 pi L)^2/3
A very thin uniform rod of length L has s² = (1/12)L². This expression can be used to calculate the maximum radius of gyration of a chain, if the length of the fully extended chain is known.
For a polyethylene chain with a molecular weight of 100,000, these two extremes for the radius of gyration are about 2.2 nm and 180 nm. They differ by two orders of magnitude. The ratio of the two radii is molecular weight dependent, because the minimum radius of gyration is proportional to M^1/3, but the maximum radius of gyration is proportional to M.

Averages

When two or more conformations are considered, the average values, <r>, <r²>, and <s²>, depend on the distribution of conformations. For several simple models of flexible chains, simple closed form expressions can be obtained. The prototype is the freely jointed chain.

Illustrative examples are available for a polyethylene chain.

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August 8, 2002

Wayne L. Mattice: wlm@polymer.uakron.edu