Concentration in a Coil

In addition to its use in the interpretation of the intrinsic viscosity, the model of the equivalent sphere lets us estimate the concentration of monomer units within the domain of a polymer in dilute solution. This model assumes solvent is trapped within the interior of the polymer coil, and the trapped solvent moves with the polymer. Although the instantaneous conformation of a polymer chain does not have spherical symmetry, for the sake of simplicity we will assume that the "particle" (polymer + trapped solvent) moving through the solution can be approximated by a sphere. Furthermore, we assume that the radius of this equivalent sphere is proportional to the root-mean-square radius of gyration.

R_E = k<s²>^1/2

If the chain is long enough, the end-to-end distance can be substituted for the radius of gyration, using

<r²> = 6 <s²>

For a Theta solvent, we can introduce the number of segments using the model of the equivalent chain

<r²> = N_{equiv. chain}L²_{equiv. chain}

The volume pervaded by the molecule is

V_E = (4/3) pi R_E³

Combining all of the above, the local concentration within the domain of the coil is

N_{equiv. chain} / V_E = (3 / 4 pi) (6^1/2 / kL_{equiv. chain})³ (1/N_{equiv. chain}^1/2)

Some general conclusions:

The concentration within the unperturbed coil decreases as the degree of polymerization increases, because the volume increases faster than the number of segments.
Using the best estimates for the proportionality constant k (which are close to 0.85), the internal concentration of unperturbed polystyrene is about
- 1 molar when the molecular weight is about 100,000.
- 1% (on a volume basis) when the molecular weight is about 1 million.

The concentrations would be lower within the random coil if it is in a good solvent, due to the expansion of the chain.

The concentration, expressed as a volume fraction, within the random coil can also be estimated from the actual radius of the chain and the minimum radius of gyration for a molecule with the same M and v. Taking the volume pervaded by each model to be proportional to the cube of its radius of gyration, and ignoring any differences in the proportionality constants, we have

volume fraction = s_min³ / <s²>^3/2

which is often on the order 10^-2 for flexible uncharged chains with M around 10⁵.

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November 14, 2001

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