# The Zimm Plot: Molecular Weight and Radius of Gyration

An easy graphical way to perform data fitting corresponding to the description given in the section on the Rayleigh Ratio, is the so-called Zimm plot. For small $$q$$ and $$w$$ we can equivalently write \begin{gather} \frac{K w}{\mathcal{R}_{\mathrm{ex}}} = \frac{1}{M}\left(1 + \frac{R_g^2}{3}q^2\right)\left(1 + 2\bar{B}_2\right) \end{gather} From this equations we see that, with data for $$q = 0$$, we could plot $$K w / \mathcal{R}_{\mathrm{ex}}$$ as a function of the mass concentration. In a linear interpolation the molar mass corresponds to the inverse of the intercept while the virial coefficient is given by the slope. However, such data is not available, but can be obtained by extrapolating to $$q = 0$$ the data available at $$q \ne 0$$ for each concentration.

The series obtained can then be linearly interpolated to obtain the molecular weight (molar mass) and the second virial coefficient as mentioned before. The same reasoning applies for the angle dependent series: for each angle one extrapolates the zero-concentration value from the series at such angle, but at varying concentrations. The angle dependent series can be plotted against $$q^2$$ so that a linear interpolation enables us to estimate the molecular mass, again from the inverse of the intercept.

The radius of gyration is obtained from the square root of the slope multiplied by 3.

Instead of creating two plots, one for the concentration series, and one for the angle series, one can plot $$K w / \mathcal{R}_{\mathrm{ex}}$$ as a function of $$q^2 + k' w$$, where $$k'$$ is a constant to be chosen arbitrarily. The result is shown in Fig. 1, where the thin green lines are the linear interpolation on the angle series at different concentration which yield the data series at zero angle (green circles) and varying concentration. The linear interpolation of such series in turn leads to the second virial coefficient and the molecular mass. The thin red lines represent the linear interpolation on the concentration series at different angles which yield the angle series at zero concentration whose linear interpolation in turn gives the radius of gyration.

Figure 1:  Zimm Plot

LS Instruments has developed an easy to use Zimm Plot software that can treat data obtained from suitable static light scattering experiments to automatically compute molecular weight, the radius of gyration, and the second virial coefficient.