# The Excess Rayleigh Ratio

In the section on "scattering from macromolecules" we  introduced the Rayleigh Ratio for light scattered by a solvent. However the equation for the scattered intensity suggests that also the contribution of the polymer can be scaled in a similar fashion. We can then define the excess Rayleigh ratio, $$\mathcal{R}_{\mathrm{ex}}$$ as the contribution of the polymer chains to the scattered light intensity scaled by the factor $$A f(\theta) I_0 V_s/R_0^2$$. According to this definition we can write for small $$q$$ and $$w$$

\begin{gather} \mathcal{R}_{\mathrm{ex}}(q,w) = w M K \left[1 - \frac{(q R_g)^2}{3} \right] \left(1 - 2\bar{B}_2 w \right), \end{gather}

where $$\bar{B}_2 = B_2 N_A/M$$.

Related products:

LS Spectrometer

3D LS Spectrometer

From an experimental point of view, we see that if we measure at several angles and at different polymer mass concentrations, we can fit the previous equation to the experimental data and obtain an estimate of the molar mass, $$M$$ the radius of gyration, $$R_g$$ and the second virial coefficient, $$\bar{B}_2$$. In order to measure $$\mathcal{R}_{\mathrm{ex}}$$ , however, we need to measure first the pure solvent to obtain $$\mathcal{R}_{\mathrm{sol}}$$. The main difficulty in this procedure lies in the fact that it is actually extremely difficult to reliably and accurately measure the factor necessary to obtain the Rayleigh ratios. Luckily some scientists have performed very complicated and accurate measurements of Rayleigh ratios on a small set of pure liquids. Thus, if we measure the scattered intensity, for one such standard liquid, we can calculate the instrument specific proportionality constant. This constant transforms measured intensities into absolute scattering ratios, as follows \begin{gather} \frac{A I_i V_s}{R_0^2} = \frac{C_{\mathrm{std}}}{\mathcal{R}_{\mathrm{std}}}, \end{gather} where $$\mathcal{R}_{\mathrm{std}}$$ is the Rayleigh ratio of the so-called "standard" solution. This value is tabulated. However, the standard solution has to be measured on the same angle set used for molar mass determination. In this way we are able to calculate absolute Rayleigh ratios. Then one has to proceed further to measure the intensity, $$C_{\mathrm{sol}}$$ scattered by a sample containing the pure solvent which is used to dissolve the polymer we want to analyze. From these values we can calculate $$\mathcal{R}_{\mathrm{sol}}$$ as \begin{gather} \mathcal{R}_{\mathrm{sol}} = C_{\mathrm{sol}}\frac{R_0^2}{A I_i V_s} = C_{\mathrm{sol}} \frac{\mathcal{R}_{\mathrm{std}}}{C_{\mathrm{std}}}, \end{gather} This step is required in order to be able to subtract the solvent contribution from the measurement performed on the samples containing the polymer, to obtain the excess Rayleigh ratios. Indeed, in the same fashion as for the "solvent" we can calculate the total Rayleigh ratio, from the intensity, $$C_s$$ scattered by the polymer solutiuons as \begin{gather} \mathcal{R} = C_s\frac{R_0^2}{A I_i V_s} = C_s \frac{\mathcal{R}_{\mathrm{std}}}{C_{\mathrm{std}}}, \end{gather} Eventually we calculate the excess Rayleigh ratio from the collected experimental data as \begin{gather} \mathcal{R}_{\mathrm{ex}} = \mathcal{R}- \mathcal{R}_{\mathrm{sol}} = \mathcal{R}_{\mathrm{std}} \frac{C_s - C_{\mathrm{sol}}}{C_{\mathrm{std}}} , \end{gather} and estimate $$M$$, $$R_g$$, and $$\bar{B}_2$$.