# Scattering from Macromolecules

The Rayleigh-Gans-Debye theory is valid for solid dielectric objects such as colloidal particles, whose dielectric properties are well known. For macromolecules, however, dielectric properties are not as well defined because continuum theories don't strictly apply. Small modifications, however, allow for an equally useful application of the theory. First we assume valid the following simple mixing rule for the solution's refractive index, $$n_t$$

\begin{gather} n_t = \phi n + (1 - \phi) n_s = n_s \left[ 1 + \frac{c M}{N_A \rho} (m - 1)\right], \end{gather}

where $$\phi = c M /(N_A \rho)$$ is the partial volume fraction and $$c$$ is number concentration.

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This relationship is asymptotically valid for small solute concentrations. By means of a simple refractometer we can measure the solution's refractive index as a function of the solute concentration. Linear interpolation at for $$c \to 0$$ yields the so-called refractive index increment, which in light of the relationship above is expressed as \begin{gather} \left . \frac{d n_t}{d c} \right|_{c = 0} = \frac{n_s M}{N_A \rho}(m - 1). \end{gather} In what follows we will drop the subscript $$c = 0$$ understanding that we are talking about the refractive index increment at zero polymer concentration. Solving for the molecular relative refractive index yields a relationship that allows obtaining the relative refractive index from the measured refractive index increment: \begin{gather} m = \frac{d n_t}{d c} \frac{N_A \rho}{n_s M} + 1. \end{gather} If we take into consideration that Rayleigh-Gans-Debye theory requires $$m \simeq 1$$ the scattering contrast can be recast as follows in terms of the refractive index increment: \begin{gather} \Delta \rho = \frac{3 \pi n_s^2}{\lambda^2} \frac{m^2 - 1}{m^2 + 2} \simeq \frac{6 \pi n_s^2}{\lambda^2} \frac{m - 1}{3} = \frac{2\pi n_s}{\lambda^2} \frac{N_A \rho}{M} \frac{d n_t}{d c}. \end{gather} By means of this expression for the scattering length density difference and working in terms of mass concentration, $$w = c M/N_A$$ the measured scattered intensity can be recast as \begin{gather} I_s(\mathbf{q},w) = I_i \frac{f(\theta)}{R_0^2} V_s w M \frac{2\pi^2 n_s^2}{N_A \lambda^4} \left(\frac{d n_t}{d c}\right)^2 P(\mathbf{q}) S(\mathbf{q},w). \end{gather} Two issues concerning the scattered intensity formula relevant to the scattering from polymer solutions have been neglected so far. First, it is not possible to measure directly "absolute" intensities. In fact any hardware implementations gives a quantity, let us call it $$C(\mathbf{q},w)$$, which is, at best, proportional to an unknown constant $$A$$ to the scattered intensity. Second, due to density fluctuations, pure liquids scatter light as well. While in the case of colloidal particle suspensions this contribution can be safely neglected, in polymer systems it must be taken into account. The ability to scatter of a pure solvent is measured by the so-called Rayleigh ratio $$\mathcal{R}_{\mathrm{sol}}$$ defined as the scattered intensity per unit solid angle, scattering volume, and incident intensity. Taking into account the previous remarks, the expression for the measured intensity takes the following form \begin{gather} C(\mathbf{q},w)= A I_i \frac{f(\theta)}{R_0^2} V_s \left[w M K P(\mathbf{q}) S(\mathbf{q},w) + \mathcal{R}_{\mathrm{sol}} \right],\\ K = \frac{2\pi^2 n_s^2}{N_A \lambda^4} \left(\frac{d n_t}{d c}\right)^2 \end{gather} Some points worth noting about this formula:

1. If we were able to perform absolute measurements (i.e. if we knew the factor $$A V_s/R_0^2$$), in a series of experiments at different angles and polymer concentrations the intercept at $$w = 0$$ and $$q = 0$$ of the measured intensity normalized by the polymer mass concentration allows for the determination of the polymer molecular weight, granted the sample characteristics (i.e. $$K$$) are known.
2. Extrapolation at $$w = 0$$ of the same dataset yields a plot whose slope allows for the determination of the radius of gyration (Guinier Plot)
3. The same procedure but for $$q = 0$$ yields the second virial coefficient