Static light scattering

The scattered intensity can be plotted as a function of the angle to give information on the R_g which can simply be calculated using the Guinier approximation as follows:

${\displaystyle \ln(\Delta R(\theta ))=1-(R_{g}^{2}/3)q^{2}}$

where ln(ΔR(θ)) = lnP(θ) also known as the form factor with q = 4πn₀sin(θ/2)/λ. Hence a plot of the corrected Rayleigh ratio, ΔR(θ) vs sin²(θ/2) or q² will yield a slope R_g²/3. However, this approximation is only true for qR_g < 1. Note that for a Guinier plot, the value of dn/dc and the concentration is not needed.

The Kratky plot is typically used to analyze the conformation of proteins but can be used to analyze the random walk model of polymers. A Kratky plot can be made by plotting sin²(θ/2)ΔR(θ) vs sin(θ/2) or q²ΔR(θ) vs q.

Evaluation of the static light-scattering measurement using the Zimm plot

For polymers and polymer complexes that are monodisperse (${\displaystyle \scriptstyle \mu _{2}/{\bar {\Gamma }}^{2}<0.3}$) as determined by static light scattering, a Zimm plot is a conventional means of deriving the parameters such as R_g, molecular mass M_w and the second virial coefficient A₂.

One must note that if the material constant K is not implemented, a Zimm plot will only yield R_g. Hence implementing K will yield the following equation:

${\displaystyle {\frac {Kc}{\Delta R(\theta ,c)}}={\frac {1}{M_{w}}}\left(1+{\frac {q^{2}R_{g}^{2}}{3}}+O(q^{4})\right)+2A_{2}c+O(c^{2})}$

The analysis performed with the Zimm plot uses a double-extrapolation to zero concentration and zero scattering angle resulting in a characteristic rhomboid plot. As the angular information is available, it is also possible to obtain the radius of gyration (R_g). Experiments are performed at several angles, which satisfy the condition ${\displaystyle qR_{g}<1}$ and at least 4 concentrations. Performing a Zimm analysis on a single concentration is known as a partial Zimm analysis and is only valid for dilute solutions of strong point scatterers. The partial Zimm however, does not yield the second virial coefficient, due to the absence of the variable concentration of the sample. More specifically, the value of the second virial coefficient is either assumed to equal zero or is inputted as a known value in order to perform the partial Zimm analysis.

Calculation of molecular mass by static light scattering measurement using Debye plot[7]

If the measured particles are smaller than λ/20, the form factor P(θ) can be neglected (P(θ)→1). Therefore, the Zimm equation is simplified to the Debye equation, as follows:

${\displaystyle {\frac {Kc}{\Delta R(\theta ,c)}}={\frac {1}{M_{w}}}+2A_{2}c}$

Note that this is also the result of an extrapolation to zero scattering angle. By acquiring data on concentration and scattering intensity, the Debye plot is constructed by plotting Kc/ΔR(θ) vs. concentration. The intercept of the fitted line gives the molecular mass, while the slope corresponds to the 2^nd virial coefficient.

As the Debye plot is a simplification of the Zimm equation, the same limitations of the latter apply, i.e., samples should present a monodisperse nature. For polydisperse samples, the resulting molecular mass from a static light-scattering measurement will represent an average value. An advantage of the Debye plot is the possibility to determine the second virial coefficient. This parameter describes the interaction between particles and the solvent. In macromolecule solutions, for instance, it can assume negative (particle-particle interactions are favored), zero, or positive values (particle-solvent interactions are favored).[8]

Static light scattering assumes that each detected photon has only been scattered exactly once. Therefore, analysis according to the calculations stated above will only be correct if the sample has been diluted sufficiently to ensure that photons are not scattered multiple times by the sample before being detected. Accurate interpretation becomes exceedingly difficult for systems with non-negligible contributions from multiple scattering. In many commercial instruments where analysis of the scattering signal is automatically performed, the error may never be noticed by the user. Particularly for larger particles and those with high refractive index contrast, this limits the application of standard static light scattering to very low particle concentrations. On the other hand, for soluble macromolecules that exhibit a relatively low refractive index contrast versus the solvent, including most polymers and biomolecules in their respective solvents, multiple scattering is rarely a limiting factor even at concentrations that approach the limits of solubility.

However, as shown by Schaetzel,[9] it is possible to suppress multiple scattering in static light scattering experiments via a cross-correlation approach. The general idea is to isolate singly scattered light and suppress undesired contributions from multiple scattering in a static light scattering experiment. Different implementations of cross-correlation light scattering have been developed and applied. Currently, the most widely used scheme is the so-called 3D-dynamic light scattering method,.[10][11] The same method can also be used to correct dynamic light scattering data for multiple scattering contributions.[12]

Samples that change their properties after dilution may not be analyzed via static light scattering in terms of the simple model presented here as the Zimm equation. A more sophisticated analysis known as 'composition-gradient static (or multi-angle) light scattering' (CG-SLS or CG-MALS) is an important class of methods to investigate protein–protein interactions, colligative properties, and other macromolecular interactions as it yields, in addition to size and molecular weight, information on the affinity and stoichiometry of molecular complexes formed by one or more associating macromolecular/biomolecular species. In particular, static light scattering from a dilution series may be analyzed to quantify self-association, reversible oligomerization, and non-specific attraction or repulsion, while static light scattering from mixtures of species may be analyzed to quantify hetero-association.[13]