Static light scattering
Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight "M" of a macromolecule like a polymer or a protein in solution. Measurement of the scattering intensity at many angles allows calculation of the root mean square radius, also called the radius of gyration "R". By measuring the scattering intensity for many samples of various concentrations, the second virial coefficient "A", can be calculated. Static light scattering is also commonly utilized to determine the size of particle suspensions in the sub-μm and supra -μm ranges, via the Lorenz-Mie (see Mie scattering) and Fraunhofer diffraction formalisms, respectively. For static light scattering experiments, a high-intensity monochromatic light, usually a laser, is launched in a solution containing the macromolecules. One or many detectors are used to measure the scattering intensity at one or many angles. The angular dependence is required to obtain accurate measurements of both molar mass and size for all macromolecules of radius above 1–2% the incident wavelength. Hence simultaneous measurements at several angles relative to the direction of incident light, known as multi-angle light scattering (MALS) or multi-angle laser  light scattering (MALLS), is generally regarded as the standard implementation of static light scattering. Additional details on the history and theory of MALS may be found in multi-angle light scattering. To measure the average molecular weight directly without calibration from the light scattering intensity, the laser intensity, the quantum efficiency of the detector, and the full scattering volume and solid angle of the detector needs to be known. Since this is impractical, all commercial instruments are calibrated using a strong, known scatterer like toluene since the Rayleigh ratio of toluene and a few other solvents were measured using an  absolute light scattering instrument. Theory. For a light scattering instrument composed of many detectors placed at various angles, all the detectors need to respond the same way. Usually detectors will have slightly different quantum efficiency, different gains and are looking at different geometrical scattering volumes. In this case a normalization of the detectors is absolutely needed. To normalize the detectors, a measurement of a pure solvent is made first. Then an isotropic scatterer is added to the solvent. Since isotropic scatterers scatter the same intensity at any angle, the detector efficiency and gain can be normalized with this procedure. It  is convenient to normalize all the detectors to the 90° angle detector. formula_1 where "I(90)" is the scattering intensity measured for the Rayleigh scatterer by the 90° angle detector. The most common equation to measure the weight-average molecular weight, "M", is the Zimm equation (the right hand side of the Zimm equation is provided incorrectly in some texts, as noted by Hiemenz and Lodge): formula_2 where formula_3 and formula_4 with formula_5 and the scattering vector for vertically polarized light is formula_6 with "n" the refractive index of the solvent, λ the wavelength of the light source, "N  Avogadro's number (6.022x10), c" the solution concentration, and d"n"/d"c" the change in refractive index of the solution with change in concentration. The intensity of the analyte measured at an angle is "I(θ)". In these equation the subscript A is for analyte (the solution) and T is for the toluene with the Rayleigh ratio of toluene, "R" being 1.35x10 cm for a HeNe laser. As described above, the radius of gyration, "R", and the second virial coefficient, "A", are also calculated from this equation. The refractive index increment "dn/dc" characterizes the change of the refractive index "n" with the  concentration "c", and can be measured with a differential refractometer. A Zimm plot is built from a double extrapolation to zero angle and zero concentration from many angle and many concentration measurements. In the most simple form, the Zimm equation is reduced to: formula_7 for measurements made at low angle and infinite dilution since "P(0) = 1". There are typically a number of analyses developed to analyze the scattering of particles in solution to derive the above named physical characteristics of particles. A simple static light scattering experiment entails the average intensity of the sample that is corrected for the  scattering of the solvent will yield the Rayleigh ratio, "R" as a function of the angle or the wave vector "q" as follows: Data analyses. Guinier plot. The scattered intensity can be plotted as a function of the angle to give information on the "R" which can simply be calculated using the Guinier approximation as follows: formula_8 where "ln(ΔR(θ)) = lnP(θ)" also known as the form factor with "q = 4πnsin(θ/2)/λ". Hence a plot of the corrected Rayleigh ratio, "ΔR(θ) vs sin(θ/2)" or "q" will yield a slope "R/3". However, this approximation is only true for "qR < 1". Note  that for a Guinier plot, the value of "dn/dc" and the concentration is not needed. Kratky plot. 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 "sin(θ/2)ΔR(θ) vs sin(θ/2)" or "qΔR(θ) vs q". Zimm plot. For polymers and polymer complexes which are of a monodisperse nature (formula_9) as determined by static light scattering, a Zimm plot is a conventional means of deriving the parameters such as "R", molecular mass "M" and the second virial coefficient "A . One must note that if the material constant K" is not implemented, a Zimm plot will only yield "R". Hence implementing "K" will yield the following equation: formula_10 Experiments are performed at several angles, which satisfy condition formula_11 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. Multiple scattering. 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, 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. The same method can also be used to correct dynamic light scattering data for multiple scattering contributions. Composition-gradient static light scattering. 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. Static light