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# Static light scattering

Static light scattering is a technique in physical chemistry that uses the intensity traces at a number of angles to derive information about the radius of gyration $\ R_g$, molecular mass $\ M_w$ of the polymer or polymer complexes, and the second virial coefficient $\ A_2$, for example, micellar formation (1-5).

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:

$\ R(\theta_{sample}) = R(\theta_{solvent})I_{sample}/I_{solvent}$

yielding the difference in the Rayleigh ratio, $\ \Delta R(\theta)$ between the sample and solvent:

$\ \Delta R(\theta) = R(\theta_{sample})-R(\theta_{solvent})$

In addition, the setup of the laser light scattering is corrected with a liquid of a known refractive index and Rayleigh ratio e.g. toluene, benzene or decalin. This is applied at all angles to correct for the distance of the scattering volume to the detector.

One must note that although data analysis can be performed without a so-called material constant $\ K$ defined below, the inclusion of this constant can lead to the calculation of other physical parameters of the system.

$\ K=4\pi^2 n_0^2 (dn/dc)^2/N_A\lambda^4$

where $\ (dn/dc)$ is the refractive index increment, $\ n_0$ is the refractive index of the solvent, $\ N_A$ is Avogadro's number (6.023x1023) and $\ \lambda$ is the wavelength of the laser light reaching the detector. This equation is for linearly polarized light like the one from a He-Ne gas laser.

## Data Analyses

### Guinier plot

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

$\ ln(\Delta R(\theta)) = 1 - (R_g^2/3)q^2$

where $\ ln(\Delta R(\theta))=lnP(\theta)$ also known as the form factor with $\ q = 4\pi n_0 sin(\theta/2)/\lambda$. Hence a plot of the corrected Rayleigh ratio,$\ \Delta R(\theta)$ versus $\ sin(\theta/2)$ or $\ q^2$ will yield a slope $\ -R_g^2/3$. 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.

### 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 $\ sin^2(\theta/2)\Delta R(\theta)$ versus $\ sin(\theta/2)$ or $\ q^2\Delta R(\theta)$ versus $\ q$.

### Debye plot

This method is used to derive the molecular mass and 2nd virial coefficient,$\ A_2$, of the polymer or polymer complex system. The difference to the Zimm plot is that the experiments are performed using a single angle. Since only one angle is used (typically 90o), the $\ R_g$ cannot be determined as can be seen from the following equation:

$\ Kc/\Delta R(\theta) = 1/M_w + 2A_2c$

### Zimm plot

For polymers and polymer complexes which are of a monodisperse nature $\ PDI<0.3$ as determined by dynamic 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 $\ A_2$.

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

$\ Kc/\Delta R(\theta)=1/{M_wP(\theta)}+A_2c = 1/M_w(1+q^2(R_g^2/3))+2A_2c$

Experiments are performed at several angles 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.

## References

1. A. Einstein, Ann. Phys. 33 (1910), 1275

2. C.V. Raman, Indian J. Phys. 2 (1927), 1

3. P.Debye, J. Appl. Phys. 15 (1944), 338

4. B.H. Zimm, J. Chem. Phys 13 (1945), 141

5. B.H. Zimm, J. Chem. Phys 16 (1948), 1093