# User:Nadiezda Fernandez-Oropeza/Notebook/Notebook/2011/06/28

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## Dynamic Light Scattering (DLS)

• This entry contains information about DLS. Some of the theory might seem to basic or too detailed, but I think it is always good to refresh the memory and have the definitions clear. The “Experiment” section will be further developed as we progress on the experiment.

### Introduction

Dynamic Light Scattering (DLS) is a technique used to determine the size of particles in suspension. It measures the laser light that is scattered from dissolved macromolecules or suspended particles. Due to the Brownian motion of the molecules and particles in solution, constant fluctuations in time of the scattering intensity can be observed. Or in other words, “when the light hits the moving particle, it changes the wavelength of the incoming light. Hence, the rate of decay of the fluctuations of scattered light is indicative of the rate of diffusion of the scattering species in the sample. This change is related to the size of the particle. It is possible to compute the sphere size distribution and give a description of the particle’s motion in the medium, measuring the diffusion coefficient of the particle and using the autocorrelation function.”

This technique has many applications; it is used to obtain molecular weight, radius of gyration, Translational diffusion constant and others. To determine the rates of decay of the intensity of the scattered light, the time correlation function of this intensity is analyzed. Ideally, distribution functions in aggregating protein solutions would have two peaks: one corresponding to the protein monomers and the other corresponding to the aggregates.

### Theory

Light will be scattered by a molecule in solutions if the molecule has polarizability different from its surroundings. Intensity is the energy, in this case the energy carried by a light beam, deposited on a one-squared meter area during one second. When more than one molecule is responsible for the light scattering, we must calculate the scattered electric field from each scattering molecule and get the total. If all the molecules are identical, the electric field contributions due to the individual molecules will have the same magnitude but the phases will be different. These phases are constantly changing in time due to the movement of the molecules in the solution. For our purposes we need to get the average scattered intensity $\displaystyle{ \lt I(t)\gt }$.

The viscous friction coefficient (ζ) for a spherical object is related to its size (radius R) by a simple relation known Stokes formula:

$\displaystyle{ \zeta=6\pi*\eta*R }$

where is η the viscosity of the fluid.

The relation that states that the fluctuations in a particle’s position are linked to the dissipation or frictional drag that it is subject to is the Einstein relation:

$\displaystyle{ \zeta*D=k_B*T }$

where D is the particle’s diffusion constant, $\displaystyle{ k_{B} }$ Boltzmann constant and T the temperature.

The combination of both this relations gives the Stoke-Einstein’s relation:

$\displaystyle{ R=\frac{k_{B}*T}{6\pi*\eta*D} }$
• Correlation and cross-correlation

The following definition was obtained from Wikipedia:

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or inner-product. It is commonly used for searching a long duration signal for a shorter, known feature. It also has applications in pattern recognition, single particle analysis, electron tomographic averaging, cryptanalysis, and neurophysiology. For continuous functions, f and g, the cross-correlation is defined as:

$\displaystyle{ (f*g)(t)= \int_0^{+\infty}f^{*}(\tau)g(t+\tau)d(\tau) }$

Autocorrelation is the cross-correlation of a signal with itself. The intensity autocorrelation function that is defined as:

$\displaystyle{ G(t)=\int_{-\infty}^{+\infty}I(\tau)I(t+\tau)d(\tau) }$

The simplest approach is to treat the first order autocorrelation function as a single exponential decay:

$\displaystyle{ G(t)={\lt I(t)\gt }^{2}(1+\alpha*e^{-\Gamma*t}) }$

Where α is an instrument constant and Γ is the decay constant. This last one is determined by:

$\displaystyle{ \Gamma=q^2D }$

The term D is the diffusion constant discussed previously and q is the magnitude of the momentum transfer vector for the scattering process and it is given by:

$\displaystyle{ q=\frac{4\pi*n}{\lambda_0}sin{\frac{\theta}{2}} }$

Where n is the index of refraction of the medium, $\displaystyle{ \lambda_{0} }$ is the wavelength of the incident light in vacuum and θ is the angle of detection with respect to the incident beam direction.

Therefore, if we know the decay constant we can get the value of D and using Stokes-Einstein’s relation we can deduce the radius R of the particles in solution.

When there are collisions between particles in fluid, there are chances that particles will attach to each other and become larger particle. This is known as aggregation. The ultimate goal of our experiment is to determine how fast the proteins will aggregate when placed in different mediums.

All this theory is valid under the following assumptions:

• The Intensity of the light scattered by the liquid and the small ions is negligible compared to that scattered by the particles.
• The suspension is sufficiently transparent.
• The particles are small and spherical.

### Experiment

The DLS equipment we are going to use is a Varium DynaPro Titan D2S. This belongs to Dr. Osinski’s Lab.

Here is the user manual

### References

1. Dynamic Light Scattering. Aplications of Photon Correlation Spectroscopy. Pecora (1985)
2. [2]
3. [3]
4. [4]
5. [5]
6. [6]
7. [7]
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