S14_DLS_Theory
Theory
Light that is scattered off a collection of particles]] experiences fluctuations as the particles move around in random directions. The probability that a particle will be at some distance after a time t is:
dP(r,t)=[((4πDt)^(-3/2))*e^(-r^2\4Dt))]d3r (1)
where r is the distance, with respect to the initial position, D is the diffusion coefficient, and d3r is the volume element in which the particle can move.
For spherical particles suspended in a fluid, the diffusion coefficient is expressed by the Stokes-Einstein equation:
D=(kT)/(6πηR) (2)
where k is boltzmann’s constant, T is the temperature, η is the viscosity of the fluid and R is the radius of the spheres.
A viscous fluid will create drag on the spheres and cause them to diffuse at a slower rate. Similarly, a large sphere will have greater drag on it due to it’s large surface area, thus lowering the diffusion rate.
The intensity of the signal at the photodiode is roughly proportional to the number of coherence areas that are incident upon it. By reducing the amount of coherence areas, more fluctuations can be viewed, but the signal will be weak. The reverse is also true; by increasing the amount of coherence areas, the signal can be strengthened, but less fluctuations will be seen because those fluctuations will be averaged over the number of coherence areas. The number of coherence areas can be adjusted by adjusting the area of scattered light that exits the scattering cell.
If Eqn (1) correctly models the motion of the spheres, then the intensity spectrum, S(f) in the frequency domain, will have the form:
S(f) ∝ Γ^2 + (2f)^2 (3)
This form is the shape of a Lorentzian curve. The width of this line, Γ, is dependent on D and the scattering vector, K, as shown in Eqn (4).
Γ=2DK2 (4)
The scattering vector is the difference between incident and scattered wave vectors of the laser. The magnitude of K is
K=4πλn sin (θ/2) (5)
where θ is the scattering angle, λ is the wavelength of the laser, and n is index of refraction of the fluid. The half width at half max of the intensity Lorentzian line Δf is
(Δf)1/2=DKτ (6)
where τ is the time it takes for any superposition of phases to become a new uncorrelated superposition of phases. This occurs when the phase has changed by a factor π, which occurs when a particle moves a distance π/K in the direction of K; thus, the HWHM can be expressed as
<Tex> (f)_(1/2)=DK^2 (7)
For this experiment, θ was 90o, allowing D and η to be calculated from Eqn (7) and Eqn (2) directly.