Post on 30-Dec-2015
description
Particle Sizing by DLS
DLS by Particles of Different Sizes
particle of radius R1exp(−Γ1τ)
exp(−Γ2τ)particle of radius R2 Γ2 =kBT
6πηsR2k2
Γ1 =kBT
6πηsR1k2 G1
G2
g1(τ) = Gi exp(−Γiτ)i∑ = G(Γ)exp(−Γτ) dΓ∫total
M M M M
distribution function of weightedby the scattering intensity
intensity
Analysis of Autocorrelation Functions
1. Cumulant expansion (Unimodal analysis)
2. Inverse-Laplace transform (SDP analysis)
Cumulant Expansion (Unimodal analysis)
ln g1(τ ) =− τ +
12!
Δ2 τ 2 −13!
Δ3 τ 3 +L
where = G(Γ)dΓ∫Δ2 = Γ − Γ( )
2G(Γ)dΓ∫
1st cumulant
2nd cumulant
Curve fitting by a second -order polynomial yields the coefficients.
(polydispersity)
Inverse-Laplace Transform (SDP Analysis)
g1(τ) = G(Γ)exp(−Γτ) dΓ∫= ΓG(Γ)exp(−Γτ) dlnΓ∫
g1(τ) is the Laplace
transform of G().
Examples of Inverse-Laplace Transform
monodisperse
unimodaldistribution
bimodaldistribution
Relationship between Unimodal Analysis and SDP Analysis
=kBT
6πηRHk2
1
RH=
G1
R1+
G2
R2
harmonic average weighted by the scattering intensity
=G1Γ1 +G2Γ2
Example of a Bimodal Distribution
What is the average radius (estimated by DLS) for an equal
mass mixture of spheres of two radii R1 and R2?
Assume R1 = 10 nm and R2 = 100 nm.
G1 ∝ R13P(kR1)
G2 ∝ R23P(kR2 )
The average depends on k.
Plot <R> as a function of .
Plot G2/G1 as a function of .
Diffusion vs. Internal Relaxation
Examples of Internal Relaxation
Rotation of a rodlike molecule Rouse normal modes
Elastic motions of a gel Reacting system