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