Static Light Scattering. Outline of Static Light Scattering FMeasurement system FRayleigh scattering FStatic structure factor FForm factors FPractical.

Static Light Scattering

Outline of Static Light Scattering

Measurement system

Rayleigh scattering

Static structure factor

Form factors

Practical problems

Light Scattering Measurement System

Scattering Wavevector

top view

k =k =4πnλ

sinθ2

scattering wavevector

wavevector

ki =ks =2πλ

(in vacuum)

=2πλ /n

(in solution)

Lengths Probed by Light Scattering

Light scattering probes the length of ~1/k.

~ 33 nm

~ 100 nm

Scattering Volume

depends on the focusing of the laser.

specified by the two pinholes.

The scattering volume is an open system.

Rayleigh Scattering by a Small Particle

Why is the sky blue?Why is the sunset reddish?

Polarization in the particle changes in phase with the incoming light.

The particle is now a broad-casting station, emanating radiation in all directions.

Rayleigh Scattering

II0

=π2

λ4α2

ε02

sin2 ′ θ r2

Rayleigh scatteringby a particle in vacuum

: polarizability of the particle particle volume

I maximizes at ´ = 90°.Usually, LS is detected in the horizontal plane.

Scattering by a Chain Molecule (in Vacuum)

The beams scattered by the two particles interfere.Two parts of a large molecule interfere more or less constructively.Therefore, a large molecule scatters the light more strongly than many small particles do.

II0

=π2

λ4α2

ε02

1r2 exp[ik⋅(ri −rj )]

i, j=1

N

∑

Static Structure Factors

S(k) =1nP

exp[ik⋅(ri −rj )]i, j =1

nP

∑ =nP exp[ik⋅(ri −rj )]

suspension of small particles

single large molecule

S1(k) =1N

exp[ik⋅(ri −rj )]i, j=1

N

∑

many large molecules

S(k) =1

nPNexp[ik⋅(rmi−rnj)]

i, j=1

N

∑m,n=1

nP

∑

=S1(k)+nPN

exp[ik⋅(r1i −r2j )]i, j=1

N

∑

Structure Factor of a Polymer Chain

I ∝1

1+k2Rg2 /3

low-angle scattering

Rg

radius of gyration

high-anglescattering

Form Factors P(k)=I(k)I(0)

Angular dependence of P(k) allows us to determine the shape of the molecule.

Form Factor of a Sphere

Rayleigh-Gans formula

EXCEL problems

1. Plot P as a function of kR.2. Plot P as a function of for R = 10, 30, 100, 300, and 1000 nm. Assume specific values of n and .

Psphere(k) =1

Vsp2 dr

Vsp∫ d ′ r

Vsp∫ exp[ik⋅(r − ′ r )]=

1Vsp

drVsp∫ exp(ik⋅r)

2

Psphere(x) =[3x−3(sinx−xcosx)]2 withx =kR

Light Scattering of a Solution

The formula derived for a molecule in vacuum can be used just by replacing with ex.

αex =αmolecule−αsolvent

II0

=π2

(λ / n)4αex

2

(ε0n2)2

1r2 =

π2

λ4αex

2

ε02

1r2 ′ θ =90°

αex

ε0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

= λ ⋅2ndndc

⎛ ⎝

⎞ ⎠

2 cVNA

Iex

I0=

1NA

2πnλ2

dndc

⎛ ⎝

⎞ ⎠

2 cVr2

A more convenient expression

Light Scattering of Polymer Solutions

• Measure I(k) for pure solvent.

• Measure I(k) for solutions of a

given polymer at different

concentrations.

• Calculate Iex(k).

Iex(k)I0

=1

NA

2πnλ2

dndc

⎛ ⎝

⎞ ⎠

2 cVr2 P(k)

Zimm Plot

Iex(k)I0

=1

NA

2πnλ2

dndc

⎛ ⎝

⎞ ⎠

2 cVr2 P(k)

1M

+2A2c+L⎡ ⎣ ⎢

⎤ ⎦ ⎥

−1

Iex

I0≡

RθVr2

H ≡1

NA

2πnλ2

dndc

⎛ ⎝

⎞ ⎠

2

P(k)= 1+k2Rg2 / 3( )

−1

Example of Zimm Plot

Polyguanidine in THF

Differential Refractive Index

dndc

≅(npolymer−nsolvent)vsp

Δn=nsolution−nsolventΔn=

dndc

ΔcAt low concentrations,

Often, we can approximate dn/dc as

Concentration Effect on Scattering Intensity

Iex(k)I0

=1

NA

2πnλ2

dndc

⎛ ⎝

⎞ ⎠

2 cMVr2 P(k) 1−2A2Mc+L[ ]

scattering at low concentrations

Scattering by a Suspension of Spheres

I(kR)= I(0)P(kR)

I(0) ∝cM=ρM2

NA

c =ρMNA

mass/volume

At constant c, I(0) ∝ M ∝ Vsp∝ R3

At constant ρ, I(0) ∝ M2 ∝ Vsp2 ∝ R6 I(kR)∝ R6P(kR)

I(kR)∝ R3P(kR)

number/volume

Scattering by Spheres at Constant c

EXCEL problems

Plot R3P(kR) as a function of for R = 10, 30, 100, 300, and 1000 nm. Assume specific values of n and .

At constant c, I(0) ∝ M ∝ Vsp∝ R3 I(kR)∝ R3P(kR)

Scattering by Spheres at Constant ρ

At constant ρ, I(0) ∝ M2 ∝ Vsp2 ∝ R6 I(kR)∝ R6P(kR)

EXCEL problems

Plot R6P(kR) as a function of for R = 10, 30, 100, 300, and 1000 nm. Assume specific values of n and .

Changes in the Scattering Intensity

I2I1=

R2

R1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3P(kR2 )P(kR1)

Spheres aggregate into larger spheres:

Porous spheres become nonporous without changing R:

(n porous spheres form 1 nonporous sphere)

Inonporous

Iporous=1n

n2 =n

Nonporous spheres become porous without changing the mass:

I2I1

=R2

R1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

6P(kR2)P(kR1)