static light scattering. outline of static light scattering fmeasurement system frayleigh scattering...
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Outline of Static Light Scattering
Measurement system
Rayleigh scattering
Static structure factor
Form factors
Practical problems
Scattering Wavevector
top view
k =k =4πnλ
sinθ2
scattering wavevector
wavevector
ki =ks =2πλ
(in vacuum)
=2πλ /n
(in solution)
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
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)