static light scattering
DESCRIPTION
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. wavevector. scattering wavevector. - PowerPoint PPT PresentationTRANSCRIPT
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Static Light Scattering
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Outline of Static Light Scattering
Measurement system
Rayleigh scattering
Static structure factor
Form factors
Practical problems
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Light Scattering Measurement System
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Scattering Wavevector
top view
k =k =4πnλ
sinθ2
scattering wavevector
wavevector
ki =ks =2πλ
(in vacuum)
=2πλ /n
(in solution)
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Lengths Probed by Light Scattering
Light scattering probes the length of ~1/k.
~ 33 nm
~ 100 nm
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Scattering Volume
depends on the focusing of the laser.
specified by the two pinholes.
The scattering volume is an open system.
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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.
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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.
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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
∑
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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
∑
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Structure Factor of a Polymer Chain
I ∝1
1+k2Rg2 /3
low-angle scattering
Rg
radius of gyration
high-anglescattering
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Form Factors P(k)=I(k)I(0)
Angular dependence of P(k) allows us to determine the shape of the molecule.
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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
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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
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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)
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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
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Example of Zimm Plot
Polyguanidine in THF
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Differential Refractive Index
dndc
≅(npolymer−nsolvent)vsp
Δn=nsolution−nsolventΔn=
dndc
ΔcAt low concentrations,
Often, we can approximate dn/dc as
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Concentration Effect on Scattering Intensity
Iex(k)I0
=1
NA
2πnλ2
dndc
⎛ ⎝
⎞ ⎠
2 cMVr2 P(k) 1−2A2Mc+L[ ]
scattering at low concentrations
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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
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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)
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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 .
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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)