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Transcript
Static & Dynamic Light Scattering
• First quantitative experiments in 1869 by Tyndall
(scattering of small particles in the air – Tyndall
effect)
• 1871 – Lord Rayleigh started a quantitative
study and theory
• Basic idea: incident monochromatic linearly
polarized light beam incident on a sample.
Assume
– No absorption
– Randomly oriented and positioned scatterers
– Isotropic scatterers
– Independently scattering particles (dilute)
– Particles small compared to wavelength of light
We’ll remove some of these restrictions later
Classical Wave description
• The incident electric field is
E = Eocos(2px/l – 2pt/T)
• Interaction with molecules drives their
electrons at the same f to induce an
oscillating dipole
pinduced = a Eocos(2px/l – 2pt/T) - a = polarizability
• This dipole will radiate producing a
scattered E field from the single molecule
a Eo 4p 2 sin 
Escattered (r , t ) 
cos  2p x / l  2p t / T 
2
rl
dipole

r
Obs. Pt.
Static (or time-average)Rayleigh
scattering
1. E ~ 1/r so I ~ 1/r2 - necessary since I
~energy/time/area and A ~ r2
2. E ~ 1/l2 dependence so I ~ 1/l4 – blue
skies and red sunsets (sunrises)
3. Elastic scattering – same f
4. sin  dependence – when  = 0 or p – at
poles of dipole – no scattering – max in
horizontal plane
5. a related to n , but how?
Polarizability and index of refraction
dn
n  1 c
dc
• Note that if n ~ 1
where c is the weight concentration
dn
dn
2
2
• Then n  1  2 c  ... so n  1  2 c  4p Na
dc
dc
where N = number concentration
• So,
dn
dn
c
a  dc
2p N
or a 
M
dc
2p N A
• For a particle in a solvent with nsolv, we have n2 –
n2solv = 4pNa so
a  (n  nsolv )(n  nsolv ) / 4p N ~
2nsolv dn c
n
dn
 a  solv
M
4p dc N
2p N A dc
Scattered Intensity
• Detect intensity, not E, where
2
 a Eo 4p sin  


2
4 2
2
r
l
16
p
a
sin





2
Eo
r 2l 4
2
I scatt 

I o 1 particle
• Substituting for a, we have
2
I scatt 

I o 1 particle
 dn 
4p M n   sin 2 
dc 


r 2l 4 N A2
2
2
2
solv
Scattered Intensity II
• If there are N scatterers/unit volume and
all are independent with N = NAc/M, then
2
I scatt 
I scatt 
N



I o  per unit volume
I o 1 particle
4p sin  n
2
2
2
solv
 dn 
  Mc
 dc 
N Ar 2l 4
• We define the Rayleigh ratio Rq:
2
 dn 
4p n   Mc
2
I scatt ,q r
dc 

Rq 

 KMc
2
4
I o sin 
N Al
2 2
solv
Basic Measurement
• If the intensity ratio Iq/Io, nsolv, dn/dc, l, c, , and r
are all known, you can find M.
• Usually write Kc/Rq = 1/M
• Measurements are usually made as a function of
concentration c and scattering angle q
• The concentration dependence is given by
Kc 1

 2 Bc
Rq M
where B is called the thermodynamic virial – same as
we saw before for c dependence of D (but called A)
q
Angle Dependence
• If the scatterers are
small (d < l/20), they
are called Rayleigh
scatterers and the
above is correct – the
scattering intensity is
independent of
scattering angle
• If not, then there is
interference from the
light scattered from
different parts of the
single scatterer
• Different shapes give
different particle
scattering factors P(q)
qR~q
From P(q), we can get a Radius
of Gyration for the scatterer
Analysis of LS Data
• Measure I(q, c) and plot
Kc/Rq vs sin2(q/2) + (const)c
– Extrapolations: c
q
0
0
Final result
Slope~RG
intercept
Slope~B
Problems: Dust, Standard to measure Io, low angle measurement
flare
Polydispersity
• If the solution is polydisperse – has a mixture
of different scatterers with different M’s - then
we measure an average M – but which
ci M i
average?
Rq  K
c  KM wc
ci
• So the weight-averaged M is measured!
Possible averages:
M i N i
Number-average M N 
N i
M i ci M 2i Ni
Mw 

Weight-average
ci
Ni M i
2
3

M
c

M
N
i i
i i
Z-average
Mz 

M i ci
Ni M i 2
Dynamic Light Scattering
- Basic ideas – what is it?
- The experiment – how do you do it?
- Some examples systems – why do it?
Double Slit Experiment
Coherent beam
Extra path length
screen
+
=
+
=
Light Scattering Experiment
Scatterers in solution (Brownian motion)
Scattered light
Laser at fo
Narrow line incident laser
Doppler broadened
scattered light
Df
0 is way off scale
fo
Df ~ 1 part in
f
1010
-
1015
More Detailed Picture
detector
q
Inter-particle interference
Detected
intensity
Iaverage
time
How can we analyze the fluctuations in intensity?
Data = g(t) = <I(t) I(t + t)>t = intensity autocorrelation function
Intensity autocorrelation
• g(t) = <I(t) I(t + t)>t
t
For small t
For larger t
t
g(t)
tc
t
What determines correlation time?
• Scatterers are diffusing – undergoing Brownian
motion – with a mean square displacement
given by <r2> = 6Dtc (Einstein)
• The correlation time tc is a measure of the time
needed to diffuse a characteristic distance in
solution – this distance is defined by the
wavelength of light, the scattering angle and the
optical properties of the solvent – ranges from
40 to 400 nm in typical systems
• Values of tc can range from 0.1 ms (small
proteins) to days (glasses, gels)
Diffusion
• What can we learn from the correlation time?
• Knowing the characteristic distance and
correlation time, we can find the diffusion
coefficient D
• According to the Stokes-Einstein equation
kBT
D
6ph R
where R is the radius of the equivalent
hydrodynamic sphere and h is the viscosity of
the solvent
• So, if h is known we can find R (or if R is known
we can find h
Why Laser Light Scattering?
1.
2.
3.
4.
5.
Probes all motion
Non-perturbing
Fast
Study complex systems
Little sample needed
Problems: Dust and
best with monodisperse samples
Aggregating/Gelling Systems
Studied at Union College
• Proteins:
– Actin – monomers to polymers and networks
Study monomer size/shape,
polymerization kinetics,
gel/network structures formed,
interactions with other actin-binding
proteins
Why?
Epithelial cell under fluorescent
microscope
Actin = red, microtubules = green,
nucleus = blue
Aggregating systems, con’t
– BSA (bovine serum albumin)
– beta-amyloid +/- chaperones
– insulin
what factors cause or promote
aggregation?
how can proteins be protected
from aggregating?
what are the kinetics?
• Polysaccharides:
– Agarose
– Carageenan
Focus on the onset of gelation –
what are the mechanisms causing gelation?
how can we control them?
what leads to the irreversibility of gelation?
Current Projects
1.b-amyloid – small peptide that aggregates
in the brain – believed to cause Alzheimer’s
disease-
Current Projects
2. Insulin
aggregation
•
EFFECTS OF ARGININE
ON THE KINETICS OF
BOVINE INSULIN
AGGREGATION STUDIED
BY DYNAMIC LIGHT
SCATTERING
•
•
By
Michael M. Varughese
•
*********
•
•
•
Submitted in partial fulfillment
of the requirements for
Honors in the Department of
Biological Sciences
and done in conjunction with the
Department of Physics and
Astronomy
UNION COLLEGE
June, 2011
•
•
•