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Applications of
Dynamic Light Scattering
to Particle SizingOnofrio Annunziata
Department of ChemistryTexas Christian University
Fort Worth, TX, USA
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Outline
Brief summary on dynamic light scattering (DLS)
Particle sizing and particle-particle interactions(monodisperse systems)
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LASER
DETECTOR
CORRELATOR
COMPUTER
THERMOSTATEDCELL HOLDER
OPTICAL FIBER
IRIS
Scatteredintensity
Correlationfunction
DiffusionCoefficient
sample
DLS INSTRUMENT SCHEME
Scattering at 90°
( )Si t
( )g τ
D
IRIS
(test tubewith filtered
solution)
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Scattering Vector
0kSk
02| | | |
( / )Sk kn
πλ
= =
wave vector ofincident light
θ
λ Wavelength of incident light in vacuum
n Refractive index of the sample
(Elastic Scattering)
wave vector ofscattered light
0 Sq k k= −Scattering Vector
02| | | | 2sin
( / ) 2Sq q k kn
π θλ
⎛ ⎞= = − = ⎜ ⎟⎝ ⎠
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kiq rS
kE M e ⋅∑∼
( )2 2 2| | j kiq r rS
j kS E M ei M N⋅ −= < >∑∑∼ ∼
The scattered electric field ES of N identical particles must take into accountinter-particle interference.
Rayleigh Scattering of many particles
Scattered field
SE
N = number of particles
Phasedifference
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Dynamic light Scattering
Scattered field
SE
N = number of particles
Phasedifference
Brownian motion
time
S Si i− < >Si< >
2| |SSi E=
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Dynamic Light Scattering
Dynamic Structure Factor
[ (0) ( )]1( , ) (0) ( ) j kiq r rS S
j kF q E E e
Nττ τ ⋅ −=< ⋅ > < >∑∑∼
How the solution structure at time τ correlateswith the solution structure at time 0?
Field autocorrelation function
( )(1) 2( , )( , ) exp( ,0)
F qg q qF q
Dττ τ= = −
D = diffusion coefficient of particles
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Field autocorrelation function
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
q 2 D τ
g(1
) ( τ
) ( )(1) 2( , ) expg q q Dτ τ= −
Strong correlationτ = 0, g(1) =1
No correlationτ → ∞, g(1) = 0
Shortτ
Longτ
(1) ( ) (0) ( )S Sg E Eτ τ< ⋅ >∼
(1) ( ) 1g τ ≈ (1) ( ) 0g τ ≈
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22[ ( ) (0)](1) [ (0) ( )] 6( , )
q r riq r rg q e eτττ
− < − >⋅ −= < >=
2[ ( ) (0)]6
r rD ττ
< − >=
( )r τ
(0)r
( )(1) 2( , )( , ) exp( ,0)
F qg q q DF q
ττ τ= = −
(Gaussian Random variable)
Field autocorrelation function
DiffusionCoefficient
(Dilute solution)
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45
46
47
48
49
50
0 20 40 60 80 100
t (s)
i S (
kcou
nts/
s )
1
2
0 1 2 3
q 2 D τ
g(2
) ( τ
)( ) 2(2) 2( , ) 1 expg q q Dτ α τ⎡ ⎤= + −⎣ ⎦
The DLS Detector probes iS( t )This is a stochastic function
Intensity autocorrelation function
(2) ( ) (0) ( )S Sg i iτ τ< ⋅ >∼
The correlator calculatesIntensity autocorrelation function
(2) (1) 2( ) 1 | ( ) |g gτ β τ= +(Siegert equation)
1β < Coherence factor
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bk TDf
=
Stokes-Einstein Equation
Einstein Equation Stokes Equation
6 hf Rπ η=
η Fluid viscosity
hR Particle hydrodynamic radius
6b
h
k TDRπ η
=
bk Boltzmann constant
T Temperature
(spherical particles)
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Einstein Equation
Langevin Equation(Equation of motion)
( ) (0) ( ) (0)d v vm f v vdτ τ
τ< ⋅ > = − < ⋅ >
2( ) (0) (0)fmv v v e
ττ
−< ⋅ > =< >
21 3(0)2 2 bm v k T< > =
2
0
0
6 [ ( ) (0)] 2 (0) ( )
6 6f
b m b
r r v v d
k T e d
D
kfmTτ
τ τ τ τ τ
τ τ τ
∞
∞ −
= < − > = < ⋅ > =
= =
∫
∫
Solution of Langevin Equation
Equipartition Theorem
f friction coefficientm particle mass
kb Boltzmann constantT Temperature
bk TDf
=
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ba
ρ =
2 1/ 2
2 1/ 22 /3
(1 )1 (1 )ln
haR ρ
ρρρ
−=⎡ ⎤+ −⎢ ⎥⎣ ⎦
2 1/ 2
2 /3 2 1/ 2
( 1)arctan ( 1)haR ρ
ρ ρ−=⎡ ⎤−⎣ ⎦
Hydrodynamic radius and particle shape
b
a
b
a
ProlateEllipsoid
OblateEllipsoid
with
SphereR hR R=
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Hydrodynamic radius of attached spheres
hR R=
1.38hR R=
1.71hR R=
2.00hR R=
De la Torre et al. Quart. Rev. Biophys., 14, p.81 (1981)
Hydrodynamic radius of thin rods of radius R
L
( / 2)ln( / ) 0.31h
LRL R
=−
I. Teraoka, Polymer solutions: an introduction to physical properties, Wiley (2002)
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Hydrodynamic radius of linear polymers
0.665h gR R=Polymer in θ solvent
Polymer in good solvent 0.641h gR R=
Polymer in bad solvent 1.29h gR R=
2
2 1
1
n
i ii
n
ii
gRm r
m
=
=
=∑
∑
Radius of gyration
r position relative to center of mass
m mass
I. Teraoka, Polymer solutions: an introduction to physical properties, Wiley (2002)
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Particle Sizing
6b
h
D k TRπ η
=
( )(1) 2( , ) expg Dq qτ τ= −
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Correct use of Stokes-Einstein Equation
6b
h
k TDRπ η
≈
10mg/mLc <
Some salt should be added in the case of charged particles
Particle concentration should be kept small
[NaCl] 0.01M>
η is the viscosity of solvent. It should include the effect of other small solutesbut not that of large particles
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6b
h
D k TRπ η
=
Applications of Stokes-Einstein Relation
Denaturation of Agglutinin
Tetramer
( )(1) 2( , ) expg Dq qτ τ= −
Sinha et al., Biophysical J., 88, p. 4243 (2005)
1. Protein denaturation
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Fluorescence studies
DLS studies
Denaturation of Agglutinin
Sinha et al., Biophysical J., 88, p. 4243 (2005)
λ m
ax(n
m)
Rh
(nm
)
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Dispersion-Stabilization of Cerium Oxide Nanoparticleswith Poly(acrylic acid)
Sehgal et al., Langmuir, 21, p. 9359 (2005)
General limitation for utility of nanoparticles is their colloidal stability
Stabilization is reached by the adsorption of molecules on the surface resulting in steric or electrostatic barriers.
2. Dispersion Stabilization of nanoparticles
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DIA
MET
ER
Hysteresis loop illustrates that colloidal particles can exist in two different states under the same physicochemical conditions.
pH titration and Dispersion-Stabilization behavior
Sehgal et al., Langmuir, 21, p. 9359 (2005)
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Nanoblossoms
Plamper et al., Nano Lett. (2007), p. 167.
Scheme of the structure of the polyelectrolyte star ( poly{[2-(methacryloyloxy)ethyl] trimethylammonium iodide} ) and the photochemical reaction (photoaquation) leading from trivalent to divalent ions
3. Polyelectrolyte espansion
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Ionic strength 0.1 M NaCl, c = 0.5 mg/mL
The vertical arrow demonstrates the principle of photostretching
Effect of divalent and trivalent counterions on polyelectrolyte radius
[Ni(CN)4] 2-
[Co(CN)6] 3-
Plamper et al., Nano Lett. (2007), p. 167.
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Photoinduced stretching measured by DLS (nanoblossoming)
illumination time (min)
Plamper et al., Nano Lett. (2007), p. 167.
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Synthesis of Janus Discs, Based on the Selective Crosslinkingof PB Domains of an SBT Terpolymer with Lamellar Morphology
Walther et al. JACS (2007), ASAP
PolystyrenePolybutadienePoly(tert-butyl methacrylate)
4. Monitoring Sonication
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Transmission electron micrograph of an ultrathin section of SBT-1 films after staining with OsO4. The ultrathin film was imaged using standard TEM grids. The white bar indicates the long period of the periodicity of the structure.
Self-assembly in lamellar structures
Walther et al. JACS (2007), ASAP
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Dependence of Hydrodynamic radius on the sonication power and duration for differently crosslinked block terpolymer templates.
Low sonication power
High sonication power
Monitoring sonication by dynamic light scattering
Walther et al. JACS (2007), ASAP
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5. Vesicle association
Reversible metal-induced assembly of clusters of vesicles
Small amount of terpy-funcionalized phospholipidis introduced into normal phospholipid vesicles.
Constable et al., Chem. Com. (1999), p. 1483.
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Metal-inducedclustering
Control(normal vesicles)
Effect of [Fe2+] on the hydrodynamic radius
Constable et al., Chem. Com. (1999), p. 1483.
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DLS coupled with SE-HPLC
Wyatt Technology
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Probing particle-particle interactions
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Diffusion-coefficient dependence on particle concentration
0 ) ( )(H cD SD c=
Hydrodynamic Factor ( ) 1 ...DH c k c= + +
Thermodynamic Factor(structure factor)
( ) 1 2 ...S c B M c= + +
[ ]0 1 ( 2 ) ...DD D k B M c= + + +
Trace diffusion coefficient 0 6b
h
k TDRπ η
=
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Probing particle-particle interactions
B > 0 particle-particle net repulsion
*S Si i< *
S Si i>B < 0
particle-particle net attraction
[ ]0 1 ( 2 ) ...DD D k B M c= + + +
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1. Protein – Protein Interactions
25
lysozyme
Repulsion
Attraction
[ ]0 1 ( 2 ) ...DD D k B M c= + + +
Kuehner et al., Biophysical J., 73, p. 3211 (1997)
0 5 10 15 20c (mg/mL)
NaCl (M)D/D
0
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2. Fast reversible oligomerization of proteins
….. …..
6< > = app
B
h
TDR
kπη
Definition of apparent hydrodynamic radius
Chemical equilibrium Monodisperse system
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βB1-Crystallin
βB1
βB1ΔN41
AGEN-terminalextensions
C2
NN
C2C2
NN
NN
Annunziata et al., Biochemistry, 44, p. 1316 (2005)
Age-related truncation at the N-terminal
Cataract-related proteinsCATARACT
EYE-LENS
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Effect of truncation on protein oligomerization
3
4
5
6
7
0 5 10 15 20 25 30
Rhap
p (n
m)
c1 (mg/m l)
β B1Δ N41β B1
T = 280 KT = 287 KT = 300 K
Rh(nm)
K2 (287 K)(M-1)
ΔH2(kJmol-1)
Tc(K)
βB1 3.9 945 -24±1 250-260βB1ΔN41 3.0 1400 -29±2 280
6B
apph
k TDRπη
=
+K2
apph
DR
= diffusion coefficient= appar. hydrod. radius
Truncation enhances oligomerization (potential cataractogenic modification)
c (mg/mL)
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Polydisperse particles ( )(1) 2( , ) expj jj
g q W q Dτ τ= −∑
Polydisperse systems
Example 1 Example 2
jj jW c M∼