Applications of

Dynamic Light Scattering

to Particle SizingOnofrio Annunziata

Department of ChemistryTexas Christian University

Fort Worth, TX, USA

Outline

Brief summary on dynamic light scattering (DLS)

Particle sizing and particle-particle interactions(monodisperse systems)

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)

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

⎛ ⎞= = − = ⎜ ⎟⎝ ⎠

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

Dynamic light Scattering

Scattered field

SE

N = number of particles

Phasedifference

Brownian motion

time

S Si i− < >Si< >

2| |SSi E=

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

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 ≈

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)

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

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)

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

=

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=

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)

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)

Particle Sizing

6b

h

D k TRπ η

=

( )(1) 2( , ) expg Dq qτ τ= −

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

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

Fluorescence studies

DLS studies

Denaturation of Agglutinin

Sinha et al., Biophysical J., 88, p. 4243 (2005)

λ m

ax(n

m)

Rh

(nm

)

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

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)

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

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.

Photoinduced stretching measured by DLS (nanoblossoming)

illumination time (min)

Plamper et al., Nano Lett. (2007), p. 167.

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

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

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

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.

Metal-inducedclustering

Control(normal vesicles)

Effect of [Fe2+] on the hydrodynamic radius

Constable et al., Chem. Com. (1999), p. 1483.

DLS coupled with SE-HPLC

Wyatt Technology

Probing particle-particle interactions

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

=

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= + + +

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

2. Fast reversible oligomerization of proteins

….. …..

6< > = app

B

h

TDR

kπη

Definition of apparent hydrodynamic radius

Chemical equilibrium Monodisperse system

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

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)

Polydisperse particles ( )(1) 2( , ) expj jj

g q W q Dτ τ= −∑

Polydisperse systems

Example 1 Example 2

jj jW c M∼