UGC-DAE Consortium for Scientific Research, Mumbai

UGC-DAE Consortium for Scientific Research, Mumbai

Light Scattering studies on Colloids & Gels

Goutam Ghosh

Elastic Light Scattering

Brownian motionBrownian motion


scattering angleq = Iks – kiI = scattering vector IksI = IkiI = 2 /

q =

)sin(4 2n

- +

I ~ <n> (p – s) V2 P(q) S(q)

Case 1

I ~ I(t) ~ n(t) ~ (t)

Case 2Dynamic Light Scattering (DLS)

Static Light Scattering (SLS)

ki

k s

Count scattered intensity as a function of q or time (t)

d = / q

t ~ 50 nSec

I ~ <n> (p – s) V2 P(q)

ki

ki

k s

q


y

x

z

Scattering vector, q

ki

ks

Incident

Scattered

Basic scattering geometry Angle between direction of polarization and scattering plane,

Scattering angle ,

Scattering vector, q = ki - ks

Source: Vertically polarized, monochromatic ( = 532 nm) laser light.

Classical theory of Classical theory of ScatteringScattering


Oscillating electric field )(2

ctyCosEE o

Induced dipole moment )(2

ctyCosEp o

Scattered Electric field

Scattered intensity 2

42

24' 16

Sinr

II o Rayleigh law

Sin

t

p

rcE

2

2

2' 1

True for particles whose size less than /20


- +


Polarization dependence

II I I

rCosII o( )

( ) ( )( )

2

81

4 2

2 42

I For perpendicular polarization, = 90, for all

II For parallel polarization, = 0 when = 90 and = 90 when = 0 and 180

III For unpolarized light

RI

I

Sin r

Cosmeas

( )

( ).

.

( )0 1

2

2

Rayleigh ratio

Angular distribution independent of size



LS P1 L S PD PM

P2

CPMT

Light Scattering Set upLight Scattering Set up

LS → Laser source

(100 mW, He-Ne, 532 nm vertically polarized)

P1 and P2 → Linear Polarizers

L → Lens used to focus the incident beam at the sample

PD → Photo-Diode

PM → Power meter

PMT → Photo-multiplier Tube detector

C → Computer for data collection

→ Scattering angle

VVVH

Indirect (reciprocal space) Low resolution (DLS) Interference (dust)

Average structure Interactions Liquid phase Fast and wide range (DLS) Non-destructive


Light Scattering method


Light, neutron and X-ray scattering

Size range of different scattering methods

Dimension (nm)10 100 1000 100001

SANS, SAXS USANS

SLSRGD MIE

DLS

Comparison (Static scattering)- Different length scales- scattering powerlight (refractive index)x-rays (electron density)neutrons (scattering length density)

Works in a range where optical microscopy fails!

Systems studied using Light Scattering method

Colloidal solutions

- Surfactants- Polymers- Drugs- Nanoparticles

Gels


In a colloidal solution particles execute Brownian motion in the entire volume.

In a gel a macroscopic network is formed and no Brownian motion exist.

UGC-DAE Consortium for Scientific Research, MumbaiStatic Light Static Light

ScatteringScattering I ~ <n> (p – s) V2 P(q) S(q)

qr

qrSine

q

riq .

Small limit, qr <<1,

6

)(1

2qr

qr

qrSin

dVqr

VqF )

6

)(1(

1)(

2

6

2 22

6

)(1)(

gRqg e

qRqF

]1

[ 22 dVrV

Rg where

2.)(

1 dverv

riq

Form factor for non-interacting particles, S(q) = 1

P(q) = F(q) 2 = ,

50 nm

I(q)= I(0) exp(-q2Rg2/3)

Guinier law

0.010 0.015 0.020 0.025 0.030

I(q

)

For sphere

22

5

3RRg

For non-interacting particles, S(q) = 1

• Non-aqueous solution

• Infinitely dilute solution in water

• Moderate concentration (vol. frac. < 10-2) in water with

salt concentration > 10 mM


P(q) for a large sphere & S(q) = 1 2

3)(

)cos()sin(3)(

qR

qRqRqRqP

0.010 0.015 0.020 0.025 0.030

100

1000

I (a.

u.)

q (nm-1)

220 nm

0.010 0.015 0.020 0.025 0.030

100

1000

q (nm-1)

450 nm

Static Light Static Light ScatteringScattering

Interparticle Structure factor, S(q)

Static Light Static Light ScatteringScattering

When the positions are correlated

drqr

qrSinrrgNqS

)()1)((41)( 2

0

S(q), Concentration dependence

•Vol. Fraction•Charge•Ionic strength

g(r) represents the probability of finding another particle at a distance r and r+dr

1E-3 0.01 0.10.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

=0.1

=0.01

S(q

)

q (Å-1)

Small particle limit

As concentration increases, peak develops at q ~ /d

I(q) ~ <n> (p – s) V2 P(q) S(q)


Dynamic Light Scattering Dynamic Light Scattering (DLS)(DLS)

Also known as

Photon Correlation Spectroscopy (PCS)

and

Quasi-Elastic Light Scattering (QELS)

UGC-DAE Consortium for Scientific Research, MumbaiDynamic Light Scattering (colloidal system)Dynamic Light Scattering (colloidal system)

Time Resolved Experiment

• Number density changes with time• Net intensity changes with time• Diffusion rate depends on particle size, medium viscosity, temperatureAuto-correlation

function

dt

2)0(

τ)I(t).I(t2T1

lim)((2)gt I

“large” slow moving particles

“small” fast moving particles

I(t)

t (S) S

g(2) (

)

qr(t)

1 2 3 4

I ~ I(t) ~ n(t) ~ (t)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

10 100 1000 10000 100000

Delay time () / S

g1(

)

Siegart’s relation: g2() = 1 + |g1()|2 , 0 < ≤ 1

dt

2)0(

τ)E(t).E(t2T1

lim)(1gt E

)( )( eee

2

)()(1

22 ee

g e( ) ( )( )1 22

12

Polydispersity index =22

Method of Cumulant

=

Exponential fit

Relaxation time dist.

N

j

M

ijiij bg

1 1

2)1(2 )]exp()([ NNLS

2

1

22)1(22 )(])()()[/1(

N

jji LGdeGg

CONTIN

=1/TR= q2 DI ~ < E

>2

DLS on colloidsDLS on colloids


Monodisperse spheres (single exponential decay)

translational

g1() = A exp (- Dq2 )

100

101

102

103

104

105

0.0

0.2

0.4

0.6

0.8

1.0

g1 ()

(-sec)

1 2 3 4

Do - diffusion coefficient, T - temperature- viscosity, Rh - hydrodynamic radius

Do = kT / (Rh)

Stokes-Einstein relation

2.5 nm (1)

54 nm (2)

214 nm (3)

422 nm (4)


For translational diffusion, = Dq2

SDS micelles in presence of additive

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

(1

04 s

-1)

q2 (1010cm-2)

D (sl

ope)

size

1



Spherical particles

A2

rotational

A1

Atranslational

2

1)1( )( DqeAg

q2

D

kT

qL > 3

D = 3L

F(p)

F(p) – shape factor

Isotropic Anisotropic

Non- (rotational diffusion) )6(

2

2RDDqeA


q2

DR

VV

VH



Sphere-to-Rod transition of SDS micelles with addition of TBABr


S. No. Frictn. Dirn. Cylindrical (L>>r) Ellipsoid (b>>a) Sphere (r)

1 fparallel

2 fperpendicular

3 frotational

4 faxial rotation

2.0)2/ln(2

rLL

84.0)2/ln(4

rLL


66.0)2/ln(

331

rLL

Lr 24

5.0)/2ln(4

abb

5.0)/2ln(8

abb

5.0)/2ln(

338

abb

ba23

16

r6

38 r

38 r

r6

Stokes-Einstein relationD = kBT / f



Polymer solutions

0.01 0.1 1 10 100 1000 10000 100000 1000000

0.0

0.2

0.4

0.6

0.8

1.0

g1 ()

(s)

Semi-dilute regime

Number density > 4Rg3

Fast mode – Diffusion

Slow mode – stress relaxation


DLS on GelWhat is a gel ?

A gel is a physically or chemically cross-linked three dimensional network which can hold liquid; therefore, visco-elastic in nature.

Colloidal solution Polymer gel



Characterization of the Sol-gel transition : Gelation kinetics

Gelation mechanismCharacterization of the

gel phase : Morphology

Dynamics

What is measured on gel using DLS ?

DLS on Gel

Gelation kinetics :- The time taken by the polymer solution to transform to a macroscopic gel phase is called the gelation time (tgel). The inverse of tgel is the gelation rate (tgel

-1), or the gelation kinetics.

Measurement methods (gelation kinetics):● Test tube tilt (TT)

● Light Scattering (LS)

NO FLOW

Sca

tter

ed c

ou

nts

Measured time


DLS on Gel

A gel is a non-ergodic system, as its dynamics are restricted by bonds. Therefore, a time-averaged measurement does not represent the complete structural and dynamical aspects of a gel system.


DLS on Gel

So, how to measure a Gel using DLS ?

Method 1 (Pusey): T = E [S (q,t ) – S (q, ∞)]

Non-ergodic ratio: Y E / T2/12)2( ][

11),( TTgYY

YtqS

1])(/)([222 T

TT qIqIwhere

T

E

Method 2 (Xue): In this method, the detection area has to be such that multiple speckles can be seen at a time. The sample (gel) is either rotated or translated to average over multiple orientation of the sample.


DLS on Gel

Therefore, directly measures g1(t) or S(q,t)


DLS on Gel

Dynamic structure factor, S(q,t ) of a gel phase has two modes, namely, fast mode and slow mode, i.e.,

S(q,t ) = Sf(q,t ) + Ss(q,t )

Fast mode : The fast mode relaxation which gives rise to the initial exponential decay of S(q,t ) arises due to the diffusive mode of segmental dynamics in polymer chains between two cross-link points.

2

2 ),(

t

txut

u

2

2

x

uE

2

2

x

uE

t

u

Df = E /

Stokes-Einstein’s equation : Df = E / = kBT / 6


DLS on Gel

Slow mode : The origin of the slow mode is not very clear. Two

models are reported.

(1)Gel mode plus inhomogeneity (GMPI) – gel is viewed as an elastic medium with overdamped modes describing the density fluctuations. Coupled with some static inhomogeneities this picture can qualitatively describe the initial decay of the correlation function (fast mode) and its saturation at long time (slow mode).

(2)Harmonically bound Brownian particle (HBBP) – at short time the particles (chain segments) undergo simple diffusion, but at longer time they find that they are restricted to a maximum displacement when the elastic energy equals the thermal energy, i.e.,

kx 2/ 2 = kBT/ 2


DLS on Gel

How to determine to dynamics of the slow mode ?The two models can be distinguished by studying S(q,t) at different wavevector (q). For example, (1) in GMPI model, the fast mode of S(q,t) is q dependent, but the slow mode is independent of q, and (2) in HBBP model, both modes are q dependent.


Thank Thank youyou

Reference: Dynamic Light Scattering: Application of Photon Correlation Spectroscopy, Ed. Robert Pecora, Plenum Press, New York and London, 1985.

Goutam Ghosh

[email protected]

[email protected]

Ph: 2550 5327

UGC-DAE Consortium for Scientific Research

http://www.csr.ernet.in

UGC-DAE Consortium for Scientific Research, Mumbai

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