UGC-DAE Consortium for Scientific Research, Mumbai Light Scattering studies on Colloids & Gels Goutam Ghosh
UGC-DAE Consortium for Scientific Research, Mumbai
Light Scattering studies on Colloids & Gels
Goutam Ghosh
Elastic Light Scattering
Brownian motionBrownian motion
UGC-DAE Consortium for Scientific Research, Mumbai
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
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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
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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
Classical theory of Classical theory of ScatteringScattering
- +
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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
Classical theory of Classical theory of ScatteringScattering
UGC-DAE Consortium for Scientific Research, Mumbai
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
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Light Scattering method
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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
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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
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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
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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)
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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)
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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
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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)
DLS on colloidsDLS on colloids
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
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DLS on colloidsDLS on colloids
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
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q2
DR
VV
VH
DLS on colloidsDLS on colloids
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Sphere-to-Rod transition of SDS micelles with addition of TBABr
DLS on colloidsDLS on colloids
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
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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
DLS on colloidsDLS on colloids
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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 colloidsDLS on colloids
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
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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
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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.
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DLS on Gel
So, how to measure a Gel using DLS ?
Method 1 (Pusey): <I (q )>T = <I (q )>E [S (q,t ) – S (q, ∞)]
Non-ergodic ratio: Y <I (q )>E / <I (q )>T2/12)2( ][
11),( TTgYY
YtqS
1])(/)([222 T
TT qIqIwhere
<I (q )>T
<I (q )>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.
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DLS on Gel
Therefore, directly measures g1(t) or S(q,t)
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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
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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
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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.
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Thank Thank youyou
Reference: Dynamic Light Scattering: Application of Photon Correlation Spectroscopy, Ed. Robert Pecora, Plenum Press, New York and London, 1985.
Goutam Ghosh
Ph: 2550 5327
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http://www.csr.ernet.in