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Optical properties of colloidal systems
Optical properties of colloidal systems
Rubén G. Barrera*Instituto de Física, UNAM
Mexico
Rubén G. Barrera*Instituto de Física, UNAM
MexicoTonanzintla, 2006
* Consultant at Centro de Investigación en Polímeros, Grupo Comex
Electrodynamics in material mediaElectrodynamics in material media SI units
Conventional approach
ω
= ∇×
= −M
P
J M
J i P
material fields
closed
open
0M P outside= =
= +ext indJ J Jmaterial
= +indM PJ J J
ρ∇ ⋅ =
∂∇× = +
∂
ext
ext
D
DH Jtµ
= +
= −0
1
D E P
H B M
AverageAverage (macroscopic level)
,E J= + flucE E E
10 20 30 40
-0.6-0.4
-0.2
0.20.4
0.6microscopic level
= AVE P Ecoherent Length scale
= −flucE E E 2 0 4 0 6 0 8 0 1 0 0
- 2
- 1
1
2
diffuse
Continuum ElectrodynamicsContinuum Electrodynamics
Linear approximation≈E E
ε χ
µ µ
=
⎛ ⎞= −⎜ ⎟⎝ ⎠
0
0
1 1
EP E
M B
Local approximation
= + fluc
fluc
S S S
S S
χ ωµ ω
( )( )E
Power
Light in material mediaLight in material media
Plane wavesPlane wavespolarization
( )ω= ⋅ −0 ˆexpi iE E ik r t eindex of refraction
ε ω µ ωωε ω µ ω
=0 0
( ) ( )( )( ) ( )
nω=0k
cε ω ε χ ω= +0( ) [1 ( )]E
ω= 0 ( )k k n
induced currents
Inhomogeneous systemsInhomogeneous systems
Fresnel
( )ε µ1 1,
i
r
tθi
( )ε µ2 2,
= r
i
ErE
( ),TE TM
reflectionamplitude
polarization
ε µ ε µ θ1 1 2 2( , , , , )ir
(1820)
=2R r
non-magnetic1 2n n θ1 2( , , )ir n n
GUSTAV MIE (1905)
⊥ ⊥
⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠ ⎝ ⎠
2
1
00
s incikr
s inc
SE EeSikrE E
Scattering matrix
θθ( )jS( )ε µ1 1,
( )ε µ2 2,
-1 0 1 2 3 4 5 6 7-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0 TiO2/resina = 0.10 µm
dCsca/dΩ
ΩscadC
d
θ
SCATTERING
Colloidal SystemsColloidal Systems
Inhomogeneous phasedispersed within a
homogeneous phase
colloidal particles
homogeneous phase
ExamplesExamples
continuousphase
dispersephase
name examples
liquid solid sol paints, blood, tissues
liquid liquid emulsion milk, water in benzene
liquid gas foam foam, whipped cream
solid solid solid sol composites, policrystals, rubys
solid liquid solid emulsion opals, milky quartz, …
solid gas solid foam porous media,…
gas solid solid aereosol smoke, powder,…
gas liquid liquid aereosol fog,…
Optical propertiesOptical properties ( )R λ
( )T λ
Propagation
Transmission
Reflectión optical spectroscopy400 800λ≤ ≤ nm
METAMATERIALSINVERSE PROBLEM
( )R λ
( )T λstructure of the colloid
Effective medium
ExampleExample reflectionspectroscopyreflection
spectroscopy
critical angle
milk1 1.50n =
2n2
1
sin cnn
θ =
effective index of refraction
2nδ state of aggregationparticle sizing
ColloidsColloids
2 ax πλ
=size parameter: 400 800λ≤ ≤ nm
SMALL 1 100x ∼ nm diffuse cohS S
BIG 1x ≥
diffuse cohS S≥
turbid
scattering
Modelo para el coloideModelo para el coloide
esferas idénticas en suspensión
vacío
En promedio: homogéneo e isotrópico
aparámetrosparámetros
parámetros geométricos
parámetros ópticos no-magnéticas
0 0
,S SS S
ε µε µ
ε µ≡ ≡ S Sn ε=1Sµ =
índice de refracción
3(2) (3)
12 1 2 34 , ( ), ( , , ),...
3N af r r r rV
π ρ ρ=parámetros estadísticos
Cálculo de propiedades ópticasCálculo de propiedades ópticas
propiedadesópticasTEORIAmodelo
TEORIA
ECUACIONES DE MAXWELL
Esparcimiento múltipleDifusión de “fotones”Transferencia radiativa
El medio efectivoEl medio efectivo“homogéneo”
eff effε µ¿posible?
medio efectivosuspension coloidal
Propiedadesópticas
PropiedadesópticasElectrodinámica
continuaElectrodinámica
continua
AntecedentesAntecedentes
0
2 ax ka πλ
= =parámetro de tamaño
partículas pequeñasel medio efectivo siempre existe!
1x
ópticos1effµ = microestructura
eff effn ε=
(no-magnético) (2) (3)0( , , ; , , , ,...)eff Sn n a fε λ ρ ρ
“reglas de mezclado”
ExampleExamplenon-magnetic
E local
0( , )p pε µ µ=Isolated sphere
( )ω= −1
a
J i pv
++
+
- - -J
Induceddipole
ε α= 0localp E
Inducedcurrent
3 14
2p
pa
εα π
ε−
=+
ε ω = −( ) 2presonancia
extEColloidColloid random locations
exti ij j
jp E T pα
⎡ ⎤= + ⋅⎢ ⎥
⎣ ⎦∑
LOCAL FIELD
NP pV
== ∑1i
ip p
N
0 effP Eε χ=[ ]extE E
JC Maxwell GarnettJC Maxwell Garnett (1905)Mean-field approximation
0effµ µ= non-magnetic
ip p≈ 1 2 1 31
eff
s
f ff
ε α αε α
+= ≈ +
−
α ω =1( )f
dilute limit343
N afV
π=
filling fraction
( ) 1( ) 2
p
p
ε ωα
ε ω−
=+ shift of resonances
optical propertiesdimensionless polarizability
*α renormalized polarizability (1988-94)
21 1*2
e
e
ff
ααα
− −=1 2 *
1 *eff
s
ff
ε αε α
+=
−(2) (3)
e e ef f f= +
(2)(2)
40
(2 )3eaxf f dx
xρ∞
= ∫
(3) 2 (3) 3 312 23 1 2 3 2 32
9 ˆ ˆ ( , , )4ef f q T T q R R R d R d Rρπ
= ⋅ ⋅ ⋅ ∆∫(3) (3) (2) (2)
1 2 3 1 2 2 3( , , ) ( , ) ( , )R R R R R R Rρ ρ ρ ρ∆ = −
Expansion in kaExpansion in ka
CA Grimes and DM Grimes, PRB 43, 10780 (1991)
( ),p pε µspheres
Mean-field approximation
1ka 0p p pk a k a unrestrictedε µ=
2 22
2
6 41 3( ) ...2 10( 2)
p p p p
pA ka
µ ε ε εεε ε
+ − +−≈ + +
+ +1 21eff
fAfA
ε +=
−
2 22
2
6 41 3( ) ...2 10( 2)
p p p p
pB ka
ε µ µ µµµ µ
+ − +−≈ + +
+ +1 21eff
fBfB
µ +=
−
Non-magnetic caseNon-magnetic case
1pµ =2
22
1 2 6 43( ) ...
2 10( 2)p p p
p pA ka
ε ε εε ε
− − +≈ + +
+ +1 21eff
fAfA
ε +=
−
2 13( ) ...
90pB kaε −
≈ +1 21eff
fBfB
µ +=
−
4 2 24 2eff
fAfA
ε +=
−Photonic crystals
4 2 24 2eff
fBfB
µ +=
−Close packed fcc
The powerThe power
∝ = +2 2 2
AV flucPower E E E
~ 1ka
0 2 40.0
0.2
0.4
0.6
0.8
1.0
diffuse
coherent
penetration
trans
mitt
ed p
ower
γ= +1 (0)eff
S
n i Sn
AttemptsAttempts
effnδ1f
complex
0effµ µ=Van de Hulst
Teoría de MIEγ = 3
32
fx límite diluido
Scattering matrix
⊥ ⊥
⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠ ⎝ ⎠
2
1
00
s incikr
s inc
SE EeSikrE E
sphere
= =1 2(0) (0) (0)S S S
TemptationTemptation
Reflection amplitude Use neff in Fresnel´s relations
Isotropic effective medium model (IEMM)
2 2
2 2
cos sincos sin
i ieffTE
i ieff
nr
nθ θ
θ θ
− −=
+ −
γ = 3
32
fx
γ= +1 (0)eff
S
n i Sn
C Bohren J. Atmos Sci. 43, 468 (85)
1 (0)effn i Sγ= +transmissionNormal incidence
11 ( )effn i Sγ π= +reflection
1
1
1 (0) ( )1 (0) ( )
eff
eff
i S Si S S
ε γ π
µ γ π
= + +⎡ ⎤⎣ ⎦= + −⎡ ⎤⎣ ⎦
r µ εµ ε−
=+
n = 1
…It might be expected that a composite medium is nonmagneticif its components are, but this is not correct… which was recognizedas long ago as 1909 by Gans and Happel…
Coherent scattering approachCoherent scattering approach
Multiple scattering
iθ
iθ
Single thinslab
transfer equations
RG Barrera and A García-ValenzuelaJOSA 20,296 (2003)A García-Valenzuela and RG BarreraJQSRT 79-80, 627 (2003)
Reflection amplitudeReflection amplitude3
32
fx
γ =
12 1/2
( 2 )/cos(cos [cos 2 (0)] ) (0)/cos
TE i ihs
i i i
Sri i S S
γ π θ θθ θ γ γ θ
−=
+ + −
(0)S
( 2 )iS π θ−
-1 0 1 2 3 4 5 6 7-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0 TiO2/resina = 0.10 µm
dCsca/dΩ
iθ
ValidationValidation
Theory
Extension toinclude the matrix
Experiment
Internal reflectionConfiguration
great sensitivityA García-Valenzuela, RG Barrera,C. Sánchez-Pérez, A. Reyes-Coronado,E Méndez, Optics Express, 13, 6723 (2005)
ProblemProblem
Can one describe the behavior of the coherentbeam with an effective medium?
effective medium (incomplete)colloidal system
eff
eff
εµ
Effective mediumEffective medium
Assume( )( )
eff
eff
ε ωµ ω
then use rCSM to determine them
( )
(1)
2
(1) (1) 2
( ) 1cos
( ) 1 2 ( ) ( )tan
TEieff
iTE
i i i ieff
i S
i S S
γµ θθ
ε θ γ θ θ θ
−
+ −
= +
= + −
MAGNETIC
(1)1
1 (0) ( 2 )2 iS S S π θ+ = + −⎡ ⎤⎣ ⎦
(1)1(0) ( 2 )iS S S π θ− = − −
Index of refractionIndex of refraction
0 20 40 60 80 100
0.800.850.900.951.001.051.101.151.201.25
a = 0.5λ
n = 1.00np = 1.50f = 0.10 Re(εTM)
Re(µTM)
Re(εTE)
Re(µTE)
Rea
l par
t of o
ptic
al c
onst
ants
angle of incidence (deg.)
( ) ( , ) ( , ) 1 (0)TE TEi ieff eff effn i Sω ε θ ω µ θ ω γ= ≈ + Van de Hulst
Limiting casesLimiting cases
small particles
1TEeffµ =
1(0) ( 2 )iS S π θ= − NON-MAGNETIC
normal incidence
1
1
1 (0) ( )1 (0) ( )
eff
eff
i S Si S S
ε γ π
µ γ π
= + +⎡ ⎤⎣ ⎦= + −⎡ ⎤⎣ ⎦
0iθ =
BOHREN
ResultsResults
A. Reyes-Coronado, A. García-Valenzuela.C. Sánchez-Pérez and RG BarreraNew Journal of Physics 7 (2005) 89
AgainAgain
QuestionQuestion
Can one describe the behavior of the coherentbeam with an effective medium?
AnswerAnswer
YES, but
The effective medium is magnétic
The effective medium is non local
µ µ≠ 0eff
( , ) ( , )eff effp pε ω µ ω
spatial dispersion
Non local responseNon local response
0
1εε χε
= = +
χ ω
ω ωχ ω
=
= − = −
( )
( )
P E
J i P i E
3( ; ) ( ; ) ( ; )P r r r E r d rω χ ω ω′ ′ ′= − ⋅∫TF
ω χ ω ω= ⋅( , ) ( , ) ( , )P p p E p
( , )r t
( , )r t′ ′
susceptibilidad
local
non-local
The T matrixThe T matrix
ESingle sphere
3
0
1( ) ( , ) ( )indJ r T r r E r d riωµ
′ ′ ′= ⋅∫
A collection of spheres
ωµ′ ′ ′= − − ⋅∫ 3
0
1( ) ( ; ) ( )ind Ep p p pJ r T r r r r E r d r
i
DRIVING FIELD
Effective-field approximationEffective-field approximation
( ) ( )EpE r E r≈ dilute limit
TAKING A CONFIGURATIONAL AVERAGE
3
0
1( ; ) (| |) ( ; )ind NJ r d r T r r E ri V
ω ωωµ
′ ′ ′= − ⋅∫
Nonlocal response
1(| |) ( , ; )
N
p pp
N T r r T r r r rV
ω=
′ ′− = − −∑
Effective mediumEffective medium
ε ωω
ε= + 0
20 0
( , ) 1 ( , )Leff p n T pk
( )µ ω
µ ω ω=
− −002
( , ) 1
1 ( , ) ( , )
eff
T L
pn T p T pp
induced currents
Magnetic response Amperianmagnetism
00
1( ; ) ( , ; )indJ p T p p Ei
ω ωω µ
= ⋅
single, isolated sphere
ResultsResults
longitudinal
20
1
( ) ( )4( , ) 1 3 ( 1)(2 1)3
L L n S n iS n
n S i
j x j xT p x a n n n dx x
πω ς χ∞
=
⎡ ⎤= + + +⎢ ⎥
⎣ ⎦∑
transverse
20
20 2 1 1 2
1
4( , ) (1 )3
12 (2 1) ( , ) ( , 1) ( 1, ) ( 1, 1)
TS
S n nn i i
T p x a
n nx a n c I n n d I n n I n n I n nx x
πω χ ξ
π χ ξ∞
=
= −
⎧ ⎫⎡ ⎤+⎪ ⎪+ + + − + + − + −⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
∑
Ag (radius = 0.1 µm)
2 4 6 8 100
1
2
3
4
Re[(ε(p,ω)/ε - 1)/f ] 0
pa
0.224 µm0.331 µm
0.400 µm 0.577 µm
0.887 µm
Electromagnetic modesElectromagnetic modes
transverse
( )T pωω ε ω µ ω=2
22 ( , ) ( , )eff effp p p
c
longitudinal
( )L pω( , ) 0eff pε ω =
Index of refraction
2 ( ) ( , ) ( , )effn p pc cω ωω ε ω µ ω≈ = = Van de Hulst
0.2 0.4 0.6 0.8 1
l0 @mm D
0.99
1
1.01
1.02
Re mH p=k0,wL
mH p=k0,wL for Ag with radius =0.1Ag (radio=0.1 µm)
λ [µm]0
0.2 0.4 0.6 0.8 1.00
0.99
1.00
1.01
1.02
Re[µ(ω/c,ω) ]
f = 0.005
Actual researchActual research
We are working on the non local properties of theelectromagnetic response of colloidal systems andWe are also performing refraction and reflectionexperiments in order to design an instrument to measurethe particle-size distribution of colloidal particles usingthe information stored in the coherent beam.
We are building up a group on optical propertiers at anindustrial lab of the paint industry, where we are applyingsome of the results obtained in our basic-researchstudies.