Congreso Internacional de Metalurgia y Materiales
14 SAM- CONAMET
XIII Simposio Materia
Optofluidic Characterization of
Porous Materials
Escuela de Materiales porosos
Nanoestructurados
Raúl Urteaga
Santa Fe - Octubre 2014
Optofluídica
Integración de la óptica y la microfluídica para obtener
información de ambas en simultaneo [1].
Caracterización óptofluídica de materiales:
_ Longitud de onda incidente mucho mayor
que el tamaño de las estructuras:
Índice de refracción efectivo
_ Interferencia coherente de haces:
Capas delgadas
_ Imbibición capilar de líquidos en matriz
porosa:
Fluidodinámica
[1] Psaltis et al. “Developing optofluidic technology through the fusion of microfluidics and optics.” Nature 2006, 442,
381−386.
Thin film optical interference
For P- type
polarization
F.L. Pedrotti, L.S. Pedrotti “Introduction to optics” 2nd Ed. Prentice-Hall inc. NJ (1993)
Thin film optical interference
Boundary Conditions:
Tangential components of magnetic and electric
field are continuos across the interface
The relation between fields at the interfaces can
be expressed in a matrix form
Which depends upon
the phase difference:
and the ‘‘admitance’’
of the film
F.L. Pedrotti, L.S. Pedrotti “Introduction to optics” 2nd Ed. Prentice-Hall inc. NJ (1993)
Transfer Matrix formalism
For a multilayer
we have:
The reflection and
transmission coefficients are
Can be calculated as
F.L. Pedrotti, L.S. Pedrotti “Introduction to optics” 2nd Ed. Prentice-Hall inc. NJ (1993)
Results for Normal Incidence
For this case we
have:
d
1
t
T
where
R
n0
The total reflectance
will be:
Here the phase
difference is:
𝛿=
2𝜋𝑛1 𝑑
𝜆0
n1
n2
Results for Normal Incidence
𝛿=
2𝜋𝑛1 𝑑
𝜆0
d
=> Periodic!
Important to note:
1
t
(If δ is real )
T
R
n0
𝑛1 𝑑 =
𝜆0
4
𝑛1 𝑑 =
𝜆0
2
n1
n2
Results for Normal Incidence
If reflectance is periodic, then the Fourier transform in
K space is discrete, and have peaks at 2nd
2𝑛1 𝑑
If there is absortion
(refractive index complex)
Optical Properties of Mixtures
How to obtain the effective
dielectric constant?
To find these variables Maxwell's
equations must be solved for
electrostatic :
Geometry must be known and
generally requires an expensive
calculation
Effective Medium Theories
• Maxwell-Garnet
𝜀𝑙
Cavidad [2]
𝑬 paralelo al eje
𝜀𝑗
𝑬 perpendicular
Esfera
⅓
⅓
Cilindro
0
½
Placa
1
0
Elipsoide de ejes
𝑙𝑎 , 𝑙𝑏 y 𝑙𝑐
[2] O. Stenzel, The Physics of Thin Film Optical Spectra Springer Series in Surface
Sciences Volume 44 (2005)
Effective Medium Theories
• Lorentz- Lorenz [2]
𝜀𝑙 = 1 𝜀𝑗
• Bruggeman [2]
𝜀𝑙 = 𝜀 𝜀𝑗
• Looyenga-Landau-Lifshithz [3]
𝜀𝑙 ~𝜀𝑗
𝜀𝑗
[2] O. Stenzel, The Physics of Thin Film Optical Spectra Springer Series in Surface Sciences Volume 44 (2005)
[3] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, 2nd. Edition (vol. 8), Elsevier, Burlington,
1984.
Effective Medium Theories
Limits
𝑬
paralell
𝑬
Perpendicular
Esfer
⅓
⅓
Cylinder
0
½
Plate
1
0
Cavidad [2]
Effective Medium Theories
7
Silicon nreal
6
Silicon nimag
Alumina
5
Refractive index of silicon
and alumina [4].
4
Alumina is transparent!
3
𝑛= 𝜀
2
1
0
200
300
400
500
600
700
 [nm]
[4]
800
900
1000
1100
1200
Effective Medium Theories
Example case I:
4
Porous Silicon + Air
3.5
Effective refractive index
@𝜆 = 1000𝑛𝑚
L=1/3
Maxwell-Garnet
Bruggeman
Looyenga
3
2.5
2
1.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
Porosity
0.7
0.8
0.9
1
Effective Medium Theories
@𝜆 = 1000𝑛𝑚
L=1/2
1.7
1.6
Effective refractive index
Example case II:
Porous Alumina +
Isopropyl Alcohol
Depolarization factor L: 0.5
1.8
1.5
1.4
1.3
Maxwell-Garnet dry alumina
Maxwell-Garnet wet alumina
Bruggeman dry alumina
Burggeman wet alumina
1.2
1.1
1
0
0.1
0.2
0.3
0.4
0.5
0.6
Porosity
0.7
0.8
0.9
1
Optical Characterization
Spectroscopic reflectance of porous Silicon single layer
Optical Characterization
Spectroscopic reflectance of porous Silicon Bragg reflector
Spectroscopic Liquid Infiltration method
(SLIM)
1) Using FFT of the reflectance
spectrum we can obtain an
estimation of the optical width
of the dry sample
Porous Alumina + Isopropyl Alcohol
Depolarization factor L: 0.5
1.8
1.7
𝑛𝑑𝑟𝑦 𝑑
Effective refractive index
1.6
1.5
2) After infiltration we can
measure the optical width of the
wetted sample
1.4
1.3
1.1
1
𝑛𝑤 𝑑
Maxwell-Garnet dry alumina
Maxwell-Garnet wet alumina
Bruggeman dry alumina
Burggeman wet alumina
1.2
0
0.1
0.2
0.3
0.4
𝑛𝑤 = 𝑓 𝑃, 𝑛𝐴𝑙 , 𝑛𝑙𝑖𝑞
0.5
0.6
Porosity
0.7
0.8
0.9
1
𝑛𝑑𝑟𝑦 = 𝑓 𝑃, 𝑛𝐴𝑙 , 𝑛𝑎𝑖𝑟
3)The system can be
solved to obtain P and d
M. J. Sailor Porous Silicon in Practice Preparation, Characterization and Applications
Wiley-VCH Verlag & Co. (2012)
Fluid Mechanics at low Re
I) Capillary filling of uniform closed channel
Laplace Pressure
Straight channel
2r
∆𝑃 =
Contact Angle
𝜎 𝑐𝑜𝑠𝜃
𝑟
Hagen-Poisseuille flow
Tortuous channel:
L
𝜏=
L’
In nondimensional form:
𝐿
𝐿′
were
L.N. Acquaroli, R. Urteaga, C.L.A. Berli, R.R. Koropecki, Langmuir 27, 2067 (2011)
Fluid Mechanics at low Re
I) Uniform closed channel
Solving the differential equation:
Lucas-Washburn Dynamics
1 𝜎𝑟
𝑙2 =
𝑡
2 𝜇
𝑖𝑓 𝛼 → 0 then 𝑥 ∗ → 𝑡 ∗
The final position defines the value of 𝛼
Edward W. Washburn “THE DYNAMICS OF CAPILLARY FLOW”
The Physical Review Vol. XVII, 3, (1921)
Posición Menisco
𝛼
𝑖𝑓 𝑡 → ∞ then 𝑥 ∗ → 1+𝛼
L
0
Tiempo
tfill
Optoflidic Characterization
Porous Silicon
Lateral view
Top view
Dynamic SLIM
Experimental setup
What is measured (twice)
Normalized reflectance during the imbibition of
a PS layer (15 μm thick, fabricated using a
current density of 13 mA/cm2), with isopropyl
alcohol at room temperature and the pressure
of 1 atm.
L.N. Acquaroli, R. Urteaga, C.L.A. Berli, R.R. Koropecki, Langmuir 27, 2067 (2011)
Relation between reflexion extremes and
liquid infiltration
An extreme in reflectance will occur
each time the optical path is a multiple
of 𝜆/4:
𝜆0
𝑒0 = 𝑚
4
𝐿
𝑒0 =
0
𝑛𝑒𝑓𝑓 𝑥 𝑑𝑥
𝑒0 = 𝑛2 (𝐿 − 𝑥) + 𝑛3 𝑥
𝑑𝑒0
= 𝑥(𝑛3 − 𝑛2 )
𝑑𝑥
Between extremes in reflectance
the interface moves Δ𝑥 ∶
𝑛1 𝑑 =
𝛥𝑥 =
𝜆
4(𝑛3 − 𝑛2 )
𝜆0
4
𝑛1 𝑑 =
𝜆0
2
Results
L.N. Acquaroli, R. Urteaga, C.L.A. Berli, R.R. Koropecki, Langmuir 27, 2067 (2011)
Results
_Obtaining this values from the fit, it is posible
to calulate the mean hidraulic radius and the
tortuosity.
_ The porosity and layer thicknes it is also
obtained by SLIM
𝜏=
𝐿
~2,6
𝐿′
Fluid Mechanics at low Re
I) Variable section open channel
Poisseuille flow
𝜕𝑝 8𝜇𝑄
−
=
𝜕𝑥 𝜋𝑟 4𝑥
Laplace Pressure
𝛥𝑝 =
2𝛾 cos𝜃
𝑟𝑥
Direct problem:
𝑢
𝑙
=𝛼
𝑟 −3
𝑙
𝑙 −4
𝑟 d𝑥
0 𝑥
were
𝛼=
𝛾cos(𝜃)
4𝜇
R. Urteaga, L.N. Acquaroli, R.R. Koropecki, A. Santos, M. Alba, J. Pallares, L.F. Marsal,
C.L.A. Berli, Langmuir 29, 2784 (2013)
Fluid Mechanics at low Re
I) Variable section open channel
Inverse problem [5]:
𝑢
𝑟
𝑙
𝑙
as input
= 𝑟
𝐿
𝑢
1
𝐿
3
1
+
3𝛼
𝐿
𝑢
𝑙
4
3
𝑙
dl
−1 3
𝑢𝑙
𝑄 𝑣 as input
𝑟
𝑣
= 𝑟
5
5
𝑄
𝑉
𝑉
5
+ 2
𝜋 𝛼
𝑉
𝑣
𝑄6𝑣 dv
1
5
𝑄−1
𝑣
Has multiple solutions!!
[5] E. Elizalde, R. Urteaga, R.R. Koropecki, C.L.A. Berli, Phys. Rev. Lett. 112, 134502 (2014)
Optoflidic Characterization
Porous Alumina
Top side
Bottom side
Experimental setup
E. Elizalde, R. Urteaga, R.R. Koropecki, C.L.A. Berli, Phys. Rev. Lett. 112, 134502 (2014)
Optoflidic Characterization
Porous Alumina
Results
Radii at the ends coincide
with SEM photographs
E. Elizalde, R. Urteaga, R.R. Koropecki, C.L.A. Berli, Phys. Rev. Lett. 112, 134502 (2014)
Thanks
Relation between reflexion extremes and
liquid infiltration
𝐿
𝑒0 =
0
𝑛𝑒𝑓𝑓 𝑥 𝑑𝑥
𝑒0 = 𝑛2 (𝐿 − 𝑥) + 𝑛3 𝑥
𝑑𝑒0
= 𝑥(𝑛3(x) − 𝑛2(𝑥) )
𝑑𝑥
An extreme in reflectance will occur
each time the optical path is a multiple
of 𝜆/4:
𝜆0
𝑒0 = 𝑚
4
Between extremes in reflectance
the interface moves Δ𝑥 ∶
𝜆
𝜆
𝛥𝑥 =
=
4(𝑛3(𝑥) − 𝑛2(𝑥) ) 4(𝑛𝑤(𝑥) − 𝑛𝑑𝑟𝑦(𝑥) )
𝑛1 𝑑 =
𝜆0
4
𝑛1 𝑑 =
𝜆0
2
The linear approximation
Porous Alumina + Isopropyl Alcohol
Relative difference from linear
Depolarization factor L: 0.5
1.8
Modelo de Bruggeman. Depolarization factor: 0.5
0
1.7
-0.01
Relative diference
Effective refractive index
1.6
1.5
1.4
1.3
-0.03
-0.04
Maxwell-Garnet dry alumina
Maxwell-Garnet wet alumina
Bruggeman dry alumina
Burggeman wet alumina
1.2
1.1
1
-0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
Porosity
-0.05
0.7
0.8
0.9
0
0.2
0.4
0.6
Porosity
1
𝑛𝑤 − 𝑛𝑑𝑟𝑦 = 𝑓 𝑃 ~(𝑛𝑙𝑖𝑞𝑢𝑖𝑑 −1)𝑃
𝛥𝑥 =
𝜆
4(𝑛3(𝑥) − 𝑛2(𝑥) )
0.8
1