optofluidic characterization of porous materials escuela de materiales porosos nanoestructurados...

Optofluidic Characterization of Porous Materials

Escuela de Materiales porosos Nanoestructurados

Raúl UrteagaSanta Fe - Octubre 2014

Congreso Internacional de Metalurgia y Materiales14 SAM- CONAMET

XIII Simposio Materia

Optofluídica

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

Integración de la óptica y la microfluídica para obtener información de ambas en simultaneo [1].

[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

Transfer Matrix formalismFor a multilayer

we have:

The reflection and transmission coefficients are

Can be calculated as


Results for Normal IncidenceFor this case we

have:

where

The total reflectance will be:

n0 n1 n2

1

T

R

d

t

𝛿=2𝜋 𝑛1𝑑𝜆0

Here the phase difference is:

Results for Normal Incidence

Important to note:(If δ is real )

n0 n1 n2

1

T

R

d

t

𝛿=2𝜋 𝑛1𝑑𝜆0

=> Periodic!

𝑛1𝑑=𝜆02

𝑛1𝑑=𝜆04

Results for Normal IncidenceIf reflectance is periodic, then the Fourier transform

in K space is discrete, and have peaks at 2nd

2𝑛1𝑑

If there is absortion (refractive index complex)

Optical Properties of Mixtures

To find these variables Maxwell's equations must be solved for electrostatic :

How to obtain the effective dielectric constant?

Geometry must be known and generally requires an expensive calculation

Effective Medium Theories

• Maxwell-Garnet

[2] O. Stenzel, The Physics of Thin Film Optical Spectra Springer Series in Surface Sciences Volume 44 (2005)

Cavidad [2] paralelo al eje perpendicular

Esfera ⅓ ⅓

Cilindro 0 ½

Placa 1 0

Elipsoide de ejes ,

𝜀 𝑗𝜀𝑙

Effective Medium Theories• Lorentz- Lorenz [2]

[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.

• Bruggeman [2]

• Looyenga-Landau-Lifshithz [3]

𝜀 𝑗

𝜀 𝑗

𝜀𝑙=1

𝜀𝑙=𝜀

𝜀 𝑗𝜀𝑙 𝜀 𝑗


LimitsCavidad [2]

paralell PerpendicularEsfer ⅓ ⅓

Cylinder 0 ½

Plate 1 0


Refractive index of silicon and alumina [4].Alumina is transparent!

[4]

200 300 400 500 600 700 800 900 1000 1100 12000

1

2

3

4

5

6

7

[nm]

Silicon nreal

Silicon nimag

Alumina

𝑛=√𝜀


Example case I: Porous Silicon + Air

@ L=1/3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

3.5

4

Porosity

Eff

ectiv

e re

frac

tive

inde

x

Maxwell-Garnet

Bruggeman

Looyenga


Example case II: Porous Alumina + Isopropyl Alcohol

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Porosity

Eff

ectiv

e re

frac

tive

inde

x

Depolarization factor L: 0.5

Maxwell-Garnet dry alumina

Maxwell-Garnet wet aluminaBruggeman dry alumina

Burggeman wet alumina

@ L=1/2

Optical CharacterizationSpectroscopic reflectance of porous Silicon single layer

Optical CharacterizationSpectroscopic 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

𝑛𝑑𝑟𝑦𝑑

2) After infiltration we can measure the optical width of the wetted sample

𝑛𝑤𝑑

𝑛𝑤= 𝑓 (𝑃 ,𝑛𝐴𝑙 ,𝑛𝑙𝑖𝑞 )

Porous Alumina + Isopropyl Alcohol

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)

𝑛𝑑𝑟𝑦= 𝑓 (𝑃 ,𝑛𝐴𝑙 ,𝑛𝑎𝑖𝑟 )0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Porosity

Eff

ectiv

e re

frac

tive

inde

x





Fluid Mechanics at low ReI) Capillary filling of uniform closed channel

2r

Straight channel

Tortuous channel:L

L’

𝜏=𝐿𝐿 ′

In nondimensional form: were

L.N. Acquaroli, R. Urteaga, C.L.A. Berli, R.R. Koropecki, Langmuir 27, 2067 (2011)

Hagen-Poisseuille flow

Contact Angle

∆ 𝑃=𝜎 𝑐𝑜𝑠𝜃

𝑟

Laplace Pressure

Fluid Mechanics at low ReI) Uniform closed channel

Solving the differential equation:

The final position defines the value of

then

Lucas-Washburn Dynamics

𝑙2=12𝜎 𝑟𝜇𝑡

0

Tiempo

Posi

ción

Men

isco

tfill

L then

Edward W. Washburn “THE DYNAMICS OF CAPILLARY FLOW”The Physical Review Vol. XVII, 3, (1921)

Lateral view Top view

Optoflidic CharacterizationPorous Silicon

Dynamic SLIM


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.

Experimental setup What is measured (twice)

Relation between reflexion extremes and liquid infiltration

𝑒0=𝑛2(𝐿−𝑥 )+𝑛3𝑥

An extreme in reflectance will occur each time the optical path is a multiple of :

𝑛1𝑑=𝜆02

𝑛1𝑑=𝜆04

𝑒0=𝑚𝜆04

𝑒0=∫0

𝐿

𝑛𝑒𝑓𝑓 (𝑥 )𝑑𝑥

𝑑𝑒0𝑑𝑥

=𝑥 (𝑛3−𝑛2)

Between extremes in reflectance the interface moves

𝛥𝑥=𝜆

4 (𝑛3−𝑛2)

Results


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=𝐿𝐿 ′

2,6

Fluid Mechanics at low ReI) Variable section open channel

Direct problem:

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)

𝑢( 𝑙 )=𝛼𝑟 (𝑙 )

−3

∫0

𝑙

𝑟 (𝑥 )− 4d 𝑥 𝛼=¿¿were

−𝜕𝑝𝜕 𝑥

=8𝜇𝑄𝜋𝑟 (𝑥 )

4

Poisseuille flow

𝛥𝑝=2𝛾 cos𝜃𝑟 (𝑥 )

Laplace Pressure

Fluid Mechanics at low ReI) Variable section open channel

Has multiple solutions!!

Inverse problem [5]:

as input

as input

𝑟 ( 𝑙 )=[𝑟 (𝐿 )𝑢(𝐿 )

13 + 13𝛼∫𝑙

𝐿

𝑢( 𝑙 )

43 dl ]𝑢 (𝑙 )

− 13

[5] E. Elizalde, R. Urteaga, R.R. Koropecki, C.L.A. Berli, Phys. Rev. Lett. 112, 134502 (2014)

𝑟 (𝑣 )=[𝑟❑ (𝑉 )

5 𝑄(𝑉 )5 + 5

𝜋 2𝛼∫𝑣

𝑉

𝑄 (𝑣 )6 dv ]

15𝑄 (𝑣 )

−1

Experimental setup

Optoflidic CharacterizationPorous Alumina

Top side Bottom side

E. Elizalde, R. Urteaga, R.R. Koropecki, C.L.A. Berli, Phys. Rev. Lett. 112, 134502 (2014)

Results

Optoflidic CharacterizationPorous Alumina

E. Elizalde, R. Urteaga, R.R. Koropecki, C.L.A. Berli, Phys. Rev. Lett. 112, 134502 (2014)

Radii at the ends coincide with SEM photographs

Thanks

Relation between reflexion extremes and liquid infiltration

𝑒0=𝑛2(𝐿−𝑥 )+𝑛3𝑥

An extreme in reflectance will occur each time the optical path is a multiple of :

𝑛1𝑑=𝜆02

𝑛1𝑑=𝜆04

𝑒0=𝑚𝜆04

𝑒0=∫0

𝐿

𝑛𝑒𝑓𝑓 (𝑥 )𝑑𝑥

𝑑𝑒0𝑑𝑥

=𝑥 (𝑛3(x )−𝑛2(𝑥 ))

Between extremes in reflectance the interface moves

𝛥𝑥=𝜆

4 (𝑛3 (𝑥)−𝑛2 (𝑥))=

𝜆4 (𝑛𝑤 (𝑥)−𝑛𝑑𝑟𝑦 (𝑥))

𝑛𝑤−𝑛𝑑𝑟𝑦= 𝑓 (𝑃 ) (𝑛¿¿ 𝑙𝑖𝑞𝑢𝑖𝑑−1)𝑃 ¿

The linear approximation

Porous Alumina + Isopropyl Alcohol

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Porosity

Eff

ectiv

e re

frac

tive

inde

x





Relative difference from linear

0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

Porosity

Rel

ativ

e di

fere

nce

Modelo de Bruggeman. Depolarization factor: 0.5

𝛥𝑥=𝜆

4 (𝑛3 (𝑥)−𝑛2 (𝑥))

optofluidic characterization of porous materials escuela de materiales porosos nanoestructurados...

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