José S. Andrade Jr.

Universidade Federal do CearáDepartamento de Física

Flow and heat transport in irregular channels

Collaborators:

Ascânio Dias Araújo (UFC)

Raimundo N. Costa Filho (UFC)

Murilo P. Almeida (UFC)

Marcel Filoche (Ecole Polytechnique, France)

Bernard Sapoval (Ecole Polytechnique, France)

Screening EffectsScreening Effects

Sapoval, Phys. Rev. Lett. (1994)Filoche and Sapoval, Phys. Rev. Lett. (2000)Andrade, Filoche and Sapoval, Chem. Eng. Sci. (2001)

temperature heat flux

Laplacian system

Makarov TheoremMakarov Theorem

Makarov theorem (1985): “The information dimension of the harmonic measure is equal to 1 in d=2.”

Meaning: The set where the activity takes place has a dimension equal to 1. The length of the active zone is proportional to the system size.

2) Laplace equation

1) Dirichlet BC

Diffusion Cell:

.1 withLLa

Makarov TheoremMakarov Theoremsubstrate

“alive” interface

active interface

SLL pa /

is the “screening efficiency”1S

L

LLS p /

pL

Makarov TheoremMakarov Theorem

1

1

2

PL

iiaL ),1( Pat LL

Active Length Square Koch Curve

02 C

0C

SCC

./ jii qq

pa LL equal partition of fluxes

1aL

with

strongly “localized”

),/1( iLpi

The value La=22.9 is compatible with the prediction of the Makarov theorem, La ≈ L=27.

Screening in flow through fractal Screening in flow through fractal channels channels Laplace & Stokes

Screening in flow through fractal Screening in flow through fractal channels channels

Laplace & Stokes

LS

highly heterogeneous!!

Evertsz & Mandelbrot,J. Phys. A (1992)

Andrade, Araújo, Filoche & Sapovalaccepted PRL (2007)

Screening - Inertial Effects Screening - Inertial Effects

High Reynolds

Low Reynolds

0 u

uuρpuμ

2continuity

Navier-Stokes

Permeability & Active LengthPermeability & Active Length

Permeability

w

PKV

12/20

hK smooth channel

Vh

Re Reynolds number

Darcy`s Law

Screening in flow through fractal Screening in flow through fractal channels channels

Random Fractal Wall active length & position in the channel

u

Convective Heat Transport Convective Heat Transport

Heat transport between (self-similar) rough

walls: Constant properties μ, ρ, and α. Steady state, ∂T/∂t=0. Diffusion-convection equation,TTu 2.

0T),( yxu

),( yxT

wT

y

x

Convective Heat Transport Convective Heat Transport

Temperature fields

Temperature increases from

blue to red.

25.0Pe

200Pe

510Pe

710Pe10Pe

D

VPe

Péclet number

Convective Heat Transport Convective Heat Transport

collapse

roughness effect

200Pe

0PeHeat Flux & Péclet

smooth

g3g2

g1

Andrade et al., Physica A (2004)

Convective Heat Transport Convective Heat Transport

rough and smooth showthe same behavior

roughness effect

Activity length & Péclet

smooth

g2

g1

g3

Recent papers on the subject:

[1] B. Sapoval, J.S. Andrade Jr. and M. Filoche, Chem. Engng. Sci. 56, 5011 (2001). (catalysis)

[2] J. S. Andrade Jr., M. Filoche and B. Sapoval, Europhys. Lett. 55, 573 (2001). (catalysis)

[3] M. Filoche, J. S. Andrade Jr. and B. Sapoval, Physica A 342, 395 (2004). (catalysis)

[4] J. S. Andrade Jr., H. F. da Silva, M. B. da Silva and B. Sapoval, Phys. Rev. E 68, 049802 (2004). (catalysis, Knudsen diffusion)

[5] B. Sapoval, M. H. A. S. Costa, J. S. Andrade Jr. E M. Filoche, Fractals 12, 381 (2004). (catalysis)

[6] J. S. Andrade Jr., E. A. A. Henrique, M. P. Almeida e M. H. A. S. Costa, Physica A 339, 296 (2004). (heat transport)

[7] M. Filoche, J. S. Andrade Jr. and B. Sapoval, AIChE Journal 51, 998 (2005). (catalysis)

[8] B. Sapoval et al., Physica A 357, 1 (2005). (review)

[9] J. S. Andrade Jr., A. D. Araújo, M. Filoche e B. Sapoval, accepted PRL (2006). (screening)

[10] M. Filoche, D. Grebenkov, J. S. Andrade Jr. E B. Sapoval, submitted (2006). (catalysis)

NanopercolationAndrade, Azevedo, Costa Filho and Correa

Filho, Nano Letters (2005)

Motivation

Nanotechnology

Material design

Functional polymers:Drug delivery Improved catalyst supportsSupramolecular structures

Dendrimers

Branching molecules

Functional polymers

Fractal dimension ≈ 2.5

Dendrimers are real molecules!!

SFM images of (1) individual molecules, and (2) thin films.

[Frauenrath H., Prog. Polym. Sci. (2005)]

Dendrimers

Percolating Molecules: Generation

Square or honeycomb lattices of size L

Spanning cluster at p=pc

Sites are carbon atoms connected by single bonds

The valence is adjusted to 4 with hydrogen atoms

Fractional stoichiometry → CxHy

Molecular Mechanics

Vbnd

Force fields from Classical Mechanics → potential energy V

Vang

MM+ force field → optimized geometry

Comparison with semi-empirical PM3 → 3% difference

Simulations: square lattice

C

H

Simulations

carbon nanosheet

square lattice

honeycomb lattice

Simulations Size, Sampling and

Properties

Critical square and honeycomb lattices

Square lattice

Honeycomb lattices

Radius of Gyration (Rg)

L 12 15 18 21 24 27 30

Nrea 300 300 150 150 80 80 50

L 10 14 18 22 26 30

Nrea 300 150 150 150 80 80

a

a

N

ii

a

N

ii

g rrwithN

rrR

10

20

1

||

ResultsFractal Dimension

fdg MR /~ 1

)(50.2 squared f )(63.2 honeycombd f

WdW LV ~

)(84.1 squaredW )(87.1 honeycombdW

Nanopercolation: Is it possible?

SEM of carbon nanosheets grown on Si substrate [Wang et al., Carbon (2004)]

Carbon Nanosheets Self-Organized Percolation (SOP)

Typical cluster grown under an SOP rule [Andrade et al., Physica A (1997), Alencar et al., PRE (1997)]

( 1) ( ) [ ( )]Tp t p t k N N t