Laplacian transport
towards irregular surfaces:
the physics and applications
Bernard Sapoval,
Condensed Matter Laboratory, Ecole Polytechnique
Palaiseau, France.
Physicists:
J. N. Chazalviel, P. Meakin, M. Rosso,
M. Filoche, M. Felici, D. G. Grebenkov
Electrochemists:
E. Chassaing, M. Keddam, H. Katenouti
Mathematicians:
J. Peyrière, P. Jones, N. Makarov
There exist an important class of physico-chemical
phenomena which are governed by Laplace equation

Electrostatics V = 0

Steady state heat transportT = 0



Steady State electric Potential in
an electrochemical cell
V = 0
Steady state concentration of reactants in
heterogeneous chemistryCreactant = 0
Absorption of oxygen in respiration Poxygen = 0
Electrochemical cell, fractal electrodes…
V 0
Working
electrode
Counter
electrode
V 1
jn  V (x ) /r

r is the surface
resistivity
1
j (x )   V (x )

V  0


 is the electrolyte
resistivity
Ohm’s
law
Electrochemical cell
Current conservation at the surface:
V (x ) 1
  nV (x )
r


Fourier
or Robin
boundary condition:

V / nV  r /   
Heterogeneous Catalysis: AA* on the surface
Catalyst
surface
 n (x )  kCA (x )
Reactant
source
C 1
A

k is the surface
reactivity
A (x )  DCA (x )
CA  0
Current conservation at the surface

Fourier or Robin boundary condition:
D is the reactant
diffusivity
Fick’s
law
C / nC  D/k  
A catalyst grain:
Role of  in 2d
Lperimeter = Lp
D
L = diameter
Conductance to cross a part of the boundary
with perimeter Lp:
Ycross ~ k Lp
Conductance to reach a part of the boundary:
Yreach ~ D
Yreach  Ycross  Lp  
Different regimes
Yreach  D , Ycross  k L p
Lp    Yreach  Ycross
Lp
the surface works uniformly
Lp    Yreach  Ycross
less accessible regions are not reached: there exist
diffusion limitations or screening effects.
→ A region with a perimeter :
1-works uniformly!
2- presents an admittance D to be reached
Experimental decoration of
the active zone on an irregular electrode
Huttel, Rosso, Chassaing, Sapoval, J. of the Electrochemical Society (1997)
Anode, positive potential
2d
electrochemical
cell
Copper sulfate solution
Cathode, negative potential
Where does
the current
reach?
Result:
Cu++ ions are deposited where the current reach.
Exact mapping of the diffusion problem
anode
CuSO4 solution in water
Cu++depletion region
cathode
An exact result (Grebenkov):
The probability to be absorbed within a distance r
from the impact on a flat line (2d):
2r  e t dt
P (r) 1
0 2
2
 t  r /  
45% of the particles
are absorbed
within -/2 and + -/2
The probability to be absorbed in a circle of radius r
from the impact on a flat surface (3d):
P (r) 1  0

te t dt
t  r /  
2
2 1/ 2
46% of the particles
are absorbed
in a disk of radius /2
This Robin problem ( finite) is not solved in
general for an irregular surface
But the Dirichlet problem ( =0) has
been extensively studied in 2D.
Important result: Makarov’s theorem
Makarov’s theorem (1985)
The Makarov’s theorem: Dirichlet boundary condition : C=0
In 2D, the information dimension of the harmonic
measure of any simply connected set is equal to 1.
Translation:
The activity (flux or current) takes place on a
subset of the boundary whose size scale as a
power 1 of the system size.
Raw translation:
The activity takes place on a subset whose size if of
order the size of the system.
Harmonic measure
If a scalar field V satisfy the Laplace equation or “harmonic ”
equation :
V = 0
with Dirichlet boundary condition V = 0 on a surface 
then | dV/dn | is the harmonic measure of the surface.
1- In electrostatics the harmonic measure is the local
surface charge on the metallic electrode of a capacitor.
2- In electrochemistry the harmonic measure is the local
current proportional to the local electric field.
3- In heterogeneous catalysis on a surface with infinite
reaction rate, the harmonic measure is the probability of
arrival of particles.
4- In diffusion towards a perfectly absorbing surface, the
harmonic measure is the probability of arrival of particles.
This measure is distributed along the surface,
and the question is how to characterize the distribution?
Classically, a (normalized) distribution can be described
by its entropy :
S = - i p i . Lnp i
or S = -  p. Lnp ds
Examples :
1. Uniform distribution on a length L :
p
p = 1/L
1/L
S=-
(1/L)Ln (1/L) dx = Ln L
2. Mass 1 on one point : p = 1
L
x
S = -1Log1 = 0
3. But 2 masses 1/2 on 2 points have an entropy :
S = - [(1/2) Ln(1/2)+(1/2) Ln(1/2)] = Log 2
which is the same as the entropy of a uniform distribution on a
length L = 2 : S = Ln 2
INFORMATION DIMENSION :
Number which gives a geometrical information on the
distribution (of probability, of charge, of current, ...).
Take a square box of side count the quantity p  of
probability, of charge, of current, …, in the box...).
DI  lim 0
 boxes  p Lnp
Ln 
Examples :
1- uniform distribution between 0 and L : p = 1/L
p
1/L
L
DI  lim 0
x
L

Ln
 L
L 1
Ln 
and 1 is the dimension of the segment which supports the
distribution

2- distribution on two points with p1 and p2 with p1 + p2 =1
DI  lim O
p1 Lnp1  p2 Lnp2
O
Ln 
and 0 is the geometrical dimension of any finite number of points.
3- uniform
 distribution on a square of side L : p = 1/L2
L     
    Ln  
  L  L 
2
Ln 
2
DI  lim 0
 boxes 
2
2
and 2 is the dimension of a surface.
4- the probability
is spread between p1 on a line and p1 on a

surface such that p1 + p2 =1  DI = 2 - p1
Makarov :
The information dimension describing the dimension of the set
which supports the harmonic measure (current, arrival
probability…) is equal to 1
1. This region is called in this context, the active zone
If the curve is a prefractal with smaller cutoff l
L
Lactive zone ≈ l(L/l)DI = l(L/l)1 = L
Screening efficiency h
fraction of the surface which is active
h = L / Lp True whatever the geometry
If the prefractal has a dimension Df its perimeter length is
Lperimeter ≈ l.(L/l)Df
The land surveyor method:
coarse-graining in real space.
System size L
 is the perimeter of a grain
Lg is the size of a grain that works
uniformly and has a conductance D
A grain works uniformly and presents a
conductance
Elementary unit of access impedance D
An (approximate) analytical expression for the conductance
The perimeter of the coarse-grained line Lper,cg= g Lg = Ng < Lg >
The admittance of the perimeter of the coarse-grained line is
that of these Ng grains is Ng.D.
But this has to be multiplied by the screeening efficiency (Makarov)
Y ≈ Ng.D. (L/ Lper,cg) = Ng.D.L/(Ng<Lg>) = D.L/<Lg>
Sapoval, Physical Review Letters, 1994
An (approximate) analytical expression for the conductance
Lg is the diameter of a part
of the surface with a perimeter equal to 
For a prefractal electrode
L
L
Y   D 
D
1
Lg  
 D
 
 
f
In electrochemistry, the surface may
behave locally as a capacitor.
The surface resistance r is really
a surface capacitance 1/jgwand 1/jgw.
1
Df
Historically:
constant phase element

  1 
Z (w )  

L j gw 
For a prefractal electrode
Y   D 
L
L
D
1
Lg  
 D
 
 
f
 but
1
Df
 
Lg    
 
is valid for both a random and a deterministic fractal.
They should have approximately the same response
 have same dimension, size and smaller cut-off…
if they
Fractality: a paradigm of irregularity
Filoche, Sapoval, Physical Review Letters, 2000.
Fractality: a paradigm of irregularity
Filoche, Sapoval, Physical Review Letters, 2000.
==55
ll
Inhomogeneous
electrode:
the surface
resistivity is not
constant
 = 25l
A more precise description …
One could define the impedance
from the heat dissipated in the surface
Z heat I   rj ds
2
2
electrode
j 
2
  r  ds   rj N ds
electrode I 
electrode
2
Z heat

If Z

j 
r
2
  r  ds   rj N ds is written as
then
electrode I 
electrode
Lact
2
heat


2
Lact    jN ds
electrode 
1
Different from L
Lattice representation of diffusion and absorption.

call  the absorption probability and  = 1-  the
probability to bounce back
Th. Halsey
Pj   pij
sites i
Distribution of
arrival probabilities
vector
The Brownian Self-transport Operator

Probability of absorption at site j, vector
Pj  Pj (1     Pk q kj (1    Pk  2 q kl q lj (1    ...
sites k
sites k,l
~
 Self-transport operator : Q  ( qij )1i  N
1 j  N
˜ P(1   2 Q
˜ 2 P...
P   (1  P  (1   Q
n
1
˜
˜
˜
  (1   Q P  (1  I   Q) P
n
~
Q : Symmetric real operator
n
System Impedance

measured impedance
Z spectroscopic  r PN .PN 

0
where the vectors are normalised



Using the eigenmodes of the brownian
self-transport operator
 0 1 

 1    
0
Z spect. (   rPN .
PN  r  
C

˜
 ˜I   Q
 eigenvectors 1  q 
The C are the projection of the harmonic measure
on the eigenvectors  of Q
M. Filoche, B. Sapoval., Eur. Phys. J. B. 9, 755-763 (1999)
Application to the respiration
of mammals
Bronchial system of mammals:
dichotomic tree
Rat
Man
At the end of each of about 30.000
bronchioles: one acinus
Planar cut
1/4 mm
pulmonary
alveoli
(300 millions)
oxygen
erythrocytes
Acinus : diffusion cell where
Oxygen diffuses towards and
across the memebrane.
Acinus du physicien
The mathematical model
• At the acinus entry:
Diffusion source
• In the alveolar air:
Diffusion obeys Fick's law
CO2  C0
JO2  DO2 CO2
 2CO2 0
• At the air/blood interface:
Membrane of permeability WM

blood
J
W
(
C

C
)
n
M
O
O
2
2
42
The role of the length  in 3d
Conductance to reach a region of the surface:
Yreach ~ D.LA (LA: diameter of that region)
Conductance to cross that region:
Ycross ~ W.A (A: area of the region)
LA
if Yreach> Ycross
 the surface works uniformly.
if Yreach< Ycross
=> less accessible regions are not reached and there
exists diffusion screening: the surface is partially
passive.
The crossover is
obtained for:
Yreach = Ycross => A/LA  
What is the geometrical significance of the length A/LA=Lp ?
Lp is the perimeter of an “average planar cut” of the surface.
Examples:
Sphere: A=4R2; LA=2R; A/LA=2R.
Cube: A=6a2; LA ≈ a; A/LA≈ 6a.
Self-similar fractal with dimension d: A =
l2(L/ l)d, LA=L;
A/LA= l(L/l) d-1 (Falconer)
The crossover is
obtained for:
Yreach = Ycross => A/LA  
What is the geometrical significance of the length A/LA=Lp ?
Lp is the perimeter of an “average planar cut” of the surface.
Experiment…
Perimeter length of a planar cut of the
acinus: total red length.
• Lp< => Yreach>Ycross
=> the surface works uniformly
•
LP> => Yreach<Ycross
=> less accessible regions are not reached
 DIFFUSION SCREENING
For the human subacinus:
A = 8.63 cm2
L = 0.29cm
LP  30 cm
D = 0.2 cm2 s-1
W = 0.79 10-2 cm s-1
 = 28 cm
  LP
!
47
And this is true of other mammals :
Mouse
Rat
Rabbit
Human
Acinus volume
(10-3 cm3)
0.41
1.70
3.40
23.4
Acinus surface
(cm2)
0.42
1.21
1.65
8.63
Acinus
diameter (cm)
0.074
0.119
0.40
0.286
5.6
10.2
11.0
30
Membrane
thickness (.
m)
0.60
0.75
1.0
1.1
 (cm)
15.2
18.9
25.3
27.8
Acinus
perimeter,
Lp(cm)
B. Sapoval, Proceedings of “Fractals in Biology and Medecine”, Ascona, 1993
B.Sapoval, M. Filoche, E.R. Weibel, Proc.Nat.Acad.Sc. 99, 10411 (2002)
Result:
Cu++ ions are deposited where the current reach.
Exact mapping of the diffusion problem
anode
CuSO4 solution in water
Cu++depletion region
cathode
Morphometric data on 8 real sub-acini
B. Haefeli-Bleuer, E.R. Weibel, Anat. Rec. 220: 401 (1988)
Does the irregularity or apparent randomness
of the acinar tree really plays a role?
Comparison between the flux in an average symmetrized acinus
and the real acinus of Haefeli-Bleuer and Weibel:
Exact analytical calculation of a finite tree:
No difference
between the total flux in the 8
Haefeli-Bleuer, Weibel
sub-acini and 8 times
a symmetric 1/8 sub-acinus:
The symmetric dichotomic
model of Weibel is sufficient
D. Grebenkov, M. Filoche, and B. Sapoval, Phys. Rev. Lett. 94, 050602-1 (2005)
Passivation
Process in which
in parallel with the useful reaction
A+ X A* +X
there exists a reaction which destroys the catalytic sites
A+ X  0
The passivation process
C A
0
n
C A C A

n 
CA  0
 CA  0
2
CA  C
ent
+ iterations
The passivation process
C A
0
n
C A C A

n 
CA  0
 CA  0
2
What happens next?
CA  C
ent
+ iterations
Passivation sequence
At each iteration, one selects the most active part of the
boundary that carries a given portion p of the total flux