Invasion of a sticky random solid:
Self-established potential gradient,
phase separation and criticality at
dynamical equilibrium
S. B. SANTRA
Department of Physics
Indian Institute of Technology Guwahati
Vimal Kishore, Santanu Sinha and Jahir Abbas Ahmed
Bernard Sapoval, Ph. Barboux, F.Devreux
Introduction
Fluid invasion and its interface motion in disordered
systems have taken a lot of interest in statistical
physics.
Many related problems are looked into in the recent past
such as : driven interface in disordered media, crack
propagation in solid, domain wall propagation in
magnets, motion of interface in multiphase flow, etc.
Consider a new problem: invasion of sticky random solid
Introduction to a random solid
Consider borosilicate glass: mainly composed of Boron, oxygen and Silicon
Simulated Glass Structure
Glass is a multi-component vitreous
system.
The environment around Si is very
different at different places and can
be considered as random.
The strength or binding energy of Si to the
rest of the solid can be considered as
randomly distributed.
Large: Si, Medium: O, Small: B
Construction of a Random Solid
r4
r8
r3
r7
r2
r6
r1
r5
ra is a random
number between
[0,1]
ra
Interest is to study
Invasion of such a
solid by a fluid, say
water
Glass Water Interaction
• Glass is a multi-component vitreous system
interacts in a complex manner with water.
• Silica dissolves in water and forms Silicic
acid.
• Silicic acid breaks spontaneously into Silica
and Hydroxyl ions.
• Silica re-deposits back on the surface of the
undissolved solid.
Interaction of a random solid
and a solution
• Say, the random solid is represented by R and
the solution is represented by S.
• The reaction of dissolution and re-deposition
then given by : R+SRSR+S
• One needs to study the invasion of a random
solid by a fluid following the above chemical
reaction.
Invasion of a porous medium
Invasion percolation (IP) is a dynamical percolation process to study the flow of two
immiscible fluids in porous media.
t=0
Invasion percolation on 100x200 square lattice
as given in Fractals by J, Feder.
IP is studied with trapping and without trapping. IP
without trapping belongs to the universality class of
percolation whereas IP with trapping does not.
t=5
Our interest is to develop and study models for invasion of sticky random solid (SRS)
Modeling invasion of sticky random solid
Model-I
• A semi-infinite random solid elongated along y-axis.
• The bottom surface is in contact with water.
• The volume of water is infinitely large.
Model-II
• A finite random solid of square shape.
• All four sides are in contact with water.
• The volume of water is infinitely large.
Model-III
• A bi-dispersed system with finite volume of water.
Widely different features are observed in the two models.
The Model of Semi-infinite Solid
ra
A block of material with
binding energy ra
R : Random solid, S : Solution
R+SRSR+S
Diffusion is assumed to be very fast in comparison to dissolution.
• This constitutes one MC step of invasion of
a sticky random solid by a solution.
• One MC step is one time unit.
No dissolution before re-deposition.
System Morphology
L=64
Water
Solid
Interface
Random solid
t=29
Random solid
Solution
t=212
Random solid
Re-deposited solid
- Invasion percolation cluster
Solution inside the solid
- finite percolation clusters
Existence of both the IP
and percolation clusters in
the same model
t=211
Growth of re-deposited
solid at the bottom
Solution
Solution
Solution Profile
Nw :Number of water Molecules per row (y)
L=64
The water profile moves like a Gaussian packet into the solid.
Characterization of Solution Profile
Profile position
1
yp 
N
N
 y (s
j
Profile width
1
/
2
max )
j 1
y
max
1i

2




y

y
i
P
 
y
i

0
max

L  64
L  128
L  256
L  64
L  128
L  256
yp ~ t
Water invades the solid
at a constant speed
Data collapse: t’=t/L
 ~ t 2/5
Dissolution and re-deposition
determine the width .
For large L, it is a slow moving solution profile with a constant drift velocity.
Dissolution threshold
Distribution of interface energy:
PI (r, t )  nr (t ) / I (t )
PI (r , 0)
PI (r , t ) 
 (r  pc )
1  pc
Dissolution threshold is exactly at the
percolation threshold.
The system on its own reaches to
the dissolution threshold at rc=pc in
the steady state.
Self-organized criticality?
Both percolation and IP are demonstrated as self-organizing systems.
Redeposited solid
Fractal dimension:
N ~
df
df=1.880.01,
Close to that of percolation &IP.
512
Chemical dimension:
M () ~  d 
dl=1.690.02
 ~
256
d d f
d d f  0.89  0.01
For percolation backbone :0.87
Self-established Potential Gradient and
Phase separation
1
r
hard

 rP' (r )dr
0
• There is a self-established
potential gradient.
• The solid system is phase
separated into hard and soft
solid.
• The solution profile is just in
front of the potential gradient.
1
r
soft

 rP (r )dr
0
0
Plot of average random number per row.
Self-clustering of solution molecules
The solution molecules pushed by the potential gradient form clusters and move
collectively. It is a process of self-clustering during the motion of solution
molecules within the dynamically evolved energy landscape.
Cluster growth:
S
L
Evolution of interface length:
t
L
and
NB
L
t
L
Diffusive growth
Self clustering of solution molecules through a diffusive dynamics.
Very similar to clustering of passive sliders in stochastically evolving surfaces.
Criticality
Dynamical cluster size distribution:
Percolation: Power law distribution
of cluster size at an equilibrium.
P( s) ~ s 
Power law distribution with
=2.010.06.
Invasion of SRS: Power law
distribution of cluster size at a
spontaneously evolved nonequilibrium steady state.
Self organized criticality
Summary of model-I
A new model of invasion of a sticky random solid by
water is studied here.
 A self-established potential gradient drifted the water
molecules at a constant speed into the solid.
 Diffusive dynamics is observed for the interface and
cluster growth.
 In long term evolution, the cluster size distribution
shows power law behavior.
 The system evolved into a self-organized critical state
driven by a self-established potential gradient.
Phys. Rev. E 78, 061135 (2008).
Modeling of invasion of finite random solid
• A finite solid is in contact with the solution.
• Solution interacts with all the available
solid surface.
• The volume of the solution is taken to be
infinite.
Model for finite random solid
Step 1:
(a) Find the perimeter
(b) Search for the lowest
ra
Step 2:
(a) Dissolve the lowest r
a
(b) Modify the perimeter
Step 3:
(a) Redeposit on the
random surface site
(b) Find the new perimeter
Constitutes one MC step – One time unit
Morphology of the solid
On a 64 by 64 square lattice
Solid
t2
13
Rough
External Perimeter
t2
20
Anti percolation
Solution
t  224
Equilibrium?
Roughening transition
Number of externally accessible perimeter sites h is counted
H saturates in time
Constant chemical
potential
RT: maximum time
rate of change of H
5
t  tr /(2  L )
'
r
3
Evolution of surface
energy
Pseudo equilibrium
before transition
This interface evolution is similar to Bak-Snappen model of biological evolution.
Anti-percolation
Cluster size saturates
In long time limit
Average cluster size
td'  td /( L2  2L / 8 )
APT: maximum time rate
of change of cluster size.
APT is very similar to
fragmentation of brittle
solid
Total number of clusters
APT: maximum time rate
of change of cluster number.
Dynamical equilibrium
Evolution of average energy
Average energy becomes
constant after APT
Critical slowing down
Prob. to have a sample
with all sites dissolved
at least once.
te= Maximum change in Pe
Logarithmic difference
in te and td
The difference
vanishes at L=210
The system is like a single fluidized particle phase.
Fragmentation and coagulation occurs at a constant rate.
Criticality?
This is demonstrated by power law behavior.
Ps (t  te ) ~ s

  1.67
The steady is then a critical state.
Distribution of fragments brittle solid =1.5
SOC: A slowly driven system evolves into a non-equilibrium steady
state characterized by long range spatio-temporal correlations.
This is an evidence of SOC at a dynamical equilibrium state.
Summary of model-II
 Dissolution of finite solid occurs after passing through
roughening and anti-percolation transitions.
 The cluster size distribution remains invariant after
complete dissolution.
 The system evolves to a dynamical equilibrium state
through critical slowing down.
 The dynamical equilibrium is characterized by constant
chemical potential, average cluster size and cluster size
distribution.
 A self-organized critical state at a dynamical equilibrium
is a new phenomenon.
Euro.Phys.Lett.71, 632 (2005).
Invasion of bi-dispersed solid
(a)
(d)
(b)
(f)
A
B
pB  p0e EB / kBT , EB  nA J BA  nB J BB
(c)
Morphology of bi-dispersed solid
The dynamics
Pseudo equilibrium ?
EPL 41, 297 (1998), C.R. Acad. Sci. Paris 326,129 (1998), Physica A 266, 160 (1999)
Conclusion
 Invasion
of a sticky as well as bi-dispersed random
solid by an aqueous solution has been studied.
 There
are features of non-equilibrium as well as
equilibrium critical phenomena.
 In
the long term evolution, the solid dissolves and
attains a self-organized critical state.
 The steady state corresponds to an equilibrium state in
the case of finite solid whereas it is a non-equilibrium (or
pseudo equilibrium state in the case of semi infinite
solid.
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