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
Transcript
Slide 1
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
Slide 2
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
Slide 3
Introduction to a random solid Large: Si, Medium: O, Small: B
Glass is a multi-component vitreous system. Simulated Glass
Structure Consider borosilicate glass: mainly composed of Boron,
oxygen and Silicon 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.
Slide 4
Construction of a Random Solid r1r1 r2r2 r3r3 r4r4 r5r5 r6r6
r7r7 r8r8 rr r is a random number between [0,1] Interest is to
study Invasion of such a solid by a fluid, say water
Slide 5
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.
Slide 6
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.
Slide 7
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 t=5 IP is studied with trapping and
without trapping. IP without trapping belongs to the universality
class of percolation whereas IP with trapping does not. Invasion
percolation on 100x200 square lattice as given in Fractals by J,
Feder. Our interest is to develop and study models for invasion of
sticky random solid (SRS)
Slide 8
Modeling invasion of sticky random solid 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-I Model-II A
finite random solid of square shape. All four sides are in contact
with water. The volume of water is infinitely large. Widely
different features are observed in the two models. Model-III A
bi-dispersed system with finite volume of water.
Slide 9
The Model of Semi-infinite Solid rr A block of material with
binding energy r R : Random solid, S : Solution R+S RS R+S This
constitutes one MC step of invasion of a sticky random solid by a
solution. One MC step is one time unit. Diffusion is assumed to be
very fast in comparison to dissolution. No dissolution before
re-deposition.
Slide 10
System Morphology WaterSolidInterface L=64 Random solid
Solution t=2 9 Random solid Solution t=2 11 Solution Random solid
t=2 12 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 Growth of
re-deposited solid at the bottom
Slide 11
Solution Profile N w :Number of water Molecules per row (y) The
water profile moves like a Gaussian packet into the solid.
L=64
Slide 12
Characterization of Solution Profile Water invades the solid at
a constant speed Profile position Profile width Dissolution and
re-deposition determine the width . Data collapse: t=t/L For large
L, it is a slow moving solution profile with a constant drift
velocity.
Slide 13
Dissolution threshold Distribution of interface energy:
Dissolution threshold is exactly at the percolation threshold. The
system on its own reaches to the dissolution threshold at r c =p c
in the steady state. Self-organized criticality? Both percolation
and IP are demonstrated as self-organizing systems.
Slide 14
Redeposited solid 256 512 Fractal dimension: d f =1.88 0.01,
Close to that of percolation &IP. Chemical dimension: d l =1.69
0.02 For percolation backbone :0.87
Slide 15
Self-established Potential Gradient and Phase separation Plot
of average random number per row. 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.
Slide 16
Self-clustering of solution molecules Self clustering of
solution molecules through a diffusive dynamics. Cluster growth:
Evolution of interface length: Diffusive growth 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. Very similar to clustering of passive sliders in
stochastically evolving surfaces.
Slide 17
Criticality Dynamical cluster size distribution: Power law
distribution with =2.01 0.06. Self organized criticality
Percolation: Power law distribution of cluster size at an
equilibrium. Invasion of SRS: Power law distribution of cluster
size at a spontaneously evolved non- equilibrium steady state.
Slide 18
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).
Slide 19
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.
Slide 20
Model for finite random solid Step 1: (a) Find the perimeter
(b) Search for the lowest Step 3: (a) Redeposit on the random
surface site (b) Find the new perimeter Step 2: (a) Dissolve the
lowest (b) Modify the perimeter Constitutes one MC step One time
unit
Slide 21
Morphology of the solid RoughAnti percolation Equilibrium? On a
64 by 64 square lattice SolidExternal PerimeterSolution
Slide 22
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 Evolution of surface energy
Pseudo equilibrium before transition This interface evolution is
similar to Bak-Snappen model of biological evolution.
Slide 23
Anti-percolation Average cluster size APT: maximum time rate of
change of cluster size. Cluster size saturates In long time limit
Total number of clusters APT: maximum time rate of change of
cluster number. APT is very similar to fragmentation of brittle
solid
Slide 24
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. t e = Maximum
change in P e Logarithmic difference in t e and t d The system is
like a single fluidized particle phase. Fragmentation and
coagulation occurs at a constant rate. The difference vanishes at
L=2 10
Slide 25
Criticality? SOC: A slowly driven system evolves into a
non-equilibrium steady state characterized by long range
spatio-temporal correlations. This is demonstrated by power law
behavior. The steady is then a critical state. This is an evidence
of SOC at a dynamical equilibrium state. Distribution of fragments
brittle solid =1.5
Slide 26
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).
Slide 27
Invasion of bi-dispersed solid A B (a) (b) (c) (d) (f)
Slide 28
Morphology of bi-dispersed solid
Slide 29
The dynamics
Slide 30
Pseudo equilibrium ? EPL 41, 297 (1998), C.R. Acad. Sci. Paris
326,129 (1998), Physica A 266, 160 (1999)
Slide 31
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. 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. In the long term
evolution, the solid dissolves and attains a self-organized
critical state.