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

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

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.

4. Construction of a Random Solid r1r1 r2r2 r3r3 r4r4 r5r5 r6r6 r7r7 r8r8 rr r  is a random number between [0,1] Interest is to study Invasion of such a solid by a fluid, say water

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.

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.

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)

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.

9. The Model of Semi-infinite Solid rr 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.

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

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

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.

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.

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

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.

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.

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.

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).

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.

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

21. Morphology of the solid RoughAnti percolation Equilibrium? On a 64 by 64 square lattice SolidExternal PerimeterSolution

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.

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

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

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

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).

27. Invasion of bi-dispersed solid A B (a) (b) (c) (d) (f)

28. Morphology of bi-dispersed solid

29. The dynamics

30. Pseudo equilibrium ? EPL 41, 297 (1998), C.R. Acad. Sci. Paris 326,129 (1998), Physica A 266, 160 (1999)

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.

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