S. B. SANTRA Department of Physics Indian Institute of Technology Guwahati

1. 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 Invasion of a sticky random solid: Self-established potential gradient, phase separation and criticality at dynamical equilibrium

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 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

4. Construction of a Random Solid ra is a random number between [0,1] r4 r8 r3 r7 r2 r6 ra Interest is to study Invasion of such a solid by a fluid, say water r1 r5

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. t=5 t=0 Invasion of a porous medium Invasion percolation (IP) is a dynamical percolation process to study the flow of two immiscible fluids in porous media. 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. Our interest is to develop and study models for invasion of sticky random solid (SRS)

8. 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.

9. ra A block of material with binding energy ra The Model of Semi-infinite Solid 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.

10. Water Solid Interface Random solid t=212 Random solid t=29 Random solid t=211 Solution Solution Solution System Morphology L=64 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 Nw :Number of water Molecules per row (y) L=64 The water profile moves like a Gaussian packet into the solid.

12. Characterization of Solution Profile Profile position Profile width Data collapse: t’=t/L Water invades the solid at a constant speed Dissolution and re-deposition determine the width . 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 rc=pc in the steady state. Self-organized criticality? Both percolation and IP are demonstrated as self-organizing systems.

14. 512 256 Redeposited solid Fractal dimension: df=1.880.01, Close to that of percolation &IP. Chemical dimension: dl=1.690.02 For percolation backbone :0.87

15. Self-established Potential Gradient and Phase separation • 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. Plot of average random number per row.

16. 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: Evolution of interface length: Diffusive growth Self clustering of solution molecules through a diffusive dynamics. Very similar to clustering of passive sliders in stochastically evolving surfaces.

17. Criticality Dynamical cluster size distribution: 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. Power law distribution with =2.010.06. Self organized criticality

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. Step 1: (a) Find the perimeter (b) Search for the lowest Step 2: (a) Dissolve the lowest (b) Modify the perimeter Model for finite random solid Step 3: (a) Redeposit on the random surface site (b) Find the new perimeter Constitutes one MC step – One time unit

21. Solid External Perimeter Solution Morphology of the solid On a 64 by 64 square lattice Equilibrium? Rough Anti percolation

22. Roughening transition Number of externally accessible perimeter sites h is counted Evolution of surface energy RT: maximum time rate of change of H H saturates in time Constant chemical potential Pseudo equilibrium before transition This interface evolution is similar to Bak-Snappen model of biological evolution.

23. Anti-percolation Cluster size saturates In long time limit APT: maximum time rate of change of cluster size. Average cluster size APT is very similar to fragmentation of brittle solid APT: maximum time rate of change of cluster number. Total number of clusters

24. Dynamical equilibrium Evolution of average energy Prob. to have a sample with all sites dissolved at least once. Logarithmic difference in te and td Average energy becomes constant after APT The difference vanishes at L=210 Critical slowing down te= Maximum change inPe The system is like a single fluidized particle phase. Fragmentation and coagulation occurs at a constant rate.

25. Criticality? This is demonstrated by power law behavior. The steady is then a critical state. Distribution of fragments brittle solid t=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.

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) (d) (b) (f) B A (c)

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. • 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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