1. Numerical Simulation of Colloidal Interaction Dr P. E. Dyshlovenko Ulyanovsk State Technical University, Russia E-mail: pavel@ulstu.ru WWW: http://people.ulstu.ru/~pavel/

2. 2 Numerical Simulation of Colloidal Interaction Introduction Numerical Method Results Conclusion

3. 3 The Poisson-Boltzmann equation

4. 4 Particle-particle geometry Suitable for free particles and particles confined in a cylindrical pore

5. 5 Particle-wall geometry

6. 6 The domain for the particle-particle or particle-wall problem. Suitable for both particle-particle and particle-wall problems.

7. 7 Dimensionless Poisson-Boltzmann equation ( 1:1 electrolyte)

8. 8 Units (1:1 electrolyte)

9. 9 Adaptive mesh enrichment process Beginning End Mesh Generator Numerical Solution Program No Yes

10. 10 Numerical method Galerkin finite-element method Irregular 2D mesh Triangular elements Quadratic approximation (six nodes on an element) Quasi-Newton method for the system of non- linear algebraic equations The sparse matrix technique

11. 11 Mesh generator The mesh is a Delaunay Triangulation Irregular mesh Triangular elements Any number of straight-line or round boundaries. Freely available at http://people.ulstu.ru/~pavel/

12. 12 Error evaluation (1)

13. 13 Error evaluation (2)

14. 14 Error evaluation (3)

15. 15 Meshes Germ mesh, 11 cells Initial mesh, 147 cells Final mesh, 15588 cells (after 8 steps)

16. 16 Steps of the adaptive process

17. 17 Long-range electrostatic attraction between confined like-charged particles Observed experimentally: G. M. Kepler and S. Fraden, Phys. Rev. Lett. 73, 356 (1994). J. C. Crocker and D. G. Grier, Phys. Rev. Lett. 77, 1897 (1996). M. D. Carbajal-Tinoco, F. Castro-Román and J. L. Arauz-Lara, Phys. Rev. E 53, 3745 (1996). A. E. Larsen and D. G. Grier, Nature 385, 230 (1997). Observed numerically (BP theory): W. R. Bowen and A. O. Sharif, Nature 393, 663 (1998). Rigorous theoretical analysis proves pure repulsive interaction (BP theory): J. C. Neu, Phys. Rev. Lett. 82, 1072 (1999). J. E. Sader and D.Y.C. Chan, J. Colloid Interface Sci. 213, 268 (1999).

18. 18 Two identical colloidal particles confined in a like-charged cylindrical pore

19. 19 Two identical colloidal particles confined in a like-charged cylindrical pore Positive values of the force mean repulsion. Dotted line schematically represents the non- existent, in the framework of the PB theory, long- range attraction. Method of the present report demonstrates the repulsive interaction at any separation distances.

20. 20 A particle near a charged plane

21. 21 A particle near a charged plane

22. 22 A particle near a charged plane

23. 23 Constant total charge model of a colloidal particle, ctc-model The total charge of the particle is kept constant. The charge can move freely over the surface of the particle. Potential is uniform over the surface of the particle. The difference between the ctc- and cp- models is that the total charge rather than the potential is kept constant.

24. 24 A particle near a charged plane

25. 25 Prospects Different boundary conditions. –Variety of the electrical models of the particles. –The interior structure of the particles. –Different surrounds. Many-particles systems. –Colloidal crystals. –Many-particles effects. 3D geometry.