Numerical Simulation of Colloidal Interaction

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

3. The Poisson-Boltzmann equation

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

5. Particle-wall geometry

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

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

8. Units(1:1 electrolyte)

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

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. 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. Error evaluation (1)

13. Error evaluation (2)

14. Error evaluation (3)

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

16. Steps of the adaptive process

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. E53, 3745 (1996). • A. E. Larsen and D. G. Grier, Nature385, 230 (1997). • Observed numerically (BP theory): • W. R. Bowen and A. O. Sharif, Nature393, 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. Two identical colloidal particles confined in a like-charged cylindrical pore

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. A particle near a charged plane

21. A particle near a charged plane

22. A particle near a charged plane

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. A particle near a charged plane

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.