Numerical Simulation of Colloidal Interaction

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Numerical Simulation of Colloidal Interaction

• Dr P. E. Dyshlovenko
• Ulyanovsk State Technical University, Russia
• E-mail pavel_at_ulstu.ru
• WWW http//people.ulstu.ru/pavel/

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Numerical Simulation of Colloidal Interaction

• Introduction
• Numerical Method
• Results
• Conclusion

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The Poisson-Boltzmann equation
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Particle-particle geometry

• Suitable for free particles and particles
  confined in a cylindrical pore

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Particle-wall geometry
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The domain for the particle-particle or
particle-wall problem.

• Suitable for both particle-particle and
  particle-wall problems.

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Dimensionless Poisson-Boltzmann equation(11
electrolyte)
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Units(11 electrolyte)
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Adaptive mesh enrichment process
Beginning
Mesh Generator
Numerical Solution Program
Mesh Generator
No
Yes
End
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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

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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/

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Error evaluation (1)
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Error evaluation (2)
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Error evaluation (3)
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Meshes
Germ mesh, 11 cells
Initial mesh, 147 cells
Final mesh, 15588 cells (after 8 steps)
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Steps of the adaptive process
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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-Romбn 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).

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

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A particle near a charged plane
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A particle near a charged plane
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A particle near a charged plane
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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.

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A particle near a charged plane
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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.