Electrostatics interactions in condensed matter, theory and simulation
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1 Electrostatics interactions in condensed matter, theory and simulation A.C. Maggs CNRS + ESPCI + ParisTech + Paris Sciences et Lettres June 2016
2 Institutes CNRS: 10,000 researchers fundamental and applied research ESPCI historic school, Curie, Langevin, de Gennes... Today: Soft matter, chemical physics, acoustics... Advanced undergraduate (age 20-24), 90 students/year Phd 30/year ParisTech (External student schemes) including SJTU-ParisTech Students work in major chemical/physical chemistry eg Total Recrutement of foreign students after first degree Paris Science et Lettres includes ENS (Ecole Normale), institut Curie. Dauphine University. Originally from London (formally in Europe)
3 Gulliver group at ESPCI Mixed theory+experimental Elie Raphael Hydrodynamics and polymers Olivier Dauchot non-equilibrium and driven matter, powders to propelled nanoparticles Patrick Tabeling Microfluidics Yannick Rondalez reaction networks and information in chemical systems
4 Overview Importance of electrostatic interactions Mathematics tools and notation Effective theories, Poisson Boltzmann Duality and exact transformations φ or D Simulation of charged systems Local simulation algorithms despite long-ranged 1/r interaction Applications: fluctuations, finite volume effects coupled to electrostatics, dielectrics
5 Topics in electrostatics Electrolytes Solvation of polymers Screening, surface properties surface tension, capacitance Conductivity Dipoles - van der Waals, Casimir, Lifshitz Biophysics, proteins and DNA, free energies Strong and weak coupling in electrostatics Born energy, solvation
6 Mathematical and Theoretical Tools Convexity Legendre transforms Minimization or Maximization Monte-Carlo simulation Landau theory for dielectrics
7 Electrostatics in soft matter physics Interactions are long ranged - electrical neutrality Non-extensive statistics (δn) V 1/6 rather than V 1/2 U = e0 2 /2C, C = 4πɛa cutting argument (δn) A 1/2 Simple virial theory diverges Sum rules S ZZ (q) = 0 + q 2 l 2 D Screening with the Debye length Conductivity Treating the long ranged part incorrectly breaks these properties, numerically or analytically
8 Understanding Surface capacitance battery technology solubility of salts Born energies U = e0 2/(8πɛ 0a) confirmation of polyelectrolyes Protein and DNA interactions, hydrophobicity
9 Fluctuations Since van der Waals - know about long-ranged forces Keesom guessed dipolar fluctuations 1/r 6, thermally activated London quantum fluctuations Casimir + Lifshitz full formulation in terms of coupled matter-electrodynamics fields Kirkwood-Shumaker Also closely linked to electrostatic properties Often dominate in soft-matter contexts
10 History of dispersion interactions Van der Waals 1873 :- Mixture of long and short range interactions explain phase diagram of gases. Keesom 1921, classical fluctuations 1/r 6 London 1930, quantum fluctuations 1/r 6 Casimir, Polder /r 7 Lifshitz et al formulation in terms of dielectric properties Often presented in terms of mode counting (with metalic boundary conditions) or Green functions, Simple formulation as a partition function with constrained measure
11 Why simulation vs why continue to do theory Gold standard in terms of model, however be careful of convergence... Many advanced theories also lead to painful numerics too However, no free energies
12 Conventional numerics Fourier based: 2 φ = ρ/ɛ solved by fast Fourier transform on an interpolating grid No-truncation, estimates of errors, need to interpolate to hundreds of grid points.
13 Other numerical methods Ewald too slow for large N ( > 1000 ), but very high precision Multipole methods Greengard Linear scaling with N but large prefactors Will talk of a local grid based algorithm this week
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