Hi Mark --
I'm Cc'ing the mailing list since this might be of broader interest.
Solutions to the PBE for a sphere are given in
* Jackson JD. Classical Electrodynamics -- this covers a sphere with no
ions (Poisson equation)
* Bockris and Reddy. Modern Electrochemistry 1 -- this covers the LPBE;
this solution is also implemented in the APBS boundary conditions (see
src/mg/vpmg-setup.c) and written down in LaTeX form there
Solutions to the PBE for two spheres are available in semi-analytic form in
a variety of places. In fact, if you pick up a copy of Spencer and Moon,
Field Theory Handbook, you can see how to solve it yourself in a bispherical
coordinate system. Other sources are:
* Sushkin NV, Phillies GDJ. 1995. Charged dielectric spheres in electrolyte
solutions: Induced dipole and counterion exclusion effects. J Chem Phys 103
(11): 4600-4612. <http://dx.doi.org/10.1063/1.470647>
* Feng JQ. 2000. Electrostatic interaction between two charged dielectric
spheres in contact. Phys Rev E 62 (2): 2891-2897.
Solutions to the PBE for cylindrical geometries are presented in a number of
places; one of the best points to start would be Manning's papers:
* Manning GS. 1978. Limiting laws and counterion condensation in
polyelectrolyte solutions v. Further development of the chemical model.
Biophys Chem 9 (1): 65-70. <http://dx.doi.org/10.1016/0301-4622(78)87016-1>
You can also find many useful examples and references in
* Holm C, Kekicheff P, Podgornik R, editors. 2001. Electrostatic effects in
soft matter and biophysics. Boston: Kluwer Academic Publishers.
* Lamm G. 2003. The Poisson-Boltzmann equation. In: Lipkowitz K.B., R.
Larter and T.R. Cundari, editors. Reviews in computational chemistry.
Hoboken, N.J.: John Wiley and Sons, Inc. p 147-366.
Nathan A. Baker, Assistant Professor
Washington University in St. Louis School of Medicine
Dept. of Biochemistry and Molecular Biophysics
Center for Computational Biology
700 S. Euclid Ave., Campus Box 8036, St. Louis, MO 63110
Phone: (314) 362-2040, Fax: (314) 362-0234
> -----Original Message-----
> From: Mark Engelhardt [mailto:marke@...]
> Sent: Wednesday, October 13, 2004 8:51 PM
> To: baker@...
> Cc: Todd Dolinsky
> Subject: Re: analytical solutions to the pbe
> The easiest direct comparison for me to make is to a sphere,
> as I can represent that simply with a pdb file. A cylinder
> would also be useful for (rough) comparisons to DNA results
> and for possible extension of the potential and charge
> density information outside the area of the numerical solutions.
> Solutions to the npbe are best, but it would probably be
> instructive for me to evaluate the lpbe solutions to the
> sphere and cylinder as well, if possible.
> I hope that helps,