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Poisson_boltzmann equation

  1. Mar 21, 2007 #1
    The problem is going from the Poisson-Boltzmann equation

    [tex]\nabla (e(r)*\nabla \phi(r)) - \kappa^2(kT/q)*sinh(q*\phi(r)/kT) = -4*\pi \rho(r)[/tex]

    The equation is than rewritten in terms of a reduced potential u

    [tex]\nabla (e(r)*\nabla u(r)) - \kappa^2 sinh(u(r)) = -4\pi*\rho(r)[/tex]

    The reduced potential is defined as [tex]u(r) = q*\phi / (kT)[/tex] - I can see that term q/kT is multiplied on the right side but nothing changes on the left side?

    Have I totally misunderstood the equation and the approximation of the PBE?
    Any help or advice appreciated. Thanks in advance.

    best regards
    Last edited by a moderator: Mar 21, 2007
  2. jcsd
  3. Mar 22, 2007 #2
    I think I have it here; the electrostatic potential \phi can be written as the reduced potential u. If one again assumes that q*u / kT << 1 than the hyperbolic function can be approximated as
    sinh(q*u/kT) \approx q*u/kT
    which than reduces to the equation.
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