# Poisson_boltzmann equation

1. Mar 21, 2007

### greisen

The problem is going from the Poisson-Boltzmann equation

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

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

$$\nabla (e(r)*\nabla u(r)) - \kappa^2 sinh(u(r)) = -4\pi*\rho(r)$$

The reduced potential is defined as $$u(r) = q*\phi / (kT)$$ - 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?

best regards

Last edited by a moderator: Mar 21, 2007
2. Mar 22, 2007

### greisen

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.