How Does the Reduced Potential u(r) Simplify the Poisson-Boltzmann Equation?

[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?
Any help or advice appreciated. Thanks in advance.

best regards

 

[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.