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

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SUMMARY

The discussion centers on the simplification of the Poisson-Boltzmann equation through the introduction of the reduced potential u(r). The original equation is transformed into a more manageable form by defining the reduced potential as u(r) = q*φ / (kT). The key insight is that under the assumption q*u / kT << 1, the hyperbolic sine function can be approximated as sinh(q*u/kT) ≈ q*u/kT, leading to a simplified equation that retains the essential physical characteristics while making calculations more straightforward.

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greisen
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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
 
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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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