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.

 

[Blueshift5]

The Poisson-Boltzmann equation is a fundamental equation in electrostatics that describes the distribution of ions in a solution around a charged molecule or surface. It is a key tool in understanding and predicting the behavior of biological systems, such as the interaction between proteins and DNA.

In this equation, the first term on the left side represents the electrostatic potential created by the charges in the system, while the second term takes into account the effects of thermal energy on the distribution of ions. The right side represents the charge density in the system.

The reduced potential u(r) is a useful approximation that simplifies the equation by incorporating the term q/kT, which represents the thermal energy of the ions. This term is multiplied on the right side of the equation, but it does not change the left side because it is a constant factor that can be absorbed into the definition of u(r).

It is important to note that the reduced potential is an approximation and may not accurately describe the behavior of the system in all cases. However, it is a useful tool for simplifying the equation and gaining insights into the behavior of the system.

I hope this helps to clarify any confusion and provides a better understanding of the Poisson-Boltzmann equation and its approximation. Keep exploring and learning!