# 6.021/Notes/2006-10-04

## Ion Transport

• Major constituents of cells
• important functions
• charge is substrate for neural communication
• every charged particle (in principle) affects every other ion
• more complicated than other mechanisms

## Mechanisms

2 distinct mechanisms of diffusion and drift

### Diffusion

Given by Fick's law $\phi_n = -D_n\frac{\partial c_n}{\partial x}$

### Drift

• Effect of electrical forces on montion of charged particles.
• Electric Field (vector field) $E(x,t)$
• force on particle $f_p = QE(x,t) = z_neE(x,t)$ where $z_n$ is valence and $e\approx 1.6\cdot 10^{-19}$ C.
• Motions of small particles in water are viscosity dominated (Stokes 1855)
 Forces Size scale Time scale inertial (F=ma) $radius^3$ acceleration viscosity $radius$ velocity

$v\propto f_p = u_p f_p = u_n f$ where $u_p$ is mechanical mobility in units of velocity/force, $u_n$ is the molar mechanical mobility and $f$ becomes the force on a mole of particle.

For charged particles: $v=u_nz_neN_AE(x,t)=u_nz_nFE(x,t)$ ($F=eN_A$ which is Faradya's number) = charge/mole about 96500 C/mol.

$D_n=u_nRT$: Einstein's relation

Flux due to drift: $\phi_n = \lim_{A\rightarrow 0 ; \Delta t\rightarrow 0}\frac{c_n(x,t)A\Delta x}{A\Delta t} = vc_n(x,t)$

$\phi_n = c_n(x,t)u_nz_nFE(x,t) = -c_nu_nz_nF\frac{\partial\psi}{\partial x}$ where $E=-\frac{\partial\psi}{\partial x}$ (electric field depends on the potential gradient)

The flux of ions is the current density given by $J_n = z_nF\phi_n$ This is in units of current/area and is easier to measure than flux.

### Combined transport

Combining diffusion and drift to get Nernst-Planck Equation:

$J_n = -z_nFD_n\frac{\partial c_n}{\partial x}-u_nz_n^2F^2c_n\frac{\partial\psi}{\partial x}$

Note that this is really just a combination of Fick's and Ohm's Laws.

Continuity: (needed to solve equations just like in other transport mechanisms)

$\frac{\partial\phi_n}{\partial x} = -\frac{\partial c_n}{\partial t}$ or equivalently

$\frac{\partial J_n}{\partial x} = -z_nF\frac{\partial c_n}{\partial t}$

Unlike diffusion, also need one more equation for $\psi$ but this electric potential depends on all particles.

From Gauss' law: $\frac{\partial E}{\partial x} = \frac{1}{\epsilon}\rho(x,t) = \frac{1}{\epsilon}\sum_n z_nFc_n(x,t)$ where $\epsilon$ is the permitivity and $\rho$ is the charge density.

This leads to Poisson's Equation $\frac{\partial^2 \psi}{\partial x^2} = -\frac{1}{\epsilon}\sum_n z_nFc_n(x,t)$

## Electroneutrality

• In a solution with some charge, after some time, all charges go to the edges away from each other.
• $\tau_r$ is the relaxation time and is on the order of nanoseconds for physiological salines
• Similarly, in space, a region around the charge is formed that negates the charge. This is known as the Debye layer and has a thickness of around a nanometer.
• Thus for times much greater than the relaxation time and distances much greater than the Debye distance, we can assume electroneutrality of the solution.