- 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
2 distinct mechanisms of diffusion and drift
Given by Fick's law
- Effect of electrical forces on montion of charged particles.
- Electric Field (vector field) E(x,t)
- force on particle fp = QE(x,t) = zneE(x,t) where zn is valence and C.
- Motions of small particles in water are viscosity dominated (Stokes 1855)
|Forces||Size scale||Time scale|
where up is mechanical mobility in units of velocity/force, un is the molar mechanical mobility and f becomes the force on a mole of particle.
For charged particles: v = unzneNAE(x,t) = unznFE(x,t) (F = eNA which is Faradya's number) = charge/mole about 96500 C/mol.
Dn = unRT: Einstein's relation
Flux due to drift:
where (electric field depends on the potential gradient)
The flux of ions is the current density given by Jn = znFφn This is in units of current/area and is easier to measure than flux.
Combining diffusion and drift to get Nernst-Planck Equation:
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)
Unlike diffusion, also need one more equation for ψ but this electric potential depends on all particles.
From Gauss' law: where ε is the permitivity and ρ is the charge density.
This leads to Poisson's Equation
- In a solution with some charge, after some time, all charges go to the edges away from each other.
- τr 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.