2.4.7 Fick’s Law

By expressing the speed through the Stokes-Einstein relation, see Eq.(2.36)

$$\frac{dV}{dt}=\frac{D\ast R\ast R\ast T}{q\ast F\ast F}\ast \frac{d^{2}C}{d{x}^2}=\left(\frac{T\ast R^2}{q\ast F^2}\right)\ast D\ast \frac{d^{2}C}{d{x}^2}$$

(2.21)

Or, alternatively,

$$\frac{dV}{dt}={\left(\frac{T\ast R}{F}\right)}^2\frac{k}{6\ast q\ast \pi \ast \eta \ast a}\ast \frac{d^{2}C}{d{x}^2}$$

(2.22)

Given that

$$\frac{dV}{dt}=D\ast \frac{d^{2}C}{d{x}^2}$$

(2.23)

expresses Fick’s Second Law of Diffusion, we can derive the ratio between the electric and thermodynamic temporal gradients. Using values $T=300K$, $q=1$, $F=96495A\ast s/mol$, $R=8.31446261815324J\ast K^{-1}\ast mo{l}^{-1}$

$$\frac{dV}{dt}=2.23\ast {10}^{-6}\ast D\ast \frac{d^{2}C}{d{x}^2}=\mathbf{2.23}\ast {\mathrm{𝟏𝟎}}^{-\mathrm{𝟔}}\frac{dC}{dt}$$

(2.24)

We can provide a rough (experimental) estimation of the value. The time course of the gradient (half-width) in Figure 2.2 in [2] was produced in $0.3\ast {10}^{3}s$. Reciprocally, a voltage gradient needs $0.1\ast {10}^{-3}s$ to produce the same concentration gradient, see Fig. 2d in [117]. Their ratio is $3\ast {10}^{-6}$. The other way round, when we assume that the propagation of the electric field in the solution is $2\ast {10}^{8}m/s$ and multiply it by the factor above, we arrive at the speed of ion’s speed in the range around ${10}^{2}m/s$.

Given that we can assume that the electrical interaction speed is about the respective speed of light, we arrive at that the speed of material transport (the speed of the mechanical wave) is about $100[\mathrm{m}/\mathrm{s}]$. Given that the mechanical wave carries an electric charge, it induces an electrical charge on the other side of the axon. This way, the particles move under the combined effect of a thermodynamic force and an electrical force. The amount of induced charge depends on the specific capacity of the axon (the thickness of the myelin layer on it), so the resultant force (and, due to this, the Stokes-Einstein speed) of the axonal current depends on the thickness of the myelin layer.