Fick's Laws of Diffusion

Fick's Laws of Diffusion describe the movement of particles due to concentration gradients, fundamental to processes like electrochemical reactions in polarography (e.g., Ilkovič equation derivation). Below are the two laws, their mathematical formulations, and their significance.

Fick's First Law

Fick's First Law relates the diffusive flux to the concentration gradient under steady-state conditions. It states that the flux of particles is proportional to the negative gradient of concentration.

\[ J = -D \frac{\partial c}{\partial x} \]

where:

• \( J \): Diffusive flux (mol/cm²·s), the rate of particle transfer per unit area.
• \( D \): Diffusion coefficient (cm²/s), a measure of how fast particles diffuse.
• \( \frac{\partial c}{\partial x} \): Concentration gradient (mol/cm⁴), with \( c(x) \) as concentration (mol/cm³) at position \( x \).

In electrochemical contexts, the current \( i \) due to ion diffusion at an electrode is related to the flux:

\[ i = n F A J = -n F A D \frac{\partial c}{\partial x} \]

where:

• \( n \): Number of electrons transferred per ion.
• \( F \): Faraday's constant (\( 96485 \, \text{C/mol} \)).
• \( A \): Electrode surface area (cm²).

This equation assumes a steady-state diffusion layer, often approximated in polarography for simplicity.

Fick's Second Law

Fick's Second Law describes non-steady-state diffusion, where concentration changes with both time and position. It is derived from the continuity equation and Fick's First Law.

\[ \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2} \]

where:

• \( \frac{\partial c}{\partial t} \): Rate of change of concentration with time (mol/cm³·s).
• \( \frac{\partial^2 c}{\partial x^2} \): Second spatial derivative of concentration, indicating curvature of the concentration profile.

Boundary Conditions (e.g., in polarography):

• At the electrode surface (\( x = 0 \), \( t > 0 \)): \( c(0, t) = 0 \) (complete depletion under limiting potential).
• Far from the electrode (\( x \to \infty \), \( t \geq 0 \)): \( c(x, t) = c_0 \) (bulk concentration).
• Initially (\( t = 0 \), \( x \geq 0 \)): \( c(x, 0) = c_0 \).

Solving this partial differential equation (e.g., via Laplace transforms) yields the concentration profile, used to derive currents like in the Cottrell equation.

Application to Polarography

Fick's Laws are central to polarography, particularly in deriving the Ilkovič equation for the diffusion-limited current at a dropping mercury electrode (DME). Fick's First Law gives the flux at the electrode surface, while Fick's Second Law accounts for the time-dependent diffusion layer growth, adjusted for the DME's expanding spherical geometry.

Example: For a planar electrode, solving Fick's Second Law gives the Cottrell equation:

\[ i_d = n F A \sqrt{\frac{D}{\pi t}} \, c_0 \]

This forms the basis for further corrections in the Ilkovič equation to account for the DME's growth.

Read also Derivation of the Cottrell Equation