Fick Membrane Diffusion

Passive transport case study: flux, permeability, membrane resistance, and equilibrium dynamics between two compartments.

Input parameters

Species preset

Diffusion coefficient

Intrinsic species diffusivity

cm2/s

1.000e-73.000e-5

Membrane thickness

Diffusion distance between compartments

0200

Membrane area

cm2

0.1100

Time: 0 / 5 tau

Equilibrium progress: 0%Instantaneous flux: 8.000e-7 mol/(cm2*s)Direction: Direction: 1 -> 2Physical time: 0 s

Compartment 1

Compartment 1 concentration: 120 mM

Left density:: 85.71%

Compartment 2

Compartment 2 concentration: 20 mM

Right density:: 14.29%

Net flux: 8.000e-7 mol/(cm2*s)

Note: The animation represents concentration density, not physical volume or total mass.

Results

Initial flux J

8.000e-7

mol/(cm2*s)

Permeability P

0.008

cm/s

Time to equilibrium (5tau)

6510

Membrane resistance Rm

125

s/cm

System regime

High gradient

Equilibrium concentration: 70 mM

Fundamentals & Explanation

What are Fick's Laws?

Fick's Laws provide the mathematical foundation for passive diffusion, the physical process by which molecules move randomly from regions of high concentration to regions of low concentration.

They were derived by the German physician and physiologist Adolf Fick in 1855, who drew heavy inspiration from Fourier's prior laws on heat conduction (1822) and Ohm's law of electrical conduction (1827), applying the same phenomenological principles to mass transport.

First Law: Molar Flux (J)

The First Law states that the molar flux per unit area ( $J$ ) is directly proportional to the concentration gradient. For a thin membrane, it is simplified using permeability ( $P$ ):

J = -D \frac{dc}{dx} \approx -P\,(C_1 - C_2)

P = \frac{D}{\Delta x}

Where $D$ is the intrinsic diffusion coefficient of the molecule and $\Delta x$ is the membrane thickness. The negative sign indicates that the flux occurs "downhill", opposing the gradient.

Assumptions and Limitations

For these equations to perfectly describe the system in this simulator, we assume certain ideal conditions that are not always met in real biology:

•Ideal dilute solutions: There are no electrostatic interactions or repulsive forces between solute molecules.
•Purely passive transport: No carrier proteins (facilitated diffusion) or energy-driven pumps (active transport) are involved.
•Perfect mixing: Concentration within each compartment is entirely homogeneous; the only gradient exists inside the membrane's thickness.

Time Scale and Equilibrium

\tau = \frac{V_1 \cdot V_2}{P \cdot A \cdot (V_1 + V_2)}

The system relaxes towards the equilibrium concentration ( $C_{eq}$ ) following an exponential curve. The speed of this decay is dictated by the time constant $\tau$ .

In physics, an exponential system is considered to reach operational equilibrium at $5\tau$ . At this point, 99.33% of the theoretical mass transfer has been completed.

Reference Coefficients

Intrinsic diffusivity ( $D$ ) in water at 25°C. Units are in $10^{-5}\text{ cm}^2/\text{s}$ . Values from Cussler (2009) and BioNumbers (Harvard).

Solute	Mass (Da)	D	Ref
Oxygen (O₂)	32	2.10	↗
Ethanol	46	1.20	↗
Glucose	180	0.67	↗
Serum Albumin	66,500	0.06	↗

Note how a massive increase in molecular weight (Albumin vs O₂) drastically reduces the diffusion coefficient due to the larger hydration radius.

Scope and Limits

If concentrations exceed physiological ranges or non-ideal membrane effects appear, results should be treated as a first-order approximation. This simulator represents a case study of passive diffusion based on classical Fick's law.