MAT-30

Fick first-law flux

J = −D (c2−c1)/Δx. Steady diffusion down a concentration gradient.

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DiffusionFick 1855

Governing equation

J=-D\dfrac{c_2-c_1}{\Delta x}

$\Delta x \in \left[0.01, 20\right]$

where

D: Diffusivity (m²/s)
c_1: Conc. 1 (mol/m³)
c_2: Conc. 2 (mol/m³)
\Delta x: Distance (mm)
J: Flux toward 2 (µmol/(m²·s))

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

J=-D\dfrac{c_2-c_1}{\Delta x}

$\Delta x \in \left[0.01, 20\right]$

$\Delta x = 0.01$ $J = 0.9$

Along the curve

②

Domain of the variable — blue

\Delta x_{\min} = 0.01,\quad \Delta x_{\max} = 20

$\Delta x \in \left[0.01, 20\right]$

$\Delta x = 0.01$ $\Delta x = 0.01$

Along the curve

③

Response curve — green

J=-D\dfrac{c_2-c_1}{\Delta x}

$\Delta x \in \left[0.01, 20\right]$

$\Delta x = 0.01$ $J = 0.9$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}J}{\mathrm{d}\Delta x}

$\Delta x \in \left[0.01, 20\right]$

$\Delta x = 0.136$ $J^{\prime} = -3.442$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-30 — Fick first-law flux) is the form associated with Fick 1855. Working symbols: $D$ , $c_1$ , $c_2$ , $\Delta x$ $\rightarrow$ $J$ . Fick's first law is the constitutive statement of diffusion: flux proportional to minus the gradient, with diffusivity D.

Purpose

Purpose: compute

J

from

D

c_1

c_2

\Delta x

in Materials via

J=-D\dfrac{c_2-c_1}{\Delta x}

J = −D (c2−c1)/Δx. Steady diffusion down a concentration gradient. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

D = 1.000e-12\,\mathrm{m^{2}/s}

c_1 = 10.000\,\mathrm{mol/m^{3}}

c_2 = 1.000\,\mathrm{mol/m^{3}}

\Delta x = 1.000\,\mathrm{mm}

, the governing relation

J=-D\dfrac{c_2-c_1}{\Delta x}

yields

J = 0.0090\,\mathrm{\mu mol/(m^{2}·s)}

. A slab, two concentrations, a flux arrow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Diffusivity D1e-12 m²/sConc. 1 c_110 mol/m³Conc. 2 c_21 mol/m³Distance \Delta x1 mm

Outputs

Flux toward 2 J0.0090 µmol/(m²·s)

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Narration of this film

A slab, two concentrations, a flux arrow.

Fick's first law is the constitutive statement of diffusion: flux proportional to minus the gradient, with diffusivity D.

Reading speed

Watch on YouTube