Physics Foundations · Topic 5 of 6
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Diffusion
Why mixing at the microscale is slow, and how to estimate how long it takes.
What you’ll learn
- What diffusion is and how far and fast it acts
- Why microscale mixing is diffusion-limited
- How to estimate a mixing time
The concept
Diffusion is the spreading of molecules by random thermal motion, from high concentration toward low. In the absence of turbulence, it is the only way two laminar streams mix.
The characteristic distance a species diffuses in time t scales as L ≈ √(2Dt), so the time to diffuse a distance L is t ≈ L²/(2D). The square dependence is the crucial point: doubling the distance quadruples the time.
Because channels are narrow, diffusion across the short dimension can be quick, but mixing across wide channels — or achieving uniformity — is slow. That is why serpentine and herringbone mixers exist: they fold the streams to shorten the diffusion distance.
Why it matters
Estimating the diffusion time tells you how long a channel (or how much residence time) you need to mix two streams — or, conversely, how to keep them separate.
The equation
A characteristic-scaling estimate, not an exact profile.
Variables
| Symbol | Variable | Unit |
|---|---|---|
| t | Diffusion time | s |
| L | Diffusion distance | m |
| D | Diffusion coefficient | m²/s |
Worked example
A small molecule (D ≈ 1×10⁻⁹ m²/s) mixing across a 100 µm channel:
t ≈ (100×10⁻⁶)² / (2 × 1×10⁻⁹) = 1×10⁻⁸ / 2×10⁻⁹ = 5 s
About 5 seconds to mix across 100 µm — and roughly 20 seconds across 200 µm, because of the square dependence.
Try it yourself
Put these numbers into the diffusion time calculator and see the result for your own channel.
Open the Diffusion time calculator →Common mistakes
Watch out for:
- Treating t ≈ L²/2D as exact; it is an order-of-magnitude estimate of a characteristic time.
- Assuming stirring will help — at low Reynolds number there is no turbulence to stir with.
Keep going
Related concepts
Related tools
Further reading
- Introduction to Microfluidics — Patrick TabelingSections on diffusion and mixing. Verify the current edition.
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Diffusion
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