Capillary Flow Simulator: Washburn Dynamics in Microchannels

simulator beginner ~8 min
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L ≈ 12.5 mm — flow front position after 1 second

Water (μ = 1 mPa·s, γ = 72 mN/m) in a 50 μm wide channel with 30° contact angle fills approximately 12.5 mm in the first second, following Washburn's square-root-of-time law.

Formula

ΔP = 2γ cos θ / w (capillary pressure in slit)
L(t) = √(γwt cos θ / (3μ)) (Washburn law for slit)
v(t) = γw cos θ / (6μL) (instantaneous velocity)

Capillary Action at the Microscale

When a liquid with a contact angle less than 90° meets a narrow channel, the curved meniscus at the air-liquid interface creates a capillary pressure that spontaneously draws the liquid forward. This capillary pressure scales inversely with channel width — a 50 μm channel generates roughly 20 times more driving pressure than a 1 mm channel. At the microscale, capillary forces are powerful enough to move fluids without any external pump, enabling entirely passive microfluidic devices.

Washburn Dynamics

Edward Washburn derived the fundamental law of capillary filling in 1921: the flow front advances as the square root of time, L ~ √t. This characteristic scaling arises from the balance between the constant capillary driving pressure and the linearly increasing viscous resistance as the liquid column grows. The initial filling is rapid — a water meniscus in a 50 μm glass channel travels the first millimeter in milliseconds — but progressively decelerates.

Surface Chemistry and Wettability

The contact angle θ determines the capillary driving force through the cos θ term. Hydrophilic surfaces (θ < 90°) drive flow forward; hydrophobic surfaces (θ > 90°) resist it. Precise control of surface chemistry is crucial: plasma treatment can make PDMS temporarily hydrophilic, while silane coatings create stable hydrophobic or hydrophilic surfaces. Patterning wettability gradients enables directional fluid transport and capillary valving.

Applications in Point-of-Care Diagnostics

Capillary-driven microfluidics underpins some of the most widely used diagnostic devices. Lateral flow immunoassays — from pregnancy tests to COVID-19 rapid antigen tests — rely entirely on capillary wicking through nitrocellulose membranes. More advanced capillary microfluidic circuits incorporate stop valves, delay channels, and sequential delivery of reagents, enabling multi-step assays on passive chips that require only a single drop of blood and no instrumentation.

FAQ

What drives capillary flow in microchannels?

Capillary flow is driven by the surface tension at the curved liquid-air meniscus inside the channel. When the liquid wets the channel walls (contact angle < 90°), the meniscus curves inward, creating a pressure difference (Laplace pressure) that pulls fluid into the channel without any external pump.

What is the Washburn equation?

The Washburn equation describes how far a liquid front has penetrated into a capillary tube or channel as a function of time: L(t) = √(γr cos θ t / (2μ)). The square-root dependence on time means the flow starts fast and progressively decelerates as viscous resistance increases with the growing liquid column length.

Why is capillary flow important for diagnostics?

Capillary flow enables pump-free microfluidic devices — critical for point-of-care diagnostics in resource-limited settings. Lateral flow assays (like pregnancy tests) and paper-based diagnostic devices rely entirely on capillary action to transport samples through detection zones without electricity or moving parts.

How does channel geometry affect capillary flow?

Narrower channels generate higher capillary pressure but also more viscous resistance. The net effect is that the Washburn filling rate scales as √(w) — wider channels fill faster despite lower capillary pressure. Channel aspect ratio, surface roughness, and corner geometry also influence the effective capillary driving force.

Sources

Embed

<iframe src="https://homo-deus.com/lab/microfluidics/capillary-flow/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub