THE DOUBLE LAYER

Charge on the surface, counter-charge in the liquid

A charged surface attracts ions of the opposite sign from the solution. The result is two sheets of charge, one on the surface and a screening cloud in the liquid, and the potential falls from ψ₀ back to zero across the cloud.

The double layer is how a solution hides a charged surface from far away.

THREE ZONES

From surface to bulk

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Surface

Fixed charge from site binding, at potential ψ₀

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Stern layer

A compact layer of ions held within a fraction of a nanometer

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Diffuse layer

Mobile ions spread out by thermal motion, thinning with distance

The potential drops across the Stern layer first, then fades through the diffuse layer.

DEBYE LENGTH

How far the screening reaches

λ_D ≈ 0.3 nm ÷ √c

The Debye length λ_D is the distance over which the potential falls to 1/e of its value. For a simple salt in water at 25 °C, λ_D is about 0.3 nm divided by the square root of the concentration c in mol/L. More salt means more mobile ions, so screening is quicker and the reach is shorter.

The Debye length shrinks as the square root of the concentration grows.

DEBYE LENGTH IN PRACTICE

Salt sets the reach

1 mM about 10 nm

10 mM about 3 nm

100 mM about 1 nm

150 mM about 0.8 nm, like physiological saline

In body-like salt, the double layer is under a nanometer thick.

SERIES CAPACITORS

The double layer is stiff

C_dl ≫ C_ox

In typical salt, the double layer holds tens of µF/cm², against about 0.35 µF/cm² for a 10 nm oxide. Capacitors in series are dominated by the smaller one, so the gate stack behaves almost like the oxide alone, and the interface potential passes through to the channel.

The liquid side adds little to the gate capacitance, so the oxide sets it.

UNIT 10 STUDY COMPLETE

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Ready for the Fab Challenge?

You've covered how ions in the solution screen the surface charge, and how far the screening reaches.