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Capture threshold and differential susceptibility

Estimate of why the capture threshold is expected close to the orientation reversal of the interfacial water.

Langevin model

The mean orientation of permanent dipoles in a field E is described by the Langevin function. What matters for the response to an additional weak field is its derivative, the differential susceptibility L′(x):

L(x) = coth x − 1/x · L′(x) = 1/x² − 1/sinh² x · x = μE / k_BT

Fig. 3L′(0) = 1/3. For the compact layer (E ≈ 10⁹ V/m, μ = 1.85 D, T = 298 K) this gives x ≈ 1.5 and L′ ≈ 0.224, i.e. about 67 % of the value at x = 0.

Consequences

L′(x) decreases monotonically with |x|; the sensitivity to an additional field is largest at x = 0.
Even in the compact layer the orientation polarisation is not saturated (L(1.5) ≈ 0.45).
The capture threshold should therefore lie close to the orientation reversal. This can be tested by comparison with an independently determined pme (experiment).

Limitations

The Langevin and Booth approaches are continuum models; the compact layer comprises only one or two molecular layers.
ε_r ≈ 6 for the compact layer is a model value fitted to capacitance data, not a direct measurement.
Electrostriction, cooperativity via hydrogen bonds and (bi)sulfate adsorption are not taken into account.
Whether the external field reaches the interface is open (Debye length ≈ 1.2 nm).