Moving contact lines and Langevin formalism

In previous work [J.-C. Fernández-Toledano, T. D. Blake^, J. De Coninck, J. Colloid Interface Sci. 540 (2019) 322-329], **we used molecular dynamics (MD) to show that the thermal oscillations of a contact line formed between a liquid and a solid at equilibrium may be interpreted in terms of an overdamped 1-D Langevin harmonic oscillator. The variance of the contact-line position and the rate of damping of its self-correlation function enabled us to determine the coefficient of contact-line friction ζ and so predict the dynamics of wetting. We now propose that the same approach may be applied to a moving contact line.

We use the same MD system as before, a liquid bridge formed between two solid plates, but now we move the plates at a steady velocity U_plate in opposite directions to generate advancing and receding contact lines and their associated dynamic contact angles θ^d. The fluctuations of the contact-line positions and the dynamic contact angles are then recorded and analyzed for a range of plate velocities and solid-liquid interaction.

We confirm that the fluctuations of a moving contact line may also be interpreted in terms of a 1-D harmonic oscillator and derive a Langevin expression analogous to that obtained for the equilibrium case, but with the harmonic term centered about the mean location of the dynamic contact line x^d, rather than its equilibrium position x⁰, and a fluctuating capillary force arising from the fluctuations of the dynamic contact angle around θ^d, rather than the equilibrium angle θ⁰. We also confirm a direct relationship between the variance of the fluctuations over the length of contact line considered L_y, the time decay of the oscillations, and the friction ζ. In addition, we demonstrate a new relationship for our systems between the distance to equilibrium x^d-x⁰ and the out of equilibrium capillary force γ_Lcosθ⁰-cosθ^d, where γ_L is the surface tension of the liquid, and show that neither the variance of the fluctuations nor their time decay depend on U_plate. Our analysis yields values of ζ nearly identical to those obtained for simulations of spreading drops confirming the common nature of the dissipation mechanism at the contact line.