(469d) Spreading and Retraction as a Function of Drop Size

At the contact line, a drop on a solid surface exhibits an equilibrium contact angle (θ_e) at rest and a dynamic contact angle (θ_D) while in motion. The dynamic contact angle and the time rate of change of drop radius (the contact line velocity) are often related by Tanner's law: θ_D³ - θ₀³ = kCa , where θ₀ is the static contact angle (often assumed equal to θ_e), and Ca is the contact line velocity scaled by the ratio of the interfacial tension  γ to the fluid viscosity μ.  This relationship has been derived in asymptotic analyses of advancing1(1,2) and receding contact lines(3) in a hydrodynamic theory is based on a small parameter, the ratio of the length of the microscopic region near the corner of the drop to the macroscopic scale of the drop itself; k is proportional to the logarithm of this small parameter.  Tanner's law has been adopted as the boundary condition for continuum simulations of drop spreading dynamics (4-9) and in studies involving moving contact lines in other geometries (10,11) disregarding the finite size of the drop.It is logical that drop size should be of paramount importance in the recovery of Tanner's law, as its derivation is predicated on a separation of length scales between the size of the macroscopic drop and the microscale behavior near the contact line.  Yet, there is significant lack of studies that probe into this size effect systematically. Experimental studies are prone to artifacts associated with contact line pinning and inaccessibility to within 10 micrometers from the contact line. Molecular dynamic simulations suffer inherent limitations of small system size (~ 10nm) , short timescales (nanoseconds) and evaporative losses, which may challenge their ability to recover bulk hydrodynamic behavior.(12)

In this study, continuum simulations of the spreading and retraction of a two-dimensional drop on a homogeneous solid surface are performed in the lubrication limit as a function of the drop size. The solid surface is covered with a thin liquid film. Drop motion is initiated by an impulsive change in surface wettability.  For small drops of the order of magnitude of the thin liquid film, disjoining pressure gradients are significant everywhere under the drops.  Such drops do not obey Tanner's law. For large drops with heights at least an order of magnitude greater than the thin liquid film, disjoining pressure gradients are isolated near the apparent contact line at all times.  After initial film-dominated dynamics, Tanner's law is recovered.  Drop retraction occurs in three regimes: rapid rim formation near the apparent contact line, followed by rim propagation toward the bulk drop, and culminating in drop recovery.  Drops retract rapidly, significantly faster than a drop would spread for same differences in wetting conditions.  We also simulate drop spreading and retraction for the case in which a drop moves over a bare solid substrate with a Navier slip length.  Drop retraction proceeds with the same regimes in qualitative agreement with the predictions for the drop retraction over the wetting film.   These results should be relevant in applications in which drop motion is controlled e.g. in microfluidic and lab-on-chip devices.

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