A possible way to extract the dynamic contact angle at the molecular scale from that measured experimentally.

HYPOTHESIS: The maximum velocity of dewetting encodes sufficient information on the hydrodynamics of the wetting process to enable the local dynamic contact angle at the molecular scale, θ, to be determined from the apparent contact angle measured experimentally at much larger scales, θapp. METHODS: Effective models of wetting dynamics need to account for differing channels of dissipation. One such model was recently verified by large-scale molecular dynamics (MD). It combines the 2-parameter molecular-kinetic theory of dynamic wetting (MKT), which attributes the velocity-dependence of θ to dissipation at the contact line, with the Cox-Voinov hydrodynamic (HD) model. The latter attributes the difference between θ and θapp to viscous bending of the interface and contains an additional, non-predictable, logarithmic parameter. Crucially, the MD simulations indicated that viscous bending may play a minor role during wetting, but dominates dewetting. This observation suggested that by applying the MKT to the advancing contact angle only and combining the results with the maximum velocity of dewetting, it might be possible to extract the value of the logarithmic parameter and so determine θ and, hence, the relative significance of the two channels of dissipation. A simple iterative procedure has been developed to achieve this. FINDINGS: Data available to test the procedure are sparce, but comparisons with the MD results and those from three experimental studies are encouraging. Near perfect agreement is achieved with the simulations, where both θ and θapp are known, and plausible results are obtained for the experimental systems. Moreover, the procedure appears to be more effective than simply fitting θapp to the 3-parameter model.

Taking a closer look: A molecular-dynamics investigation of microscopic and apparent dynamic contact angles.

HYPOTHESIS: Molecular dynamics (MD) may be used to investigate the velocity dependence of both the microscopic and apparent dynamic contact angles (θm and θapp). METHODS: We use large-scale MD to explore the steady displacement of a water-like liquid bridge between two molecularly-smooth solid plates under the influence of an external force F0. A coarse-grained model of water reduces the computational demand and the solid-liquid affinity is varied to adjust the equilibrium contact angle θ0. Protocols are devised to measure θm and θapp as a function of contact-line velocity Ucl. FINDINGS: For all θ0, θm is velocity-dependent and consistent with the molecular-kinetic theory of dynamic wetting (MKT). However, θapp diverges from θm as F0 is increased, especially at the receding meniscus. The behavior of θapp follows that predicted by Voinov: (θapp)3 = (θm)3 + 9Ca·ln(L/Lm), where Ca is the capillary number and L and Lm are suitably-chosen macroscopic and microscopic length scales. For each θ0, there is a critical velocity Ucrit and contact angle θcrit at which θapp→0 and the receding meniscus deposits a liquid film. Setting θapp=0, θm=θcrit and Ucl=Ucrit in the Voinov equation yields the value of L/Lm. The predicted values of θapp then agree well with those measured from the simulations. Since θm obeys the MKT, we have, therefore, demonstrated the utility of the combined model of dynamic wetting proposed by Petrov and Petrov.

Controlling the pinning time of a receding contact line under forced wetting conditions.

HYPOTHESIS: The contact line pinning that appears in a flow coating process over substrates patterned with chemical or physical heterogeneities has been recently applied to deposit micro- and nanoparticles with great precision. However, the mechanism underlying pinning of a receding contact line at the nanoscale is not yet well understood. In the case of a contact line pinned at a chemical heterogeneity, we hypothesise that it is possible to establish a relation between the pinning time, the contact line velocity and the liquid/plate/heterogeneity affinity that can help to optimize particle deposition. METHODS: We use large-scale molecular dynamic (MD) simulations of a finite liquid bridge formed between two parallel, non-identical, smooth solid plates. The top plate slides relative to the bottom plate inducing a displacement of the four different contact lines of the liquid bridge. The introduction of a chemical heterogeneity on the bottom plate by modifying locally the liquid-solid affinity provokes the transient pinning of the contact line in contact with the bottom substrate. By means of this simple MD simulation, we can study the mechanism of contact line pinning and its relation with the liquid/heterogeneity affinity and the contact line velocity. Additionally, we compare this mechanism with the case of the receding contact line pinned on a physical heterogeneity (a simple step discontinuity). FINDINGS: We propose an analytical model that predicts the values of the dynamic contact angles in the general case of a capillary liquid bridge confined between two parallel plates with different wettabilities versus the relative velocity of the top plate. These predictions are successfully validated by the results of the large-scale MD simulations. The model allows thus to predict the value of the dynamic contact angles for the different contact lines of the system versus the relative speed of the moving plate. Once the chemical heterogeneity is introduced in the bottom plate, we show that when the receding contact line reaches the patch it remains temporarily pinned while the receding contact angle evolves with time. Once the receding angle reaches the value of the equilibrium contact angle of the patch, the receding contact line overcomes pinning. A geometrical model able to predict the pinning time is proposed and validated by our MD simulations. The pinning time depends not only on the relative plate velocity and plate wettability properties but also on the separation distance between the plates confining the capillary bridge. The model can consequently be used to select the substrate wettability or meniscus geometry suitable to impose the pinning time required for specific applications.

Moving contact lines and Langevin formalism.

HYPOTHESIS: 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. METHODS: 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 Uplate in opposite directions to generate advancing and receding contact lines and their associated dynamic contact angles θd. 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. FINDINGS: 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 xd, rather than its equilibrium position x0, and a fluctuating capillary force arising from the fluctuations of the dynamic contact angle around θd, 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 Ly, 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 xd-x0 and the out of equilibrium capillary force γLcosθ0-cosθd, where γL is the surface tension of the liquid, and show that neither the variance of the fluctuations nor their time decay depend on Uplate. 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.

A molecular-dynamics study of sliding liquid nanodrops: Dynamic contact angles and the pearling transition.

HYPOTHESIS: That the behavior of sliding drops at the nanoscale mirrors that seen in macroscopic experiments, that the local microscopic contact angle is velocity dependent in a way that is consistent with the molecular-kinetic theory (MKT), and that observations at this scale shed light on the pearling transition seen with larger drops. METHODS: We use large-scale molecular dynamics (MD) to model a nanodrop of liquid sliding across a solid surface under the influence of an external force. The simulations enable us to extract the shape of the drop, details of flow within the drop and the local dynamic contact angle at all points around its periphery. FINDINGS: Our results confirm the macroscopic observation that the dynamic contact angle at all points around the drop is a function of the velocity of the contact line normal to itself, Ucmsinϕ, where Ucm is the velocity of the drop's center of mass and ϕ is the slope of the contact line with respect to the direction of travel. Flow within the drop agrees with that observed on the surface of macroscopic drops. If slip between the first layer of liquid molecules and the solid surface is accounted for, the velocity-dependence of the dynamic contact angle is identical with that found previous MD simulations of spreading drops, and consistent with the MKT. If the external force is increased beyond a certain point, the drop elongates and a neck appears between the front and rear of the drop, which separate into two distinct zones. This appears to be the onset of the pearling transition at the tip of a macroscopic drop. The receding contact angle at the tip of the drop is far removed from its equilibrium value but non-zero and approaches a more-or-less constant critical value as the transition progresses.

Contact-line fluctuations and dynamic wetting.

HYPOTHESIS: The thermal fluctuations of the three-phase contact line formed between a liquid and a solid at equilibrium can be used to determine key parameters that control dynamic wetting. METHODS: We use large-scale molecular dynamics simulations and Lennard-Jones potentials to model a liquid bridge between two molecularly smooth solid surfaces and study the positional fluctuations of the contact lines so formed as a function of the solid-liquid interaction. FINDINGS: We show that the fluctuations have a Gaussian distribution and may be modelled as an overdamped one-dimensional Langevin oscillator. Our analysis allows us to extract the coefficients of friction per unit length of the contact lines ζ, which arise from the collective interaction of the contact-line's constituent liquid atoms with each other and the solid surface. We then compare these coefficients with those obtained by measuring the dynamic contact angle as a function of contact-line speed in independent simulations and applying the molecular-kinetic theory of dynamic wetting. We find excellent agreement between the two, with the same dependence on solid-liquid interaction and, therefore, the equilibrium contact angle θ0. As well as providing further evidence for the underlying validity of the molecular-kinetic model, our results suggest that it should be possible to predict the dynamics of wetting and, in particular, the velocity-dependence of the local, microscopic dynamic contact angle, by experimentally measuring the fluctuations of the contact line of a capillary system at equilibrium. This would circumvent the need to measure the microscopic dynamic contact angle directly.

On the cohesion of fluids and their adhesion to solids: Young's equation at the atomic scale.

Using large-scale molecular dynamics simulations, we model a 9.2nm liquid bridge between two solid plates having a regular hexagonal lattice and analyse the forces acting at the various interfaces for a range of liquid-solid interactions. Our objective is to study the mechanical equilibrium of the system, especially that at the three-phase contact line. We confirm previous MD studies that have shown that the internal pressure inside the liquid is given precisely by the Laplace contribution and that the solid exerts a global force at the contact line in agreement with Young's equation, validating it down to the nanometre scale, which we quantify. In addition, we confirm that the force exerted by the liquid on the solid has the expected normal component equal to γlvsinθ0, where γlv is the surface tension of the liquid and θ0 is the equilibrium contact angle measured on the scale of the meniscus. Recent thermodynamic arguments predict that the tangential force exerted by the liquid on the solid should be equal to the work of adhesion expressed as Wa0=γlv(1+cosθ0). However, we find that this is true only when any layering of the liquid molecules close to liquid-solid interface is negligible. The force significantly exceeds this value when strong layering is present.

Young's Equation for a Two-Liquid System on the Nanometer Scale.

We use large-scale molecular dynamics simulations to study the Lennard-Jones forces acting at the various interfaces of a liquid bridge (liquid 1) between two realistic solid plates on the scale of few nanometers when the two free surfaces are in contact with a second immiscible liquid (liquid 2) with an interfacial tension of γ12. Each plate comprises a regular square planar lattice of atoms arranged in three atomic layers. To maintain rigidity while allowing momentum exchange with the liquid, solid atoms are allowed to vibrate thermally around their initial positions by a strong harmonic potential. By varying the solid-liquid coupling, we investigate a range of nonzero contact angles between the liquid-liquid interface and the solid. We first compute the forces when the plates are stationary (equilibrium case), from the perspectives of both the liquid and the solid. Our results confirm that the normal and tangential components of the computed interfacial forces at each contact line are consistent with Young's equation on this small scale. In particular, we show that the tangential force exerted by the liquid-liquid interface on the plates is given by the difference in the individual works of adhesion of the two liquids and equal to γ12 cos θ1,20, where θ1,20 is the equilibrium contact angle measured through liquid 1. This result, which differs from that expected for a single liquid, is relevant to the interactions and behavior of two liquid-solid systems in nanotechnology. We then study the forces when the plates are translated at equal speeds in opposite directions over a range of steady velocities (dynamic case) and repeat the measurements of the force exerted by the liquid-liquid interface on the solid. We find that the normal and tangential components of this force are still correctly predicted by the normal and tangential components of the interfacial tension, provided only that the equilibrium contact angle is replaced by its dynamic analogue θ1,2D. Usually assumed without proof, this result is significant for our proper understanding of dynamic wetting at all scales.