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.

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.

The temperature-dependence of the dynamic contact angle.

HYPOTHESIS: The wetting of a solid by a liquid is a thermally-activated molecular rate process, which may be investigated by studying the temperature-dependence of the dynamic contact angle at standard experimental scales. EXPERIMENTS: We use the plunging-tape method and a low-powered microscope to measure the dynamic contact angle of di-n-butyl phthalate (DBP) on poly(ethylene terephthalate) (PET) tape over a very wide speed range of 0.003-100 cm/s at 5 temperatures from 15 °C to 55 °C. The molecular-kinetic theory of dynamic wetting (the MKT) is then used to interpret the data, which span angles from 8° to near 180°. FINDINGS: The MKT successfully accounts for the temperature-dependence of the dynamic contact angle, yielding rational values for key parameters including the activation free energy of wetting. Arrhenius-like behavior is also demonstrated. These results would appear to confirm that, at the molecular-scale, dynamic wetting is a thermally-activated rate process and that the influence of temperature is not restricted simply to its effect on surface tension and viscosity. The data show a discontinuity at dynamic contact angles between 60° and 90° that implies a velocity-dependent change in the wetting mechanism. We attribute this to the chemical heterogeneity of the PET surface, which contains both polar and non-polar groups. The parameters obtained by applying the MKT suggest that the interactions of DBP with these groups determine, respectively, the dynamics observed below and above the transition. The sum of the activation free energies of wetting on either side of the transition is close to the total thermodynamic work of adhesion of DBP to PET.

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.

Forced wetting of a reactive surface.

The dynamic wetting of water on gelatin-coated poly(ethylene terephthalate) (GC-PET) has been investigated by forced wetting over a wide speed range and compared with earlier data obtained with unmodified PET. The results were analysed according to the molecular-kinetic theory of dynamic wetting (MKT). Both substrates show complex behaviour, with separate low- and high-speed modes. For the GC-PET, this is attributed to a rapid change in the wettability of the substrate on contact with water, specifically a surface molecular transformation from hydrophobic to hydrophilic. This results in a smooth wetting transition from one mode to the other. For the PET, the bimodal behaviour is attributed to surface heterogeneity, with the low-speed dynamics dominated by interactions with polar sites on the substrate that become masked at higher speeds. In this case, the transition is discontinuous. The study has general ramifications for the investigation of any wetting processes in which a physicochemical transformation takes place at the solid surface on contact with the liquid. In particular, it shows how forced wetting, combined with the MKT, can reveal subtle details of the processes involved. It is unlikely that similar insight could be gained from spontaneous wetting studies, such as spreading drops.

Predicting the wetting dynamics of a two-liquid system.

We propose a new theoretical model of dynamic wetting for systems comprising two immiscible liquids, in which one liquid displaces another from the surface of a solid. Such systems are important in many industrial processes and the natural world. The new model is an extension of the molecular-kinetic theory of wetting and offers a way to predict the dynamics of a two-liquid system from the individual wetting dynamics of its parent liquids. We also present the results of large-scale molecular dynamics simulations for one- and two-liquid systems and show them to be in good agreement with the new model. Finally, we show that the new model is consistent with the limited data currently available from experiment.

Experimental evidence of the role of viscosity in the molecular kinetic theory of dynamic wetting.

We report an experimental study of the dynamics of spontaneous spreading of aqueous glycerol drops on glass. For a range of glycerol concentrations, we follow the evolution of the radius and contact angle over several decades of time and investigate the influence of solution viscosity. The application of the molecular kinetic theory to the resulting data allows us to extract the coefficient of contact-line friction ζ, the molecular jump frequency κ(0), and the jump length λ for each solution. Our results show that the modified theory, which explicitly accounts for the effect of viscosity, can successfully be applied to droplet spreading. The viscosity affects the jump frequency but not the jump length. In combining these data, we confirm that the contact-line friction of the solution/air interface against the glass is proportional to the viscosity and exponentially dependent on the work of adhesion.

Experimental investigation of the link between static and dynamic wetting by forced wetting of nylon filament.

Forced wetting experiments with various liquids were conducted to study the dynamic wetting properties of nylon filament. The molecular-kinetic theory of wetting (MKT) was used to interpret the dynamic contact angle data and evaluate the contact-line friction zeta0 at the microscopic scale. By taking account of the viscosity of the liquid, zeta0 could be related exponentially to the reversible work of adhesion. This clearly establishes an experimental link between the static and dynamic wetting properties of the material. Moreover, statistical analysis of the equilibrium molecular displacement frequency K0 and the length of the displacements lambda reveals that these two fundamental parameters of the MKT are strongly correlated, not only in the linearized form of the theory (valid close to equilibrium) but also when the nonlinear form of the equations has to be considered at higher wetting speeds.

Dynamics of dewetting at the nanoscale using molecular dynamics.

Large-scale molecular dynamics simulations are used to model the dewetting of solid surfaces by partially wetting thin liquid films. Two levels of solid-liquid interaction are considered that give rise to large equilibrium contact angles. The initial length and thickness of the films are varied over a wide range at the nanoscale. Spontaneous dewetting is initiated by removing a band of molecules either from each end of the film or from its center. As observed experimentally and in previous simulations, the films recede at an initially constant speed, creating a growing rim of liquid with a constant receding dynamic contact angle. Consistent with the current understanding of wetting dynamics, film recession is faster on the more poorly wetted surface to an extent that cannot be explained solely by the increase in the surface tension driving force. In addition, the rates of recession of the thinnest films are found to increase with decreasing film thickness. These new results imply not only that the mobility of the liquid molecules adjacent to the solid increases with decreasing solid-liquid interactions, but also that the mobility adjacent to the free surface of the film is higher than in the bulk, so that the effective viscosity of the film decreases with thickness.

Contact-line friction of liquid drops on self-assembled monolayers: chain-length effects.

The static and dynamic wetting properties of self-assembled alkanethiol monolayers of increasing chain length were studied. The molecular-kinetic theory of wetting was used to interpret the dynamic contact angle data and evaluate the contact-line friction on the microscopic scale. Although the surfaces had a similar static wettability, the coefficient of contact-line friction zeta0 increased linearly with alkyl chain length. This result supports the hypothesis of energy dissipation due to a local deformation of the nanometer-thick layer at the contact line.

Dynamics of imbibition into a pore with a heterogeneous surface.

We use large-scale molecular dynamics simulations to study the dynamics of liquid penetration into a cylindrical pore having a randomly heterogeneous surface comprising areas of differing wettability. Our results confirm that the equilibrium contact angle in the heterogeneous pore is well described by Cassie's law. As in the case of the uniform pore studied previously, the dynamics of penetration can be described by the Lucas-Washburn equation corrected to include the effect of a dynamic contact angle. The dissipation at the three-phase line, which gives rise to the dynamic contact angle, may be characterized in terms of a friction coefficient. Interestingly, the wetting-line friction on the heterogeneous surface also turns out to be a linear function of the fractional concentration of the areas of different wettability, analogous to Cassie's law. These results can be interpreted in terms of an independent random walk mechanism.

Spreading dynamics of chain-like monolayers: a molecular dynamics study.

Using large-scale molecular dynamics simulations, we have shown previously that the spreading dynamics of sessile drops on solid surfaces can be described in detail using the molecular-kinetic theory of dynamic wetting. Here we present our first steps in extending this approach to investigate the spreading dynamics of Langmuir-Blodgett monolayers. We make use of a monolayer model originally developed by Karaborni and Toxvaerd, but somewhat simplified to facilitate large-scale simulations. Our preliminary results are in good agreement with recent experimental observations and also support a molecular-kinetic interpretation in which the driving force for spreading is the lateral pressure in the monolayer. Away from equilibrium, initial spreading rates are constant and logarithmically dependent on pressure. However, near equilibrium, spreading is pseudo-diffusive and follows the square root of time. In both regimes the controlling factor is the equilibrium frequency of molecular displacements within the monolayer.

The possibility of different time scales in the dynamics of pore imbibition.

To model the imbibition of liquids into porous solids, use is often made of the Lucas-Washburn equation, which relates the distance of penetration of a liquid at a given time to the pore radius, the viscosity and surface tension of the liquid, and the effective contact angle between the liquid and the solid. In this paper, we extend previous large-scale molecular dynamics simulations to show how this tool can be used to study the details of liquid imbibition, including the impact of the contact angle on the dynamics of penetration and the evolution of the internal flow field. In particular, we show that the asymptotic behavior of the contact angle versus time for a completely wetting liquid is given by approximately t(-1/4).

Influence of the dynamic contact angle on the characterization of porous media.

It has been shown recently that the classical Lucas-Washburn equation, often used to model the dynamics of liquid penetration into porous media, should be modified to take account of the dynamic contact angle between the liquid and the pore. Here we show how neglect of this effect can lead to significant errors in estimation of the effective pore radius.