With aim of finding mathematical criteria for contact line depinning from sharp corners, we have studied the equilibrium and stability of a semi-infinite planar liquid layer pinned at the vertex of a wedge. Equilibrium is compatible with a fan of apparent contact angles $θ_0$ bracketed by the equilibrium contact angles of both flanks of the wedge, so the contact line could remain pinned if $θ_0$ is within this fan. However, the analysis of the stability of these solutions, studied exploiting the variational structure of the problem through turning-point arguments, shows that the equilibrium becomes unstable at critical depinning advancing $θ_0^a$ and receding $θ_0^r$ contact angles, which are found as subcritical saddle-node bifurcations. Equilibrium is thus possible (stable) within the interval $θ_0^a < θ_0 <θ_0^a$ but the contact line depins from the vertex beyond these critical angles if they occur within the equilibrium fan.
We propose an equation that describes the shape of the driven contact line in dynamics in the presence of an arbitrary (possibly random) distribution of the surface defects. It is shown that the triple contact line depinning differs from the depinning of interfaces separating two phases; the equations describing these phenomena have an essential difference. The force--velocity dependence is considered for a periodical defect pattern. It appears to be strongly nonlinear both near the depinning threshold and for large contact line speeds. This nonlinearity is comparable to experimental results o…
In this paper, the pinning and depinning mechanism of the contact line during droplet evaporation on chemically stripe-patterned surfaces is numerically investigated using a thermal multiphase lattice Boltzmann (LB) model with liquid-vapor phase change. A local force balance in the context of diffuse interfaces is introduced to explain the equilibrium states of droplets on chemically patterned surfaces. It is shown that, when the contact line is pinned on a hydrophobic-hydrophilic boundary, different contact angles can be interpreted as the variation of the length of the contact line occupied …
The influence of solidification on the spreading of liquids is addressed in the situation of an advancing liquid wedge on a cold substrate at $T_p < T_f$, of infinite thermal conductivity, where $T_f$ is the melting temperature. We propose a model derived from lubrication theory of contact-line dynamics, where an equilibrium between capillary pressure and viscous stress is at play, adapted here for the geometry of a quadruple line where the vapour, liquid, solidified liquid and basal substrate meet. The Stefan thermal problem is solved in an intermediate region between molecular and mesoscopic…
When a film of a liquid suspension of nanoparticles or a polymer solution is deposited on a surface, it may dewet from the surface and as the solvent evaporates the solute particles/polymer can be deposited on the surface in regular line patterns. In this paper we explore a hydrodynamic model for the process that is based on a long-wave approximation that predicts the deposition of irregular and regular line patterns. This is due to a self-organised pinning-depinning cycle that resembles a stick-slip motion of the contact line. We present a detailed analysis of how the line pattern properties …
We present a fluid dynamics video showing the adhesion of a drop to a superhydrophobic surface. We use environmental scanning electron microscopy to observe depinning events at the microscale. As the drop moves along the surface, the advancing portion of the contact line simply lies down onto the upcoming roughness features, contributing negligibly to adhesion. After measuring the local receding contact angle of capillary bridges formed on a micropillar array, we find that these depinning events follow the Gibbs depinning criterion. We further extend this technique to two-scale hierarchical st…
We describe the formation of deposition patterns that are observed in many different experiments where a three-phase contact line of a volatile nanoparticle suspension or polymer solution recedes. A dynamical model based on a long-wave approximation predicts the deposition of irregular and regular line patterns due to self-organised pinning-depinning cycles corresponding to a stick-slip motion of the contact line. We analyze how the line pattern properties depend on the evaporation rate and solute concentration.
A molecular-dynamics type simulation method, which is suitable for investigating the dewetting dynamics of thin and viscous liquid layers, is discussed. The efficiency of the method is exemplified by studying a two-parameter depinning-like model defined on inhomogeneous solid surfaces. The morphology and the statistical properties of the contact line is mapped in the relevant parameter space, and as a result critical behavior in the vicinity of the depinning transition is revealed. The model allows for the tearing of the layer, which leads to a new propagation regime resulting in non-trivial c…
The so-called coffee stain effect has been intensively studied over the past decades, but most of the studies are focused on sessile droplets. In this paper, we analyse the origin of the difference between the deposition of suspended particles in a sessile drop and in an axisymmetric drop deposited on a fibre. First, we model the shape of a drop on a fibre and its evaporative flux with some approximations to derive analytical calculations. Then, for pinned contact lines, we solve the hydrodynamics equations in the liquid phase under the lubrication approximation to determine the flow velocity …
We develop a model which predicts the contact angle hysteresis introduced by smooth micropatterned defects. The defects are modeled by a smooth function and the contact angle hysteresis is explained using a tangent line solution. When the liquid micro-meniscus touches both sides of the defect simultaneously, depinning of the contact line occurs. The defects are fabricated using a photoresist and experimental results confirm the model. An important point is that the model is scale-independent, i.e. the contact angle hysteresis is dependent on the aspect ratio of the function, not on its absolut…
During the spreading of a liquid over a solid substrate, the contact line can stay pinned at sharp edges until the contact angle exceeds a critical value. At (or sufficiently near) equilibrium, this is known as Gibbs' criterion. Here, we show both experimentally and theoretically that for completely wetting volatile liquids there also exists a dynamically-produced critical angle for depinning, which increases with the evaporation rate. This suggests that one may introduce a simple modification of the Gibbs' criterion for (de)pinning, that accounts for the non-equilibrium effect of evaporation.
We reconcile two scaling laws that have been proposed in the literature for the slip length associated with a moving contact line in diffuse interface models, by demonstrating each to apply in a different regime of the ratio of the microscopic interfacial width $l$ and the macroscopic diffusive length $l_D= (Mη)^{1/2}$, where $η$ is the fluid viscosity and $M$ the mobility governing intermolecular diffusion. For small $l_D/l$ we find a diffuse interface regime in which the slip length scales as $ξ\sim(l_Dl)^{1/2}$. For larger $l_D/l>1$ we find a sharp interface regime in which the slip length …
The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact line problem are studied by asymptotic analysis and numerical simulations. The effects of the {mobility} number as well as a phenomenological relaxation parameter in the boundary condition are considered. In asymptotic analysis, we focus on the case that the {mobility} number is the same order of the Cahn number and derive the sharp-interface limits for several setups of the boundary relaxation parameter. It is shown that the sharp interface limit of the phase field model is t…
For super-heated water on a substrate with hydrophobic patches immersed in a hydrophilic matrix, one can choose the temperature so that micro-bubbles will form, grow and merge on the hydrophobic patches and not on the hydrophilic matrix. Until covering a patch, making a pinned macro-bubble, a bubble has a contact angle $π-θ_2$, where $θ_2$ is the receding contact angle of water on the patch material. This pinned macro-bubble serves as the initial condition of a quasi-static growth process, à la Landau, leading to detachment through the formation of a neck, so long as depinning and dewetting of…
We investigate velocity variations inside of and surrounding a gravity driven drop impacting on and moving through a confining orifice, wherein the effects of edge geometry (round- vs. sharp-edged) and surface wettability (hydrophobic vs. hydrophilic) of the orifice are considered. Using refractive index matching and time-resolved PIV, we quantify the redistribution of energy in the drop and the surrounding fluid during the drop's impact and motion through a round-edged orifice. The measurements show the importance of a) drop kinetic energy transferred to and dissipated within the surrounding …
A solid-liquid-gas moving contact line is considered through a diffuse-interface model with the classical boundary condition of no-slip at the solid surface. Examination of the asymptotic behaviour as the contact line is approached shows that the relaxation of the classical model of a sharp liquid-gas interface, whilst retaining the no-slip condition, resolves the stress and pressure singularities associated with the moving contact line problem while the fluid velocity is well defined (not multi-valued). The moving contact line behaviour is analysed for a general problem relevant for any densi…
We developed a sharp interface level-set approach for two-phase immiscible flow with moving contact lines. The Cox-Voinov model is used to describe the moving contact line. A piecewise linear interface method is used to construct the signed distance function and to implement the contact angle boundary condition. Spurious currents are studied in the case of static and moving fluid interfaces, which show convergence behavior. Pressure and the surface tension force are balanced up to machine precision for static parabolic interfaces, while the velocity error decreases steadily with grid refinemen…
The behaviour of a solid-liquid-gas system near the three-phase contact line is considered using a diffuse-interface model with no-slip at the solid and where the fluid phase is specified by a continuous density field. Relaxation of the classical approach of a sharp liquid-gas interface and careful examination of the asymptotic behaviour as the contact line is approached is shown to resolve the stress and pressure singularities associated with the moving contact line problem. Various features of the model are scrutinised, alongside extensions to incorporate slip, finite-time relaxation of the …
