Disjoining Pressure Model Research Articles

Dewetting of a corner film wrapping a wall-mounted cylinder

In this study, we investigate the stability of a film that is attached to a corner between a cylinder and a substrate, using a combination of theoretical and numerical approaches. Notably, we place our focus on flat and thin films where the contact line is almost perpendicular to the cylinder wall whereas a small angle forms between the contact line and the substrate, and the film size is smaller than the cylinder radius. The film stability, which depends on the film size and the wall wettability, is first predicted by a standard linear stability analysis (LSA) within the long-wave theoretical framework. We find that the film size plays the most important role in controlling the film stability. Specifically, the thicker the film is, the less sensitive it becomes to the large-wavenumber perturbation. The wall wettability mainly impacts the growth rates of perturbations and slightly influences the marginal stability and postinstability patterns of wrapping films. We compare the LSA predictions with numerical results obtained from a disjoining pressure model (DPM) and volume-of-fluid (VOF) simulations, which provide more insights into the film breakup process. At the early stage there is a strong agreement between the LSA predictions and the DPM results. Notably, as the perturbation grows, thin film regions connecting two neighbouring satellite droplets form which may eventually lead to a stable or temporary secondary droplet, an aspect which the LSA is incapable of capturing. In addition, the VOF simulations suggest that beyond a critical film size, merging between two neighbouring drops becomes involved during the breakup stage. Therefore, the LSA predictions are able to provide only an upper limit on the final number of satellite droplets.

Equilibrium contact angles and dewetting in capillaries

In this work, we extend the model of contact angles that we have previously developed for sessile drops on a wetted surface to the case of a meniscus in a capillary. The underlying physics of our model describe the intermolecular forces between the fluid and the surface of the capillary that result in the formation of a thin, non-removable fluid layer that coats the capillary wall. We describe the shape of the meniscus using a Young–Laplace equation and an incompressible, two-phase, computational fluid dynamics (CFD) calculation, both modified to take into account intermolecular forces using the disjoining pressure model. We find that our numerical solutions of the Young–Laplace equation and equilibrium meniscus shapes obtained by CFD agree well with each other. Furthermore, for capillaries that are sufficiently larger than the thickness of the non-removable film, our numerical solutions agree well with the effective contact angle model that we previously developed for sessile drops. Finally, we observe that it is possible to tune the disjoining pressure model parameters so that the intermolecular forces between the liquid and solid molecules become so strong compared to the surface tension that our formula for effective contact angle gives an imaginary solution. We analyze this situation using CFD and find that it corresponds to dewetting, where the bulk liquid detaches from the walls of the capillary leaving behind the non-removable thin liquid film.

Equilibrium contact angle at the wetted substrate

We construct a novel model for the steady-state contact angles of liquid droplets at the wetted substrate. The non-removable, thin liquid film covering the substrate is governed by the intermolecular forces between molecules of liquid and solid, which we describe using the standard disjoining pressure approximation. Balancing the disjoining pressure against the surface tension, we find the smooth shape of the surface of the liquid. We show that we can extract an effective contact angle from the region where the film and the droplet meet. Crucially, we find that for large droplets the contact angle is independent of the droplet size. Instead, the contact angle is determined by the surface tension and the disjoining pressure parameters through a simple formula that works for both small and large contact angles. We suggest that comparing predictions of our model to experimentally measure contact angles will enable constraining the parameters of the disjoining pressure models.

Numerical modelization of contact angle hysteresis of falling droplet under enhanced lubrication approximation

Moving contact lines are involved in several engineering applications: in in-flight icing phenomenon, the eventual transition from droplet to rivulet or continuous film regime is crucial for the prediction of ice accretion over the aircraft surface; absorption process through structured packing is also characterized by a thin film flowing over the corrugated sheets. Disjoining pressure together with the assumption of a thin precursor film is largely used in numerical simulations of thin films and moving droplets in order to model the dynamics of moving contact lines and the surface wettability properties, in terms of imposed static contact angle. The disjoining pressure model was largely validated in case of falling films with the well known Voinov-Hoffman-Tanner law. On the other side, the capability of the disjoining pressure to model the contact angle hysteresis, which is a crucial parameter for predicting moving droplets behavior, has not been discussed yet. Here, numerical simulations of both falling films and moving droplets under lubrication approximation are conducted and the disjoining pressure model is used to predict the contact line dynamics. After verification of the full curvature implementation for a 1D falling film, the effective contact angle hysteresis is estimated for a moving droplet under different flow conditions and the transition from droplet to rivulet regime detected.

On the effect of structural forces on a condensing film profile near a fin-groove corner

Estimation of condenser performance of two-phase passive heat spreaders with grooved wick structures is crucial in the prediction of the overall performance of the heat spreader. Whilst the evaporation problem in micro-grooves has been widely studied, studies focusing on the condensation on fin-groove systems have been scarce. Condensation on fin-groove systems is actually a multi-scale problem. Thickness of the film near the fin-groove corner can decrease to nanoscale dimensions, which requires the inclusion of nanoscale effects into the modeling. While a few previous studies investigated the effect of dispersion forces, the effect of structural forces has never been considered in the thin film condensation modeling on fin-groove systems. The present study utilizes a disjoining pressure model which considers both dispersion and structural forces. The results reveal that structural forces are able to dominate dispersion forces in certain configurations. Consequently, by intensifying the disjoining pressure, structural forces lead to a sudden change of the film profile (slope break) for subcooling values which are relevant to engineering applications.

Numerical simulation of film instability over low wettability surfaces through lubrication theory

A computational study of thin liquid films over a solid surface is reported. The lubrication equation is numerically solved using an in-house code, which implements the finite volume method. Small slope approximation is abandoned, and a more accurate model for capillary pressure estimation is presented, allowing us to correctly investigate higher contact angles, when compared to the maximum value allowed by small slope approximation. Disjoining pressure is used for modeling substrate wettability. The in-house solver is first validated: a 1D flowing film driven by gravity is simulated and the disjoining pressure model is verified for contact angles up to 60°; replicating literature experimental investigations, a uniform film covering an inclined plate is perturbed, inducing the generation of a large dry patch; rivulet buildup is simulated; and the numerical results are compared with fully 3D computations found in the literature and verified with analytical evidences. Then, a film flowing over an inclined plate bounded by lateral walls, which is a complex configuration commonly used for studying liquid behavior in structured packing, is investigated and relevant parameters are reported.

Characterization of Porous Silica Materials with Water at Ambient Conditions. Calculating the Pore Size Distribution from the Excess Surface Work Disjoining Pressure Model

AbstractCharacterization of mesoporous materials today is routinely performed recording the adsorption isotherm of nitrogen at 77 K. The pore size distribution (PSD) can subsequently be determined using either thermodynamics or statistical mechanics based approaches. While nitrogen has evolved as the reference adsorptive, other gases are often used to extract more information from the isotherm. This contribution is focused on the calculation of a PSD from the water vapor isotherm. It is demonstrated that water adsorption provides accurate results comparable to the standard nitrogen adsorption.

Effect of concentration-dependent disjoining pressure on drainage process of vertical liquid film

For the drainage under the gravity of a vertical foam film containing insoluble surfactant, an improved concentration-dependent disjoining pressure model is formulated based on the published experimental results. The lubrication theory is used to establish the evolution equations of the film thickness, the surface concentration of insoluble surfactant, and the surface velocity, and the evolution characteristics of the film under different disjoining pressures are simulated numerically. The results show that the drainage process of a vertical liquid film generally undergoes two stages:the first stage is the thick film stage and the gravity plays a leading role in the drainage process; the subsequent stage is the thin film stage, the effects of capillary pressure and disjoining pressure increase gradually, and the disjoining pressure dominates the evolution of the film. The disjoining pressure effect is closely related to surfactant type and the correlation strength between the surfactant concentration and electrostatic repulsion force of disjoining pressure. For the ionic surfactant, electrostatic repulsion force increases with the increase of the surfactant concentration, but it is opposite for the nonionic surfactant. It is likely that the free hydroxide ions, which are considered to render the surface negatively charged, are partly adsorbed by the nonionic surfactant. So the surface charge of the foam film decreases as the concentration of the nonionic surfactant increases, resulting in a decrease in electrostatic repulsion. Therefore, some ionic surfactants can improve the stability of liquid film drainage and slow down the drainage process, while the effects of some nonionic surfactants are opposite. When the disjoining pressure is positively correlated with surfactant concentration, with the increase of correlation strength coefficient α, the thinning and drainaging processes of the film tend to slow down, hence the stability of the film is enhanced. When the disjoining pressure is negatively correlated with surfactant concentration, with the increase of the absolute value of α, the drainage process of the film is accelerated and the risk of film rupture is augmented. The results obtained in this paper are consistent with some of the experimental results, indicating that the concentration-dependent disjoining pressure is indeed an important factor in maintaining the stability of foam film containing some certain anionic or nonionic surfactants. The improved concentration-dependent disjoining pressure model established in this paper could not explain the phenomena of parts of cationic nor non-ionic surfactant film in drainage experiments. It can be inferred that the structure of surfactant molecule, the more detailed disjoining pressure model and the coupling of the disjoining pressure and surface elasticity should be considered in the future work.

Droplet on an elastic substrate: Finite Element Method coupled with lubrication approximation

Sessile liquid droplets deposited on soft substrates may cause substrate deformation. The substrate elasticity affects the apparent contact angle and the spreading dynamics of the sessile drop. In the present work, a model for description of a droplet on an elastic substrate is developed using the disjoining pressure concept. A Finite Element Method for the elasticity problem in a substrate is coupled with lubrication approximation for the modelling of droplet spreading. The disjoining pressure model allows an independent definition of contact angle and the thickness of the adsorbed layer. Simulations are performed to quantify the influence of mechanical properties, the substrate size and the parameters of the disjoining pressure model on the substrate deformation and apparent contact angle. It is shown that the dimensionless substrate deformation peak height scales with the dimensionless parameter σsinθ/(Ea), and the solid angle appearing in the deformed substrate near the apparent cotact line scales with σ/(Ea), where σ denotes the surface tension, E is the Young modulus, θ is the apparent contact angle and a is a width of the region in which the disjoining pressure plays an important role.

A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading

The motion of a contact line is examined, and comparisons drawn, for a variety of models proposed in the literature. Pressure and stress behaviours at the contact line are examined in the prototype system of quasistatic spreading of a thin two-dimensional droplet on a planar substrate. The models analysed include three disjoining pressure models based on van der Waals interactions, a model introduced for polar fluids, and a liquid–gas diffuse-interface model; Navier-slip and two non-linear slip models are investigated, with three microscopic contact angle boundary conditions imposed (two of these contact angle conditions having a contact line velocity dependence); and the interface formation model is also considered. In certain parameter regimes it is shown that all of the models predict the same quasistatic droplet spreading behaviour.

Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory

We examine the nanoscale behavior of an equilibrium three-phase contact line in the presence of long-ranged intermolecular forces by employing a statistical mechanics of fluids approach, namely, density functional theory (DFT) together with fundamental measure theory (FMT). This enables us to evaluate the predictive quality of effective Hamiltonian models in the vicinity of the contact line. In particular, we compare the results for mean field effective Hamiltonians with disjoining pressures defined through (i) the adsorption isotherm for a planar liquid film, and (ii) the normal force balance at the contact line. We find that the height profile obtained using (i) shows good agreement with the adsorption film thickness of the DFT-FMT equilibrium density profile in terms of maximal curvature and the behavior at large film heights. In contrast, we observe that while the height profile obtained by using (ii) satisfies basic sum rules, it shows little agreement with the adsorption film thickness of the DFT results. The results are verified for contact angles of 20°, 40°, and 60°.

Stability of a liquid ring on a substrate

AbstractWe study the stability of a viscous incompressible fluid ring on a partially wetting substrate within the framework of long-wave theory. We discuss the conditions under which a static equilibrium of the ring is possible in the presence of contact angle hysteresis. A linear stability analysis (LSA) of this equilibrium solution is carried out by using a slip model to account for the contact line divergence. The LSA provides specific predictions regarding the evolution of unstable modes. In order to describe the evolution of the ring for longer times, a quasi-static approximation is implemented. This approach assumes a quasi-static evolution and takes into account the concomitant variation of the instantaneous growth rates of the modes responsible for either collapse of the ring into a single central drop or breakup into a number of droplets along the ring periphery. We compare the results of the LSA and the quasi-static model approach with those obtained from nonlinear numerical simulations using a complementary disjoining pressure model. We find remarkably good agreement between the predictions of the two models regarding the expected number of drops forming during the breakup process.

Thin film evaporation in microchannels with slope- and curvature-dependent disjoining pressure

Physics of thin film evaporation plays a critical role in designing highly efficient micro-scale thermal management systems. In such systems, envelope of the evaporating thin film is an extended meniscus beyond the apparent contact line at a liquid/solid interface, which experiences strong intermolecular forces because of micro-scale interactions. Traditionally, such forces are represented by disjoining pressures that are postulated solely on the basis of the local film thickness, disregarding the local slope and curvature of the interfacial profile. In the present study, an improved model for evaporation in thin liquid films in microfluidic channels is developed, which explicitly takes into account the slope and curvature dependence of the disjoining pressure, under the assumption of a steady meniscus in existence with its own vapor. Such an improved formalism essentially prevents the contact line movement without slip, and allows an asymptotic merging of the evaporating interfacial profile to an infinitesimally thin non-evaporating adsorbed film. Implications of this rigorous disjoining pressure model on the thermo-physics of thin film evaporation in a kinetically controlled limit are investigated in details, and contrasting confluences as compared to the traditional descriptions of the disjoining pressure are emphatically pinpointed.

New boundary conditions for the evaporating thin-film model in a rectangular micro channel

The augmented Young–Laplace equation has been validated for modeling evaporating thin-films in the literature. However, physical conflicts still exist when it is used with the traditional rules to model the entire meniscus. New boundary conditions are developed for this model. The new conditions ensure that the center of the intrinsic meniscus is on the axis of the micro channel. The calculations indicate that the meniscus radius of the intrinsic region increases with the superheating of the heater wall. This implies a reduction of the capillary suction and a reduction of the critical heat flux for the whole micro channel. A polar disjoining pressure model is used and compared with the non-polar model. A narrow, but high heat flux, sub-region is also found at the thinner end of the intrinsic region. By considering the new boundary conditions and this sub-region, the thin-film region can be integrated smoothly with the bulk liquid region.

Evaporation of Ultra-thin Liquid Films into Air

We develop a mathematical model of evaporation of a thin liquid film into air under the action of disjoining pressure. The rate of evaporation is found from the numerical solution of the diffusion equation with specified local vapor concentration near the film surface, assumed equal to its equilibrium value. Evolution of the film is studied for two commonly used disjoining pressure models using a simplified formulation which neglects thermal gradients present in the system. The results are then compared to numerical simulations of coupled systems of equations describing heat conduction in the liquid and solid phases and diffusion of vapor through air. Conditions are formulated at which disjoining pressure suppresses evaporation.

Static and dynamic contact angles of evaporating liquids on heated surfaces

We studied both static and dynamic values of the apparent contact angle for gravity-driven flow of a volatile liquid down a heated inclined plane. The apparent contact line is modeled as the transition region between the macroscopic film and ultra-thin adsorbed film dominated by disjoining pressure effects. Four commonly used disjoining pressure models are investigated. The static contact angle is shown to increase with heater temperature, in qualitative agreement with experimental observations. The angle is less sensitive to the details of the disjoining pressure curves than in the isothermal regime. A generalization of the classical Frumkin–Derjaguin theory is proposed to explain this observation. The dynamic contact angle follows the Tanner’s law remarkably well over a range of evaporation conditions. However, deviations from the predictions based on the Tanner’s law are found when interface shape changes rapidly in response to rapid changes of the heater temperature. The Marangoni stresses are shown to result in increase of the values of apparent contact angles.

Disjoining Pressure Effects in Ultra-Thin Liquid Films in Micropassages—Comparison of Thermodynamic Theory With Predictions of Molecular Dynamics Simulations

The concept of disjoining pressure, developed from thermodynamic and hydrodynamic analysis, has been widely used as a means of modeling the liquid-solid molecular force interactions in an ultra-thin liquid film on a solid surface. In particular, this approach has been extensively used in models of thin film transport in passages in micro evaporators and micro heat pipes. In this investigation, hybrid μPT molecular dynamics (MD) simulations were used to predict the pressure field and film thermophysics for an argon film on a metal surface. The results of the simulations are compared with predictions of the classic thermodynamic disjoining pressure model and the Born-Green-Yvon (BGY) equation. The thermodynamic model provides only a prediction of the relation between vapor pressure and film thickness for a specified temperature. The MD simulations provide a detailed prediction of the density and pressure variation in the liquid film, as well as a prediction of the variation of the equilibrium vapor pressure variation with temperature and film thickness. Comparisons indicate that the predicted variations of vapor pressure with thickness for the three models are in close agreement. In addition, the density profile layering predicted by the MD simulations is in qualitative agreement with BGY results, however the exact density profile is dependent upon simulation parameters. Furthermore, the disjoining pressure effect predicted by MD simulations is strongly influenced by the allowable propagation time of injected molecules through the vapor region in the simulation domain. A modified thermodynamic model is developed that suggests that presence of a wall-affected layer tends to enhance the reduction of the equilibrium vapor pressure. However, the MD simulation results imply that presence of a wall layer has little effect on the vapor pressure. Implications of the MD simulation predictions for thin film transport in micro evaporators and heat pipes are also discussed.

Effects of the liquid polarity and the wall slip on the heat and mass transport characteristics of the micro-scale evaporating transition film

A mathematical model is developed to describe the micro-/nano-scale fluid flow and heat/mass transfer phenomena in an evaporating extended meniscus, focusing on the transition film region under non-isothermal interfacial conditions. The model incorporates polarity contributions to the working fluid field, a slip boundary condition on the solid wall, and thermocapillary stresses at the liquid–vapor interface. Two different disjoining pressure models, one polar and one non-polar, are considered for water as the working fluid so that the effect of polar interactions between the working fluid and solid surface can be exclusively examined on heat and mass transfer from the thin film. The polar effect is examined for the thin film established in a 20-μm diameter capillary pore. The effect of the slip boundary condition is separately examined for the thin film developed in a two-dimensional 20-μm slotted pore. The analytical results show that for a polar liquid, the transition region of the evaporating meniscus is longer than that of a non-polar liquid. In addition, the strong polar attraction with the solid wall acts to lower the evaporative heat transfer flux. The slip boundary condition, on the other hand, increases evaporative heat and mass flux and lowers the liquid pressure gradients and viscous drag at the wall. The slip effect shows a more pronounced enhancement as superheat increases. Another thing to note is that the slip effect of elongating the transition region can counteract the thermocapillary action of reducing the region and a potential delay of thermocapillary driven instability onset may be anticipated.

Fingering phenomena for driven coating films

A theoretical and numerical model is formulated to describe the instability and the long-time evolution of both gravity-driven and surface-shear-stress-driven thin coating films. A single evolution equation, of higher-order diffusive type, models the flow for either problem. It is derived using the lubrication approximation. For partially wetting systems, the effect of finite contact angle is incorporated in the equation using a particular disjoining pressure model. The base state, in each case, is a two-dimensional steadily propagating capillary front. Slight perturbations of the base state, applied along the front, initiate the fingering instability. Early-time results accurately reproduce the wavelengths of fastest growth and the corresponding eigenmodes as reported in published linear stability analyses. As time proceeds, depending on parameter values, various fingering patterns arise. For conditions of perfect wetting with the substrate downstream of the moving front covered with a thin precursor layer, predicted nonlinear finger evolution agrees well with published experiments. The ultimate pattern, in this case, is a steadily translating pattern of wedge-shaped fingers. Alternatively, for partially wetting systems that exhibit sufficiently large static contact angles, long straight-sided fingers or rivulets are formed. Finally, for larger contact angles, or at relatively low speeds, we predict that the flowing rivulets will become unstable and break up into strings of isolated droplets.