Kinetic Effects in Dynamic Wetting

The maximum speed at which a liquid can wet a solid is limited by the need to displace gas lubrication films in front of the moving contact line. The characteristic height of these films is often comparable to the mean free path in the gas so that hydrodynamic models do not adequately describe the flow physics. This Letter develops a model which incorporates kinetic effects in the gas, via the Boltzmann equation, and can predict experimentally-observed increases in the maximum speed of wetting when (a) the liquid's viscosity is varied, (b) the ambient gas pressure is reduced or (c) the meniscus is confined.

The maximum speed at which a liquid can wet a solid is limited by the need to displace gas lubrication films in front of the moving contact line. The characteristic height of these films is often comparable to the mean free path in the gas so that hydrodynamic models do not adequately describe the flow physics. This Letter develops a model which incorporates kinetic effects in the gas, via the Boltzmann equation, and can predict experimentally-observed increases in the maximum speed of wetting when (a) the liquid's viscosity is varied, (b) the ambient gas pressure is reduced or (c) the meniscus is confined.
Understanding the physical mechanisms determining the maximum speed U max at which a liquid-gas freesurface can wet a solid substrate is a fundamental problem with applications to a range of natural and technological phenomena. For example, if drops of rain (or pesticide) spreading across plant leaves exceed U max then a splash is generated which reduces the retention of liquid [1] whilst coating processes must operate below U max in order to avoid product-destroying gas entrainment [2].
Despite the small gas-to-liquid density and viscosity ratios (ρ g /ρ l ≈ 10 −3 and µ g /µ l ≈ 10 −2 for air-water), the importance of gas dynamics has been established both in coating flows and in impact events, for the collisions of solid bodies with liquids [3] and liquid drops with solids. In particular, recent activity in liquid drop impact has been aimed at understanding the gas' role using novel experimental techniques, see [4] and references therein. Whilst a full characterisation of drop splashing remains an open problem, recent experiments in [5] highlight the critical role of gas films durin the wetting phase.
The gas' dynamics become relevant when thin films are formed that generate lubrication effects, with experimental observations in both coating [6] and drop impact [7] showing that the height h of these films is in the range ≈ 1 − 10µm as U max is approached [8]. At atmospheric pressure P atm (atm will denote atmospheric values), the mean free path in the gas is ≈ 0.1µm, so that the Knudsen number Kn = /h ≈ 0.01 − 0.1. Consequently, it has been suggested [6,9] that kinetic effects in the gas should be built into models for moving contact line phenomena.
Recent models [6,9] account for kinetic effects by allowing for 'slip', i.e. a jump in the velocity tangential to the boundary, at the gas-liquid and gas-solid interfaces, with a slip length proportional to [10] and the usual equations of hydrodynamics remaining in the bulk. These models can qualitatively explain experimental observations in coating [11] and drop impact [12], that reductions in the ambient gas pressure P can suppress gas entrainment and splashing [13]: as = atm P atm /P increases with reduced P , slip is enhanced and gas is more easily removed from the thin film.
Research in kinetic theory has established that these 'first-order' slip models are only accurate for Kn 0.1.
Technically, they can be derived from the Boltzmann equation for small Kn [14]. Physically, they represent the case where the non-hydrodynamic effects are confined to a boundary layer of width ≈ , the so-called Knudsen layer, which is small relative to the channel height ( h) so that this additional physics can be incorporated into boundary conditions. These models are on the edge of their applicability for dynamic wetting at atmospheric pressure, where Kn ≈ 0.01 − 0.1, so that when P is reduced they will be outside their limits of validity. This has been confirmed in [9] where it has been shown that the situation is even more severe, as decreases in P also lead to reductions in h, so that Kn can easily exceed unity in experimentally-realisable conditions.
In this Letter, methods originally developed to predict rarefied lubrication flows in MEMS [15] are used to derive a dynamic wetting model which is valid for all Kn. As demanded by the physics, this model combines kinetic theory in the gas film described by the Boltzmann equation with hydrodynamics in the liquid phase governed by the Navier Stokes equations.
Flow configuration. -The steady dynamic wetting geometry in Figure 1 allows us to consider both a coating flow, where a solid is continuously driven through a liquid bath whose free-surface is flattened by gravity, as well as the steady propagation of a meniscus confined to a microchannel of width 2L. These cases correspond, respectively, to L L σ = σ/(ρ l g) and L L σ , where L σ is the capillary length with σ the liquid-gas surface tension and g the acceleration due to gravity.
The liquid's flow is described by the steady incompressible Navier-Stokes equations. At the liquid-solid interface conditions of impermeability and Navier-slip are used, which circumvent the moving contact line problem, choosing a fixed slip length of l s = 10 nm which is well within the range of experimentally observed values. At the liquid-gas free-surface, the kinematic condition is combined with a balance of stress and capillarity. The contact angle at which the free-surface meets the solid is assumed to be a constant θ e . By construction, the simplest possible dynamic wetting model (formulated mathematically in the Supplementary Material) has been chosen to allow us to focus attention on the dynamics of the gas without additional parameters coming into play. Having established the importance of the gas, more complex models for the wetting process, such as dynamic contact angles, reviewed in [16], can be built on top of this basic model.
As the gas flow is only strong enough to effect the liquid when it is thin, the lubrication equations can be used to describe its dynamics, see [17]. Simulations comparing results from this formulation to full computations of the gas phase confirm the accuracy of this approach (Supplementary Material) and indicate that the gas only influences the liquid through the pressure term in the normal stress boundary condition; one may expect for µ g /µ l 1 the gas' contribution to the tangential stress condition is negligible compared to the liquid's. Consider then, in the lubrication framework, three different models for the gas phase: • No slip: conventional model, with a non-zero slip length (fixed at 10 nm) at the solid boundary only to circumvent the moving contact line problem. • Slip: current state of the art, with slip at the gassolid and gas-liquid boundaries proportional to . • Boltzmann: the model developed in this Letter, with the gas phase described by kinetic theory. Thin film gas dynamics. -As the process is steady, a pressure-driven Poiseuille flow forms to remove the gas dragged into the the contact line region by a boundarydriven Couette flow, caused by the motion of the solid moving at constant speed U and the liquid at U f s (x) tangential to the free surface ( Figure 1). Such arguments are routine in hydrodynamics, formalised in the Reynolds equation, but more recently have been generalised for the Boltzmann equation [15,18]. There, it is possible to identify Poiseuille and Couette flow components, but the Boltzmann equation must be solved to evaluate the respective contributions to the mass flux. Assuming diffuse reflection of molecules from each boundary, which is a sensible starting point, due to symmetry the mass flux from the Couette flow m C remains the same for all models whilst the plane Poiseuille flow contribution m P is model-dependent where Q(Kn) are the so-called 'flow coefficients' [19] that depend on the model used ( Figure 2a) and p is the local pressure. For this problem it is reasonable to assume incompressible flow (see Figure 4b for confirmation), although the extension to compressible flow is not difficult, see [20]. Notably, only the Boltzmann equation captures the famous 'Knudsen minimum' (Figure 2a) in the mass flux of gas through a channel of fixed h (so that ρgh 2 √ πµg is a constant) which is driven by a constant pressure gradient.
Conservation of mass (m C = −m P ) then gives where it is noted that h, U f s and Kn all vary along the film. The form of (2) suggests that an effective viscosity could be defined as µ eff g = rµ g , as considered in [6,11], in order to absorb kinetic corrections into a hydrodynamic framework, and this idea has recently been pursued in pioneering drop collision simulations [21]. Notably, at Kn = 0.1, 1, 10 it is found that µ eff g /µ g = 0.57, 0.096, 0.0071, showing the rapid drop in resistance as the film height is reduced. Although this gives us a picture of the role that r plays, it can confuse matters, as pointed out in [22], as µ eff g is problem specific and dependent on the ambient pressure, whilst µ g is not. Therefore, r(Kn) is retained in the formulation.
For Kn 1, the Knudsen layer is small relative to the channel height and the results of the Boltzmann equation are equivalent to using the Navier Stokes equations with Navier slip at the boundary accounting for the non-hydrodynamic effects. For the slip model r = (1 + 6αKn) −1 , where α is the slip length and α ≈ 1 is a parameter which depends on both the collision model in the Boltzmann equation and the accommodation coefficient of the surface [22]. For Kn = 0, r = 1 [23].
Methods for obtaining r(Kn), or more typically Q(Kn), from the Boltzmann equation are reviewed in [19] and often involve simplifications from a BGK approximation and/or linearisation. Experimental data is well captured by most variants and so here the simplest possible approach of linearised BGK (where α = 1.15 [24]) is used. To solve this model a variational method proposed in [25] is implemented, which is shown in [19] to be the simplest method for accurately approximating Q (giving results within 2% of the exact Boltzmann solution). This leads to the curves for Q, r labelled Boltzmann in Figure 2. Whilst the slip model appears satisfactory at first glance of Figure 2b, the inset shows that the relative error of r from the Boltzmann solution becomes unacceptable for Kn 0.1.
Simulations. -The problem is solved using a multiscale finite element framework developed in [26], and first applied to gas entrainment phenomena in [9], where it was benchmarked with a similar code [27]. As there are length scales of nanometres (l s ), micrometres ( ) and millimetres (L σ ) in the problem, the computational mesh, based on an arbitrary Lagrangian Eulerian description, has to be specially designed to capture all of the physical effects. The main output from this code is the maximum speed of wetting U max , past which no steady two-dimensional solutions exist [9] and gas entrainment is expected to occur. As noted in [28], predictions of flow transitions are ideal candidates for comparing models for moving contact line phenomena, as they are easily observed experimentally, in contrast to measurements of the contact angle.
The extension of this code to allow for a thin film description of the gas flow is relatively straightforward, and as suggested in [29], and developed in [17], it is assumed that dx ≈ ds (Figure 1) in order to circumvent regions where the thin film approximation is not strictly valid. Benchmark simulations in the Supplementary Material show the scheme is highly accurate, giving values for U max that are indistinguishable from those obtained when solving the full problem in the gas. Values for r(Kn) obtained from the Boltzmann equation could either be calculated 'on the fly', i.e. when required by the code (a 'concurrent' approach), or before the code is run (a 'sequential' method). For simplicity, the sequential method is chosen and the Supplementary Material provides the code used to generate r(Kn).
Atmospheric pressure. -Standard dip coating experiments measure the air ( atm = 70 nm, µ g = 18µPa s) entrainment speed U max for different liquids on a range of solid substrates. In Figure 3, this data is shown for water-glycerol solutions where σ = 65 mN m −1 is approximately constant, so that the effect of varying the liquid's viscosity (dimensionlessly µ g /µ l ) can be isolated. Despite no attempt to fit the data (θ e = 90 • is fixed), the theoretical predictions are in good agreement with the experiments and support the validity of the approach.
Remarkably, for µ g /µ l < 10 −5 kinetic effects become prominent at atmospheric pressure, as the gas film's height shrinks becomes comparable to atm . This creates a dependence on µ g /µ l which diverges from the no-slip model, with the slow logarithmic increase of µ l U max /σ blown away by a rapid power-law-type increase. Interestingly, this is supported by experiments in [30] that for high viscosity liquids U max → 0.1 ms −1 , corresponding to µ l U max /σ → (µ g /µ l ) −1 , and similar scalings are in [6]. Clearly, further experimental analysis at µ g /µ l < 10 −5 is required to properly elucidate the new trends.
Reduced ambient pressure. -Consider dip coating experiments performed in [11] (see their Figure 9) with silicone oil (viscosity µ l = 112 mPa s, density ρ l = 985 kg m −3 , surface tension σ = 17.9 mN m −1 , equilibrium contact angle θ e = 19.5 • ) as the coating liquid and helium as the ambient gas, inside a pressure-controlled chamber. Helium's mean free path atm = 190 nm at atmospheric pressure P atm = 10 5 Pa is three times larger than that of air, so kinetic effects will be enhanced, whilst its viscosity is similar µ g = 19µPa s.
In Figure 4, the computational results are compared to experimental data. The main results are that (a) the no-slip model completely misses the qualitative trend of U max as the pressure is reduced, (b) the Boltzmann model diverges from the slip model once the ambient pressure has been reduced by a factor of ten, and (c) the Boltzmann model appears to slightly better describe the experimental data. Such agreement between theory and experiment is remarkably good, when allowing for the simple dynamic wetting model implemented and the fact there are no parameters to fit. However, the key message is that kinetic effects play a role in moving contact line phenomena in experimentally-accessible regimes. Typical profiles in Figure 4b show how the model chosen changes the pressure distribution in the film. It is clear that no-slip drastically over-predicts the pressure, peaking at p = 0.07P whilst the slip model's prediction is just 4% of this value. The Boltzmann model gives further substantial reductions. Variations in pressure along the film are consistent with gas incompressibility.
Confined menisci. -By considering the effect of U max on flow dimension L (Supplementary Material), full kinetic effects are also shown to be critical for 'microfluidic flow', such as for a meniscus confined to a microchannel.
Physical mechanisms. -In [31], careful simulations identified that entrainment occurs when the capillary forces at the free surface can no longer sustain the pressure gradients required to pump gas away (via a Poiseuille flow) from the contact line region. Global balances of capillary and viscous forces have also been used in unsteady processes to predict splashing [3,32] and microdrop emission [33], where exceeding U max results in the contact line being left behind the advancing liquid front [34].
Interestingly, computations here, and in [31], show that the free surface shape is relatively insensitive to the gas dynamics. Its shape is determined by the balance of viscous forces in the liquid with capillary forces at the interface. Wettability then enters the model as a boundary condition for the free surface shape, as does confinement when the channel is sufficiently narrow. Therefore, given this profile we can compare the pressure build up in the film for the different models of the gas.
Isolating the Poiseuille flow component, equation (1) shows that the pressure gradient required to drive a given mass flux m P is inversely proportional to Q(Kn), so that pressure increases will be least for the Boltzmann model, where Q(Kn) is largest (Figure 2a), as confirmed by Figure 4b. Physically, the increased Q predicted by the Boltzmann model occurs for Kn > 0.1 as nonequilibrium effects not only cause slip at the wall, but also drive a non-Newtonian bulk flow. As expected, from Figures 3 and 4a, it is the models with the smallest increases in pressure, i.e. the largest Q(Kn), which predict the largest U max .
Encouragingly, it appears that the free surface shape could be calculated independently of the gas phase and used to infer the maximum pressure gradients which the free surface can sustain. This is where the dependencies on the capillary number, wettability and confinement would enter the model. With this information U max could be calculated from lubrication theory for the gas phase, where kinetic effects would enter. However, as no simple method currently exists for characterising the required free surface profiles, a less computationally intensive model based on these ideas remains an open problem.
Discussion. -Simulations have identified situations where conventional approaches fail to predict U max due to an inadequate description of the flow physics in the gas film. Whilst the slip model captures the qualitative behaviour, the Boltzmann equation is required for quantitative predictions and thus deserves further attention. Incorporating this new physics into existing codes is relatively simple and could play a role in a wide range of free surface flows where gas microfilms appear such as in the collisions of liquid drops [35]; the formation of tip-singularities in free-surfaces [36]; the stability of nanobubbles on solids [37]; the impact of projectiles on liquid surfaces [38]; and the creation of anti-bubbles from air films [39]. Furthermore, these findings motivative new directions of research, such as (a) understanding how gas molecules interact with moving liquid-gas free surfaces, from experiments or molecular dynamics simula-tions, and (b) developing non-lubrication formulations of the gas flow, such as moment methods approaches [40].
The author thanks Duncan Lockerby and Alex Patronis for useful discussions and the reviewers for their constructive criticism. This work was supported by the Leverhulme Trust (Research Project Grant) and EPSRC (grant EP/N016602/1).