Detailed modelling of contact line motion in oscillatory wetting.

The experimental results of Xia and Steen for the contact line dynamics of a drop placed on a vertically oscillating surface are analyzed by numerical phase field simulations. The concept of contact line mobility or friction is discussed, and an angle-dependent model is formulated. The results of numerical simulations based on this model are compared to the detailed experimental results of Xia and Steen with good general agreement. The total energy input in terms of work done by the oscillating support, and the dissipation at the contact line, are calculated from the simulated results. It is found that the contact line dissipation is almost entirely responsible for the dissipation that sets the amplitude of the response. It is argued that angle-dependent line friction may be a fruitful interpretation of the relations between contact line speed and dynamic contact angle that are often used in practical computational fluid dynamics.

Introduction

A liquid spreading over a dry surface is a phenomenon that is crucial to many natural processes and important in technology. However, the detailed description and understanding of dynamic wetting is still a complex and challenging problem[1,2]. The fact that the continuum equations of fluid mechanics exhibit a non-integrable singularity[3] of the viscous stress at the contact line (CL) shows that the detailed microscopic and nanoscopic features of the liquid and the surface will be important for the macroscopic flow. This introduces a host of different processes and phenomena that need to be understood to predict and control wetting processes.

In technology, in addition to such examples as spray painting, coating, etc., one particularly important field is microfluidics[4]. A common challenge is to handle small volumes of liquid, often in the form of small droplets, and one means for achieving this is to use wetting phenomena. Another area where surface tension and wetting become dominant is microgravity[5]. In the absence of gravity, surface tension becomes dominant, and wetting will be important in any fluid handling, from liquid fuel to many daily activities and needs of the astronauts.

Dynamic wetting driven by vibration is both of practical importance and a convenient way to study the phenomenon. The dynamic wetting on a glass plate dipping into a tank and oscillated vertically was studied in ref. [6], and damping of surface waves in a rectangular tank was investigated in ref. [7], revealing complex dependencies of damping rates on oscillation amplitude. The dynamics of a sessile droplet on a vertically oscillating surface will be sensitive to the detailed conditions at the CL, such as the presence of hysteresis or CL dissipation[8-10]. Xia and Steen[9] made careful experiments using droplets on a polydimethylsiloxane (PDMS)-covered substrate, which was oscillated vertically at frequencies near drop resonance. The resulting dynamics was examined through phase plots of the dynamic contact angle, the CL position, and the CL speed. In particular, Xia and Steen used this information to measure the CL mobility.

On earth, the size of droplets that can be used is limited to a radius in the order of millimeters, but in microgravity, a larger parameter space can be investigated. In microgravity a droplet size in the order of centimeters can be used instead, which will be advantageous in several respects; one is that spatial dimensions are larger, and the resonance frequency much lower, allowing for higher both spatial and temporal resolution. The larger droplet size also implies that the droplet and CL dynamics are even more dominated by inertia than in a millimeter-sized droplet on earth. It was the intention of Paul Steen to perform such experiments[11], and these have now been carried out.

From a strictly thermodynamic point of view, a moving CL should be associated with energetic losses of some kind[12]. The idea of a localized dissipation at the CL has been invoked in different ways over the years. Following Hocking[13], Xia and Steen introduce the concept of a CL mobility M as a phenomenological parameter that relates the deviation of the dynamic contact angle from equilibrium θ − θe to the CL speed UCL,see also refs. [8,13]. In computational fluid dynamics, more elaborate phenomenological relations between contact angle and CL speed have been devised[14], which take the form .

In Molecular Kinetic Theory (MKT)[15,16], dynamic wetting is described as an activated process on the molecular scale, and the line friction ζ is given a phenomenological interpretation on the molecular scale. In its simplest linearized form, this can be written aswhere γ is the surface tension and ζ is the coefficient of wetting-line friction, which in MKT is estimated in terms of molecular quantities and thermal fluctuations.

In the phase field method, the fluid is viewed as a mixture of two immiscible species. The governing equations are derived from the thermodynamic potentials of such a system to yield typically the Cahn–Hilliard equations[17,18]. The interface now becomes a diffuse region separating the two species, which has a definite width ε. The line friction appears as an energy dissipation associated with the CL displacement. Yue and Feng[18] derive the resulting equivalent condition relating CL speed and dynamic contact angle ashere Γ is introduced as a rate parameter in the relaxation of the dynamic contact angle boundary condition.

It is noted that the parameters in Eqs. (1)–(3) in the approximation of can be identified by setting

In the last equality, we have introduced the line friction μ, which is essentially the same as the coefficient of wetting-line friction used in ref. [16], except that it also absorbs the factor sinθe. We note that ζ and μ have the dimensions of viscosity and that γ/μ is a velocity.

In many classical treatments, notably the Cox–Voinov law[2], it is presumed that the static contact angle applies right at the solid boundary and that the angle variations with the speed that are often observed are an “apparent” contact angle, which is attained a short distance away from the wall. It is also often assumed that there is a fluid slip on the wall at the CL, which helps regularize the singularity in stresses. In MKT, and inherent in the introduction of a CL mobility, the contact angle is assumed to be different from the static value right at the wall, when viewed on molecular length scales. In the phase field model, mass diffusion will help regularize the CL and there is no need for a fluid slip. The introduction of dissipation related to CL movement will cause the contact angle at the wall to deviate from the static value.

It is far from clear what the actual conditions are for a given liquid spreading on a particular surface. For a system of decane spreading on a surface covered with a thin layer of PDMS, it was demonstrated experimentally[19] that the microscopic dynamic contact angle is velocity dependent and deviates substantially from the equilibrium value also at very small but nonzero capillary numbers. In ref. [20], a theoretical model is developed that links the distribution of assumed nanoscopic geometrical surface defects to the line friction dissipation. Recent molecular dynamics (MD) simulations have also shown that for water molecules and a wall with hydrogen bonds with the water molecules, the first layer of water molecules are effectively bound to the surface, a no-slip condition is appropriate, and the contact angle deviates from the equilibrium value[21,22]. It is known that the local molecular arrangements will be different in electrowetting and that this will alter the line friction[23,24]. Also, a microscopic geometrical structure on the surface will change the dynamic wetting and can be described through an effective line friction[25]. In an oil–water system, Rondepierre et al.[26] demonstrated that CL friction was responsible for a decrease in CL speed of three orders of magnitude, as a certain surfactant was added.

The actual nanoscopic cause of local dissipation at the CL can thus be very different depending on the properties of the surface, surface roughness and structure, the liquid properties, the surface chemistry of the wet surface, etc. We conclude that we should not expect any universal answer to the question of what is causing CL friction. Many different nanoscopic or microscopic processes can no doubt have this effect. However, whatever the origin, the effect can be described as a single parameter, the CL friction.

So far, the dependence of line friction on the contact angle has received limited attention, but there is clearly every reason for line friction to vary with the dynamic contact angle. The nonlinear MKT theory gives one possibility and based on MD simulations Johansson and Hess[22] formulated an expression for the angle dependence of line friction for water molecules on a surface with hydrogen bonds to the water. Other surfaces and liquids may certainly show different behavior.

On this note, we evaluate the detailed experimental results of Xia and Steen[9], using phase field simulations, with the aim of determining precisely what is required in the mathematical model, in order to faithfully reproduce the experiments. We will study how the angle dependence of the CL friction should be chosen. The model and the simulation are then used to draw conclusions on the source of the dissipation that is evident in the experiment.

Methods

The simulations are made using the Navier–Stokes–Cahn–Hilliard equations[18,27,28]. These describe the two-phase system as two immiscible species and are motivated by the thermodynamics of such a mixture. A phase field variable is introduced that has different values in the two species, and the fluid interface is identified as the steep but continuous transition between those values.with the standard choices: , and .

Here u, C, and ϕ are the fluid velocity, the phase field variable, and the associated chemical potential, respectively. C is +1 in the liquid and −1 outside the droplet. S is a reduced pressure such that is the actual pressure. μ and ρ are the local viscosity and density, respectively, and κ is the phase field mobility. σ and ε are the phase field parameters, where ε denotes the interface thickness, and σ gives the surface tension γ through . The function represents the standard choice in phase field methods and gives a qualitatively reasonable thermodynamic behavior representative of an immiscible mixture.

The simulations are performed in a cylindrical coordinate system that follows the oscillating substrate, giving rise to the acceleration term on the right-hand side of the momentum equation, with denoting the vertical position of the substrate.

The boundary conditions on the solid wall express that the fluid cannot penetrate through the wall and does not slip on the wall.

One additional boundary condition is needed for the phase field variable, which expresses the wetting conditions.Here σ and ε are the phase field parameters as above, giving surface tension as . γ1 and γ0 are the surface energies of the dry and wet solid surface, respectively, so that the equilibrium contact angle θe is given by . The form of the function is chosen in relation to the function f in Eq. (4).

Yue[29] developed a phase field treatment of contact angle hysteresis, where advancing and receding contact angels are introduced in a piecewise continuous function on the right-hand side of the equation corresponding to Eq. (11). The CL is then allowed to move according to whether the value of the dynamic contact angle exceeds (subceeds) the advancing (receding) angle. A treatment inspired by this is also developed for level-set methods[30].

The left-hand side represents the dissipation associated with CL motion, quantified by the parameter μ, which we will call CL friction. This has dimensions of viscosity.

By considering solutions to the Cahn–Hilliard equations near equilibrium at the CL, an explicit relation between CL speed UCL and dynamic contact angle θ can be derived from Eq. (11), see ref. [18] and Eq. (3):

The dynamic contact angle is equal to the equilibrium angle if μ = 0, in which case there is also no dissipation at the CL. The last approximate equality holds if θ is near θe.

The above equations are made nondimensional using R as length reference, the radius of the half-sphere (approximately the initial wet footprint radius), and an inertial capillary velocity .

The nondimensional parameters that appear are

In addition to these, the Cahn–Hilliard equations use the following parameters:

The simulations are carried out using an adaptive finite element solver, as in ref. [31]. The adaptive grid is automatically refined as needed down to an element size of 0.001. The air viscosity and density are taken larger than those of air but much smaller than the values for the liquid. They are deemed small enough for the air to have a negligible influence on the flow inside the droplet.

Oh=μργR	Ohnesorge number
Oh_f=μ_fργR	Line friction Ohnesorge number
Bo=ρgR^2γ	Bond number, g=9.81m/s^2<?tpb5pt?>
ω=2πfρR^3γ	Nondimensional oscillation angular frequency. f is the oscillation frequency in Hertz
A = Ad/R	Nondimensional oscillation amplitude

Pe=URD=γρRR223κγε	Peclet number related to phase field mobility. Set to 100 in the simulations. Reported here
Cn=εR	Cahn number, nondimensional interface width. Kept at 0.01 in the simulations. Reported here
ρ_airρ_liq	Ratio between air and liquid density. Set to 0.01 here, for computational convenience
μ_airμ_liq	Ratio between air and liquid viscosity. Set to 0.03 here, for computational convenience