Marco
Rivetti
a,
Thomas
Salez
b,
Michael
Benzaquen†
b,
Elie
Raphaël
b and
Oliver
Bäumchen
*a
aMax Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077 Göttingen, Germany. E-mail: oliver.baeumchen@ds.mpg.de
bLaboratoire de Physico-Chimie Théorique, UMR CNRS 7083 Gulliver, ESPCI ParisTech, PSL Research University, 10 rue Vauquelin, 75005 Paris, France
First published on 6th October 2015
The relaxation dynamics of the contact angle between a viscous liquid and a smooth substrate is studied at the nanoscale. Through atomic force microscopy measurements of polystyrene nanostripes we simultaneously monitor both the temporal evolution of the liquid–air interface and the position of the contact line. The initial configuration exhibits high curvature gradients and a non-equilibrium contact angle that drive liquid flow. Both these conditions are relaxed to achieve the final state, leading to three successive regimes in time: (i) stationary contact line levelling; (ii) receding contact line dewetting; (iii) collapse of the two fronts. For the first regime, we reveal the existence of a self-similar evolution of the liquid interface, which is in excellent agreement with numerical calculations from a lubrication model. For different liquid viscosities and film thicknesses we provide evidence for a transition to dewetting featuring a universal critical contact angle and dimensionless time.
In some cases, due to the finite size of the film, the interaction of the fluid with the substrate is mediated by the presence of a three-phase contact line where liquid, solid and vapor phases coexist. This is a common situation for instance in the case of liquid droplets supported by a solid surface. It is of primary importance in many applications, e.g. ink-jet printing, to know whether the droplets will spread or not on the substrate. In these wetting problems the movement of the three-phase contact line is often related to the equilibrium contact angle θY, which follows Young's construction at the contact line: cosθY = (γsv − γsl)/γ, where γsv, γsl and γ are the surface tensions of the solid–vapor, solid–liquid and liquid–vapor interfaces, respectively.14,15 In the vicinity of a moving contact line, however, classical fluid dynamics approaches based on corner flow predict a divergence of the viscous dissipation that would require an infinite force to move the line, as first pointed out by Huh and Scriven.16 This apparent paradox can be solved by including microscopic effects like, for instance, slip at the solid–liquid boundary,17 the presence of a precursor film ahead the line18 or a height dependence of the interfacial tension.19
A well-established system is that of a liquid droplet spreading onto a completly wettable substrate. The so-called Tanner's law predicts the growth of the drop radius with a power law evolution.20 Surprisingly, this law is valid for every liquid that wets the substrate and this universality is related to the presence of a thin precursor film ahead of the contact line.15,18 Recently, it has been proven that the power law changes in case of spreading on a thicker liquid layer.21 Different evolutions are known for droplets on partially wettable substrates. In particular, an exponential relaxation to the equilibrium contact angle is observed when θY is small and the system is close to equilibrium.22–24 Tanner's law as well as other investigations of wetting and contact-line dynamics are limited to spherical and cylindrical droplets, i.e. configurations in which the curvature of the liquid–air interface and therefore the liquid pressure are constant. More intriguing and complex phenomena may appear in the presence of non-constant film curvatures. Following the pioneering work of Stillwagon and Larson,25,26 there has been an increased interest in the capillary-driven dynamics of the relaxation of stepped and trench-like film topographies in the last few years.27–33 In these situations Laplace pressure gradients drive a capillary flow mediated by viscosity, leading to the levelling of the interface. All these studies are, however, limited to pure liquid–air interfaces in which a three-phase contact line is absent. Nevertheless, we note that the latter aspect has been recently considered in the context of studies about residual stress.34
Here, we study the relaxation dynamics of the contact angle at the nanoscale by tracking the evolution towards equilibrium of a viscous nanofilm in the presence of a three-phase contact line. The interesting aspect of this system is the simultaneous relaxation of both the contact angle and the curvature of the liquid-vapor interface. The resulting liquid dynamics is driven by both Laplace pressure gradients and the balance of forces at the three-phase contact line, while mediated by viscosity. We carry out experiments involving polystyrene films on silicon wafers, a common system that represents a partial wetting situation.35 In our experiments the thin films are invariant in the y direction and exhibit a rectangular cross section (cf.Fig. 1). The spatial and temporal dynamics of the liquid–air interface z = h(x,t) is monitored and distinct regimes corresponding to different mechanisms of relaxation are observed. We show that in general the advancing or receding of the contact line can not be predicted by the simple observation of the initial contact angle, as in a spherical or cylindrical droplet, and propose a geometrical approach to describe the evolution of the interface.
Once the preparation is completed, a nanostripe exhibiting straight edges is identified with an optical microscope and scanned with an atomic force microscope (AFM, Bruker, Multimode) in tapping-mode. The sample is then annealed above the glass transition temperature of PS on a high-precision heating stage (Linkam, UK) to induce flow. The annealing temperatures are set to 110 °C for the 3.2 kg mol−1 molecular weight and to 140 °C or 150 °C for the two others. Note that for these experimental parameters, the viscosity of PS is expected to be of the order of η ∼ 103–105 Pa s while the surface tension γ is around 30 mN m−1.37 After quenching the PS at room temperature the height profile is scanned with the AFM. This procedure is repeated several times, while keeping track of the time that the system has spent in the liquid state in order to record the temporal evolution of the profiles. As an alternative to this ex situ technique, we also performed in situ measurements in which the sample is annealed directly on a high-temperature scanner while the liquid interface is probed by the AFM tip. This way it has been safely ensured that the quenching has neither an influence on the shape of the profiles nor on the dynamics of the three-phase contact line.
For the next profile, at t = 90 min, it is clearly visible that both contact lines have receded. We also observe that the bumps have grown due to the accumulation of the liquid that has moved. The retraction of the contact lines continues in the successive scans and the liquid keeps accumulating in the rims that become higher and larger. The velocity of the contact lines during this stage is roughly constant, in agreement with earlier observations by Redon et al.38 in the presence of a no-slip boundary condition.
Around t ≃ 700 min the front retraction reaches the unperturbed region in the middle of the film and the rims start to merge. The portion of positive curvature disappears and the interface slowly converges to a cylindrical cap. The velocity of the contact lines slows down during the merging process. Note that at a more advanced stage of the process the system might eventually undergo a Plateau–Rayleigh instability and lose its invariance in the third dimension.
To summarise, three different regimes can be identified in this experiment: (i) the initial stationary contact line (SCL) regime is followed by a dewetting transition, as evidenced by (ii) the receding contact line (RCL) regime in which the two sides of the nanostripe retract independently, and eventually by (iii) the coalescence regime where the two rims merge to form the cylindrical droplet. Note that during the entire process the shape of the system remains symmetric.
At first glance the apparent absence of early spreading, the fixed position of the contact line at the beginning, and the following dewetting process may appear counterintuitive and surprising. However, the perfect reflection symmetry of the profile around x = 0 in Fig. 1(c) strongly suggests that these features are general, as opposed to pinning of the contact line at random defects. In order to check the validity of the previous observations, a series of experiments has been carried out involving films with various thicknesses and viscosities. Due to the symmetry of the profiles, in the next paragraphs we focus on one single edge of the film and discuss in detail the dynamics of the SCL regime and the dewetting transition.
The formation of a bump in the presence of a corner as well as the relaxation of the interfacial slope have been previously studied in the levelling of a stepped interface in a thin viscous film.29,32 The relaxation of the rectangular interfaces in Fig. 2 can be understood in terms of the flow generated by Laplace pressure gradients. As a consequence of the high viscosity of PS and the small film thickness, the Reynolds and Weber numbers of the flow are very small (Re ∼ We ∼ 10−11 if the capillary velocity γ/η is used as the velocity scale) and inertial effects can be neglected. The liquid dynamics can be safely described using the Stokes equation ∇p = η∇2v, where v is the liquid velocity and the pressure p is related to the curvature of the liquid–air interface by the Laplace equation p = −γ∂2h/∂x2, which is valid only for a 2D interface within the small slopes approximation.
Following the theoretical framework summarized in Oron et al.,39 the Stokes equation can be further simplified by introducing the lubrication approximation. The equation governing the evolution of the liquid interface h(x,t), in the presence of a no-slip boundary condition at the solid–liquid interface, a no-shear boundary condition at the free interface, and in the absence of disjoining forces can be deduced:
(1) |
In Fig. 3(a), the horizontal axis is rescaled by applying the transformation x → x/t1/4 and we observe that in each experiment the different profiles collapse on a single curve. This collapse demonstrates that the self-similar dynamics h(x,t) = h(x/t1/4) is valid even for an interface with a contact line, provided that the system is in the SCL regime where the line does not move.
Fig. 3 (a) Self-similar profiles for each of the experiments shown in Fig. 2 as obtained by plotting the vertical position h of the interface as a function of x/t1/4. The experiments are shifted horizontally for clarity. (b) A universal profile appears when the vertical axis is rescaled by h0 and the horizontal axis is non-dimensionalized by applying a lateral stretching in each experiment. The numerical solution (orange dashed line) is in excellent agreement with the experiments. |
In the following the experimental profiles are compared to the numerical solution of the thin film equation, see eqn (1). The numerical profile is computed using a finite difference method.44,45 The dimensionless self-similar variable X/T1/4 is introduced, where X = x/0 and T = γth03/(3η04), and thus:
(2) |
Thus, a general picture can be obtained by rescaling the vertical axis of the experimental profiles with the initial thickness, i.e. h → h/h0, and by stretching the horizontal axis x → (3η/γh03)1/4x/t1/4. This lateral stretch is a fitting parameter that depends on the experiment and has a clear physical interpretation.32,45 Applying this rescaling to all experimental profiles leads to a perfect collapse of all the profiles on a single master curve (Fig. 3(b)). This curve represents a universal profile of the SCL regime valid for all the parameters involved in these experiments (annealing time, film thickness, molecular weight and temperature). Note that the rescaled height of the bump is equal to 22 ± 2% of the initial thickness of the film.
The excellent agreement between the experiments and the thin film model (Fig. 3(b)) suggests that the thin film equation with a fixed contact line is sufficient to capture the physics of the SCL regime. The values of the resulting fitting parameters are used to compute the capillary velocity γ/η. Based on γ = 30.8 mN m−1,37 the viscosity η of the PS is also evaluated (see Table 1) and is found to be in excellent agreement with the values reported in the literature.37,46
M w [kg mol−1] | T [°C] | η [Pa s] |
---|---|---|
3.2 | 110 | 5.8 × 103 |
19 | 140 | 7.8 × 103 |
19 | 150 | 1.6 × 103 |
34 | 140 | 2.9 × 104 |
34 | 150 | 3.1 × 103 |
Fig. 4(b) shows a typical evolution of the contact angle. During the SCL regime the angle θ monotonically decreases due to the relaxation of the interface and the stationary position of the three-phase contact line. Interestingly, the SCL regime extends even when the angle is smaller than θY. Eventually, the receding motion of the line takes place when the angle reaches a critical value θ* < θY. As soon as the contact line retracts, θ rapidly increases to a receding contact angle which stays roughly constant in the course of dewetting.
In all the experiments θ decreases following a t−1/4 power law (see Fig. 4(b), inset) in the SCL regime. This power law is a direct consequence of the self-similar evolution of the liquid interface. Indeed, as shown above, the horizontal length scales evolve as ∼t1/4, which implies for a constant vertical length scale h0 that tanθ ∼ t−1/4, and thus θ ∼ t−1/4 at small angles. This relation leads to a fast decrease of the angle at short times and θ generally drops to small values within a few minutes.
Let us now introduce the dimensionless time τ = tγ/(h0η). Using the values of h0 recorded with the AFM and those of η/γ extracted from the comparison between the self-similar profile and the numerical solution, θ is plotted as a function of τ (see Fig. 5). The values of the contact angles for all the experiments collapse on the same master curve, a significant result because the experiments involve different heights of the film and different capillary velocities, since the viscosity changes by more than one order of magnitude. An important observation is the fact that in all the experiments the retraction of the line precisely appears at the same value of the contact angle θ* = 4.5° ± 0.5° < θY. From the master curve we, hence, deduce that the dimensionless dewetting time is also universal and equal to τ* ≃ 105. After the dewetting transition an exponential relaxation of the contact angle might be expected, although the experimental uncertainty of the contact angle measurement together with the time resolution here can not provide a precise validation. Note that the rescaling with τ is valid in the SCL regime, but does not necessarily apply to the RCL regime.
Aside from the general features outlined above, a prominent and interesting observation is the fact that a stationary position of the contact line is observed for values of the contact angle smaller than the equilibrium one, but larger than the critical one, i.e. θ* < θ < θY. This observation has been confirmed even for scan sizes in proximity of the contact line and in the presence of small defects that can be used as reference points, although the limited lateral resolution of the AFM can not accurately detect displacements in the order of a few nanometers. The dynamics of thin liquid films is governed by short-range as well as long-range forces, e.g. originating from van-der-Waals interactions, between the substrate and the liquid.9 In principle these long-range forces are negligible for a film thickness larger than ∼30 nm, which is always the case in the experiments discussed here. However, in the region close to the three-phase contact line the thickness of the film decreases monotonically to zero and the extent of the zone where h < 30 nm grows as soon as the contact angle decreases. In previous work it has already been shown that long-range intermolecular forces might affect the shape of the liquid interface close to the contact line of nanometric droplets.35 In order to test if long-range forces play a role in this zone and in particular if they affect the onset of the retraction of the line experiments on Si wafers exhibiting a thick (150 nm) oxide layer have been carried out. This choice is motivated by the fact that the presence of a thick oxide layer considerably changes the effective interface potential comprising short- and long-range forces and represents a well-established model system.9,35 One separate experiment has been performed on thick oxide layer Si wafers (see Fig. 5), and it has not shown any significant difference with respect to the experiments on a native Si oxide layer: the values of the contact angles for this experiment perfectly collapse on the master curve in Fig. 5 and the contact angle at the transition is preserved. Hence, we conclude that long-range forces affect neither the relaxation dynamics nor the onset of motion of the contact line. A possible alternative explanation would be the presence of a contact angle hysteresis related to a uniform intrinsic pinning potential of the polymer molecules on the Si wafers, yet to be substantiated in future experimental work.
The occurrence of dewetting, despite the fact the initial contact angle is much larger than the equilibrium one, is a common feature in all the experiments and might appear counterintuitive at first glance. Indeed, for a liquid interface with constant curvature (a spherical drop, or a cylindrical drop in the 2D case) the advancing or receding motion of the contact line can be ultimately predicted from the value of the contact angle: in particular the situation θ0 > θY, which characterizes all our experiments, would have lead to the monotonic spreading of the liquid. In fact it is easy to prove that a spherical or cylindrical interface in the absence of gravity reaches its minimum of energy when the forces at the contact line are at equilibrium, which is precisely the foundation of Young's construction of the equilibrium contact angle. However, in the more general case of a non-constant curvature interface, the equilibrium of the forces at the contact line given by θ = θY does not necessarily correspond to the minimum of the energy. The system has to adjust the contact angle and to relax the liquid interface at the same time in order to achieve the global minimum and, thus, it is not possible to predict spreading or dewetting ab initio only from the value of the contact angle.
A simple geometrical argument based on the comparison of the initial and the final state of the system can be introduced to anticipate the occurrence of spreading or dewetting. In the 2D configuration, the initial state is a rectangular interface featuring a width 20 and a thickness h0 (see Fig. 1(b)). The final state is a circular cap with the contact line radius r and the equilibrium contact angle θY. Invoking volume (area, in 2D) conservation between the two states, the contact line radius can be deduced as a function of h0, 0 and θY. We define the wetting parameter = r/0 and show that:
(3) |
In future work, we envision to explore whether and how the robust features observed in our experiments can be generalized to different types of substrates exhibiting different surface energies and/or a variation of the hydrodynamic boundary condition between liquid and solid. These experiments might provide new fundamental insights on the contact-line dynamics at the nanoscale.
Footnotes |
† Present address: Capital Fund Management, 23 rue de l'Université, 75007 Paris, France. |
‡ Even for the smallest scan sizes of 2 × 2 μm and in the presence of a unique reference point, such as a defect on the substrate, no significant movement of the contact line has been detected. Nevertheless, given the limited lateral resolution of the AFM, a displacement of the contact line on the scale of a few nm can not be safely excluded. |
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