Moving contact lines and Langevin formalism

Journal Pre-proofs Moving Contact Lines and Langevin Formalism J-C. Fernández-Toledano, T.D. Blake, J. De Coninck PII: DOI: Reference:

S0021-9797(19)31457-2 https://doi.org/10.1016/j.jcis.2019.11.123 YJCIS 25740

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Journal of Colloid and Interface Science

Received Date: Revised Date: Accepted Date:

4 October 2019 29 November 2019 30 November 2019

Please cite this article as: J-C. Fernández-Toledano, T.D. Blake, J. De Coninck, Moving Contact Lines and Langevin Formalism, Journal of Colloid and Interface Science (2019), doi: https://doi.org/10.1016/j.jcis. 2019.11.123

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Moving Contact Lines and Langevin Formalism J-C. Fernández-Toledano*‡, T. D. Blake‡, and J. De Coninck ‡ Laboratory of Surface and Interfacial Physics (LPSI), University of Mons, 7000 Mons, Belgium

*

[email protected]

‡

These authors contributed equally.

1

Abstract Hypothesis: In previous work [J-C. Fernández-Toledano, T. D. Blake, J. De Coninck, J. Colloid Interface Sci. 540 (2019) 322–329], we used molecular dynamics (MD) to show that the thermal oscillations of a contact line formed between a liquid and a solid at equilibrium may be interpreted in terms of an overdamped 1-D Langevin harmonic oscillator. The variance of the contact-line position and the rate of damping of its self-correlation function enabled us to determine the coefficient of contact-line friction 𝜁 and so predict the dynamics of wetting. We now propose that the same approach may be applied to a moving contact line. Methods: We use the same MD system as before, a liquid bridge formed between two solid plates, but now we move the plates at a steady velocity 𝑈𝑝𝑙𝑎𝑡𝑒 in opposite directions to generate advancing and receding contact lines and their associated dynamic contact angles 𝜃𝑑. The fluctuations of the contact-line positions and the dynamic contact angles are then recorded and analyzed for a range of plate velocities and solid-liquid interaction. Findings: We confirm that the fluctuations of a moving contact line may also be interpreted in terms of a 1-D harmonic oscillator and derive a Langevin expression analogous to that obtained for the equilibrium case, but with the harmonic term centered about the mean location of the dynamic contact line 𝑥𝑑, rather than its equilibrium position 𝑥0, and a fluctuating capillary force arising from the fluctuations of the dynamic contact angle around 𝜃𝑑, rather than the equilibrium angle 𝜃0. We also confirm a direct relationship between the variance of the fluctuations over the length of contact line considered 𝐿𝑦, the time decay of the oscillations, and the friction 𝜁. In addition, we demonstrate a new relationship for our systems between the distance to equilibrium 𝑥𝑑 ― 𝑥0 and the out of equilibrium capillary force 𝛾𝐿(cos 𝜃0 ― cos 𝜃𝑑), where 𝛾𝐿 is the surface tension of the liquid, and show that neither the variance of the fluctuations nor their time decay 2

depend on 𝑈𝑝𝑙𝑎𝑡𝑒. Our analysis yields values of 𝜁 nearly identical to those obtained for simulations of spreading drops confirming the common nature of the dissipation mechanism at the contact line. Keywords: molecular-dynamics, solid-liquid interactions, dynamic contact angles, Langevin process, molecular-kinetic theory. 1. Introduction The wetting of a solid by a liquid is such a familiar process that one would expect it to be well understood. A drop of liquid placed on a flat solid surface either spreads completely or, more often, comes to rest with its meniscus subtending a finite contact angle 𝜃 with the solid surface. But what is extremely frustrating about this apparently simple spreading process is that we cannot predict its dynamics from first principles. We have to perform an experiment and then use one theory or another [e.g., 1-4] to fit the resulting data. Since these data usually cover a limited period of time or rate of spreading, perhaps just a few decades, it is not very surprising that all the fits are often acceptable, irrespective of the underlying model [5-7]. In addition, all current theories have parameters that we cannot determine with sufficient accuracy a priori or independently verify. Thus, any prediction is likely to be flawed. Nevertheless, an understanding of the dynamics of wetting is of key importance to a wide range of processes both industrial and naturally occurring. In this paper we consolidate a new approach to solving this problem by establishing a direct link between the dynamics of a moving contact line and classical Langevin theory. Using large scale MD simulations, we show, for the first time, that the moving contact line is, in critical respects, nothing more than a Langevin oscillator that can be described completely using two parameters. Moreover, these two parameters are the same as those that can be ascertained by studying the contact line at equilibrium. The corollary is that,

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independent of any theory, the dynamics of wetting is controlled by the equilibrium properties of the solid, liquid and vapor interfaces as already stated in Ref. [8]. Thus, we can make use of the results already established for Langevin processes to investigate the moving contact line problem in an entirely novel way. 2. Theory In a recent publication [8], we have used molecular-dynamics (MD) simulations to show that at equilibrium, the thermal fluctuations of the contact line position 𝑥𝑐𝑙(𝑡) with time 𝑡 may be modelled as an over-damped one–dimensional Langevin oscillator [9,10] confined around its equilibrium position 𝑥𝑒𝑞 by an harmonic potential 𝑉(𝑟) = 0.5𝑘(𝑥𝑐𝑙(𝑡) ― 𝑥𝑒𝑞)2, where 𝑘 is the stiffness of the oscillator. This leads to 𝑑𝑥𝑐𝑙(𝑡)

𝐿𝑦𝜁

𝑑𝑡

= ―𝑘(𝑥𝑐𝑙(𝑡) ― 𝑥𝑒𝑞) + 𝑓,

(1)

where 𝑓 is a random force acting on the contact line, 𝐿𝑦 is the length of the contact line considered and 𝜁 is the coefficient of contact-line friction, (friction divided by contact-line length). The friction arises from the interaction of the liquid molecules within the three-phase zone, that constitutes the contact line at the molecular scale, with the atoms of the solid surface and the resulting dissipation. By dividing Eq. (1) by 𝐿𝑦 we obtain the Langevin equation per unit of the contact line. 𝑑𝑥𝑐𝑙(𝑡)

𝜁

𝑑𝑡

= ― 𝑘(𝑥𝑐𝑙(𝑡) ― 𝑥0) + 𝑓,

(2)

where 𝑘 = 𝑘/𝐿𝑦 and 𝑓 = 𝑓/𝐿𝑦. The random force 𝑓 is due to the random fluctuations of the contact angle 𝜃(𝑡) about its equilibrium value 𝜃0, which induce a very fast variation in the capillary force 𝛾𝐿(cos𝜃0 ― cos𝜃(𝑡)). Here, 𝛾𝐿 is the surface tension of the liquid. This force is uncorrelated at

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very short times, has a zero mean 〈𝑓〉𝑡 = 0 and must satisfy 〈𝑓(𝑡)𝑓(𝑡0)〉𝑡0 = 2𝜁𝑘𝐵𝑇𝛿(𝑡 ― 𝑡0)/𝐿𝑦 where 𝑘𝐵 is the Boltzmann constant, 𝑇 is the absolute temperature, 𝐿𝑦 is the length of contact line used to compute 𝑥𝑐𝑙(𝑡), and 𝛿(𝑡 ― 𝑡0) is the classic Dirac delta distribution. This means that there is a relation between 𝑘 and the temporal evolution of the signal 𝑥𝑐𝑙(𝑡), which allows us to compute 𝑘: 𝑘𝐵𝑇

𝑘 = 𝜎2𝐿 ,

(3)

𝑦

Here, 𝜎2 = 〈𝑥2𝑐𝑙(𝑡0)〉𝑡0 is the variance of the contact-line position. For a given system, the product of the variance and contact-line length 𝜎2𝐿𝑦 is invariant with 𝐿𝑦. Furthermore, the contact-line friction 𝜁 may be determined from the time decay of the self-correlation function

〈𝑥𝑐𝑙(𝑡0 + 𝑡)𝑥𝑐𝑙(𝑡0)〉𝑡0: 〈𝑥𝑐𝑙(𝑡0 + 𝑡)𝑥𝑐𝑙(𝑡0)〉𝑡0 = 𝜎2exp ( ― 𝜁𝑡) = 𝜎2exp ( ―𝑏𝑡). 𝑘

(4)

The parameter 𝑏 = 𝑘 𝜁 determines the rate of decay. The larger the value of 𝑏, the more rapidly the system becomes uncorrelated. The approach outlined above was successfully used to determine the coefficient of contact-line friction of a simulated liquid bridge at equilibrium between two solid surfaces for a range of equilibrium contact angles. The resulting frictions were identical to those found in MD simulations of spreading liquid drops and obtained by fitting the contact angle data to 𝛾𝐿

𝑈𝑐𝑙 = 𝜁 (cos𝜃0 ― cos𝜃𝑑),

(5)

where 𝑈𝑐𝑙 is the instantaneous velocity of the contact line as the drop spreads. Equation (5) is the linear limit of the principal equation of the molecular–kinetic theory (MKT) of dynamic wetting

5

[1,11]. Thus, we were able to predict the full dynamics of wetting simply by studying the fluctuations of the contact line at equilibrium. In new work presented here, we aim to analyze the contact-line fluctuations when the contact line is already moving and, therefore, when the dynamic contact angle deviates from its equilibrium value. A successful outcome would demonstrate that the application of the Langevin equation to modeling contact-line friction is very general and independent of contact-line velocity. To do this, we use the same liquid bridge geometry as in the equilibrium case [8], but now we induce flow within the liquid by moving the top and bottom plates in opposite directions at equal steady velocities 𝑈𝑝𝑙𝑎𝑡𝑒. The liquid bridge is the stationary frame of reference; hence, 𝑈𝑝𝑙𝑎𝑡𝑒 = ―𝑈𝑐𝑙. As well as inducing advancing and receding dynamic contact angles 𝜃𝑑 that differ from 𝜃0, the flow displaces the mean position of the contact line from its equilibrium position 𝑥0 to some new position 𝑥𝑑, as illustrated in Fig. 1. However, thermal fluctuations still occur and the resulting instantaneous capillary force can then be written as the sum of two terms: 𝐹𝑐𝑙 = 𝛾𝐿(cos𝜃0 ― cos𝜃(𝑡)) = 𝛾𝐿(cos𝜃0 ― cos𝜃𝑑) + 𝛾𝐿(cos𝜃𝑑 ― cos𝜃(𝑡)).

(6)

The first term is the standard averaged capillary force, which has a constant value for given 𝑈𝑐𝑙. The second term represents the rapidly fluctuating force that arises as 𝜃(𝑡) fluctuates around its new mean value 𝜃𝑑 and now plays role of the random force 𝑓 in the Langevin equation, which becomes 𝑑𝑥𝑐𝑙(𝑡)

𝜁

𝑑𝑡

= ― 𝑘(𝑥𝑐𝑙(𝑡) ― 𝑥0) + 𝛾𝐿(cos𝜃0 ― cos𝜃𝑑) + 𝑓.

(7)

As before, 〈𝑓〉𝑡 = 0.

6

Figure 1. Schematic representation of a liquid bridge between two solid surfaces separated by a distance 𝐻 = 8.85 nm and moving in opposite directions at velocities 𝑈𝑝𝑙𝑎𝑡𝑒, showing the displacement of the liquid-vapor interface from its equilibrium position (dashed line) to some dynamic position (continuous line). The inset shows the equilibrium (𝑥0) and dynamic (𝑥𝑑) locations of the contact line and the associated contact angles 𝜃0 and 𝜃𝑑. The dimensions of the simulation box are 𝐿𝑥 × 𝐿𝑦, × 𝐿𝑧 = 69.6 × 12.25 × 11.1 nm, with periodic boundary conditions in the 𝑥 and 𝑦 directions. Once the system reaches a steady state at a given plate velocity 𝑈𝑐𝑙 = ―𝑈𝑝𝑙𝑎𝑡𝑒 and the mean locations of the contact lines do not change with time. Hence, the time–averaged value of the harmonic term in the new Langevin equation must be exactly compensated by the time–averaged value of the capillary force:

〈𝑘(𝑥𝑐𝑙(𝑡) ― 𝑥0)〉𝑡0 = 〈𝛾𝐿(cos 𝜃0 ― cos 𝜃𝑑) + 𝑓〉

(8)

𝑘(𝑥𝑑 ― 𝑥0) = 𝛾𝐿(cos 𝜃0 ― cos 𝜃𝑑).

(9)

and

7

Eq. (9) establishes a new relationship between 𝑘(𝑥𝑑 ― 𝑥0) and the capillary driving force 𝛾𝐿(cos𝜃0 ― cos𝜃𝑑) consistent with a linear theory. Furthermore, by substituting Eqs. (8) and (9) in (7) we can obtain 𝑑𝑥𝑐𝑙(𝑡)

𝜁

𝑑𝑡

= ― 𝑘(𝑥𝑐𝑙(𝑡) ― 𝑥𝑑) + 𝑓

(10)

This is identical to the overdamped Langevin equation proposed for the fluctuation of the contact line at equilibrium, but now with the harmonic term centered about the mean location of the dynamic contact line 𝑥𝑑. 3. Methods and results Full details of the model and procedures used to locate the contact lines and measure the static and dynamic contact angles may be found in our earlier publications [8,18]. A few specifics are given here. The system studied is illustrated in Fig. 1. Each solid plate comprises a 3-layer square planar lattice of 16,275 atoms. The liquid bridge contains 48,840 atoms arranged in 8-atom molecular chains. The MD simulations are performed within the LAMMPS software package [12], which has been widely used to study wetting phenomena at the nanoscale [e.g., 13–17]. The mean displacement of the atoms between each time step of 5 fs is about ± 0.001 nm, which is much less than their diameter and defines the spatial resolution of the simulation. The separation between the plates is H = 8.85 nm, which is defined as the average distance between the surfaces of the atoms of the top and the bottom plates. The surface tension of the liquid is 𝛾𝐿 = 2.84 ± 0.56 mN/m and its density 384.6 ± 1.2 kg/m3. While these values are untypical of everyday experience, we are, nevertheless, dealing with model liquids that genuinely behave as real liquids. The values arise because there are limitations to the timescales currently accessible in MD simulations, which restrict the properties of the liquids that may be modelled. For example, their viscosity must be sufficiently low to enable the system to reach equilibrium by diffusion within an 8

acceptable computational timeframe. In general, the model parameters are chosen to minimize computational demand while yielding realistic behavior, which has been confirmed in previous studies by verifying, for example, the Laplace and Young equations [20–23] and demonstrating Poiseuille and Couette flow [18,24,25]. 3.1. Contact-line position and variance At the start of each simulation, the liquid bridge is equilibrated between the plates for 2 × 106 time steps. This is sufficient to achieve a system with time-independent values for its energy and equilibrium contact angle. We then restart the simulation and move each plate at constant velocity 𝑈𝑝𝑙𝑎𝑡𝑒 ∈ (0,8) m/s in opposite directions for an additional 2 × 106 time steps, which is long enough to reach a stationary state characterized by constant mean values of the advancing and receding dynamic contact angles. During these initial phases, the temperature is kept constant by velocityscaling both liquid and solid atoms, but subsequently we remove the thermostat from the liquid and apply velocity-scaling only to the solid. To determine the locations of each of the four contact lines formed by the liquid bridge (see Fig. 1) over the width of the system 𝐿𝑦, we first select a slice of liquid in the x-y plane adjacent to each SL interface and determine the density profile in the z direction. The density is constant in the bulk but becomes layered in proximity to the plates. We define the first layer of liquid as the slice containing the first peak in the density profile. We run the simulation for another 20 × 106 and save the density profiles of this first layer in the 𝑥 direction. Each saved density profile is constant across the center of the S–L interface, but decays to zero at the contact lines. We define the contactline location 𝑥𝑐𝑙(𝑡) at time 𝑡 as the value of 𝑥 at which the density profile decays to 50% of its central value.

9

Once we have located the position of the contact lines at each time step, we compute their mean location 〈𝑥𝑐𝑙(𝑡)〉𝑡. This will be equal to 𝑥0 when 𝑈𝑝𝑙𝑎𝑡𝑒 = 0 or to 𝑥𝑑 when 𝑈𝑝𝑙𝑎𝑡𝑒 ≠ 0. Finally, we subtract the averaged positions from the instantaneous positions 𝑥𝑐𝑙(𝑡) to determine the distribution of 𝑥𝑐𝑙(𝑡) ― 〈𝑥𝑐𝑙(𝑡)〉𝑡. The distribution can be fitted with the Gaussian function exp ( ―

(𝑥𝑐𝑙(𝑡) ― 〈𝑥𝑐𝑙(𝑡)〉)2 𝜎2

)/√2𝜋𝜎2, where 𝜎 is the standard deviation, i.e. half the width of the

distribution between the two inflection points. This is illustrated in Fig. 2 for S–L coupling 𝐶𝐿𝑆 = 0.6 and 𝑈𝑝𝑙𝑎𝑡𝑒 = 0, 1, 3 and 5 m/s. From the Gaussian fits we can obtain the variance of the contact line position 𝜎2. In our previous paper [8], we showed that the product of the variance and contact-line length 𝜎2𝐿𝑦 was constant for a given S-L coupling 𝐶𝑆𝐿. In Fig. 3a we plot 𝜎2𝐿𝑦 versus plate velocity. As it can be seen, for a given coupling, the value of 𝜎2𝐿𝑦 is also essentially independent of the plate velocity within the uncertainty indicated by the error bars, although there is perhaps a slight upward trend that may require further investigation. This important result was not obvious a priori. The average values of 𝜎2𝐿𝑦 at each coupling are plotted in Fig. 3b and show an exponential decline to a plateau at high couplings. The value at 𝐶𝑆𝐿 = 0.9 is for 𝑈𝑝𝑙𝑎𝑡𝑒 = 0 only, but helps to illustrate the trend to a limiting value of about 0.57 nm3.

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Figure 2. Probability density function (PDF) of the contact line position about its mean location computed from Gaussian fits (lines) to the density histograms (symbols) for a solid-liquid coupling 𝐶𝑆𝐿 = 0.6 and plate velocities 𝑈𝑝𝑙𝑎𝑡𝑒 = 0 m/s, 1 m/s, 3 m/s and 5 m/s. The distributions shown are for the length of contact line considered 𝐿𝑦. From the values of 𝜎2𝐿𝑦, we can compute 𝑘 using Eq. (3). In Fig. 4 we show the mean capillary force 𝛾𝐿(cos𝜃0 ― cos𝜃𝑑) plotted versus the resulting values of 𝑘(𝑥𝑑 ― 𝑥0) for all couplings and plate velocities considered in our simulations. Every point lies close to the line 𝑦 = 𝑥, which confirms the relationship predicted by Eq. (9), which will always be true for small local displacements of the contact line.

This result also has implications for the more general

interpretation of spreading data and requires further investigation.

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Figure 3. a) Dependence of the product of the variance and contact-line length 𝜎2𝐿𝑦 on 𝑈𝑝𝑙𝑎𝑡𝑒 for the different S–L couplings 𝐶𝑆𝐿. b) Mean value of 𝜎2𝐿𝑦 at each coupling. The value at 𝐶𝑆𝐿 = 0.9 is for 𝑈𝑝𝑙𝑎𝑡𝑒 = 0.

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Figure 4. Capillary force 𝛾𝐿(cos𝜃0 ― cos𝜃𝑑) versus the computed value of 𝑘(𝑥0 ― 𝑥𝑑) for all S– L couplings and plate velocities considered in our simulations. The dashed line has unit slope. 3.2. Self-correlation function and damping coefficient Figure 5a shows examples of the decay of the self-correlation function normalized by the variance of the contact-line position, 〈𝑥𝑐𝑙(𝑡 + 𝑡0)𝑥𝑐𝑙(𝑡0)𝑡0〉 𝜎2, for three different S–L couplings: 𝐶𝑆𝐿 = 0.4, 0.7 and 0.8 and the same value of the plate velocity 𝑈 = 3 m/s. It is apparent that the rate of decay of the correlation decreases as we increase the coupling, but is insensitive to plate velocity, as revealed by Fig. 5b for 𝐶𝐿𝑆 = 0.4 and different values of 𝑈𝑝𝑙𝑎𝑡𝑒. This can be made more explicit by plotting the damping coefficient 𝑏, obtained by fitting Eq. (4) to the decay data for the different couplings, versus the plate velocity 𝑈𝑝𝑙𝑎𝑡𝑒, as demonstrated in the insert. The inset

13

in Figure 5a shows the average value of the 𝑏 parameter for the different couplings. The decrease in 𝑏 with increasing coupling is consistent with the increase in contact-line friction.

Figure 5. a) Exponential decay of the self–correlation function for the same plate velocity 𝑈𝑝𝑙𝑎𝑡𝑒 = 3 m/s and three different S–L couplings: 𝐶𝑆𝐿 = 0.4, 0.7 and 0.8. The lines are fits to Eq. (4) and the inset shows the 𝑏 parameter for each coupling averaged for all plate velocities. b) Exponential decay of the self–correlation function for 𝐶𝑆𝐿 = 0.4 and different plate velocities. Here, the inset shows the 𝑏 parameter obtained by fitting Eq. (4) to the decay for each velocity studied. 3.3. Contact-line friction

14

Once we have measured the values of 𝜎2𝐿𝑦 and 𝑏, we can obtain the contact-line friction 𝜁 for the different simulations using Eq. (12), which is derived from Eqs. (3) and (4): 𝑘𝐵𝑇

𝜁 = 𝑏𝜎2𝐿

𝑦

(12)

In Fig. 6a we plot the resulting values of 𝜁 versus the plate velocity 𝑈𝑝𝑙𝑎𝑡𝑒 for all S–L couplings investigated. The value of 𝜁 is sensibly independent of 𝑈𝑝𝑙𝑎𝑡𝑒 within the error bars, but clearly increases with coupling. If we average the values of the frictions for each 𝑈𝑝𝑙𝑎𝑡𝑒 and plot them versus 𝐶𝐿𝑆, we obtain values almost identical with the coefficients of friction measured for simulations of spreading drops [19]. This is demonstrated in Fig. 6b. In Fig. 6c, we plot the same data versus the equilibrium contact angle.

These figures also include the results obtained

previously from the fluctuations of equilibrium contact lines [8]. Evidently, the fluctuations contain the same information on dynamic wetting, irrespective of whether or not the contact line is moving. This is an important result of our work and suggests that Langevin approach to dynamic wetting is quite general and not confined to systems at equilibrium. Nevertheless, the impact of much greater plate velocities remains to be determined.

15

Figure 6. a) Contact-line friction 𝜁 calculated via Eq. (12) versus 𝑈𝑝𝑙𝑎𝑡𝑒 for each coupling studied. b) Average value of 𝜁 versus coupling compare with those derived by fitting Eq. (5) to dynamic contact angle data from simulations of spreading drops [19]. We also include the values of 𝜁 obtained previously [8] for a contact line at equilibrium. c) The same data plotted versus the equilibrium contact angle. 4. Conclusions Dynamic wetting is a complex subject. Even when the systems are comparatively well defined, it proves impossible to be certain which theory is the most relevant and what factors determine the observed behavior [5–7]. This makes accurate prediction equally impossible and one must take recourse to experiment. Because of this, new experimental and theoretical approaches are needed, especially ones that might shed light on wetting at the nanoscale where much of the uncertainty lies. In previous work [8], we used large-scale molecular dynamics to study the fluctuations of the 16

contact lines formed by of an equilibrium liquid bridge between two solid surfaces. We showed that the fluctuations about the mean position of the contact line 𝑥0 may be interpreted in terms of an overdamped 1-D Langevin harmonic oscillator of stiffness 𝑘 and demonstrated a relationship between the variance of the fluctuations 𝜎2 over the length of contact line considered 𝐿𝑦, the time decay of the oscillations and the contact-line friction 𝜁. Here, we have extended this work to study the fluctuations when the contact lines are moving, yielding dynamic advancing and receding contact angles 𝜃𝑑 that differ from their equilibrium values 𝜃0. A steady dynamic state is achieved by moving the plates in opposite directions at constant velocities 𝑈𝑝𝑙𝑎𝑡𝑒. Under these conditions, we obtain an identical Langevin expression to that found at equilibrium, but now with the harmonic term centered about the mean location of the dynamic contact line 𝑥𝑑 and a fluctuating capillary force arising from the fluctuations of the dynamic contact angle 𝜃(𝑡) around its mean value. We have also demonstrated for our system the validity of a new linear relation: 𝑘(𝑥𝑑 ― 𝑥0) = 𝛾𝐿(cos 𝜃0 ― cos 𝜃𝑑), where 𝑘 = 𝑘/𝐿𝑦 and 𝛾𝐿 is the surface tension of the liquid. This has implications for the more general interpretation of spreading data. Significantly, we have shown that neither the variance of the fluctuations nor the time decay of their self-correlation 〈𝑥𝑐𝑙(𝑡0 + 𝑡)𝑥𝑐𝑙(𝑡0)〉𝑡0 depend on 𝑈𝑝𝑙𝑎𝑡𝑒 over the range of plate velocities studied. Hence, the contact-line frictions derived from the time decay are also independent of 𝑈𝑝𝑙𝑎𝑡𝑒 and identical with those found at equilibrium. Moreover, this friction maps directly onto the coefficients of contact-line friction obtained by fitting the MKT to dynamic contact angle data from simulations of spreading drops, yielding nearly identical values over the whole range of equilibrium contact angles investigated. We therefore submit that the contact-line fluctuations are

17

the same irrespective of whether the contact line is at equilibrium or moving and contain all the information necessary to predict the dynamics of wetting. This result was not obvious a priori and is perhaps the most important conclusion to be drawn from our work. In addition, it has not escaped our attention that Eq. (12) for the contact line friction has the same 𝑘𝐵𝑇

form as the corresponding expression derived via the MKT: 𝜁 = 𝜅0𝜆3,

where

𝜅0

is

the

characteristic frequency of molecular displacements within the three-phase zone (TPZ) and 𝜆 is their characteristic length. The latter closely approximates the spacing of the energy wells in the solid surface, i.e. the sites of solid-liquid interaction responsible for contact-line friction. Comparing this relationship with Eq. (12) and (13), we see that both 𝑏 and 𝜅0 have dimensions of reciprocal time and both 𝜎2𝐿𝑦 and 𝜆3 define a volume. Essentially, 𝜎2𝐿𝑦 (which is constant for a given S-L coupling) defines the accuracy with which we can specify the location of the contact line: if 𝐿𝑦 is large, then 𝜎 is small and we know the location with precision; conversely, if 𝐿𝑦 is small, then 𝜎 is large and the location is known with less certainty. In the framework of the MKT, 𝜆3 defines the effective space that the individual atoms explore within TPZ as they escape from one energy well and fall into the next. We aim to investigate the possible relationships between 𝑏 and 𝜅0 in a later publication. Acknowledgements The authors thank the European Space Agency (ESA) and the Belgian Federal Science Policy (BELSPO) for their support in the framework of the PRODEX Programme. Computational resources have been provided by the Consortium des Equipements de Calcul Intensif (CECI), funded by the Fonds de la Recherche Scientique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11

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Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Contribution J-C. Fernández-Toledano, T. D. Blake, and J. De Coninck contributed equally in the elaboration of this manuscript.

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