The discussion up to this point only covers the conditions under which pinned droplets exist.
These results can now be combined with the results from full dynamical simulations to give a full picture of the depinning/repinning process of droplets on heterogeneous substrates.
With BEM simulations for the case of k=10 and a low slip length, the depinning point from the static limit was recovered. That means at the driving force where no more stationary solutions exist, the first moving droplet solutions were found. For driving forces higher than the depinning force, the classic stick-slip motion was observed.
5.2.1 Introduction: Sniper bifurcation
A classical model for a stick-slip motion of a droplet on a periodically patterned substrate[85]
is a point mass in a sinusoidal potential with over-damped dynamics and a volume force acting on it. The governing equation of motion that describes the motion of the particle positionx over timet, also known as Adler equation[2, 79], takes the following form:
x(t)0=µ+sin(x(t))
Fig. 5.8: Illustration of a particle on a periodically structured substrate. The driv-ing force points down, so an increased slope represents an increased driving force. In the over-damped limit and for small heterogeneities, the observed dynamics can be approximated by the Adler equation.
with the prime denoting the derivative with respect to time. This system can be thought of as a particle on a sinusoidal substrate that gets tilted with a constant volume force pointing down, as illustrated in Fig. 5.8. As the inclination increases, the local energy minimum
becomes more and more shallow, until the stable static solution (red) and the unstable static solution (blue) annihilate and a periodically moving solution (green) appears, when no more local minima exist.
The periodically moving solution features a stick-slip motion where the system slows down in the region where the stable and the unstable solution branch annihilated. The velocity of the solution (i.e. the inverse of the periodicity of the motion τ) after depinning, for µ>0 scales asu∝(µ−µcrit)0.5, withµcrit=1 for the Adler equation. The occurring bifurcation is called “Saddle-Node Infinite PERiod” (SNIPER) bifurcation, as the period diverges close to the critical point.
5.2.2 Observations
Using the BEM code, the time evolution of droplets driven on substrates with different pe-riodicities and different wetting heterogeneities was modeled. The simulations did not only display the classical stick-slip motion associated with a single Adler equation. Depending on the periodicity of the substrate, either a synchronized stick-slip motion can be observed or a desynchronized motion, where the maximum of the front and the back contact line are phase shifted by approximately half a period. This is shown in Fig. 5.9. The strength of the hetero-geneity and the driving force were kept constant, but the periodicity of the heterohetero-geneity was changed by a quarter of a period fromk=10 tok=10.25.
This desynchronized motion is especially pronounced when the droplet baselength matches the periodicity of the substrate, i.e. the previously discussed case in the static limit where the critical depinning force is reduced. In this case the velocity variation is significantly lower than in the synchronized case. A small increase in the velocity of the other contact line occurs whenever a contact line has maximum velocity. That means, while they are moving desynchronized, a coupling of the dynamics of the two contact lines takes place.
Fig. 5.10(a) shows the average velocity of the droplet over the driving force for different slip lengths. While the droplet velocity increases with increasing slip length and increasing driving force acting on the droplet, another phenomenon is observed: For higher slip lengths, the point where no more depinned state is observed shifts to lower driving forces. That means there is not one critical driving force where the system changes from pinned to periodically moving solutions, but rather a range of bistability where both periodically moving and pinned solutions coexist.
Together with this, a discontinuity in the droplet velocity at the transition was observed, indicating that the square root scaling of the velocity does not hold anymore. These imply a change in the bifurcation scenario away from the SNIPER bifurcation.
To show how the dynamics change, Fig. 5.10(b) displays the minimum and maximum observed baselength over one period of motion for periodically moving states, together with the static states. As the driving force, and thus the droplet velocity increases, the change in baselength decreases, as the droplet has less time to respond to the varying substrate wetting energy.
With increasing slip length, the variation in the baselength decreases too, i.e. the droplet deforms less over the course of one period. Also, the lowest driving force where a periodically moving solution is observed decreases. As the static energy landscape and therefore the static solutions not change with the slip length an increased range of bistability can be seen. This implies that two competing time scales play a role, one determining the deformation of the droplet and the second one determining the periodicity of the motion.
6.0 6.1 6.2 6.3 6.4 6.5
Fig. 5.9: Velocity of the front and the back contact line fork=10 (left), k= 10.25 (right), showing desynchronized and synchronized motion with a small change in the periodicity of the substrate forw0=0,∆w=0.2, µ=0.35
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 driving forceµ
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 driving forceµ
Fig. 5.10: (a) Observed average droplet velocity over driving force for different slip lengths showing varying range of bistability forw0=0,∆w=0.2, k=10
(b) Observed minimum and maximum droplet baselength in one pe-riod (symbols) plus static solution branches (full lines for stable, dot-ted lines for unstable) over driving force for different slip lengths, w0=0,∆w=0.2,k=10.
5.2.3 Mechanism
To get a better understanding for the dynamics, it is instructive to observe the interfacial energies and contact line velocities over the position of the droplet on the substrate. This might provide an insight in the change of the dynamics. The curves were reconstructed from dumps of the droplet shape taken from multiple periods and smoothed with an interpolating spline for presentation.
Fig. 5.11(a) shows how the droplet baselength varies over over period in space. While the average baselength is quite similar for both slip length, the oscillation is significantly smaller for the high slip length case as indicated by the minimal/maximal baselength plot presented previously.
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 substrate position of center of massxc
1.57
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 substrate positionxc
Fig. 5.11: (a) Baselength of the moving solution over center of mass for different slip lengths, showing increased variation in baselength with reduced slip. System parameters:w0=0,∆w=0.2,k=10
(b) Contact line velocity over the center of mass for different slip length, showing the alternating oscillation and decreased average ve-locity for decreased slip length forw0=0, ∆w=0.2, k=10. The total fluctuation amplitude is changed only slightly compared to the change in the average velocity.
When comparing the graph to the contact line velocities in Fig. 5.11(b), it becomes clear that the velocity of the front or the back contact line is maximal when the baselength is close to the average baselength. The desynchronized contact line motion is observed again, with the second small maximum synchronized with the maximum of the other contact line.
It is interesting to observe that the difference between the minimal and maximal contact line velocities is similar for both slip lengths. The total velocity is significantly higher for the high slip length case, giving the droplet less time to adapt to underlying substrate by deformation.
One of the hypotheses for the origin of the observed bistability is that a faster moving droplet is strongly deformed due to hydrodynamic stresses. This allows the droplet to store in the configuration of the free interface to overcome energy barriers. To test this hypothesis, Fig. 5.12 shows the total energy (excluding the gravitational energy) of the configuration at different substrate positions. While it displays the same oscillating behavior, the absolute
change is very small with less than 0.5% of the total configurational energy.
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 substrate position xc
2.505 2.506 2.507 2.508 2.509 2.510 2.511
sumofsubstrateandfreeenergyEsub+Efree
ls=7 ls=3
Fig. 5.12: Sum of wetting and free interface energy over the center of mass for different slip length, showing slightly decreased fluctuation for higher slip forw0=0,∆w=0.2,k=10
This observation indicates that the droplet does not so much store energy to cross the en-ergy barriers, as it bypasses them by not going through each local minimum in the enen-ergy landscape. This might be explained by the increased translational velocity with increasing slip length, with only a smaller change in the velocity variation over one period. The question is which underlying process is responsible for this behavior and if there is a reduced model capable of reproducing this change in the bifurcation scenario.