Statics and dynamics of a cylindrical droplet under an external body force

Dynamics of Droplets Moving on Lubricated Polymer Brushes

Understanding the dynamics of drops on polymer-coated surfaces is crucial for optimizing applications such as self-cleaning materials or microfluidic devices. While the static and dynamic properties of deposited drops have been well characterized, a microscopic understanding of the underlying dynamics is missing. In particular, it is unclear how drop dynamics depends on the amount of uncross-linked chains in the brush, because experimental techniques fail to quantify those. Here we use coarse-grained simulations to study droplets moving on a lubricated polymer brush substrate under the influence of an external body force. The simulation model is based on the many body dissipative particle dynamics (MDPD) method and designed to mimic a system of water droplets on poly(dimethylsiloxane) (PDMS) brushes with chemically identical PDMS lubricant. In agreement with experiments, we find a sublinear power law dependence between the external force F and the droplet velocity v, F ∝ vα with α < 1; however, the exponents differ (α ∼ 0.6-0.7 in simulations versus α ∼ 0.25 in experiments). With increasing velocity, the droplets elongate and the receding contact angle decreases, whereas the advancing contact angle remains roughly constant. Analyzing the flow profiles inside the droplet reveals that the droplets do not slide but roll, with vanishing slip at the substrate surface. Surprisingly, adding lubricant has very little effect on the effective friction force between the droplet and the substrate, even though it has a pronounced effect on the size and structure of the wetting ridge, especially above the cloaking transition.

Role of Interaction Range and Buoyancy on the Adhesion of Vesicles

Vesicles on substrates play a fundamental role in many biological processes, ranging from neurotransmitter release at the synapse on small scales to the nutrient intake of trees by large vesicles. For these processes, the adsorption or desorption of vesicles to biological substrates is crucial. Consequently, it is important to understand the factors determining whether and for how long a vesicle adsorbs to a substrate and what shape it will adopt. Here, we systematically study the adsorption of a vesicle to planar substrates with short- and long-range interactions, with and without buoyancy. We assume an axially symmetric system throughout our simulations. Previous studies often considered a contact potential of zero range and neutral buoyancy. The interaction range alters the location and order of the adsorption transition and is particularly important for small vesicles, e.g., in the synapse. Whereas even small density differences between the inside and the outside of the vesicle give rise to strong buoyancy effects for large vesicles, e.g., giant unilamellar vesicles, as buoyancy effects scale with the fourth power of the vesicle size. We find that (i) an attractive membrane-substrate potential with nonzero spatial extension leads to a pinned state, where the vesicle benefits from the attractive membrane-substrate interaction without significant deformation. The adsorption transition is of first order and occurs when the substrate switches from repulsive to attractive. (ii) Buoyancy shifts the transversality condition, which relates the maximal curvature in the contact zone to the adhesion strength and bending rigidity, up/downward, depending on the direction of the buoyancy force. The magnitude of the shift is influenced by the range of the potential. For upward buoyancy, adsorbed vesicles are at most metastable. We determine the stability limit and the desorption mechanisms and compile the thermodynamic data into an adsorption diagram. Our findings reveal that buoyancy, as well as spatially extended interactions, are essential when quantitatively comparing experiments to theory.

Wettability gradient-driven droplets with an applied external force

On homogeneous substrates, droplets can slide due to external driving forces, such as gravity, whereas in the presence of wettability gradients, sliding occurs without external forces since this gradient gives rise to an internal driving force. Here, we study via molecular dynamics simulations the more complex behavior when droplets are driven under the combined influence of an external and internal driving force. For comparison, the limiting cases of a single driving force are studied as well. During a large part of the sliding process over the borderline of both substrates, separating both wettabilities, the velocity is nearly constant. When expressing it as the product of the effective mobility and the effective force, the effective mobility mainly depends on the mobility of the initial substrate, experienced by the receding contact line. This observation can be reconciled with the properties of the flow pattern, indicating that the desorption of particles at the receding contact line is the time-limiting step. The effective force is the sum of the external force and a renormalized internal force. This renormalization can be interpreted as stronger dissipation effects when driving occurs via wettability gradients.

Statics and dynamics of polymeric droplets on chemically homogeneous and heterogeneous substrates

We present a molecular dynamics study of the motion of cylindrical polymer droplets on striped surfaces. We first consider the equilibrium properties of droplets on different surfaces, we show that for small stripes the Cassie-Baxter equation gives a good approximation of the equilibrium contact angle. As the stripe width becomes nonnegligible compared to the dimension of the droplet, it has to deform significantly to minimize its free energy; this results in a smaller value of the contact angle than the continuum model predicts. We then evaluate the slip length and thus the damping coefficient as a function of the stripe width. For very small stripes, the heterogeneous surface behaves as an effective surface, with the same damping as a homogeneous surface with the same contact angle. However, as the stripe width increases, damping at the surface increases until reaching a plateau. Afterwards, we study the dynamics of droplets under a bulk force. We show that if the stripes are large enough the droplets are pinned until a critical force. The critical force increases linearly with stripe width. For large enough forces, the average velocity increases linearly with the force, we show that it can then be predicted by a model depending only on droplet size, contact angle, viscosity, and slip length. We show that the velocity of the droplet varies sinusoidally as a function of its position on the substrate. However, for bulk forces just above the depinning force we observe a characteristic stick-slip motion, with successive pinnings and depinnings.

Slip of polymer melts over micro/nano-patterned metallic surfaces

The slip behavior of high-density polyethylenes (HDPEs) is studied over surfaces of different topology and surface energy. Laser ablation has been used to micro/nano-pattern the surface of dies in order to examine the effect of surface roughness on slip. In addition, fluoroalkyl silane-based coatings on smooth and patterned substrates were used to understand the effect of surface energy on slip. Surface roughness and surface energy effects were incorporated into the double reptation slip model (Ebrahimi et al., J. Rheol., 2015, 59, 885-901) in order to predict the slip velocity of studied polymers on different substrates. It was found that for dies with rough surfaces, polymer melt penetrates into the cavities of the substrate (depending on the depth and the distance between the asperities), thus decreasing wall slip. On the other hand, silanization of the surface increases the slip velocity of polymers in the case of smooth die, although it has a negligible effect on rough dies. Interestingly, the slip velocity of the studied polymers on various substrates of different degrees of roughness and surface energy, were brought into a mastercurve by modifying the double reptation slip velocity model.

Hierarchical multi-scale simulations of adhesion at polymer–metal interfaces: dry and wet conditions

Multi-scale simulations are performed to study the adhesion properties of different polymer–metal interfaces in the absence and presence of water.

Directed transport of polymer drops on vibrating superhydrophobic substrates: a molecular dynamics study

Using molecular dynamics simulations of a coarse-grained polymer liquid we investigate the transport of droplets on asymmetrically structured (saw-tooth shaped), vibrating substrates. Due to a continuous supply of power by substrate vibrations and the asymmetry of its topography, the droplets are driven in a preferred direction. We study this directed motion as a function of the size of the droplets, the linear dimensions of the substrate corrugation, and the period of vibrations. Two mechanisms of driven transport are identified: (i) one that relies on the droplet's contact lines and (ii) in a range of vibration periods, the entire contact area contributes to the driving. In this latter regime, the set-up may be used in experiments for sorting droplets according to their size. Additionally, we show that the linear dimension of the substrate corrugation affects the flux inside the droplet. While on a substrate with a fine corrugation droplets mostly slide, on a more coarsely corrugated substrate the flux may exhibit an additional rotation pattern.

Sliding drops across alternating hydrophobic and hydrophilic stripes

We perform a joint numerical and experimental study to systematically characterize the motion of 30 μl drops of pure water and of ethanol in water solutions, sliding over a periodic array of alternating hydrophobic and hydrophilic stripes with a large wettability contrast and a typical width of hundreds of microns. The fraction of the hydrophobic areas has been varied from about 20% to 80%. The effects of the heterogeneous patterning can be described by a renormalized value of the critical Bond number, i.e., the critical dimensionless force needed to depin the drop before it starts to move. Close to the critical Bond number we observe a jerky motion characterized by an evident stick-slip dynamics. As a result, dissipation is strongly localized in time, and the mean velocity of the drops can easily decrease by an order of magnitude compared to the sliding on the homogeneous surface. Lattice Boltzmann numerical simulations are crucial for disclosing to what extent the sliding dynamics can be deduced from the computed balance of capillary, viscous, and body forces by varying the Bond number, the surface composition, and the liquid viscosity. Beyond the critical Bond number, we characterize both experimentally and numerically the dissipation inside the droplet by studying the relation between the average velocity and the applied volume forces.

Do liquid drops roll or slide on inclined surfaces?

We study the motion of a two-dimensional droplet on an inclined surface, under the action of gravity, using a diffuse interface model which allows for arbitrary equilibrium contact angles. The kinematics of motion is analyzed by decomposing the gradient of the velocity inside the droplet into a shear and a residual flow. This decomposition helps in distinguishing sliding versus rolling motion of the drop. Our detailed study confirms intuition, in that rolling motion dominates as the droplet shape approaches a circle, and the viscosity contrast between the droplet and the ambient fluid becomes large. As a consequence of kinematics, the amount of rotation in a general droplet shape follows a universal curve characterized by geometry, and independent of Bond number, surface inclination and equilibrium contact angle, but determined by the slip length and viscosity contrast. Our results open the way toward a rational design of droplet-surface properties, both when rolling motion is desirable (as in self-cleaning hydrophobic droplets) and when it must be prevented (as in insecticide sprays on leaves).

Surfactant influence on rivulet droplet flow in minitubes and capillaries and its downstream evolution

During our investigations of two-phase flow in long hydrophobic minitubes and capillaries, we have observed transformation of the main rivulet into different new hydrodynamic modes with the use of different kinds of surfactants. The destabilization of rivulet flow at air velocities <80 m/s occurs primarily due to the strong branching off of sub-rivulets from the main rivulet during the downstream flow in the tube. The addition of some surfactants of not-so-high surface activity was found to increase the frequency of sub-rivulet formation and to suppress the Rayleigh and sinuous instabilities of the formed sub-rivulets. Such instabilities result in subsequent fragmentation of the sub-rivulets and in the formation of linear or sinuous arrays of sub-rivulet fragments (SRFs), which later transform into random arrays of SRFs. In the downstream flow, SRFs further transform into large sliding cornered droplets and linear droplet arrays (LDAs), a phenomenon which agrees with recent theories. At higher surface activity, suppression of the Rayleigh instability of sub-rivulets with surfactants becomes significant, which prevents sub-rivulet fragmentation, and only the rivulet and sub-rivulets can be visualized in the tube. At the highest surface activity, the bottom rivulet transforms rapidly into an annular liquid film. The surfactant influence on the behavior of the rivulets in minitubes is incomparably stronger than the classic example of the known surfactant stabilizing influence on a free jet. The evolution of a rivulet in the downstream flow inside a long minitube includes the following sequence of hydrodynamic modes/patterns: i) single rivulet; ii) rivulet and sub-rivulets; and iii) rivulet, sub-rivulets, sub-rivulet fragments, cornered droplets, linear droplet arrays, linear arrays of sub-rivulet fragments and annular film. The formation of these many different hydrodynamic patterns downstream is in drastic contrast with the known characteristics of two-phase flow, which demonstrates one mode for the entire tube length. Recent achievements in fluid mechanics regarding the stability of sliding thin films and in wetting dynamics have allowed us to interpret many of our findings. However, the most important phenomenon of the surfactant influence on sub-rivulet formation remains poorly understood. To achieve further progress in this new area, an interdisciplinary approach based on the use of methods of two-phase flow, wetting dynamics and interfacial rheology will be necessary.

Molecular transport and flow past hard and soft surfaces: computer simulation of model systems

The equilibrium and flow of polymer films and drops past a surface are characterized by the interface and surface tensions, viscosity, slip length and hydrodynamic boundary position. These parameters of the continuum description are extracted from molecular simulations of coarse-grained models. Hard, corrugated substrates are modelled by a Lennard-Jones solid while polymer brushes are studied as prototypes of soft, deformable surfaces. Four observations are discussed. (i) If the surface becomes strongly attractive or is coated with a brush, the Navier boundary condition fails to describe the effect of the surface independently of the strength and type of the flow. This failure stems from the formation of a boundary layer with an effective, higher viscosity. (ii) In the case of brush-coated surfaces, flow induces a cyclic, tumbling motion of the tethered chain molecules. Their collective motion gives rise to an inversion of the flow in the vicinity of the grafting surfaces and leads to strong, non-Gaussian fluctuations of the molecular orientations. The flow past a polymer brush cannot be described by Brinkman's equation. (iii) The hydrodynamic boundary condition is an important parameter for predicting the motion of polymer droplets on a surface under the influence of an external force. Their steady-state velocity is dictated by a balance between the power that is provided by the external force and the dissipation. If there is slippage at the liquid-solid interface, the friction at the solid-liquid interface and the viscous dissipation of the flow inside the drop will be the dominant dissipation mechanisms; dissipation at the three-phase contact line appears to be less important on a hard surface. (iv) On a soft, deformable substrate like a polymer brush, we observe a lifting-up of the three-phase contact line. Controlling the grafting density and the incompatibility between the brush and the polymer liquid we can independently tune the softness of the surface and the contact angle and thereby identify the parameters for maximizing the deformation at the three-phase contact.

Structure and flow of droplets on solid surfaces

The structure and flow of droplets on solid surfaces is investigated with imaging and scattering techniques and compared to simulations. To access nanostructures at the liquid-solid interface advanced scattering techniques such as grazing incidence small-angle x-ray scattering (GISAXS) with micro- and nanometer-sized beams, GISAXS and in situ imaging ellipsometry and GISAXS tomography are used. Using gold nanoparticle suspensions, structures observed in the wetting area due to deposition are probed in situ during the drying of the droplets. After drying, nanostructures in the wetting area and inside the dried droplets are monitored. In addition to drying, a macroscopic movement of droplets is caused by body forces acting on an inclined substrate. The complexity of the solid surfaces is increased from simple silicon substrates to binary polymer brushes, which undergo a switching due to the liquid in the droplet. Nanostructures introduced in the polymer brush due to the movement of droplets are observed.

Stability and dynamics of droplets on patterned substrates: insights from experiments and lattice Boltzmann simulations

The stability and dynamics of droplets on solid substrates are studied both theoretically and via experiments. Focusing on our recent achievements within the DFG-priority program 1164 (Nano- and Microfluidics), we first consider the case of (large) droplets on the so-called gradient substrates. Here the term gradient refers to both a change of wettability (chemical gradient) or topography (roughness gradient). While the motion of a droplet on a perfectly flat substrate upon the action of a chemical gradient appears to be a natural consequence of the considered situation, we show that the behavior of a droplet on a gradient of topography is less obvious. Nevertheless, if care is taken in the choice of the topographic patterns (in order to reduce hysteresis effects), a motion may be observed. Interestingly, in this case, simple scaling arguments adequately account for the dependence of the droplet velocity on the roughness gradient (Moradi et al 2010 Europhys. Lett. 89 26006). Another issue addressed in this paper is the behavior of droplets on hydrophobic substrates with a periodic arrangement of square shaped pillars. Here, it is possible to propose an analytically solvable model for the case where the droplet size becomes comparable to the roughness scale (Gross et al 2009 Europhys. Lett. 88 26002). Two important predictions of the model are highlighted here. (i) There exists a state with a finite penetration depth, distinct from the full wetting (Wenzel) and suspended (Cassie-Baxter, CB) states. (ii) Upon quasi-static evaporation, a droplet initially on the top of the pillars (CB state) undergoes a transition to this new state with a finite penetration depth but then (upon further evaporation) climbs up the pillars and goes back to the CB state again. These predictions are confirmed via independent numerical simulations. Moreover, we also address the fundamental issue of the internal droplet dynamics and the terminal center of mass velocity on a flat substrate.

Drop dynamics on hydrophobic and superhydrophobic surfaces

We investigate the dynamics of micron-scale drops pushed across a hydrophobic or superhydrophobic surface. The velocity profile across the drop varies from quadratic to linear with increasing height, indicating a crossover from a sliding to a rolling motion. We identify a mesoscopic slip capillary number which depends only on the motion of the contact line and the shape of the drop, and show that the angular velocity of the rolling increases with increasing viscosity. For drops on superhydrophobic surfaces we argue that a tank treading advance from post to post replaces the diffusive relaxation that allows the contact line to move on smooth surfaces. Hence drops move on superhydrophobic surfaces more quickly than on smooth geometries.

Dynamics of nanodroplets on topographically structured substrates

Mesoscopic hydrodynamic equations are solved to investigate the dynamics of nanodroplets positioned near a topographic step of the supporting substrate. Our results show that the dynamics depends on the characteristic length scales of the system given by the height of the step and the size of the nanodroplets as well as on the constituting substances of both the nanodroplets and the substrate. The lateral motion of nanodroplets far from the step can be described well in terms of a power law of the distance from the step. In general the direction of motion depends on the details of the effective laterally varying intermolecular forces. But for nanodroplets positioned far from the step it is solely given by the sign of the Hamaker constant of the system. Moreover, our study reveals that the steps always act as a barrier for transporting liquid droplets from one side of the step to the other.

Temperature dependence of the slip length in polymer melts at attractive surfaces

Using Couette and Poiseuille flows, we extract the temperature dependence of the slip length, delta, from molecular dynamics simulations of a coarse-grained polymer model in contact with an attractive surface. delta is dictated by the ratio of bulk viscosity and surface mobility. At weakly attractive surfaces, lubrication layers form; delta is large and increases upon cooling. Close to the glass transition temperature Tg, very large slip lengths are observed. At a more attractive surface, a sticky surface layer is built up, giving rise to small slip lengths. Upon cooling, delta decreases at high temperatures, passes through a minimum, and grows for T-->Tg. At strongly attractive surfaces, the Navier-slip condition fails to describe Couette and Poiseuille flows simultaneously. The simulations are corroborated by a schematic, two-layer model suggesting that the observations do not depend on details of the computational model.