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Martin
Tress‡§
^{a},
Stefan
Karpitschka‡
^{b},
Periklis
Papadopoulos
^{c},
Jacco H.
Snoeijer
^{b},
Doris
Vollmer
^{a} and
Hans-Jürgen
Butt
*^{a}
^{a}Max Planck Institute for Polymer Research, Ackermannweg 10, D-55128 Mainz, Germany. E-mail: butt@mpip-mainz.mpg.de
^{b}Physics of Fluids Group, MESA+ Institute, University of Twente, NL-7500AE Enschede, The Netherlands
^{c}Department of Physics, University Ioannina, P.O. Box 1186, GR-45110 Ioannina, Greece

Received
2nd March 2017
, Accepted 21st April 2017

First published on 4th May 2017

Motivated by the development of lubricant-infused slippery surfaces, we study a sessile drop of a nonvolatile (ionic) liquid which is embedded in a slowly evaporating lubricant film (n-decane) on a horizontal, planar solid substrate. Using laser scanning confocal microscopy we imaged the evolution of the shape of the liquid/liquid and liquid/air interfaces, including the angles between them. Results are compared to solutions of the generalized Laplace equations describing the drop profile and the annular wetting ridge. For all film thicknesses, experimental results agree quantitatively with the calculated drop and film shapes. With the verified theory we can predict height and volume of the wetting ridge. Two regimes can be distinguished: for macroscopically thick films (excess lubrication) the meniscus size is insensitive to changes in film thickness. Once the film is thin enough that surface forces between the lubricant/air and solid/lubricant interfaces become significant the meniscus changes significantly with varying film thickness (starved lubrication). The size of the meniscus is particularly relevant because it affects sliding angles of drops on lubricant-infused surfaces.

Different morphologies of drop and lubricant are possible, depending on whether the lubricant fully wets and cloaks the drop or whether it forms an intervening film between drop and solid substrate.^{7,18} Aiming for a quantitative understanding of the shape of the drop and the wetting ridge we focus on the underlying prototype: a liquid drop in direct contact with a smooth planar solid substrate surrounded by a lubricant film (Fig. 1). For this model system we present experimental data and compare it to numerical and analytical calculations which coincide quantitatively in the relationship between film thickness, drop shape, meniscus height and the angles formed between interfaces.

In each experiment, a drop of IL of ≈0.2 nL was placed on a dry microscope cover slide with a thin syringe needle. The drop forms a static contact angle of ≈50°–80° with the cover slide. Here we restrict ourselves to drops which are much smaller than their capillary length, (g = 9.81 m s^{−2} is the gravitational acceleration), so that the effect of gravitation on the drop shape remains negligible. Rather than the drop/air interface one can also consider the capillary length related to the drop/lubricant interface: .

After depositing the drop, decane was added. It spreads and fully wets the cover slide, but does not underspread the drop of IL. We always started with films of more than 200 μm thickness so that initially the drop was fully covered by decane (Fig. 2). During the gradual evaporation of decane (vapor pressure 195 Pa at 20 °C) the whole range of film thicknesses could be explored down to zero.

Drop and lubricant were imaged with a laser scanning confocal microscope (Leica SP8, HCX PL APO 40× water objective), in the following termed confocal microscopy. The resolution was <400 nm in horizontal and <1 μm in vertical direction. The dye PMI is soluble in both decane and IL. The emission spectra of the dye PMI in decane and IL are sufficiently different, so that we could detect both liquids in two channels simultaneously (Fig. S1, ESI†). During the evaporation of decane, vertical cross sections (xz-plane) through the drop center were recorded. Images were processed to extract the contours of the drop and the lubricant (open source image analysis software FIJI, Fig. 2b and Fig. S2, ESI†). In the fitting procedure of the contours, the refractive indices of decane and the ionic liquid were taken into account to correct for a slight distortion of the images in z-direction.

As decane gradually evaporated, at some point the top of the drop pierced through the decane/air interface. A drop/lubricant/air three-phase contact line (fluid contact line) formed. We define t = 0 s as the point where the lubricant film ruptured on top of the droplet and a drop/air interface formed. As the evaporation continued, the radius r_{m} of the fluid contact line defined in Fig. 1 increased, exposing more and more free surface of the drop (Fig. 2c). During the whole evaporation process, the drop remained pinned to the substrate; i.e. the contact radius r_{s} of the drop/lubricant/solid three phase line (solid contact line) remained constant, while the contact angle Θ_{S} with the solid, changed. Here, Θ_{S} is the angle of the drop/lubricant interface with the solid/drop interface inside the drop.

The quantitative evaluation of a representative experiment is shown in Fig. 3. The film thickness h_{∞} (height far away from the droplet) decreased linearly with time as expected for film evaporation (Fig. 3a, blue circles).^{19,20} In contrast to the linear decrease of h_{∞}, the maximum height of the wetting ridge of lubricant h_{m} (Fig. 3a) decreased by less than 10 μm within the first 490 s. Only during the final phase of evaporation h_{m} decreased rapidly.

Fig. 3 Parameters of drop and meniscus shape during evaporation. (a) Film thickness at large distance to the drop, h_{∞} (blue), and height of the fluid three-phase contact line, h_{m}, (black squares). (b) Temporal development of the angles between the phase boundaries at the fluid three-phase contact line, Θ_{a}, Θ_{d}, Θ_{l}, i.e. the Neumann angles. The solid lines represent the values for the Neumann angles according to eqn (1) with independently measured surface tensions. (c) The contact angle of the drop at the solid-TPCL, Θ_{s}, and the angle of the drop-air interface at the liquid TPCL with respect to the horizontal, Θ, which quantifies the rotation of the Neumann triangle. (d) Young–Laplace pressure in the drop as calculated from the radii of curvature of the air/drop (black) and the drop/lubricant (blue) interfaces. In all cases values measured on the right and left sides of the drop were averaged. Errors are smaller than or equal to symbol size unless they are indicated. |

(1) |

Indeed, we observed that during the whole process of lubricant evaporation, the Neumann angles remained constant (Fig. 3b). The solid lines in Fig. 3b are the Neumann angles as calculated from the measured surface tensions according to eqn (1). The deviations between measured and calculated angles reflect the experimental error in measuring the angles at the fluid contact line.

Although the Neumann angles remain constant, the whole Neumann triangle rotates. We quantify the rotation of the triangle by the angle Θ between the drop/air interface and the horizontal (Fig. 1). Just before decane had fully evaporated (t ≈ 500 s) the Neumann triangle rotated from 12° to 43°. While r_{s} remained constant, the capillary action of the lubricant meniscus reduced the overall aspect ratio of the drop.

In addition to the rotation of the Neumann triangle, Θ_{S} weakly increased during the first 490 s, followed by a fast increase in the last seconds of evaporation. This is in contrast to the “inverse” situation of a volatile drop and a non-volatile lubricant; here the drop evaporates keeping the solid contact angle Θ_{S} constant.^{25}

After a fluid contact line was formed, the drop/air interface at the top of the drop is shaped like a spherical cap with a radius of curvature r_{d}. The Laplace pressure inside the drop is

(2) |

(3) |

The drop/lubricant interface is shaped similar to a part of a torus. Due to symmetry, the radii of the two principal curvatures, r_{dl1} and r_{dl2}, represent the radii of curvature in the xz-plane and the corresponding orthogonal direction, respectively. The former of these curvatures is directly imaged in the cuts of the xy-plane while the latter is determined at the same spot but in perpendicular direction. Since in equilibrium the Laplace pressure in the drop is the same throughout the whole drop, these radii are related by^{26}

(4) |

During the evaporation of the lubricant, the Laplace pressure inside the drop gradually increases (Fig. 3d). The pressure increases by approximately 200 Pa during the first 490 s. The subsequent increase by 500 Pa reflects the strong deformation of the drop during the last stages of evaporation. Eventually, all lubricant has evaporated and the final radius of the drop is equal to the initial radius; both radii are equal because the drop is shaped like a spherical cap and the solid contact line is pinned. The final Laplace pressure is P = 2γ_{d}/r_{0} = 1523 Pa, while the hydrostatic pressure in the droplet remained below 1 Pa and can safely be neglected.

To describe the shape of the lubricant film we consider the Young–Laplace equation in radial symmetry. The thickness of the lubricant film is described as a function of the radial coordinate r by h_{l}(r). In this axisymmetric case, additional pressure terms need to be taken into account to obtain the desired shape at large distance, namely h_{l}(r → ∞) = h_{∞}. For relatively thick lubricant layers this will be provided by gravity. However, for the case where the lubricant film is thinner than ≈100 nm, the complete wetting is ensured by the disjoining pressure. Taking disjoining pressure into account, the extended Young–Laplace equation is:^{27–29}

(5) |

(6) |

h_{m} = h_{d}(r = r_{m}) = h_{l}(r = r_{m}) = h_{dl}(s = 0), r_{dl}(s = 0) = r_{m} | (7) |

γ_{d}cosarctanh_{d}′(r = r_{m}) − γ_{dl}r_{dl}′(s = 0) = γ_{l}cosarctanh_{l}′(r = r_{m}) | (8a) |

γ_{d}sinarctanh_{d}′(r = r_{m}) + γ_{dl}h_{dl}′(s = 0) = −γ_{l}sinarctanh_{l}′(r = r_{m}) | (8b) |

−arctanh_{l}′(r = r_{m}) = Θ_{r} | (9) |

The slopes of the other profiles follow from eqn (8). Further, the slopes of both the drop contour at the center of symmetry as well as the lubricant contour in infinite distance from the drop shall vanish, h_{d}′(r = 0) = 0 and h_{l}′(r → ∞) = 0. Finally, the boundary condition of the drop/lubricant interface at the solid substrate is, in case of a drop pinned to the substrate, given by the corresponding pinning radius, r_{dl}(s = s_{max}) = r_{s} and h_{dl}(s = s_{max}) = 0. Together with these relations and boundary conditions the equations for the pressure drop across the interfaces between drop/air (eqn (2) and (3)), drop/lubricant (eqn (6)) and lubricant/air (eqn (5)) are solved numerically to determine the film profile. For that, the initial values of Θ_{r} = 0 and P_{l} are selected such that the remaining boundary conditions are obeyed for an entirely flat lubricant film. Then, one or more parameters are changed, and new values of Θ_{r} and P_{l} are obtained by a variable-order predictor–corrector algorithm (“shooting”).

Fig. 4 Comparison of experimental (black open squares) and numerical results (red solid lines) for different parameters. (a) Height of the fluid contact line h_{m}vs. film thickness h_{∞} both scaled with respect to the pinned radius of the drop r_{s}. (b) Difference between h_{m} and h_{∞}vs. h_{∞} both scaled to r_{s}. (c) Contact angle of the drop at the solid TPLC vs. h_{∞} scaled to r_{s}. (d) Angle between the drop-air boundary and the horizontal Θ vs. (h_{m} − h_{∞})/r_{s}. Panel (d) also depicts analytical solutions for the asymptotic cases of macroscopic (blue dashed line),^{32} and microscopic film heights h_{∞} (green dotted).^{28} |

Fig. 4 shows that two regimes can be distinguished. Plotting Θ as a function of h_{m} − h_{∞}, two branches that correspond to distinct physical regimes become even more obvious (Fig. 4d). For macroscopically thick films, h_{m} − h_{∞} and Θ increase up to a maximum with decreasing film thickness. We shall call this regime “excess lubrication”. Hereafter, the second regime starts: as lubricant continues to evaporate, the meniscus size decreases more steeply. To the precision of the confocal measurement, the lubricant appears to completely wet the substrate. Θ increases while changes in h_{∞} are below the resolution limit and cannot be used as a reliable independent variable. We will refer to this as “starved lubrication”. The transition is at a film thickness where surface forces, due to van der Waals interactions, and the hydrostatic pressure difference between meniscus and film become equally important for leveling the lubricant at large r:

(10) |

(11) |

(12) |

h_{l}(r) = r_{m}sinΘ_{r}·K_{0}(r/κ_{l}) | (13) |

In the regime of starved lubrication, the apparent radius of the macroscopic part of the liquid meniscus is smaller than the capillary length and the hydrostatic term in eqn (5) can be neglected. An analytical solution of the remaining equation is again unavailable. Once more we use matched asymptotic expansions to obtain a functional relation between h_{m} − h_{∞} and Θ_{r}. Here, we follow the approach of Renk et al.:^{28} the outer regime is again dominated by capillarity, while the inner regime is dominated by the disjoining pressure. In the starved regime, the pressure gauge is

(14) |

h_{l}(r*) = 0, h_{l}′(r*) = 0, | (15) |

(16) |

(17) |

H′(R_{m}) = −tanΘ_{r} | (18) |

(19) |

Here, F(ϕ,k) and E(ϕ,k) are the elliptic integrals of first and second kind, respectively, and K(k) and E(k) are the corresponding complete elliptical integrals. The condition h

(20) |

(21) |

(22) |

Two regimes can be distinguished. In the excess lubrication regime (h_{∞} ≫ 200 nm) the height of the wetting ridge h_{m} is relatively insensitive to changes in film thickness. The meniscus height h_{m} − h_{∞} decreases roughly linearly with increasing film thickness h_{∞}. In the starved lubrication regime, i.e. for thin lubricant films, the height of the wetting ridge h_{m} − h_{∞} increases with increasing film thickness and most lubricant is localized near the edge of the droplet.

The presence of the meniscus is expected to reduce the sliding angle of small drops because the effective mass of the drop is increased (r_{S} ≪ κ_{d}). These results can have practical consequences regarding the drainage and depletion of lubricant by sliding drops. Likely, lubricant in the wetting ridge is taken along if a drop slides down-hill.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm00437k |

‡ Equal contribution. |

§ Present address: Department of Chemistry, University of Tennessee Knoxville, 1420 Circle Dr, Knoxville TN-37919, USA. |

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