Siddarth
Srinivasan
a,
Wonjae
Choi
b,
Kyoo-Chul
Park
c,
Shreerang S.
Chhatre
a,
Robert E.
Cohen
*a and
Gareth H.
McKinley
*c
aDepartment of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA. E-mail: recohen@mit.edu
bDepartment of Mechanical Engineering, University of Texas at Dallas, USA
cDepartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA. E-mail: gareth@mit.edu
First published on 25th April 2013
We estimate the effective Navier-slip length for flow over a spray-fabricated liquid-repellent surface which supports a composite solid–air–liquid interface or ‘Cassie–Baxter’ state. The morphology of the coated substrate consists of randomly distributed corpuscular microstructures which encapsulate a film of trapped air (or ‘plastron’) upon contact with liquid. The reduction in viscous skin friction due to the plastron is evaluated using torque measurements in a parallel plate rheometer resulting in a measured slip length of bslip ≈ 39 μm, comparable to the mean periodicity of the microstructure evaluated from confocal fluorescence microscopy. The introduction of a large primary length-scale using dual-textured spray-coated meshes increases the magnitude of the effective slip length to values in the range 94 μm ≤ bslip ≤ 213 μm depending on the geometric features of the mesh. The wetted solid fractions on each mesh are calculated from free surface simulations on model sinusoidal mesh geometries. The trend in measured values of bslip with the mesh periodicity L and the computed wetted solid-fraction rϕs are found to be consistent with existing analytic predictions.
A growing body of work has attempted to utilize ‘superhydrophobic’ textured surfaces with regular microfabricated patterns or hierarchical textures11–21 to amplify the effective fluid slip at the interface. On such non-wetting surfaces, the liquid layer sits on a composite solid–air interface (or Cassie–Baxter interface22,23), by entrapping pockets of air between the individual topographical features. The composite interface can robustly resist pressure-induced wetting transitions over a range of liquid surface tensions and externally imposed pressure differences by careful design of the fabricated surface morphology.24,25 In facilitating the establishment of an air layer or ‘plastron’ that is stable to externally imposed pressure differences, such surfaces can reduce the frictional dissipation associated with laminar flows in microfluidic devices,17,26 in rheometers,20,27–29 in pipes,21,30 over coated spheres31 and in turbulent flows in channels.32 The reduction in viscous skin friction due to the composite micro-textured interface can be significant in confined flows; in a seminal study, Watanabe et al. demonstrated a 14% reduction of drag in a 16 mm diameter pipeline textured with a superhydrophobic surface.21 The vast majority of subsequent investigations on superhydrophobic surfaces have involved precisely fabricated and regularly patterned geometries which help develop a systematic understanding of the influence of the wetted solid-fraction and surface periodicity in promoting large effective slip lengths and associated friction reduction. It is less clear whether substrates with randomly deposited micro-structures, which are cost-effective to manufacture and more readily applicable to large coated areas, would exhibit similar dramatic reduction in drag.5,33 Sbragaglia and Prosperetti34 and Feuillebois et al.35 propose theoretical models to investigate how random textures can enhance the effective slip at a fluid–solid interface. In the present work, we use parallel-plate rheometery to determine the effective slip length for flow over spray-fabricated corpuscular microtextures that are randomly deposited over both flat substrates and on woven wire meshes.
In Fig. 1, we illustrate conceptually the effective slip present at the interface for flow over a spherically textured non-wetting substrate in the presence of an unconfined or pressure driven flow (Fig. 1a) and also in a laminar Couette flow (Fig. 1b). In each of these cases, the conventional no-slip condition is valid on the top of the wetted features, while the local fluid velocity at the liquid–air interface is determined by a tangential stress-balance. The net dissipative interaction of the fluid with this textured surface can be expressed using an area-averaged effective slip velocity 〈Vw〉, or alternatively in terms of an effective slip length 〈bslip〉, again averaged over the periodicity (L) of the textured surface. As indicated in Fig. 1, the slip length 〈bslip〉 can be greater than the characteristic scale of the texture (2R); a larger value of slip length indicates higher friction-reducing ability of the corresponding textured surface. For the laminar Couette flow shown in Fig. 1b, the velocity in the fluid varies linearly and the resulting shear rate is constant, except in the immediate vicinity of the surface texture. The apparent shear rate in the Couette flow (with the assumption of a no-slip boundary) a = Vplate/h can be related to the true shear rate t that is established in the fluid in the presence of slip as ah = t(h + 〈bslip〉), where h is the gap height.36 For a Newtonian liquid with viscosity η and shear stress τ = η, measurement of the frictional forces or torques (in a torsional rheometer) due to Couette flow at a fixed height between two flat parallel rigid surfaces (with no slip) and textured non-wetting surfaces (with slip) enables the effective slip length on the latter to be directly related to the measured viscous friction using a rheometer.20,27,28 The viscous stress for a linear Couette flow in the proximity of the top plate can be expressed as τslip = ηVplate(h + 〈bslip〉)−1 = ηa(1 + 〈bslip〉/h)−1, where Vplate is the velocity of the upper plate. In a parallel-plate rheometer with disc radius R, the total torque M = ∫2πr2τdr measured by the instrument for a Newtonian fluid is37M = (πηaR3)/2. Therefore, for a fixed upper plate velocity, the ratio of the apparent viscosities (or measured torques) between (i) two flat surfaces with no slip and (ii) a flat surface and a textured non-wetting surface with slip can be directly related to the average slip as,
(1) |
Fig. 1 (a) A schematic diagram showing a liquid flow on a textured non-wetting surface, possessing an effective velocity 〈w〉 that is averaged over the texture period L. (b) A laminar Couette flow with average shear rate a = Vplate/h between a non-wetting textured bottom surface exhibiting an average slip length 〈bslip〉 and a conventional flat solid top surface possessing a no-slip boundary condition. |
Eqn (1) implies that in order for fluid slip in confined laminar flows to be manifested as a significant effect, the magnitude of the slip length 〈bslip〉 should be comparable to the length scale of the flow h (i.e., b/h ∼ O(1)). This can be readily achieved in a rheometer because gaps in the range h ≲ O(100 μm) can be attained reliably.38 A more universal measure of fluid slip is the fractional extent of drag reduction (DR) associated with the reduction in the measured apparent viscosities and can thus be written for a torsional Couette flow as:
(2) |
Rheometric torque measurements can therefore be usefully employed as a macroscopic measurement technique that provides a systematic method to probe area-averaged microscopic liquid-slip phenomenon over a large, random (and possibly anisotropic) surface morphology. Care must however be taken to ensure that edge effects are eliminated and systematic errors are minimized.27,39,40
Fig. 2 (a) SEM image of spray-coated superhydrophobic surface displaying corpuscular micro-structured morphology. (b) Schematic of plate–plate fixture mounted on a controlled stress rheometer used to quantify apparent slip lengths using a 50 vol% glycerol–water mixture as probe liquid. |
Fig. 3 Apparent viscosities of 50 vol% glycerol–water mixture measured for a gap separation of h = 1000 μm on (i) flat spin-coated hydrophobic surface; (ii) spray-coated superhydrophobic silicon wafer; (iii) spray-coated superhydrophobic mesh with wire radius R = 254 μm and mesh spacing D = 805 μm. |
Fig. 4 Ratio of shear viscosity measured in contact with a flat surface to the apparent viscosity measured on the spray-coated superhydrophobic surface plotted against the inverse gap height. The effective slip length is extracted from a linear regression using eqn (1), and the corresponding 95% confidence bands are plotted as dashed lines. |
The effective slip on randomly deposited textures is described by a tensorial quantity that varies depending on the local flow and the geometry specific to that local region.45 Therefore, the numerical value of 〈bslip〉 ≈ 39 μm we obtain from rheometry is a macroscopic averaged representation of the more complex velocity profiles that are established close to the rough hydrophobic texture shown in Fig. 2a. In order to understand the relationship between the mean spacing 〈L〉 of the beaded microstructures (Fig. 2a) and the values of 〈bslip〉 obtained from torque measurements on our sprayed surface, we consider an equivalent periodic model geometry with the same wetted solid fraction as the sprayed substrate. It is convenient to use the Cassie–Baxter equation written in the form, cos θ* = rϕscos θE − 1 + ϕs, to determine the wetted area fraction rϕs. Using a model of hexagonally packed spheres for the corpuscular structures, we have previously estimated53 the total wetted fraction as rϕs ≈ 0.1, where θ* = 160° is the macroscopic apparent contact angle on the sprayed substrate and θE = 124° is the equilibrium contact angle on a flat spin-coated substrate with the same chemical composition as the POSS–PMMA mixture that is sprayed onto the substrate.
In order to use hydrodynamic models to evaluate the predicted slip present on a composite textured surface in which the wetted fraction is only rϕs ≃ 10% it is necessary to determine the characteristic length scale (denoted 〈L〉) of the random sprayed texture. We calculate the mean periodicity between the individual corpuscular features in Fig. 2b by incorporating fluorescent red dye (Nile red) in the fluorodecyl POSS–PMMA solution. The sample was then illuminated with a Helium/Neon (He/Ne) 543 nm laser, and the fluorescence was imaged using a Zeiss LSM 510 confocal microscope. The presence of the red dye, which is embedded in the microstructures produced on spraying, allows for mapping of the three-dimensional surface morphology (for details see ESI Fig. S3†). In Fig. 5, we show a 142 μm × 142 μm planar cross-section of the spray-coated corpuscular substrate imaged at a depth of 25 μm, which corresponds to the vertical midplane of the sprayed surface morphology. The confocal image was thresholded and converted to a binary image using the freely available ImageJ software package.58 The light regions in Fig. 5 correspond to the voids between the corpuscular microstructure, and the dark regions indicate domains of fluorodecyl POSS–PMMA. The location of the centroids of each of these domains was determined by particle analysis, and the mean periodicity was obtained from the centroids using Delauney triangulation to be 〈L〉 ≈ 32 μm. The calculation of 〈L〉 and rϕs for the spray-coated corpuscular morphology allows us to compare our experimentally measured slip length with analytical predictions that have been obtained for model periodic geometries. Ybert et al.47 demonstrate that for a 2D array of solid patches in a square lattice, the slip length scales as:
(3) |
Fig. 5 Confocal microscope slice image of the spray-coated surface tagged with a red Nile dye and imaged at the mid-plane. The overlaid lines correspond to the Delaunay triangulation of the centroids of the individual microtextured features (determined from the ImageJ software), and is used to determine the mean periodicity 〈L〉. |
Mesh | R (μm) | D (μm) | D* | rϕ s | A* | P b (Pa) | 〈bslip〉 (μm) | b mesh (μm) |
---|---|---|---|---|---|---|---|---|
I | 70 | 229 | 4.3 | 0.17 | 4.99 | 190 | 92 ± 10 | 122 |
II | 127 | 191 | 2.5 | 0.24 | 11.1 | 481 | 94 ± 7 | 108 |
III | 127 | 326 | 3.6 | 0.25 | 4.4 | 180 | 157 ± 14 | 149 |
IV | 254 | 452 | 2.8 | 0.40 | 4.1 | 112 | 194 ± 24 | 157 |
V | 254 | 805 | 4.2 | 0.41 | 1.5 | 56 | 213 ± 15 | 219 |
VI | 127 | 720 | 6.7 | 0.24 | 1.0 | 28 | — | 287 |
Fig. 6 (a and b) Scanning electron micrographs at different magnifications of the dual-textured spray-coated superhydrophobic mesh surfaces; (c) image of the solid–liquid–vapor composite interface for a 50 vol% glycerol–water solution resting between a spray-coated mesh and the upper rotating plate of the rheometer. The dark regions indicate the wetted area of the mesh and the light regions correspond to reflection of incident light off the liquid meniscus. The inset shows the structure of the spray-coated mesh in the dry state; (d) the composite interface on a model sinusoidal woven mesh, simulated using Surface Evolver FEM software. The dark blue, light blue, and red colors represent the wetted solid, liquid–air meniscus and the dry mesh surfaces respectively. The inset depicts the structure of the periodic mesh in the dry state. |
The reduction in the total wetted area fraction on this hierarchical structure is expected to further reduce the viscous friction. However, a number of factors, including the meniscus curvature, gravity, inertia, and other body forces, can give rise to a pressure difference that drives the liquid–air interface into the air pockets entrapped within the mesh.24,61 The main source of external pressure in the plate–plate rheometer system is the Laplace pressure from the curvature of the meniscus as it sits on the wires. The Laplace pressure of the liquid sample in the rheometer is inversely related to the gap separation h as:
(4) |
(5) |
The breakthrough pressure Pb is thus a function of the apparent contact angle θ, the characteristic length scale L and a dimensionless geometrical spacing ratio D* = (R + D)/R. Therefore, as long as the external applied pressure difference ΔP < Pb, the composite liquid–air interface is still stable and when ΔP ≃ Pb, a wetting transition occurs in the large pockets of air trapped between the wires. The coated surface of the individual wires are themselves strongly non-wetting (i.e. θ = 160°) and continue to maintain their non-wetting characteristic (even after breakthrough in the large air pockets) due to the much smaller length scale of the microtextures formed by the spraying process.
In Fig. 6c, we show an image of the composite interface that is established upon depositing the glycerol–water probe liquid between the spray-coated mesh and a transparent upper plate. The image is obtained using a CCD camera focused through the upper plate of the rheometer at an oblique angle (for schematic, see ESI Fig. S4†). The dark regions in Fig. 6c correspond to the solid–liquid–air interface resting on the corpuscular structures that have been sprayed on the wires of the woven mesh, while the light regions correspond to incident light reflecting off the large pockets corresponding to the liquid–air interface. The position of the liquid–air interface depends on the externally applied pressure difference ΔP. A comparison between the composite interface that develops on the meshes in the plate–plate rheometer and the simulated interface at a pressure differential of ΔP = 100 Pa is provided in Fig. 6c and d. The meniscus configuration is calculated using the public domain Surface Evolver package.64 The simulation captures the essential details of the composite interface system, including the wetted solid fraction rϕs (dark blue) and the air–liquid fraction ϕa (light blue).
Such calculations can be combined with experimental measurements at different gap separations (and thus, from eqn (4), corresponding to different applied pressures) to understand the progressive decrease in the friction reduction that is achievable. Eqn (5) serves as a framework to allow for a rational selection of non-wetting meshes. The equation can be non-dimensionalized by a reference pressure Pref = 2γlv/lcap (where is the capillary length of the liquid) to obtain a dimensionless scaled breakthrough pressure A* = Pb/Pref. The reference pressure corresponds to the capillary pressure for a large drop of size lcap, and is a measure of the pressure differential across a millimetric scale liquid droplet. Therefore, A* ≲ 1 is the appropriate criterion for the spontaneous wetting transition of the composite interface for a liquid drop on a freely suspended mesh. Six meshes with varying wire radii R and mesh spacing D satisfying A* ≳ 1 were spray-coated and the breakthrough pressure Pb for each sprayed mesh was experimentally determined by vertical immersion into a 50 vol% glycerol–water solution using a dynamic tensiometer and analysis of the resultant force curves (see ESI Fig. S5†). In order to prevent an irreversible wetting transition during the rheometry experiments, the underlying flat surface on which the mesh was horizontally affixed was also spray-coated with the corpuscular microtexture to make it superhydrophobic. The experimentally measured values of Pb are shown in Table 1, along with the values of D* and A*. The measured breakthrough pressures are qualitatively consistent with the simple cylindrical model of eqn (5). As higher pressures are applied across the composite liquid–air interface, the liquid meniscus descends into the weave of the mesh until the meniscus between the wires collapses and rests on the bottom surface. Therefore, the numerical value of A* can also be interpreted as a robustness factor, i.e., a measure of the susceptibility of the meniscus to exhibit this sagging behaviour. Meshes with A* ≫ 1 are robustly metastable, while meshes exhibiting A* ≈ 1 are more prone to a wetting transition.24,25,63 The presence of the flat spray-coated bottom substrate and the connectivity of the air pockets allow the meniscus to reversibly recover to its original location upon removal of the pressure difference acting on the plastron.
Davis and Lauga65 have analytically studied liquid flow along the principal direction of an ideal flat 2D continuous mesh substrate, and obtain an asymptotic estimate for the slip length in the limit of a widely spaced mesh of thin rungs (D* = (R + D)/R ≫ 1) as
(6) |
Fig. 7 The design chart for ideal flat non-wetting mesh surfaces, with contours of fixed slip length plotted as solid red lines (from eqn (6)) and contours of fixed robustness factor A* drawn as dashed black lines as a function of the dimensionless spacing ratio D* = (R + D)/R and the mesh periodicity L = 2D + 2R. The shaded region indicates parts of the design space where A* < 1 and the liquid meniscus will penetrate into the mesh. The data points correspond to the location in the design space of the various superhydrophobic coated meshes used in this study. |
It is important to note that a number of assumptions in the ideal model system considered by Davis and Lauga are not strictly valid during parallel plate rheometry over the spray-coated mesh surfaces. These include the existence of a continuously connected solid–liquid wetted region, a flat liquid–air interface, and the uniform directionality of the imposed flow. Despite the continuous nature of the fibrous woven meshes shown in Fig. 6a, the real contact regions between the liquid and the textured solid surface consists of arrays of discrete elliptical wetted patches as shown in Fig. 6d. In order to calculate the expected frictional drag of these mesh surfaces it is necessary to evaluate the wetted surface fraction as it evolves with the imposed pressure. A set of simulations computing the deformation of the liquid–air interface on model sinusoidal woven meshes under imposed pressure differentials were performed using Surface Evolver. In Fig. 8a and b, we illustrate the computed shape of the composite liquid–air meniscus on Mesh II (with robustness factor A* = 11.1) at pressure differentials of ΔP = 100 Pa and ΔP = 500 Pa respectively. The total wetted solid fraction calculated from the simulation increases from rϕs = 0.24 at ΔP = 100 Pa to rϕs = 0.78 at ΔP = 500 Pa. At a fixed pressure differential of ΔP = 100 Pa, the wetted fraction rϕs also depends on the geometry of the woven mesh. The more open Mesh IV (Fig. 8c) has a reduced robustness factor (A* = 4.1), and consequently the liquid–air interface penetrates deeper into the mesh, corresponding to a larger wetted solid fraction of rϕs = 0.40. In Fig. 8d, we show the variation of the total liquid–solid wetted area fraction rϕs and liquid–air area fraction ϕa with the imposed pressure differential ΔP. The woven topography of the meshes allows for the sum rϕs + ϕa > 1 in general.42,43 The liquid–air interface visibly distorts or ‘sags’ with increasing pressure differences, considerably increasing the wetted solid fraction rϕs on the given texture and correspondingly weakening the overall friction reduction that can be expected. In Table 1, we present the wetted solid fractions on each mesh used in this study calculated at an intermediate pressure differential of ΔP = 100 Pa.
Fig. 8 The simulated composite interface and wetted solid fractions (rϕs) on sinusoidal woven wire meshes of diameter R = 127 μm and spacing D = 191 μm (corresponding to Mesh II), calculated using Surface Evolver at imposed pressure differentials of (a) ΔP = 100 Pa resulting in rϕs = 0.24 and (b) ΔP = 500 Pa resulting in rϕs = 0.40. (c) The simulated composite interface corresponding to Mesh IV (R = 254 μm; D = 452 μm) at ΔP = 100 Pa with rϕs = 0.40; (d) the variation of the wetted solid fraction rϕs and the air–liquid area fraction ϕa with increasing pressure differential (on Mesh II) arising from the Laplace pressure associated with decreasing plate–plate height (ΔP ∝ 1/H, see eqn (4)). |
On an ideal flat mesh (with r = 1), the wetted solid fraction rϕs is directly related to the dimensionless geometrical spacing ratio as . While it is clearly evident from the surface evolver simulations that the model of a continuously connected flat mesh is not strictly correct for robustly non-wetting meshes, the simulated values of rϕs can be used to obtain an alternative estimate for the slip length bmesh on a woven texture by eliminating D* from eqn (6) to obtain
(7) |
Eqn (7) enables us to evaluate the expected value of the slip length that can be generated for the spray-coated woven meshes, where the value of rϕs is estimated for a mesh of given geometrical dimensions using Surface Evolver as shown in Fig. 8. We present predictions of the effective slip length in Table 1 along with values obtained experimentally from rheometry.
In Fig. 9, we plot the ratio of the averaged viscosity measured on a flat surface to that of the superhydrophobic mesh flat/mesh against the inverse gap height 1/h for a series of four meshes. A linear least squares fit of eqn (1) was performed to determine the mean slip lengths which are shown in Table 1, along with the predicted slip lengths (eqn (7)) using the simulated wetted solid fractions calculated for each mesh. The dual textured meshes I to IV show a larger decrease in the apparent viscosity for a given height in the rheometer when compared to the sprayed corpuscular structures discussed in the first section. For a fixed wire radius R = 127 μm, the slip length is observed to increase with mesh spacing, from 〈bslip〉 ≈ 94 μm for Mesh II (D = 191 μm) to 〈bslip〉 ≈ 157 μm for Mesh III (D = 326 μm). Upon further increasing the mesh periodicity (Mesh V: R = 254, D = 805 μm, A* = 1.5), the effective slip length increases to 〈bslip〉 ≈ 213 μm but the mesh exhibits a lower robustness to the wetting transition. The measured slip lengths on these spray-coated meshes are consistent with the analytical prediction for bmesh obtained from eqn (7). The effect of meniscus sagging is most evident for Mesh VI (A* = 1.0), for which flat/mesh < 1 for h ≤ 1250 μm signifying enhanced form drag when the liquid meniscus lies fully in between the features of the mesh. As shown in Fig. 9d, a systematic decrease in the ratio of viscosities corresponding to enhanced frictional dissipation is observed as the Laplace pressure drives the liquid further into the mesh features at lower gap heights.
Fig. 9 Ratio of measured viscosity on a flat surface to the apparent viscosity measured on the friction reducing mesh plotted against the inverse gap height on the following spray-coated meshes surface: (a) Mesh II with R = 127 μm and D = 191 μm (b) Mesh III with R = 127 μm and D = 326 μm (c) Mesh V with R = 254 μm and D = 805 μm (d) Mesh VI with R = 127 μm and D = 720 μm. On the latter mesh, little or no friction reduction is observed due to a wetting transition; in fact the fully wetted Wenzel state results in an enhanced form drag. The slip lengths for each mesh were extracted from a linear regression to eqn (1), and the dotted lines on each plot correspond to the 95% confidence bands. |
The variation of the effective slip lengths of the different geometries depends on the periodicity of the mesh L and the wetted solid fractions viaeqn (7). In Fig. 10, we plot the ratio of 〈bslip〉/L against values of the wetted solid fraction rϕs (obtained from Surface Evolver simulations) for each of the meshes. The solid line corresponds to the prediction obtained from eqn (7) and the dashed line is an alternate estimate of the slip length from eqn (3) which approximates the ellipsoidal wetted regions as a series of discrete circular patches.47,50 Despite the non-ideal topographic features of the woven mesh surface (i.e., fabrication tolerances, waviness and form drag), the model of Davis and Lauga given by eqn (7) captures the slow variation of the experimentally obtained slip values with the wetted solid fraction (calculated from Surface Evolver simulations) over the range of woven meshes used in our study. We are able to generate maximum slip lengths of 〈bslip〉 ≃ 213 μm corresponding to a friction reduction of 30% for a gap of 500 μm on Mesh V. Our results indicate that giant liquid slip can only be obtained as rϕs → 0, consistent with the work of Lee and Kim15 who study 2D post surfaces (with very low area fractions ϕs ≈ 1%). These large slip lengths can be achieved with woven meshes by judicious choice of D* and L, as well as by increasing the intrinsic non-wettability of the mesh coating (i.e. increasing the value of the contact angle θE). However, as we demonstrate in eqn (5) and Fig. 10, such an increase in the slip length necessitates a tradeoff in the robustness of the mesh against wetting transitions.
Fig. 10 Ratio of bslip/L plotted against the solid area fraction rϕs for Meshes I–V. The solid blue line and dashed red line correspond to the predictive curves for bmesh (from eqn (7)) and bdiscrete (from eqn (3)) respectively. |
The larger intrinsic periodicity L in dual-scale sprayed superhydrophobic meshes can greatly increase the achievable slip length. We determined the effective slip length for a series of spray-coated meshes and the extracted effective slip length on the spray-coated meshes ranges between 〈bslip〉 ≈ 94 μm to 〈bslip〉 ≈ 213 μm (see Table 1). The slow variation of the slip length is consistent with the prediction of Davis and Lauga, with the wetted solid fraction rϕs for each mesh determined from Surface Evolver simulations. By comparing the governing equations for the slip length with a dimensionless expression for robustness of the slip-inducing composite interface, we also have shown that the slip lengths of such composite interfaces have a strong inverse coupling with the robustness of the plastron film. This inverse correlation occurs because both the liquid slip and the robustness of the composite interface are scale-dependent properties, unlike static measures of superhydrophobicity such as the effective advancing contact angle θ*. The simplicity of the solution spraying process used in the present study is particularly helpful in facilitating rapid and cheap production of friction-reducing coatings that can be applied over large areas. When combined with periodic textures (such as woven meshes or cylindrical post arrays) that can help support the composite vapour–liquid–solid interface (or plastron), very large slip lengths can be established.
Footnote |
† Electronic supplementary information (ESI) available: Viscosity and torque measurements of n-decane calibration liquid, side view of pinned liquid meniscus, array of confocal microscopy images at various depths, schematic of imaging setup, dynamic force trace of spray-coated mesh from tensiometry. See DOI: 10.1039/c3sm50445j |
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