Article pubs.acs.org/Langmuir
Scaling Hydrodynamic Boundary Conditions of Microstructured Surfaces in the Thin Channel Limit Georgia A. Pilkington, Rohini Gupta, and Joel̈ le Fréchette* Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21218, United States ABSTRACT: Despite its importance in many applications and processes, a complete and unified view on how nano- and microscale asperities influence hydrodynamic interactions has yet to be reached. In particular, the effects of surface structure can be expected to become more dominant when the length scale of the asperities or textures becomes comparable to that of the fluid flow. Here we analyze the hydrodynamic drainage of a viscous silicone oil squeezed between a smooth plane and a surface decorated with hexagonal arrays of lyophilic microsized cylindrical posts. For all micropost arrays studied, the periodicity of the structures was much larger than the separation range of our measurements. In this thin channel geometry, we find the hydrodynamic drainage and separation forces for the micropost arrays cannot be fully described by existing boundary condition models for interfacial slip or a no-slip shifted plane. Instead, our results show that the influence of the microposts on the hydrodynamic drag exhibits three distinct regimes as a function of separation. For large separations, a no slip boundary condition (Reynolds theory) is observed for all surfaces until a critical (intermediate) separation, below which the position of the no-slip plane scales with surface separation until reaching a maximum, just before contact. Below this separation, a sharp decrease in the no-slip plane position then suggests that a boundary condition of a smooth surface is recovered at contact.
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INTRODUCTION Understanding and controlling fluid flow on small scales is important in many industrial applications, particularly due to the continual drive toward the miniaturization of systems and development of micro- and nanoscale devices.1−3 In recent years there has been much focus on understanding the flow characteristics of rough or structured surfaces.4 This interest has been mainly motivated by observations of superhydrophobicity on surfaces with hierarchical nano- and microscale roughness such as, and most famously, the lotus leaf.5 As a result, many experimental and theoretical studies have focused on investigating the flow of polar fluids on hydrophobic surfaces bearing surface structure or roughness.6−19 Almost unanimously, these studies have revealed that the presence of nonwetting asperities gives rise to interfacial slip and attributed their effect to the presence of air bubbles or layers within the structure of surface asperities. As a result, a large number of theoretical models have been developed for predicting the magnitude of the slip, parametrized by the effective slip length, bs, on a wide variety of superhydrophobic textures based on their characteristic dimensions.14−19 In the case of wetting surfaces, the consequences of surface roughness or structure on fluid flow are less clear, with relatively few studies reported.20−26 Primarily these studies have involved measuring the hydrodynamic force between a smooth sphere and a rough or structured flat using colloidal probe atomic force microscopy (CP-AFM)20,21,24,25,27 or the surface force apparatus (SFA).26 In general, the presence of wetting © XXXX American Chemical Society
structures has led to a reduction in the hydrodynamic force acting on an approaching surface, similar to that observed for slip. For this reason, a number of studies have similarly employed existing theoretical models for hydrodynamic slip to quantify the effects of wetting structures or roughness.20,21,25 However, concerns due to the measurement of unphysically large slip lengths,20,21,25 as well as notably large deviations at near-contact separations, has led to questioning of the appropriateness of slip models on wetting surfaces.20 In particular, Vinogradova and coworkers have discussed this problem.22,23 For wetting surfaces with nanometer-scale roughness, they demonstrated that the hydrodynamic force exhibits notably different asymptotic behavior with decreasing surface separation, leading to a weaker force at near-surface separations than can be predicted by invoking slip.24 Previously, it has been theoretically argued by Richardson28 that a more realistic boundary condition for wetting surfaces with microscale structure would be that of no-slip, but with the effective boundary position located at a position intermediate between the top and bottom of the asperities. More recently, Vinogradova and coworkers have adopted this approach and proposed a new theoretical model for predicting the hydrodynamic forces for wetting surface asperities.24 Instead of allowing for slip, the shifted plane model corrects the Received: November 9, 2015 Revised: February 15, 2016
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DOI: 10.1021/acs.langmuir.5b04134 Langmuir XXXX, XXX, XXX−XXX
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Langmuir
Figure 1. (a) Schematic diagram of the typical SFA experimental configuration employed in this work. A surface decorated with micron-sized SU-8 posts orientated with their long axes perpendicular to an underlying SU-8 substrate interacts with a smooth hydrophobized silver surface across a viscous silicone oil. The surfaces are mounted on crossed cylindrical lenses (radius of curvature R ≈ 2 cm) to achieve a point contact with contact geometry equivalent to that of a sphere and a flat. The hydrodynamic drainage force is measured as a function of the separation distance, h, which is defined as the point of closest separation between the surfaces. (b) Bright-field microscopy top-view image of an SU-8 micropost array. (c) Schematic of characteristic dimensions of micropost arrays. The SU-8 posts of diameter, d, are arranged in hexagonal arrays of channel width, w, and depth, D. The periodicity of the arrays is defined as L = w + d.
hydrodynamic force for separation by shifting the position of the no-slip plane from the top of the surface structure to a position located at a distance (rs) within the surface structure. By employing the shifted model, they then demonstrated that the hydrodynamic forces for wetting surfaces of varying degrees of roughness, obtained from both theoretical calculations and direct force measurements, could be well described.22,24 To further test the shifted plane model, the same group has also measured the hydrodynamic drainage of a viscous oil over lyophilic surfaces textured with micron-sized grooves.23 For surface separations, h, larger or comparable than the periodicity of the microgrooved textures, L, it was demonstrated that the hydrodynamic force could be similarly well described. However, for separations much smaller than the periodicity of the microcorrugations (h ≪ L), the hydrodynamic forces were noted to deviate from those predicted by the shifted plane model. By performing supplemental theoretical calculations of the average velocity across the gap between a sphere and a microcorrugated surface in the limit of a thin channel (h < L), the origin of the observed deviation for thin channel separations was explained in terms of a scaling shifted plane position (rs) with decreasing separation h. In this limiting geometry, the scaling for rs was shown to play a role at surface separations smaller than or comparable to depth, D, of the microcorrugations and vary with the porosity or liquid fraction of the textures. However, a model for predicting the specific scaling of rs with respect to h for different texture parameters was not obtained. In this study, we analyze the hydrodynamic drainage and adhesion forces between a smooth plane and surfaces decorated with hexagonal arrays of lyophilic micrometer-sized cylindrical posts of different dimensions (diameter, d, depth, D, width, w, periodicity, L; cf. Figure 1 and Table 1) in a viscous silicone oil. The measurements were performed over a wide range of velocities (v ≈ 25−700 nm s−1) using a surface forces apparatus (SFA). For all micropost arrays studied, the periodicity of the structures was much larger than the separation range of our measurements. Therefore, we consider the geometry of our measurements to be within the thin channel limit (h < L).
Table 1. Characteristic Dimensions of the SU-8 Micropost Arrays Studied in This Work surface
depth D (μm)
diameter d (μm)
width w (μm)
periodicity L = w + d (μm)
solid fraction ϕs = d/L
A B C D E
0.30 0.42 0.80 1.26 1.21
65.0 6.5 6.5 65.0 65.0
6.5 6.5 3.0 6.5 6.5
71.5 13.0 9.5 71.5 71.5
0.91 0.50 0.68 0.91 0.91
To quantify the drainage and separation forces measured in this regime, we first compare our results to theoretical models for both interfacial slip and a no-slip shifted plane boundary condition; however, we find that neither the drainage nor the separation forces can be fully quantified at all separations by either of these existing boundary condition models. Instead, by iteratively fitting the hydrodynamic forces as a function of surface separation, we show a limit of validity for the shifted plane model. For each of the microstructures studied, the position of the no-slip plane rs is found to scale with the surface separation, h, depending on both the surface depth and porosity. Such hydrodynamic interactions in limited geometries are particularly relevant to microfluidics and other systems involving the confinement of fluid flow.
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EXPERIMENTAL METHODS
Surface Force Apparatus (SFA). The experimental details of these measurements have been previously reported.26 In brief, the hydrodynamic drainage and separation forces (denoted here as FH) were measured between two surfaces mounted on two rigid crossedcylinders with a radius R ≈ 2 cm using the MK II SFA.29 For all measurements, the surfaces were submerged in a Teflon bath filled with silicone oil (Xiameter PMX 200 Silicone Fluid, η = 48 mPa·s or 50 cSt). To perform the dynamic force measurements, we drove the bottom surface (mounted on a soft cantilever spring; spring constant k = 165 N m−1) toward the (immobile) top surface at a constant velocity (v) using microstepping motors. The approach velocities ranged from v = 25 to 700 nm s−1. During each approach or retraction, the surface separation, h, was monitored by recording the fringes of equal chromatic order (FECO)30−32 resulting from the multiple beam interferometry B
DOI: 10.1021/acs.langmuir.5b04134 Langmuir XXXX, XXX, XXX−XXX
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Langmuir (MBI) between the surfaces.33,34 The parabolic shape of the FECO, which reflects the crossed-cylindrical geometry of the surfaces, was used to calculate the radii of curvature (R1 and R2) of the crossed cylinders. For the structured surfaces, the microscale features were also observable as discontinuities in the FECO.35 The surface separation was then calculated from the wavelength of the FECO vertices, corresponding to the point of closest approach, using the multilayer matrix method36,37 and a fast spectral correlation algorithm.38 For the structured surfaces, the microposts were orientated with their longest axis perpendicular to the underlying substrate. Therefore, the point of closest approach was at the top of the microposts. During our measurements no efforts were taken to orientate the point of contact with respect to the top of the posts as the radius of the region of interaction r ≈ 2Rh is expected to be larger than the diameters of the posts up to short surface separations. Surface Fabrication. Muscovite mica (Ruby, ASTM V-1, S&J Trading) was chosen as a substrate for both the structured and smooth surfaces. The mica pieces were cleaved to a thickness between 2 to 8 μm in a laminar flow hood and placed on a freshly cleaved larger, thicker (backing) sheet. The pieces were then coated with a 50 nm film of silver (99.999% purity, Alfa Aesar) via thermal evaporation (Kurt J. Lesker Nano38). The mica pieces are not part of the interferometer but act as a support for the silver films, which were further functionalized as follows. Smooth or Structured SU-8 Layers (Top Surface). For the fabrication of both the smooth or microstructured SU-8 layer a silvered mica piece (supported by a backing sheet) was taped to a glass slide and a negative photoresist (SU-8 2007, MicroChem) was spincoated onto the silver layer at 5000 rpm for 1 min. After spin-coating, the mica piece was transferred onto a fresh backing sheet and baked on a hot plate according to the SU-8 suppliers guidelines. The smooth SU-8 layer was then exposed to an energy of 140 mJ/cm2 and baked again following an identical procedure to that of the pre-exposure baking step. For the microstructured surfaces, an additional thinner SU-8 layer (SU-8 200.5 or 2002, MicroChem) was spin-coated on top of the existing smooth SU-8 layer to a desired thickness, exposed to an energy of 100 mJ/cm2 through a chrome-on-glass mask, and baked. After the postexposure baking step the SU-8 coated surfaces, smooth or structured, were immersed in SU-8 developer (MicroChem) for a 3 min. Any excess SU-8 was then removed by rinsing with excess developer. The SU-8 surfaces were then rinsed with isopropanol and dried with compressed air. For each experiment, an SU-8 coated surface was glued (EPON 1004 epoxy), mica side down, onto a cylindrical silica lens. Previously McHale et al. have shown silicone oil to completely wet similar surfaces of similar SU-8 structures surface after an initial wetting phase.39 Therefore, to ensure the structures were completely wetted a small amount of oil was spin coated onto the structured surface at 100 rpm for 30 s, then 200 rpm for 1.5 min before mounting the cylindrical lens in a (immobile) top surface holder inside the SFA. Hydrophobized Silver (Bottom Surface). The silvered mica pieces (supported by a larger, thicker backing sheet) were rendered hydrophobic (water contact angle ∼107°) by immersing them overnight in a 1 mM hexadecanethiol (HDT; 92% purity, SigmaAldrich) solution in ethanol (200 proof, Pharmco-Aaper). After immersion, the silvered mica pieces were rinsed thoroughly with ethanol to remove any excess thiol from the surfaces and dried with a gentle flow of N2 gas. For each experiment, a hydrophobized silvered mica piece was removed from the backing support sheet and glued, mica side down, onto a cylindrical silica lens. The cylindrical lens was then mounted on a bottom surface holder, which is suspended on the cantilever spring inside the SFA.
FH = −
6πηRHR G dh h dt
(1)
where η is the dynamic viscosity of the intervening fluid, RH = [(2R1R2)/R1 + R2], and RG = (R1R2)1/2 are the harmonic and geometric means of the crossed cylinder radii, and dh/dt is the change in relative separation distance as a function of time t. To obtain the hydrodynamic force between the surfaces we employed the method proposed by Chan and Horn.40 For each approach or retraction, the interaction between the surfaces was calculated using the instantaneous surface separation, h, which was recorded as a function of time t. Because of drag force resulting in a deflection of the spring, the rate of change in surface separation (dh/dt) is different from the separation predicted by Reynolds lubrication theory. In this case, the quasi-static force balance between the hydrodynamic drainage force and the restoring force of the cantilever spring supporting the bottom surface can be given by40 k(h − h initial − vt ) = −
6πηRHR G dh h dt
(2)
where hinitial is the surface separation at t = 0 s and v is the constant drive velocity of the bottom surface (negative for approach and positive for retraction). To obtain the theoretical predictions for dh/dt, we solved eq 2 numerically for the different velocities by employing the fourth- and fifth-order Runge−Kutta method (ode45 in MATLAB). Existing Boundary Condition Models. To quantify the deviations from Reynolds lubrication theory that we observe for the micropost surfaces, we compare our experimental results to existing boundary condition models for slip and a shifted plane. In both cases, eq 1 is modified to incorporate a correction factor f * FH = −
6πηRHR G dh f* h dt
(3)
For slip, we invoke a partial slip boundary condition, for which we assume slip at only the microstructured surface (cf. Figure 2a). In this model, the correction f * to the hydrodynamic force takes the form41 f* =
⎧ ⎤⎫ ⎪ 4bs ⎞ 1⎪ 3h ⎡⎛ h ⎞ ⎛ ⎢⎜1 + ⎥⎬ ⎨ ⎟ ⎜ 1 ln 1 1 + + − ⎟ ⎪ ⎥⎦⎪ 4⎩ 2bs ⎢⎣⎝ 4bs ⎠ ⎝ h ⎠ ⎭
(4)
where the degree of interfacial slip at the structured interface is quantified by the effective slip length bs (cf. Figure 2a), which is used as a fitting parameter. For the shifted plane model, we assume a no-slip boundary condition, as expected for completely wetting smooth surfaces, but at an equivalent smooth plane located at a distance rs (cf. Figure 2a) between rs = 0 (at the top of posts) and rs = D (at the bottom of the posts).24 In this case, the correction factor f * is given by
f* =
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h h + rs
(5)
At very large separations, both models are equivalent and if separated far enough reach Reynolds limit; however, at shortrange separations the shifted plane model results in a smaller force compared with that predicted for slip (cf. Figure 2b). The reason is that for the shifted plane model the surfaces are essentially further apart but with the same flow profile as the one between smooth surfaces (with a no slip boundary
DATA ANALYSIS Hydrodynamic Drainage Force Measurement and Analysis. For small separations (h ≪ R), the hydrodynamic force FH acting between two crossed cylinders of radii R1 and R2 across a Newtonian fluid can be predicted for no-slip boundary condition by Reynolds lubrication equation C
DOI: 10.1021/acs.langmuir.5b04134 Langmuir XXXX, XXX, XXX−XXX
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Figure 2. (a) Schematic diagram illustrating different possible boundary conditions at a structured surface. The black solid line represents a Poiseuille flow velocity profile for a no-slip boundary condition at the top of the structures. The blue dashed line represents a slip boundary condition, characterized by the effective slip length bs. The red dashed-dotted line corresponds to a shifted plane boundary condition where the no-slip plane is located at a distance rs from the tops of the structures. (b) Hydrodynamic force FH plotted as a function of inverse separation (1/h) for: a smooth surface with a noslip boundary condition (Reynolds theory; black solid line), a partial slip boundary condition (blue dashed line), and a shifted plane boundary condition (red dashed-dotted line). For comparison, the magnitudes of bs and rs are equal.
Figure 3. (a) Normalized hydrodynamic drainage and (b) separation forces (FH/RGRH) between a smooth hydrophobized silver surface and an SU-8 micropost surface (Surface D) in a viscous silicone oil at a relative drive velocity of 245 nm s−1 plotted as a function of surface separation h. For comparison, the insets show the (a) normalized hydrodynamic drainage and (b) separation forces for a smooth hydrophobized silver surface and a smooth SU-8 surface at a similar relative velocity (252 nm s−1). The solid and dashed-dotted lines represent predicted force curves for a smooth surface positioned at the top and bottom of the microposts, calculated using Reynolds theory (eq 1), for the same relative velocities as the experiment curves. The radii of curvature of the interacting surfaces in the main figures were R1 = 2.00 cm and R2 = 1.86 cm. In the insets, R1 = 1.81 cm and R2 = 2.13 cm.
condition). Conversely, the slip model alters the hydrodynamic forces by changing the flow profile in the gap, leading to a decrease in the forces compared with Reynolds theory due to the breakdown of the no-slip boundary condition, but the surfaces are still effectively at the same separation. To obtain theoretical fits to the experimental data, the values of bs and rs were systematically adjusted in the two correction factors for each model (cf. eqs 4 and 5). Best fits to the data were determined from the bs or rs values that provided the lowest mean squared error (MSE) between the raw experimental data (h versus t; cf. Figure 3 inset) and the theoretical curves.
For all surfaces, separations of
