Influence of interfacial rheology on drainage from curved surfaces

Thin lubrication ﬂ ows accompanying drainage from curved surfaces surround us ( e


Introduction
In this work, we examine the drainage of liquids from curved surfaces and focus on an important aspect of this problem that has not received previous attention: the interaction of rheologically complex, insoluble lipid layers at the surface of the draining liquid. One application of our work is drainage and thinning of the tear lm atop silicone hydrogel contact lenses.
Osbourne Reynolds originally formulated the problem of drainage between two approaching rigid parallel surfaces in 1886. 1 In his seminal paper, Reynolds proposed the lubrication theory equations that govern thin lm ow and showed that the draining lm thickness decayed as a square-root in time (h $ t À1/2 ), also known as Reynolds thinning law. Today, drainage of thin lms nds diverse applications from drop-drop coalescence, [2][3][4][5] to drainage of foams, [6][7][8] and biologically relevant applications, such as the stability of the tear lm. [9][10][11] Important developments on the problem of drainage from curved surfaces came about a century later, when Hartland 12 reported on experiments measuring the transient thickness of a thin liquid lm squeezed beneath an approaching solid sphere. In his experiment, the thin lm is of a lighter liquid and is sandwiched between the sphere at the top and a denser, immiscible liquid at the bottom. Hartland offered a simple theory assuming a uniform thickness of the draining lm, and his prediction for the diminishing lm thickness was identical to Reynolds thinning law. Interestingly though, Hartland observed that as the thin, lighter liquid was squeezed out beneath the sphere, changes in the dynamic pressure led to the formation of a dimple, which he revealed by impressive photographic evidence. These dimples in the free surface were a result of the pressure eld arising from the interplay of gravity, viscous stresses, and interfacial tension. Not surprisingly these dimples were unexplained by his simple theoretical model. However, in a later paper, 13 he offered a numerical analysis that predicted the observed dimple formation by assuming a nonuniform lm thickness.
Over the years, various other researchers have contributed to the understanding of drainage from a curved surface: Jones and Wilson 14 employed matched asymptotic expansions to predict the shape of the free interface of the thin liquid lm sandwiched underneath a translating solid sphere; Smith and Van De Ven 15 considered the long time stages of lm drainage; Leal and coworkers 16 considered the possibility that a solid sphere might "break through" the uid interface and drag a long tail of liquid behind its wake; and more recently, Dietrich et al. 17 offered recent experimental results for conditions where impacting solid spheres drag long tails, adding further physical insight into the complex interfacial phenomenon involved in drainage from curved surfaces.
In addition to the drainage of liquid lms sandwiched between solid spheres and free interfaces, researchers have investigated the case of advancing drops and bubbles as these problems are linked to important questions concerning emulsion and foam stability. In the literature, there are numerous studies on lm drainage associated with advancing droplets, 4,18-21 as well as advancing bubbles. 6,22,23 Chan et al. 24 have compiled a critical review covering literature up to 2010, on lm drainage associated with advancing drops and bubbles.
However, the problem of drainage in the presence of rheologically complex uid-uid interfaces has received scant attention. Park 25 considered the problem of dip coating in the presence of insoluble surfactants. In his analysis, Park models the process of Langmuir-Blodgett deposition, where a planar substrate is withdrawn through a layer of insoluble surfactant that is initially deposited on a liquid interface (normally water). The associated coating ow is the classic Landau-Levich-Derjaguin (LLD) dip-coating process, 26,27 which was originally developed for a Newtonian liquid in the absence of surfactants. Park does not explicitly incorporate interfacial shear rheology into his model but rather employs a convection-diffusion equation for surfactant concentration at the interface, which leads to a prediction of Marangoni interfacial ows. An important outcome of Park's analysis is that the layer of the coating liquid is thickened in the presence of insoluble surfactants. More recently, Scheid et al. 28 improved on this analysis, but in the limit where Marangoni stresses are absent and where the surfactant layer is characterized by a surface viscosity. Those authors also demonstrated that the coating layer thickness increases with the interfacial viscosity of the surfactant layer. A complete review of the literature on drainage in the presence of rheologically active species at the interface has been recently compiled by Langevin. 29 This work is aimed at understanding drainage from a curved surface that approaches the air-liquid interface in the presence of viscoelastic, insoluble monolayers and multilayers. Unlike the Landau-Levich-Derjaguin coating problem, where steady state proles can be assumed, drainage from a spherical cap is inherently transient, which poses an additional layer of complexity. With this background, we rst discuss the theoretical description using a model that describes the thinning of a lm coated with an insoluble but Newtonian lipid layer. We compare the results of our model with drainage experiments on lms coated with dipalmitoylphosphatidylcholine (DPPC) and meibomian lipids, which are known to be viscoelastic. [30][31][32] Finally, a discussion on the consequences of surface rheology on lm drainage is presented with some ideas for future work.

Theoretical description
Consider a hemispherical dome that is raised through a bath of liquid resulting in the entrainment of a liquid lm of thickness h(q, t), which is shrouded by an insoluble monolayer (see Fig. 1). The surface of the hemispherical dome is covered with a hydrogel (so contact lens). By design, our experimental platform enables the analysis of osmotic pressure ow through the hydrogel, which is worth considering and will be addressed in a future publication. However, our focus in this current work is on the possible inuence of the viscoelastic monolayer on drainage atop this hydrogel. Consequently, we assume that the lower boundary of the porous layer (lens) is impermeable and stationary.
Our theory is based on the following ve conditions.
(1) The draining ow at the surface of the lens (r ¼ R), is axisymmetric (i.e., all the quantities are independent of the azimuthal coordinate).
(2) The draining liquid layer is thin compared with the dome radius, i.e., the aspect ratio is small, h 0 /R $ 10 À3 , which enables the application of lubrication theory (h 0 is the initial lm thickness at q ¼ 0).
(3) The Bond number, Bo ¼ rgR 3 /(gh 0 ) $ 10 4 , is large, so that the effects of surface tension can be neglected during drainage ow. The surface tension of the layer is g, and r is the liquid's density.
(4) Marangoni stresses are neglected in our simple model. Justication for this assumption, based on the relative Gibbs elasticities of the insoluble surfactants reported in this work, is presented in the Discussion section.
(5) The insoluble surfactant layer is characterized by a constant surface shear viscosity (m s ).
In the lubrication limit, the equation of motion in the aqueous layer is where y ¼ r À R, m is the bulk viscosity, u is the velocity in the q direction and g is the magnitude of gravitational acceleration. We assume that m and r are constant. Eqn (1) is subject to boundary conditions u| y¼0 ¼ 0 and u| y¼0 ¼ u s (q, t). The velocity at the surface, u s (q, t), is unknown and depends on the viscoelasticity of the interface. In the case of a simple interface that is only characterized by surface tension, this upper boundary condition would be the familiar zero stress requirement at the air-liquid interface. However, since we are concerned with interfaces laden with insoluble layers that are potentially viscoelastic, a rheological surface constitutive equation must be prescribed. The solution to the velocity eld within the draining liquid is This result is then combined with the following equation of mass conservation: where the volumetric ow rate, Q, at q is Introducing the dimensionless variables, It is le to specify the interfacial stress balance from which the surface velocity eld, u s , can be evaluated. In general, this requires a balance of interfacial stresses with tractions exerted on the interface by the bulk uid. Assuming the interface is Newtonian and only characterized by a surface shear viscosity, u s , this is given by the tangential stress balance, which can be re-written in the following dimensionless form, where Dh v vq and Bq ¼ m s h 0 /(mR 2 ) is a modied Boussinesq number. This dimensionless group gauges the relative strength of interfacial stress to bulk stresses. In the asymptotic limit that Bq ¼ 0, we have the simple case of an air-liquid interface that amounts to forcing gradients in the subphase velocity to be zero at the interface and is equivalent to the nostress condition on the surface. On the other hand, as Bq / N, the interface resists lateral deformation (tangentially immobile), and this leads to a no-slip boundary condition at the interface. Intermediate values of the Boussinesq number (0 < Bq < N) produce drainage dynamics between the limiting cases of a zero stress and no-slip condition at the free surface. The solutions of the lm drainage problem for the limiting cases of either a zero stress condition or the no-slip condition at the free surface are easily obtained and that analysis is not elaborated here. The nal solutions for the lm thickness at the apex of a hemispherical dome H(q ¼ 0, s) in these two limits are of the form where a ¼ 1/3 for the case of zero-stress (Bq ¼ 0) and a ¼ 1/12 when the no-slip condition is applied to the free surface (Bq / N). Eqn (8) is an exact solution obtained using the method of characteristics. For intermediate values of Bq, this function form is used to t the data using a as a tting parameter. It is also noted from eqn (8) that the height decays with a squareroot dependence on time, similar to the Reynolds thinning law.
Predictions of the dimensionless lm thickness as a function of dimensionless time are shown in Fig. 2 for several values of the Boussinesq number. Shown in this gure are numerical solutions to eqn (5) and (7) along with ts of those numerical predictions to eqn (8) with a used as a tting parameter. As expected, when Bq ¼ 0, the zero stress condition yielding a ¼ 1/3 is reproduced. Likewise, as Bq / N (the values Bq ¼ 100, 1000 in the gure), the expected response for a tangentially immobile surface (a ¼ 1/12) is reproduced. Interestingly, when intermediate values of Bq are chosen, the simple square-root dependence of eqn (8) can be used successfully to t the numerical results.
It must be emphasized that the initial lm thickness captured at the apex, h 0 , is simply specied and is not a prediction of this simple model. As the experimental results will reveal, this captured thickness is a strong function of interfacial rheology and its prediction remains an important, unsolved problem.   Inc. (Alabaster, AL), and diluted to a concentration of 1 mg mL À1 in chloroform (Sigma-Aldrich, St. Louis, MO). Animal ethics were approved for the collection of meibomian lipids. Bovine meibomian lipids were harvested from cow eyelids obtained from a local abattoir. The eyelids were incubated at 37 C, and the lipids squeezed out by applying force on the eyelid margins following the protocol of Nicolaides et al. 33 The meibomian lipids were then collected on a glass slide using a spatula and stored in an amber jar at À20 C until use. Prior to experimental use, the meibomian lipids were dissolved in chloroform to a concentration of 1 mg mL À1 . Henceforth, meibomian lipids will be referred to as "meibum".
Bovine meibum is a complex mixture of waxy esters, cholesterol esters, polar lipids and fatty acids. Unlike DPPC, the viscoelastic properties of meibum will vary slightly from source to source. We have therefore carefully measured the viscoelastic properties of the meibum samples used in this study. Furthermore, an appreciation of how this viscoelastic material inuences drainage was essential for one of the central applications of these studies, which concerns the stability of the tear lm.
3.1.2 Silicone hydrogel lens. A single type of commercial silicone hydrogel so contact lens was used as the substrate in this study: AirOptix Aqua (Lotralcon B, CIBA Vision, Duluth, GA). For our experiments, lenses with a low dioptric power of À0.5 were chosen to minimize undulations in the lens thickness. The lenses were obtained in commercial blister packs which typically contain buffered saline with surface active agents. To leach out blister-pack surfactants, the lenses were soaked overnight in 5 mL phosphate buffer solution (PBS, Gibco Life Technologies, NY) at room temperature with gentle agitation. This procedure effectively eliminated any osmotic uxes. Aer soaking, the lenses were gently rinsed with fresh PBS and delicately transferred to the experimental setup using teoncoated tweezers. As documented, the AirOptix Aqua lens is a silicone hydrogel with 33% water-content and 67% principal monomers that include trimethylsiloxy silane, siloxane monomers and N,N-dimethylacrylamide. The surface of the AirOptix lens additionally has a plasma coating which renders the surface hydrophilic.

Interfacial rheology
An interfacial shear rheometer (KSV NIMA Ltd., Helsinki, Finland) was used to measure the interfacial shear rheology of the insoluble materials. 34 A detailed analysis of the device is available elsewhere. 35,36 The protocol that was followed to make these measurements is described in detail in Leiske et al. 30 Briey, insoluble materials dissolved in chloroform were spread droplet wise on a clean water subphase (Milli-Q) using a clean Hamilton syringe. Chloroform was allowed to evaporate for 15 minutes, and the interface was compressed using symmetric teon barriers at 1.5 cm 2 min À1 . Dynamic interfacial moduli of the insoluble surfactants were measured by imposing interfacial, oscillating strains. Strain amplitudes of 0.029 and 0.0174 were used for DPPC and meibum, respectively, which fall within the linear viscoelastic regimes of the respective materials. The data was measured at a frequency of 1 Hz.

Film thickness measurements
The aqueous lm thickness measurements were made using a Filmetrics F70 thin-lm measurement system (Filmetrics, Inc.) with a High-Brightness White LED light source over a wavelength of 400-720 nm. The incident light was reected normally at the apex of the hemispherical interface. The working distance was 3 mm and the spot size was 10 mm. Due to the similar refractive indices of the contact lens (1.42) and the aqueous layer (1.33), the interference data was t to a single-layer model to obtain the combined thickness of the contact lens and the aqueous lm, with a combined refractive index (1.42) chosen to accurately reproduce the known thickness of the contact lens. The possible error on the thickness measurements of the aqueous lm is less than 10%. As the aqueous lm drained, the thickness measurements as a function of time would monotonically decay to a constant value, which represents the thickness of the contact lens. This thickness was then subtracted from the measurements to obtain the actual thickness of the draining aqueous lm. The average thickness at the center of the lens using our measurement technique is 100 AE 10 mm, which is in good agreement with the values reported by the manufacturer. 37

Drainage apparatus
The instrument consists of several main elements (Fig. 3). A Teon mini-Langmuir trough is xed onto a stationary support structure. This trough allows one to spread an insoluble layer of material (DPPC, meibum) on top of an aqueous subphase at a controlled surface pressure. This control is important since it is well established that the interfacial rheology is a strong function of surface pressure. For most insoluble amphiphiles, increasing the surface pressure will increase the surface viscosity and surface moduli. 30,31 The surface pressure is monitored using a paper Wilhelmy balance connected to a surface pressure sensor (KSV NIMA Ltd., Helsinki, Finland). Surrounding the trough is a moving platform that can elevate a hemispherical surface (supporting a contact lens) from an initial position slightly beneath the interface and li it at computer-controlled speeds (10 mm s À1 ). The curvature of the hemispherical surface can potentially diminish the surface pressure due to dilatational effectsthe consequences of which will be discussed in the Results section. However, the change in surface area as a result of elevation of the dome is small (15%) and this minimizes the possible inuence of Marangoni ows and dilatational stresses. Additionally, the surface pressure is continuously monitored during the experiment and only a small deviation of AE0.5 mN m À1 is observed during the duration of the experiment. So, the surface pressure remains practically constant during the experiment. The inset photograph on the le side of Fig. 3 shows the lens placed onto a small titanium dome, which has the same base curvature as a commercial silicone hydrogel contact lens. Attached to the moving platform is a high speed interferometer that is focussed on the apex of the lens. The schematic in Fig. 1 shows a contact lens having captured a thin layer of draining uid. On top of the draining layer, an insoluble lipid layer may have been deposited. Once the elevation motion is commenced, the interferometer begins measuring the thickness of the aqueous layer on top of the lens as a function of time. The top of the lens is located at q ¼ 0 (Fig. 1).

Results and discussion
The surface pressure versus mean molecular area (MMA) isotherms for the two insoluble materials are shown in Fig. 4. The isotherm measures the surface pressure of the insoluble layer as a function of decreasing area (proceeding right to le on the horizontal axis). As the layer is compressed, DPPC shows a characteristic transition from a liquid-expanded state to a liquid-condensed state at 6 mN m À1 , evidenced by the presence of a plateau in the curve. Upon further compression, DPPC molecules are packed closely together, resulting in a steep rise in the surface pressure. Meibum, on the other hand, does not exhibit any distinctive plateaus or the corresponding phase transitions.
Furthermore, from the slopes of the isotherms in Fig. 4, the Gibbs modulus (E Gibbs ¼ ÀvP/vln A) can be estimated, where A is the mean molecular area and P is the surface pressure. As shown in Fig. 5, both the insoluble materials exhibit relatively constant Gibbs moduli at surface pressures greater than 10 mN m À1 . At 15 mN m À1 , E Gibbs,DPPC $ 100 mN m À1 and is an order higher than E Gibbs,meibum $ 10 mN m À1 .
The interfacial rheology for the insoluble materials at room temperature is presented in Fig. 5. Over the range of surface pressures accessible to our instrument, the viscous interfacial modulus of DPPC remains greater than its elastic modulus. Additionally, the elastic modulus for DPPC becomes measurable only above 25 mN m À1 . Compared with DPPC, meibum exhibits high surface elasticity, with the surface elastic modulus larger than the surface viscous modulus. The surface moduli for meibum increase over four orders of magnitude, from 0.008 mN m À1 to 20 mN m À1 . Both the elastic and viscous moduli of meibum are one and two orders higher than DPPC, respectively. It is important to note that even though both the materials demonstrate viscoelastic behavior, DPPC is viscous-dominated, whereas meibum is highly elastic. From these measurements, the surface shear viscosity for DPPC at 15 mN m À1 is 0.005 mN s m À1 and is in good agreement with values published in the literature. 32 Also, at surface pressures below 20 mN m À1 , the surface rheology of DPPC is independent of frequency. 38 It is important to note that  The results of several drainage experiments are shown in Fig. 6 as a function of dimensionless time, as dened previously. Four curves are shown: the blue (diamond) symbols are for a layer of water in the absence of an insoluble layer, the green (squares) and teal (triangle) symbols are the result of depositing DPPC on top of the water at surface pressures of 15 mN m À1 and 20 mN m À1 , respectively, and lastly, the red (circles) for meibum on top of water at a surface pressure of 15 mN m À1 .
As shown in Fig. 5, the interfacial viscoelasticity of DPPC monotonically increases with surface pressure. Increasing the surface pressure of DPPC produces two noticeable effects, both of which are benecial to the retention of aqueous lms above the lens: (1) the presence of viscoelastic layers allows the capture of much thicker aqueous lms; (2) the time scale for thinning of the layers is increased substantially with the presence of the insoluble layers. In the same gure, the drainage dynamics of bovine meibum are compared to those of DPPC at the surface pressure of 15 mN m À1 . The meibum layer is observed to capture a substantially thicker lm of water and the time taken for drainage is also much longer.
The solid curves that accompany the data in Fig. 6 are ts to eqn (8) using a as a tting parameter. This simple square root law ts the data remarkably well. The data for an uncoated lm of water (blue) is well-t using a ¼ 1/3, which is the value expected for a stress-free air surface. The surface pressure values for water were 0 AE 0.5 mN m À1 . The possibility that there is some minor contamination in the water cannot be ruled out, although this effect is minor.
The values of a required to t the DPPC drainage data depend on the surface pressure of the surfactant. At a surface pressure of 15 mN m À1 (the green symbols), a ¼ 0.2 ts the data quite well. As the surface pressure is increased to 20 mN m À1 (the teal colored symbols) and the surface viscous modulus increases by approximately 50%, the data is best t using a value of a ¼ 0.13. Thus, increasing the viscoelasticity of DPPC requires a progressively smaller value of a. The data in Fig. 6 also reveal that the captured, initial lm thickness increases substantially when the surface pressure of DPPC is increased.
The value used to t the bovine meibum coated surface (red), however, is a ¼ 0.05, which is substantially less that the value of a ¼ 1/12 $ 0.083 that is appropriate for an innite Boussinesq number and a no-slip air surface. This suggests that the highly viscoelastic response of bovine meibum presents qualitatively different dynamical responses and these are explored below.
The parameter a provides a convenient means of comparing experiment and theory and Fig. 7 provides such a comparison. From the numerical results (black), it is observed that, as the Boussinesq number increases, a decreases from 1/3 to 1/12 with a strong transition when a ¼ 1 These results indicate the dampening of drainage dynamics due to pure surface shear viscosity effects. The best empirical t in the transition region is a $ Bq À2/5 . On the same plot, experimental values are shown as well. The drainage of water is t using a value of a ¼ 1/3 and Bq / 0. The experimental Boussinesq values for DPPC are 0.1 and 0.9 at surface pressures of 15 mN m À1 and 20 mN m À1 , respectively. These experimental values are on the same order of magnitude as the corresponding numerical predictions, however, the actual values are slightly smaller, which may be due to the assumption of a simple Newtonian interface to model these viscoelastic materials. For meibum, a high Boussinesq  value (15) is observed but the a value is smaller than 1/12. Thus, the experimental values are in good, qualitative agreement with the model and match the transition in the drainage dynamics at a critical Boussinesq number on the order of unity. A quantitative comparison, however, would require the use of more sophisticated interfacial rheological constitutive models, which is not within the scope of the present paper.
Two possible sources for the slower-than-expected drainage are both linked to the deformation of the interface as the hemispherical domed substrate is elevated. As the hemisphere is pushed upward, it will stretch the interface, leading to two possible outcomes: (1) elongational stretching of the viscoelastic meibum interface may result in a recoil of this interface and a subsequent back ow at the surface, which will slow down drainage; (2) stretching of the interface could result in a polar gradient in the surfactant from the apex of the dome that can induce an upward Marangoni ow that will also slow down drainage.
To quantify the existence of back ow kinematics retarding drainage, surface ow in meibum coated lms was monitored using a CCD camera, looking normally down on to the curved draining surface. At a moderately high surface pressure of 15 mN m À1 , meibum forms a waxy viscoelastic skin at the airwater interface which is translucent and is highly textured. These textures have been imaged in past studies using Brewster Angle Microscopy (BAM), 39 and provide a means of tracking the time dependent position of identiable features in the insoluble meibum layer. Using this imaging system, the displacement vectors joining the initial locations of microstructures at 4 s and their locations aer 44 s of drainage are shown in Fig. 8. The result is a family of displacement vectors that clearly indicate a non-zero, upward ow towards the apex is, which is against the direction of gravity-driven ow.
The presence of a non-zero upward surface ow qualitatively explains the slower dynamical response for meibum. We suspect that this upward ow is a consequence of the surface shear elasticity of meibum and not due to Marangoni ows or dilatational rheological effects. In fact, if the upward ow was a manifestation of surface tension gradients, then this ow would be greater in DPPC, since it has a larger Gibbs modulus than meibum. However, this is clearly not the case as meibum retards drainage more substantially than DPPC. Furthermore, the Gibbs modulus for DPPC is largely independent of surface pressure whereas the drainage dynamics are a strong function of this variable.
Our experimental results indicate that the presence of an insoluble surfactant layer has a two-fold effect on lm drainage: (1) the initial capture thickness is higher with increases in surface viscosity and even more pronounced in the presence of surface elasticity, and (2) the lm drainage rates are retarded as the surface viscoelasticity is increased.
Interfacial viscoelasticity plays a major role in foam stability and many authors have investigated the effect of both the surface shear and dilatational viscosity, on drainage of thin lms. 2,5,[40][41][42][43] These studies have shown that increasing the interfacial shear viscosity has a retarding effect on the lm thinning rate. However, most of these investigations treat the interface as either rigid plane or a Newtonian interface. Tambe and Sharma, 5 were the rst to offer a numerical analysis that attributed viscoelastic properties to the interface bounding plane-parallel lms. Using a simple Maxwell constitutive model for their interface, they revealed that the rate of drainage was strongly inuenced by the surface rheology of the bounding layer. Moreover, they found that the retarding inuence of surface viscosity was more pronounced if the interface possessed nite elasticity. The predictions of these authors are in good qualitative agreement with our experimental ndings.
It is important to underscore that these enhancements in lm stability are a consequence of enhanced interfacial  rheology as surface pressure is increased and not a lowering of surface tension. This was established with experiments using simple, soluble surfactants (SDS) at a concentration of 0.003 mol L À1 , where it was found that drainage was not retarded (data not shown). Thus, this work eliminates the complication of bulk-to-interface diffusion since we employ insoluble surfactants. An important implication of our ndings is to suggest that the interfacial rheology of meibomian lipids on our eyes may play an extremely important role in enhancing tear lm stability and reduced drainage on our eyes.

Conclusions
We have shown that increased lm stability during drainage on a curved surface can occur as a consequence of enhanced surface rheology. Specically, we nd that insoluble surfactants such as DPPC with nite interfacial viscosity stabilize drainage. Moreover, the rate of drainage decreases as the surface pressure of the surfactant layer is increased. Importantly, the retarding inuence is most pronounced when the insoluble surfactant has signicant elasticity, as in the case of meibum. We present a simple theoretical model which enables qualitative comparison of the inuence of surface viscoelasticity on drainage from curved surfaces.