Flow Reduction in Microchannels Coated with a Polymer Brush

Aug 30, 2012 - Laminar flow drag reduction on soft porous media ... Lionel Bureau , Gwennou Coupier , Frank Dubois , Alain Duperray , Alexander Faruti...
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Flow Reduction in Microchannels Coated with a Polymer Brush Luca Lanotte,†,‡,§ Stefano Guido,‡,§ Chaouqi Misbah,† Philippe Peyla,† and Lionel Bureau*,† †

University of Grenoble 1/CNRS, LIPhy UMR 5588, BP 87, 38041 Grenoble France Department of Chemical Engineering, University of Napoli Federico II, Piazzale Tecchio 80, 80125 Napoli, Italy § CEINGE, Advanced Biotechnologies, via G. Salvatore 486, 80145 Napoli, Italy ‡

ABSTRACT: We report on the design of microchannels made of glass capillary coated with polymer brushes elaborated by the so-called “graftingfrom” technique. We present measurements of velocity profiles for pressure-driven flows of water in such “hairy” capillaries. We show that the flow reduction induced by the presence of the brush is unexpectedly greater than what could be anticipated from simple geometric arguments on the reduction of the effective capillary diameter or from predictions by models describing the brush layer as a poro-elastic boundary.

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preserved or degraded by enzymatic cleavage.22 Such experiments that do suggest that the glycocalyx is likely to be responsible for the observed differences between in vivo and in vitro hemorheology are, however, extremely challenging to perform, whereas control over the flow parameters and the actual state and thickness of the glycocalyx is limited. Still, a better knowledge of the hydrodynamic role of the endothelial surface layer is important (i) in order to understand how glycocalyx dysfunctions are involved in vascular diseases21 and (ii) from the perspective of developing microfluidic-based blood assays that account properly for the interactions between walls and blood constituents. This prompted us to adopt a bioinspired approach to this issue and to investigate the flow in microchannels lined with a synthetic brush of macromolecules of well-controlled thickness. In this article, we report on the design of channels made of glass capillary of 10 μm inner diameter, coated with polymer brushes elaborated by the so-called “grafting-from” technique. We present measurements of velocity profiles for pressuredriven flows of water in such “hairy” capillaries. We show that the flow reduction induced by the presence of the brush is unexpectedly greater than what could be anticipated from simple geometric arguments on reduction of the effective capillary diameter or from predictions by models describing the brush layer as a poro-elastic boundary, as proposed in a previous theoretical study.24

INTRODUCTION Polymer brushes (i.e., ultrathin layers made of macromolecules tethered by one end to an underlying substrate) are increasingly used to tailor surface physicochemical properties.1,2 Such brushes can be elaborated with tight control over their molecular structure and chemical composition. They thus offer an extremely versatile way to tune surface or interface properties in applications such as wetting,3 friction,4,5 adhesion of biological objects (cells, proteins),6 and the manipulation of flow in micro- or nanofluidics.7−10 In the latter situation, which has been studied both experimentally10−12 and numerically13−15 in recent years, brushes have a thickness comparable to the typical size of the nanochannel section and basically act as molecular gates controlling transport. As far as microfluidics is concerned, the use of polymer brushes has been viewed mostly as a means to alter hydrodynamic boundary conditions.7,16 The question of the overall resistance to flow of a microchannel bearing a polymer brush on its walls has, paradoxically, received rather little attention.9,17,18 Such a question, which is of obvious interest for the design of advanced microfluidic devices, is also deeply connected to the problem of blood flow in the microvascular system. Indeed, it is well established that the lumen of blood vessels is lined with a layer made of glycopolymers bound to the membrane of the endothelial cells forming the vessel walls.19−22 This layer, referred to as the glycocalyx, has a thickness estimated in the range of 100−1000 nm and is exposed to the flow of plasma and blood cells. Beyond its biochemical and mechanotransduction functions,21 the glycocalyx is also expected to play a hydrodynamic role and to affect the resistance to flow, in particular, that of microvessels with dimensions comparable to the size of red blood cells (i.e., ≲10 μm). Starting from early observations that the in vivo blood flow resistance of small vessels was significantly larger than expected from in vitro studies,23 in vivo measurements of blood velocity have been performed in microvessels in which the glycocalyx was either © 2012 American Chemical Society

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MATERIALS AND METHODS

Materials. Fused silica capillary tubing of 10 μm internal diameter was obtained from Polymicro Technologies (USA) in rolls of 10 m and cut to the desired length. Flat substrates for ex situ brush characterization were silicon wafers bearing a native oxide layer (ACS, Received: May 29, 2012 Revised: August 2, 2012 Published: August 30, 2012 13758

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Characterization Techniques. Polymer brushes were systematically characterized by ellipsometry measurements of their dry thickness on flat silicon wafers. Their swollen thickness in water was inferred from surface forces measurements as described below. In situ brush growth inside microcapillaries has been checked to yield similar results as on silion wafer samples from electron microscopy images of capillary cross sections and elemental analysis. Ellipsometry. Ellipsometry measurements were performed on a rotating quarterwave plate home-built instrument working at a 70° angle of incidence and a 632 nm wavelength. Data analysis was done by assuming a multilayer model including a silicon substrate, a silicon oxide layer (thickness of 2 nm and refractive index 1.46), and an outermost polymer layer of refractive index 1.51 and thickness to be determined. SEM and EDX Elemental Analysis. Small fragments of the PHEMA brush-coated capillaries (2 to 3 mm in length) were cut to conduct observations by means of a scanning electron microscope (SEM, FEI Inspect). One of their ends was cut roughly perpendicular to the channel axis and used to image the sample cross section, and the other end was crushed in order to obtain a transverse section exposing the internal surface of the coated channel. The samples were placed horizontally on an adhesive support and then coated with a thin layer of gold by plasma-assisted deposition. The support was then placed into the vacuum chamber of the microscope for imaging. We have also performed elemental analysis using the energy-dispersive X-ray (EDX) spectroscopy microanalysis facility of the SEM. The electron beam was directed to the inner surfaces of the microcapillaries to check for the quantities of silicon, oxygen, and carbon. Surface Forces Apparatus. Surface forces apparatus (SFA) experiments were performed on a home-built instrument,26 according to a protocol described in detail in ref 25. Briefly, a pair of freshly cleaved mica sheets (∼1−5 μm in thickness) were glued onto cylindrical lenses of 1 cm radius of curvature. Following the grafting protocol described above, a PHEMA brush was grown on one mica sample after plasma activation of its surface.25 The brush-bearing mica sheet was then mounted into the SFA, facing the bare mica sample, and the gap between the two surfaces was filled with ultrapure water. The surfaces were then approached at low velocity (∼1 nm s−1) while recording the force and the distance between the mica substrates by means of multiple beam interferometry, as described in ref 25. Experimental Setup for Flow Measurements. Pressure-driven flow inside brush-coated capillaries was investigated using a home-built flow cell (Figure 1). The flow cell is composed of a sheet of

France) and freshly cleaved mica sheets (JBG Metafix, France). The monomer, hydroxyethylmethacrylate (HEMA 97%, Sigma-Aldrich), was passed through a column packed with activated microbeads (Inhibitor Remover, Sigma) in order to remove the polymerization inhibitor prior to use. Copper chloride (CuCl, 97%, Alfa Aesar), 2,2′bipyridyl (Bpy, 99+%, Acros Organics), 3-aminopropyl-triethoxysilane (APTES, 98%, Merck), triethylamine (TEA), 2-bromo-2-methylpropionyl bromide (BMPB, 98%, Acros Organics), and dichloromethane (DCM) were used as received. Aqueous solutions were prepared in ultrapure water (18.2 M Ω). Sample Preparation. PHEMA brushes were grown inside microcapillaries and on silicon wafers by the so-called grafting-from technique via surface-initiated atom transfer radical polymerization (ATRP), according to a protocol akin to that described in a previous work.25 PHEMA was chosen for the present study because it is a hydrophilic polymer, and the flow of water in channels coated with such a polymer is expected to correspond to a situation where the macromolecules of the brush are swollen by the flowing liquid. (i) A 4-cm-long piece of microcapillary was cut, and one of its ends was fitted inside and glued to a luer-type connector using a UV setting glue (NOA 81, Norland). This allowed us to connect the capillary to a membrane pump that was used to flush the needed solutions into the sample in the various steps described below. (ii) Capillaries and silicon substrates were first cleaned in a 1 M sodium hydroxide aqueous solution for 5 min and rinsed with water. (iii) An aqueous solution of APTES (10−3 M) was prepared and stirred for 2 h in order to hydrolyze the ethoxysilane groups. The solution was subsequently passed through a 0.2 μm membrane filter in order to remove possible large APTES aggregates. The filtered solution was then pumped into the capillary for 30 s, after which pumping was stopped and the APTES solution was left inside the capillary for 10 min. In parallel, a clean oxidized silicon sample was immersed in the APTES solution for the same amount of time. Samples were then rinsed with water and dried. (iv) The APTES-treated surfaces were then placed in a solution of 2-bromo-2-methylpropionyl bromide (250 μL) and triethylamine (1.25 mL) in dichloromethane (20 mL) for 10 min and then successively rinsed with pure dichloromethane, ethanol, and water before being dried. This step leads to bromineterminated surfaces from which atom-transfer polymerization can be initiated. (v) A solution of HEMA (4 mL) in ultrapure water (10 mL) was placed in a round flask. A second flask was used to mix 20 mg of CuCl and 50 mg of Bpy. The two flasks, sealed with rubber septae, were separately purged with argon gas for 30 min. The monomer solution was then transferred to the flask containing the copper catalyst and stirred for 5 min under an argon atmosphere. The dark-brown solution obtained after mixing was then transferred to a sealed flask containing a Brfunctionalized flat silicon sample and a capillary, with the later being connected to the membrane pump through the sealing septum. The aqueous solution was pumped into the capillary for 1 to 2 min and then left inside without flow for a prescribed amount of time and at a temperature fixed in the range between 25 and 100 °C. After polymerization, samples were rinsed with water, dried, and characterized as described below. Steps iii and iv above determine the areal density of initiator and hence the brush grafting density (σ), whereas step v controls the polymerization index (N, the number of monomer units per chains) and thus the length of the grafted chains. These two parameters determine the dry thickness (hdry) of a brush as hdry/σ = Na3, with a being the monomer size. All of the results presented below have been obtained with different brushes exhibiting the same grafting density and various chain lengths.

Figure 1. Sketch of the flow cell components. poly(dimethylsiloxane) (PDMS) of 1 mm thickness and 24 × 60 mm2 lateral dimensions. A 5-cm-long channel, exhibiting a central constriction as represented in Figure 1, is cut into this PDMS sheet. Two pieces of microcapillaries (one bare and one coated with a brush), 3 to 4 mm in length, are placed on each side of the central constriction and held by adhesion. The PDMS slab holding the capillary samples is then sandwiched between a 24 × 60 mm2 glass coverslip and a 5-mm-thick polycarbonate plate of the same lateral dimensions. The sandwich is then lightly pressed in an aluminum frame to ensure proper sealing of the channel. A dilute aqueous suspension of fluorescent nanoparticles (FluoroMax, Thermo Scientific, 200-nm-diameter latex beads) was fed into 13759

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the flow cell by means of flexible tubing connected to a 50 mL glass syringe, used as an upper reservoir, and collected at the exit in a lower glass reservoir. The pressure drop was fixed by changing the height difference between the liquid free surfaces in the two reservoirs, which can be set in the range between 150 and 400 mm. Velocity profiles inside the microchannels were measured by tracking the fluorescent nanoparticles as follows. The flow cell was placed on the stage of an inverted epifluorescence microscope (Olympus IX 70) equipped with a 100× oil immersion objective (NA = 1.3). The objective was focused at the midplane of the circular microcapillaries, and the fluorescence intensity coming from particles flowing in the midplane was detected with a cooled CCD camera (Sensicam, PCO Imaging). This setup allowed us to acquire time-lapse sequences of flowing particles at a frame rate of up to 50 Hz over a region of 15 × 90 μm2 along the channel axis, with a resolution in space of 0.5 μm (the latter value was determined by the pixel binning required to obtain a good signal-tonoise ratio using short exposure times). Image analysis of the acquired sequences was performed with ImageJ freeware27 using the manual tracking plugin28 to determine the particles' velocity as a function of their lateral position in the capillary midplane.

Figure 3. (a) SEM image of a bare microcapillary. Observation of a broken cross section with a finite angle of incidence with respect to the capillary axis. (b) SEM image of a brush-coated capillary. The polymer layer, not visible in panel a, is indicated by arrows. The scale bars in a and b are 5 μm. (c) Evolution of the C/Si ratio as a function of the ellipsometric brush thichkness.

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RESULTS Sample Characterization. Brush Growth. We have probed the influence of polymerization conditions on brush growth by monitoring their dry thickness, measured by ellipsometry, as a function of polymerization time and temperature. Figure 2 shows that, at ambient temperature,

polymer layer can be estimated to be around 200 nm from this image. Such a thickness is in excellent agreement with the ellipsometric thickness measured ex situ on a silicon wafer sample elaborated along with the capillary, which we find to be 195 nm. This strongly suggests that brush growth occurs similarly inside the capillaries and on the flat silica surfaces and that we can safely use the thickness measured by ellipsometry as the reference value for each brush elaborated. Elemental analysis inside the capillaries was, in contrast to direct imaging, easier to perform, and several samples have thus been analyzed. This provides qualitative information: spectra acquired inside bare capillaries were found to display only silicon and oxygen peaks, whereas brush-coated samples show also a peak associated with the presence of carbon. This is consistent with the presence of an added organic layer inside the glass capillaries. We have computed the ratio of the carbon to silicon peak intensities for different brushes, as presented in Figure 3b. With the depth of analysis of the EDX probe being about 1 μm, the acquired spectra are averaged over the brush thickness and the underlying glass substrate. An increase in the C/Si ratio thus points to a larger amount of carbon, which is qualitatively consistent with the increasing brush thickness measured ex situ. Brush Swelling in Water. A proper analysis of the effect of a brush on the flow resistance of a microchannel requires a knowledge of the thickness of the brush when it is in the presence of the surrounding liquid (i.e., water in the present study). The swelling of the brushes immersed in water was estimated from the force−distance curves measured with the SFA. First, the dry thickness of brushes grown on plasma-activated mica was measured in the SFA and checked to be the same as that obtained under the same grafting conditions on silicon wafers. This allowed us to control the fact that PHEMA brushes grown on mica had the same density as those grafted on silicon oxide. Next, the facing surfaces were immersed in water and approached quasi-statically while measuring the distance between the two mica substrates and the interaction force. It can be seen in Figure 4 that repulsive forces become measurable below a given intersurface separation and

Figure 2. Brush growth kinetics obtained at room temperature (○), T = 55 °C (●), and T = 90 °C (□).

there is a fast initial increase in the thickness within the first tens of minutes, followed by a slowdown of the kinetics and a plateau, at a value of about 75 nm, reached after typically 3 h of polymerization. Such a trend is qualitatively consistent with earlier results concerning the growth of PHEMA brushes in pure water and is associated with the poor “living” character of ATRP of HEMA in water.29 To reach larger brush thicknesses, we have increased the polymerization temperature. We thus find that, as the temperature is increased, the initial brush growth displays a broader linear increase with time, typically over the first hour of reaction, which allows us to obtain, at T = 90 °C, a brush thickness of hdry ≳ 200 nm. We have also used SEM to perform a qualitative characterization of brush growth inside the microcapillaries. We have found that the direct observation of the polymer brushes on sections of the glass capillaries was extremely difficult, most likely because of the damage caused to the soft polymer films when fracturing or polishing the samples. We have succeeded only once in obtaining a satisfactory image of a capillary crosssection showing a clearly visible internal coating. In Figure 3b, we present such a picture in which the internal polymer coating can be seen. (Such a coating is absent from images of bare capillaries, as shown in Figure 3a). The dry thickness of the 13760

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Figure 5. Normalized velocity profiles v(r)/Vmax as a function of the radial position r in the midplane of the capillary. (Line) Theoretical Poiseuille profile; (•) measured profile.

Figure 4. Force vs distance curve measured with a brush immersed in water (hdry = 110 nm). (Inset) Force νs distance normalized by hdry, showing the onset of repulsion for two different brushes. (●) hdry = 110 nm and (○) hdry = 40 nm.

measured capillary length L, and ΔP = 0.135ΔP0. It can be seen that the experimental profile displays the expected parabolic shape and that the value of the measured maximum velocity is, to within 5%, in agreement with the theoretical value. This confirms that our velocity measurement technique is reliable. Furthermore, in order to be as accurate as possible in identifying the effect of a brush on the flow profile in each experimental run, the pair of capillaries placed in the flow cell is composed of a brush-coated and a bare sample, and a velocity profile is systematically remeasured in the bare sample and used as the reference for comparison. This ensures that the velocity measurements in both treated and bare samples have been obtained under exactly the same imposed pressure drops and that the observed differences in flow profiles do not arise from slight variations in ΔP from one run to the other. Velocity Profiles in Brush-Coated Capillaries. We now report on the effect of the grafted brushes on the measured velocities. In Figure 6, we have plotted velocity profiles

progressively increase as the surfaces are further approached. Such forces are of steric origin and correspond to the repulsion due to compression of the brush.25,30,31 The onset of repulsion, as indicated in Figure 4, therefore indicates the point at which the bare mica surface comes into contact with the facing polymer bush. The distance between the mica substrates at the onset of repulsion thus provides an estimate of the brush swollen thickness. Figure 4 presents a force−distance curve measured with a brush of dry thickness hdry = 110 nm. It can be seen that the onset of repulsion lies at hswell ≃ 225 nm. The swelling ratio for this brush is therefore α = hswell/hdry ≃ 2. Moreover, we have checked that the value of the swelling ratio does not depend on the length of the grafted chains. We have measured a force−distance curve with a brush of hdry = 40 nm (same nominal grafting density but shorter chains than the 110-nm-thick brush). It can be seen in the inset of Figure 4 that this thinner brush displays the same value of hswell/hdry ≃ 2 at the onset of repulsion. In what follows, we will therefore use α = 2 in order to estimate the swollen thickness of the various brushes from their dry thickness measured by ellipsometry. Flow Cell Calibration. We have validated our velocity measurement technique by comparing the measured velocity profiles in bare glass capillaries with those expected from Poiseuille theory. For that purpose, we first had to calibrate the flow cell. Indeed, as described above, the cell design is such that the pair of capillaries under investigation is placed parallel to the central constriction of the main macroscopic channel cut into the PDMS part. These three channels are in turn placed in series with the inlet and outlet connection tubes and channels (Figure 1). The pressure drop between the capillary ends (ΔP) is therefore different from the externally imposed pressure drop (ΔP0). The relationship between these two pressure drops is given by ΔP = ΔP0Req/Rtot, where Req is the hydraulic resistance of the two capillaries in parallel with the central constriction and Rtot is the total resistance of the flow cell + capillaries + connection tubes. We have measured Rtot = (1.77 × 1010) ± (0.04 × 1010) Pa s m−3 and computed, given the constriction geometry, Req = (2.35 × 109) ± (0.65 × 109) Pa s m−3, which yields ΔP/ΔP0 = 0.135 ± 0.04. We are then able to compare our measurements, obtained in bare glass capillaries, with the velocity profiles calculated from v(r) = ΔPR2/(4Lη) × (1 − (r2/R2)), with R being the radius, L being the length of the capillaries, and η being the fluid viscosity. In Figure 5, we have plotted the measured and calculated velocities as a function of coordinate r inside the capillary. Both profiles have been normalized by the same theoretical value of Vmax = ΔPR2/ (4ηL), computed using R = 5 μm, η = 10−3 Pa s, the optically

Figure 6. Normalized velocity profiles v(r)/Vmax as a function of the radial position r in the midplane of the capillary. (●) Bare capillary; (○) brush-swollen thickness of 240 nm; (▲) brush-swollen thickness of 390 nm. Lines are parabolic fits of the data points.

obtained for bare and brush-coated capillaries. For the sake of comparison, each profile has been normalized, as already done above, by the maximum velocity expected from Poiseuille theory in a channel of 10 μm diameter: Vmax = ΔPR2/(4ηL). We thus observe an overall downward shift of the normalized velocity profiles, Ṽ (r) = V(r)/Vmax, in the presence of brushes, with the magnitude of the flow reduction being larger for thicker brushes. The flow profiles measured in brush-coated channels appear to retain their parabolic shape in the central part of the microchannel. However, the space and velocity resolutions of our setup do not allow us to measure velocities in the vicinity of the channel walls, where deviations from the parabolic shape are expected to be more pronounced. We have determined, for brushes of swollen thicknesses ranging from 45 to 400 nm, the relative variation ΔṼ /Ṽ max, 13761

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where Ṽ max is the normalized central velocity measured in the reference bare capillary and ΔṼ is the difference in normalized maximum velocities between bare and brush-coated samples. The variation of ΔṼ /Ṽ max with hswell is presented in Figure 7. It

Figure 8. ΔṼ /Ṽ max as a function of hswell. Symbols are experimental measurements, the continuous line is the prediction assuming no flow in a layer of thickness hswell, and the dashed line is the best linear fit to the data obtained with heff = 1.8hswell (see the text). Figure 7. ΔṼ /Ṽ max as a function of hswell. Error bars correspond to the uncertainty in the velocity measurements.

systematic underestimation of the flow reduction. It follows that considering a layer of finite resistivity, as is the general case in Damiano’s model,24 will lead only to stronger underestimates of the measurements. Such a discrepancy raises the question of the brush thickness in the presence of shear flow. Indeed, we estimate hswell from SFA measurements that are made under quasi-static conditions in the absence of flow. Now, in order to account for the unexpectedly large values of ΔṼ /Ṽ max that we measure, one has to think of an increase in hswell under shear. This issue has already been a matter of debate in the past. Theoretically, there is no consensus on whether shear flow will increase32,33 or leave unchanged34,35 the thickness of a polymer brush. However, experiments using neutron reflectivity to determine the brush height indicate that no significant change in the polymer layer thickness can be detected in the presence of shear.36,37 This is confirmed by numerical simulations38 that indicate that, as proposed initially by Rabin and Alexander,35 chain tilting and stretching occurring under shear compensate to leave the brush height globally unaffected. On this basis, we rule out the possibility of brush swelling under shear to account for our measurements. To go further in the interpretation of the results, we propose two possible mechanisms that may be at the origin of our observations. The first one is associated with the distribution of chain length in the brush. Indeed, the grafted brushes are likely to display polydispersity and be such that a small number of long chains are present in the layers. The contribution of such long chains is difficult to detect in the quasi-static compression experiments performed using the surface forces apparatus because of force sensitivity limitations. However, it is plausible that, when exposed to shear flow, these longer chains contribute significantly to the drag, over an “effective” brush thickness larger than that determined by SFA. To account for our measurements of ΔṼ /Ṽ max on the basis of such an effective thickness heff and by assuming as above that the brush behaves as an impenetrable layer in which no flow occurs, one needs to use heff ≃ 1.8hswell (Figure 8). Such an estimate of heff, which is almost twice the value of the thickness determined by SFA, seems difficult to reconcile with the overall shape of the force− distance curves shown in Figure 4 that display a rather welldefined onset of repulsion, as predicted theoretically by Alexander and deGennes30,31 for monodisperse brushes. Therefore, if brush polydispersity is likely to play a role in the enhanced frictional drag that we observe, we think that this effect alone is not sufficient to explain the experimental data fully.

can be seen that the flow reduction varies roughly linearly with the brush thickness and that a decrease in the maximum velocity of up to 35% can be reached for the thickest brushes studied here, which have a swollen thickness representing about 8% of the nominal capillary radius.

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DISCUSSION The results reported in Figures 6 and 7 are, to the best of our knowledge, the first ones showing the quantitative effect on the flow velocity of grafted brushes of various thicknesses. A slowdown of the flow, for a given pressure gradient, in the presence of a grafted polymer layer is a priori not surprising. One indeed expects that adding a thin layer to the microchannel walls will reduce its effective diameter, thus inducing a velocity drop. In that respect, our results are qualitatively consistent with a recent study9 of flow in chemically modified capillaries. Now the main question raised by our observations is how to account quantitatively for the effect reported in Figure 7. We first attempt to interpret our results within the framework proposed in a previous theoretical study24 in which a grafted polymer brush is described as a thin poroelastic layer coating the lumen of the microchannel. Indeed, a brush is made of elastic chains, for which the flowing fluid is a good solvent, and one can anticipate that flow in this porous and deformable medium will affect the overall velocity profile. Such a picture is at the heart of a model developed by Damiano et al.24 in order to address theoretically the issue of red blood cells flowing inside a microvessel bearing a glycocalyx layer. In this model, the overall flow rate and the shape of the velocity profile inside the brush depend essentially on the ratio (R − d)/R, where R is the bare channel radius and d is the thickness of the poro-elastic layer, and on the parameter δ2 = η/(R2K), where η is the viscosity of the fluid inside the porous layer and K is the hydraulic resistivity of the layer. A limiting case of the theory corresponds to K → ∞, which amounts to assuming that no flow occurs inside the wall layer of thickness d. In such a limiting case, the velocity profile is simply that predicted by Poiseuille theory in a channel of radius R − d. In Figure 8, we compare our measurements of ΔṼ /Ṽ max with the flow reduction expected from a radius reduction of d = hswell (i.e., ΔṼ /Ṽ max ≃ 2ΔR/R = 2hswell/R). It can be seen that such a simple “no flow” assumption inside the brush is not enough to account for our experimental observations and leads to a 13762

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Langmuir The second possible mechanism relies on recent observations obtained in simulations of the molecular dynamics of polymer brushes under shear flow.39,40 These simulations show that, under the combined effects of advection in the flow and thermal fluctuations, the macromolecules of the brush individually undergo a cyclic motion that consists of alternating stretching and recoil phases. Such a stretching/recoil motion produces a net backflow in the region occupied by the brush (i.e., creates a region close to the channel wall where the flow is opposite to that induced by the applied pressure drop). Such a molecular mechanism, and the associated counterflow, could also contribute to the flow reduction. More generally, we believe that near-wall secondary flows, as observed in recent simulations of nanociliated surfaces,41,42 or oscillatory motions of the elastic chains submitted to shear flow43 are in part responsible for the observed net flow reduction. We close the discussion by connecting our experimental results with earlier measurements of in vivo blood flow in microvessels.22 In vessels of 20 μm nominal diameter, Pries et al.22 have measured variations in flow velocities of up to 30% between an intact vessel and a venule treated in order to remove the glycocalyx layer. If we consider the results obtained here with the brush thickness that is closest to a plausible glycocalyx thickness (i.e., hswell ≃ 400 nm), then we note that a velocity reduction of about 35% is obtained. Our results are therefore consistent, in terms of the order of magnitude, with in vivo measurements. This is all the more striking because our study focuses only on the flow of pure water whereas in vivo experiments have been done with whole blood. This suggest that a large part of the flow reduction due to the presence of the glycocalyx arises from a slowdown of the fluid in which red blood cells (RBCs) are suspended and that direct interactions between RBCs and vessel walls do not entirely control the blood flow resistance in microvessels.

ACKNOWLEDGMENTS

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REFERENCES

We acknowledge the French-Italian University for providing a Ph.D. scholarship to L.L. under the 2009 Vinci program and the Regione Campania (MICROEMA project, 220 APQ-RT02 2008) for financial support. We are grateful to CNES (Centre d’Etudes Spatiales) and ESA (European Space Agency) for financial support. We are indebted to Salima Rafai, who provided us with full access to her microscopy facilities to perform the present study. We thank Sabatino Russo for his help with the SEM experiments and Benjamin Doppagne for his valuable contribution during flow cell calibration.

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CONCLUSIONS We have reported on the design of microcapillaries coated with hydrophilic polymer brushes. We have performed an extensive study of the growth kinetics and swelling of the brushes using ellipsometry, the surface forces apparatus, and in situ electron microscopy and elemental analysis in order to provide the best possible characterization of our systems. We have then probed the effect of brushes on the flow of water inside such microchannels. We have thus shown that a grafted polymer brush induces a flow reduction that is larger than expected from a mere reduction of the channel internal diameter. We propose that such a large brush-induced drag is controlled by the brush polydispersity and by the nontrivial motion of the grafted chains submitted to shear, which creates flow perturbations in the near-wall region, as observed in numerical simulation.40 The present study, which has been motivated by the issue of blood flow in the microvascular system, opens the way to the design of synthetic microchannels that account for the effect of the glycocalyx for in vitro studies of hemodynamics.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 13763

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