LETTER pubs.acs.org/Langmuir
Controlling Viscoelastic Flow in Microchannels with Slip M. E. Bravo-Gutierrez,† M. Castro,‡ A. Hernandez-Machado,§ and E. Corvera Poire*,† †
Departamento de Física y Química Teorica, Facultad de Química, Universidad Nacional Autonoma de Mexico, Mexico DF 04510, Mexico ‡ GISC and Grupo de Dinamica No Lineal (DNL), Escuela Tec. Sup. de Ingeniería (ICAI), Universidad Pontificia Comillas, E-28015 Madrid, Spain § Departament ECM, Facultat de Física, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain ABSTRACT: We show that viscoelastic flow in a microchannel under a dynamic pressure gradient dramatically changes with the value of the apparent slip. We demonstrate this by using classical hydrodynamics and the Navier boundary condition for the apparent slip. At certain driving frequencies, the flow is orders of magnitude different for systems with and without slip, implying that controlling the degree of hydrophobicity of a microchannel can lead to the control of the magnitude of the flow. We verify this for viscoelastic fluids with very different constitutive equations. Moreover, we demonstrate that flow, given a value of the apparent slip, is a non-monotonic function of the driving frequency and can be increased or reduced by orders of magnitude by slightly changing the frequency of the driving pressure gradient. Finally, we show that, for dynamic situations, slip causes and effectively thicker channel whose effective thickness depends on frequency. We have calculated relevant quantities for blood and a polymeric fluid in order to motivate experimental studies.
S
lippage is an old subject in macroscopic hydrodynamics, for both Newtonian and complex fluids.1-12 The subject has been revisited in recent years due to its importance in microfluidic applications.13-26 In microfluidics, the nature of the interaction between the fluid and the confining media determines largely the system response to a pressure gradient since the areato-volume ratio is very large. In this context, the concept of an apparent slip has emerged, and even though its origin is controversial,2,20-23,27-35 it is particularly relevant when the interaction between the fluid and the substrate is hydrophobic; the larger the hydrophobicity, the larger the apparent slip. The apparent slip is theoretically consistent with a nonzero velocity at the microchannel walls. Recently, it has been experimentally demonstrated that the degree of hydrophobicity of a polydimethylsiloxane (PDMS) microchannel can be reduced or even suppressed and the microchannel can become hydrophilic.36 Similarly, it has also been demonstrated that selected pathways in a hydrophilic microchannel can be turn into hydrophobic.37 More impressively, it has recently been shown that slip can largely be increased or decreased by nanopatterning of the microchannel surfaces.17,18 Both chemical treatment and nanopatterning of the surfaces imply the possibility of experimentally controlling the value of the apparent slip in a microchannel. Understanding the theoretical consequences of an apparent slip in a system both is a fundamental problem and might also provide insight into new phenomena that can help to design new technological applications.38 One of the promises of microfluidic devices has to do with biological systems, in order to produce cheap and scalable medical testing r 2011 American Chemical Society
systems. However, most of the theoretical studies have focused on the properties of Newtonian fluids in those microdevices, even though most of the relevant biological fluids have viscoelastic properties. Pulsatile or oscillatory flows in microfluidics have recently been explored experimentally39 and theoretically40 for water. However, they are of particular importance whenever the fluids are viscoelastic as they are in many microfluidic devices where plasma or whole blood is analyzed.41,42 Even when water is used as a solvent, many microfluidic devices carry cells and chemical reactants that cause the fluid solution to have viscoelastic properties. The effect of an apparent slip in the dynamic response of viscoelastic flow to a pulsatile pressure gradient has not been treated in literature. Moreover, this response might be critical in biological systems where transport or pumping of biofluids with complex rheological properties is often oscillating or periodic functions of time. In this Letter, we show that viscoelastic flow in a microchannel under an oscillatory pressure gradient at a given frequency can be dramatically increased or decreased by changing the value of the apparent slip. This happens for viscoelastic fluids governed by very different constitutive equations. Moreover, we demonstrate that flow is a non-monotonic function of the driving frequency. This implies that, for a given value of the slip, flow can be increased Received: September 2, 2010 Revised: December 2, 2010 Published: February 15, 2011 2075
dx.doi.org/10.1021/la103520a | Langmuir 2011, 27, 2075–2079
Langmuir
LETTER
or reduced by orders of magnitude by slightly changing the frequency of the driving pressure gradient. Our predictions are applicable to a wide range of viscoelastic materials when the amplitude of the oscillations is small. We provide a systematic method to apply our predictions to real fluids. The continuum description of a viscoelastic incompressible fluid is given by the conservation laws of mass, r 3 v = 0, and momentum: ð1Þ F Dt vþv 3 rv ¼ -rpþr 3 τ where v is the velocity, p is the pressure, F is the density of the fluid, and τ is the stress tensor.43 This equation needs to be supplemented with a proper constitutive equation for the stress tensor τ. All the physics of the viscoelastic material is contained in the constitutive equation. We consider small amplitude oscillatory flow in a rectangular microchannel. The separation between walls 2l , in the z-direction, is considered to be much smaller than any other dimension in the system, so that the effect of lateral walls in the y-direction can be neglected. Because of the low Reynolds number involved in this kind of flow, we assume that convective terms are negligible. Consequently, the only nonzero component of the velocity is parallel to the microchannel walls (on the x-direction) and, from the continuity equation, it is a function of the coordinate transverse to the flow, z, and time, t, that is, u(z,t). We consider the apparent slip in the microchannel through the boundary condition for the velocity at the confining walls, namely, du ð2Þ uðz ¼ (l Þ ¼ -λ dz (l
λ is called the slip length and was introduced by Navier (when λ = 0, one recovers the no-slip condition for the velocity at the walls). Equation 2 interpolates between the classical no-slip boundary condition and a condition of full slip (dvx/dz = 0) which would take place if the fluid is in contact with a vapor phase.28 We consider a large family of linearized constitutive equations with constant viscosity (small amplitude flow leads to small shear rates in the system and assures the absence of shear thinning or shear thickening effects). This family of viscoelastic constitutive equations includes the models of Maxwell, Kelvin-Voight, Jeffreys (linear Oldroyd-B), among others, and may be written as Z t ð3Þ τðz, tÞ ¼ gðt - t 0 ÞDz uðz, t 0 Þdt 0 -¥
where the kernel g(t - t0 ) is called the relaxation modulus and contains all the rheological properties of the fluid. Linearity allows us to Fourier transform the problem to the frequency domain. In this domain, eq 1, together with eq 3, takes the simple form: -iωF~u ¼ -Dx~pþ~g ðωÞ Dz 2 ~u
ð4Þ
where the tildes indicate that functions are on the frequency domain (note that velocity and pressure are now functions of space and frequency). For a Newtonian fluid, ~g (ω) = η; for a Maxwell fluid, which is built as a mathematical interpolation between a Newtonian fluid and an elastic solid, ~g (ω) = η(1 itrω)-1, where tr is a time-scale given by the ratio of the viscosity and the elastic modulus of the material; for a linear Oldroyd-B fluid, ~g (ω) = η(1 - it2ω)(1 - itrω)-1, where t2 is a second time scale of the fluid. For more complex fluids, such as polymeric
fluids, ~g (ω) might involve several characteristic times. These ones might be interpreted in the context of the Rouse model for polymer physics.44 For this model, polymers are conceived as monomers attached to each other by springs, and the normal modes of vibration of the chain are associated to the different characteristic times of the fluid. As the equations are linear, we can obtain the fluid response to any time-dependent pressure gradient as a linear superposition of sinusoidal modes. For a single-mode pressure gradient ∂xp = (dp0/dx)cos(ω0t), eq 4 can be solved with the boundary condition, eq 2, in the frequency domain. By inverse Fourier transformation of the velocity in frequency domain, we can have the velocity in the time domain. The volumetric flow, which is the velocity averaged over the cross-sectional area A times that area, is then given by h idp 0 Q ðtÞ ¼ -A ReK λ cosðω0 tÞþImK λ sinðω0 tÞ ð5Þ dx where the complex response function Kλ(ω) is evaluated at ω0; it is an inverse measure of the resistance to flow, and it is given by ! 1 1 1K λ ðωÞ ¼ ð6Þ iωF Γ cotðΓl Þ-ðΓl Þ2 d Here, Γ2l 2 = iWo2 i[ωFl 2/~g (ω)] is a nondimensional complex number, written in terms of Wo, defined as a generalized Womersley number. Note that, for Newtonian fluids, Wo2 = ωFl 2/η relates the angular frequency, the kinematic viscosity, and a characteristic length scale of the system. For complex fluids, it also involves the different relaxation times. d = λ/l relates the new length scale introduced by slip relative to the microchannel thickness. This result (eqs 5 and 6) is consistent with the steady state results that are well-known in literature. Namely, when ω f 0, slip causes an enhancement of the net flow relative to the system without slip, that is, dp Aλl 0 ð7Þ Q ¼ Q λ¼0 þ η dx This zero frequency behavior, has led to the common belief that slip enhances the net flow. Equation 6 is of main importance. It is a universal function for the family of constitutive equations represented by eq 3. Different viscoelastic materials enter through ~g (ω). It is also worth noticing that Kλ(ω) does not depend on a particular driving pressure gradient; it depends on fluid and wall properties, microchannel geometry45 and says how the system would respond to each frequency involved in the driving pressure gradient. For Newtonian fluids, ~g (ω) is real, and the real part of the response function (eq 6) decays monotonically with angular frequency. On the other hand, when elastic effects are present, ~g (ω) is complex and causes the real part of the response function to have peaks at certain resonance frequencies.46 Like any system with elastic character that, when subject to external driving, behaves resonantly at certain frequencies, the viscoelastic fluids present resonances. The peaks in the response function tell us at which frequencies (that enter the system through the driving pressure gradient) the fluid would have a resonance. Figure 1 shows that the curves for the real part of the response function with and without slip are non-monotonic functions of frequency and cross each other multiple times. Therefore, for 2076
dx.doi.org/10.1021/la103520a |Langmuir 2011, 27, 2075–2079
Langmuir
LETTER
Figure 2. Relaxation modulus data for a polystyrene solution in chlorinated diphenyl at 33.5. Circles stand for experimental data adapted from ref 48, and solid line is a nonlinear fit to g(t) = Σ4k = 1gk exp(-t/λk), with g1,2,3,4 = 500, 900, 1300, and 1600 Pa and λ1,2,3,4 = 450, 100, 30, and 5 s. The experimental values are taken from two experiments at ∂zu = 0.41 and 1.87 s-1, showing that the response is independent of forcing.
Figure 1. Real part of the response function, K, with slip (black solid line) and without slip (dashed line) (in m3s/kg) versus normalized angular frequency for (a) a Maxwell fluid with parameters η = 5.5 10-3 kg/(m s), F = 1050 kg/m3, and tr = 0.5 s in an 8 μm microchannel with slip lengths λ = 0.5 μm and λ = 0 μm. Note that the vertical axis is in logarithmic scale. Circles stand for K with λ = 0 but using the effective channel width defined in eq 9. Inset: Resonance frequency (in rad/s on the scale at the left and in Hz on the scale at the right) versus slip length λ (in m). (b) Polystyrene fluid with relaxation modulus shown in Figure 2. The density of the fluid is F = 1454 kg/m3.47
certain frequencies, the real part of the response function with slip is larger than the real part of the response function without slip. For other frequencies, the opposite happens. Figure 1a shows this behavior for a simple Maxwell fluid. Figure 1b shows a similar behavior for the more complex polymeric fluid of Figure 2 for which we have used the experimental curve for the relaxation modulus in our eq 6. For both cases, it is easy to see that maxima occur at different frequencies in the absence or presence of slip; therefore, at the resonance frequencies of the system with slip, the system without slip is out of resonance and vice versa. In the inset of Figure 1a, we show the dependence of the first resonance frequency when the slip length, λ, changes from a few nm up to 1 μm. The discussed effect for the real part of the response function, leads to a striking effect on the magnitude of the flow. In Figure 3a, |Q(t)|, from eq 5 for a Maxwell fluid, is plotted as a function of time at the first resonance frequency for the system without slip ωres. The two curves show that the amplitude of the oscillatory flow of the system without slip is orders of magnitude larger than the flow magnitude of the system with slip. The opposite happens when the system is driven at the first resonance frequency for the system with slip ωλres as shown in Figure 3b. This result is of main importance: the effect of slip is not always that of enhancing the flow magnitude, rather it has a complex
Figure 3. Magnitude of the volumetric flow (in m3/s) versus time (in s) for a Maxwell fluid with parameters η = 5.5 10-3 kg/(m s), F = 1050 kg/m3, and tr = 0.5s in an 8 μm thick and 100 μm wide microchannel with slip lengths λ = 0.5 μm (black solid lines) and λ = 0 μm (red dashed lines), driven by a one-mode pressure gradient of amplitude -5 Pa/m and angular resonance frequencies: (a) ωres = 1271 rad/s and (b) ωλres = 1131 rad/s. Note that the vertical axes are in logarithmic scale.
effect on the dynamics of the system, causing either an enhancement or a reduction of the magnitude of the flow, depending on how the system is driven. The fluid flow in response to an arbitrary time-dependent pressure gradient can be calculated by superposition of harmonics with amplitudes given by Kλ(ω).49 Saying that a channel with slip has a larger effective thickness is a way of referring to the virtual thickness that would be needed in order to make the local velocity zero at the walls. In the steady state case, the local response function with slip can be written as the local response function without slip but with an effective thickness l 0 = l þ λ. For dynamic situations, the local response function KL, before averaging over the cross-sectional area to obtain eq 6, is given by 1 cos Γz 1 λ 1KL ðz, ωÞ ¼ ð8Þ iωF cos Γl 1-Γλ tan Γl We can associate the effect of slip to an effective channel thickness l 0 , using the following equation for the effective thickness cos Γl 0 ¼ ½1-Γλ tan Γl cos Γl
ð9Þ
Equation 9 shows that l 0 , for dynamic situations, is a function of the fluid parameters, the actual channel thickness, and, most notably, frequency. We have numerically solved eq 9, for a Maxwell fluid, to 2077
dx.doi.org/10.1021/la103520a |Langmuir 2011, 27, 2075–2079
Langmuir
Figure 4. Effective thickness, l 0 , in units of the actual thickness, l , versus angular frequency (in rad/s) of a fluid with parameters η = 5.5 10-3 kg/(m s), F = 1050 kg/m3, and tr = 0.5 s and slip length λ = 0.5 μm. Fluid parameters are as in the rest of the figures.
LETTER
forms for the boundary condition that might include the substrate roughness, the constitutive equation for the complex fluid, the energy interactions between the fluid and the walls, and the presence of a depletion layer close to the walls51,52 could be considered for refined experimental comparisons in particular systems. The possibility of controlling the interaction between the fluid and the solid boundary opened by micro- and nanopatterning of the substrates17,18 as well as by chemical treatment of the surface36,37 offers the possibility of causing changes in the hydrodynamic behavior by controlling the degree of slippage. This is precisely what our Letter presents: a method for dynamically modifying the magnitude of the viscoelastic flow in a microchannel by changing the slippage.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. obtain the effective thickness of an 8 μm thick microchannel with slip, in the range of angular frequencies of 0-6500 rad/s (see Figure 4). Our results indicate that the effective thickness (l 0 ) is at all frequencies, larger than the actual microchannel thickness (l ), and it can be up to 20% larger than l . As the figure indicates, the effective thickness is a non-monotonic function of frequency whose maxima and minima do not coincide with resonances of either system. In order to better illustrate our calculations, we plot the response function (eq 6), for a Maxwell fluid, without slip (λ = 0) in a microchannel with l 0 (according to eq 9), overimposed to the curve for the response function (eq 6) of a Maxwell fluid in a microchannel with finite slip length λ and actual thickness l. This can be seen in Figure 1a (circles). Regarding a possible experimental comparison, we have calculated relevant quantities for two real fluids. We find that the Maxwell model with the parameters measured by Thurston and Henderson for blood50 in a range of shear rates where there is no shear thinning and a practically constant Maxwell relaxation time, for a 30 100 μm2 microchannel, gives a resonance frequency in the absence of slip of 1029 Hz and of 996 Hz in the presence of a 500 nm slip length. For the later frequency, the maximum magnitude of the volumetric flow at a pressure gradient amplitude of 5 kPa/m is 2.9 nL/min in the presence of slip and 2.56 nL/min in the absence of slip, which is 13% smaller. The effects are more dramatic in the case of the polystyrene sample used in Figure 2. In this case, using a 30 100 μm2 microchannel, the resonance frequency is 5 MHz and the shift in the presence of a slip length of 100 nm is 34 kHz. Frequencies of MHz are in the ultrasound range and could be applied by using standard equipment. With a pressure gradient of amplitude equal to 5 kPa/m at the resonance frequency of the system with slip, the volumetric flow is 247 mL/min in the presence of slip and only 0.43 mL/min in the absence of slip which implies a difference between 2 and 3 orders of magnitude. Even for a slip length of 50 nm, the frequency shift is 17 kHz and the ratio of volumetric flows with and without slip at the resonance frequency of the former is 106, a difference of 2 orders of magnitude. A device based on measurement of volumetric flow differences could in principle be used in both cases as the precise form of the velocity profile is not needed experimentally. The Navier hypothesis (eq 2) is the simplest form for an effective boundary condition with which one can work in order to study an unexplored hydrodynamic effect. More complicated
’ ACKNOWLEDGMENT We thank Patrick Tabeling, Lyderic Bocquet, and Romen Rodríguez-Trujillo for useful and stimulating discussions. We acknowledge financial support from MICINN, Spain, through Projects FIS2009-12964-C05-03, FIS2009-12964-C05-02; from DGAPA, UNAM, through PAPIIT Project IN101907; and from CONACYT, Mexico, through Project 83149. ’ REFERENCES (1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (2) Thompson, P.; Troian, S. Nature 1997, 389, 360. (3) Schowalter, W. R. J. Non-Newtonian Fluid Mech. 1988, 29, 25. (4) Leger, L.; Raphael, E.; Hervet, H. Adv. Polym. Sci. 1999, 138, 185. (5) Denn, M. M. Annu. Rev. Fluid Mech. 2001, 33, 265. (6) Barnes, H. A. J. Non-Newtonian Fluid Mech. 1995, 56, 221. (7) Achilleos, E. C.; Georgiu, G.; Hatzikiriakos, S. G. J. Vinyl Addit. Technol. 2002, 8, 7. (8) Pearson, J. R. A.; Petrie, C. J. S. In Polymer Systems: Deformation and flow; Wetton, R. E., Whorlow, R. W., Eds.; Macmillian: London, 1968; Vol. 163. (9) Richardson, S. J. Fluid Mech. 1973, 59, 707. (10) Denn, L. M. Annu. Rev. Fluid Mech. 1990, 22, 13. (11) Brochard-Wyart, F.; de Gennes, P. G. Langmuir 1992, 8, 3033. (12) de Gennes, P. G. C. R. Acad. Sci., Ser. B 1979, 288, 219. (13) Reyes, D.; Iossifidis, D.; Auroux, P.-A.; Manz, A. Anal. Chem. 2002, 74, 2623. (14) Auroux, P.-A.; Iossifidis, D.; Reyes, D.; Manz, A. Anal. Chem. 2002, 74, 2637. (15) Meldrum, D.; Holl, M. Science 2002, 297, 1197. (16) Squires, T.; Quake, S. Rev. Mod. Phys. 2005, 77, 977. (17) Cottin-Bizonne, C.; Barrat, J.-L.; Bocquet, L.; Charlaix, E. Nat. Mater. 2003, 2, 237. (18) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. Rev. Fluid Mech 2004, 36, 381. (19) Ho, Ch.-M.; Tai, Y.-Ch. Annu. Rev. Fluid Mech. 1998, 30, 579. (20) Lauga, E.; Brenner, M. P.; Stone, H. A. Handbook of experimental Fluid Dynamics; Tropea, C., Yarin, A., Foss, J. F., Eds.; Springer: Berlin, 2007. (21) Bocquet, L.; Tabeling, P.; Manneville, S. Phys. Rev. Lett. 2006, 97, 109601. (22) Bocquet, L.; Barrat, J.-L. Soft Matter 2007, 3, 685. (23) Bouzigues, C. I.; Tabeling, P.; Bocquet, L. Phys. Rev. Lett. 2008, 101, 114503. (24) Huang, D. M.; Sendner, Ch.; Horinek, D.; Netz, R. R.; Bocquet, L. Phys. Rev. Lett. 2008, 101, 2261001. 2078
dx.doi.org/10.1021/la103520a |Langmuir 2011, 27, 2075–2079
Langmuir
LETTER
(25) Khair, A. D.; Squires, T. M. Phys. Rev. Lett. 2010, 105, 156001. (26) Mason, T. G.; Ganesan, K.; van Zanten, J. H.; Wirtz, D.; Kuo, S. C. Phys. Rev. Lett. 1997, 79, 32823. (27) Granick, S.; Zhu, Y.; Lee, H. Nat. Mater. 2003, 2, 221. (28) Lauga, E.; Stone, H. J. Fluid Mech. 2003, 489, 55. (29) Joseph, P.; Tabeling, P. Phys. Rev. E 2005, 71, 35303. (30) Neto, C.; Evans, D.; Bonaccurso, E.; Butt, H.; Craig, V. Rep. Prog. Phys. 2005, 68, 2859. (31) Zhu, Y.; Granick, S. Phys. Rev. Lett. 2001, 87, 96105. (32) Lauga, E.; Brenner, M. Phys. Rev. E 2004, 70, 26311. (33) de Gennes, P. Langmuir 2002, 18, 3413. (34) Tretheway, D.; Meinhart, C. Phys. Fluids 2002, 14, L9. (35) Choi, C.-H.; Westin, K. J. A.; Breuer, K. S. Phys. Fluids 2003, 15, 2897. (36) Yu, L.; Li, C. M.; Liu, Y.; Gao, J.; Wang, W.; Gan, Y. Lab Chip 2009, 9, 1243. (37) Zhao, B.; Morre, J.; Beebe, D. Science 2001, 291, 1023. (38) Whitesides, G. Nature 2006, 442, 368. (39) Leslie, D.; Easley, C.; Seker, E.; Karlinsey, J.; Utz, M.; Begley, M.; Landers, J. Nat. Phys. 2009, 5, 231. (40) Castro, M.; Bravo-Gutierrez, M.; Hernandez-Machado, A.; Corvera Poire, E. Phys. Rev. Lett. 2008, 101, 224501. (41) Kim, D.; S.H., L.; Ahn, C. H.; Lee, J.; Kwon, T. Lab Chip 2006, 6, 794. (42) Yang, A.; Under, S.; Zahn, J. Lab Chip 2006, 6, 871. (43) Morrison, F. Understanding rheology; Oxford University Press: New York, 2001. (44) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of polymeric liquids. Vol. 1: Fluid mechanics; John Wiley and Sons Inc.: New York, 1987. (45) If a cylindrical tube is considered, one obtains a function similar to that in eq 6 but containing Bessel functions instead of trigonometric tangents. (46) At microscales, the imaginary part of the response function, at the resonance frequencies, is orders of magnitude smaller than the corresponding real part. (47) Osaki, K.; Nishimura, Y.; Kurata, M. Macromolecules 1985, 18, 1153. (48) Einaga, Y.; Osaki, K.; Kurata, M.; Kimura, S.; Yamada, N.; Tamura, M. Polym. J. 1973, 5, 91. (49) Collepardo Guevara, R.; Corvera Poire, E. Phys. Rev. E 2007, 76, 026301. (50) Thurston, G. B.; Henderson, N. M. Biorheology 2006, 43, 729– 746. (51) Doshi, D. A.; Watkins, E. B.; Israelachvili, J. N.; Majewski, J. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 9458. (52) Mezger, M.; Reichert, H.; Sch€oder, S.; Okasinski, J.; Schr€oder, H.; Dosch, H.; Palms, D.; Ralston, J.; Honkim€aki, V. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 18401.
’ NOTE ADDED AFTER ASAP PUBLICATION This article was published ASAP on February 15, 2011. Equation 8 has been modified. The correct version was published on February 21, 2011.
2079
dx.doi.org/10.1021/la103520a |Langmuir 2011, 27, 2075–2079
