Langmuir 2008, 24, 4209-4214
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Competitive Spreading versus Imbibition of Polymer Liquid Drops in Nanoporous Membranes: Scaling Behavior with Viscosity H. Haidara,*,† B. Lebeau,‡ C. Grzelakowski,†,‡ L. Vonna,† F. Biguenet,§ and L. Vidal† Institut de Chimie des Surfaces & Interfaces, ICSI-CNRS/UHA, 15 rue Jean Starcky-B.P. 2488, 68057 Mulhouse Cedex, France, Laboratoire de Mate´ riaux a` Porosite´ Controˆ le´ e, ENSCMu/UHA, 3 rue A. Werner - 68093, Mulhouse Cedex, France, and Laboratoire de Physique des Mate´ riaux Textiles, ENSISA/UHA, 11 rue Alfred Werner, 68093 Mulhouse Cedex, France ReceiVed NoVember 13, 2007. In Final Form: January 17, 2008 The way a liquid drop that is in contact with a nanoporous substrate evolves essentially depends on the competition between imbibition and spreading. Although the scaling behavior of this competitive process with liquid viscosity is important for various applications requiring the filling of nanoporous substrates (template-assisted fabrication, storage, and controlled release of liquids), they appear to be poorly investigated and insufficiently understood. We developed a model study to investigate the wetting and spontaneous imbibition of silicon oil drops of viscosities ranging from 1 to 100 Pa s on nanoporous alumina membranes (pore size of 200 nm). Our results show that the drop radius essentially follows the power law t1/10 time dependence as expected by Tanner’s law. However, the scaling of the spreading velocity with the viscosity (∼η-n) was found to display an exponent that is comparable on both the reference (impermeable) and nanoporous substrates (n ) 0.55) but notably higher than theoretically expected (0.1). More surprisingly, we show that despite the confinement, the rate of imbibition into the nanopores displays a weaker dependence on the viscosity, as compared to the spreading velocity on both the reference and nanoprous substrates. On the basis of Darcy’s law for capillary-driven imbibition, this result was discussed in the context of the scaling behavior of the contact angle with the viscosity.
Introduction The flow of liquids on porous substrates has consistently attracted great interest from a fundamental point of view with regard its relevance to a broad range of phenomena and technological processes (transport through tree capillaries, paper and textile coating and printing, etc.).1-4 However, it is the use of nanoporous and textured materials in the template-assisted fabrication of nanostructured polymers and as fluidic devices on one hand and the crucial role that capillary effects play in these processes on the other hand that essentially accounts for the renewed interest that today characterizes this field.5-9 Those recent applications, in particular, require not only a general description of the impregnation but also the characterization of its dependence on the fluid viscosity and topological effects of the substrate, which appear to be both poorly investigated and understood today.9,10 In this work, we developed model studies based on purposely designed experiments to characterize the viscosity dependence of the spreading and spontaneous imbibition of sessile drops on nanoporous substrates. The two questions that we want to answer are the following: * Corresponding author. E-mail:
[email protected]. † Institut de Chimie des Surfaces & Interfaces, ICSI-CNRS/UHA. ‡ Laboratoire de Mate ´ riaux a` Porosite´ Controˆle´e, ENSCMu/UHA. § Laboratoire de Physique des Mate ´ riaux Textiles, ENSISA/UHA. (1) Kohonen, M. M. Langmuir 2006, 22, 3148. (2) Le, H. P. J. Imaging Sci. Technol. 1998, 42, 49. (3) Kannangara, D.; Zhang, H.; Shen, W. Colloids Surf., A 2006, 208, 203. (4) Baffoun, A.; Viallier, P.; Dupuis, D.; Haidara, H. Carbohyd. Polym. 2005, 61, 103. (5) Zhang, M.; Dobriyal, P.; Chen, J-T.; Russell, T. P.; Olmo, J.; Merry, A. Nano Lett. 2006, 6, 1075. (6) Gang, O.; Alvine, K. J.; Fukuto, M.; Pershan, P. S.; Black, C. T.; Ocko, B. M. Phys. ReV. Lett. 2005, 95, 217801. (7) Xu, J.; Li, M.; Zhao, Y.; Lu, Q. Colloids Surf., A 2007, 302, 136. (8) Greiner, A.; Wendorff, J. H.; Yarin, A. L.; Zussman, E. Appl. Microbiol. Biotechnol. 2006, 71, 387. (9) Lim, J. M.; Yi, G. R.; Moon, J. H.; Heo, C-J.; Yang, S. M. Langmuir 2007, 23, 7981.
(i) How do the spreading and the impregnation rates vary with the liquid viscosity on nanoporous substrates? (ii) How is the spontaneous imbibition rate affected by the competitive spreading process? Our results show that the drop radius essentially follows the power law t1/10 time dependence as expected by Tanner’s law.11-13 However, the scaling of the spreading velocity with the viscosity (∼η-n) was found to display an exponent n that is comparable with regard to both the reference (impermeable) and nanoporous substrates but notably higher than theoretically expected (0.1). More surprisingly, we show that despite the confinement, the rate of imbibition into the nanopores displays a weaker dependence on the viscosity as compared to the spreading velocity on both the reference and nanoprous substrates. Finally, we show that a wetting inversion appears for low-viscosity drops that accounts for the transition between the spreading that dominates in the early stage and the imbibition that rapidly overcomes the spreading and drives the retraction of the contact line. In the following, we discuss these results on the basis of both Tanner’s law for spreading and Darcy’s law for capillary imbibitions highlighting the complex scaling behavior of this competitive process with the viscosity in particular.
Basic Relations For enough small sessile drops where inertial and hydrostatic effects can be dropped, as we will show in our experiments, the imbibition is essentially governed by the balance (eq 1) between the driving Laplace capillary pressure pc and the resisting viscous shear ση,10 (10) Alava, M.; Dube´, M.; Rost, M. AdV. Phys. 2004, 53, 83. (11) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (12) Pe´rez, E.; Scha¨ffer, E.; Steiner, U. J. Colloid Interface Sci. 2001, 234, 178. (13) Rafaı¨, S.; Bonn, D.; Boudaoud, A. J. Fluid Mech. 2004, 513, 77.
10.1021/la703538g CCC: $40.75 © 2008 American Chemical Society Published on Web 02/27/2008
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pc - ση )
2γL cos θ η - vz ) 0 Rp κ
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(1)
In eq 1, γL and θ represent the surface tension of the fluid and its contact angle on the substrate material, Rp is the average radius of the nanopores, κ ) (Rp2/8) is the permeability of the porous substrate, η is the viscosity of the fluid, v is the average flow (imbibition) velocity, and z is the length along the flow direction. The imbibition velocity thus follows from eq 1 as5,10
vimb )
κ 2γL cos θ η Rpz
(2)
which represents the linear Darcy equation for the flow in porous media. As stated above, eqs 1 and 2 hold for our experiments where small drop volumes (∼3 µL) of high-viscosity liquids are used. Indeed, the resulting characteristic drop size, LD ≈ (drop volume)1/3 ≈ 1.4 mm, and mean flow velocity v are in this case such that LD is smaller than the capillary length lC ) x(γL/Fg) ≈ 1.5 mm and the Reynolds numbers are Re ) (FVLD/η) ,1. However, for the spreading of the drop, both the drop radius R and spreading velocity vspread are expected to follow the universal Hoffman-Tanner law.11 The Tanner relation predicts a power law scaling of the spreading with both the viscosity and time, which can be expressed either in terms of the drop radius (eq 3.1) or its dynamic contact angle θd (eq 3.2):11-13
γL R10 ≈ Ω3 t η vspread ≈
γL 3 θ η d
(3.1) (3.2)
In eq 3.1, Ω represents the drop volume, and t is the spreading time. These equations are derived for thin spherical drops and small dynamic contact angles (θd , 1 rad). In particular, even in the limit of small θd, eq 3.2, which has a more complete expression given by (γL/η)θd(θd2 - θe2), strictly holds only for complete wetting, as in our experiments where θd < 25° and θe < 5°. Whereas eq 3.1 directly and explicitly provides the scaling of the spreading with time (eq 3.3) and its dependence R(η) or (dR/dt)η on the viscosity (eq 3.4), the dependence on both t and η is implicit in the contact angle-dependent expression (eq 3.2). But because both expressions of the Tanner spreading law are strictly equivalent, their scaling behavior either with the time or viscosity is identical in eqs 3.2 and 3.4, giving the scaling of the contact angle with viscosity, θd3 ≈ η0.9 (θd ≈ η0.3).
R(t) ≈ ct1/10 vspread )
dR ≈ Ω3/10(γη-1)1/10t-9/10 ≈ η-0.1 dt
(3.3) (3.4)
Equations 2, 3.3, and 3.4 are used hereafter to discuss the experimental results and especially the influence of the competing spreading-imbibition process on the expected scaling behavior of the flow over the nanoporous membranes. From an experimental standpoint, the relevant physical parameter that appears in both the Darcy and the contact angledependent equations is the characteristic velocity of the liquid, (γL/η). Interestingly, if one can make (γL/η) vary over a wide range while keeping γL and pc ≈ γLcos θ/Rp fixed, then the experiments would essentially account for the viscosity (molecular
size) dependence of the spontaneous imbibition of the drop in the nanoporous substrate. The conditions regarding (γL/η) and γL are satisfied by polydimethylsiloxane (PDMS) oils whose viscosity can span a few orders of magnitude at constant surface tension (∼21 mN/m).12-14 The softer wetting condition requiring a low θ and the constancy of γL cos θ/Rp is achieved by the choice of the nanoporous alumina membranes that are used in this work (Figure 1 Experiments and Methods). Experiments and Method The nanoporous alumina membranes (Anodisc25) were supplied by Whatman, England. The main features of these 60-µm-thick nanoporous membranes are shown on the MEB images of Figure 1. From side A up to ∼95% of the membrane thickness, the porosity is composed of isolated channels that are aligned (unidirectionally) parallel to the surface normal. This porosity becomes interconnected, denser, and free of directional order ∼2 µm toward the side B. All of our experiments were performed on side A of the membranes. For PDMS oils, five viscosities covering two orders of magnitude and presenting a surface tension of 20.5 ( 1 mN/m14 were studied: 1, 5, 10, 30, and 100 Pa s. These PDMS oils were supplied by ABCR-Roth GmbH, Karlsruhe, Germany. Basically, the experiments consisted of capturing the shape evolution of 3 to 4 µL drops that were gently placed on the nanoporous membrane. Before each experiment, the membrane was cleaned with ethanol in an ultrasonic bath for 20 s, rinsed with ethanol, and fully dried under nitrogen flow. Images of the membranes before and after cleaning are shown in Figure 1d,e. All of the experiments were carried out at ambient temperature (22 °C) using the Kru¨ss automatic contact angle analyzer (G2) equipped with a CCD video camera operating at a frame rate of 24 images/s and drop shape analysis software (height, contact diameter, and angle). A second video/monitor system settled in the top view configuration was used to check the axisymmetry of the drop shape. The acquired video movies were used to recalculate the time variation of the drop volume using the shape parameters, height h and diameter D (or radius R), and the spherical cap approximation:12 Vdrop )
πh(3D2 + 4h2) πh(R2 + h2/3) ) 24 2
(4)
Equation 4 was used to check the volume conservation of the drops on imbibition-free substrates (smooth reference aluminum foil) under ambient conditions (22 °C, 33%) and for the less favorable case of the 1 Pa s viscosity drops, confirming experimentally the evaporationfree assumption. For each viscosity, an average of three experimental runs were performed. The raw data, D(t), h(t), and V(t)drop, for each run were fitted to power-law functions whenever the convergence was reasonably acceptable. To homogenize the time origin, the points were recalculated from the fit function for each run and viscosity. The kinetic curves that are shown below represent the best fit over these averaged recalculated data. A series of as-acquired raw data of three experimental runs corresponding to 1, 10, and 100 Pa s drops on the nanoporous membrane are given in Supporting Information.
Results and Discussion The imbibition kinetics were characterized for the different viscosities and discussed in terms of both the evolution of the normalized drop parameters (h(t)/h0, D(t)/D0) and of the infiltrated drop volume, Vi ) (V0 - V(t))/V0, which was determined from eq 4. The time variation of drop parameters (h/h0, D/D0) and the related power law fit equations are shown in Figure 2a for the smooth reference substrate and in Figure 2b,c for the nanoporous membrane. The corresponding kinetics for the infiltrated volume, (14) Haidara, H.; Vonna, L.; Schultz, J. Langmuir 1998, 14, 3425.
Spreading Vs Imbibition of Polymer Liquid Drops
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Figure 1. MEB images of the nanoporous alumina membranes showing (a) a cross-sectional view of the pore structure that is characterized by unidirectional and isolated channels that are aligned parallel to the surface normal, (b) top views of the investigated side (A) of the membrane showing the pore size, shape, and density, (c) top view of side B, and (d, e) top views of as-received unclean membranes showing micrometersized dust particles vs clean membranes.
Vi ) (V0 - V(t))/V0, was plotted for different PDMS viscosities and is given in Figure 2d. As expected from Tanner’s law (eq 3.3), both the drop diameter and height appear to follow a power-law scaling with time (∼tn), with exponents that clearly show a deviation from the predicted value of 0.1, depending on both the substrate and the linear drop parameter (D, h). Indeed, on the smooth reference alumina substrate the spreading exponent has a theoretical value of 0.1 for the drop diameter D versus 0.2 for the drop height h. On the nanoporous membrane (Figure 2b), this dependence of the observed scaling exponent on the linear size of the drop is accompanied by an overall depression of these exponents (D ≈ t0.06, h ≈ t0.14), compared to the impermeable reference substrate. This general depression accounts for the smooth dependence of the spreading on the nanoporous membranes that shows up visually in “equilibration times” that are on average notably lower in Figure 2b compared to those in Figure 2a. This latter result is a priori consistent with the intuitive expectation that the imbibition, by sucking the liquid, creates a depression inside the drop that slows down and stops the spreading on the nanoporous membrane earlier. Figure 2c is an enlarged view that highlights this phenomenon that often leads to a wetting inversion with the lowest drop viscosity (1 Pa s). Although this inversion appears after the drop has reached the quasi-equilibrium plateau of the spreading, it shows that the imbibition and related fluid suction become predominant on longer time scales, causing the pinning of the contact line and eventually its retraction.15 Using
microfiltration PTFE membranes of a higher pore size (average of 450 nm) and volume, we showed that this wetting inversion could take place on a much shorter time scale (early stage), indicating in this case a regime where impregnation rapidly dominates the spreading (Figure 3a). Apparently, the drop still keeps thinning during a certain time after the inversion has started. A possible explanation for this may come from the time that is required by the drop to adopt the ideal spherical shape as the contact line retracts. During this lag time, the liquid collected from the tiny reduction in area and the resulting shape reconformation may both remain confined to the drop edge. They may therefore not contribute to thicken the drop uniformly or to compensate for its thinning that results from the imbibition. The topological effects of the substrate may also play a role, especially at the drop edge. The sketch of Figure 3b describes schematically the way in which the depression induced by the suction of the liquid acts as an additional resisting force to spreading, in the same way as the hydrostatic pressure generated by a drop volume that is too high would act as an additional driving force (∼1/2Fgh2) for spreading. Contrary to spreading, our results for the imbibition kinetics (Figure 2d) that naturally involves only the nanoporous membranes could not be properly fitted by power law functions (except for the 100 Pa s viscosity drop). These imbibition kinetics were (15) Starov, V. M.; Zhdanov, S. A.; Kosvintsev, S. R.; Sobolev, V. D.; Velarde, M. G. AdV. Colloid Interface Sci. 2003, 104, 123.
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Figure 2. Time variation of the normalized drop parameters (a) on the impermeable reference substrate and (b) on the nanoporous Al membranes. (c) Full kinetics showing an enlargement of the wetting inversion for the drop of 1 Pa s on nanoporous alumina. (d) Infiltration kinetics as given by the time variation of the infiltrated volume. To avoid unnecessary crowding and overlap of the curves, the kinetics is given only for three chosen viscosities: 1 (]), 10 (O), and 100 Pa s (4).
Figure 3. (a) Illustration of the imbibition-induced wetting inversion on a microfiltration PTFE membrane of higher pore size (450 nm in average) and pore volume as compared to the nanoporous alumina membranes used in this work and (b) picturing the imbibition-induced liquid suction (small vertical arrows) and resulting hydrostatic depression inside the drop (large arrows). This depression can overcome the spreading, causing pinning of the triple line and eventually the retraction of the drop (wetting inversion).
rather best fitted by exponential functions of the form Vimb(t) ≈ Vimb(∞)[1 - exp(-t/τ)]. Although the scaling of this impregnation volume with time is not the primary interest of this work (but rather the viscosity dependence), it may be interesting to determine whether such a scaling law exists, regardless of the texture of
the porous media. If one starts from Darcy’s law given in eq 2, we may expect the impregnated volume Vimb(t) to scale with the time as ∼t cos θ(t), which gives in the limit of low contact angles ∼t(1 - θ2(t)). Omitting the topological and confinement effects of the porous network and taking for θ the known time dependence on spreading, θ ≈ t-0.3 (Tanner’s law), we arrive at an impregnated volume that should scale in principle with time as ∼t-0.4 at short times and linearly in t at longer times: Vimb(t) ≈ (t - t0.4). More systematic studies are needed to check for this scaling of capillarydriven imbibition with time. The dynamics of the competitive spreading imbibition was studied using the viscosity dependence of the linear drop size and the infiltrated volume given in the kinetic curves of Figure 2. This study requires that we define speeds characteristic of propagation and imbibition starting from the kinetic curves, which will be plotted as a function of their dependence on viscosity. For linear drop size D(t) and h(t) that follow the Tanner’s spreading power law (Figure 2a,b), these representative velocities are taken for each viscosity at the point where the drop has spread out of half of its initial radius (i.e., D(t)/D0 ) 1.5). The time corresponding to this spreading rate is determined by setting the power law fit function equal to the value of 1.5, (D(t)/D0) ) ctn ) 1.5. This leads to a time of t(1.5)) (1.5/c)1/n from which the representative velocity follows as v(1.5)) (1.5/t(1.5)). For the imbibition that seems to follow a priori no universal scaling law with the time that can be used for a more rigorous definition, this representative velocity is simply taken at a point where the imbibition is quite stabilized on the plateau (t ) 1 h). It is worthwhile to mention here that although these representative points vary with the choice of the time scale this simply results in a shift of the plots without changing their scaling law. And this was verified for both the spreading and imbibition whenever
Spreading Vs Imbibition of Polymer Liquid Drops
Figure 4. log-log plot of the spreading and imbibition velocities as a function of the drop viscosity: (a) spreading velocities on the reference and nanoporous membrane and (b) infiltration (imbibition) rate on the nanoporous membrane. The correspondence for symbols is 2 (spreading velocity) and 9 (infiltration rate). (b) Imbibition dynamics on the nanoporous membrane.
the time scale on which the representative velocity is taken is beyond the early time of the spreading-imbibition process. The linear log-log plots of Figure 4a,c represent the viscosity dependence of these velocities that all display a power law scaling (eqs 3), v ≈ η-n. For the spreading velocity, the exponent n (slope of the log-log plot) is quite comparable for the smooth reference substrate (0.65) and the nanoporous membrane (0.51). As compared to the expected value of 0.1 predicted by Tanner’s law (eq 3.4), these exponents show a much stronger dependence (decrease) of the spreading dynamics on the viscosity, regardless of whether the substrate is porous or not. Although the observed spreading exponent does not seem to vary significantly with nanoporosity, one may reasonably expect these topological features to affect the spreading process within some range of pore size and density of the porous network. Because applications related to structured substrates have renewed the debate on the influence of these topological effects on the contact line motion,16-18 it might be interesting to search for such a dependence of the spreading exponent on the topological feature of the porous network. For the imbibition rate, power law scaling with viscosity is observed, the exponent of which (∼0.17) clearly shows a much
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weaker dependence of the imbibition on the drop viscosity, as compared to spreading. Now whether this exponent fits that predicted for capillary-driven imbibitions depends on how the contact angle is involved in the Darcy relation (eq 2). As we did above for the time dependence of cos θ (and thus of θ) in the Darcy relation, we can use the viscosity dependence of θ given by Tanner’s law (∼η0.3) to derive explicitly the scaling that is expected theoretically for imbibition with viscosity. Replacing cos θ by (1 - θ2(η)) in eq 2 in the limit of small θ then leads to an imbibition rate that should scale as ∼η-1(1 - θh2) (i.e., η-1(1 - η0.6)). As for the scaling of the imbibition with the time expected from the Darcy relation, this derivation shows that one should expect the imbition rate to follow a power law scaling in η-0.4 for low drop viscosities and η-1 for higher viscosities (two imbibition regimes). Because only capillary pressure is at work in this spontaneous imbibition experiment, we can hardly evoke a pressure or swelling-induced deformation10,19 of the porous network (alumina) to account for the observed low scaling exponent as compared to both the spreading and the aboveexpected theoretical behavior. Finally, preliminary results of ongoing experiments seem to indicate that smaller exponents (very low sensitivity to η) and, for certain systems, a qualitatively different scaling law can appear for the viscosity dependence of the imbibition, depending on the pore size and structure (volume and connectivity). This result has direct practical relevance to coating applications that involve the penetration of the formulation into the substrate (wood, for instance)20 whereas its viscosity varies with time as a result of solvent uptake and evaporation. For those applications, finding the scaling behavior of the penetration rate with the time variation of the viscosity may provide a key control parameter over these processes. Another practical issue arising from the sensitivity of the scaling exponent and its variation with the structure of porous media is that of the limits within which the techniques of porosimetry using linear Darcy’s relation are relevant and applicable.20,21 Finally, it is important to outline some of the investigations that deserve to be undertaken toward a more complete understanding and description of these systems. The first is the dependence of this competitive spreading/free-imbibition process on the topological feature of the nanoporous membrane and the way that this affects their scaling behavior with viscosity. A good starting point for this issue is here provided by the alumina membrane that presents two distinct pore structures on each side (A and B) for rigorously identical wetting properties (same material). For the difference of the isolated and parallel channels on side A that was used in this work, the porous network on side B is interconnected (Figure 1) with a pore opening of a smaller size and different shape but with an identical surface fraction. If one now develops the same investigations on that side, any difference in the imbibition would account exclusively for the impact of the pore structure on the competitive spreadingimbibition process. The preliminary experiments that we have undertaken with respect to this issue already show that both the imbibition kinetics and the rate are impeded by the interconnection (and the related size reduction of the pore at the surface), for an (16) Nosonovsky, M. Langmuir 2007, 23, 9919. (17) Anantharaju, N.; Panchagnula, M. V.; Vedantam, S.; Neti, S.; TaticLucic, S. Langmuir 2007, 23, 11673. (18) Steinberger, A.; Cottin-Bizonne, C.; Kleimann, P.; Charlaix, E. Nat. Mater. 2007, 6, 665. (19) Baffoun, A.; Haidara, H.; Dupuis, D.; Viallier, P. Langmuir 2007, 23, 9447. (20) de Meijer, M.; van de Velde, B.; Militz, H. J. Coat. Technol. 2001, 73, 39. (21) Safiana, L.; Mantalaris, A.; Bismarck, A. Langmuir 2006, 22, 3235. (22) Hong, K. H.; Kang, T. J. J. Appl. Polym. Sci. 2006, 100, 167.
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identical pore fraction. The second issue that also deserves special attention in these systems is related to the size effect that determines the dominant transport regime in the nanopores. In other words, one seeks an experimental answer to the following question: What is the scaling relation between the size of the impregnating molecules and that of the nanopores that determines the crossover between the capillary (pressure) and the diffusive transport regimes?
Conclusions We have developed model experimental studies to investigate the wetting and spontaneous imbibition of silicon oil drops of different viscosities (but equal surface tension) on nanoporous alumina membranes. Whereas the spreading kinetics was found to follow consistently the expected Tanner power law scaling with time (∼t0.1), more complex scaling was observed for the spreading rates with viscosity. The spreading velocity that is
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expected from Tanner’s law to scale as η-0.1 with the viscosity was shown to have a much stronger dependence on viscosity (∼η-0.5) on both the smooth reference and nanoporous substrates. The impregnation rate on the nanoporous membranes was also found to follow a power law decay with viscosity but with an exponent that is notably depressed from that expected from the linear Darcy relation. Acknowledgment. This work was financially supported by the University of Haute Alsace, Mulhouse, France, through the BQR project. We here acknowledge this support. Supporting Information Available: A series of as-acquired raw data of three experimental runs corresponding to 1, 10, and 100 Pa s drops on the nanoporous membrane. This material is available free of charge via the Internet at http://pubs.acs.org. LA703538G
