pubs.acs.org/Langmuir © 2010 American Chemical Society
Spreading Behavior of Water Droplets on Fractal Agar Gel Surfaces Yoshimune Nonomura,*,† Yusuke Morita,† Takako Hikima,† Eri Seino,† Shigeki Chida,† and Hiroyuki Mayama‡ †
Department of Biochemical Engineering, Graduate School of Science and Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa 992-8510, Japan, and ‡Research Institute for Electronic Science, Hokkaido University, CRIS Building, N21W10, Sapporo 001-0021, Japan Received August 5, 2010. Revised Manuscript Received September 6, 2010 Agar gels with hierarchical rough surfaces, referred to as “fractal agar gels,” were prepared to model biological surfaces coated with mucus. Agar gels with rough surfaces of fractal dimension D=2.2 were synthesized by transferring a fractal surface structure of alkylketene dimer (AKD). The rough structure accelerated the spreading of water droplets and induced the appearance of a wicking front. The mechanism of acceleration of the spreading on fractal surfaces based on a semiquantitative theoretical model was also clarified.
1. Introduction In our body, there exist several hydrophilic fractal surfaces coated with mucus, which contribute either in effective nutritional absorption or in sensitization of the senses. Sapid substances, for example, are perceived by taste buds via the small projections “papillae” on the tongue.1 Villous surfaces on the nasal and esophageal mucosa increase the trapping efficiency of odorous substances while simultaneously excluding foreign substances.2 In the circular folds of the small intestine walls, the hierarchical rough surfaces consist of villuses and microvilluses that improve the efficiency of nourishment absorption.3 Researchers have accurately predicted that fractal structures on solid substrates would affect the mass transfer phenomena, especially wetting.4-17 For example, Onda et al. proposed that fractal surfaces can be super-water-repellent when these surfaces are composed of hydrophobic materials and subsequently prepared a super-waterrepellent fractal surface made of alkylketene dimer (AKD).4,5 However, there are few reports on the wetting phenomena on *Corresponding author. Tel: þ81-238-26-3164. Fax: þ81-238-26-3406. E-mail:
[email protected]. (1) Farman, A. J. Ultrastruct. Res. 1965, 12, 328–350. (2) Mygind, N. Rhinology 1975, 13, 57–75. (3) Palay, S. L.; Karlin, L. J. J. Cell Biol. 1959, 5, 363–371. (4) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125–2127. (5) Shibuichi, S.; Onda, T.; Satoh, N.; Tsujii, K. J. Phys. Chem. 1996, 100, 19512–19517. (6) Yan, H.; Kurogi, K.; Mayama, H.; Tsujii, K. Angew. Chem., Int. Ed. 2005, 44, 3453–3456. (7) Fang, W.; Mayama, H.; Tsujii, K. J. Phys. Chem. B 2007, 111, 564–571. (8) Kurogi, K.; Yan, H.; Mayama, H.; Tsujii, K. J. Colloid Interface Sci. 2007, 312, 156–163. (9) Minami, T.; Mayama, H.; Nakamura, S.; Yokojima, S.; Shen, J.; Tsujii, K. Soft Matter 2008, 4, 140–144. (10) Fang, W.; Mayama, H.; Tsujii, K. Colloids Surf., A 2008, 316, 258–265. (11) Minami, T.; Mayama, H.; Tsujii, K. J. Phys. Chem. B 2008, 112, 14620– 14627. (12) Karfar, M.; Indekeu, J. O. Phys. Rev. Lett. 1990, 65, 662. (13) Karfar, M.; Indekeu, J. O. Europhys. Lett. 1990, 12, 161–166. (14) Kwon, T. H.; Hopkins, A. E.; O’Donnell, S. E. Phys. Rev. E 1996, 54, 685– 690. (15) Broseta, D.; Barre, L.; Vizika, O. Phys. Rev. Lett. 2001, 86, 5313–5316. (16) Yu, B.; Liu, W. AIChE J. 2004, 50, 46–57. (17) Byun, D.; Hong, J.; Saputra; Ko, J. H.; Lee, Y. J.; Park, H. C.; Byun, B. K.; Lukes, J. R. J. Bionic Eng. 2009, 6, 63–70. (18) Schott, H. J. Pharm. Sci. 1971, 60, 1893–1895. (19) Shanker, R. M.; Ahmed, I.; Bourassa, P. A.; Carola, K. M. Int. J. Pharm. 1995, 119, 149–163.
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rough surfaces in human bodies owing to the difficulty of direct observations or physical evaluations.18-21 In the present study, we prepared agar gels with fractal surfaces and used a new biosurface model to observe the spreading behavior of liquids on the surfaces. The “fractal agar gel” is a hydrogel solidified by a polysaccharide and is suitable for modeling biosurfaces coated with mucus, including mucopolysaccharide. Some researchers have studied the kinetics of liquid spreading on flat gel surfaces and found that the spreading rate demonstrated a power-law behavior with time.22-26 In this study, the fractal surface was molded to a crystal of AKD. AKD undergoes fractal growth when it solidifies, thus forming super-waterrepellent fractal surfaces spontaneously.4,5 In addition, the wetting phenomena on these surfaces were evaluated by using a contact angle apparatus with a high speed camera.
2. Experimental Section Materials. AKD was supplied by Arakawa Chemical Industries (Osaka, Japan) and was used after recrystallization in n-hexane. Plaster (Sanjyo, Sapporo, Japan) and agar powder (Kanto Chemical, Tokyo, Japan) were used without further purification. Preparation. The templates of the fractal surfaces were prepared by the solidification of melted AKD. The AKD was placed on a Petri dish (86 and 12 mm in diameter and depth, respectively) and heated at 333 K on an electric hot plate. After the AKD melted, the Petri dish was cooled to room temperature on the experimental desk, where it solidified. The solidified AKD remained at room temperature for several days, which induced blooming. After the contact angle reached 150°, a plaster replica was prepared using the AKD surface as a mold. A mixture of agar (20) Gomez-Suarez, C.; Bruinsma, G. M.; Rakhorst, G.; van der Mei, H. C.; Busscher, H. J. J. Colloid Interface Sci. 2002, 253, 470–471. (21) Ranc, H.; Elkhyat, A.; Servais, C.; Mac-Mary, S.; Launay, B.; Humbert, P. Colloids Surf., A 2006, 276, 155–161. (22) Szabo, D.; Akiyoshi, S.; Matsunaga, T.; Gong, J. P.; Osada, Y.; Zrinyi, M. J. Chem. Phys. 2000, 113, 8253–8259. (23) Kaneko, D.; Gong, J. P.; Zrinyi, R.; Osada, Y. J. Polym. Sci., Part B 2005, 43, 562–572. (24) Daniels, K. E.; Mukhopadhyay, S.; Houseworth, P. J.; Behringer, R. P. Phys. Rev. Lett. 2007, 99, 124501. (25) Kaneko, D.; Furukawa, H.; Tanaka, Y.; Osada, Y.; Gong, J. P. Prog. Colloid Polym. Sci. 2008, 135, 225–230. (26) Banaha, M.; Daerr, A.; Limat, L. Eur. Phys. J. 2009, 166, 185–188.
Published on Web 09/17/2010
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Nonomura et al. powder (6 g) and deionized water (144 g) was heated and agitated until the agar powder dissolved. We then poured 150 g of a 4% agar aqueous solution in Petri dishes of 12 cm diameter. The plaster replica was then immediately placed at the center of the Petri dish to transfer the fractal structure to the agar gel surface. At 277 K, the templates took ∼12 h to set while the gel formed. Measurements. The cross-sectional fractal dimension D was determined for each prepared sample. Before microscopic observation, all agar gels were freeze-dried using an EYELA FD-5N freeze-drying apparatus (Tokyo Rikakikai, Tokyo, Japan). From scanning electron microscope (SEM) images (JEOL JCM-5000, JEOL, Tokyo, Japan), the real cross sections were extrapolated. Using the real cross sections, D was determined by the boxcounting method.4,5 The contact angles of liquids on the gel surfaces were estimated using a DM-501 contact angle meter
Article (Kyowa Interface Science, Tokyo, Japan) based on a sessile dropmeasuring method with a water drop volume of 0.5 μL. The water purified by a water deionizing unit (DX-15 from Kurita Water Industries, Tokyo, Japan) was used for contact angle measurements. The dye rhodamine B was added to the water only when we took the image of water drops on agar gels. From the results of preliminary studies, we determined that the dye does not affect the spreading behavior of water significantly. The measurements were conducted in air at 298 K. All contact angles were measured at 10 different points and were averaged.
3. Results and Discussion a. Morphological Characterization of Fractal Agar Gels. Figure 1 illustrates microscopic images of the gel surfaces. There are many projections having a scale of several tens of micrometers. We found a hierarchy of small projections existing over larger ones on the surface. The microscopic images predict that the surface morphology of the AKD crystals is roughly transferred on the agar gels. Shibuichi et al. reported that there are two types of rough structures on the AKD crystals.5 One possesses a spherical shape with a roughness scale of ∼30 μm, whereas the other is a flake-like structure with ∼1 μm roughness. We observed cross sections of the freeze-dried gels to determine D by the boxcounting method, as per common practice.4,5,27 Self-similarity and D can be evaluated by the following relationship NðrÞ µ r - D
Figure 1. Surface microscopic images of a fractal gel surface at different magnification.
where r is the size of boxes, N(r) is the number of boxes to cover the object, and D is the fractal dimension of the object. D can be obtained from the slope of the log N(r) versus log r plot. The fractal dimension of cross section Dcross (1 < Dcross < 2) has been measured by the box-counting method because direct measurement of the fractal dimension D (2 < D < 3) is difficult. The
Figure 2. SEM images and trace curves of the cross section of a fractal gel surface (a-c) and a flat gel surface (d-f ). Langmuir 2010, 26(20), 16150–16154
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Figure 4. Water drops spreading on a fractal gel surface.
Figure 3. Number of boxes to cover the object, N(r), as a function of the size of boxes, r for a fractal agar gel (a) and a flat agar gel (b). Insets are d(log N)/d(log r) as a function of r.
fractal dimension D (2 < D < 3) of the surface has been evaluated as D=Dcross þ 1. Figure 2a-f shows typical SEM images of cross sections of the fractal and the flat agar gels at different magnifications. Trace curves of the solid surfaces were drawn from the cross-sectional SEM images and are shown in these Figures. The box size r was changed from 1 to 200 μm. Figure 3 shows the log N(r) versus log r plot for the fractal and the flat agar gel surfaces. The slopes of the straight lines indicate the fractal dimensions of the surface structures. We show the slopes d(log N )/d(log r) for the flat and fractal surfaces as two insets in Figure 3 to clarify their difference. As a result, the slope is almost constant at -1 in the flat surface. However, the discrete change in the slope is found in the fractal surface. Below ca. 4 μm and above ca. 30 μm, the slopes are constant around -1, but in the intermediate region, it is obviously smaller than -1. Because the discreteness is relatively larger than data scattering and the width of the scale range is ca. 1 decade, it can be concluded that this discreteness is a significant difference. The evaluation of the slopes based on the least-mean-square method showed that the slope is -1.2 between the two critical (27) Mayama, H.; Tsujii, K. J. Chem. Phys. 2006, 125, 124706.
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sizes. Therefore, the cross section of the surface is a fractal with the dimension Dcross = 1.2, and self-similarity is found to hold between r=4 and 30 μm. The fractal dimension of the surface, D, was thus evaluated to be 2.2. b. Wettability of Fractal Agar Gels. When water droplets come in contact with fractal agar gel surfaces, the droplets spread along the surfaces, as shown in Figure 4. The images show three distinct wetting regimes. At early times (time t = 1 to 2 ms), the surface of the droplet deforms significantly, as a capillary wave travels along it. In this regime, the capillary forces due to the curved surface rapidly drive the spreading process, whereas the inertia of the fluid resists the deformation.28,29 In the second regime (t g 5 ms), the spreading droplet takes the form of a spherical cap with a small dynamic contact angle. Between the two regimes (3 e t e 4 ms), the amplitude of the surface deformation is large enough to separate the drop into two parts. One part remains suspended by the needle and retracts to form a sphere, whereas the other continues to spread on the surface. These three regimes were observed in the spreading processes for both the fractal and the flat agar gels. We found significant differences in wetting dynamics between the fractal and the flat agar gels. To quantify the effects of these wetting properties, the time-dependent change of contact angle θ during the spreading processes was measured. Figure 5 shows the time evolution of the θ of water on the fractal and the flat gel (28) Bird, J. C.; Mandre, S.; Stone, H. A. Phys. Rev. Lett. 2008, 100, 234501. (29) Courbin, L.; Bird, J. C.; Reyssat, M.; Stone, H. A. J. Phys.: Condens. Matter 2009, 21, 464127.
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Figure 5. Time evolution of the contact angle θ of water on a fractal gel surface (9) and on flat gel surfaces (0).
Figure 7. Schematic illustration of the present theoretical model (a) and dependence of 3(1 þ ε)/(10 þ ε) on ε (b) within the range of ε e 1.
volume is constant, its spreading size and apparent contact angle are related to each other. Assuming the volume of the drop Ω as Ω ¼ Figure 6. Photograph of 0.5% aqueous solution of rhodamine B on a fractal surface.
surfaces. The log-log plot of the flat agar gels is a straight line with a slope of -0.27. This result roughly follows the Tanner’s law in which θ is proportional to t-0.3 on smooth and clean surfaces.30 For fractal agar gels, the slope was -0.39. These results show that the spreading velocity of water droplets on fractal gels is faster than that on flat gel surfaces. Another interesting phenomenon was the appearance of a wicking front. Figure 6 is an image of a 0.5% rhodamine 6B aqueous solution on a fractal agar gel. A bright area was observed around the red droplet, which corresponds to water flowing under capillary action into the groove network on the fractal surface. Higher magnification images taken in the area of the wicking front clearly show that the water is present only in the grooves. The wicking velocity was, therefore, greater than that of the spreading velocity. Dussaud et al. studied liquid transport in the networked microchannels of the skin surface and detected the wicking flows from an initially placed reservoir drop.31Adapting a theory for fracture networks together with a treatment of capillary flow in a single V groove that has a cuneate cross-section, they described a model to account for the observed flows dependent on the surface tension and viscosity of liquids, and the groove shape, depth, and density of the grooves.
4. Discussion Why is the spreading velocity of water droplets on fractal gels faster than that on flat gel surfaces? Let us briefly discuss this topic along the usual scenario of Tanner’s law.30,32 Because the drop (30) Tanner, L. H. J. Phys. D 1979, 12, 1473–1484. (31) Dussaud, A. D.; Adler, P. M.; Lips, A. Langmuir 2003, 19, 7341–7345.
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π 3 R θD 4
ð1Þ
where R is the radius of the spreading drop on the agar gel surface and θD is the contact angle during spreading (Figure 7a). By differentiation -
1 dθD 3 dR ¼ θD dt R dt
ð2Þ
where dR/dt reflects the energy dissipation rate arising from viscous phenomena, as discussed below. Originally, the entropy generation of fluid is caused by heat transfer and internal friction (viscosity). The entropy generation rate is33 dS ¼ S_ dt Z Z Kðgrad TÞ2 η Dvi Dvk 2 Dvl 2 dV þ þ dV δ ¼ ik 2T Dxk Dxi 3 Dxl T2 Z þ
ς ðdiv vBÞ2 dV T
ð3Þ
where T is temperature, κ, η, and ζ are the thermal conductivity, the usual, and the second viscosity coefficients, respectively, vi, vk, and vl are the tensor notation of velocities, xi, xk, and xl are the tensor notation of coordinating axes, δik is the Kronecker delta, and B v is the velocity of fluid. The first and other terms in eq 3 are due to thermal conduction and viscous friction in fluid, respectively. Under the conditions of a thermally isotropic system, an (32) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillary and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; Springer: 2002; pp 142-150. (33) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics, 2nd ed.; ButterworthHeinemann: Boston, 1998; p 195.
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incompressible ideal fluid, and the symmetry of the spreading drop, the energy dissipation rate TS_ is described as32 Z L Z e dvx ðzÞ 2 T S_ dx η dz ¼ FV ð4Þ dz 0 0 where vx(z) is the spreading velocity of the fluid at height z, e is θDx, x is the distance from the contact line, and F and V are the force to spread the drop and the spreading velocity of macroscopic contact line, respectively. On the flat surface, the velocity gradient is usually approximated as dvx ðzÞ V dz θD x
ð5Þ
It is easy to imagine that hydrophilic rough surface would enhance the spreading velocity because the surface touches with liquid at different height. Actually, the spreading velocity on the fractal gel surface was faster than that on the flat gel surface. Assuming that the fluid velocity at the gas-liquid interface of the liquid on the rough surface V 0 is V 0 ≈ V1þε based on the experimental findings, the effective velocity gradient on the rough surface could roughly be described as dvx ðzÞ V0 V 1þε dz θD x θD x
ð6Þ
where V 0 is measured along the same axis where V was determined and ε is phenomenologically introduced to describe that V 0 on the rough surface (ε > 0) was faster than V on the flat surface (ε=0). Therefore, eq 4 becomes Z ηV 2þ2ε L dx ηV 2þ2ε l ð7Þ T S_ ¼ θD θD a x where a is size of a molecule, L is the characteristic size Rdetermined from the volume of a water drop (L=Ω1/3), and l= La (1/x) dx. When FV on the flat surface is modified as FV1þε on the rough surface, we get ηlV 1þε ð8Þ F ¼ θD where F is determined by the balance between the surface tensions of the liquid on the macroscopic drop and the wicking film (the thin film in Figure 7a): γ θD F F~ ¼ γL - γL cos θD L 2 Therefore, V
2
1=ð1 þ εÞ V θD 3=ð1þεÞ l
ð9Þ
ð10Þ
where V* = γL/2η and V is equal to dR/dt in eq 2. By the substitution of eq 10 into eq 2, we get dθD ðV Þ1=ð1þεÞ ð13 þ 4εÞ=3ð1 þ εÞ ¼ θD dt L
ð11Þ
where R ≈ LθD-1/3. The temporal change of θD can be obtained as θD t - 3ð1 þ εÞ=ð10 þ εÞ
ð12Þ
Equation 12 is consistent with Tanner’s law for the flat surfaces (ε=0). By considering the deviation of the velocity gradient arising from surface roughness, the power-law behavior of the spreading liquid including Tanner’s law can thus be discussed. Figure 7b shows the dependence of the exponent of -3(1 þ ε)/(10 þ ε) within 16154 DOI: 10.1021/la103123d
the range of ε e 1 on ε, which gradually decreases with an increase in ε. In our experiments, ε for the fractal surface is estimated to be 0.4. Considering V in terms of t based on eqs 10 and 12 VðtÞ µ t - 9=ð10 þ εÞ
ð13Þ -0.9
-0.87
and µ t , For the flat and fractal surfaces, V(t) µ t respectively, which means that the decay of V on the fractal surface is slower than on a flat surface, that is, faster spreading velocity on the fractal surface. Here we have semiquantitatively discussed the spreading behaviors of liquids on flat and rough surfaces. The correlation between ε and surface roughness (fractal dimension, characteristic sizes, etc.), however, will be discussed in further studies. Acceleration of the spreading of water on rough surfaces has been observed by some researchers. Particularly fascinating is the spreading behavior of liquids over micropillars fabricated by patterning a layer of photoresist.29,34-36 McHale et al. reported a power-law behavior described by θD µ t-3/4∼-3/10 and contact angle oscillations that are caused by the microscopic stick-slip motion of the liquid droplet at the edges of the aligned pillars.34 Courbin et al. showed that the liquid droplets form not only circles but also square and octagonal shapes in spreading.36 In the present study, we report the spreading characteristics of water droplets on fractal agar gel surfaces. We do not observe the contact angle oscillations because specific oscillations are not enhanced when contact angle hysteresis occurs at the edges of a random rough surface. In addition, we find that the contact lines of water droplets on the agar gel surfaces are serpentine curved, as shown in Figure 6. Such unstable spreading behavior has been reported for some gel surfaces and is caused by the energy transfer from the mean flow to the surface fluctuation because of the work done by the mean flow at the interface.25,34 When a thin fluid layer flows over a gel surface much thicker than the fluid, unstable modes appear above a critical liquid velocity. These characteristic spreading phenomena might also occur on the human tongue, the nasal mucus, the esophageal mucus, and the small intestine wall because they are hydrophilic fractal surfaces similar to fractal agar gels.
5. Conclusions We successfully prepared agar gels with hierarchical rough surfaces by transferring fractal surface structures of AKD. Boxcounting measurements confirmed the surface structures to be fractals. We suppose that they are suitable for a new biosurface model because their dimensions are roughly similar to known fractal dimensions of biosurfaces; D = 2.2 in human cortical surfaces and 2.3 in human skin surfaces,37,38 although values of D for the tongue, the villous surface, and the small intestine walls have not yet been reported. The rough structure accelerated the spreading of water droplets. For the fractal and flat agar gel surfaces, the contact angle θ of water droplets was proportional to t-0.39 and t-0.27, respectively. The experimental results of the wetting phenomena of a rough surface were verified theoretically. Acknowledgment. This research was supported by a Grant-inAid for Scientific Research on Innovative Area (no. 21106504) from the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT). (34) McHale, G.; Shirtcliffe, N. J.; Aqil, S.; Perry, C. C.; Newton, M. I. Phys. Rev. Lett. 2004, 93, 0361021–0361024. (35) Ishino, C.; Reyssat, M.; Reyssat, E.; Okumura, K.; Quere, D. Europhys. Lett. 2007, 79, 56005-1–56005-5. (36) Courbin, L.; Denieul, E.; Dressaire, E.; Roper, M.; Ajdari, A.; Stone, H. A. Nat. Mater. 2007, 6, 661–664. (37) Ha, T. H.; Yoon, U.; Lee, K. J.; Shin, Y. W.; Lee, J. M.; Kim, I. Y.; Ha, K. S.; Kim, S. I.; Kwon, J. S. Neurosci. Lett. 2005, 384, 172–176. (38) Honda, H.; Imayama, S.; Tanemura, M. Fractals 1996, 4, 139–147.
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