Article pubs.acs.org/Langmuir
Anomalous Spreading with Marangoni Flow on Agar Gel Surfaces Yoshimune Nonomura,*,† Shigeki Chida,† Eri Seino,† and Hiroyuki Mayama‡ †
Department of Biochemical Engineering, Graduate School of Science and Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa 992-8510, Japan ‡ Research Institute for Electronic Science, Hokkaido University, CRIS Building, N21W10, Sapporo 001-0021, Japan S Supporting Information *
ABSTRACT: We have experimentally observed anomalous spreading of aqueous alcohol solutions on flat and rough fractal agar gel surfaces. On flat agar gel surfaces, extremely fast spreading [θD(t) ∝ t−0.92] that differs from Tanner’s law [θD(t) ∝ t−0.3] was observed when the liquid contained over 9 wt % of 1-propanol in which strong Marangoni flow was observed as a fluctuation on the liquid surface. However, on fractal gel surfaces, different spreading dynamics [θD(t) ∝ t−0.58] were observed, although Marangoni flow still occurred. We found the surfacedependent spreading can be discussed in terms of competition between Marangoni flow and the pinning effect due to surface roughness.
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topography and chemical patterning,26 and a theoretical model on two-dimensional droplet spreading on random topographical surfaces.27 In contrast, we approximately rule out any dependence of the spreading dynamics on surface roughness.17 On the other hand, to acquire a deep understanding of the spreading dynamics, it is crucial to not only observe the spreading but also to find experimental and theoretical strategies to control it. In general, the spreading dynamics of liquids depend on the interfacial tension and the viscosity of the liquid.28 Hoffman showed that the characteristic moving velocity of threephase contact lines can be described by γ/η, where γ and η are the surface tension and viscosity of liquid, respectively.29 We focus herein on Marangoni flow (or the Marangoni effect) as the driving force behind water spreading over a gel surface. Marangoni flow is a well-known mechanism for mass transfer along an interface between two phases from one with lower γ to another with higher γ. Roughly speaking, there are two methods to induce Marangoni flow. One method is to form a surface tension gradient between two phases, causing the fingering instability of thin spreading films, fast drop movements, and accelerated dissolution of solid particles.30−34 The surface tension gradient is induced by adsorption of surfactants at interfaces. Troian et al. found that a drop of aqueous surfactant solution placed on a glass surface spreads by propagating fingers the velocity and shape of which depend on the thickness of the ambient water layer and on the surfactant concentration.35 Stoebe et al. and Nikolov et al. reported the spontaneous spreading of surfactant aqueous solutions over hydrophobic solid surfaces.36−38 Cachile and Cazabat investigated the spontaneous spreading of solutions of nonionic surfactants and observed an increase in the
INTRODUCTION Wetting dynamics of liquids on gel surfaces is important for understanding mass transfer phenomena at biological interfaces because spreading processes contribute to effective nutritional absorption through the small-intestinal wall or to sensitization of the tongue surface.1−5 In general, a liquid is expected to spread and cover a solid−vapor interface only if the spreading coefficient S = γS − (γSL + γL) is positive, where γS, γSL, and γL are the surface tensions between the solid−vapor, solid−liquid, and liquid−vapor interfaces, respectively.6 Indeed, the spreading velocity is not always directly related to the magnitude of the spreading coefficient.7−9 A well-established law that describes the spreading of nonvolatile liquids on flat and clean surfaces is θD ∝ t−0.3, which is called Tanner’s law.10,11 Here, θD and t are the contact angle at three-phase contact lines and the dynamic spreading process time, respectively. Previous studies report that the spreading flow on flat gel surfaces follows power-law kinetics.12−16 We prepared agar gels with rough surfaces, here referred to as “fractal agar gels”, to model biological surfaces coated with mucus.17 The rough structure of the fractal agar gels mimics the hierarchical structure of the small-intestinal wall, which consist of circular folds, villuses, and microvilluses. We found that the rough structure accelerated the spreading of water droplets and induced the appearance of a wicking front. We also discuss the spreading dynamics in terms of surface roughness as an extension of Tanner’s law, whereas many previous studies reported that the spreading behaviors of liquids is controlled by the surface morphology of solids.18,19,21,20,22 Some recent studies showed the effects of surface morphology on wettability and spreading behaviors: for example, behaviors of impinging droplets on hydrophobic textured surfaces,23,24 the effect of the remaining liquid film on the pillars’ tops,25 spreading and pinning of oil droplets on © 2012 American Chemical Society
Received: September 10, 2011 Published: January 26, 2012 3799
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spreading dynamics of the liquids: the exponent x when we assume a power law for θD(t) of the form θD(t) ∝ t−x was roughly constant at the initial volume of 0.3−0.7 μL. Furthermore, we investigated how volume change of the spreading droplet by evaporation affects the time evolution of the contact angle change during spreading. As a result, we confirmed that the contact angle change can be neglected within 1 s at least. This is discussed in the Supporting Information. We also checked the top view of the initial droplet before measuring spreading velocity. The top views of liquid droplets were circle shaped on both flat and fractal agar gel surfaces. The fitting procedures of the temporal change in contact angles were achieved by the least-squares method. The agreement of regression curves with experimental values was evaluated by the correlation coefficient r, which was defined as the covariance of the two variables divided by the product of their standard deviations. This coefficient ranges from −1 to 1. A value of 1 implies that a linear equation describes the relationship between X and Y perfectly, with all data points lying on a line for which Y increases as X increases. A value of −1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is no linear correlation between the variables. In the present study, good agreements with r > 0.9 were obtained for all alcohol aqueous solutions. On flat agar gel surfaces, the spreading phenomena with power law were observed in 5−1500 and 300−1500 ms for water and alcohol aqueous solutions, respectively. On the other hand, on fractal agar gel surfaces, the power law was observed in 5−1500 ms for both water and alcohol aqueous solutions.
spreading velocity and the emergence of contact line instabilities around the critical micelle concentration.39 Another method to induce Marangoni flow is by evaporating mixtures containing volatile liquids. A well-known example is “tears of wine” on glass surfaces.40−42 Because water has a higher surface tension than alcohol, the concentration gradient of alcohol creates a surface tension gradient. Pesach and Marmur experimentally studied the spreading of binary liquid mixtures on glass.43 Binary mixtures of completely spreading components are expected to spread faster than individual components when Marangoni flow coincides with the spreading direction. Many mixtures of two incompletely spreading components are expected to spread completely when Marangoni flow coincides with the spreading direction. Fanton and Cazabat showed that the spreading kinetics are controlled by some parameters: composition, volatility, temperature, and humidity.44 Gotkis et al. reported instabilities when isopropyl alcohol is deposited on a silicon wafer. This instability is characterized by emission of drops ahead of the expanding front, with each drop followed by satellite droplets.45 In view of these findings, we expected aqueous alcohol solutions on gel surfaces to exhibit anomalously fast spreading as a result of surface roughness and Marangoni flow, because both factors play significant roles in spreading. In the present study, we observed the spreading of some aqueous alcohol solutions on flat gel surfaces by using a high-speed camera to examine the effects of alcohol on the wetting dynamics. We also studied the spreading on fractal agar gel surfaces to investigate the effects of surface roughness. Furthermore, based on a simple model, we discussed the observed spreading dynamics with Marangoni flow on flat and fractal agar gels.
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RESULTS (a). Spreading of Aqueous Alcohol Solutions on Flat Agar Gel Surfaces. We investigated the behavior of aqueous alcohol solutions that spread with Marangoni flow. In this section, we focus on how 1-propanol aqueous solution spreads by comparing it with water. When droplets of water and 9 wt % 1-propanol aqueous solution contacted flat agar gel surfaces, the droplets spread along the surfaces, as shown in Figure 1. At early times (for example, t = 30 ms), we found no significant
EXPERIMENTAL SECTION
(a). Materials. Agar powder, methanol (CH3OH), ethanol (C2H5OH), 1-propanol (C3H7OH), 1-butanol (C4H9OH), 1-hexanol (C6H13OH), 1-octanol (C8H17OH), and 1-decanol (C10H21OH) were purchased from Kanto Chemical Co. (Tokyo, Japan) and used without further purification. Water was purified by a DX-15 water deionizing unit (Kurita Water Industries Ltd., Tokyo, Japan). (b). Preparation and Method. Flat agar gels and fractal agar gels were prepared by the method discussed in a previous study.17 A mixture of agar powder (6 g) and deionized water (144 g) was heated and agitated until the agar powder dissolved. We then poured 150 g of the 4 wt % agar aqueous solution in 12-cm-diameter Petri dishes. A plaster replica with a fractal surface was placed at the center of each Petri dish to transfer its fractal structure to the agar gel surface. Alcohols were mixed with water for 2 min in a screw-cap test tube by using a vortex mixer (Vortex-genie 2, Scientific Industries Inc.). We checked the viscosity, density, and surface tension of 10 wt % 1-propanol aqueous solutions when the mixing time was 1, 10, 100, 1000, and 10 000 s. Statistically significant changes with the mixing time were not observed in these physical parameters. The contact angle of liquids on the gel surfaces θD was estimated with a DM-501 contact angle meter (Kyowa Interface Science, Tokyo, Japan) on the basis of the sessile drop measuring method with a setting volume of a water droplet = ca. 0.5 μL. The liquid droplets were dropped by an automatic dispenser for the DM-501 contact angle meter from the tip of a 22G syringe (internal diameter, 0.4 mm; outer diameter, 0.7 mm) coated with tetrafluoroethylene. We checked the apparent initial volume of the 1-propanol aqueous solutions on flat agar gel surfaces and fractal agar gel surfaces: the volume was derived from a radius and a height of the water droplet on agar gel surfaces. The averages of the apparent initial volume on flat and fractal agar gel surfaces were 0.39−0.50 and 0.44−0.63 μL, respectively. The volume decreased by about several tens of percent at 1 s after instillation. The variability on the initial volume and the temporal change have few effects on the
Figure 1. Water droplets spreading on flat agar gel surfaces: (a) water and (b) 9 wt % 1-propanol aqueous solution. 3800
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in the range 9−20 wt %, x = 0.5−0.8 in 20−100 wt %. Remarkably, at 9 wt % 1-propanol, x increased in a discrete jump from ∼0.3 to 0.92 in the second step, whereas with an increase in the 1-propanol concentration, x increased in a continuous manner from ∼0.3 to ∼0.8 in the first step. These results show that the spreading accelerated only when the liquid contained a suitable amount of 1-propanol. We observed an interesting droplet surface phenomena only when the liquid contained 9−20 wt % of 1-propanol. In particular, droplet surfaces deformed and fluctuated during spreading, whereas pure water droplets spread smoothly on the gel surfaces. Such surface deformation or fluctuation was reported for the spreading of droplets in which strong Marangoni flow occurred.35,37−39 This implies that Marangoni flow plays a significant role in determining the spreading dynamics. In addition, strong Marangoni flow is always larger (x ∼ 0.5−0.8) than that in spreading of water (x ∼ 0.3) above 20 wt %. The correlation between x, 1-propanol concentration, and strong Marangoni flow is semiquantitatively discussed later in comparison with the vapor−liquid equilibrium data of the 1-propanol−water binary system (the inset of Figure 3).46 We also evaluated the spreading velocity for other alkyl alcohol solutions containing methanol, ethanol, 1-butanol, 1-hexanol, 1-octanol, or 1-decanol. The contact angle of all alcohol solutions obeyed a temporal power law. The power-law exponents of all alcohol solutions except 1-propanol solutions were smaller than that for water; for example, x = 0.21 for 10 wt % methanol solution, 0.16 for 10 wt % ethanol solution, 0.17 for 1 wt % 1-butanol solution, and 0.12 for 0.1 wt % 1-hexanol solution (Table 1). These results show that a significant increase in spreading velocity is caused by the addition of 1-propanol, which shows its unique property. This point is briefly discussed later. (b). Spreading of Aqueous Alcohol Solutions on Fractal Agar Gel Surfaces. In this section, we focus on how 1-propanol aqueous solutions with Marangoni flow spread over rough fractal agar gel surfaces, where Marangoni flow and surface roughness accelerate the spreading of the mixture on flat surface and water on fractal surface,17 respectively. In other words, we speculated that remarkably faster spreading of the solutions would occur on fractal agar gels. When droplets of water and 10 wt % 1-propanol aqueous solution contacted fractal agar gel surfaces, the droplets spread over the surfaces, as shown in Figure 4. On the fractal surfaces, the contact angle of both water and 1-propanol solution obeyed a similar power law. The contact angle θD(t) for water was proportional to t−0.53, while θD for 10 wt % 1-propanol solution was proportional to t−0.58; these results are not significantly different. Next, the effect of the 1-propanol concentration on the spreading velocity was investigated. The power-law exponent x was ∼0.5, independent of the 1-propanol concentration, as shown in Figure 5. Remarkably, these results indicate that the power law followed by the spreading of the mixture over fractal agar gels is almost same as that of the spreading of water, even if 1-propanol is added to the liquids. Namely, the surface roughness of agar gel suppresses the spreading of the mixtures. This tendency is contradicted by the experimental results in our previous study in which the surface roughness accelerates the spreading of water.17
differences in the wetting dynamics between water and 9 wt % 1-propanol solution; the spreading droplet took the form of a spherical cap with a small dynamic contact angle. However, at t = 300 and 1000 ms, the 9 wt % solution formed a thinner spherical cap with a smaller contact angle than water. After several tens of seconds after the contact of water or the 9 wt % solution, the agar gel surfaces were wetted perfectly. To quantify the effects of these wetting properties, we measured the time-dependent change of contact angle θD(t) during the dynamic spreading processes. Figure 2 shows the time evolution of θD(t) for water
Figure 2. Time evolution of the contact angle θD(t) of 9 wt % 1propanol aqueous solution (□) and water (Δ) on flat agar gel surfaces.
and 9 wt % 1-propanol solution on a flat gel surface. The log−log plot from 5 to 1500 ms for water forms a straight line with a slope of −0.26. This result approximately follows a well-known power law, namely Tanner’s law.10,11 The addition of 1-propanol greatly accelerates the spreading. For the 9 wt % solution, θD(t) was proportional to t−0.45 and t−0.92 in 5−300 and 300−1500 ms, respectively. These results show that the spreading processes of 1-propanol solutions consist of two steps, that is, the first step of 5 ms to several hundred milliseconds and the second step beyond several hundred milliseconds. In addition, the results also show that the addition of 1-propanol increases the spreading velocity in the second step. Next, the temporal change of contact angle θD(t) was measured under 13 concentration conditions to examine the effect of 1-propanol concentration on the spreading velocity. Figure 3 shows the relationship between 1-propanol concen-
Figure 3. The relationship between the 1-propanol concentration and the exponent x on flat agar gel surfaces when θD(t) ∝ t−x: 0−300 ms (□) and 300−1500 ms (Δ). The data of vapor−liquid equilibrium (○) and boiling point (●) of 1-propanol−water binary system are shown as an inset figure.
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tration and exponent x when we assume a power law for θD(t) of the form θD(t) ∝ t−x. At 0−8 wt %, the spreading velocity was almost constant and x ∼ 0.3 in the first and second steps. However, x exceeded 0.7 in the second step for concentrations
DISCUSSION (a). Correlation between 1-Propanol Concentration and Marangoni Flow. We observed the strong Marangoni 3801
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Table 1. Exponent x of Spreading Process of Alcohol Aqueous Solutions on Flat Agar Gel Surfaces When θ(t) ∝ t−x 10% 1% 0.10% 0.01%
CH3OH
C2H5OH
C3H7OH
C4H9OH
C6H13OH
C8H17OH
C10H21OH
0.21(±0.14) 0.21(±0.14) 0.11(±0.07) 0.24(±0.10)
0.16(±0.11) 0.37(±0.12) 0.41(±0.08) 0.39(±0.09)
0.74(±0.24) 0.22(±0.09) 0.24(±0.12 0.23(±0.14)
0.17(±0.06) 0.15(±0.03) 0.26(±0.04)
0.12(±0.09) 0.17(±0.11)
0.07(±0.07)
0.11(±0.12)
(b). Peculiarity of 1-Propanol Aqueous solution. Why does the 1-propanol aqueous solution spread faster than pure water and other alcohol solutions? As mentioned in the Introduction, the spreading velocity of liquids depends on their interfacial tension and viscosity.28,29 However, we found that the power-law exponent x was not highly correlated with interfacial tension, viscosity, or the ratio of these physical factors. The correlation coefficients, which reflect statistical connections between two quantities, were 0.340, 0.544, and 0.333 for interfacial tension, viscosity, and their ratio, respectively.47−52 These results demonstrate that the anomalous fast flow observed with the addition of 1-propanol is caused by a specific driving force that is not accounted for in the common theory of liquid spreading. As mentioned in the previous section, we inferred that Marangoni flow was caused by the surface tension gradient induced by adsorption and evaporation of 1-propanol (Figure 6). Alcohols with a short or
Figure 4. Time evolution of the contact angle θD(t) of 10 wt % 1-propanol aqueous solution (□) and water (Δ) on fractal agar gel surfaces.
Figure 5. The relationship between the 1-propanol concentration and exponent x on fractal agar gel surfaces when θD(t) ∝ t‑x (0−500 ms).
Figure 6. Schematic illustration of liquid droplet with Marangoni flow.
flow in 9−20 wt % 1-propanol solutions. Here, let us consider the correlation between 1-propanol concentration and Marangoni flow. To understand this, the vapor−liquid equilibrium data of 1propanol/water binary solution, which provides us the concentrations of 1-propanol in vapor and liquid, would be helpful for rough understanding. As shown in the inset of Figure 3,46 the boiling point decreases sharply from 100 to 90 °C at 90 wt %. On the other hand, the 1-propanol concentration in vapor is always higher than that in liquid at ≲70 wt %. In particular, the concentration of vapor is several to 10 times higher than that in liquid phase in several to 20 wt %. This suggests the possibility that surface tension may fluctuate by emission of 1-propanol molecules from the droplet to air. Considering the balance between the boiling point and the concentration in vapor, it is easy to understand that the evaporation of a mixture of 10−20 wt % causes significant change in surface tension, which induces strong Marangoni flow as the fluctuation of the air−liquid interface. In addition, it should be noted that Marangoni flow always exists in spreading a mixture of 1-propanol on agar gels because there is a difference in surface tensions between the mixture and a very thin water film on agar gel. In the following discussion, we treat Marangoni flow through a difference in surface tension Δγ = γW − γM, where γW and γM are the surface tensions of the wicking front and the macroscopic droplet, respectively.
long alkyl chain may not be suitable to induce Marangoni flow because of a lack of adsorption ability onto the surface or because of volatility. In addition, they spread slower than water: this tendency is consistent with Hoffman’s finding that the characteristic moving velocity of three-phase contact lines can be described by γ/η,29 where γ and η for alcohol/water mixtures are smaller and larger than those for water, respectively. (c). Spreading on Flat and Fractal Gels with Marangoni Flow. Next, we discuss the mechanism of the observed surface-dependent spreading with Marangoni effect of 1-propanol aqueous solutions. In our previous study, we found that surface roughness of agar gel accelerates the spreading of water.17 In discrepancy, we experimentally observed quite the opposite effect for the mixture in the present studies, that is, the surface roughness strongly suppresses the spreading of the mixture. In the following section, we discuss the mechanism along the following line: We first present the governing equation to show our scheme and then discuss the detail in spreading of liquids (water and the mixtures) on flat and fractal rough surfaces. Governing Equation of Spreading. We roughly discuss a possible scenario to catch the essence of the observed spreading behaviors based on the simple framework of “hydrodynamic theory”53,54 because our experiments include the complex factors such as Marangoni flow and surface roughness. Before detailed discussion, we address the governing equation to make 3802
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height z, e is θD(t) x, and F(t) and V(t) are the spreading force and velocity of the macroscopic contact line, respectively. By considering dvx(z)/dz ≈ V(t)/θD(t)x and l = ∫ aL(1/x) dx, (a is size of a molecule) F(t) is
the theoretical scenario clear. The governing equation in the following discussion is summarized as γ Δγ θ D(t ) − fpin θ D(t ) + M θ D(t )3 V (t )1 + ε ≈ ηl 2ηl (1) where V(t) is the velocity of a macroscopic contact line at time t and other notations are explained below. Roughly speaking, the first, second, and third terms describe Marangoni flow, pinning force by physical defects in surface roughness, and viscous dissipation (Tanner’s law) in spreading on flat surface, respectively. ε is the velocity enhancement factor by surface roughness in the spreading.17 On smooth surface, ε = 0, while ε ≠ 0 on rough surfaces. For example, ε = 0 and 1.02 in θD ∝ t−0.3 (Tanner’s law) and t−0.55 (spreading of water and the mixture on fractal rough surface), respectively, because of θD ∝ t−3(1+ε)/(10+ε).17 f pin is the pinning force ( f pin ≠ 0 for a pinned state on rough surface and f pin = 0 for a nonpinned state on a flat surface). The pinning force on inhomogeneous surfaces is considerable: some experimental results predicted that the spreading is hindered by periodical energy barriers combined with physical or chemical pinning.55−57 As discussed in the following sections, eq 1 comprehensively summarizes the spreading behaviors in the different cases. Case 1: Spreading of the mixtures (≥9 wt % 1-propanol) with Marangoni flow (Δγ ≠ 0) and no pinning effect (f pin = 0) on smooth surfaces (ε = 0). Case 2: Spreading of a water droplet with no Marangoni flow (Δγ = 0) and no pinning effect (f pin = 0) on flat agar surfaces (ε = 0). Case 3: Spreading of the mixtures (≥9 wt % 1-propanol) with Marangoni flow (Δγ ≠ 0) and pinning effect (f pin ≠ 0) on rough agar surfaces (ε ≠ 0). Case 4: Spreading of a water droplet with no Marangoni flow (Δγ = 0) and pinning effect ( f pin ≠ 0) on rough agar surfaces (ε ≠ 0). Table 2 summarizes cases 1−4 with the related factors. Under the hypothetical condition of constant volume of the droplet defined by eq 6, temporal change of θD(t) in each case can be argued.
F (t ) ≈
1 2 3 4
Marangoni effect
solution mixtures (≥9 wt %) water and mixtures (
