Inertial to Viscoelastic Transition in Early Drop Spreading on Soft

Article pubs.acs.org/Langmuir

Inertial to Viscoelastic Transition in Early Drop Spreading on Soft Surfaces Longquan Chen,† Elmar Bonaccurso,*,† and Martin E. R. Shanahan*,‡ †

Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstraße 17, 64287 Darmstadt, Germany Université de Bordeaux, Centre National de la Recherche Scientifique (CNRS), and Arts et Métiers ParisTech, I2M, UMR 5295, F-33400 Talence, France

‡

S Supporting Information *

ABSTRACT: It has been known for many years that a spreading liquid droplet can be appreciably slowed on a soft, viscoelastic substrate by the appearance of a “wetting ridge” or protuberance of the solid near the triple phase contact line because of capillary forces. Viscoelastic dissipation in the solid surface can outweigh that of liquid viscosity and, therefore, dominate wetting dynamics. In this paper, we show that a short, rapid spreading stage exists after initial contact. The requisite balance determining the speed of motion is between capillary forces and inertial effects. As spreading proceeds, however, inertia lessens and the lower spreading speed allow for viscoelastic effects in the solid to increase. The transition between early inertial and viscoelastic regimes is studied with high-speed photography and explained by a simple theory.

1. INTRODUCTION The spreading of liquid drops on solid surfaces is intriguing and, perhaps surprisingly, not yet fully understood. Huh and Scriven1 were probably the first to consider the exceptionally complicated hydrodynamic problem of flow near the triple line of a liquid spreading on a solid surface. Since then, much work has been performed, of which a classic review is given by de Gennes.2 The last author developed a highly successful and elegant “hydrodynamic” model, based on Huh and Scriven’s earlier work, in which the unbalanced Young, capillary force at the three-phase contact line (TPCL) of a sessile drop parallel to the solid surface is compensated by viscous friction because of essentially Poiseuille flow within the spreading liquid. However, the component of liquid surface tension perpendicular to the solid, although often neglected, can have a significant influence on soft solids, because a “wetting ridge” or deformation of the solid at the TPCL of size of the order of γ/G arises, where γ is the surface tension of the liquid and G is the shear modulus of the solid. The size of this ridge, typically from a few tens of nanometers up to some micrometers,3−10 added to the viscoelastic nature of such a solid can seriously slow TPCL motion. The unbalanced Young force must contend with not only liquid viscosity but also “viscoelastic braking” because of dissipation within the traveling ridge associated with the TPCL.11 Viscoelastic behavior is, of course, a complex subject, but it is widely recognized since early polymer research (e.g., Flory12) that both time (strain rate) and temperature play an important role in deciding whether a given polymer behaves predominantly elastically or more viscously. The value of the relevant Deborah number,13 D = relaxation time/experimental observa© 2013 American Chemical Society

tion time, is a measure of this, and high strain rates or low temperatures can have the same effect of making a solid more resistant to change, thus “harder” or “more elastic”. The earlier work on drops spreading on viscoelastic solids considered that the substrates indeed displayed a significant viscous component.11,14−17 This work considers the very early stages of spreading on such materials, in which motion is so rapid that the viscoelastic solid behaves initially as a true, elastic substrate and inertial forces dominate spreading dynamics. As motion slows, inertia is of lesser importance and the viscous character of the substrate increases in importance. Other than on elastic substrates, on which fast spreading shows only one characteristic time governed by inertia, on viscoelastic substrates, fast spreading shows an additional characteristic time scale that is governed by the relaxation time of the substrate.18 On rigid substrates, spreading of a viscous drop proceeds in two stages, the so-called inertial, with high spreading speed, and viscous, with low spreading speed, regimes. In the first, for spreading times typically between 0.1 and 10 ms, the speed is determined by the balance of capillary and inertial forces.19,20 The spreading radius r grows with time t as r ∼ Ctα, where C is a coefficient and α is an exponent only depending upon the equilibrium contact angle of the liquid with the surface, θeq. Typically, α increases from ≈0.25 for θeq ≈ 120° to 0.5 for θeq ≈ 0.20 As spreading proceeds, for times ≳10 ms, the speed decreases and inertial effects lessen. In the second regime, viscous dissipation within the drop becomes the main source Received: November 26, 2012 Revised: January 12, 2013 Published: January 14, 2013 1893

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Table 1. Physical Properties and Equilibrium Contact Angles of Probe Liquids on PDMS and Hydrophobic Glass Surfaces θeq (deg) PDMS surfaces, G at 1 Hz (kPa) liquids

ρ (kg/m3)

γ (mN/m)

η (cP)

water 10 wt % ethanol/water 60 wt % glycerol/water

1000 980 1154

72.8 48.1 67.2

1.0 1.3 10.8

679

204

5.8

1.8

0.02

silanized silicon

110 ± 3 96 ± 1 92 ± 2

109 ± 2 94 ± 2 105 ± 3

contact angles on the PDMS and hydrophobic glass surfaces are shown in Table 1. The hydrophobic glass surfaces have similar wettability to the rigid PDMS. 2.2. Monitoring of the Wetting Process. Pendant drops with radii of 1.00 ± 0.05 mm were generated and brought to the substrates quasi-statically using a syringe pump. Both the needle and substrates were grounded to prevent the influence of any electrostatic charges on spreading.26,31 We recorded drop spreading with a high-speed camera (FASTCAM SA-1, Photron, Inc.) at 54 000 fps. 2.3. Data Analysis. The spreading radius r was extracted from the recorded images with an ad-hoc developed MATLAB (MathWorks, Inc.) algorithm. Because the early wetting dynamics is dominated by inertia, we applied the Pearson product-moment correlation coefficient, K, to check the power-law relation between r and t.

resisting spreading. The spreading radius follows Tanner’s law, r ∼ C′t0.1, where C′ is another coefficient.2,21−25 The transition from the inertial to the viscous regime has been found to take place in an interval between a characteristic inertial time TI = (ρR3/γ)1/2 and a characteristic inertial/viscous time TIV = (ργR/η2)1/8(ρR3/γ)1/2,19,20,26 where ρ is the density, η is the dynamic viscosity of the liquid, and R is the initial drop radius. Very recently,27 an additional stage at very early spreading times (t ≲ 0.1 ms) has been identified. During this stage, the radius grows according to a power law but with an exponent independent of surface wettability and always equal to 0.5 (r ∼ C″t0.5). Various types of power-law behavior have also been observed with the spreading of drops of non-Newtonian or “power-law fluids”.28 It was recently shown that inertial spreading also existed on viscoelastic substrates and that it was initially independent of surface softness.18 However, the rapid stage was followed by a slower viscoelastic spreading stage that set in at a characteristic transition time TT, which was always smaller than the inertial wetting time on rigid substrates. This time was found to depend upon surface softness as well as the surface tension of the liquid. However, a more precise analysis of the inertial-to-viscoelastic transition regime and the corresponding physical model are still missing. In this paper, we present ad-hoc measurements with drops of liquids with different properties spreading on a number of viscoelastic substrates and a simple scaling analysis based on an energy balance. The model, despite its simplicity, is capable of capturing the time scales of early inertial and viscoelastic spreading on soft substrates and, thus, describing how the characteristic transition time depends upon viscoelasticity.

⎞ ⎛ n K = ⎜⎜ ∑ (r′i − r′)(t ′i − t ′)⎟⎟ ⎠ ⎝ i=1 n n ⎞ ⎛ /⎜⎜ ∑ (r′i − r′)2 ∑ (t ′i − t ′)2 ⎟⎟, ⎠ ⎝ i=1 i=1 t ′ = log t ,

r′ =

1 n

n

∑ r′i , i=1

r′ = log r , n

t′ =

∑ t′i i=1

K can vary between −1 and 1. K = 1 (or −1 for a negative correlation) corresponds to a perfect linear relation between log r and log t, i.e., a perfect power-law relation between r and t, whereas values near zero indicate poor correlation. The transition in drop spreading occurs at the time TT, which is when K starts diverging from 1. To find the coefficient and the exponent of the power law, we fitted the experimental spreading radii for t ≤ TT by the least-squares method (LSM).

3. THEORETICAL MODELS We balance the energy contributions during drop spreading. The main driving force is capillarity, and the main braking (restraining) forces slowing spreading are the kinetic energy of the drop (i.e., inertia), viscous dissipation within the liquid, and viscoelastic dissipation in the deformed substrate. Because the Reynolds number is always notably larger than unity during the spreading process that we consider here, i.e., Re = ρUR/η ≫ 1, viscous dissipation within the drop can be neglected. The spreading velocity of the contact line is denoted U. 3.1. Inertial Wetting. The part of our model corresponding to the (early) inertial regime is a slight modification of existing work19,20 but allows some discussion on the power-law exponent to be engaged. As in the works cited, it is taken that the viscosity of the drop and the viscoelasticity of the substrate are unimportant in this regime. The dynamic balance is established between capillary and kinetic energy. Assuming that, in the initial stages, the drop remains essentially spherical but with a slightly flattened contact region with the substrate (Figure 1), the mass of liquid entrained is ≈kρr4/R. ρ is liquid density, and k is of the order π/4. Given the changing form of the traveling liquid disk, its average speed is of the order ṙ/2, leading to a kinetic (inertial) spreading force term of FK ≈ d/

2. MATERIALS AND METHODS 2.1. Surfaces and Liquids. We prepared soft polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning) samples by spin-coating the oligomer containing different cross-linker ratios on clean smooth glass slides at 500 rpm for 1 min. After curing at 70 °C overnight, we obtained soft PDMS surfaces of ≈200 μm thickness. We did not carry out experiments with thinner PDMS surfaces here, because we found that thickness had no major influence on wetting dynamics.18 By controlling the cross-linker/oligomer ratio from 1:10 to 1:100 (wt %), we made viscoelastic films with shear modulus, G, varying between ≈23 Pa and ≈680 kPa (measured at 1 Hz). For PDMS surfaces fabricated by spin coating, the root-mean-square roughness was always less than 2 nm.29,30 Thus, the effects of roughness on spreading could be neglected. Hydrophobic glass surfaces were fabricated by silanization in vapor phase. Clean glass substrates were treated with oxygen plasma (Femto, Diener Electronic GmbH, Germany) for 2 min and silanized by 1H,1H,2H,2H-perfluorodecyltriethoxysilane (Sigma-Aldrich) in a desiccator at ≈80 °C overnight. We used only liquids with a relatively high surface tension because it is only with these that the transition between inertial and viscoelastic spreading could be observed.18 Liquids used were pure water, mixtures of 10 wt % ethanol in water, and mixtures of 60 wt % glycerol in water. The physical properties of these liquids and corresponding equilibrium 1894

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⎛ φωτ r 2 ⎞ rγ 2 2πrγ ⎜1 + cos θeq − − ≈0 2⎟ ⎝ 2R ⎠ Gε (1 + ω 2τ 2)

(3)

Substituting for ω and considering sufficiently low speeds, as is the case at the transition between inertial and viscoelastic spreading, eq 3 can be simplified to 2R2(1 + cos θeq) − r 2 ≈

Figure 1. Schematic of a drop before (left side, spherical) and after (right side, truncated prolate spheroid) contact with a surface, with relevant notation. The dashed line represents a constant height from the solid surface.

1− 1+

d (πρr 4r /8 ̇ R) ≈ 0 dt

r ξ r ξ

2

2

2

= e−(2ξ / ψ )t (5)

With the reasonable approximation ξ ≪ 1, the left term of eq 5 can be written as (1 − (r/ξ))2. Thus, one finds the final expression r ≈ ξ(1 − e−(ξ / ψ )t ) = R 2(1 + cos θeq) [1 − e−((

)(

2(1 + cos θeq) 2Gε 2 / γφτR

))t ] (6)

This spreading equation predicts that, in this viscoelastic stage, spreading is faster on surfaces with larger G, with all other parameters remaining the same. 3.3. Inertial to Viscoelastic Wetting Transition Time TT. The crossover from inertial to viscoelastic spreading, although not an exact instant in reality, can be taken as when kinetic and viscoelastic spreading radii cross, i.e., when Ctα ≈ ξ(1 − e−(ξ/ψ)t). Solving this equation for t should yield an estimate of the crossover time TT. However, we must point out here that the above equations are only approximate solutions and that some of the parameters (i.e., ṙ, θ, G, and ω) vary during spreading, while others (i.e., ϕ and τ) are themselves simplifications whose value we cannot determine accurately. In fact, a whole spectrum of relaxation times is present at any given moment during drop spreading: the closer to the TPCL, the smaller the relaxation time and the faster the relaxation. Nevertheless, we believe that the equations that we derived for the two spreading stages capture the main physics involved in the spreading of a low-viscosity drop on a viscoelastic surface: (i) for inertial spreading, we have found a scaling law with an explanation why the spreading exponent α changes with the equilibrium contact angle θeq; (ii) for viscoelastic spreading, we have found an exponential decay that links the most relevant rheological parameters of the substrate material to the spreading dynamics.

(1)

In the very first instants, 1 + cos θeq ≫ r2/2R2. Equation 1 can then be simplified to 2πrγ(1 + cos θeq) −

(4)

Redefining R γωτ/2Gε = ψ and 2R (1 + cos θeq) = ξ , eq 4 can be finally written as ξ2 − r2 ≈ ψṙ. The solution of this differential equation is 2

dt(πρr4ṙ/8R). To within the multiplicative constant π/8, which is anyway only approximate, this is of the form that has been shown previously.19,20 However, we implement a slightly different form for the capillary spreading force. The drop is initially spherical, and the volume of the drop is constant during spreading. After spreading starts, the bottom of the drop becomes flattened because of contact with the surface, while the top of the drop is still not moving (see the dashed line in Figure 1). This way the initial sphere becomes a truncated prolate spheroid with a missing cap of height δ and radius r, as illustrated in Figure 1. From simple geometry, the dynamic contact angle is given by cos θ(t) ≈ (r2/2R2) − 1. Thus, the capillary force can be written as FC ≈ 2πrγ(cos θeq − cos θ(t)) ≈ 2πrγ(1 + cos θeq − (r2/2R2)). The second approximate expression is only valid for small r or, equivalently, for large θ(t). This condition applies for our analysis. Balancing FK and FC, we obtain the governing equation of inertial wetting ⎛ r2 ⎞ d 2πrγ ⎜1 + cos θeq − − (πρr 4r /8 ̇ R) ≈ 0 2⎟ ⎝ 2R ⎠ dt

R2γφτ ṙ 2Gε 2

(2) α

We recover the classic scaling law r ∼ Ct , where α ≈ 0.5. Clearly, as r increases, simple scaling becomes impossible but the effective value of α becomes smaller. The value of r/R for which α is no longer acceptable depends, of course, upon the value of cos θeq, with it occurring earlier for larger values of θeq. Taking the limit as θ → θeq, we obtain d/dt(πρr4ṙ/8R) ≈ 0, which suggests that the limiting lower value of α should be 1/5. This is, at least qualitatively, in agreement with previous work,20 including the more recent contribution,27 but was derived on different assumptions. 3.2. Viscoelastic Wetting. For the viscoelastic dissipation, Shanahan and Carré developed a model based on experimental adhesion results.11,14−17 However, this is not directly applicable with the high spreading speeds arising in our work. Instead, we adapt the standard linear model of viscoelasticity32 and obtain a viscoelastic dissipation term, FVE ≈ (rγ2/Gε)(φωτ/(1 + ω2τ2)). ε is a cutoff length near the TPCL (typically of the order 0.1−1 μm). φ ≈ (Gu − Gr)/((GuGr)1/2) is a parameter depending upon the unrelaxed (Gu) and relaxed (Gr) shear moduli of the substrate and is of the order unity. τ is the relaxation time of the viscoelastic substrate. ω is the angular frequency from dynamic measurements. The last can be related approximately to the spreading speed, with ω ≈ πU/2ε = πṙ/2ε. Balancing FC and FVE, we obtain the governing equation

4. RESULTS AND DISCUSSION Figure 2 shows the images of water drops spreading on hydrophobic glass surfaces (top) and ≈200 μm thick PDMS films with G ≈ 204 kPa (middle) and ≈23 Pa (bottom). Drops on substrates with G ≳ 204 kPa spread “continuously” as they do on undeformable hydrophobic glass substrates (panels a and b of Figure 2). Drops on substrates with G ≲ 204 kPa spread continuously at the beginning (t ≲ 1 ms) but afterward slow dramatically or even stop spreading (Figure 2c). We found similar trends with all three liquids. Figure 3a illustrates the spreading radius, r, of a water drop as a function of time, t, for hydrophobic glass surfaces and three 1895

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≤ TT or Ti. We found that the spreading radius grew according to r = Ct0.3 on both hydrophobic surfaces and PDMS with only slightly different coefficients (C1 ≈ C2 ≈ 0.34, and C3 ≈ C4 ≈ 0.35), as shown in Figure 3b. The exponent α ≈ 0.3 of a water drop spreading on PDMS is the same as that of a water drop spreading on a similarly wetting but undeformable and hydrophobic glass surface. This is proof of inertia-dominated spreading. Moreover, spreading does not follow a power law during the slower stage for soft surfaces with G ≲ 204 kPa, unlike inertial (r ∼ Ctα) and viscous (r ∼ C′t0.1) spreading on rigid surfaces.2,19−21,25,26 This second, slow stage of spreading is independent of liquid viscosity (shown for η = 1−10.2 mPa s in our case) and solely depends upon the viscoelasticity of the substrate. In summary, on the relatively soft surfaces, we observed rapid, inertial spreading followed by slower, viscoelastic spreading. In early inertial spreading, the surface free energy of the substrate or wettability and the surface tension of the liquid are the only motors driving the motion of the TPCL, while the softness of the substrate should play no role.18 Figure 4a summarizes the exponents of all three liquids on all 15 PDMS substrates. α is independent of G within the margin of experimental error. The inset in Figure 4a shows the relationship between α and θeq for PDMS surfaces and hydrophobic glass surfaces. Clearly, α has a value between 0.5 and 0.2, and it is smaller on relatively hydrophobic surfaces than on hydrophilic surfaces; e.g., α increases when θeq decreases. This is consistent with our simple model and is further confirmation that the early spreading stage is inertiadriven. Figure 4b shows the inertial to viscoelastic transition time, TT, as a function of G for all wetting experiments. On PDMS with G ≳ 204 kPa, TT is independent of the softness and has a value similar to inertial wetting time Ti on undeformable hydrophobic surfaces. On soft PDMS with G ≲ 204 kPa, with the same liquid, TT increases with G and is independent of the thickness of the substrates. For liquids with smaller surface tensions, the trend is similar, although TT becomes larger. We determined the crossover time by asymptotically matching the experimental spreading radius curves with the power law, Ctα, for the inertial stage and with eq 6 for the viscoelastic stage. In the first case, C and α were the fitting parameters of the LSM routine. In the second case, because we could not determine independently φ and τ, we chose to use the term φτ as the fitting parameter. ξ, the drop contact radius

Figure 2. High-speed video images of a water droplet spreading on (a) hydrophobic glass surface, (b) relatively “hard” PDMS substrate with G ≈ 204 kPa, and (c) relatively “soft” PDMS substrate with G ≈ 23 Pa.

PDMS surfaces. On all four surfaces, drops initially spread with a speed U = ṙ ≈ 0.5 m/s. The spreading on relatively rigid PDMS surfaces (G ≳ 204 kPa) is similar to that on the hydrophobic glass surfaces. The duration of the fast wetting is TT ≈ 8 ms, which is larger than the inertial characteristic time TI = (ρR3/γ)1/2 (≈3.7 ms for water). Similar results were also observed by Biance et al.19 and Bird et al.20 One possible explanation is that inertial spreading lasts as long as the capillary wave generated upon contact with the surface propagates along the drop.20 Following the vibration model of suspended drops proposed by Lamb,33 we found that the actual inertial time Ti ≈ 2.22TI for drops,18 which is consistent with our results here as well. Afterward, drops reach the equilibrium state. For the PDMS surface with G ≲ 204 kPa, the fast wetting lasts only TT ≈ 1 ms, depending upon surface softness. Then, spreading enters into another, slower regime. Here, U ≈ 0.007 m/s and the motion of the TPCL proceeds with stick/slip behavior (Figure 3a). Figure 3b shows the linear correlation coefficient K between log r and log t during spreading, for data from Figure 3a. K is ≈1 when t ≤ TT or Ti (TT for PDMS surfaces and Ti for a hydrophobic glass surface), which indicates a nearly perfect power-law correlation between r and t. For t > TT or Ti, K shows a sharp downward kink when the spreading drop reaches the equilibrium contact angle on hard surfaces or when spreading enters into a new regime on soft surfaces. Thus, we fitted the spreading radius r as a power law of t with LSM for t

Figure 3. (a) Spreading radius r of the water drop as a function of the spreading time t on hydrophobic glass surfaces and three PDMS substrates with a thickness of 200 μm and different shear modulus G. (b) Log−log representation of panel a and the linear correlation coefficient K of log r and log t during spreading. The dashed vertical lines show the different transition times TT. 1896

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Figure 5. (a) Spreading radii of water drops (color symbols) on different viscoelastic substrates and asymptotic fits to the inertial (solid lines) and viscoelastic (dashed lines) parts of the curves. Only each 10th data point is shown. (b) Comparison of all calculated and measured TT as a function of G.

Figure 4. (a) Exponent α of various liquids on different PDMS substrates. The inset shows the relationship between α and θeq for water drops on the PDMS substrates and hydrophobic glass surfaces. (b) Transition times TT as function of G and γ for various liquid surface tensions (left) and inertial wetting times Ti on hydrophobic glass for three different liquids (right).

approximation, the relaxation times derived from fitting are reasonable for such soft polymers. However, we should note that τ also changes continuously during spreading.

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at the end of the viscoelastic spreading stage, was also determined from a fit to the experimental curves. ωT was calculated using the transition speed, ṙT, measured at the crossover between inertial and viscoelastic spreading, and a cutoff length of ε = 1 μm. Then, we obtained the corresponding G(ωT) from independent rheological measurements. Figure 5a shows the experimentally obtained spreading radii of water drops in the transition zone, along with the fitted curves for inertial and viscoelastic spreading versus time. The fitting parameters where set so that inertial and viscoelastic curves meet at one point, i.e., at TT. This is indicated by arrows. All of the used or fitted parameters are listed in tables in the Supporting Information. Figure 5b compares experimental and calculated TT for all different liquids and substrates that we used. The experimental times are taken from Figure 4. With the same liquid, the calculated TT increased with G, which is consistent with experiments and with eq 6. TT obtained from fitting agrees within an order of magnitude or better with experimental times. The agreement is best for the softer surfaces, while for the more rigid surfaces, the calculated TT deviates from that measured. We note here that the more rigid substrates no longer behaved completely viscoelastically under the spreading drop and that the inertial spreading time was better described by the characteristic time Ti for rigid, undeformable substrates. Another clear trend in agreement with eq 6 is that TT was larger for liquids with smaller surface tensions. In the range of low viscosities that we employed, we could not find a relation between TT and the viscosity of the liquid. A last observation is about the fitting parameter φτ: its value ranged from ≈7 × 10−5 to ≈7 × 10−7 s. Taking φ to be of order unity, which is not necessarily strictly true because it changes during drop spreading but is nevertheless a reasonable

CONCLUSION In summary, even if an accurate model of the viscoelastic behavior of soft substrates is still elusive, we present a simple model at the level of a scaling law, which is able to capture the main physical processes that control the transition from rapid inertial to slower viscoelastic spreading of low-viscosity drops on PDMS substrates with shear moduli G in the range from ≈23 Pa to ≈680 kPa. Such substrates were deformed to various degrees (wetting ridge height of tens of nanometers to a few micrometers) by capillary forces generated by the drops during spreading. Measured and calculated transition times agree satisfactorily well, considering the simple assumptions and the approximations made in the model. On relatively rigid substrates, spreading proceeded as if on a completely undeformable substrate and was dominated by inertia. A likely explanation for this is as follows. The strain energy input near the wetting ridge (per unit length of TPCL) scales with γ2/G, and therefore, for a given liquid surface tension, energy decreases with an increasing G. In addition, the dissipated fraction of this energy during cycling will generally be smaller for higher G. The duration of inertial spreading on the hardest substrates could be described by a characteristic inertial time Ti ≈ 2.22(ρR3/γ)1/2, depending upon the surface tension of the liquid. On relatively soft surfaces, spreading was initially dominated by inertial forces. However, when the speed of the TPCL fell below a critical value, the stress generated by the drop was applied for a time long enough to deform considerably the substrate. Spreading beyond this critical time was controlled and further slowed by the viscoelastic properties of the substrate. The crossover between inertial and viscoelastic 1897

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(15) Shanahan, M. E. R.; Carré, A. Viscoelastic dissipation in wetting and adhesion phenomena. Langmuir 1995, 11, 1396−1402. (16) Carré, A.; Shanahan, M. E. R. Direct evidence for viscosityindependent spreading on a soft solid. Langmuir 1995, 11, 24−26. (17) Carré, A.; Gastel, J. C.; Shanahan, M. E. R. Viscoelastic effects in the spreading of liquids. Nature 1996, 379, 432−434. (18) Chen, L. Q.; Auernhammer, G. K.; Bonaccurso, E. Short time wetting dynamics on soft surfaces. Soft Matter 2011, 7, 9084−9089. (19) Biance, A. L.; Clanet, C.; Quere, D. First steps in the spreading of a liquid droplet. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2004, 69, 016301. (20) Bird, J. C.; Mandre, S.; Stone, H. A. Short-time dynamics of partial wetting. Phys. Rev. Lett. 2008, 100, 234501. (21) Tanner, L. H. Spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys 1979, 12, 1473−1484. (22) de Gennes, P. G. C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers. 1984, 298, 111−115. (23) Cazabat, A. M.; Stuart, M. A. C. Dynamics of wettingEffects of surface roughness. J. Phys. Chem. 1986, 90, 5845−5849. (24) Cazabat, A. M. How does a droplet spread. Contemp. Phys. 1987, 28, 347−364. (25) Lavi, B.; Marmur, A. The exponential power law: Partial wetting kinetics and dynamic contact angles. Colloid Surf., A 2004, 250, 409− 414. (26) Courbin, L.; Bird, J. C.; Reyssat, M.; Stone, H. A. Dynamics of wetting: From inertial spreading to viscous imbibition. J. Phys.: Condens. Matter 2009, 21, 464127. (27) Winkels, K. G.; Weijs, J. H.; Eddi, A.; Snoeijer, J. H. Initial spreading of low-viscosity drops on partially wetting surfaces. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2012, 85, 055301. (28) Liang, Z. P.; Wang, X. D.; Lee, D. J.; Peng, X. F.; Su, A. Spreading dynamics of power-law fluid droplets. J. Phys.: Condens. Matter 2009, 21, 464117. (29) Chen, L. Q.; Li, Z. G. Bouncing droplets on nonsuperhydrophobic surfaces. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2010, 82, 016308. (30) Sokuler, M.; Auernhammer, G. K.; Roth, M.; Liu, C. J.; Bonaccurso, E.; Butt, H. J. The softer the better: Fast condensation on soft surfaces. Langmuir 2010, 26, 1544−1547. (31) Chen, L. Q.; Li, C. L.; van der Vegt, N. F. A.; Auernhammer, G. K.; Bonaccurso, E. Initial electrospreading of aqueous electrolyte drops. Phys. Rev. Lett. 2013, 110, 026103. (32) Ward, I. M. Mechanical Properties of Solid Polymers; Wiley Interscience: New York, 1979. (33) Lamb, H. Hydrodynamics; Dover: New York, 1932.

spreading was described by another characteristic time, TT, depending upon not only the surface tension of the liquid and the equilibrium contact angle but also the viscoelastic properties of the substrate.

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ASSOCIATED CONTENT

S Supporting Information *

Used and fitted parameters presented in tables and mechanical properties of PDMS substrates from rheological measurements presented in a graph. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (E.B.); martin. [email protected] (M.E.R.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Andreas Hanewald and Kaloian Koynov (MPI for Polymer Research, Mainz, Germany) for the rheology measurements and Günter Auernhammer, Hans-Jürgen Butt, and Marcus Lopes for stimulating discussions. This research was supported by the German Research Foundation (DFG) within the Cluster of Excellence 259 “Smart Interfaces Understanding and Designing Fluid Boundaries”.

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dx.doi.org/10.1021/la3046862 | Langmuir 2013, 29, 1893−1898