Letter pubs.acs.org/Langmuir
Forced versus Spontaneous Spreading of Liquids A. Mohammad Karim,† S. H. Davis,‡ and H. P. Kavehpour*,† †
Department of Mechanical and Aerospace Engineering, Complex Fluids and Interfacial Physics Laboratory, University of California at Los Angeles, Los Angeles, California 90095, United States ‡ Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208,United States S Supporting Information *
ABSTRACT: Two sets of experiments are performed, one for the free spreading of a liquid drop on a glass substrate and the other for the forced motion of a glass plate through a gas− liquid interface. The measured macroscopic advancing contact angle, θA, versus the contact line speed, U, differ markedly between the two configurations. The hydrodynamic theory (HDT) and the molecular kinetic theory (MKT) are shown to apply separately to the two systems. This distinction has not been previously noted. Rules of thumb are given that for an experimentalist involve a priori knowledge of the expected behavior.
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which the capillary number is Ca = μU/σ ≤ 10−4). Here, μ and ν are the dynamic and kinematic viscosities of the liquid, and Lmac is the radius of the drop. The HDT relates the advancing dynamic contact angle θA to its microscopic value θ0 through eq 13,7
INTRODUCTION At first glance, there would be no need to study the comparison between contact-line dynamics of a spreading drop (free spreading, Figure 1a) and the motion of a contact line on a plate moving through a liquid−gas interface (forced spreading, Figure 1b) with the same liquid, gas, and solid. One might argue that at small capillary number the contact line mechanics, say, the measured advancing dynamic contact angle θA versus the contact line speed U, should be identical. There is, however, one difference that can be seen. In the case of free spreading one fixes θA (or its variations) and determines U. In the forced spreading, one imposes U and finds θA; the two problems are the inverses of one another. In addition, in both spontaneous and forced spreading, there are relaxation processes of the inner-scale physics and of the intermediate and outer-scale physics (i.e., probably the visco-capillary time for the intermediate and outer-scale physics). In forced spreading, another time scale can be added to the problem, and how that relaxation time competes with each of the other relaxation processes tells how the dynamic wetting will proceed. In spontaneous spreading, the natural relaxations control the spreading. This in part motives the present investigation. It is found that the θA−U characteristic for the free case is well predicted by the hydrodynamics theory (HDT), as is well known,1−11 whereas the forced case is not but follows the molecular kinetic theory (MKT). These two theories are sketched now. The distinction between forced and spontaneous spreading dynamics can also relate to whether the inner scale or viscous bending physics will dominate the variation of the effective macroscopic contact angle with speed. If the dominant dissipation at a specific forcing speed is in the inner scale, then MKT may work and it can switch depending on the forcing speed. There is the HDT based on slow viscous flow (Reynolds number Re = ULmac/ν ≤ 10−1 and large surface tension σ in © 2016 American Chemical Society
θA 3 − θ0 3 = 9
μU ⎛ Lmac ⎞ ln⎜ ⎟ σ ⎝ Lmic ⎠
(1)
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where Lmic is the slip length. Eggers and Stone12 have shown that Lmac/Lmic is a velocitydependent parameter. They have obtained the dependence of Lmac/Lmic on the capillary number (i.e., Lmac/Lmic = αCa2/3) for the case of perfectly wetting liquids with a consideration of the slip boundary condition over a small slip length Lmic in the region close to the liquid contact line. They assume a nearly flat interface close to the liquid contact line to apply the lubrication assumption for describing the liquid motion, and they have considered the pressure variation caused by capillary and van der Waals forces. There is the MKT13,14 that ignores bulk flow but focuses on the jumping of molecules from the drop ahead of the contact line by employing Eyring’s theory.15 This leads to eq 2 ⎡ ⎛ U ⎞⎤ 2k T θA = arccos⎢cos θ0 − B2 sinh−1⎜ ⎟⎥ ⎢⎣ σλ ⎝ 2K wλ ⎠⎥⎦
(2)
where kB is the Boltzmann constant, T is the absolute temperature, Kw is the equilibrium frequency of the random molecular displacements at the contact line, and λ is the average distance between adsorption sites on the substrate on which the random molecular displacements occur. Received: February 25, 2016 Revised: August 26, 2016 Published: September 19, 2016 10153
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Figure 1. Schematics of experimental methods. (a) Free spreading: depositing a liquid droplet with almost zero impact velocity on the substrate and measuring the diameter of a droplet and contact angle at the contact line from the side during the spreading of the droplet on the substrate. (b) Forced spreading using the force balance method with a tensiometer: moving the substrate into the liquid at constant velocity; measuring the contact angle at the contact line by the balance of forces applied to the substrate to create steady motion. (c) Forced spreading using an optical technique with ImageJ, which is the imaging-processing software.
Figure 2. Schematic plots explaining our hypothesis for predicting the more appropriate model to describe the contact line dynamics. (a) Free spreading. (b) Forced spreading. The plots of HDT and MKT shown in the illustrations of the hypothesis are obtained from the analysis for the spreading of silicone oil 100 [cSt] on glass. The strategy to predict the appropriate spreading dynamics for free spreading is for the observed dynamic contact angle, the appropriate spreading dynamics is the one that gives a higher contact-line speed for the corresponding observed dynamic contact angle. The strategy to predict the appropriate spreading dynamics for forced spreading is for given fixed contact line speed, the appropriate spreading dynamics is the one that gives a larger dynamic contact angle for the corresponding given fixed speed of the contact line. The plots are from eqs 1 and 2.
Motivation 1. Davis and Davis11 compared the two theories for two sets of free spreading, common liquids at room temperature and droplets of liquid metals at elevated temperatures. When they plotted θA versus U using the two theories, they found cases such as those in Figure 2. Note that there are finite intersections at (θc, Uc) (i.e., θc is the contact angle at the intersection of the plots of HDT and MKT and Uc is the contact line velocity at the intersection of the plots of HDT and MKT). They argue that the crossing point should separate HDT (higher U and θA) from MKT (lower U and θA). This is found out by comparing large sets of experiments in each category. Thus, in general the data follow different spreading laws in different parameter ranges. Motivation 2. The present forced-spreading experiments show different U versus θA characteristics from those of free spreading despite the use of the same materials in both sets. Thus, the following question can be explored. By switching from free to forced spreading, must one switch the underlying
theory that is used to explain the data? It is found that the answer is, indeed, positive. The MKT fits the data best for forced spreading, a conclusion that is seemingly quite surprising. Note that in the literature there have been some attempts to blend two theories, but this approach would seem to obscure rather than clarify the situation.1−12,16−39
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EXPERIMENTAL PROCEDURES AND MATERIALS
Materials. The experiments were performed for several silicone oils with different dynamic viscosities on clean glass substrates where equilibrium contact angles, θ0, are zero and for glycerin on a glass substrate where θ0 = 0.552 rad. Table 1 gives the experimentally measured physical properties of pure liquids that have been used in the experimental investigations. 10154
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(i.e., the force due to the surface tension of the liquid), and the buoyancy force, Fbuoyancy (i.e., the force due to the difference between the density of the solid plate and the density of the liquid). The measured force, Fmeasured, is the force applied by the plate holder to the plate to maintain the steady motion of the solid plate during the experiment, and its value is obtained by the force sensor connected to the plate holder from the top. It should be noted that the force of gravity is calibrated at the onset of contact between the plate and the liquid/air interface. Equation 3 has been applied by the tensiometer software for the calculation of the dynamic contact angle.
Table 1. Physical Properties of Pure Liquids Used in the Experiments liquid
density [kg/m3]
μ [Pa s]
σ [N/m]
dodecane silicone oil - 100 [cSt] silicone oil - 1000 [cSt] silicone oil - 10 000 [cSt] glycerin
746 964 969 971 1260
0.001 0.096 0.969 9.710 1.412
0.023 0.020 0.020 0.022 0.064
Fmeasured + Fcapillary + Fbuoyancy = 0
Experimental Setup. Here, free spreading experiments are done using a drop shape analyzer (DSA100, KRÜ SS) with the deposition of the liquid droplet at zero impact velocity on the solid surface, as shown. Forced spreading experiments are done using a tensiometer (K100, KRÜ SS) with a specified speed of motion of the contact line. The tensiometer uses the balance of forces applied to the solid plate during immersion/withdrawal of the plate in the pool of liquid and measures the dynamic contact angle during the motion. The forces that are considered by the tensiometer are the capillary force, Fcapillary
(3)
The capillary force and the buoyancy force are calculated on the basis of eqs 4a and 4b Fcapillary = 2σ(w + t ) cos θA
(4a)
Fbuoyancy = ρgwtx
(4b)
Figure 3. Experimental comparison between free and forced spreading. (a) Spreading of dodecane on a glass surface: θ0 = 0 [rad], α = 86 751 ± (3.54 × 104), λ = (2.6421 × 10−9) ± (1.86 × 10−10) [m], Kw = (1.5019 × 105) ± (3.92 × 104) [Hz]. (b) Spreading of silicone oil 100 [cSt] on a glass surface: θ0 = 0 [rad], α = 37 183 ± (5.96 × 103), λ = (2.0762 × 10−9) ± (1.23 × 10−10) [m], Kw = (48 172 ± 1.44) × 104 [Hz]. (c) Spreading of silicone oil 1000 [cSt] on a glass surface: θ0 = 0 [rad], α = 9767.5 ± (1.37 × 103), λ = (5.7932 × 10−10) ± (2.95 × 10−11) [m], Kw = (6.6252 × 105) ± (1.36 × 105) [Hz]. (d) Spreading of silicone oil 10 000 [cSt] on glass: θ0 = 0 [rad], α = (1.0716 × 1010) ± (9.74 × 107), λ = (1.5194 × 10−9) ± (1.17 × 10−10) [m], Kw = 50.962 ± 84.1 [Hz]. 10155
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Figure 4. Spreading of glycerine on a glass substrate: θ0 = 0.5397 ± 0.00387 [rad], Ls = (7.7421 × 10−7) ± (2.75 × 10−7) [m], λ = (1.2894 × 10−9) ± (1.8 × 10−10) [m], Kw = (1.4984 × 105) ± (1.17 × 105) [Hz].
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where w is the width of the plate, t is the plate thickness, ρ is the liquid density, g is the acceleration of gravity, and x is the depth of the immersion of the plate in the pool of the liquid. For experiments with the tensiometer, the speed of the motion of the plate (i.e., speed of the liquid contact line) is set to a specific constant value for each set of experiments. The experiments, performed at room temperature, have been done for several cycles to increase the level of confidence in the results. The tensiometer provides very accurate results for experiments on the liquids with μ ≤ 1 Pa·s. For experiments with liquids with higher μ (i.e., silicone oil 10 000 [cSt] and glycerin), the optical method also has been applied simultaneously during the experiment with a tensiometer because the viscous force along the plate can be an important force in the force balance for dynamic contact angle measurement.40 The optical method was performed using a Canon ultrasonic EOS-1 camera that is a single lens reflex (SLR) focused on the liquid contact line where the menisci formed during the motion of the solid plate during the immersion of solid substrate into the pool of the liquid; the menisci were captured and then the advancing dynamic contact angles were measured using ImageJ, which is image-processing software. Figure 1c illustrates the schematic of the optical method for measuring the dynamic contact angle during the experiment with the tensiometer for highly viscous liquids (i.e., silicone oil 10 000 [cSt] and glycerin) on glass.
RESULTS AND DISCUSSION
Figures 3 and 4 display experimental data, θA (macroscopic advancing contact angle) versus U (liquid contact line speed), for both free and forced spreading on glass of identical material systems with plot of the spreading laws given by the HDT equation (eq 1) and by MKT (eq 2). Figure 3a applies to dodecane. Figure 3b−d applies to silicone oil, and Figure 4 applies to glycerine. For the forced spreading case, the measured advancing dynamic contact angles from the optical technique match very well with concurrent measurements from the force balance method for low-viscosity fluids such as dodecane, silicone oil 100 [cSt], and silicone oil 1000 [cSt]. However, for more viscous liquids such as silicone oil 10 000 [cSt] and glycerin, only measurements obtained from the optical technique were considered in our analysis as represented in Figures 3d and 4. These experiments were conducted by controlling the speed of the glass in the pool of liquid at constant speed, and the contact angle measurements were performed long after start/stop processes because the contact angle was constant with time. 10156
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angle for the corresponding given fixed speed of the contact line.
Consider first free spreading. Clearly, the HDT fits the data unambiguously for dodecane, Figure 3a, with high capillary number departures for U ≥ 2 × 10−3 [m/s], which corresponds to Ca ≥ 1.4 × 10−4. Note that failure also is associated with the onset of evaporation. For silicone oil 100 [cSt], Figure 3b, the data agrees unambiguously with the HDT for U ≤ 7.8 × 10−4 [m/s] or Ca ≤ 3.8 × 10−2. For silicone oil 1000 [cSt], Figure 3c, the data follows the HDT up to U ≤ 2.5 × 10−4 [m/s] or Ca ≤ 1.2 × 10−2. For silicone oil 10 000 [cSt], Figure 3d, the HDT applies well up to U ≤ 1.3 × 10−5 [m/s] or Ca ≤ 5.8 × 10−3. For glycerine, Figure 4b,c, the HDT follow the data unambiguously up to U ≤ 4.1 × 10−4 [m/s] or Ca ≤ 9.0 × 10−3. The applicability to free spreading data is not surprising in its agreement with previous studies3,6 and many others. Consider now forced spreading. The dodecane data show dramatically different behavior from the free case and unambiguous agreement with the MKT, Figure 3a. The same statement applies to silicone oil 1000 [cSt], Figure 3c. For silicone oil 100 [cSt], Figure 3b, the situation is different because the data surrounds the crossing point {(Uc, θc) = (2.0 × 10−4 [m/s], 0.8 [rad])} of the two theories. Note that the free spreading case follows the MKT beyond (Uc, θc), consistent with a suggestion by Davis11 for a different system in which the spreading mechanism should change across the intersection. For silicone oil 10 000 [cSt], Figure 3d, the MKT is well descriptive. However, the speed of spreading for the free case could not reach that of the forced case, so measurements for both modes do not overlap. For the spreading of glycerine, Figure 4a,c, one sees good agreement with the MKT. Here (Uc, θc) = (1.06 × 10−2 [m/s], 2.7634 [rad]) obtained from Davis,11 consistent with the data; this Uc is too large for free spreading to reach. It should be noted that the experimental results and theoretical analyses presented in this article are not about invalidating the HDT or any other spreading dynamics model. The main point of this study is that there are discrepancies between forced and free spreading of the same fluid/solid system, which was previously thought to be governed by same physical model. Recently, Blake et al.39 have also done the numerical analysis to do a comparison between spontaneous spreading and forced spreading in the same solid/liquid system using molecular dynamics simulations. They have found out the difference between the dynamic contact angle obtained from spontaneous spreading for a given liquid contact line speed compared to the dynamic contact angle obtained from forced spreading for the same liquid contact line speed in the same solid/liquid system. In spontaneous spreading, it is very difficult to obtain the receding contact angles because the liquid droplet spreads freely (i.e., the liquid contact line advances on the horizontal solid surface spontaneously); therefore, it was not possible to compare the spontaneous spreading and forced spreading for the receding case. But if it were possible for the case of receding motion, the strategy to predict the appropriate spreading dynamics for spontaneous spreading would be similar to that for the advancing dynamic contact angle. The appropriate spreading dynamics is the one that gives a larger negative contact-line speed (i.e., since the contact line speed would be negative for the receding motion) for the corresponding observed dynamic contact angle. For forced spreading, the appropriate spreading dynamics is the one that also gives a higher dynamic contact
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CONCLUSIONS AND OUTLOOK The results outlined above for free and forced spreading reveal qualitatively different behaviors for the two cases.To explain their differences, the analysis of Davis11 has been utilized. They studied two free spreading systems, common liquids at room temperature, and liquid metals at elevated temperatures and found that the HDT and MKT curves, θA versus U, crossed at some (Uc, θc) and hypothesized then which theory applies to experiment switch applicability there. This suggests that such switches might be general in spreading systems. They suggested that for U > Uc, a fluid dynamic theory, HDT should apply where as for U < Uc the MKT should apply. This was shown to be consistent with large sets of data. In the present research, contrasts are seen between free and forced spreading of identical physical systems. Figure 2a illustrates the idea that different mechanisms are to be interpreted differently. Free spreading, θA → U (HDT) is associated with higher U for U > Uc. For forced spreading, U → θA (MKT) is associated with higher θA for U > Uc. For U < Uc, the reverse holds. It should be noted that Blake et al.39 considered free and forced spreading using molecular dynamics simulations and found that the two systems behave differently, compatible with the situation of higher θA, Figure 2b. Previous experiments in the forced spreading mechanism41−45 also support our hypothesis, which can be found in the Supporting Information section. We hope that this paper can entice scientists to study this phenomenon both experimentally and numerically to further expand on these finding regarding different modes of spreading.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b00747. Experimental results obtained from literature for the case of the dynamics of the forced spreading mechanism (DOCX)
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
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