A new method for evaluating the most-stable contact angle using mechanical vibration

Abstract

We propose a new method for the direct measurement of the most-stable contact angle, using the mechanical vibration of sessile drops from different metastable states. We relaxed sessile drops of identical volume but with different stable contact angles between advancing and receding configurations. Before the vibration, we were able to scan the range of experimentally-accessible drop configurations. In this manner, the most-stable contact angle was experimentally recognized as the observable contact angle unchanged after vibration independent of the previous history (initial state) of the system. We applied this novel strategy to paraffin wax surfaces with a wide range of roughness degree.

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In surface thermodynamics, the most-stable contact angle (MSCA) of real solid surfaces is the most meaningful angle within the range of observable contact angles. This contact angle represents the system configuration with the lowest energy and so, it is directly related to the concerning specific interfacial energies. By definition, the MSCA is held when the system attains the state of global energy minimum. This definition is not operative because it is not actually possible to measure the system energy at different configurations or metastable states. Furthermore, in practice, the energy barriers (the energy difference between a local minimum and an adjacent local maximum) around the global minimum in the free energy curve are commonly high,¹ and the system can be “trapped” into a metastable state close to the global equilibrium state. Therefore, like the theoretical values of advancing and receding angles,² the MSCA is hardly recognizable experimentally.

Since the energy barriers of a solid–liquid–vapor system are susceptible to be overcome by acoustic or mechanical vibrations, the application of vibrations to relax the contact angle of sessile drops was reported several decades ago.^3–6 Recently, several authors have proposed different strategies of liquid meniscus relaxation induced by vibration^7–11 for contact angle measurement. However, there is no conclusive evidence regarding the global equilibrium condition of the vibrated drop state. As Marmur reported,² the stable contact angle measured after vibration does not necessarily correspond to the MSCA.

Andrieu et al.⁴ proposed, as an equilibrium criterion, the agreement between the values of contact angle in advancing and receding modes after vibration. This proposal has been followed by several authors.^8,11,12 Advancing and receding angles should relax toward an intermediate value, such as the average contact angle⁴, due to the vibrations, but we found cases in which this does not happen (very rough surfaces or with low receding angle). Furthermore, we often observed that, after vibration, the advancing contact angle relaxed to a value lower than the vibrated angle from the receding mode (see Fig. S1, ESI†). Instead, we propose to relax the system using mechanical vibration but from different metastable states for a fixed drop volume. In our method, the MSCA is the contact angle that remains stable even after vibration, independent of the previous history (initial state) of the system. Following this operative definition, we were able to reproduce different configurations within the range of experimentally-accessible configurations for a given drop between the advancing and receding contact angles. Once the system was relaxed by vibration, we monitored the difference between the initial contact angle and the vibrated contact angle, and when this difference was null, i.e. when the hysteresis was mitigated, then the MSCA was found.

The aim of this work was to measure the MSCA using the vibration-induced relaxation of a solid–liquid system from different stable contact angles. We designed a setup which allows for the observation of the forced oscillations of a sessile drop caused by vertical vibrations of the substrate surface. Furthermore, with this setup, we could vary the initial contact angle of the drop by growing and shrinking it differently, up to a fixed volume. We present the values of the MSCA of water on several rough surfaces of paraffin wax.

As described in the literature,¹³ commercial paraffin wax (melting point 53–57 °C, Aldrich) was solidified and textured by casting against different polydimethylsiloxane (PDMS) templates (Sylgard 184, DowCorning). PDMS masters with randomized surface roughness were produced by template replication with a wide range of roughnesses. Finally, we produced paraffin wax surfaces with values of arithmetic roughness within the range 0.124–5.9 μm, measured with white-light confocal profilometry (model Plμ, Sensofar, Spain). These values were lower than the roughness values of the PDMS masters (R_a = 0.08–7.5 μm). This apparent surface relaxation might be explained by the out-of-equilibrium state of the paraffin wax after the tempering stage. However, the paraffin wax samples were fully stable during the course of the experiments.

Advancing and receding contact angles were measured through the addition or withdrawal of water to/from the sessile drop. In this manner, the drop volume was changed from below the sample and thus, the solid–liquid area was forced to increase or decrease, accordingly.¹⁴ We perforated the solid paraffin wax samples with a small hole (1.2 mm diameter). The volume variation was controlled with a motor-driven syringe (Hamilton^©ML500) through the hole drilled in the sample. We used Milli-Q water as the probe liquid and all the experiments were performed at 22–23 °C and 50% relative humidity. We used a 500 μl syringe and the drop volume ranged from 20 μl to 500 μl. Drop profiles were analyzed with the axisymmetric drop shape analysis-profile (ADSA-P) technique.¹⁴ We employed large drops to increase the number of metastable states, to mitigate the corrugation of the contact line and to enhance the numerical effectiveness of ADSA-P algorithm. In addition, the energy barriers of the free energy curve can be overcome more easily by large drops than by small ones, thanks to their high internal energy.

The procedure for measuring the MSCA is described as follows (see Fig. 1). The aim is to vibrate water drops of identical volume placed on the surface but with different contact angle. “Seed” sessile drops with volume arbitrarily greater than 150 μl were formed with the microinjector at 2 μl s⁻¹ and next, an amount of liquid was suctioned at the same flow rate for each drop in order to reach the final drop volume (150 μl). Following this procedure, we could create stable drops with different contact angles θ₀ within the practical hysteresis range. It should be noted that we were able to replicate those metastable states as long as the previous history of the drop was carefully reproduced. Next, the resonance frequency of the sessile drop, with volume 150 μl and initial contact angle θ₀, was estimated using the method of Strani and Sabetta.¹⁵ From its operative definition, the MSCA was identified as the initial contact angle unchanged after vibration.

Fig. 1 A scheme of the method for measuring the most-stable contact angle. Stable sessile drops were created on the surface with the same volume V but following different routes: sequences of addition (V_i) and withdrawal (V_i − V) of liquid. In this manner, different contact angles were attained {θ_rec …, θ₂, θ₁, θ_adv}. Next, each drop was vibrated at its resonance frequency as the initial contact angle.

Although the exact prediction of resonance frequency of unconstrained sessile drops with arbitrary sizes is not known yet, the drop vibration at frequencies close to the value of resonance seems to be sufficient to induce its relaxation.⁹ Unlike freely floating drops, the resonance frequency of the sessile drops takes into account the liquid adhesion on the solid surface. In our experiments, the drop radius was roughly approximated by the curvature radius of a spherical cap with volume and contact angle identical to the actual drop.⁶ The input signal (frequency, amplitude) was generated by a sound-card and amplified accordingly. The drop-substrate system was vertically vibrated with a mechanical driver (PASCO SF-9324) at the chosen frequency for 10 s and next, the final contact angle θ_vib was measured. We verified that the three-phase contact line was moved by the vibration.⁹ We observed that the final contact angle did not depend on the vibration amplitude (see Fig. S1, ESI†) when this was greater than a given value (0.28–0.46 mm), which depended on the roughness of the sample. All vibrated drops were axisymmetric and their contact angle changed. For each “seed” drop (θ₀-value), we made at least three runs and the contact angle after vibration was averaged. Contact angles measured after vibration were highly reproducible (with standard deviations ≤2°).

In our experimental approach, the input signal to the mechanical oscillator was a frequency-modulated sine wave, where we used a linear frequency scanning. We chose this scanning in order to minimize the overrelaxation of the system. Although the effectiveness of vibrations in changing contact angles depends on the vibration parameters,⁵ we found with our methodology that the type of excitation signal was not critical for relaxing energetically the system towards the MSCA. Further work is being addressed to apply stochastic vibrations⁸ with our method.

In Fig. 2, for two paraffin wax surfaces with low (Fig. 2a) and high contact angle hysteresis (Fig. 2b), we show the unsigned difference between the vibrated contact angle and the initial contact angle in terms of this angle. In both graphs, there is a well-defined minimum at the null variation of observable contact angle, which corresponds to the MSCA. In this manner, we were able to estimate the values of the MSCA of paraffin wax surfaces with different degrees of roughness. Since the drop size used in this study (150 μl) was much larger than the typical size of topography features (R_a = 0.124–5.9 μm), the values of the MSCA were independent of the drop volume (i.e. the solid–liquid area).

Fig. 2 (a) The difference between the vibrated contact angle and the initial contact angle of 150 μl-water drops on a relatively smooth paraffin wax surface (R_a = 0.124 ± 0.009 μm) as a function of the initial contact angle. The extreme points correspond to the advancing and receding contact angles. The most-stable contact angle was 100.4 ± 0.9°. (b) The difference between the vibrated contact angle and the initial contact angle of 150 μl-water drops on a rough paraffin wax surface (R_a = 4.53 ± 0.17 μm) as a function of the initial contact angle. The most-stable contact angle was 118 ± 2°.

In Fig. 3a, the non-zero contact angle hysteresis observed on the smoother paraffin wax surface (28.7 ± 2.3°) was probably due to the surface heterogeneity of the paraffin wax, as suggested by Kamusewitz et al.¹⁶ A maximum contact angle hysteresis is observed in Fig. 3a, which points out to the transition between the Wenzel and Cassie–Baxter regimes. At those values of roughness, where contact angle hysteresis was maximized, we found that the values of contact angle after vibration in advancing and receding modes disagreed (Fig. 3b).

Fig. 3 (a) Contact angle hysteresis in terms of the arithmetic roughness. The continuous line indicates the extrapolation to R_a = 0. (b) Unsigned difference of the vibrated contact angles from advancing and receding drops in terms of the arithmetic roughness.

In Fig. 4a, as expected for smooth low-energy surfaces, the value of the MSCA at low roughness values agreed with the advancing contact angle better than with the receding one. Instead of the extrapolation to the case of no hysteresis,¹⁶ we extrapolated the values of the MSCA when the Wenzel factor¹⁷r_W was equal to the unity (104 ± 4°). This value agreed with measurements reported in previous works for purified paraffin wax (103.0 ± 0.4°).^16,18 However, the values of the MSCA were mostly greater than the values predicted by the Wenzel equation.¹⁷

Fig. 4 (a) Advancing, most-stable and receding contact angles of 150 μl-water drops on rough paraffin wax surfaces as a function of the Wenzel factor. The dashed line indicates the extrapolation to r_W = 1, and the continuous line is the prediction of the Wenzel equation¹⁷. (b) The most-stable contact angle and the average contact angle (cosθ_avg = (cosθ_adv + cosθ_rec)/2) as a function of the arithmetic roughness.

The symmetry between advancing and receding contact angles in Fig. 2 is evident. When the values of |θ_vib − θ₀| and θ_vib coincide for both the advancing and receding angles, the rule of mean contact angle¹⁶ is fulfilled. As expected,¹⁸ the value of the MSCA of water drops on rough paraffin wax surfaces was mostly predicted by the arithmetic mean of the advancing and receding contact angles (see Fig. 4a). However, there are cases, such as smooth surfaces of poly(methyl methacrylate) (see Fig. S2, ESI†), where this rule did not apply. Furthermore, as expected,⁴ the average contact angle computed from the mean cosine (cosθ_avg = (cosθ_adv + cosθ_rec)/2), as an estimate of the MSCA for smooth heterogeneous surfaces, disagreed with our results at high values of arithmetic roughness (see Fig. 4a). A detailed discussion of the estimation of the MSCA from the advancing angle and the receding angle goes beyond the scope of this communication. However, rather than mean values computed from the advancing angle and the receding angle, the values of the MSCA, provided by our method, are more meaningful for the system because they are experimentally evaluated using several metastable states within the practical range of contact angle hysteresis.

To sum up, we propose a new method for the direct measurement of the MSCA using the mechanical vibration of sessile drops with fixed volume, which had previously attained different stable contact angles. The MSCA provided by our method becomes the observable (reproducible) contact angle intimately related to surface energetics. The MSCA is the closest available approximation for thermodynamically meaningful contact angles over a wide range of roughness. Hence, the MSCA might be used to compute solid surface energies from the appropriate thermodynamic theory.¹²

Acknowledgements

This work was supported by the “MICINN” (project MAT2007-66117 and contract RYC-2005-000983), Junta de Andalucía (projects P07-FQM-02517, P08-FQM-4325 and P09-FQM-4698) and the European Social Fund (ESF). The authors are grateful to F. Vereda-Moratilla for fruitful discussions.

References

1. R. E. Johnson and R. H. Dettre, J. Phys. Chem., 1964, 68, 1744–1750.
2. A. Marmur, Soft Matter, 2006, 2, 12–17.
3. T. Smith and G. Lindberg, J. Colloid Interface Sci., 1978, 66, 363–366.
4. C. Andrieu, C. Sykes and F. Brochard, Langmuir, 1994, 10, 2077–2080.
5. E. Decker and S. Garoff, Langmuir, 1996, 12, 2100–2110.
6. S. Yamakita, Y. Matsui and S. Shiokawa, Jpn. J. Appl. Phys., 1999, 38, 3127–3130.
7. T. S. Meiron, A. Marmur and I. S. Saguy, J. Colloid Interface Sci., 2004, 274, 637–644.
8. S. Mettu and M. K. Chaudhury, Langmuir, 2010, 26, 8131–8140.
9. X. Noblin, A. Buguin and F. Brochard-Wyart, Eur. Phys. J. E, 2004, 14, 395–404.
10. E. Bormashenko, R. Pogreb, T. Stein, G. Whyman, M. Erlich, A. Musin, V. Machavariani and D. Aurbach, Phys. Chem. Chem. Phys., 2008, 10, 4056–4061.
11. C. Della Volpe, D. Maniglio, S. Siboni and M. Morra, Oil Gas Sci. Technol., 2001, 56, 9–22.
12. R. Sedev, M. Fabretto and J. Ralston, J. Adhes., 2004, 80, 497–520.
13. M. Zbik, R. G. Horn and N. Shaw, Colloids Surf., A, 2006, 287, 139–146.
14. D. Y. Kwok, R. Lin, M. Mui and A. W. Neumann, Colloids Surf., A, 1996, 116, 63–77.
15. M. Strani and F. Sabetta, J. Fluid Mech., 1984, 141, 233–247.
16. H. Kamusewitz, W. Possart and D. Paul, Colloids Surf., A, 1999, 156, 271–279.
17. R. Wenzel, J. Phys. Chem., 1949, 53, 1466–1467.
18. C. Della Volpe, D. Maniglio, M. Morra and S. Siboni, Colloids Surf., A, 2002, 206, 47–67.

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Footnote

† Electronic Supplementary Information (ESI) available: Fig. S1 Graph of the vibrated contact angle in terms of the amplitude at a fix frequency for advancing and receding drops on a smooth PDMS surface. Fig. S2 Graph of the difference between the vibrated contact angle and the initial contact angle of 150 μl-water drops on a smooth surface of poly(methyl methacrylate). See DOI: 10.1039/c0sm00939c/

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