Analysis of Droplet Evaporation on a Superhydrophobic Surface

These types of surface provide superhydrophobic systems with theoretical initial Cassie-Baxter contact angles for water ..... component beneath the dr...
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Langmuir 2005, 21, 11053-11060

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Analysis of Droplet Evaporation on a Superhydrophobic Surface G. McHale,*,† S. Aqil,† N. J. Shirtcliffe,† M. I. Newton,† and H. Y. Erbil‡ School of Biomedical and Natural Sciences, Nottingham Trent University, Clifton Lane, Nottingham NG11 8NS, U.K., and Gebze Institute of Technology, Faculty of Engineering, Department of Chemical Engineering, Cayırova, Gebze 41400, Kocaeli, Turkey Received July 11, 2005. In Final Form: September 10, 2005 The evaporation process for small, 1-2-mm-diameter droplets of water from patterned polymer surfaces is followed and characterized. The surfaces consist of circular pillars (5-15 µm diameter) of SU-8 photoresist arranged in square lattice patterns such that the center-to-center separation between pillars is 20-30 µm. These types of surface provide superhydrophobic systems with theoretical initial Cassie-Baxter contact angles for water droplets of up to 140-167°, which are significantly larger than can be achieved by smooth hydrophobic surfaces. Experiments show that on these SU-8 textured surfaces water droplets initially evaporate in a pinned contact line mode, before the contact line recedes in a stepwise fashion jumping from pillar to pillar. Provided the droplets of water are deposited without too much pressure from the needle, the initial state appears to correspond to a Cassie-Baxter one with the droplet sitting upon the tops of the pillars. In some cases, but not all, a collapse of the droplet into the pillar structure occurs abruptly. For these collapsed droplets, further evaporation occurs with a completely pinned contact area consistent with a Wenzel-type state. It is shown that a simple quantitative analysis based on the diffusion of water vapor into the surrounding atmosphere can be performed, and estimates of the product of the diffusion coefficient and the concentration difference (saturation minus ambient) are obtained.

I. Introduction Free evaporation of completely spherical droplets of water is an apparently simple problem with widespread relevance to drying problems such as ink-jet printing, spraying of pesticides, and the spotting of DNA microarray data. These types of applications involve small droplets, dominated by surface tension rather than gravity, deposited on solid substrates. The involvement of the substrate complicates the evaporation process in obvious ways, such as altering the droplet shape to a spherical cap with a contact angle determined by various interfacial surface tensions. Less obviously, when the droplet evaporation is governed by the diffusion of the liquid into the surrounding atmosphere, the presence of a substrate restricts the space into which vapor may diffuse and so reduces the evaporation rate. Thus, it has been theoretically predicted that a small droplet resting on a flat solid surface but possessing a 180° contact angle so that it is completely spherical will have a rate of mass loss reduced by loge 2 compared to an identical, completely spherical droplet far removed from any solid surface.1 Investigating the free evaporation of almost spherical droplets from solid substrates has not previously been attempted because of the obvious difficulty of obtaining a model physical system with an initial contact angle higher than that of the waterTeflon system of 110-120°. As has become clear over recent years from work on superhydrophobic surfaces, nature has not refrained from creating leaves on which contact angles approaching 180° are observed.2-4 * Corresponding author. E-mail: [email protected]. Tel: +44 (0)115 8483383. † Nottingham Trent University. ‡ Gebze Institute of Technology. (1) Picknett, R. G.; Bexon, R. J. Colloid Interface Sci. 1977, 61, 336350. (2) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667-677. (3) Blossey, R. Nat. Mater. 2003, 2, 301-306. (4) Que´re´, D.; Lafuma, A.; Bico, J. Nanotechnology 2003, 14, 11091112.

Picknett and Bexon1 investigated the mass and profile evolution of a slowly evaporating liquid (methyl acetoacetate) droplet on a poly(tetrafluoroethylene) (Teflon) surface in still air. They reported three modes of evaporation: mode 1, corresponding to a constant solid-liquid contact area; mode 2, corresponding to a constant contact angle, and mode 3, corresponding to changes in both the contact angle and contact area. They also developed a diffusion-based theory to predict the evaporation rate. Since then, various authors have considered the evaporation of small droplets of liquid in constant contact area mode,5-9 constant contact angle mode,8-12 and mixed mode.8,9 In each case, models were developed to enable data on the product D∆c, where D is the diffusion constant and ∆c ) cS - c∞ is the difference in concentration of vapor at the droplet surface (assumed to equal the saturated vapor concentration) and that far removed from the droplet surface, to be estimated. We emphasize that these estimates were not intended to be accurate measurements but were used to demonstrate the consistency of theory with experiment. In a sequence of papers, Erbil and coworkers investigated how deviations from a spherical cap droplet shape might influence the evaporation rate13,14 and how the rate of evaporation might be used to determine (5) Birdi, K. S.; Vu, D. T.; Winter, A. J. Phys. Chem. 1989, 93, 37023703. (6) Rowan, S. M.; Newton, M. I.; McHale, G. J. Phys. Chem. 1995, 99, 13268-13271. (7) Rowan, S. M.; McHale, G.; Newton, M. I.; Toorneman, M. J. Phys. Chem. 1997, 101, 1265-1267. (8) Shanahan, M. E. R.; Bourge`s, C. Int. J. Adhes. Adhes. 1994, 14, 201-205. (9) Bourge`s-Monnier, C.; Shanahan, M. E. R. Langmuir 1995, 11, 2820-2829. (10) Birdi, K. S.; Vu, D. T. J. Adhes. Sci. Technol. 1993, 7, 485-493. (11) McHale, G.; Rowan, S. M.; Newton, M. I.; Banerjee, M. K. J. Phys. Chem. B 1998, 102, 1964-1967. (12) Erbil, H. Y.; McHale, G.; Newton, M. I. Langmuir 2002, 18, 2636-2641. (13) Erbil, H. Y.; Meric, R. A. J. Phys. Chem. B 1997, 101, 68676873. (14) Meric, R. A.; Erbil, H. Y. Langmuir 1998, 14, 1915-1920.

10.1021/la0518795 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/14/2005

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the initial contact angle.15,16 However, in all of these studies the contact angles involved have been lower than 110° and so have not tested the higher contact angle regime. In their work on the influence of evaporation on contact angle measurements, Bourge`s and Shanahan9 noted that various physical factors, particularly surface roughness, had still not been incorporated into the understanding of evaporation. At that time, relatively little had been published on superhydrophobic surfaces. From the work of numerous authors, we now know that rough or textured hydrophobic surfaces providing contact angles of 150180° are common in nature and can be produced artificially in a wide variety of ways. (See the review by Blossey.3) Moreover, these surfaces can be divided into two distinctive types, Wenzel17 and Cassie-Baxter,18,19 depending on whether the liquid fully penetrates the surface features or whether the liquid bridges across the surface features. Thus, the development of techniques for producing high contact angle surfaces and our understanding of the equilibrium and metastable states of droplets on these surfaces provide new regimes for the study of droplet evaporation. It is therefore the purpose of the present study to report qualitative features of the evaporation of sessile droplets of water from solid surfaces with initial contact angles for water greater than 120° and to quantitatively relate the rate of mass loss to diffusion models for the evaporation of sessile droplets. II. Theoretical Development II.a. Models of Evaporation from Solid Surfaces. The rate of change of mass of an evaporating droplet can be modeled on the basis of the diffusion of water molecules from the droplet surface into the surrounding atmosphere. For a completely spherical droplet of liquid of density FL, spherical radius RS, and volume VC, the rate of mass loss depends on the liquid-vapor surface area, ALV ) 4πRS2, the diffusion coefficient, D, and the difference, ∆c ) (cS c∞), between the vapor concentration at the droplet surface (assumed to be equal to the saturation concentration cS) and the ambient value far removed from the droplet surface (c∞). Assuming a spherical droplet far removed from any boundaries and solving the diffusion equation, the concentration of vapor is found to be inversely proportional to the radial distance from the droplet. The rate of mass loss is then given by

dVC ) -4πRSD∆c FL dt

(1)

When the completely spherical droplet is just in contact (i.e., contact angle of 180°) with a plane boundary, it is predicted that the evaporation rate will decrease because of the reduction in space into which vapor can diffuse.1 The reduction can be calculated theoretically and is given by eq 1, but with an extra factor of loge 2 on the right-hand side of the equation. By using the analogy between the diffusive flux and the electrostatic potential, Picknett and Bexon1 obtained an exact equation for the rate of mass loss that is valid for all droplets of a spherical cap shape (15) Erbil, H. Y. J. Phys. Chem. B 1998, 102, 9234-9238. (16) Erbil, H. Y. J. Adhes. Sci. Technol. 1999, 13, 1405-1413. (17) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988-994. (18) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546551. (19) Johnson, R. E.; Dettre, R. H. Contact Angle, Wettability and Adhesion; Advances in Chemistry Series 43; American Chemical Society: Washington, DC, 1964; p 112.

resting on a solid plane boundary

dVC FL ) -4πRSD∆cf(θ) dt

(2)

where f(θ) is a function of the contact angle θ. Picknett and Bexon’s solution involves a complex function (Snow’s solution), but to aid calculations, they gave two useful polynomial fits covering the angular ranges of 0-10° and 10-180°

{

2fPB(θ) ) 0.6366θ + 0.09591θ2 - 0.06144θ3 0° < θ < 10° 0.00008957 + 0.6333θ + 0.116θ2 - 10° < θ < 180° 0.08878θ3 + 0.01033θ4 (3)

where the contact angle θ is in radians. For a completely spherical droplet with a vanishingly small contact area with respect to the solid, this approximation gives 2fPB(π) ) 1.387 compared to an exact solution of 2 loge 2 ) 1.386. For a hemispherical droplet having a contact angle of 90° with respect to the solid substrate, the Picknett and Bexon value of 2fPB(π/2) ) 0.999 compares well to the exact solution of eq 1. For comparison to previous studies of the evaporation of sessile droplets on solid surfaces, it is useful to review some approximate models and relate them to eq 2. (See also ref 12.) Equation 2 recovers the evaporation of a completely spherical droplet neglecting any boundary, eq 1, by taking fBV(θ) ) 1, where fBV(θ) is the function from the work of Birdi and Vu5,10 used in their study of the evaporation of sessile water droplets from glass surfaces. The limitation of Birdi and Vu’s model for contact angles close to 90° was addressed in the work of Rowan et al.,6 who also studied the evaporation of water droplets from solid (PMMA and Teflon) substrates by taking into account the reduced liquid-vapor interfacial area of the spherical cap droplet for diffusion through a factor fR(θ) ) (1 - cos θ)/2. Although this formula provides the correct numerical result at 90°, it overestimates the evaporation rate at 180° through its neglect of the effect of the solid substrate on reducing the space for the diffusion of water vapor. Bourge`s-Monnier and Shanahan addressed this neglect in their self-consistent model of the concentration gradient occurring during evaporation.9 Assuming a radial diffusion of vapor into the surrounding atmosphere and a spherical cap droplet supported by a plane solid substrate, they derived

fBMS(θ) )

-cos θ 2 loge(1 - cos θ)

(4)

The Bourge`s-Monnier and Shanahan result is 0.5 at 90°, in agreement with the exact solution, and gives 2fBMS(π) ) 1/loge 2 ) 1.4427 compared to 2 loge 2 ) 1.386 at 180°. II.b. Constant Contact Radius Mode. In the present study, our patterned substrates enable contact angles greater than the range of 30-100° studied by Bourge`sMonnier and Shanahan to be obtained while possessing the simplifications of a spherical cap droplet and a pinned contact area; this corresponds to the stage II evaporation that they identified. Our analysis broadly follows that of Bourge`s-Monnier and Shanahan9 but uses the more accurate Picknett and Bexon1 formula, eq 3, rather than eq 4. This allows us to give an analytical form for a function of the contact angle that is linear with time rather than

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providing a form requiring a numerical integration. To do so, consider a spherical cap droplet with a volume

VC(rb, θ) )

πrb3(1 - cos θ)2(2 + cos θ) 3 sin3 θ

(5)

where rb is the contact (or base) radius of the droplet. Assuming the contact radius is constant, the rate of change of the volume can be written in terms of the rate of change of the cosine of the contact angle

-πrb3 dVC du ) dt (1 - u2)1/2(1 + u)2 dt

( )

(6)

where u ) cos θ. Because rb ) RS sin θ and sin θ ) (1 u2)1/2, substituting eq 6 into eq 2 gives

du 2D∆c 1 ) 2f(θ)(1 + u)2 dt FLrb2

( )

(7)

which cannot be integrated exactly in its present form. However, terms of the form un/(1 + u)2 are integrable, so we can develop a polynomial fit to 1/2fPB(θ), accurate to 0.04% at all angles, for our range of interest of 90-180°,

1 2fPB(θ)

3

≈

dnun ) 0.999766 + 0.481517u + ∑ n)0 0.292040u2 + 0.089118u3 (8)

Substituting eq 8 into eq 7 and integrating gives

HPB(θ) ≡

-e0 + e1 loge(1 + u) + e2u + e3u2 ) 1+u -2D∆ct + HPB(θo) (9) FLrb2

where u(t) ) cos θ(t) and the constants en are given by

-1 -2D∆ct + HBV(θo) ) 2(1 + u) FLrb2

(11)

Similarly, using the Rowan et al. approximation fR(θ) ) (1 - cos θ)/2 gives

HR(θ) ≡

u-1 -1 -2D∆ct 1 ) + HR(θo) - loge 4 u + 1 2(1 + u) FLrb2 (12)

(

)

The exact result for a completely spherical droplet just in contact with a solid surface is similar to eq 11 modified by an extra factor of 1/loge 2 in the H(θ) function. The numerical factor multiplying the term -1/(1 + u) is then 1 /2 loge 2 ) 0.7213 compared to eo ) 0.721171. II.c. Surface Texture and Initial Contact Angle. On a flat, smooth surface, the highest contact angles for droplets of water that are achievable are determined by the surface chemistry and are around θes ) 110-120° for poly(tetrafluoroethylene) (PTFE). To achieve higher contact angles, superhydrophobic surfaces can be created by roughening or texturing the surfaces.3 For a textured surface, the liquid may either bridge across surface protrusions (Cassie-Baxter case18,19) or fully penetrate into the surface features (Wenzel case17). In the first case, the droplet sits on a composite solid-air interface, and the contact angle is given by the Cassie-Baxter equation

(13)

where φs is defined as the solid fraction upon which the droplet rests. In the second case, the droplet is in contact with the whole solid surface, and the contact angle is given by the Wenzel equation

e1 ) d1 - 2d2 + 3d3 ) 0.164791 e2 ) d2 - 2d3 ) 0.113804 d3 ) 0.044559 2

HBV(θ) ≡

cos θeCB ) φs cos θes - (1 - φs)

eo ) d0 - d1 + d2 - d3 ) 0.721171

e3 )

expression involving polylogarithm functions. When this result is expanded about u ) -1, an expression similar to eq 9 is obtained, but with only the first two terms in HPB(θ) and with the coefficients eo and e1 replaced by the numerical values loge 2 ) 0.693 and (loge 2 - 1/2) ) 0.193147; this approximate expression is an accurate approximation to HPB(θ) to around 10.5% over the range of 90-180°. If the solid boundary is completely ignored for the diffusion and the Birdi and Vu approximation fBV(θ) ) 1 is used in eq 7, then the equivalent result to eq 9 is

(10)

and are valid over the contact angle range of 90-180°. The constant HPB(θo) is a constant of integration and represents the function evaluated at the contact angle for the initial time t ) 0. Thus, the evaporation of spherical cap droplets with constant contact radii and contact angles greater than 90° can be analyzed by plotting the function HPB(θ) against time, t. On such a graph, the data for any given droplet should lie on a straight line, and the slope, multiplied by -FLrb2/2, will provide an estimate of the diffusion constant-vapor concentration difference product (i.e., D∆c). Although the specific results in eqs 8-10 are based on a polynomial expansion valid in the range of 90-180°, a similar approach can be adopted for other contact angle ranges. For completeness in comparing different approaches to analyzing data on evaporation with pinned contact lines, we have evaluated the effect of different approximations to f(θ) in eq 7. Using the Bourge`s-Monnier and Shanahan function fBMS(θ) in eq 7 and integrating gives a complex

cos θeW ) r cos θes

(14)

where r is the surface roughness defined as the ratio of the true surface area to the horizontal projection of the surface area. In these approaches to modeling the contact angles for the macroscopic droplet, the effect of the substrate heterogeneities are taken into account in an overall manner. Clearly, at a more local level, each element of the triple line must lie on either the vapor or the solid component beneath the droplet. In principle, the lower of the two contact angles predicted by eqs 13 and 14 is the equilibrium value, but in practice, edges in surface features can lead to a metastable CassieBaxter state even when the Wenzel state is energetically preferred; pressure is then needed to cause a conversion from the Cassie-Baxter to the Wenzel state.4 In addition, the Cassie-Baxter state reduces the true contact area between the liquid droplet and the solid and so tends to reduce any intrinsic contact angle hysteresis arising from the substrate material and leads to a “slippy” state. In contrast, the Wenzel state increases the true contact area

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Figure 1. (a) Schematic of the surface pattern, (b) scanning electron micrograph of a textured SU-8 surface (dp ) 11.5 µm, Lp ) 18.5 µm, hp ) 25 µm), (c) side profile of a droplet of water on a flat SU-8 surface, and (d) side profile of a droplet of water on a textured SU-8 surface.

between the liquid droplet and the solid and so tends to increase contact angle hysteresis leading to a “sticky” state.4,21 The patterns for the present study consist of circular cross-sectional pillars of diameter dp and height hp arranged in a square lattice of period Lp in each of the two in-plane directions (Figure 1a). The solid fraction φs required for the Cassie-Baxter equation is then

φs )

πdp2 4Lp2

(15)

and the roughness factor for the Wenzel equation is

r)1+

πdphp Lp2

(16)

When θes < 90°, the Wenzel formula always predicts a decrease in contact angle whereas the Cassie-Baxter equation can still predict an increase in contact angle. In the specific case when the separation between pillars is equal to their diameter (i.e., Lp ) 2dp), eq 15 gives φs ) π/16 ) 0.1963 and r ) 1 + πhp/4dp. Thus, using eq 13 and θes ) 80°, typical of an SU-8 polymer surface, would give a Cassie-Baxter angle of 140°. The equivalent Wenzel contact angle is dependent on the aspect ratio of pillar height to diameter and for a ratio of unity (i.e., hp/dp ) 1) is 72°. Increasing the aspect ratio to 2 gives a Wenzel angle of 63°. III. Experimental Development SU-8 is a negative photoresist that can be deposited in a thick layer and patterned using photolithography; the vertical sides of the features in those patterns can be smooth. SU-8 is epoxybased and becomes strong, stiff, and chemically resistant after processing with the typical contact angle to water of θes ≈ 80° on a smooth, flat surface and large contact angle hysteresis. The photolithographic technique for producing these substrates has (20) Shirtcliffe, N. J.; Aqil, S.; Evans, C.; McHale, G.; Newton, M. I.; Perry, C. C.; Roach, P. J. Micromech. Microeng. 2004, 14, 1384-1389. (21) McHale, G.; Shirtcliffe, N. J.; Newton, M. I. Langmuir 2004, 20, 10146-10149.

been described in detail by Shirtcliffe et al.20 In this paper, it was shown experimentally that droplets of water on SU-8 textured surfaces with a pillar separation twice that of the pillar diameter and pillar diameters in the range 5-40 µm arranged in a square lattice obey the Cassie-Baxter equation, provided the pillar height is sufficiently high. Importantly, the observed contact angle hysteresis was around 50°, thus providing a system with both a superhydrophobic Cassie-Baxter-type contact angle and conditions for a pinned contact line during free evaporation; the contact angle hysteresis is reduced by the Cassie-Baxter effect but not sufficiently to give a truly slippy state. The pinning of the contact line corresponds to our analysis in section II.b, which is equivalent to the Picknett and Bexon mode 1 evaporation mode and so simplifies the quantitative analysis of the evaporation. We constructed surfaces with pillar diameters in the range of 5-15 µm arranged in a square lattice and with spacings between the centers of pillars in the range of 20-30 µm in each in-plane direction; the heights of the pillars ranged from 12 to 35 µm (Figure 1a and b). The surfaces used had a Cassie-Baxter solid fraction area parameter φs of between 0.064 and 0.135 predicting Cassie-Baxter angles of between 157 and 147°, respectively. Ideally, the pillars have vertical sides, but in practice, on some of the samples with the highest pillars a slight outward taper occurred, leading to a marginally larger diameter at the top of the pillar as evident in Figure 1b. This effect resulted in a larger solid fraction area parameter φs than in the design and hence a lower Cassie-Baxter angle, but it did not prevent a superhydrophobic surface from being created. All experiments were carried out at room temperature (23-26 °C) and relative humidity (45-58%) in an enclosure to shield from drafts. Contact angles, contact base diameters, heights, liquid-vapor surface areas, and volumes were obtained from side profile images of a droplet and its reflection in the substrate using a Kru¨ss DSA10 contact angle meter (video profilometry with drop shape analysis software). The drop shape analysis provides volume and surface area data using the measured profile and by assuming axial symmetry; it does not assume any particular droplet shape. In our case, top view observation suggested that the droplets we studied did possess axial symmetry during the evaporation process. It is conceivable that such symmetry could be lost for droplets of sizes more comparable to the period of the substrate pattern or in the later stages of evaporation when droplets became smaller and entered the substrate pattern; we did not pursue these aspects of small droplet evaporation or late stage evaporation. There are several subtleties in the measurement of the contact diameter in the superhydrophobic droplet on a patterned substrate system that need to be understood when analyzing and interpreting the data and forming conclusions. The first is that the accuracy of contact diameter measurements for a droplet with a high contact angle of say 160° is more limited than one for which the contact angle is low, say 60°, if the optical magnification is chosen to keep the full droplet in the image. In the latter case, the initial contact diameter can be the full field of view and so provide optimal accuracy in the measurement of the contact diameter from the digital image. In the former case, the initial contact diameter is necessarily only a fraction of the field of view (rapidly becoming less so as the droplet becomes more superhydrophobic), and the accuracy of contact diameter measurements from the digital images is therefore reduced. The second subtlety is that accurate measurements of contact diameter depend on accurate identification of the vertical position of the solid-liquid interface (i.e., substrate surface) in the image, and as the droplet becomes superhydrophobic, this becomes a more significant problem. For contact angles approaching 180°, small changes in the vertical position of the identified baseline can provide large changes in the apparent contact diameter. Similarly, if the baseline is misidentified by, say, one vertical pixel, then an evaporating superhydrophobic droplet with a decreasing contact angle will give an apparent (rather than real) decrease in contact diameter. The baseline issue has a particularly strong effect on the ability to make accurate measurements of contact diameters once the droplet enters the substrate pattern. In our case, the substrates have periods of 20-30 µm, and the heights of the pillars range from 12 to 35 µm. This means the placements of baselines are more difficult, and we strongly

Droplet Evaporation on a Superhydrophobic Surface

Figure 2. Evolution of the contact angle with time ()) and height (solid curve) and contact diameter (+) during the evaporation of a droplet of water where no obvious collapse occurs during the initial evaporation period.

Figure 3. Evolution with time of the contact angle ()) and height (solid curve) and contact diameter (+) during the evaporation of a droplet of water where an obvious collapse occurs during the initial evaporation period. emphasize that small changes in measurements of contact diameters, particularly of less than a substrate lattice period, and late stage evaporation measurements once the droplet has penetrated the substrate pattern should be interpreted cautiously and the data should not be overinterpreted. Figure 1c shows a typical side profile image of a droplet of water on a flat SU-8 substrate (θs ≈ 80°), and Figure 1d shows an equivalent image on a textured substrate (dp ) 11.5 µm, Lp ) 18.5 µm, hp ) 25 µm, θs ≈ 146°). Provided the droplet was deposited without too much pressing to detach it, the droplet tended to deposit in a Cassie-Baxter-type state bridging the gaps between pillars; the Cassie-Baxter state is visually apparent from the pattern of light penetrating between the pillars below the droplet.

IV. Results and Discussion Figures 2 and 3 show the contact angle, base diameter, and droplet height during a typical evaporation sequence for droplets on a surface with a pillar center-to-center spacing of 30 µm and pillar diameter and height of 8 and 15 µm, respectively. In Figure 2, the initial contact angle is greater than 150°, and after the brief initial period (100 s in this specific case) it steadily decreases (up to t ) 600 s in this specific case), during which time the contact diameter remains approximately constant. Between 100 and 500 s, the change with respect to contact diameter is within one period of the substrate pattern. The contact radius then clearly depins as evidenced by the beginning of a stepwise retreat shown in the contact diameter data. During this period, the contact angle shows a net decrease, but with increases and decreases accompanying each step change in contact diameter. For this particular example, the droplet becomes too small for accurate measurements after 1100 s. Figure 3 shows a second type of sequence that is qualitatively different from Figure 2 in that a very obvious discontinuity occurs in the data during the period corresponding to the pinned contact diameter evaporation (in this particular example just after 800 s). Such a

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discontinuity has been observed during the detachment of droplets from the syringe shortly after deposition (several seconds) and a long time (minutes) after deposition as shown in Figure 3. Prior to the discontinuity, the evaporation proceeds in a manner similar to that shown in Figure 2, but afterward the contact angle continues to decrease smoothly and no obvious stepwise changes occur in the contact diameter. The discontinuity appears rapidly. In these data, the time between recorded frames was 2 s for the first 2 min and subsequently 10 s, but visually the collapse occurred in less than 1 s. As in Figure 2, the droplet becomes too small for accurate measurements of contact diameter after around 1000 s. All superhydrophobic droplet evaporation sequences observed on these SU-8 pillar surfaces corresponded to one of the behaviors in Figure 2 or 3. Figure 4 shows a sequence of four images (panels a-d) illustrating the behavior in Figure 2 and four images (panels e-h) illustrating the behavior in Figure 3. For both cases, the pattern of light visible beneath the droplets indicate the initial droplets sit upon the pillars and bridge the gaps between the pillars, so that the droplets are in the Cassie-Baxter state. As the evaporation proceeds, the volumes decrease with either no change or a small change in contact area (Figure 4b and f). In the first case (Figure 4c and e), the contact diameter eventually decreases in a stepwise manner; this is visually obvious when viewing the video of the evaporation. We anticipate that the period of the steps should correlate with the period of the lattice of pillars consistent with work reported on the spreading of silicone oils on this type of surface.22 In the second case (Figure 4g and h), a discontinuity occurs in the evaporation sequence; this is a rapid process as illustrated by Figure 4g and h, which are successive images in the recording. A close examination of the image before the discontinuity (Figure 4g) appears to show a shadow beneath the droplet at the center of the contact area. The pattern of light below the droplet in the image immediately after the discontinuity (Figure 4h) appears to show that the droplet is no longer suspended upon the pillars but has penetrated into the substrate pattern. The contact angle immediately after the discontinuity remains higher than observed on the flat surface. We therefore conclude that the discontinuity represents a collapse of the droplet from the Cassie-Baxter state to a Wenzel-type state. Visual examination of the video sequences shows that subsequent evaporation appears to have a pinned nature (for around 2.5 min in the case of Figure 3) until the droplet becomes small and accurate measurement difficult. Care is needed in interpreting the contact diameter data in Figure 3 in this later stage of evaporation because a 30 µm change in contact diameter corresponds to the period of the pattern and with the lower contact angle and volume there is more sensitivity in the measurement accuracy of the diameter, which depends on the placement of the baseline in the video profile. Qualitatively, we believe that a droplet collapse also probably always occurs in droplets showing the evaporation sequence in Figure 2 but that it occurs for a such a small volume that accurate measurements to confirm it are difficult. Figure 5 shows the stepwise retreat in the later stages of evaporation for a larger initial volume droplet. The surface had a pillar center-to-center spacing of 30 µm and a pillar diameter and height of 8 and 14 µm, respectively, which resulted in an initial contact angle of 145.6°. The time period from 2500 to 3100 s clearly shows eight (22) McHale, G.; Shirtcliffe, N. J.; Aqil, S.; Perry, C. C.; Newton, M. I. Phys. Rev. Lett. 2004, 93, article 036102.

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Figure 4. Images of droplet evaporation. Panels a-d show a droplet where the collapse from the Cassie-Baxter state occurs late in the evaporation. Panels e-h show a droplet where the collapse from the Cassie-Baxter to the Wenzel state occurs at a middle stage during the evaporation.

Figure 5. Evolution with time of the contact angle ()) and contact diameter (+) during the stepwise retreat of a droplet evaporation sequence.

stepwise decreases reducing the base diameter from 857 to 600 µm, and this gives an average step of 32 µm compared to the lattice center-to-center spacing of 30 µm. In fact, the period from 1950 to 3060 s is consistent with 16 steps each of 30 µm. Each stepwise retreat event is also accompanied by a temporary increase in the contact angle. Because the stepwise retreat event occurs at a given volume, we would expect a temporary increase in the contact angle as observed. Quantitatively, this idea can be formalized by requiring the volume given by the spherical cap approximation in eq 5 to remain constant when rb changes by an amount ∆rb and the contact angle changes by an amount ∆θ (in radians),

∆θ ) -sin θ(2 + cos θ)

( ) ∆rb rb

(17)

For a 3.5% decrease in contact radius occurring at a contact angle of 131°, eq 17 predicts an increase in contact angle of around 2°, which is of the same order of magnitude as the observed changes in the data. Such an increase would reduce the horizontal (inward) component of the liquidvapor surface tension force (the reduction in force can be estimated from eq 17), thus allowing the droplet to again become temporarily pinned. Further volume reduction by evaporation would reduce the contact angle and so increase the horizontal (inward) component of the liquid-vapor surface tension force until sufficient force occurs to cause another stepwise retreat of the contact radius in a stickslip manner. It is also possible to analyze the change in surface free energy for a stepwise retreat event at constant volume,

Figure 6. Contact angle data ()) reproduced with the measured liquid-vapor surface area (divided by 3; ‚‚‚, in mm2), volume (×, in mm3), and height (+, in mm) for droplet data shown in Figure 3. The solid curves are the estimates calculated for a spherical cap using the measured contact angle and base diameter.

and Shanahan has previously considered such an analysis for stick-slip on a flat surface.23 He derived a minimum step size necessary to overcome an intrinsic energy barrier causing contact angle hysteresis on a flat surface. Applying this interpretation to our substrates, an intrinsic energy barrier would exist for the substrate material, but the pattern coupled with the Cassie-Baxter effect of a reduced solid-liquid interfacial area would then reduce this energy barrier and impose a stepwise retreat distance.21 Figure 6 reproduces the contact angle data from Figure 3 but now includes the measured data for the liquidvapor surface area (‚‚‚ and in mm2), the volume (× and in mm3), and the droplet height (+ and in mm; the liquidvapor surface area has been divided by a factor of 3 so that it fits onto the same axis as the other two parameters). The solid lines are the spherical cap values for these parameters calculated from the measured contact angle and base diameter and are clearly slight overestimates. Initially, the spherical cap calculation overestimates the surface area by 2-3%, but this decreases as time progresses toward the droplet collapse at 850 s. Similar overestimates occur in data for drops not showing the collapse (e.g., Figure 2), and in these cases the overestimate decreases as the stepwise retreat regime is approached. From the full data set of observations, we conclude that the measured liquid-vapor surface area, ALV, is well approximated by a linear change with time up to the point (23) Shanahan, M. E. R. Langmuir 1995, 11, 10411043.

Droplet Evaporation on a Superhydrophobic Surface

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Figure 7. Tests of eqs 20 and 21 for the data in Figure 2 (lowest pair of curves), Figure 3 (middle pair of curves), and Figure 5 (upper pair of curves); in each case, eq 21 is the lower one of the pair at t ) 0. The solid lines are fits over the range to when the droplet collapses or a stepwise retreat begins.

of either droplet collapse or stepwise retreat of the base diameter. Using the spherical cap approximation, we can understand why this may be the case. In this approximation, the surface area is

ALV )

2πrb2 1 + cos θ

(18)

and assuming a constant contact radius, this can be used in eq 7 to obtain

( )

dALV -8πD∆cf(θ) ) dt FL

(19)

During the initial evaporation phase when the contact angle is slowly varying, f(θ) can be approximated to a constant. For example, in Figure 6 in the period before droplet collapse the contact angle changes from 149.7 to 138.0°, and the corresponding values of fPB(θ) are 0.6723 and 0.6528 so that approximating f(θ) by 0.6626 gives a value accurate to (2% over this experimental angular range and eq 19 is expected to be a constant as observed in the experiments. To examine the theory in section II.b describing the constant contact radius mode with diffusion-controlled evaporation, we analyzed the data for the validity of eq 9 and consistency with eq 19. To compare these two equations with data in the same Figure, we use the rearrangement of eq 9 into

FLrb2HPB(θ) ) -D∆ct + k2 2

(20)

and eq 19 into

FLALV ) -D∆ct + k1 8πfav

(21)

where k1 and k2 are constants and it has been assumed that rb and f(θ) ≈ fav are approximately constant over the time period considered during a droplet evaporation; eq 20 is valid only for contact angles of 90° or greater. Figure 7 shows eqs 20 and 21 calculated from the data for rb, θ, and ALV presented in Figures 2, 3, and 5; the average of the initial and final values has been used for fav. The solid curves are straight line fits over the range before either the droplet collapsed or a stepwise retreat of the contact line began. The fits show excellent agreement with the expected linearity having R2 parameters of better than 0.999, although eq 20 gives a slightly higher estimate

than eq 21 for the diffusion constant-vapor concentration difference product (i.e., D∆c). To verify the quantitative agreement of the diffusion constant-vapor concentration difference product (i.e., D∆c), a series of experiments on two surfaces with pillar diameters of 10 µm and center-to-center separations of 20 and 30 µm were analyzed. The data range used in the analysis was such that rb is constant to within 4% of its initial value and the contact angle is greater than 90°; we have also taken fav as the average of the initial and final values. The fits of eqs 20 and 21 to the data were straight lines with regression coefficient parameters of R2 ) 0.9999 or better. Data for four droplets, including temperature and relative humidity, on each of these two surfaces are provided in Table 1. This Table also provides a comparison to values of the diffusion coefficient estimated by correcting the CRC Handbook24 value of 2.39 × 10-5 m2 s-1 at 8 °C to the measured temperatures using a T3/2 temperature dependence to provide reference values. The values of diffusion constants using both eqs 20 and 21 are to within 8% of these reference values and within the uncertainty arising from our estimates of concentration difference ∆c from the temperature and relative humidity. These quantitative estimates of diffusion coefficients are impressive and provide confidence in the theoretical interpretation of the evaporation process from superhydrophobic surfaces. The agreement in estimates of the diffusion coefficient using droplets that start with almost completely spherical shapes (i.e., contact angle approaching 180°) but with a plane boundary in close proximity supports the prediction that the evaporation rate will decrease compared to that of a spherical droplet far removed from any surfaces. The physical mechanism is the reduction in space into which vapor can diffuse.1 There are other possible mechanisms related to droplet evaporation from superhydrophobic surfaces that we have not examined in the current work. Examples include the precise triggers for the CassieBaxter to Wenzel transition, evaporation from slippy-type Cassie-Baxter surfaces, and whether heat flow from the substrate into a droplet influences evaporative cooling and if so whether this differs from the Cassie-Baxter to Wenzel states as would be expected because of different microscopic contact areas. It is also conceivable that in some superhydrophobic situations a droplet would effectively sit upon a layer of vapor. This latter effect could be analogous to the Leidenfrost effect in which a droplet placed on a very hot surface creates a layer of vapor in an initial rapid evaporation of the contact area. The layer of vapor then prevents further heat being transmitted into the droplet and hence reduces evaporation. Another possibility is that condensing a layer of vapor that exists below a droplet into a liquid layer on the substrate features below a droplet could trigger a Cassie-Baxter to Wenzel transition. However, it is not clear whether any of these mechanisms occur. Because superhydrophobic surfaces are common in nature and the conservation of water droplets or vapor near the surface of the leaves of plants is of importance, further study of evaporation from superhydrophobic surfaces could be relevant to biological systems as much as physical systems involving heat and mass transfer. V. Conclusions Free evaporation of small sessile droplets of water from textured surfaces composed of a lattice of tall pillars of (24) CRC Handbook of Chemistry and Physics, 62nd ed.; Weast, R. C., Astle, M. J., Eds.; CRC Press: Boca Raton, FL, 1981-1982.

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Table 1. Estimates of Diffusion Coefficients for Water into Air for Droplets on Two SU-8 Patterned Surfaces: Surface A with 10-µm-Diameter Pillars with 20-µm Center-to-Center Spacing and Surface B with 10-µm-Diameter Pillars with 30-µm Center-to-Center Spacing D∆c (× 10-7 kg m-1 s-1)

D (× 10-5 m-2 s-1)

surface

T (°C)

RH (%)

∆c (g m-3)

θi°

θf°

eq 20

eq 21

eq 20

eq 21

ref

A A A A B B B B

23 24 26 26 23 23 25 26

58.0 57.0 57.0 47.5 51.0 51.0 48.3 46.0

8.62 9.34 10.45 12.76 10.06 10.06 11.88 13.12

133.3 136.6 140.0 136.2 125.9 141.3 125.0 138.9

106.3 103.1 94.8 92.3 90.6 129.5 90.3 118.4

2.396 2.559 2.961 3.258 2.664 2.714 3.194 3.318

2.358 2.481 2.588 3.189 2.588 2.399 3.153 3.068

2.780 2.740 2.833 2.553 2.648 2.698 2.688 2.529

2.735 2.657 2.476 2.499 2.572 2.385 2.654 2.339

2.584 2.597 2.623 2.623 2.584 2.584 2.610 2.623

polymer photoresist SU-8 has been studied. It has been shown that such surfaces allow a starting contact angle for water droplet evaporation exceeding 150° to be achieved. On our regularly textured surfaces, evaporation initially proceeds in a pinned contact area mode followed by a steplike contact line retreat, which mirrors the underlying lattice structure of the substrate. In some cases, a sharp steplike decrease in the contact angle with an accompanying steplike increase in the contact radius occurs, and the contact area is then completely pinned for the remainder of the evaporation period. This steplike change may also be occurring late in the evaporation process for all droplets, but this has not been possible to determine because of the resolution of the experimental system. It has been suggested that the initial phase of evaporation, consisting of pinned contact line followed by steplike contact line retreat, corresponds to the droplet evaporating while suspended across the tops of the pillars (i.e., a Cassie-Baxter state). The subsequent collapse and apparent complete pinning of the contact line then

corresponds to a Wenzel state. The initial pinned contact line phase of evaporation has been analyzed quantitatively using a model depending of the diffusion of vapor into the surrounding atmosphere. This model provides a linear form analysis equation (eqs 9 and 20) that takes into account a loge 2 correction factor due to the solid boundary restricting the space into which vapor may diffuse. The experiments are well described by the model, and this allows a simple method for the estimation of the diffusion constant-vapor concentration difference product (i.e., D∆c) and hence the coefficient of diffusion D. Acknowledgment. We acknowledge the financial support of the UK EPSRC and MOD/Dstl (grant GR/ R02184/01) and of the EU-COST Chemistry Action D19, WG-007, and TUBITAK-MISAG-COST.D19 projects. LA0518795