ARTICLE pubs.acs.org/Langmuir
Dependence of Volatile Droplet Lifetime on the Hydrophobicity of the Substrate M. E. R. Shanahan,† K. Sefiane,*,‡ and J. R. Moffat‡ †
Universite de Bordeaux, Institut de Mecanique et d'Ingenierie de Bordeaux (I2M), CNRS UMR 5295, B^at. A4, Cours de la Liberation, 33405 TALENCE Cedex, France ‡ School of Engineering, The University of Edinburgh, The Kings Buildings, Mayfield Road, Edinburgh EH9 3JL, United Kingdom ABSTRACT: In this Article, we demonstrate the dependence of the lifetime of a volatile droplet on the hydrophobicity of the substrate. Ethanol droplets placed on the molecularly smooth surfaces of three polymers, applied to substrates by spin-coating, showed distinct types of behavior depending on the hydrophobicity of the latter. High contact angles, θ, lead to fairly regular recession of the triple line during liquid evaporation at essentially constant θ, whereas low contact angle caused pinning, θ decreasing with time. The latter case leads to shorter drop lifetimes.
’ INTRODUCTION Droplet evaporation is of great importance in many wide ranging areas, such as spray cooling, thin film coating, detergency, as well as in biological areas such as DNA stretching.1 Colloidal deposition is also used to produce nanocrystals.2 The evaporation of colloidal suspensions can also be used to produce precise patterns such as those used in bioassays. Resulting deposit patterns left after complete evaporation can be quite varied and complex. The interaction between droplets and surfaces on which they rest is key in understanding and developing biomimetics applications.3 It is well-known that during the evaporation of a colloidal suspension, particles are drawn to the drop periphery. This accumulation leads to what is known as the coffee ring effect, where rings of dried particles are left after complete evaporation.4 Marangoni stresses, internal convection and advection, conduction through the substrate, temperature effects of the fluid, substrate, and ambient vapor, mass transfer through diffusion, and various other factors, such as electrostatic attraction between the particles and substrate, all combine to complicate the process of drop evaporation.5,6 Clearly all of these interact with evaporation rate and therefore affect drop lifetime. However, this aspect seems not to have received much attention. Here, we consider drop lifetime as related to solid surface properties, in particular hydrophobicity. The results have possible impact in cooling applications, because evaporation is found to be dependent on intrinsic contact angle. r 2011 American Chemical Society
’ EXPERIMENTAL SECTION The aim of these experiments was to investigate the evaporative behavior of droplets of ethanol on surfaces of varying hydrophobicity. Drops of known volume (5 ( 0.5 μL: it is difficult to be more precise due to the drop/syringe separation process) were deposited on a given substrate, and the evaporative process was followed visually by camera. An experimental setup was constructed, which consisted of a drop shape analyzer (Kruss, DSA 100), that could record the evolution of drop profile (contact angle, drop radius, drop volume) with time, using high speed imaging. The analyzer provides high-precision dosing and positioning of liquid drops and permits recording and evaluation of video images through accompanying PC controlled software. A special chamber was designed in which the drop would be deposited and the experiments conducted. By carrying out the experiment in a contained environment, it was possible to control the surrounding vapor composition and pressure. The chamber was also helpful in preventing sample contamination from atmospheric particulates such as dust. The droplets investigated in this study are pure ethanol (>99.9% pure from Aldrich). The presence of traces of water is believed to have very little effect on the results and conclusions drawn from the study. All runs were conducted using a dry nitrogen atmosphere. The chamber itself was constructed from stainless steel, with borosilicate glass viewing windows embedded in the side walls. Separate connections to the chamber were Received: February 2, 2011 Revised: March 1, 2011 Published: March 24, 2011 4572
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Figure 1. Contact radius, R, versus time, t, for ethanol on PTFE, Cytop, and C4F8. Inset shows extrapolation to estimate total time to evaporation. also constructed, for the nitrogen supply, a connection line to a vacuum pump to remove the ambient air in the chamber, and for an electrical cable connected to the horizontal plate. The plate was attached to a vertical axis, enabling it to be raised or lowered to control drop deposition. Three different solid surfaces were studied. They correspond to polymeric coatings deposited onto silicon substrates by spin-coating. These solids are molecularly smooth. The polymers were C4F8 (a fluorocarbon polymer), Cytop (perfluorinated polymer consisting of CC, CF, and CO bonds), and PTFE (Teflon), in increasing order of hydrophobicity. Experiments were performed at ca. 20 °C.
’ RESULTS The results presented correspond to drops of pure ethanol, as the probe liquid, having typical static contact angles of ca. 18°, 45°, and 57° on C4F8, Cytop, and Teflon, respectively. Results are presented in Figure 1 of base radius, R, versus evaporation time, t, on each of the three substrates. From the graph, it can be seen that the base radius, R, recedes quite homogeneously, and monotonically, as time progresses, in the case of Cytop and Teflon. However, initially R remains essentially constant on the less hydrophobic C4F8 surface. There is even a slight increase, apparently. In contrast, the contact angle, θ, is observed to remain almost constant during the evaporative lifetime on the Cytop and Teflon surfaces, while it decreases steadily on C4F8, as shown in Figure 2. The evaporative behavior of ethanol on Cytop and Teflon is consistent with that which might be expected on an (almost) ideal surface. There is very little change in contact angle during evaporation, and what there is seems to be random. Consequently, drop radius constantly decreases, apart from slight stochastic changes corresponding to those of the radius. This indicates that there is little pinning of the contact line. In contrast, the behavior of the ethanol drop on the C4F8 surface is of almost total pinning of the contact line, similar to what would be observed on a non-ideal surface. Similar results were obtained on repeated experimental runs (35 per surface used in the following). Unfortunately, absolute drop lifetimes, that is, until total evaporation occurred, were not accessible with the present experimental apparatus, because precision was poor for drops just before disappearing. Notwithstanding, the ranges observed correspond to the order of 95% of total evaporation in volume, and the last stage is, anyway, recognized as being somewhat
ARTICLE
Figure 2. Contact angle, θ, versus time, t, for ethanol on PTFE, Cytop, and C4F8.
stochastic,7 and therefore corresponds to a different regime (poorly understood, but possibly with long-range van der Waals forces playing a role, due to very low drop height). The droplet, in its very last moments (ca. last 5% of its lifetime), becomes very thin and, indeed, behaves like a very thin evaporating film. Longrange interactions between interfaces are the dominant forces, and “bulk” evaporation is no longer relevant. The evaporation of very thin liquid films has been shown to exhibit instabilities dominated by the interactions between the solid substrate and the liquid.8 Looking at Figure 2, we see that pure ethanol evaporates on PTFE at a virtually constant contact angle of ca. 57° (0.995 rad) and that the decrease of contact radius is virtually linear, although slightly concave toward the abscissa. On Cytop, similar behavior is observed, but this time with θ = ca. 45° (0.785 rad). Any possible concavity is barely discernible. In both cases, the form of the curves and drop lifetime, not directly measurable, will be discussed below. In the case of C4F8, triple pinning occurs until near the end of drop lifetime. Drop lifetime, tf, is ca. 110 s, of which the large majority is in the pinning mode: mean of three runs, 107 ( 3 s.
’ INTERPRETATION AND DISCUSSION Contact Angle and Pinning. Let us consider the essential differences between the behavior of Cytop and Teflon, on one hand, and that of C4F8, on the other. The last has by far the lowest (typical) value of contact angle, θ (whereas the other two are similar). We may write Young’s equation as:
0 ¼ γSL γSðVÞ þ γ cos θo
ð1Þ
where γSL, γS(V), and γ are, respectively, solid/liquid, solid (in the presence of vapor), and liquid surface tension, and θo is equilibrium contact angle. If the contact angle decreases by a small amount, δθ, following evaporation, we have a force (per unit length of triple line), δF B, “attempting” to move the triple line (to cause recession) (see Figure 3): δF B ¼ γSL γSðVÞ þ γ cosðθo δθÞ γ sin θo 3 δθ ð2Þ If there exists an intrinsic energy (or force) barrier, U, opposing triple line motion (wetting hysteresis),9,10 then for a 4573
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Figure 3. Schematic representation of triple line region with liquid surface at its equilibrium contact angle, θo, and at a slightly smaller angle, θo δθ.
given liquid, a small value of θo implies a small value of δF B unless δθ is larger. Thus, a larger deviation from equilibrium is required before the triple line unpins. Presumably, with the results obtained, θo is sufficiently large for ethanol on both Cytop and Teflon for unpinning to occur virtually continuously, or for very small δθ, leading to an approximately constant value of θ, at least at the observable scale. (The slight variations seen in Figure 1 could well be related, at least partially, to small scale unpinning.) However, ethanol on C4F8, with a much lower value of θo, cannot presumably “generate” enough force to overcome the hysteretic barrier, at least until θ has decreased to ca. 12°, toward the end of drop lifetime. Evaporation and Drop Lifetime. Returning to Figures 1 and 2, in which it can be seen that for pure ethanol, contact angle, θ, remains essentially constant during the entire process of evaporation for the PTFE and the Cytop substrates, we may attempt to J 3 nB (the explain drop lifetime, tf, as follows. Evaporation flux, B vector notation with nB as unit normal to the surface is simply to “formalize” evaporation, |JB|, perpendicular to the liquid surface), may be expressed as: !λðθÞ r2 ð3Þ BJ 3 n B ¼ Jo 1 R 2 where Jo represents flux perpendicular to a planar liquid surface, r is radial distance, and R is drop contact radius. The exponent λ(θ) is equal to (π 2θ)/(2π 2θ), according to Deegan et al.,11 whereas an apparently improved version is given by λ(θ) equal to 0.5 θ/π, suggested by Hu and Larson.12 We adopt the latter in the following. In either case, evaporation rate is exacerbated near the triple line, and increasingly so for small values of contact angle. Assuming the above, we may calculate overall evaporation rate, · M(θ), or rate of mass loss, as a function of θ from: Z R qffiffiffiffiffiffiffiffiffiffiffiffiffi _ 2πr 1 þ y0 2 3 BJ 3 n MðθÞ ¼ B dr 0
Z
2πR 2 Jo 0
1
z½1 þ ðz2 θ2 =2Þ ð1 z2 Þλ
dz
ð4Þ
where z = r/R and spherical drop geometry of sufficiently low θ may be approximated by height, y ≈ (R2 r2)θ/2R. The term in 2 2 the right-hand integral, in (z θ /2), corresponds to the first term 0 2 1/2 in the expansion of (1 þ y ) . It is appropriate to truncate after the (z2θ2/2) term, because even this only contributes ca. 22% to the integral in the “worst” case considered, that of θ ≈ 1 rad. We therefore retain it below, but neglect further (smaller) terms because they complicate the algebraic expressions for little gain in precision. Integration of expression 4 leads to: " # πR 2 Jo θ2 _ 1þ ð5Þ MðθÞ 1λ 2ð2 λÞ
In eqs 3 and 5, we have ignored any functional dependence of evaporation flux, Jo. As it stands, eq 5 suggests that M ∼ R2. There is, however, reason to believe that Jo ∼ R1, as shown by Hu and · Larson,12 leading to M ∼ R, and as found by various 4,11,12 A similar conclusion of Jo ∼ R1 may be reached authors. using a simplified, less rigorous approach, as follows. To do this, it is assumed that an averaged value of Jo will suffice for the argument [i.e., not allowing for differences of flux on the drop surface, cf., see eq 3]. We use the steady-state simplification of Fick’s laws of diffusion to account for evaporation: Jo ¼ D
DC ;~r > R D~r
1 D 2 DC rC¼ 2 ~r 0 D~r ~r D~r 2
ð6Þ ð7Þ
where ~r is radial distance from the spherical center of symmetry (i.e., below the solid surface) of the drop (but treating the latter as a “point”, with respect to the environmental chamber, which is much larger), D is the coefficient of diffusion of ethanol in air (nitrogen), and C is the vapor concentration (kg m3), a function of ~. r Assuming saturation in ethanol vapor, CSAT, at the drop surface,12 corresponding to a radius of curvature of ca. R/sin θ, and vapor concentration C = 0 at infinity, it is readily shown from eqs 6 and 7 that: Jo
DCSAT sin θ R
ð8Þ
~ ), using the CSAT may be calculated from CSAT = MWPV/(RT law for ideal gases, where MW is the liquid molecular weight (46 g mol1), PV is the saturated vapor pressure (ca. 6 102 bar), and R ~ and T are the ideal gas constant and absolute temperature, giving a value of ca. 0.11 kg m3. With D = ca. 1.1 105 m2 s1, and R = ca. 1 mm, we obtain Jo ≈ (410) 104 kg m2 s1, depending on the contact angle. Returning to eq 5, in the general case, θ = θ(t), which would · make integration of M(θ(t)) very difficult, because λ = λ(θ(t)) also. Fortunately, we have two limiting cases, at least to a good approximation, which allows considerable simplification. As discussed above, for both Cytop and Teflon, contact angle, θ, remains virtually constant throughout evaporation, while for C4F8, contact radius, R, does not change significantly until near the end of drop lifetime. a. Constant Contact Angle. In the case of constant contact angle, we approximate drop volume, V, by V = M/F ≈ πR3θ/4, where M is drop mass and F is liquid density, leading to: ! 3 d FπθR 3FπθR 2 dR _ : MðθÞ ¼ dt dt 4 4 2π2 R 2 Jo
ð3π þ 2θ þ πθ2 Þ ð3π2 þ 8πθ þ 4θ2 Þ
ð9Þ
Whether Jo ∼ R1 or whether Jo ∼ constant (Kelvin’s equation suggests the latter, given the weak curvature of the drop surface) is still not entirely clear, and so we consider the two possibilities (realizing that maybe something between the two corresponds to reality). The former expression was developed for drops with a pinned contact line. For both PTFE and Cytop substrates, the contact line is continually moving, and this may have some influence. 4574
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We have then two possibilities for the expression for Jo, Jo ≈ constant or expression 8. For the former case, with θ constant (=initial value θo), direct integration of eq 9, with the boundary condition of R(t = 0) = Ro (Ro is initial contact radius) leads to: ðRo RÞ
8πJo t ð3π þ 2θo þ πθ2o Þ 3Fθo 3 ð3π2 þ 8πθo þ 4θ2o Þ
ð10Þ
For the latter case, with Jo given by expression 8, we obtain: ðRo2 R 2 Þ
16πDCSAT t ð3π þ 2θo þ πθ2o Þ 3 ð3π2 þ 8πθ þ 4θ2 Þ 3F o o
ð11Þ
If Jo is considered constant, a linear relationship between R and t (eq 10) is predicted, as more or less shown by Figure 1 for Cytop and Teflon. However, with Jo ∼ R1, a parabolic relationship (eq 11) is to be expected. In Figure 4, we show the same data given in Figure 1, but as R2 versus t, which, of course, should give a straight line if eq 11 is obeyed, rather than eq 10. As can be seen, the linearity of the plot in Figure 1 seems to be rather better than that in Figure 4, suggesting that eq 10 is a better approximation. In reality, given the slight concavity in Figure 1, Jo is probably between the two extremes, for reasons unknown at present. We therefore assume linearity in the following. The extrapolated overall drop lifetime, tf, for Cytop in Figure 1 is 560 s, and that for Teflon is 590 s. Taking the mean over five experiments (for each substrate), we find tf = 600 ( 75 s (Cytop) and tf = 680 ( 105 s (Teflon). Assuming the regime given by eq 10 over the entirety of drop lifetime, tf, we obtain the expression: 2 2 3F 4V 1=3 θ2=3 o ð3π þ 8πθo þ 4θo Þ ð12Þ tf 3 ð3π þ 2θ þ πθ2 Þ 8πJo π o o
Figure 4. Square of contact radius, R2, versus time, t, for ethanol on PTFE, Cytop, and C4F8. Little qualitative difference can be seen in the features of the curve for the C4F8 polymeric surface, whereas PTFE and Cytop show convexity.
Equations 12 and 15 could, in principle, be used to estimate drop lifetime for the two cases considered, but unfortunately, as discussed above, we have poor a priori knowledge of Jo, the evaporation current flux at the drop center, although θ, F, and V are accessible. Notwithstanding, to circumvent this absence of definitive knowledge of Jo, let us define a constant, K, by: K ≈ 3F[4V/ π]1/3/8πJo, in which case with a “normalized” lifetime, ~t f = K1tf, we have:
where V refers to initial drop volume. b. Constant Contact Radius. If wetting line pinning occurs, it is R (=Ro) that remains (approximately) constant and θ that which decreases, in which case: ! 3 d FπθR FπR 3 dθ _ MðθÞ ¼ dt 4 4 3 dt ð3π þ 2θ þ πθ2 Þ ð13Þ ð3π2 þ 8πθ þ 4θ2 Þ We again assume, for consistency, that Jo ≈ constant. Assuming drop lifetime to be terminated at θ = 0 (after starting at θo), we obtain: Z θo 8πJo tf ð3π2 þ 8πθ þ 4θ2 Þ dθ FR ð3π þ 2θ þ πθ2 Þ 0 9π 2θo ln 1 þ π 4θo ð14Þ πθo 2 3π 2π2 R 2 Jo
The integral above is not straightforward, but because small values of θ are relevant (
