Edinburgh Evaporation of Droplets on Strongly Hydrophobic Substrates

: The manner in which the extreme modes of droplet evaporation (namely, the constant contact radius and the constant contact angle modes) become indistinguishable on strongly hydrophobic substrates is described. Simple asymptotic expressions are obtained which provide good approximations to the evolutions of the contact radius, the contact angle, and the volume of droplets evaporating in the extreme modes for a wide range of hydrophobic substrates. As a consequence, on strongly hydrophobic substrates it is appropriate to use the so-called “ 2/3 power law ” to extrapolate the lifetimes of droplets evaporating in the constant contact radius mode as well as in the constant contact angle mode.


INTRODUCTION
As, for example, the recent review articles by Cazabat and Gueńa, 1 Erbil, 2 and Larson 3 demonstrate, the evaporation of a fluid droplet on a solid substrate has been the subject of extensive ongoing theoretical and experimental investigations by a wide range of research groups in recent years.
As many authors have shown, after a short transient in which a droplet deposited onto a substrate rapidly adjusts to a quasiequilibrium shape with initial contact radius R 0 and initial contact angle θ 0 , the droplet can evaporate in a wide variety of modes. The most extreme modes of evaporation are the so-called "constant contact radius" (CR) mode, in which the contact line is always pinned and the contact angle θ = θ(t) decreases while the contact radius R(t) = R 0 remains constant, and the so-called "constant contact angle" (CA) mode, in which the contact line is always depinned and the contact radius R = R(t) decreases while the contact angle θ(t) = θ 0 remains constant, where t denotes time. As the pioneering studies of Picknett and Bexon 4 and Bourges-Monnier and Shanahan 5 and many subsequent works have shown, in practice, droplets often evaporate in a "stick-slide" (SS) mode, which involves some combination of "stick" (i.e., with a pinned contact line) and "slide" (i.e., with a depinned contact line) phases. Recently Nguyen and Nguyen,6,7 Dash and Garimella, 8 and Stauber et al. 9, 10 have formulated and analyzed a simple yet effective model for an idealized SS mode consisting of a single stick phase followed by a single slide phase. Nevertheless, understanding the extreme modes remains a key part of understanding the evaporation of droplets.
The main aim of the present work is to show how the extreme modes of droplet evaporation converge as the value of the initial contact angle θ 0 increases toward π, and so, in particular, to describe the manner in which they become indistinguishable on strongly hydrophobic substrates.

THE MATHEMATICAL MODEL
The widely used "diffusion-limited" model employed in the present work is based on the assumption that the evaporation from the droplet is quasi-steady and limited by the diffusion of vapor in the quiescent atmosphere above it. This model, together with the assumption that the droplet is sufficiently small that gravitational effects are negligible and hence its free surface is a spherical cap, has been the basis for work by a large number of previous authors, notably Picknett and Bexon,4 Coutant and Penski, 11 Deegan et al., 12,13 Erbil et al., 14 Hu and Larson, 15 Popov, 16 Dunn et al., 17,18 Masoud and Felske, 19 Sefiane et al., 20 Shin et al., 21 Eggers and Pismen, 22 Semenov et al., 23−25 Doganci et al., 26 Gelderblom et al., 27,28 Sobac and Brutin,29,30 Song et al., 31 Nguyen et al., 32 Nguyen and Nguyen,6,7,33,34 Talbot et al., 35 Yu et al., 36 Dash and Garimella, 8,37 Stauber et al., 9,10 Trybala et al., and Gatapova et al. 40 Referred to cylindrical polar coordinates (r,z) with origin on the substrate at the center of the droplet, its free surface, denoted by z = h(r,t), is a spherical cap with radius = t ( ) ( ≥ R ), contact radius R = R(t) (R ≥ 0), and contact angle θ = θ(t) (0 ≤ θ ≤ π) given by where sin 2 2 (1) Note that the physically relevant (i.e., the non-negative) part of h given by eq 1 is a single-valued function of r for 0 ≤ r ≤ R when 0 ≤ θ ≤ π/2 and for 0 ≤ r < R when π/2 < θ ≤ π (in which case only the "+" sign is relevant), but a double-valued function of r for ≤ < R r when π/2 < θ ≤ π (in which case the "+" and "−" signs correspond to the upper and lower hemispheres, respectively). The volume of the droplet, V = V(t), is given by Since the evaporation is assumed to be quasi-steady and limited by the diffusion of vapor in the atmosphere, the vapor concentration in the atmosphere, c = c(r,z,t), satisfies Laplace's equation, ∇ 2 c = 0, subject to the boundary conditions that at the free surface of the droplet the atmosphere is saturated with vapor and so c takes its saturation value, c sat , that far away from the droplet c takes its constant ambient value, c ∞ (0 ≤ c ∞ ≤ c sat ), and that the vapor cannot penetrate the substrate (i.e., ∂c/∂z = 0 on z = 0 for r > R). In the simplest and most widely used version of the model employed here, c sat is assumed to be constant; we will discuss the validity of this assumption in Section 5. As several previous authors (see, for example, Popov 16 ) describe, the solution for c is given by Lebedev's 41 solution for a mathematically equivalent electrostatics problem. In particular, the evaporative flux from the free surface of the droplet, J = J(r,t), defined by J = −Dn·∇c, where n is the unit outward normal to the free surface and D is the diffusion coefficient of vapor in the atmosphere, is given by where P −1/2+iτ (cosh α) denotes the Legendre function of the first kind of degree −1/2 + iτ and argument where again the "+" and "−" signs correspond to the upper and lower hemispheres, respectively, when π/2 < θ ≤ π. At leading order in the limit of small contact angle, θ → 0 + , the free surface of the droplet is a parabola, namely, h = θ(R 2 − r 2 )/ (2R), and the flux is given by Figure 1. Scaled plots of four droplets given by eq 1, each with the same volume, V, but different contact angles, namely, (a) θ = π/18 = 10°, (b) θ = π/2 = 90°, (c) θ = 17π/18 = 170°, and (d) θ = π = 180°, and different scaled contact radii, R/V 1/3 , together with the corresponding scaled evaporative flux from the free surface, JV 1/3 /(D(c sat −c ∞ )), given by eq 3 or, in the special case θ = π, by eq 7, shown by the arrows. Note that the length of the arrows is proportional to the magnitude of JV 1/3 /(D(c sat −c ∞ )), with the length of the reference arrow in panel d corresponding to JV 1/3 /(D(c sat − c ∞ )) = 1.
(see, for example, Popov 16 ). In the special case θ = π/2 the free surface of the droplet is a hemisphere with radius = R , namely, h = (R 2 − r 2 ) 1/2 , and the flux is uniform and given by (see, for example, Popov 16 ).
In the special case θ = π the free surface of the droplet is a complete sphere of radius ( ≥ 0 ) with zero contact radius, In this case the expression for the flux (eq 3) requires careful interpretation, and so it is more convenient to use Smith and Barakat's 42 solution for a mathematically equivalent electrostatics problem to obtain (after some simplification) where J 0 (·) denotes the Bessel function of the first kind of order zero. In particular, the flux from the apex of the droplet, r = 0, is given by where Catalan ≃ 0.9160 is Catalan's constant. Following Deegan et al., 12 Gelderblom et al. 28 showed that near the contact line, r = R and z = 0, the flux is given by (9) where the function A = A(θ) can, in principle, be calculated from eq 3. In particular, eq 9 shows that the behavior of the flux near the contact line depends qualitatively (rather than just quantitatively) on the value of the contact angle. Specifically, when 0 ≤ θ < π/2, then −1/2 ≤ λ < 0, and so the flux is (integrably) singular at the contact line; when θ = π/2, then λ = 0, and so the flux is finite at the contact line; and when π/2 < θ ≤ π, then λ > 0, and so the flux is zero at the contact line. Figure 1 shows scaled plots of four droplets given by eq 1 each with the same volume, V, but different contact angles, namely, θ = π/18 = 10°(typical of 0 ≤ θ < π/2), θ = π/2 = 90°, θ = 17π/18 = 170°(typical of π/2 < θ < π), and θ = π = 180°, and different scaled contact radii, R/V 1/3 , together with the corresponding scaled evaporative flux from the free surface, J V 1/3 /(D(c sat −c ∞ )), given by eq 3 or, in the special case θ = π, by eq 7, shown by the arrows. In particular, Figure 1 clearly illustrates the qualitatively different behavior of the flux near the contact line in the cases 0 ≤ θ < π/2, θ = π/2, and π/2 < θ ≤ π described above. Specifically, Figure 1 shows that the diffusion-limited model predicts that, when 0 ≤ θ < π/2, the flux is largest (theoretically infinite) at the contact line and smallest at the apex of the droplet (i.e., at r = 0), when θ = π/2 the flux is uniform and given by eq 6, and when π/2 < θ ≤ π, the flux is largest at the apex of the droplet and smallest (theoretically zero) at the contact line.

THE EVOLUTION OF THE DROPLET
Integrating the flux J given by eq 3 over the free surface of the droplet gives the total flux from the droplet at any instant, and so the rate of change of the volume of the droplet, dV/dt, is given by where V is given in terms of R and θ by eq 2, and the function g = g(θ) is defined by In particular, the function g satisfies g(0) = 16/π, g(π/2) = 2, and g ∼ (π−θ) 3 ln 2 → 0 + as θ → π − . Equation 10 determines the evolution of V, and hence the evolution of R and/or θ as the droplet evaporates, and hence, in particular, the lifetime of the droplet (i.e., the time it takes for R and/or θ and hence for V to reach zero).
As several previous authors have described, for a droplet evaporating in the CR mode, R = R 0 is constant, and so eq 10 becomes an equation for θ = θ(t), namely, which has the implicit solution while for a droplet evaporating in the CA mode, θ = θ 0 is constant, and so eq 10 becomes an equation for R = R(t), namely, which has the exact solution A key observation is that in the special case θ 0 = π (i.e., for a perfectly hydrophobic substrate) the CR and CA modes coincide at all times t. In this case, the free surface of the droplet is a complete sphere of radius = t ( ) ( ≥ 0 ) touching the substrate at the single point r = 0 with constant contact radius R ≡ R 0 = 0 and constant contact angle θ ≡ θ 0 = π throughout its entire lifetime, i.e., in this case (and only in this case) the CR and CA modes are identical throughout their entire lifetimes. Integrating the expression for the flux J in this special case given by eq 7 over the free surface of the droplet yields an equation for = t ( ) , namely, where = (0) 0 is the initial radius of the sphere, and hence To simplify the subsequent presentation it is convenient to scale time t with the maximum lifetime of a droplet evaporating in the CA mode (which occurs for θ 0 = π/2), namely, where V 0 = V(0) is the initial volume of the droplet. Setting θ = 0 in eq 13 yields an expression for the scaled lifetime of a droplet evaporating in the CR mode, namely, Setting = 0 in eq 18 or V = 0 in eq 19 (or, equivalently, taking the limit θ 0 → π − in eq 21 or 22) yields an expression for the scaled lifetime of both modes when θ 0 = π, namely, t CA = t CR = t π , where Figure 2. Evolutions of (a) the contact angle θ, (b) the scaled contact radius R/V 0 1/3 , and (c) the scaled volume V/V 0 , plotted as functions of scaled time t/ for droplets with different initial contact angles, namely, θ 0 = π/18 = 10°, θ 0 = π/2 = 90°, θ 0 = θ crit ≃ 2.5830 ≃ 148°, and θ 0 = 17π/18 = 170°, evaporating in the CR and CA modes, shown by solid and dashed lines, respectively.  10 ). Figure 2 shows the evolutions of θ, R/V 0 1/3 , and V/V 0 plotted as functions of t/ for droplets with different initial contact angles, namely, θ 0 = π/18 = 10°, θ 0 = π/2 = 90°, θ 0 = θ crit ≃ 2.5830 ≃ 148°, and θ 0 = 17π/18 = 170°, evaporating in the CR and CA modes. In particular, Figure 2 shows that in the special case θ 0 = θ crit (first identified approximately by Picknett and Bexon 4 using an approximate expression for g(θ)), the values of t CR and t CA coincide (specifically, t CR = t CA = t crit ≃ 0.9354), but the evolutions of R, θ, and V for the two modes are very different. Note that this behavior is qualitatively different from that when θ 0 = π described previously for which the two modes are identical for their entire lifetimes. Figure 3a shows the scaled lifetimes of droplets evaporating in the CR mode, t CR , given by eq 21, and in the CA mode, t CA , given by eq 22, plotted as functions of the initial contact angle θ 0 . In particular, Figure 3a illustrates the sometimes overlooked fact that the lifetime of the CR mode is (slightly) longer than that of the CA mode when θ crit < θ 0 < π (see Picknett and Bexon 4 ), and shows that both t CR and t CA approach t π from above as θ 0 → π − . In fact, analysis of the expressions for t CR and t CA given by eqs 21 and 22, respectively, in the limit θ 0 → π − reveals that and (see Stauber et al. 10 ). Figure 3b is an enlargement of Figure 3a near θ 0 = π also showing the asymptotic expressions 24 and 25, and illustrates that both asymptotic expressions, but particularly that for t CA , provide good approximations to the exact values of t CR and t CA for a reasonably wide range of values of θ 0 near π.

THE CONVERGENCE OF THE EXTREME MODES
The evolutions of θ and R/V 0 1/3 plotted in Figure 2a,b show that as the value of the initial contact angle θ 0 increases toward π both the value of the (varying) contact angle θ in the CR mode and the value of the (varying) contact radius R in the CA mode stay increasingly close to their initial values θ 0 and R 0 , respectively. In other words, the extreme modes of droplet evaporation converge as the value of θ 0 increases toward π, and so, in particular, the extreme modes become indistinguishable on strongly hydrophobic substrates.
This behavior might have been expected for the CA mode, for which θ ≡ θ 0 ≈ π necessarily remains close to π and R decreases by a small amount from its small initial value of R 0 ≈ 0 to zero, and hence necessarily remains close to its value R ≡ 0 in the special case θ 0 = π during its evolution. Indeed, analysis of eq 14 or 15 reveals that for a droplet evaporating in the CA mode, the asymptotic expression for the contact radius R is given by in the limit θ 0 → π − , which remains close to R = 0 for the entire lifetime of the droplet (i.e., until t = t π at leading order).
However, this behavior is not so immediately obvious for the CR mode, for which R ≡ R 0 ≈ 0 necessarily remains small, but θ decreases by a nonsmall amount from its initial value of θ 0 ≈ π to zero during its evolution. However, analysis of eq 12 reveals that for a droplet evaporating in the CR mode, the asymptotic expression for the contact angle θ is given by in the limit θ 0 → π − , which indeed remains close to θ = π until near the end of its lifetime (i.e., until near to t = t π at leading order). Furthermore, as the evolution of V in the case θ 0 = 17π/18 = 170°shown in Figure 2c shows, V decreases from V 0 to zero in a very similar manner for both extreme modes, and indeed for this showing the asymptotic expressions for t CR and t CA in the limit θ 0 → π − given by eqs 24 and 25, respectively.

Langmuir
Article value of θ 0 they are virtually indistinguishable. Specifically, eqs 26 and 27 reveal that, whichever mode the droplet is evaporating in, the asymptotic expression for the droplet volume V is given by The increasing accuracy of the asymptotic expressions 26−28 as θ 0 increases toward π is illustrated in Figure 4, which shows the evolutions of θ and V/V 0 for a droplet evaporating in the CR mode, and the evolutions of R/V 0 1/3 and V/V 0 for a droplet evaporating in the CA mode, plotted as functions of t/ for droplets on hydrophobic substrates with different initial contact angles, namely, θ 0 = 11π/18 = 110°, θ 0 = 13π/18 = 130°, θ 0 = 5π/6 = 150°, and θ 0 = 17π/18 = 170°, together with the corresponding asymptotic expressions in the limit θ 0 → π − given by eqs 26−28. In particular, Figure 4a,c shows that the asymptotic expressions 27 and 26 provide good approximations to the evolutions of θ and R, respectively, when θ 0 is greater than about 150°, and Figure 4b,d shows that the asymptotic expression 28 provides a good approximation to the evolutions of V in both modes for all of the values of θ 0 shown.
One important consequence of the convergence of the extreme modes is that using the exact solution for the evolution of V in the CA mode given by eq 16, which predicts the so-called "2/3 power law" that (V/V 0 ) 2/3 varies linearly with t, to extrapolate experimental data for V to estimate the lifetime of a droplet (as was done by, for example, Nguyen et al., 32 Nguyen and Nguyen, 33 and Stauber et al. 10 ), is also valid for the CR mode provided that θ 0 is sufficiently close to π. This conclusion is entirely consistent with the recent work of Nguyen and Nguyen, 34 who found that evolution of V in the CR mode is well approximated by the 2/3 power law when θ 0 is sufficiently close to π.

SUMMARY AND CONCLUSIONS
In the present work we have described the manner in which the extreme modes of droplet evaporation become indistinguishable on strongly hydrophobic substrates. In particular, we obtained simple asymptotic expressions 26−28 which provide good approximations to the evolutions of R, θ, and V, respectively, for a Figure 4. Evolutions of (a) the contact angle θ and (b) the scaled volume V/V 0 for a droplet evaporating in the CR mode, and evolutions of (c) the scaled contact radius R/V 0 1/3 and (d) the scaled volume V/V 0 for a droplet evaporating in the CA mode, plotted as functions of scaled time t/ for droplets on hydrophobic substrates with different initial contact angles, namely, θ 0 = 11π/18 = 110°, θ 0 = 13π/18 = 130°, θ 0 = 5π/6 = 150°, and θ 0 = 17π/18 = 170°, together with the corresponding asymptotic expressions in the limit θ 0 → π − given by eqs 26−28 shown with dashed lines.

Langmuir
Article wide range of hydrophobic substrates. As a consequence, on strongly hydrophobic substrates it is appropriate to use the 2/3 power law to extrapolate the lifetimes of droplets evaporating in the CR mode as well as in the CA mode.
As pointed out in Section 2, the present analysis is based on the simplest and most widely used version of the diffusion-limited model in which the saturation value of the vapor concentration, c sat , is assumed to be constant. The predictions of this version of the model have been found to be in excellent agreement with experimental results by many authors, including Picknett and Bexon, 4 Coutant and Penski, 11 Erbil et al., 14 Shin et al., 21 Doganci et al., 26 Gelderblom et al., 27 Sobac and Brutin, 29 Song et al., 31 Nguyen et al., 32 Nguyen and Nguyen, 6,33 Talbot et al., 35 Dash and Garimella, 8,37 Semenov et al., 24,25 Trybala et al., 38 and Stauber et al. 10 However, in reality, the value of c sat depends on temperature, and so in situations in which evaporative cooling of the free surface of the droplet becomes significant it may be necessary to extend the model to include variations in c sat . For example, Dunn et al. 17 and Sefiane et al. 20 used this approach to analyze the effects of the conductivity of the substrate and of the pressure in the atmosphere above the droplet. In particular, since instantaneous evaporative cooling increases with the instantaneous contact angle of the droplet (see, for example, the work of Dash and Garimella 8 ) it may become significant as the initial contact angle increases toward π (i.e., on hydrophobic and superhydrophobic substrates). In particular, two recent investigations of the evaporation of droplets of water on different superhydrophobic substrates have reached different conclusions about the significance of evaporative cooling. Whereas Gelderblom et al. 27 found good agreement between the predictions of the simplest version of the model and their experiments on a substrate with an initial contact angle of up to θ 0 ≃ 150°, Dash and Garimella 8 observed significant evaporative cooling on a substrate with an initial contact angle of θ 0 ≃ 160°a nd proposed an ad hoc modification of the simplest version of the model (namely, reducing the theoretical prediction of the total evaporative flux by an empirically determined factor of 20%) to account for it. Recently Gleason and Putnam 39 proposed another ad hoc modification of the simplest version of the model (namely, incorporating a prescribed temperature distribution of the free surface of the droplet) to account for the evaporative cooling of a droplet on a heated substrate. However, since the instantaneous evaporative cooling depends on the instantaneous geometry of the droplet, its instantaneous effect will be the same on both extreme modes. Hence we hypothesize that, even in situations such as those studied by Dash and Garimella 8 in which evaporative cooling is significant, the main result of the present work, namely, that the extreme modes become indistinguishable on strongly hydrophobic substrates, will still hold true.