Edinburgh Research Explorer On the lifetimes of evaporating droplets with related initial and receding contact angles

A physically credible relationship based on the unbalanced Young force between the initial and receding contact angles of an evaporating droplet is proposed and used to give a complete description of the lifetime of a droplet evaporating in an idealised stick-slide mode. In particular, it is shown that the dependence of the lifetime on the initial contact angle is qualitatively di ff erent from that when the relationship between the initial and receding contact angles is not taken into account. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http: // dx.doi.org / 10.1063 / 1.4935232]


I. INTRODUCTION
Droplet evaporation plays a crucial role in many practical applications (such as, for example, biochemical assays, deposition of DNA and RNA micro-arrays, deposition of pesticides, ink-jet printing, manufacture of novel optical and electronic materials, nano-wire fabrication, spray cooling, and thin film coating). As a result, the evaporation of a fluid droplet on a solid substrate has been the subject of extensive theoretical and experimental investigations by a wide range of research groups from many different countries in recent years (see, for example, the recent review articles by Cazabat and Guéna, 1 Erbil, 2 Larson, 3 and Lohse and Zhang 4 ).
One aspect of droplet evaporation that has, until recently, received relatively little attention is that of the lifetime of a droplet (i.e., the time it takes for a droplet to evaporate entirely) and, in particular, how it depends on the manner in which it evaporates. This is rather surprising, since understanding and hence optimising the lifetime of an evaporating droplet could have considerable benefits in many of the practical applications mentioned above.
After a short transient in which it rapidly adjusts to a quasi-equilibrium shape with initial contact radius R 0 and initial contact angle θ 0 , a droplet with initial volume V 0 deposited onto an ideal (i.e., perfectly smooth and chemically homogeneous) substrate at time t = 0 will, in principle, evaporate in the so-called "constant contact angle" (CA) mode in which the contact radius R = R(t) and volume V = V (t) decrease while the contact angle θ(t) = θ 0 remains constant. Of course, real substrates are never ideal, and so, in reality, the contact line of the droplet will be pinned by surface roughness and/or chemical heterogeneities for some or all of its lifetime. In the most extreme case in which the contact line always remains pinned, evaporation occurs in the so-called "constant contact radius" (CR) mode in which the contact angle θ = θ(t) and volume V = V (t) decrease while the contact radius R(t) = R 0 remains constant. As the pioneering studies of Picknett and Bexon 5 and Bourgès-Monnier and Shanahan 6 and many subsequent works have shown, in practice, the manner in which a droplet evaporates usually involves some combination of "stick" (i.e., with a pinned contact line) and "slide" (i.e., with a depinned contact line) phases. Note that we prefer to use the term "slide" rather than "slip" or "jump" to emphasise that, in general, the "stick" and "slide" phases can be of comparable duration.
A variety of "stick-slide" (SS) modes have been observed, but perhaps the most commonly reported (see, for example, the experiments described in Refs. [6][7][8][9][10][11][12][13][14] is one in which an initial stick phase is followed by a first slide phase with constant contact angle and a second slide phase in which both the contact radius and the contact angle vary. In practice, the second slide phase can be relatively short compared to the other two phases, and so, Nguyen and Nguyen, 15 Dash and Garimella, 14 and Stauber et al. 16 considered a simple but effective model for an idealised SS mode in which the second slide phase is entirely neglected and initially the droplet evaporates in a CR phase in which R = R 0 and θ(t) and V (t) decrease until θ(t) reaches the receding contact angle θ * (0 ≤ θ * ≤ θ 0 ), at which the contact line depins, and subsequently, the droplet evaporates in a CA phase in which θ(t) = θ * and R(t) and V (t) decrease to zero at time t = t SS , where t SS (which depends on both θ 0 and θ * ) denotes the lifetime of the droplet. This mode of evaporation is sketched in Figure 1. Stauber et al. 16 showed that the resulting theoretical predictions for t SS are not, as might naively have been expected, always constrained to lie between the lifetimes of the extreme (i.e., the CR and CA) modes, and, moreover, that they are in good agreement with the lifetimes measured experimentally by previous authors. In order to make this latter comparison, the values of θ 0 and θ * for each experiment (which, in general, depend of the nature of the substrate, the fluid, and the atmosphere) were taken directly from the experimental measurements. However, it is expected that in practice the values of θ 0 and θ * are related to each other, and so, in the present work, we extend FIG. 1. Sketch of the idealised stick-slide (SS) mode studied in the present work in which initially the droplet evaporates in a CR phase in which R = R 0 and θ(t) and V (t) decrease until θ(t) reaches the receding contact angle θ * (0 ≤ θ * ≤ θ 0 ) at which the contact line depins and subsequently the droplet evaporates in a CA phase in which θ(t) = θ * and R(t) and V (t) decrease to zero at time t = t SS . the analysis of Stauber et al. 16 by proposing a physically credible relationship between them based on the unbalanced Young force and use this relationship to give a complete description of t SS . In particular, we show that the dependence of t SS on θ 0 is qualitatively different from that when the relationship between θ 0 and θ * is not taken into account.

A. The diffusion-limited model
The 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 vapour in the quiescent atmosphere above it. This model, together with the assumption that the droplet is sufficiently small that gravitational effects can be neglected, is appropriate in a wide range of physical situations and so has been used with considerable success by a large number of previous authors (see, for example, Refs. 5,10,11,13,. The geometry of the mathematical model is shown in Figure 2. Referred to the cylindrical polar coordinates (r, z) shown in Figure 2, the free surface of the droplet, z = h(r,t), is a spherical cap with radius R = R(t) (R ≥ R), contact radius R = R(t) (R ≥ 0), and contact angle θ = θ(t) (0 ≤ θ ≤ π) given by (1) Note that when π/2 < θ ≤ π, the physically relevant (i.e., the non-negative) part of h given by (1) is a double-valued function of r for R ≤ r < R, with the + and − signs corresponding to the upper and lower hemispheres, respectively. The volume of the droplet, V , is given by and so, in particular, the initial volume, V 0 , is given by FIG. 2. Geometry of the mathematical model. The free surface of the droplet, z = h, is a spherical cap with radius R, contact radius R, and contact angle θ. The arrows indicate the evaporative flux from the free surface of the droplet into the quiescent atmosphere above the droplet, J .
The concentration of vapour in the atmosphere, c = c(r, z,t), satisfies Laplace's equation, ∇ 2 c = 0, subject to the boundary conditions that the atmosphere is saturated with vapour at the free surface of the droplet, i.e., c = c sat on z = h, where c sat denotes the saturation concentration, that the concentration of vapour approaches of its ambient value far from the droplet, i.e., c → c ∞ as r → ∞ for z > 0, where c ∞ (0 ≤ c ∞ ≤ c sat ) denotes the ambient concentration, and that the substrate is impenetrable to vapour, i.e., ∂c/∂z = 0 on z = 0 for r > R. In the simplest and most widely used version of the model employed here, the saturation concentration c sat is assumed to be constant, although in situations in which evaporative cooling can become significant (such as, for example, when the conductivity of the substrate is reduced, as studied by Dunn et al., 22 when the atmospheric pressure is reduced, as studied by Sefiane et al., 25 or when the contact angle becomes very large, as studied by Dash and Garimella 14 ), the model can be extended to account for the temperature dependence of c sat . As described by, for example, Popov, 21 the solution for c when c sat is constant was obtained by Lebedev, 36 who solved 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 vapour 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 with the + and − signs again corresponding to the upper and lower hemispheres, respectively, when π/2 < θ ≤ π. In particular, the diffusion-limited model predicts that when 0 ≤ θ < π/2, the flux is largest (theoretically integrably singular) 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 J = D(c sat − c ∞ )/R, and when π/2 < θ ≤ π, the flux is largest at the apex of the droplet and smallest (theoretically zero) at the contact line (see, for example, Stauber et al. 35 ).

B. The evolution of the droplet
Integrating the evaporative flux J given by (4) over the free surface of the droplet gives the total evaporative flux from the droplet at any instant, and hence (as described by, for example, Popov 21 ), the rate of change of the volume of the droplet is given by where the function g = g(θ) is defined by As the droplet evaporates, R and θ evolve according to (6) with V given in terms of R and θ by (2). In particular, Equation (6) determines the lifetime of the droplet, defined to be the time it takes for R and/or θ, and hence for V , to reach zero.

III. THE RELATIONSHIP BETWEEN θ 0 AND θ *
In this section, we propose a relationship between θ 0 and θ * based on the unbalanced Young force, specifically on the assumption that the scaled difference between the maximum pinning force and the initial pinning force is independent of both θ 0 and θ * . As we have already mentioned, a droplet deposited onto a substrate rapidly adjusts to a quasiequilibrium shape with initial contact radius R 0 and initial contact angle θ 0 . If the initial contact angle is equal to the receding contact angle, i.e., if θ 0 = θ * , then the contact line immediately depins and thereafter the droplet simply evaporates in the CA mode with θ = θ 0 = θ * constant and R decreasing. However, if the initial contact angle is greater than the receding contact angle, i.e., if θ 0 > θ * , then, as sketched in Figure 3, the contact line is pinned by a pinning force per unit length F p = F p (t) due to surface roughness and/or chemical heterogeneities of the substrate which opposes the unbalanced Young force, and the droplet begins to evaporate in a CR phase with R = R 0 constant and θ decreasing. Specifically, the horizontal force balance at the contact line reveals that F p is given by where γ, γ SV , and γ SF are the constant surface tensions of the fluid-vapour, substrate-vapour, and substrate-fluid interfaces, respectively. Note that in the special case of an ideal substrate with no pinning force, i.e., in the special case F p = 0, Equation (8) reduces to the well-known Young-Laplace equation for θ = θ 0 . As the droplet continues to evaporate, θ decreases and hence F p increases until it reaches its maximum possible value, denoted by F p max , when θ = θ * , at which instant the contact line depins, and subsequently, the droplet evaporates in a CA phase with θ = θ * constant and R decreasing. Subtracting the expression for the initial pinning force, F p (0) = γ cos θ 0 + γ SF − γ SV , from that for the maximum pinning force, F p max = γ cos θ * + γ SF − γ SV , gives a relationship between θ 0 and θ * , namely, where is the scaled difference between the maximum pinning force and the initial pinning force (hereafter simply referred to as the "maximum pinning force" for brevity). Note that, from (9), physically realisable values of f p lie in the range 0 ≤ f p ≤ 2, with the extreme values f p = 0 and f p = 2 corresponding to the case θ 0 = θ * and to the cases θ 0 = π and θ * = 0, respectively. In general, the value of f p as defined in (10) will depend on the nature of the substrate, the fluid, and the atmosphere, and so could, in principle, depend on θ 0 and/or θ * . However, in the present work, we make the simplest possible assumption regarding f p , namely, that it is independent of both θ 0 and θ * , and so, (9) provides a simple explicit expression for θ * in terms of θ 0 and f p , namely, This physically credible relationship between θ 0 and θ * is crucial to all of the results presented in the remainder of the present work, and so, Figure 4 shows θ * given by (11) plotted as a function For each value of f p , the corresponding curve is symmetric about the line θ * = π − θ 0 and intersects the θ 0 -axis at the point (θ 0min , 0) and the line θ 0 = π at the point (π, π − θ 0min ), where θ 0min = arccos(1 − f p ). For values of θ 0 smaller than θ 0min (i.e., when 0 ≤ θ 0 ≤ θ 0min ), then (11) yields θ * = 0, so that the contact line never depins and the droplet evaporates in the CR mode, while for values of θ 0 larger than θ 0min (i.e., when θ 0min < θ 0 ≤ π), then (11) yields θ * = arccos( f p + cos θ 0 ) (0 < θ * ≤ π − θ 0min ), and the droplet evaporates in the SS mode. Stauber et al. 16 (Figure 2) showed that the theoretical predictions of the present model are in rather good agreement with the lifetimes extrapolated from 29 sets of experimental data for droplets evaporating in a SS mode in which the second slide phase is smaller than 10% of the lifetime of the droplet (so that the present idealised SS mode is likely to be an appropriate description of their behaviour) obtained by previous authors. Details of these sets of experimental data are given in Table I. In order to make this comparison, the values of θ 0 and θ * for each experiment (not given by Stauber et al., 16 and so given in Table I for reference) were taken directly from the experimental measurements. In particular, in the context of the present work, this is equivalent to determining the value of f p for each experiment directly from the experimental measurements, and these values of f p (calculated from the corresponding values of θ 0 and θ * using (9)) are given in Table I for reference.

IV. THE LIFETIME OF AN EVAPORATING DROPLET
In this section, we use the relationship between θ 0 and θ * given by (11) to determine how t SS depends on θ 0 and f p . In particular, we compare t SS with the lifetimes of initially identical droplets (i.e., droplets with the same values of R 0 and θ 0 , and hence of V 0 ) evaporating in the extreme modes for the full range of all possible initial contact angles, i.e., for 0 ≤ θ 0 ≤ π.
To simplify the subsequent presentation, it is convenient to scale time t with the maximum lifetime of a droplet evaporating in the CA mode, namely, TABLE I. Details of the 29 sets of experimental data for droplets evaporating in a SS mode in which the second slide phase is smaller than 10% of the lifetime of the droplet obtained by previous authors and used in Figures 7 and 8. The values of θ 0 and θ * were taken directly from the experimental measurements and the values of f p were calculated from them using (9). In the "substrate" column, "ODTES 100 " denotes octadecyltriethoxysilane with a static contact angle of roughly 100 • for pure water, "SO 3 H" denotes 3-mercaptopropyltrimethoxysilane, "R f " denotes perfluorohexylethyltrimethoxysilane, "Oct-silicon" denotes silicon hydrophobised with octanol, "OTS-silicon" denotes silicon hydrophobised with n-octadecyltrichlorosilane in heptane, and "PDMS" denotes polydimethylsiloxane. With this scaling of time, the lifetime of a droplet evaporating in the CR mode, denoted by t CR = t CR (θ 0 ), is given by

Reference
and the lifetime of a droplet evaporating in the CA mode, denoted by t CA = t CA (θ 0 ), is given by t CA = ( 2(1 + cos θ 0 ) 2 sin θ 0 (2 + cos θ 0 ) ) 2/3 sin θ 0 (2 + cos θ 0 ) g(θ 0 ) (see, for example, Stauber et al. 16 ). Note that, as a consequence of scaling (12), t CA given by (14) attains its maximum value of unity at θ 0 = π/2. The lifetimes of droplets evaporating in the extreme modes given by (13) and (14) are, by definition, independent of θ * and hence of f p . However, as we have already seen, the lifetime of a droplet evaporating in the SS mode with θ * given by (11) depends, in general, on f p as well as on θ 0 , i.e., t SS = t SS (θ 0 , f p ).
If θ 0min < θ 0 ≤ π, then the droplet evaporates in the SS mode with θ * = arccos( f p + cos θ 0 ), and hence, its lifetime is the sum of the duration of the CR phase (i.e., the time it takes for θ to decrease from θ 0 to θ * with R = R 0 ) and the duration of the CA phase (i.e., the time it takes for R to decrease from R = R 0 to R = 0 with θ = θ * ) and hence is given by (see, for example, Stauber et al. 16 ). Note that (15) reduces to t SS = t CR when θ * = 0, and so it is, in fact, valid for all values of θ 0 . Figure 5(a) shows t SS given by (15) plotted as a function of θ 0 (0 ≤ θ 0 ≤ π) for a range of values of f p , together with the lifetimes of initially identical droplets evaporating in the extreme modes, t CR and t CA , given by (13) and (14), respectively. For clarity, Figure 5(b) shows the behaviour in the range π/2 ≤ θ 0 ≤ π in greater detail. Note that (as first pointed out by Picknett and Bexon 5 ) the lifetimes (but not the evolutions) of the extreme modes coincide at θ 0 = θ crit ≃ 2.5830, where t CR = t CA = t crit ≃ 0.9354, and that (as described by Stauber et al. 35 ) the extreme modes become indistinguishable in the limit θ 0 → π − and so, in particular, that t SS = t CR = t CA = t π = (4 1/3 log 2) −1 ≃ 0.9088 at θ 0 = π.
The most striking feature of Figure 5 is that the shapes of the curves representing t SS are qualitatively different from those obtained by Stauber et al. 16 (Figures 3 and 4) in the case when θ 0 and θ * are independent parameters. Specifically, as Figure 5 shows, whatever the value of f p , the curves representing t SS coincide with the curve representing t CR for 0 ≤ θ 0 ≤ θ 0min depart from it with vertical slope at θ 0 = θ 0min according to as θ 0 → θ 0min + , increase to a local maximum at a value of θ 0 in the range π/2 ≤ θ 0 ≤ π (marked with a dot in Figure 5(b)), and then decrease, ultimately reaching the value t SS = t π with zero slope at θ 0 = π. For contrast, recall that, unlike the present curves, the corresponding curves obtained by Stauber et al. 16 coincide with the curve representing t CA (not t CR ) when 0 ≤ θ 0 ≤ θ * and depart from it with zero (not vertical) slope at θ 0 = θ * . However, also recall that, like the present curves, the corresponding curves obtained by Stauber et al. 16 always lie below t CA (π/2) = 1 and always lie on or above the minimum of the two extreme modes (i.e., t SS ≥ min(t CR ,t CA ) for 0 ≤ θ 0 ≤ π), but may lie above the maximum of the two extreme modes when π/2 < θ 0 < π. Moreover, the total envelope of the present curves as f p varies between 0 and 2 is the same as the total envelope of the curves obtained by Stauber et al. 16 as θ * varies between 0 and π.
As Figure 5 illustrates, in the limit f p → 0 + , then θ 0min → 0 + , and the curve representing t SS approaches that representing t CA from below for 0 < θ 0 ≤ π/2 and from above for π/2 < θ 0 < π according to t SS = t CA + ( 2(1 + cos θ 0 ) 2 sin θ 0 (2 + cos θ 0 ) where a dash ( ′ ) denotes differentiation with respect to argument. Furthermore, as Figure 5 also illustrates, in the limit f p → 2 − , then θ 0min → π − , and the curve representing t SS converges to that representing t CR from above in the vanishingly small range θ 0min < θ 0 < π. In addition, as Figure 5(b) illustrates, for most values of f p , the local maximum of t SS is also its global maximum, but for values of f p sufficiently close to 2 (specifically, for values of f p in the range 1.9046 ≤ f p ≤ 2), the global maximum of t SS is t SS = t CR = t CA = t crit at θ 0 = θ crit . As Figure 5(b) also illustrates, for most values of f p , the curves representing t SS lie above the curve representing t CR near θ 0 = π, but for values of f p in the range 0 ≤ f p ≤ f p π , where f p π ≃ 0.1520, they lie below it (but still above the curve representing t CA ).
In summary, Figure 5 shows that for sufficiently small values of θ 0 (i.e., for values in the range 0 ≤ θ 0 ≤ θ 0min ), the droplet evaporates in the CR mode and has lifetime t SS = t CR (θ 0 ), while for larger values of θ 0 (i.e., for values in the range θ 0min < θ 0 ≤ π), the droplet evaporates in the SS mode and has lifetime t SS = t SS (θ 0 , f p ) which is never less than both t CR and t CA .
Note that the present curves are qualitatively much more similar (but still not identical) to that tentatively suggested by Shanahan et al. 37 than those obtained by Stauber et al. 16 In particular, Shanahan et al. 37 used an ad hoc approximation to the diffusion-limited model (see Nguyen and Nguyen 29 for further discussion of this) and hypothesised the existence of a sigmoidal curve representing "a transition between the two [extreme] regimes over a range of intermediate values of θ 0 [approximately 20 • ≤ θ 0 ≤ 45 • ], corresponding to the change between pinning and (virtually) continuous triple line [i.e., contact line] recession" but that "its position, however, is not known." In particular, while the present curves depart from the curve representing t CR at θ 0 = θ 0min and cross the curve representing t CA (albeit always at a value of θ 0 in the range π/2 < θ 0 < π), they are not sigmoidal and have a considerably more complicated structure than that envisaged by Shanahan et al. 37

V. MASTER DIAGRAM
Since the results presented in Figure 5 are fairly complicated, we follow the approach of Stauber et al. 16 ( Figure 5) and summarise all of the possible relationships between the lifetimes of initially identical droplets evaporating in the CR, CA, and SS modes in the master diagram shown in Figure 6, which shows how the θ 0f p parameter plane is divided up into regions in which the six possible orderings of t CR , t CA , and t SS occur. Four of these six regions, namely, region I, which corresponds to t CR < t SS < t CA , region II, which corresponds to t CR < t CA < t SS , region III, which corresponds to t CA < t CR < t SS , and region IV, which corresponds to t CA < t SS < t CR , appear in the corresponding diagram of Stauber et al. 16 and so are labelled in the same way. The other two regions, namely, region VII, which corresponds to t CA < t SS = t CR , and region VIII, which corresponds to t SS = t CR < t CA , correspond to θ * = 0 and hence collapse onto the θ 0 -axis in the corresponding diagram of Stauber et al. 16 and so appear here for the first time. Similarly, two additional regions found by Stauber et al., 16 namely, region V, which corresponds to t SS = t CA < t CR , and region VI, which corresponds to t CR < t SS = t CA , correspond to f p = 0 and hence collapse onto the θ 0 -axis in Figure 6 and so do not appear here. As Figure 6 shows, as f p increases from 0 to 2, region IV disappears as f p passes through the critical value f p π ≃ 0.1520, and regions I and II disappear and region VII appears as f p passes through the critical value f p crit , where f p crit is the value of f p at which θ 0min = θ crit , i.e., f p crit = 1 − cos(θ crit ) ≃ 1.8480. In particular, Figure 6 confirms that t SS is never less than both t CR and t CA and shows that regions I and IV (i.e., the regions in which t SS lies between t CR and t CA ), regions II and III (i.e., the regions in which t SS is greater than both t CR and t CA ), and regions VII and VIII (i.e., the regions in which t SS = t CA ) all occupy substantial proportions of parameter space.

VI. DISCUSSION
In Sec. III, we proposed a physically credible relationship between θ 0 and θ * based on the assumption that f p is independent of both θ 0 and θ * , and in Secs. IV and V, we used it to give a complete description of the lifetime of a droplet evaporating in the idealised SS mode. In particular, we showed that the dependence of t SS on θ 0 is qualitatively different from that described by Stauber et al. 16 when the relationship between θ 0 and θ * is not taken into account and is qualitatively much more similar (but still not identical) to that tentatively suggested by Shanahan et al. 37 As we have already mentioned, in general, the value of f p depends on the nature of the substrate, the fluid, and the atmosphere. In the light of all these, what is most immediately striking about the values of f p presented in Table I is how surprisingly similar they are, despite the fact that they are from different experiments by different authors on a variety of substrates with a range of surface roughnesses and chemical heterogeneities using several different fluids. In order to investigate this in more detail, Figure 7 shows θ * plotted as a function of θ 0 obtained from the 29 sets of experimental data listed in Table I. Figure 7 also includes a solid curve showing relationship (11) with the value of f p that best fits all 29 sets of experimental data, namely, f p = 0.2005. The corresponding value of θ 0min is θ 0min ≃ 0.6443 ≃ 37 • , and hence, the corresponding values of θ * vary between θ * = 0 • and θ * = π − θ 0min ≃ 2.4973 ≃ 143 • . The quality of the fit is confirmed by FIG. 7. The receding contact angle θ * plotted as a function of the initial contact angle θ 0 obtained from the 29 sets of experimental data listed in Table I. The values from the two experiments by Bourgès-Monnier and Shanahan 6 are denoted by inverted triangles (▼) rather than by squares (■). The solid curve shows relationship (11) with the value of f p that best fits all 29 sets of experimental data, namely, f p = 0.2005.  Table I plotted as a function of the initial contact angle θ 0 . The lifetimes from the two experiments by Bourgès-Monnier and Shanahan 6 are denoted by inverted triangles (▼) rather than by squares (■). Also shown are the theoretical predictions for the lifetimes of initially identical droplets evaporating in the CR, CA, and SS modes, t CR , t CA , and t SS , the latter calculated using f p = 0.2005.
an R 2 value of R 2 = 0.9676, and by the fact that even if the two experiments by Bourgès-Monnier and Shanahan 6 with the largest known roughness values and two of the three largest values of f p in Table I (denoted by inverted triangles rather than by squares in Figure 7) are excluded, then the value of f p that best fits all of the remaining 27 sets of experimental data decreases only slightly to 0.1858 and the corresponding R 2 value rises only slightly to R 2 = 0.9800. Figure 8 shows the lifetimes of the droplets extrapolated from the 29 sets of experimental data listed in Table I plotted as a function of θ 0 . Also shown are the theoretical predictions for the lifetimes of initially identical droplets evaporating in the CR, CA, and SS modes, t CR , t CA , and t SS , the latter calculated using f p = 0.2005. In particular, Figure 8 shows that the theoretical prediction for t SS using this single value of f p is reasonably close to all 29 of the experimentally determined lifetimes (even those from the two experiments by Bourgès-Monnier and Shanahan 6 ), and that it captures the qualitative behaviour of the experimental results surprisingly well. Despite this impressive level of agreement, we do not, of course, seek to claim that this single value of f p will be appropriate in all situations. Rather, we simply wish to point out that the unexpected insensitivity of the experimentally determined lifetimes to the surface roughness and chemical heterogeneity of the substrates revealed in the present work highlight the need for further theoretical and experimental work on the nature of contact line pinning and depinning on non-ideal substrates.