Article pubs.acs.org/Langmuir
Transient Volume of Evaporating Sessile Droplets: 2/3, 1/1, or Another Power Law? Tuan A. H. Nguyen and Anh V. Nguyen* School of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia ABSTRACT: The transient shape and volume of evaporating sessile droplets are critical to our understanding and prediction of deposits left over on the solid surface after droplet evaporation. The 2/3 power law of scaling, (V/Vo)β = 1 − t/tf with β = 2/ 3, has been widely used. The 1/1 power law of scaling with β = 1 was also obtained for vanishingly small contact angles. Here we show that β significantly deviates from 2/3 and 1 when the droplet base is pinned: β depends on both initial and transient contact angles. The 1/1 power law presents the upper limit of β = 1, while β = 2/3 is the lower limit if contact angles are smaller than 148°. Unexpectedly, β can be smaller than 2/3 if contact angles are larger than 148°. We also present a semianalytical approximation for β as a function of the initial contact angle.
1. INTRODUCTION Research into the physics of droplet evaporation has been intensively conducted over the last two decades. It is shown that the evaporation of a sessile droplet strongly depends on the physical and chemical properties of solid surfaces on which the three-phase contact line (TPCL) follows the pinned and/or receding mode. These distinctive evaporation modes were classified as the single constant contact radius (CCR) and the constant contact angle (CCA) modes by Picknett and Bexon1 in 1977. Given that solid surfaces are usually rough and that contact angle hysteresis does normally exist, the CCR mode usually takes a large portion of the droplet lifetime. Additionally, the accumulation of nonvolatile components at the droplet edge due to the well-known coffee-ring effect can pin the TPCL for an extended time.2−4 It is therefore important to better understand the evaporation kinetics by the CCR mode for a full range of contact angles. A closed-form expression for the transient residual droplet volume can be obtained for the single CCA mode. Accordingly, the droplet volume raised to the power of 2/3 linearly decreases with increasing evaporation time,1,5−7 which is referred to as the 2/3 law here. However, for the CCR mode, a closed-form expression for droplet volume versus time cannot generally be obtained because of the transcendentally complicated relationship between the transient contact angle and evaporation time.5 Furthermore, in the limit of small initial contact angles, the approximate equations5,8 show that droplet volume in the CCR mode decreases almost linearly with time (referred to as the 1/1 law here). This linear dependence was also experimentally9,10 and numerically7 confirmed for small initial contact angles. For very hydrophobic surfaces (with an initial contact angle larger than approximately 125°), it is shown that the 2/3 law predicts the change in transient volume or mass more accurately than does the 1/1 law.11,12 These © 2014 American Chemical Society
results also show that the evaporation rate is slower on the superhydrophobic surfaces because of geometric constriction at the droplet edge and the evaporative cooling effect due to the presence of air gaps. Furthermore, available experimental data for droplet evaporation on a superhydrophobic surface with a pinned TPCL13 gives reasonable agreement up to 70% and starts to deviate from the 2/3 law at about 30% droplet lifetime. Additionally, the experimental data for transient volume, located above the curve of the 2/3 law, suggest that the exponent of a power law, if it exists, is larger than 2/3.12 These disagreements in predicting transient volumes of a sessile droplet cannot be solved unless the dewetting dynamics of the triple line and the initial contact angle are considered in the modeling. In this article, we perform a theoretical analysis and present a simple power law which can cover both the CCR and CCA evaporation modes. Theoretical predictions will then be compared with available experimental results.
2. THEORETICAL ANALYSIS The volume of a spherical capped-shape sessile droplet with contact angle θ and contact base radius R is described by V = πR3f (θ ) f (θ ) =
(1)
(1 − cos θ )2 (2 + cos θ ) 3 sin 3 θ
(2)
The transient volume of sessile droplets evaporating under the diffusion-controlled condition is approximated to a power law which can generally be described by the following equation Received: October 2, 2013 Revised: May 4, 2014 Published: May 16, 2014 6544
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⎛ V ⎞β t ⎜ ⎟ =1− tf ⎝ Vo ⎠
linear dependency was experimentally,9,10 numerically,7 and semianalytically8 reported for small initial contact angles as discussed earlier in the Introduction. For the CCR mode with large contact angles, the evaporation time and droplet lifetime obtained by integrating eq 7 can be inserted into eq 5 to give
(3)
where V0 is the initial droplet volume and tf is the total evaporation time (i.e., the droplet lifetime). The exponent β of the power-law scaling in eq 3 can be established as follows. Inserting eq 1 into eq 3 gives ⎛R⎞ ⎜ ⎟ ⎝ Ro ⎠
3β
=1−
t tf
θ dx ⎤ ⎡ ∫θ o g (x) ⎢ ln⎢1 − θ dx ⎥⎥ o ∫o ⎣ g (x) ⎦ β= ⎡ f (θ) ⎤ ln⎢⎣ f (θ ) ⎥⎦ o
(CCA case) (4)
⎡ f (θ ) ⎤ β t ⎢ ⎥ =1− tf ⎣ f (θo) ⎦
Equation 11 convincingly shows that β is no longer equal to 1 or 2/3 but strongly depends on the contact angles. Further analysis of the numerical results reveals that β is confined between 1 and 2/3 only if the contact angle is smaller than 148° and can be smaller than 2/3 if the contact angle is larger than 148°. The dependence of β on the surface hydrophobicity (θ0) can be established by taking the average over the interval [0, θ0], which gives
(CCR case) (5)
where R0 and θ0 are the initial contact base radius and the initial contact angle, respectively. The changes in the transient contact angle and base radius can be established from the mass conservation and mass transfer processes, governing natural evaporation by the vapor diffusion of sessile droplets. Under the diffusion-controlled conditions, the transient contact base radius for the CCA mode and the transient contact angle for the CCR mode can be predicted as follows5−7 R2 = R o2 +
g (θ) 2D(C∞ − Cs) t ρ sin θ(2 + cos θ )
β ̅ (θ0) = (CCA case)
(CCR case)
(7)
where g(x) = (1 + cos x) {tan(x/2) + xτ)/(sinh 2πτ)) tanh[(π − x)τ] dτ}, C∞ is the constant vapor concentration far away from the droplet surface (at infinity), Cs is the vapor saturation concentration (at the droplet surface), ρ is the liquid density, and D is the vapor diffusion coefficient. For the CCA mode, using eq 6 to calculate the droplet lifetime we obtain ⎛ R ⎞2 t ⎜ ⎟ =1− tf ⎝ Ro ⎠
θ0
β dθ
(12)
(CCA case) (8)
Comparing eqs 4 and 8 gives β = 2/3. The slope of this linear decrease of (V/Vo)2/3 with normalized time, the 2/3 law, has been widely used to determine the evaporation rate.1,6,14,15 The wide use of the 2/3 law11,14,16−20 for droplets evaporating in the CCA mode has led to its erroneous use for droplets evaporating in other modes. For the CCR evaporation mode, eq 7 cannot be further analytically integrated. In the limit of a small contact angle, the right-hand side of eq 7 can be expanded into a Maclaurin series. The following predictions can be obtained if the first term of the series is obtained:
θ t =1− θo tf
Figure 1. Exponent β of the power law, (V/V0)β = 1 − t/tf, for the transient volume of sessile droplets evaporating by the CCR mode versus the transient contact angle, θ, and initial contact angle, θ0. β is described by eq 11. The curves are plotted for different initial contact angles at a step of 10°. The top left corner of the shaded area is the 1/ 1 law. The dashed line at the bottom represents the 2/3 law.
(9) 3
∫0
3. RESULTS AND DISCUSSION Figure 1 shows the numerical results of eq 11 for exponent β of the power law, (V/V0)β = 1 − t/tf. β is a function of both the
8∫ ∞ 0 ((cosh
2
1 θ0
where β is given by eq 11.
(6)
D(C∞ − Cs) dθ = g (θ ) dt ρR2
(11)
initial and transient contact angles. It increases with decreasing transient contact angle and increasing evaporation time. β = 1 for the case of a pinned contact line with a vanishingly small initial contact angle present at the upper bound of exponent β. It is noted that β approaches 2/3 in the limit as θ → θ0 = 148°. When θ0 > 148°, the β curve lightly undergoes 2/3 in the early stage of evaporation, reaching a minimum value of βmin = 0.6566 at θ0 = 158.75°, and then returns to 2/3 in the limit as θ
2
sin θo (1 − cos θ ) (2 + cos θ ) f (θ ) θ = + O[θθo] = 3 2 (2 cos ) f (θo) + θ θ sin θ (1 − cos θo) o o (10)
Comparing eqs 5, 9, and 10 shows that β = 1, corresponding to the 1/1 law, with the normalized droplet volume decreasing linearly with normalized time when the TPCL is pinned. This 6545
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→ θ0 = 180°. The special value of initial contact angle θ0 = 148° is in accordance with previous studies on the droplet lifetime.2,21,22 Picknett and Bexon1 were the first authors to identify qualitatively, although not quantitatively as they used only an approximate expression for g(θ), the special value of the contact angle of 148° at which the droplet lifetime for the CCA mode crosses the droplet lifetime for the CCR mode. The special value of θ0 = 148° is close to the critical value of approximately 150° for the transition to superhydrophobic surfaces,23 which also displays a deviation of the droplet lifetime from prediction by the 2/3 law.11,12 It is our conjecture that this special value of θ0 = 148° presents the theoretical transition to the superhydrophobicity of solid surfaces. Evidently, superhydrophobic surfaces can affect the evaporation of sessile droplets by a different mechanism with the air pockets underneath the droplet and the possible five droplet states on superhydrophobic surfaces, namely, the Wenzel state, the Cassie−Baxter state, the transitional state between the Wenzel and Cassie−Baxter states, the lotus state, and the gecko state. The future evaporation models for sessile droplets on superhydrophobic surfaces would have to take these special features into consideration. Figure 2 shows the dependence of the averaged β on the initial contact angle θ0 as described by eq 12. Accordingly, the
Figure 3. Comparison of model predictions (lines) with experimental data (points)2 for the transient volume of a water droplet (on a silicon wafer made hydrophobic by esterification) evaporating by the CCR mode. β for the analytical and approximate predictions is calculated using eqs 11 and 13, respectively. The initial contact angle of θo = 80° is extracted from the experimental data.
data can be described very well by the theoretical models for both the analytical and approximate predictions for up to 90% of the droplet lifetime. Near the end of the droplet existence, evaporation occurs by the mixed mode2 where the condition of a constant contact radius is no longer valid and therefore the experimental data deviate from the theoretical curves during the last 10% of the droplet lifetime. Finally, we note that this article deals only with the single CCA and CCR modes. The lifetime of evaporating sessile droplets in various modes is dealt with in recent papers. In particular, in the limit of small initial contact angle, Stauber et al.24 have shown that the lifetime of the CCA mode is longer than the lifetime of the CCR mode by a factor of 3/2. Furthermore, the CCA and CCR modes are identical in the limit of droplets touching the solid surface (similar to the evaporation of free droplets), i.e., the initial and transient base radii are equal to zero, and the initial and transient contact angles are equal to 180°.
Figure 2. Dependence of averaged exponent, β̅, on initial contact angle, θ0. Points show the prediction by eq 12. The line presents the ). fitting by β̅ = 2 − exp(0.0722θ1.242 0
averaged exponent of the power law is now simply a function of the initial contact angle. We propose the following semianalytical approximation for the averaged exponent: β̅ = 2 − exp[a(θ0)n], where a and n are the model parameters. The approximation approaches the exact value of β = 1 in the limit as θ0 → 0. For sessile droplets with initial contact angles being smaller than 148°, the two model parameters can be obtained by fitting, giving β ̅ = 2 − exp(0.0722θ01.242)
4. CONCLUSIONS A theoretical analysis has been performed to assess the feasibility of using the simple power law, (V/Vo)β = 1 − t/tf, in describing the transient volume of sessile droplets evaporating by either the CCR or CCA mode within the diffusion-controlled regime. Exponent β = 2/3, or the 2/3 law, is applied to evaporation by the CCA evaporation mode. For the CCR mode, exponent β generally ranges between 1 (small contact angles) and 2/3 (large contact angles but smaller than 148°). A semianalytical approximation described by eq 13 is established for the averaged exponent as a function of the initial contact angle (surface hydrophobicity). When contact angles are larger than 148°, β is smaller than 2/3 and has a minimum of βmin = 0.6566 at θ0 = 158.75°. β returns to 2/3 in the limit as θ → θ0 = 180°.
(13)
where θ0 is given in radians. This simple function is useful for engineering calculation exercises. This approximation function slightly underestimates the averaged exponent for the evaporation occurring on hydrophobic surfaces of θ0 > 148°. Shown in Figure 3 is the comparison of the experimental results with the predictions by the power law described by eq 3. During the evaporation by the CCR mode, the experimental 6546
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(19) Dash, S.; Garimella, S. V. Droplet evaporation dynamics on a superhydrophobic surface with negligible hysteresis. Langmuir 2013, 29, 10785−10795. (20) Hampton, M. A.; Nguyen, T. A. H.; Nguyen, A. V.; Xu, Z. P.; Huang, L.; Rudolph, V. Influence of surface orientation on the organization of nanoparticles in drying nanofluid droplets. J. Colloid Interface Sci. 2012, 377, 456−462. (21) Stauber, J. M.; Wilson, S. K.; Duffy, B. R.; Sefiane, K. Comment on Increased Evaporation Kinetics of Sessile Droplets by Using Nanoparticles. Langmuir 2013, 29, 12328−12329. (22) Nguyen, T. A. H.; Nguyen, A. V. Reply to Comment on Increased Evaporation Kinetics of Sessile Droplets by Using Nanoparticles. Langmuir 2013, 29, 12330. (23) Wang, S.; Jiang, L. Definition of superhydrophobic states. Adv. Mater. 2007, 19, 3423−3424. (24) Stauber, J. M.; Wilson, S. K.; Duffy, B. R.; Sefiane, K. On the lifetimes of evaporating droplets. J. Fluid Mech. 2014, 744, R2.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +61 7 336 53665. Fax: +61 7 336 54199. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
We gratefully acknowledge the University of Queensland for the international postgraduate scholarship awarded to T.A.H.N. This research was supported under the Australian Research Council’s Linkage Projects funding scheme (project number LP0989217).
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