Resolving an ostensible inconsistency in calculating

Advances in Colloid and Interface Science 243 (2017) 121–128

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Resolving an ostensible inconsistency in calculating the evaporation rate of sessile drops S.F. Chini a,⁎, A. Amirfazli b a b

Department of Mechanical Engineering, University of Tehran, Tehran 1417613131, Iran Department of Mechanical Engineering, York University, Toronto, ON, Canada

a r t i c l e

i n f o

a b s t r a c t This paper resolves an ostensible inconsistency in the literature in calculating the evaporation rate for sessile drops in a quiescent environment. The earlier models in the literature have shown that adapting the evaporation flux model for a suspended spherical drop to calculate the evaporation rate of a sessile drop needs a correction factor; the correction factor was shown to be a function of the drop contact angle, i.e. f(θ). However, there seemed to be a problem as none of the earlier models explicitly or implicitly mentioned the evaporation flux variations along the surface of a sessile drop. The more recent evaporation models include this variation using an electrostatic analogy, i.e. the Laplace equation (steady-state continuity) in a domain with a known boundary condition value, or known as the Dirichlet problem for Laplace's equation. The challenge is that the calculated evaporation rates using the earlier models seemed to differ from that of the recent models (note both types of models were validated in the literature by experiments). We have reinvestigated the recent models and found that the mathematical simplifications in solving the Dirichlet problem in toroidal coordinates have created the inconsistency. We also proposed a closed form approximation for f(θ) which is valid in a wide range, i.e. 8° ≤θ ≤131°. Using the proposed model in this study, theoretically, it was shown that the evaporation rate in the CWA (constant wetted area) mode is faster than the evaporation rate in the CCA (constant contact angle) mode for a sessile drop. © 2016 Elsevier B.V. All rights reserved.

Available online 4 June 2016 Keywords: Evaporation Sessile drops Micro-liter drops Electrostatic analogy Evaporation flux variation Diffusion

1. Introduction Evaporation of micro-liter drops at room temperature, normal atmospheric condition and in quiescent environment, is considered not to be limited by the transfer rate of molecules across the liquid-vapor interface (phase change), but by the transfer from the drop surface to the surrounding (vapor transport) [1]. Transport of vapor from drop surface to the surrounding may potentially be attributed to diffusion, convection, or both [2,3]. Solving the mass balance equations, Langmuir [4] showed for micro-liter drops, assuming a pure-convective driven evaporation results in a linear relationship between the evaporation flux and drop radius squared; while assuming a pure-diffusive driven evaporation results in a linear relationship between the evaporation flux and drop radius. As through experiments, Langmuir [4] and proceeding researchers observed the latter case, they came to the conclusion that evaporation of a non-volatile micro-liter sessile drop in room temperature is governed by diffusion and not convection e.g. [2,3,5–11]. It was also assumed that such evaporation is a steady-state process. The steady-state and non-convective assumptions (i.e. Maxwell assumptions [12]) are widely used in literature for finding the evaporation rate of sessile drops [6,7,9,10,13–23]. As shown in our previous study [24], evaporation of micro-liter drops is neither steady-state, nor purely-diffusive, in general. As shown in [24], the error in using Maxwellian model depends on the value of wv⋄eq (ratio of molar vapor density to molar air-vapor mixture density in a vapor saturated air). It should be noted that for drops in this study the value of wv⋄eq is b 0.01 which means using Maxwellian model results in b 1% error in calculating the evaporation flux on the free surface of a suspended drop [24]. Adapting Maxwellian models to sessile drops, but for now ignoring the solid substrate effect (i.e. evaporation flux on free surface of a sessile drop is assumed equal to the evaporation flux of a suspended drop with an equal radius of curvature), and knowing that the surface area of a spherical cap 2

drop (S) can be calculated as 1þ2πa cosθ, the total evaporation from a sessile drop (J) would be [21]: J ¼ ρ⋄L

sin θ ∂V ¼ J⋄v : S ¼ −2πaρ⋄ D w⋄v eq −w⋄v ∞ 1 þ cos θ ∂t

⁎ Corresponding author. E-mail addresses: [email protected] (S.F. Chini), [email protected] (A. Amirfazli).

http://dx.doi.org/10.1016/j.cis.2016.05.015 0001-8686/© 2016 Elsevier B.V. All rights reserved.

ð1Þ

122

S.F. Chini, A. Amirfazli / Advances in Colloid and Interface Science 243 (2017) 121–128

); V is the drop volume (m3); t is time (s); J⋄v is the molar flux of vapor (mol m2 s ); S is drop surface area (m2); a is the drop wetted contact radius (m); ρ⋄ is the molar density of air-vapor mixture (mol m3 ); D is the binary diffusion constant of vapor into air . 2 Þ; wv⋄eq and wv⋄∞ are ρ⋄v/ρ⋄ in a thin shell surrounding the liquid and at far afield, respectively; ρ⋄v is the molar density of vapor (mol m3 ), and θ is the (m where ρ⋄L is the liquid molar density (mol

m3

s

drop's contact angle. It can be assumed that wv⋄eq is equal to the density ratio of vapor to air-vapor mixture in a vapor saturated air [25]. It is argued that the presence of a solid substrate changes the evaporation rate values from Maxwellian models (e.g. [10,20]) by f(θ), see Eq. (2). J ¼ ρ⋄L

sin θ ∂V f ðθÞ f ðθÞ ¼ J⋄v :Sf ðθÞ ¼ −2πaρ⋄ D w⋄v eq −w⋄v ∞ 1 þ cos θ ∂t

ð2Þ

Literature studies have used different approaches for finding the value of f(θ), e.g. differential mass balance [9,21], transforming into an electrostatic problem [20], or fitting experimental values [26]. Picknett and Bexon [20] is the only study in the literature which provides an exact solution for f(θ). They solved an electrostatic analogy (evaluating the capacitance of an isolated conducting body with the same size and shape as a drop) using Snow's series solution, and showed that in presence of a solid substrate evaporation rate changes as [20]:

f ðθÞPicknett & Bexon ¼

8 > > > > > > > > >
> > πθ 2 πθ 3 πθ 4 πθ > > > þ 0:1160 180 0:00008957 þ 0:6333 −0:08878 180 þ 0:01033 180 > > 180 > : ; 10° ≤ θ ≤ 180° 1− cos θ

ð3Þ

where θ is in degrees, and the total evaporation rate (J) can be found by substituting Eq. (3) for f(θ) in Eq. (2). For other approximate relations for f(θ) see [9,10,21]. Note that in [20] or other studies mentioned above, there was no explicit reference to variation of evaporation flux along the surface of a sessile drop; f(θ) was thought of a term to modify the total evaporation rate while evaporation flux was deemed uniform along the drop free surface (similar to a suspended spherical drop). A number of studies, starting approximately 15 years ago, found or argued that for sessile drops, the evaporation flux changes from a maximum value at the contact line to a minimum value at the drop apex, see Fig. 1, e.g. [27–36]. In the meantime, many have tried to capture this variation and include it into their evaporation models. The problem here is that the recent models which capture the evaporation variation along the drop surface (e.g. [35]) provide results which are inconsistent with Eqs. (2) and (3) (e.g. older models given in [20]). This has divided the literature into two streams, and of course the majority of these works support the recent models (e.g. in [28,29]), with the logic that the evaporation flux variation was not explicitly mentioned in the early literature models e.g. [20]. The aim of this paper is to investigate the cause(s) that have created the inconsistencies between the modeling approaches in the literature. Models which quantify the evaporation variation along the drop surface are based on electrostatic analogies (see Appendix I). In electrostatic analogies, the vapor concentration and evaporation flux are related to electrostatic potential and electrostatic field, respectively [35]. The two types of electrostatic analogy found in literature are: (i) the problem of finding the electrostatic fields and charge densities in two-dimensional corners and along the edges [37], and (ii) the Dirichlet problem for a domain bounded by two intersecting spheres, or capacitance of an equiconvex lens [38]. The first electrostatic analogy (mentioned in [36]) is for flat conducting surfaces. As such, its use is limited to small contact angles, i.e. when the drop surface is nearly flat. The second electrostatic analogy (used in [2,3,27–35,39]) is valid for spherical cap geometries. The spherical cap geometry is not a major limitation for small drops (b2 mm), since drops smaller than 2 mm have spherical cap shape (capillary length for pure water at standard temperature and pressure is approximately 2 mm). As such, the Dirichlet analogy is an appropriate analogy for the evaporation of micro-liter sessile drops. This analogy problem is relatively complicated [38] and so far, it has been solved with some approximations and simplifications e.g. [35] or [41]. For example, Popov [41] solved the electrostatic analogy assuming that the contact line of the drop remains pinned during the evaporation, the initial contact angle is small, and the contact angle linearly decreases in time.

Fig. 1. Variation of evaporation flux along the drop surface is shown. J⋄v(x) is the evaporation flux at x. Evaporation flux is at its maximum at the contact line. It is assumed that a drop has a spherical cap shape with radius R.


123

With a closer look one can understand that the electrostatic analogy problem in [20] is fundamentally similar to the Dirichlet problem in [35], as both study the electric field of two intersecting spherical capacitances. The difference is that they are presented in different coordinate systems and in [20], the analogy is used to find a relation for f(θ), see Eq. (3); whereas in [35] it is used to model the variation of evaporation flux along the drop free surface. So, the two models should provide consistent evaporation rate results; however, they do not. It will be shown in this study that the causes for the inconsistency between the two streams in the literature are the simplifications and assumptions taken to solve the Dirichlet problem in toroidal coordinates.

2. Dirichlet electrostatic analogy Considering a steady state process and assuming a constant D, continuity equation takes the form of Laplace's equation, i.e. Δw⋄v = 0. The Dirichlet problem for solving the Laplace's equation consists of finding a solution on a domain with a known boundary condition, as a known function. The case where vapor concentration on the surface of drop is constant, fits the description of the Dirichlet problem. To investigate the drop evaporation using the Dirichlet problem (in an electrostatic analogy), one can use toroidal coordinates for mathematical convenience. For this case the toroidal coordinates coincide with the orthogonal coordinates that match the shape of the drop (one of the coordinate surfaces coincides with the surface of the drop). The equivalent electrostatic analogy for this problem is a conductor of the shape of a double-convex lens (analogue to the drop plus its reflection in the plane of the substrate). Surface of the lens is kept at a constant potential (analogue to the constant vapor concentration on the surface of the drop). As we need to calculate the vapor concentration gradient outside of the domain, the exterior Dirichlet problem should be considered. The exterior Dirichlet problem in toroidal coordinates is extensively discussed in [38]. Fig. 2 shows the toroidal coordinates (α,β,z) with symmetry in z. In 3D, the representation is similar to a “donut” around the z axis (difficult to represent in a 2D plot); Fig. 2 also shows the cylindrical coordinates (r , φ, z) with symmetry in azimuthal (φ) direction. In the exterior Dirichlet problem the domain of β is: β2 b β b β1 + 2π. For two equal intersecting lenses the values for β1 and β2 are: β1 = π −θ and β2 = π+ θ (see Fig. 2). By relating the analogous evaporation parameters, one has the following relation for the vapor concentration at different locations in the domain (Laplace's equation in toroidal coordinates has solutions of the following form) [38]:

Z

w⋄v ðα; βÞ ¼ ∞ 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosh α−2 cos β w⋄v eq −w⋄v ∞

coshððπ−β1 ÞτÞ sinhððβ−β2 Þτ Þ þ coshððπ−β2 Þτ Þ sinhðð2π þ β1 −βÞτÞ P −1 =2 þiτ ð cosh α Þdτ coshðπτ Þ sinhðð2π þ β1 −β2 ÞτÞ

ð4Þ

where τ is a dummy variable, α is a toroidal coordinate (α ¼ ln dd12, where d1 and d2 are distances from the two foci, see Fig. 2) and it is zero on the z-axis, and infinity on the edge points of the spheres [38]; P −1 =2 þiτ ð coshαÞ is the Legendre function of the complex degree with the argument of the hyperbolic function and may be found using Eq. (5) [38]:

P −1 =2 þiτ ð cosh α Þ ¼

Z ∞ Z ∞ 2 sinðpτ Þ 2 cosðpτ Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dp ¼ coshðπτ Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dp cothðπτÞ π π 2 cosh p−2 cosh α 2 cosh p þ 2 cosh α α 0

ð5Þ

Fig. 2. Schematic of the cylindrical coordinates (r,φ,z) with symmetry in azimuthal (φ) direction and the toroidal coordinates (α,β,z) with symmetry in z direction is shown. Toroidal coordinates system results from rotating the 2D bipolar coordinate system about the z axis. β increases counterclockwise from its zero value on r axis. The inset shows two equal intersecting lenses where β1 =π−θ and β2 =π+θ.

124


and the value of coshα can be found as [38]:

cosh α ¼

( x2 a

12 ) x2 x2 cos θ þ 1− sin2 θ = 1− a a

ð6Þ

where x is the distance measured from apex (see Fig. 1). By taking the derivative of Eq. (4) in toroidal coordinates, one has the following relation for vapor concentration gradient on the surface of drop [29]: cosh α− cos β ∂w⋄v ðα; βÞ a ∂β β¼β1

∇w⋄v ðα; β ¼ β1 Þ ¼

ð7Þ

coshα− cosβ is called the metric coefficient and appears when taking derivation in toroidal coordinates. As β is periodic with period 2π, we a ∂w⋄ ðα;βÞ β = β1 + 2π. The relation for ð v∂β Þ can be found by simple derivation with respect to β at β = β1 + 2π. Taking the derivative of β¼β1 þ2π

The term can choose

Eq. (4) with respect to β gives:

sin θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosh α þ 2 cos θ

Z

∞ 0

∂wv ðα; βÞ ¼ w⋄v eq −w⋄v ∞ ∂β β¼3π−θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ∞ coshðθτÞ coshðθτÞ P −1 =2 þiτ ð cosh α Þdτ þ 2 cosh α þ 2 cos θ tanhððπ−θÞτ Þ P −1 =2 þiτ ð cosh α Þdτ τ coshðπτ Þ coshðπτÞ 0 ð8Þ

Using J⋄v = ρ⋄v u⋄ − ρ⋄ D∇w⋄v [24] and neglecting the convection (i.e. u⋄ = 0, or Maxwell's assumption) along with using Eqs. (7) and (8), one has: cosh α þ cos θ ⋄ wv eq −w⋄v ∞ J⋄v ðα; θÞ ¼ −ρ⋄ D a Z ∞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ∞ coshðθτ Þ sin θ coshðθτ Þ P −1 =2 þiτ ð cosh α Þdτ þ 2 cosh α þ 2 cos θ tanhððπ−θÞτ Þ P −1 =2 þiτ ð cosh α Þdτ τ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coshðπτÞ 2 cosh α þ 2 cos θ 0 coshðπτ Þ 0

ð9Þ where J⋄v(α, θ) is the evaporation flux at α for a sessile drop with contact angle of θ. It should be noted that α is a toroidal coordinate, using Eq. (6), the value of evaporation flux at different locations can be presented in cylindrical coordinates, i.e. J⋄v(ℵ, θ) where ℵ (¼ ax, see Fig. 3). Finding the value of J⋄v(ℵ, θ) requires finding two double integrals (one integral is for the Legendre functions, see Eq. (5)). Deegan et al. [35] proposed the following approximation for finding the value of J⋄v(ℵ. θ): −λ

J⋄v ðℵ; θÞ ≈ J⋄0 ð1−ℵm Þ

ð10Þ

where m and λ are fitting parameters. Deegan et al. [35] used 2 for m and used the λ in [37], i.e. λ = (180 ° − 2θ)/(360 ° − 2θ). But they did not suggest any particular value for J⋄0. One may infer from their paper [35] that J⋄0 is equal to the evaporation flux of a suspended drop. Hu and Larson [32] using a finite element model showed that J⋄0 in [35] is a function of θ; and λ =(180° − 2θ)/(360° − 2θ) is for a different electrostatic problem (i.e. electroθ static field near a sharp edge on a conductor, e.g. a lightning rod); and λ ¼ 12 − 180 ° yields better results. In the remaining of this section, Eq. (9) will be solved without simplification and the proposed model will be compared with the model in [20]. As shown in [40], the following relation holds for Legendre integration: Z

∞ 0

coshðθτÞ 1 P −1 =2 þiτ ð cosh α Þdτ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coshðπτÞ 2 cosh α þ 2 cos θ

ð11Þ

Fig. 3. Schematic of one half of a drop viewed from the side. The thick solid line represents the drop free-surface and the dashed-dotted line shows the plane of symmetry. The evaporation flux is shown at x away from the apex.


Using Eqs. (5), (9) and (11), and knowing that R ¼ calculated as (see Appendix II for details):

(

3=2

1 2ð cosh α þ cos θÞ þ 2 π sin θ

J⋄v ðα; θÞ ¼ −ρ⋄ D

Z

∞ 0

125

a sinθ (see Fig. 1), the value of evaporation flux at different locations along the drop surface can be

w⋄v eq −w⋄v ∞

R ) Z τ coshðθτÞ tanhððπ−θÞτ Þ ∞ sinðpτÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpdτ sinhðπτ Þ cosh p− cosh α α

ð12Þ

The value of α, which is a toroidal coordinate, can be converted to the cylindrical coordinates using Eq. (6), e.g. J⋄v(ℵ.θ). Integrating the J⋄v(ℵ.θ) over the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a 2 Þ −x2 −a cot θ, entire drop surface where the area element is 2πx ðdxÞ þ ðdyÞ (which is found using some geometry and knowing that yðxÞ ¼ ð sinθ see Fig. 3) gives the total evaporation rate (J) as: Z J¼

a 0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Z 1 dy ℵ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J⋄v ðℵ:θÞdℵ 2πx 1 þ J⋄v ðℵ:θÞdx ¼ 2πa2 2 dx 0 1−ℵ2 sin θ

ð13Þ

The relation in Eq. (13) has three integrations (two for J⋄v(ℵ,θ), see Eq. (12)). Popov [41] simplified Eqs. (12) and (13) to find a relation for evaporation of sessile drops. However, his model was based on the following assumptions: (i) the contact line of the drop liquid remains pinned during the evaporation; (ii) the initial contact angle is small (drops with small contact angles are thin, and for such drops, slope of the drop interface is negligible, i.e. dy in Eq. (13) vanishes); and (iii) the contact angle linearly decreases in time. As such, his developed model is valid only for drops evapodx rating in constant wetted area (CWA) mode and when the initial contact angle is less than or up to 45°. To compare Eq. (13) (the proposed model) with other literature models, let's rearrange Eq. (13) similar to Eq. (2): J ¼ −2πaρ⋄ D w⋄v eq −w⋄v ∞

sin θ f ðθÞproposed model 1 þ cos θ

ð14Þ

where f(θ)proposed model is the ratio of J⋄J:S which has three integrations (see Eq. (13)); as shown in [40] for a similar problem, these three integrals can be v

reduced to one:

f ðθÞproposed model ¼

Z ∞ 1 1 þ cos θ 1 þ coshð2θτÞ þ2 tanhððπ−θÞτÞdτ 2 sin θ sinhð2πτÞ 0

ð15Þ

The f(θ)proposed model shows the evaporation rate value change by including the evaporation flux variation along the drop surface. Eq. (14) gives the total evaporation from a sessile drop taking into account the variation of evaporation flux along the drop surface. The value of f(θ)proposed model (i.e. Eq. (15)) was found numerically using the “right Riemann sum” method; we used the implementation of the “right Riemann sum” method, in Matlab (MathWorks R2010b) for numerical calculations. Results are presented in Fig. 4. Results from literature are also shown in the Fig. 4: Hu and Larson [32] used the Deegan et al. [35] simplification (i.e. Eq. (10)); Rowan et al. [21] model assumes that the solid surface has no effect on evaporation of drops; and Bourgès-Monnier and Shanahan [10] model is only suggested for CWA (constant wetted area) mode of evaporation and assumes the vapor concentration gradient is proportional to the surface area (however, the suitability of this assumption was not discussed or justified). As shown in Fig. 4, the simplification in Eq. (10) has led to inconsistent results in the literature, e.g. vis-à-vis [20]. The value of f(θ)proposed model is consistent with Eq. (3) (i.e. [20]). This means that Picknett and Bexon model [20] inherently captures the evaporation variation along the drop surface; also the simplifications done in [35] (i.e. using Eq. (10)) and in [41] are not recommended.

Fig. 4. The value of f(θ) found from Eq. (15) is compared with the values found in Rowan et al. [21], Bourgès-Monnier and Shanahan [10], Picknett and Bexon [20], and Hu and Larson [32]. It should be noted that the model in [10] has a singularity (jump) at θ=90° and the value represented here is its limit calculated using L'Hôpital's rule.

126


Fig. 5. The symbols show the value of f(θ)proposed model found by solving Eq. (15) numerically. The solid line (f(θ)simplified =56.561 θ−0.892 ) is a fit to f(θ)proposed model values and is accurate (within 5% error) for 8° ≤ θ ≤ 131°. The accuracy for θ values larger than 131° will be higher than 5%.

It was also found that for the vast majority of sessile drops, given the range of applicability between 8° and 130° (within this range the error of using Eq. (16) in lieu of Eq. (15) is b5%), Eq. (16) can provide accurate estimates for Eq. (15 Fig. 5). f ðθÞsimplified ≈ 56:561 θ−0:892 ; 8°≤ θ ≤ 131°

ð16Þ

So, combining Eqs. (16) and (2), the total evaporation of many sessile drops can be described as: J ¼ −113:122 πaρ⋄ D w⋄v eq −w⋄v ∞

sin θ θ−0:892 ; 8° ≤ θ ≤ 131° 1 þ cos θ

ð17Þ

where θ is in degrees. and the drop surface area (S) as a function of drop wetted radius (a) and contact angle (θ) is:

3. Evaporation time for sessile drops Eq. (17) provides a formula to calculate the evaporation rate (and evaporation time accordingly). However, Eq. (17) has three unknown variables i.e. J, a and θ. From Eq. (1), one may find a relation for J in terms of drop volume, i.e. J ¼ ρ⋄L ∂V . The volume of a sessile drop with ∂t a spherical cap shape has two independent parameters (i.e. two of wetted radius, a; drop height, h; and contact angle, θ, are independent). Using a and θ as independent variables to calculate the drop volume, J can be written as a function of a and θ. Therefore, now Eq. (17) has two independent parameters. As such, to calculate the evaporation rate, another equation is needed. To find the second equation, we use the physical behavior of sessile drops during the evaporation. According to [20], for evaporation of pure sessile drops, the following two drop shape changes or a combination of them (one at a time) is expected: (i) decrease of “a” with “θ = constant”; and (ii) decrease of “θ” with “a = constant”. Each of these drop evolution cases is called an evaporation mode [10,13,17,18]. It should be noted that for the evaporation of drops of liquid mixtures which is not the subject of this study, other modes may also occur e.g. see [13,16]. So, if the evaporation mode is known, by using Eq. (17), one may calculate the evaporation rate and evaporation time. In the next two subsections, a formula for calculating the evaporation rate and evaporation time in each evaporation mode is derived.

S¼

2πa2 1 þ cos θ

ð19Þ

Using Eqs. (2), (16), (17), (18) and (19) and bearing in mind that a R ¼ sin θ , one may find the following relation for the wetted radius a as a function of time:

a2 −a0 2 ¼ −4

cos θ þ 1 ρ⋄ D ⋄ ⋄ 56:561 θ−0:892 t w −w v v eq ∞ ρ⋄L cos θ þ 2

ð20Þ

where a0 is the initial wetted radius. This equation can be used to find the evaporation time by setting a = 0 and solving for t. It should be mentioned that neglecting the solid surface effect on evaporation rate, i.e. no variation in evaporation flux, using Eq. (1), one may find the following relation for the wetted radius [9,22,23]:

a2 −a0 2 ¼ −4

cos θ þ 1 ρ⋄ D ⋄ t wv eq −w⋄v ∞ ⋄ ρL cos θ þ 2

ð21Þ

3.1. Constant contact angle (CCA) mode 3.2. Constant wetted area (CWA) mode Assuming that the evaporating drop has a spherical cap shape, drop volume (V) as a function of drop wetted radius (a) and contact angle (θ) can be found as:

V¼

πð1− cos θÞð cos θ þ 2Þa3 3 sin θð1 þ cos θÞ

ð18Þ

The volume of a sessile drop with a spherical cap shape can also be calculated as a function of the spherical cap height (h) and wetted radius (a) as: V¼

πh 2 2 3a þ h 6

ð22Þ


127

and, the surface area as a function of the spherical cap height (h) and wetted radius (a) is:

5. Conclusions

2 S ¼ π a2 þ h

The earlier evaporation models for sessile drops relied on a correction factor as a function of the contact angles i.e. f(θ), to account for the presence of the solid substrate. Recent models citing the variation of the evaporation flux along the free surface of a sessile drop, claimed that earlier models were inadequate. Using an electrostatic analogy, i.e. the Dirichlet problem for Laplace's equation, this variation was captured in recent models; however, in this process simplification were used. In this work we showed that the earlier models (i.e. Picknett and Bexon analogy) and the Dirichlet problem (recent modeling approach) are fundamentally equivalent. The confusion stems from the fact that different variables are used and solutions are done in different coordinate systems. We reinvestigated the Dirichlet problem and showed that the mathematical simplifications in solving the Dirichlet problem in toroidal coordinates created the inconsistency. This means that Picknett and Bexon model inherently captures the evaporation flux variation along the drop surface. In this study, also, we presented a simple f(θ) for practical range of contact angels that are formed on everyday surfaces (i.e. 8 ° ≤ θ ≤ 131 ° ). The developed f(θ) is in closed form and much simpler than that provided in Picknett and Bexon's study. Using the mathematical insights developed, it was demonstrated that the evaporation rate for sessile drops in CWA (constant wetted area) is faster than that in CCA (constant contact angle). This was previously observed in experimental studies in the literature, but its theoretical foundation was unclear.

ð23Þ

Using Eqs. (2), (16), (17), (22) and (23), one has: ⋄ ⋄ 1 1 1 ρ⋄ D wv eq −wv ∞ dθ ¼ −2 ⋄ dt sin θ 1 þ cos θ 56:561 θ−0:892 ρL a2

ð24Þ

Integration of Eq. (24) with respect to θ and t, gives: Z

θf θi

⋄ ⋄ θ0:892 ρ⋄ D wv eq −wv ∞ dθ ¼ −2 ⋄ t ρL 56:561 sin θð1 þ cos θÞ a2

ð25Þ

where θi and θf are the initial and final contact angles (at the beginning and end of evaporation). For finding the evaporation time, the relation in Eq. (25) should be solved. 4. Comparing the evaporation rate at different modes One of the classical questions in evaporation field is: “In what mode is evaporation faster?” In this section using a case study, it will be shown that evaporation is faster in CWA mode. Consider a water drop with wetted radius of a = 1 mm, relative humidity of 20%, wv⋄eq = 0.01789, ρ⋄L = 5.54 × 104 mol/m3, ρ⋄ = 37.9 mol/m3, and D = 2.04 × 10− 5 m2/s. If this drop evaporates in CCA mode, with an initial contact angle of θi = 40°, using Eq. (20) the evaporation time is 580 s. If the same drop evaporates in CWA mode, using Eq. (25) the evaporation time is 435 s. As shown in Fig. 6 evaporation time is shorter in the CWA mode (triangle symbols) which means that evaporation in CWA mode is faster compared to CCA mode (square symbols). Note that in Fig. 6, hollow symbols present the evaporation times calculated using Picknett and Bexon model [20], and filled symbols show the evaporation times found using the proposed model here (i.e. Eqs. (20) and (25)). These findings provide a theoretical foundation for the observation and models reported in the literature, e.g. [20,42]. Another observation from Fig. 6 is that the difference between time of evaporation for the two modes increases by increasing the initial contact angle. All this can have practical ramifications; for example, if the idea is to increase the evaporation rate, surfaces should have low contact angle and high contact angle hysteresis (to facilitate the CWA mode of evaporations).

Acknowledgement This research was partly funded by Iran National Science Foundation (INSF) under the contract No. 94011975. Authors would like to thank INSF for their financial support. Appendix I. Why electrostatic analogy was used to show the evaporation variation along the drop free surface? There are two categories of studies supporting the variation of evaporation flux from apex to contact line: (i) flow visualization techniques and (ii) solutions based on electrostatic analogies. Flow visualization techniques, as done, are not proper to find the evaporation flux distribution across the drop surface. In flow visualization techniques, the flow inside an evaporating drop is attributed to the Marangoni flow [1]. From that, it was concluded that the evaporation flux increases from apex to the contact line [2]. The explanation is as follows. Surface tension is temperature dependent and decreases by increasing the temperature. Therefore, the higher evaporation rate near the contact line may cause a lower temperature at the contact line and Marangoni flow, accordingly. Flow visualization techniques cannot provide the magnitude of the evaporation flux distribution on the drop surface. Also, due to the following two reasons flow visualization techniques cannot even provide a complete explanation for the higher evaporation rate near the contact line. First, the flow inside the drop may have a different source e.g. capillary flow due to seed particles used for visualization [3]. Second, the temperature decrease may be attributed to the thermal conductivity of the substrate [4]. References

Fig. 6. Evaporation time of water drop evaporating in CWA (triangle symbols) and CCA modes (square symbols) is compared at different contact angles (contact angles at the beginning of evaporation). Hollow and filled symbols show the evaporation times found using Picknett and Bexon model [20]; and the proposed model here (i.e. Eqs. (20) and (25)), accordingly. Used parameters for the drop are: a = 1 mm, relative humidity of 20%, wv⋄eq =0.01789, ρL =998.2 kg/m3, ρ=1.097 kg/m3 and D=2.04×10−5 m2/s.

[1] C. Xu, J.L. Prince, Snakes, shapes, and gradient vector flow, IEEE Trans. Image Process. 7 (1998) 359–369. [2] H.Y. Erbil, G. McHale, M.I. Newton, Drop evaporation on solid surfaces: constant contact angle mode, Langmuir. 18 (2002) 2636–2641. [3] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Capillary flow as the cause of ring stains from dried liquid drops, Nature. 389 (1997) 827–829.

128


[4] G.J. Dunn, S.K. Wilson, B.R. Duffy, S. David, K. Sefiane, The strong influence of substrate conductivity on droplet evaporation, J. Fluid Mech. 623 (2009) 329–351. Appendix II. Derivation of Eq. (12) By inserting Eq. (11) into Eq. (9), one has: cosh α þ cos θ ⋄ sin θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wv eq −w⋄v ∞ a 2 cosh α þ 2 cos θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 cosh α þ 2 cos θ 2 cosh α þ 2 cos θ ! Z ∞ coshðθτÞ tanhððπ−θÞτÞ P −1 =2 þiτ ð cosh α Þdτ τ coshðπτÞ 0


ðAII 1Þ which simplifies to: cosh α þ cos θ ⋄ sin θ wv eq −w⋄v ∞ a 2 cosh α þ 2 cos θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ∞ coshðθτÞ tanhððπ−θÞτÞ τ þ 2 cosh α þ 2 cos θ coshðπτÞ 0 !


P −1 =2 þiτ ð cosh α Þdτ

ðAII 2Þ

Further simplification and knowing that a = Rsinθ gives: w⋄v eq −w⋄v ∞

J⋄v ðα; θÞ ¼ −ρ⋄ D Z ×

∞ 0

1 þ 2

R

pffiffiffi 3=2 2ð cosh α þ cos θÞ sin θ

coshðθτÞ tanhððπ−θÞτÞ P −1 =2 þiτ ð cosh α Þdτ τ coshðπτÞ

!

ðAII 3Þ Using Eqs. (5) and (AII-3), one has: J⋄v ðα; θÞ ¼ −ρ⋄ D Z ×

∞

0 Z ∞

× α

w⋄v eq −w⋄v ∞ R

1 þ 2

pffiffiffi 3=2 2ð cosh α þ cos θÞ sin θ

ðAII 4Þ

coshðθτÞ 2 tanhððπ−θÞτÞ cothðπτ Þ coshðπτÞ π ! sinðpτÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpdτ 2 cosh p−2 cosh α

τ

which simplifies to:

(

3=2

1 2ð cosh α þ cos θÞ þ 2 π sin θ


Z

∞ 0

w⋄v eq −w⋄v ∞

R ) Z τ coshðθτÞ tanhððπ−θÞτÞ ∞ sinðpτÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpdτ sinhðπτÞ cosh p− cosh α α

ðAII 5Þ

References [1] Poulard C, Guéna G, Cazabat A. Diffusion-driven evaporation of sessile drops. J Phys Condens Matter 2005;17:S4213–27. [2] Guéna G, Poulard C, Cazabat A. The leading edge of evaporating droplets. J Colloid Interface Sci 2007;312:164–71. [3] Guena G, Poulard C, Cazabat A. The dynamics of evaporating sessile droplets. Colloid J 2007;69:1–8. [4] Langmuir I. The evaporation of small spheres. Phys Rev 1918;12:368–70. [5] Poulard C, Guéna G, Cazabat A, Boudaoud A, Ben Amar M. Rescaling the dynamics of evaporating drops. Langmuir 2005;21:8226–33.

[6] Birdi KS, Vu DT, Winter A. A study of the evaporation rates of small water drops placed on a solid surface. J Phys Chem 1989;93:3702–3. [7] Fang X, Li B, Petersen E, Ji Y, Sokolov JC, Rafailovich MH. Factors controlling the drop evaporation constant. J Phys Chem B 2005;109:20554–7. [8] Dunn GJ, Wilson SK, Duffy BR, David S, Sefiane K. The strong influence of substrate conductivity on droplet evaporation. J Fluid Mech 2009;623:329–51. [9] Erbil HY, McHale G, Newton MI. Drop evaporation on solid surfaces: constant contact angle mode. Langmuir 2002;18:2636–41. [10] Bourgès-Monnier C, Shanahan MER. Influence of evaporation on contact angle. Langmuir 1995;11:2820–9. [11] Dhavaleswarapu HK, Migliaccio CP, Garimella SV, Murthy JY. Experimental investigation of evaporation from low-contact-angle sessile droplets. Langmuir 2010;26: 880–8. [12] Maxwell JC. Anonymous collected scientific papers 1st ed. ; 1890 628 [Cambridge]. [13] Shi L, Shen P, Zhang D, Lin Q, Jiang Q. Wetting and evaporation behaviors of waterethanol sessile drops on PTFE surfaces. Surf Interface Anal 2009;41:951–5. [14] Soolaman DM, Yu HZ. Water microdroplets on molecularly tailored surfaces: correlation between wetting hysteresis and evaporation mode switching. J Phys Chem B 2005;109:17967–73. [15] Song H, Lee Y, Jin S, Kim H, Yoo JY. Sessile drop evaporation on surfaces of various wettability, 2008. Proceedings of the ASME Micro/Nanoscale Heat Transfer International Conference, MNHT 2008. Part A; 2008. p. 445–51. [16] Liu C, Bonaccurso E, Butt H. Evaporation of sessile water/ethanol drops in a controlled environment. Phys Chem Chem Phys 2008;10:7150–7. [17] Shanahan MER, Bourgès C. Effects of evaporation on contact angles on polymer surfaces. Int J Adhes Adhes 1994;14:201–5. [18] Shin DH, Lee SH, Jung J, Yoo JY. Evaporating characteristics of sessile droplet on hydrophobic and hydrophilic surfaces. Microelectron Eng 2009;86:1350–3. [19] Cioulachtjian S, Launay S, Boddaert S, Lallemand M. Experimental investigation of water drop evaporation under moist air or saturated vapour conditions. Int J Therm Sci 2010;49:859–66. [20] Picknett RG, Bexon R. The evaporation of sessile or pendant drops in still air. J Colloid Interface Sci 1977;61:336–50. [21] Rowan SM, Newton MI, McHale G. Evaporation of microdroplets and the wetting of solid surfaces. J Phys Chem 1995;99:13268–71. [22] Furuta T, Sakai M, Isobe T, Nakajima A. Evaporation behavior of microliter- and subnanoliter-scale water droplets on two different fluoroalkylsilane coatings. Langmuir 2009;25:11998–2001. [23] McHale G, Rowan SM, Newton MI, Banerjee MK. Evaporation and the wetting of a low-energy solid surface. J Phys Chem B 1998;102:1964–7. [24] Chini SF, Amirfazli A. Understanding the evaporation of spherical drops in quiescent environment. Colloids Surf A Physicochem Eng Asp 2013;432:82–8. [25] Tonini S. Heat and mass transfer modeling of submicrometer droplets under atmospheric pressure conditions. Atomization Sprays 2009;19:833–46. [26] Shahidzadeh-Bonn N, Rafaï S, Azouni A, Bonn D. Evaporating droplets. J Fluid Mech 2006;549:307–13. [27] Gelderblom H, Marín ÁG, Nair H, Van Houselt A, Lefferts L, Snoeijer JH, et al. How water droplets evaporate on a superhydrophobic substrate. Phys Rev E Stat Nonlinear Soft Matter Phys 2011;83. [28] Nguyen TAH, Nguyen AV. On the lifetime of evaporating sessile droplets. Langmuir 2012;28:1924–30. [29] Nguyen TAH, Nguyen AV, Hampton MA, Xu ZP, Huang L, Rudolph V. Theoretical and experimental analysis of droplet evaporation on solid surfaces. Chem Eng Sci 2012; 69:522–9. [30] Sobac B, Brutin D. Triple-line behavior and wettability controlled by nanocoated substrates: influence on sessile drop evaporation. Langmuir 2011;27:14999–5007. [31] Barash LY, Bigioni TP, Vinokur VM, Shchur LN. Evaporation and fluid dynamics of a sessile drop of capillary size. Phys Rev E 2009;79. [32] Hu H, Larson RG. Evaporation of a sessile droplet on a substrate. J Phys Chem B 2002; 106:1334–44. [33] Semenov S, Starov VM, Rubio RG, Agogo H, Velarde MG. Evaporation of sessile water droplets: universal behaviour in presence of contact angle hysteresis. Colloids Surf A Physicochem Eng Asp 2011;391:135–44. [34] Deegan RD. Pattern formation in drying drops. Phys Rev E 2000;61:475–85. [35] Deegan RD, Bakajin O, Dupont TF, Huber G, Nagel SR, Witten TA. Contact line deposits in an evaporating drop. Phys Rev E 2000;62:756–65. [36] Berteloot G, Pham C, Daerr A, Lequeux F, Limat L. Evaporation-induced flow near a contact line: consequences on coating and contact angle. EPL 2008;83. [37] Jackson JD. Anonymous classical electrodynamics. 3rd ed. John Wiley& Sons Inc.; 1998 75. [38] Lebedev NN. Anonymous special functions and their applications. 1st ed. PrenticeHall Inc.; 1965 227. [39] Widjaja E, Harris MT. Numerical study of vapor phase-diffusion driven sessile drop evaporation. Comput Chem Eng 2008;32:2169–78. [40] Prudnikov AP, Brychkov YA. Integrals and series, Vol. 4; 1992. [41] Popov YO. Evaporative deposition patterns: spatial dimensions of the deposit. Phys Rev E Stat Nonlinear Soft Matter Phys 2005;71 [036313/1-036313/17]. [42] Ous T, Arcoumanis C. Visualisation of water droplets during the operation of PEM fuel cells. J Power Sources 2007;173:137–48.