PHYSICAL REVIEW E 96, 062803 (2017)

Simple scaling laws for the evaporation of droplets pinned on pillars: Transfer-rate- anddiffusion-limited regimes

Ruth Hernandez-Perez,1 José L. García-Cordero,1 and Juan V. Escobar2,*

1Unidad Monterrey, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Vía del Conocimiento 201,Parque PIIT, Apodaca, Nuevo León, CP 66628, Mexico

2Instituto de Física, Universidad Nacional Autónoma de México, PO Box 20-364, Mexico City, 04510, Mexico(Received 30 September 2017; published 12 December 2017)

The evaporation of droplets can give rise to a wide range of interesting phenomena in which the dynamics of theevaporation are crucial. In this work, we find simple scaling laws for the evaporation dynamics of axisymmetricdroplets pinned on millimeter-sized pillars. Different laws are found depending on whether evaporation islimited by the diffusion of vapor molecules or by the transfer rate across the liquid-vapor interface. For thediffusion-limited regime, we find that a mass-loss rate equal to 3/7 of that of a free-standing evaporating dropletbrings a good balance between simplicity and physical correctness. We also find a scaling law for the evaporationof multicomponent solutions. The scaling laws found are validated against experiments of the evaporation ofdroplets of (1) water, (2) blood plasma, and (3) a mixture of water and polyethylene glycol, pinned on acrylicpillars of different diameters. These results shed light on the macroscopic dynamics of evaporation on pillars as afirst step towards the understanding of other complex phenomena that may be taking place during the evaporationprocess, such as particle transport and chemical reactions.

DOI: 10.1103/PhysRevE.96.062803

I. INTRODUCTION

The dynamics of the evaporation of droplets can be verycomplex and is dependent on many factors [1]. For example,it is known to depend on the thermal properties of boththe liquid and substrate [2] the degree of hydrophobicityof the substrate [3], the interaction between the liquid andthe solid at the contact line (i.e., whether the contact line isperfectly pinned [4,5] or if its movement is hysteresis-free [6]),the substrate’s stiffness [7], and even whether the substrateis soluble or not [8]. Thorough reviews on subject can befound in Ref. [2] and more recently in Ref. [1]. Furtherunderstanding of a droplet’s evaporation process [9] maycontribute to accelerate the discovery of new applications thatrange from aerosol production and ink absorption [10], to eyehealth [11], DNA mapping [12], formation of nanotube-basedconductive microtracks [13], and bioassays [14]. For example,Hernandez-Perez et al. [15] recently showed that the detectionof glucose and proteins is enhanced if microliter droplets of thesolutions evaporate on millimeter-sized pillars [see Fig. 1(a)],as compared to the case when evaporation happens at constantcontact angle on an effectively infinite plane. While the exactmechanisms responsible for this enhancement are not yet fullyunderstood, it is very likely driven by currents generated insidethe droplet as it evaporates, which, in turn, are known todepend on the dynamics of the evaporation process. In thisrespect, Deegan et al. explained the transport of colloidalparticles towards the edge of pinned droplets (the so-called“coffee-ring” effect) as being the result of currents arisingfrom the nonuniform spatial evaporation that line pinningnecessarily brings about [4,5].

They also showed that the experimental conditions underwhich evaporation happens—which determine the functional

*Author to whom correspondence should be addressed:escobar@fisica.unam.mx

form of the evaporation rate (ER) as we review below—arecrucial for the presence or absence of particle transport. Ingeneral, the specific dynamics of the evaporation process cangive rise to different microscopic phenomena, of which theenhancement of chemical reactions mentioned above serve as asecond example. Thus, as a first step towards the understandingof these complex phenomena, it is useful to investigate theevaporation dynamics that each of these experimental condi-tions leads to. There are mainly two experimental conditionswhich correspond to two distinct regimes leading to differentfunctional forms of the ER. In the first one, the diffusion-limited regime, evaporation is limited by the time neededfor the diffusion of vapor molecules. In the second one, thetransfer-rate-limited regime, vapor molecules are consideredto move away instantaneously from the liquid-vapor interfaceand are limited only by the transfer rate of molecules across theliquid-vapor interface. In both regimes, evaporation is assumedto be a quasistatic process. Figure 1 shows an example of thevapor concentration gradients in each of these regimes for aperfectly spherical droplet evaporating in free space, which isthe simplest possible system.

In this work, we find closed-form expressions for variablesthat characterize the evaporation process as well as scalinglaws for the evaporation dynamics of axisymmetric dropletspinned on pillars for each of these regimes. In finding thesesolutions, we construct dimensionless variables assuming thatboth the line pinning and spherical cap approximations arevalid throughout the evaporation process. We show that auseful dynamic quantity to study the system is the availablenormalized surface area of the droplets, which is equal tothe square of the height of the drop normalized by thepillar’s diameter. We then find closed-form expressions forthe time dependence of this variable and find remarkablysimple universal curves onto which experimental data forthe evaporation of water on pillars with different diameterscollapse. Driven by our experimental results for the diffusion-limited regime, we find that a mass-loss rate proportional to

2470-0045/2017/96(6)/062803(13) 062803-1 ©2017 American Physical Society

HERNANDEZ-PEREZ, GARCÍA-CORDERO, AND ESCOBAR PHYSICAL REVIEW E 96, 062803 (2017)

FIG. 1. (a) Evaporation of a droplet resting on a 1-mm acrylic pillar, evidencing that the triple line remains pinned. (b, c) Idealized vaporconcentration profiles for the evaporation in free space of a perfectly spherical drop of radius c∞ for the transfer-rate- and diffusion-limitedregimes, respectively. The vapor concentrations at the liquid-vapor interface and at infinity are denoted by �∇2c = 0 and c∞, respectively.

the drop’s height times a geometric factor is equivalent toa rate proportional to the drop’s radius, yielding a simplescaling law for the available normalized area. We then proposethat, for this system, a mass-loss rate equal to dm/dt =−(12/7)πDcs(1 − R.H.)r(t) brings a good balance betweensimplicity and physical correctness, implying that the pillarseffectively hinder evaporation by a factor of 7/3 as comparedto the case of a complete spherical drop evaporating in freespace. We also find a scaling law for the evaporation of binarymixtures in the diffusion-limited regime from physically soundmodifications of the equations found for pure substances. Thismodel fits extremely well to experimental data obtained forpolyethylene glycol (PEG) dissolved in water, as well as forblood plasma.

This paper is organized as follows. In the two nextsubsections, we briefly review the two main evaporationregimes mentioned above. In Sec. II we present the modelsfor evaporation leading to differential equations for thecorresponding mass-loss rates and find closed-form solutionsfor the experimentally measurable quantities. Materials andmethods are described in Sec. III. In Sec. IV we test the modelsagainst experimental data of both pure and mixed substances,including a solution of PEG and water as well as blood plasma.Conclusions are presented in Sec. V.

A. Diffusion-limited evaporation

When evaporation is limited by the time needed forvapor molecules to diffuse, the vapor concentration c isin general assumed to obey the Laplace equation, �∇2c = 0,throughout the evaporation process [16]. This condition holdsif the process is assumed to be quasisteady, so that theconcentration field adjusts quickly to changes in the dropletsize as compared to the time needed for evaporation. In general,the concentration gradient is obtained given the particularboundary conditions as imposed by the experimental setup,

from which the evaporation flux �j (r,t) = −D �∇c (Fick’s law)is then calculated, where D is the diffusion constant of thevapor. The drop’s mass-loss rate is then obtained as [17]

dm/dt = −∫

�j (r,t) · d A, (1)

where A is the drop’s surface area. As the concentrationgradient is the source of the mas flux �j , it is important to statethe boundary conditions for c: at the liquid-vapor interface, thevapor saturation value, cs , is assumed, while at infinity (i.e., farfield), c = c∞ ≈ (R.H.)cs , where R.H. stands for the relativehumidity. In addition, the flux normal to any substrate must bezero. The clear analogy between the evaporation variables, c

and �j , and the potential and electric field from electrostaticswas first noted and exploited by Maxwell [16]. The geometricalconstraints of the problem (e.g., line pinning) are then used tosolve the integral in Eq. (1) from which a differential equationfor the mass loss is obtained. For example, in the case ofthe evaporation of a completely spherical suspended dropletof radius rs , the diffusion-limited regime [Fig. 1(c)] yieldsc = (cs − c∞)rs/r + c∞, and a time rate of the drop’s massloss given by [16,18]

dm/dt = −4πrD(cs − c∞). (2)

If instead, a droplet of radius smaller than the capillarylength is sessile on a flat substrate of comparatively largedimensions, a spherical cap is formed and diffusion is hinderedby the presence of the plane, slowing down evaporation [19].In this case, evaporation can occur in different modes: constantcontact radius or constant contact angle θ . Often, evaporationhappens on flat surfaces in a stick-slip fashion, alternatingbetween these modes [19,20]. If the contact line is pinned sothat the wetting area is constant (constant contact radius mode),as mentioned above, Deegan et al. [4,5] demonstrated that it isprecisely the line pinning that is responsible for the transport

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FIG. 2. Schematic of a droplet of height h resting on a pillar ofradius a (and diameter d = 2a) making a contact angle θ . The radiusof the spherical cap is denoted by r .

of insoluble particles to the edge of the drop, in what is knownas the “coffee ring” effect. Furthermore, they showed that itis the diffusion-limited nature of the evaporation process thatmakes the transport of particles possible. Deegan et al. basedtheir results on an approximation of the solution to the ERfor sessile droplets derived by Picknett and Bexon [19] andLevedev [21] and later corroborated by numerical simulationsand experiments by Hu and Larson [22]. In this evaporationmodel, the mass-loss rate is no longer proportional to the drop’sspherical radius as in the case of the free-standing drop, but tothe contact radius a, so that [19,22]

dm/dt = −πaD(cs − c∞)f (θ ), (3)

where the function f (θ ) gives the so-called “evaporation-reduction factor” [2]. In experiments with pinned droplets,θ is constantly changing and integration of the mass-loss ratein a closed analytical form is not possible. Consider now anevaporating droplet pinned on a circular pillar whose radiusis comparable to the drop’s radius as shown in Figs. 1(a) and2. In this case, clearly evaporation will no longer be hinderedas much as in the case of a droplet sessile on a plane, and themodel outlined above is likely to be invalid. On the other hand,finding the exact expression for the concentration gradientwould not only be very challenging (due to the particularboundary conditions of the substrate), but it would also leadto an equivalent function to f (θ ), which will not allow for aclosed-form solution, just as in the case of a sessile dropleton a plane. Recent work by Sáenz et al. [9] has suggestedthat diffusion-limited evaporation of drops pinned on pillarsobeys a universal scaling law that depends on the area-averagedmean interface curvature, which may be particularly relevantfor nonaxisymmetric droplets. As we show in Sec. II B, whenan axisymmetric droplet is considered, the rate predicted bytheir model for the early stages of the evaporation processreduces to

dm/dt ∝ −hD(cs − c∞) = −hDcs(1 − R.H.), (4)

where h is the height of the spherical cap. A mass-loss rateproportional to h was also obtained by Rowan et al. [17] butthey proposed this model for the evaporation of water dropletsresting on effectively infinite substrates. We will show belowthat the model proposed by Rowan et al. is actually bettersuited for the case of pinned droplets on pillars than on planes.While this model is based on simplifying assumptions, it leadsto a closed-form analytical expression for the evaporationdynamics that would not be possible to obtain if an exactexpression for the diffusion was used instead. Furthermore,as we will explore in the discussion section, adding a simplegeometric factor to the mass-loss rate allows us to find a scalinglaw for this regime that fits well the experimental data. Thisgeometrical factor will turn out to make the mass-loss rateproportional to the drop’s radius, r .

B. Evaporation limited by the transfer rate across theliquid-vapor interface

The Knudsen-Hertz equation (KHE) [23,24] is typicallyinvoked to study evaporation in cases where vapor moleculescan be considered to move away instantly from the liquid-vaporinterface so that the vapor concentration is equal to cs at theinterface and to c∞ everywhere else [see example in Fig. 1(b)].Experimentally, this condition can be achieved by studyingevaporation in vacuum or by allowing a flow of neutral gas inthe direction of the drop. By construction, the KHE considersevaporation to be a quasistatic process based on the Maxwell-Boltzmann velocity distribution, in which there exists a netflux of vapor molecules at the liquid-vapor interface drivenby the difference between the atmospheric and saturation (Ps)vapor pressures. The KHE gives a mass-loss rate proportionalto the liquid-vapor surface area A [25]:

dm/dt = −φAA, (5)

where

φA ≡√

mm/2πkbT Ps(σe − σcR.H.), (6)

kb is the Boltzmann constant and mm is the mass in kg ofa molecule of the evaporating liquid. In this model, it isassumed that both the liquid and the vapor are at temperatureT . The symbols σe and σc are the so-called evaporation andcondensation coefficients, respectively. These coefficients aresmaller than 1 and represent the probability that a moleculestriking the interface will change from the vapor to thecondensed phase and from the condensed to the vapor one,respectively. The nature of these coefficients, i.e., whether ornot they are constant as a function of temperature, molecularstructure, or interfacial curvature, is a matter of currentdebate, and suitable modifications of the original KHE are thesubject of ongoing research [25,26]. For example, the so-calledStatistical Rate Theory (SRT) gives an alternative expressionfor the mass-loss rate. SRT gives an expression for the flux thatis identical to Eq. (6), but with the important difference thatthe sticking coefficients are a function of temperature and haveno fitting parameters associated to them. A thorough reviewof both KHE and SRT can be found in Ref. [25]. However, inthis work we will stick to the usual KHE [Eqs. (5) and (6)]since it provides simplicity and yields a model that comparesvery well with the experimental data.

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II. EVAPORATION MODELS AND SCALING LAWS

A. Assumptions and dimensionless variables

First, as in other studies [22], we assume that the evap-oration process is slow enough as for changes in the drop’stemperature not to alter the diffusion constant D. Second,we assume that the three-phase contact line remains pinnedall throughout the evaporation process. In this respect, it hasbeen shown that as pinned droplets evaporate their free energyincreases enough to possibly unpin the line towards anotherlocal energy minimum [27]. However, the sharp edges of thepillars make this energy barrier high enough [28] so that theline does remain pinned during the entire evaporation process,as is corroborated experimentally [Fig. 1(a)]. Third, sincethe dimensions of the droplets are smaller than the capillarylength for water, gravitational effects can be neglected andthe droplets are considered as spherical caps (Fig. 2). Forour analysis, it is convenient to write the droplet’s volume,spherical radius, and height from the center of the pillar asa function of the liquid-vapor surface area A and of thepillars radius a as V = [(A + 2πa2)

√(A/π ) − a2]/6, r =

(A/π )/[2√

(A/π ) − a2], and h =√

(A/π ) − a2 respectively.We now use the pillar’s dimensions to define the dimensionlesssphere’s volume β, radius ς , and height η, and write them asa function of the dimensionless area α ≡ A/4πa2 so that therelations above become

β ≡ V/(4πa3/3) =(

α + 1

2

)√α − 1

4, (7)

ς ≡ r/a = α√α − 1

4

, (8)

and

η ≡ h/2a =√

α − 1

4. (9)

In terms of dimensionless area, the contact angle θ betweenthe droplet and the pillar is

cos(θ ) = 1

2α− 1. (10)

Complete evaporation of the droplet happens when A isequal to the area of the pillar, or, in terms of the dimensionlessvariables, when α = 1/4, at which point β = 0.

B. Liquid-vapor interface transfer-rate-limited evaporation:Mass-loss rate proportional to liquid-vapor surface area

In this case, the transfer rate across the liquid-vaporinterface is the rate-limiting step. As mentioned in theIntroduction, in the construction of the KHE it is assumed thatthe evaporation rate of solvent molecules (at constant baseradius a) is proportional to the surface area,

dm/dt = −φAA, (11)

where φA is a constant that in principle can depend on thecorresponding vapor and ambient pressure and temperature. Inprinciple, the vapor pressure of the solvent can also depend onthe drop’s curvature [29], but the constant pillars’ width ontowhich the contact line is pinned actually prevents this effect

from being relevant. The mass-loss rate is then obtained byassuming a constant density of solvent molecules inside theliquid, ρ ≡ (V/m), and substituting into Eq. (11). Using thedimensionless variables defined above, the volume change canbe written as a function of the dimensionless area α(t) only,

d

dtβ(t) = d

dt

{[α(t) + 1

2

]√α(t) − 1

4

}= −α(t)ϕA, (12)

where the scaled evaporation rate, ϕA ≡ 3φA/aρN , has unitsof s−1. Defining α0 ≡ α(0), the solution of Eq. (12) is

α(t) = α0 − 1

3tϕA

√4α0 − 1 + 1

9t2ϕ2

A. (13)

The time for complete evaporation is then found to be tmax ≡3√

4α0 − 1(2ϕA)−1 = h0ρ(2φA)−1, which we now use to writeEq. (13) in complete dimensionless form after defining thedimensionless time τA ≡ (t/tmax):

α(τ ) = α0 + (τA

2 − 2τA

)(α0 − 1/4). (14)

We now define the “available normalized area” as

ξA(τA) ≡ α(τA) − 1/4

α0 − 1/4, (15)

which is equal to 1 when evaporation begins, and 0 when thedroplet has evaporated completely [α(τ = 1) = 1/4]. FromEq. (9), we have α = η2 + 1/4, which after substituting intoEq. (15) leads to

ξA(τA) ≡ [η2(τA)/η02] = [h(τA)/h0]2 = (1 − τA)2, (16)

where the solution for α(τ ) was substituted. Thus, the droplets’time-dependent squared normalized height should collapseonto the universal curve given by Eq. (16), regardless of thepillars´width. Information about the pillars’ width is indirectlyintegrated into the value of h0 if the volume of the droplet isknown. If necessary, the rest of the important variables usuallymeasured in evaporation experiments could be all recoveredfrom α(τ ), namely, the contact angle and the drop’s radius r ,through Eqs. (8)–(10).

C. Diffusion-limited evaporation: Mass-loss rate proportionalto drop’s height

As mentioned in the introduction, it is possible to obtainan approximate solution that circumvents dealing with anequivalent function f (θ ) in the case of diffusion-limitedevaporation. To this end, we consider as a starting pointa hybrid model in which the solution of the Laplaceequation is taken as that of a free-standing sphericaldroplet while simultaneously imposing that the actualshape remains a spherical cap throughout the evaporationprocess. This was precisely the approach that Rowanet al. [17] undertook, in good agreement with theirexperimental data for droplets on large flat surfaces.The free-standing droplet simplification implies that the

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flux points radially, which leads to a straightforwardsolution to the mass-loss rate equation, dm/dt =−2πD

∫ h

0 (cs − c∞)√

1 + (dh/dz)2zdz = −2πhD(cs − c∞)(where h is the height of the spherical cap) [1,17]. In thelast integral, the height h(z) is constrained to be that ofa spherical cap, and z is the horizontal coordinate fromthe center of the pillar. Note that, while this model wasoriginally developed for droplets sessile on a plane, it isactually better suited for modeling sessile droplets on pillars,since in this case evaporation is hindered only by the pillar’sarea instead of the whole plane. Therefore, modeling theconcentration gradient as that of a free-standing sphericaldroplet is a reasonable first approximation to this problem.In this respect, Sáenz et al. [9] have recently found thatdiffusion-limited evaporation on pillars follows a universalscaling law that depends on the area averaged mean interfacecurvature. They found that the mass-loss rate that bestfits their data is dm/dt = −2.24D(cs − c∞)[(Aκ̄)0.53/κ̄],where κ̄ is the area-averaged mean interface curvature.For an axisymmetric droplet (spherical cap) for which thecurvatures are constant and κ̄ = 1/r , it is straightforwardto show that the mass-loss rate they propose reducesto dm/dt = −4.28[(a/h)2 + 1]0.47hD(cs − c∞), whichapproximates Rowan´s solution, −2πhD(cs − c∞), if(a/h) ≈ 1, i.e., for the first stages of the evaporationprocess. Thus, the results found by Sáenz et al. validates thesimplifying assumption we propose here: for axisymmetricdroplets resting on pillars, the concentration gradient can,to a first order, be approximated by that of a free-standingspherical drop during the initial stages of the evaporationprocess. As h decreases, the evaporation rate will increase andthe full expression suggested by Sáenz et al. must be used.However, Rowan’s model serves as a good first approximationto gain insight about this system. Thus, the evaporation ratewe first propose in this model is

dm(t)/dt = −h(t)φh = −2πh(t)D(cs − c∞), (17)

where φh ≡ 2πD(cs − c∞). This equation can be written as afunction of the dimensionless volume as

dβ(t)/dt = −η(t)ϕh, (18)

where η(t) ≡ h/2a = √α(t) − 1/4 is the dimensionless

droplet height, ϕh ≡ (3φh/2πρa2), and ρ ≡ m/V is thevolumetric mass density. Writing both the volume and theheight as functions of the area, we obtain the following,relatively simple, differential equation:

d

dtα(t) = ϕh

6

[1

α(t)− 4

], (19)

whose solution is

α(t) = 14

{1 + W

[e−1+4α0− 8tϕh

3 (−1 + 4α0)]}

, (20)

where W(x) is the Lambert omega function [30]. This equationcan be recast to give the square normalized drop’s height ξ as

FIG. 3. Diffusion-limited evaporation model: rate proportional todrop’s height. Square of the normalized drop’s height, Eq. (21), forwater droplets (ρ = 1000 kg/m3) of different initial relative heights.

we defined it above [Eq. (15)], which in this case is equal to

ξ (τ ) = W(xexe−τ )

x, (21)

where

x ≡ (4α0 − 1) = 4η20 = h2

0/a2, (22)

and τ ≡ 8tϕh/3 = t(4φh/a2πρ). Note that x serves as the

initial condition for the evaporation process. The same so-lution is found, for example, for the flow front of debrisavalanches [31], and the depletion of substrate concentrationin the Michaelis–Menten model of enzyme kinetics [32,33].Equation (21) predicts that total evaporation will happenasymptotically, and therefore, it is not possible to assign thetime needed for total evaporation as a scaling factor as wedid in the previous section. Furthermore, note that the explicitdependence of Eq. (21) on x means that the time evolution of ξ

depends on the pillars’ diameters through α0, in contrast withthe case of transfer-rate-limited evaporation examined in theprevious section. Indeed, Fig. 3 shows the time evolution of ξ

for different values of x. This dependence holds in particularif the initial volumes of the droplets are constant (as in theexperiments we perform), which yields very different valuesfor x for different a’s. If this model is correct, we expect thatthe square of the normalized height will follow Eq. (21) witha constant φh for different pillars’ diameters. However, as wewill justify in the discussion section, a mass-loss rate equalto dm/dt = −(12/7)πDcs(1 − R.H.)r(t) will turn out to bemore accurate for this system and lead to a scaling law of theform ξ = (1 − τ ), independent of the pillars’ size.

D. Diffusion-limited evaporation: Binary mixtures

The evaporation of a solution composed of twosubstances—one of which evaporates at a much lower rate(or does not evaporate at all) than the other—can be studied

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following the time evolution of a modified expression for thesquare of the normalized height. To do this, first note that thefinal height of such an evaporating droplet is not zero, andfurthermore, this height will vary for different pillars’ widths.Second, we make the following physical argument: given thatξ (τ ) = [h(τ )/h0]2 [Eq. (15)] is the available normalized heightfor total evaporation, then, if hf is the final height of the drop,the available normalized height for partial evaporation will begiven by

ξh(τ )∗ ≡[h(t) − hf

h0 − hf

]2

=[η(t) − ηf

η0 − ηf

]2

=[√

W(xexe−τ ) −√

W(xexe−τf )√x −

√W(xexe−τf )

]2

, (23)

where τf is the dimensionless time needed for the height ofthe droplet to reach hf . We note that this is a first orderapproximation, since it is known that buckling can occurleading to nonspherical shapes of the solute [34]. A naturaltime scale for this system is that needed for the evaporationrate to decelerate, or, in mathematical terms, the point in timeat which the first derivative of ξ (τ ) has an inflection point. Thistime is τdec ≡ x − log(

√e/2x). Using this time scale, ξh(τh)∗

can be approximated (see Sec. A 2) as

ξh(τh)∗ ≈ (1 − τh)2 = (1 − τ/τdec)2, (24)

where now τh ≡ (τ/τdec) ∈ [0,1]. Note that this equation isindependent of x, i.e., it should hold for any initial conditionor relative droplet height and therefore serves as a scalinglaw. It is the same function we found before for the liquid-vapor interface transfer-rate-limited evaporation, but witha different characteristic time scale. We will show in theexperimental section that Eq. (24) fits remarkably well datafor the evaporation of both a binary mixture of water and PEGand blood plasma.

III. MATERIALS AND METHODS

A. Fabrication of pillars

Submillimeter-size pillars were designed in Dr. Engrave(Roland DG, Germany) and fabricated on a 1.5-mm-thick poly(methyl methacrylate) sheet using a high-precision millingmachine (MDX-40A, Roland DG, Germany). Pillars wereengraved with a 2-mm tungsten carbide drill bit at 6000 rpm.Grooving velocities were 6 mm/s for the x and y axes and3 mm/s for the z axis with a z step of 100 µm. Circularpillars had diameters ranging from 400 to 1800 µm witha height of 500 µm. Pillar borders were inspected with aninverted microscope (Axiovert, Zeiss) using a 40× objective.To remove residues, pillars were blow-dried with compressedair and difluoroethane, propane-butane (e-dust, PC-030300PERFECT CHOICE, Mexico). When necessary, plastic pillarswere washed with neutral soap (Low-Foaming Phosphate-FreePowdered Detergent No. 2204, Tergajet, Alconox) at 1 g/l.

B. Experimental section

Humidity-controlled chamber. Droplet evaporation experi-ments were carried out inside a custom-built humidity chamber

FIG. 4. (a) Scanning electron microscopy image of one of thepillars fabricated. (b) Close-up to the pillar’s border; the rim’sroughness allows droplets to remain pinned.

made of 1-inch-thick acrylic. The chamber dimensions were30 × 30 × 50 cm. A 500-lumens white light source (902-838,Ecosmart, USA) was placed outside the chamber and behindthe pillars to increase the contrast when taking photographs,but also to keep a constant temperature inside the chamber.The chamber also contained a 20 kHz ultrasonic humidifier(AIR- 200, Steren, Mexico), and a 5-V fan (TFD-8025M12S,Titan, Taiwan). Pellets of calcium chloride anhydrous wereused to reduce humidity when necessary.

Electronic control. An on-off control programmed in anArduino UNO board was used to maintain a constant humidityin the chamber and to control the fan speed for the durationof the experiments. The humidity and temperature inside thechamber were monitored using a capacity sensor (SHT75,Sensirion, Switzerland). A graphical user interface pro-grammed in LabView was designed to display both variablesin real time.

Image acquisition and analysis. High-resolution images ofthe pillars were acquired with a scanning electron microscope(Hitashi-SU5000; Fig. 4). For the evaporation experiments aUSB digital microscope (AM2111, Dino Lite, Taiwan) wasused at a 50× and 230× magnifications to take photographsof the pillars sideways and from the top, respectively. Paperfilters and black background were used when necessary toavoid glare. Images were captured automatically and convertedto 8-bit images. The contact angle and height were analyzedusing the Drop Analyze (LBADSA) plugin from ImageJ.A dispersion x-y graph and linear regression or polynomialsecond order analysis was performed on the data usingGraphPad (Prism). Figure 5 shows photographs of four ofthe pillars fabricated with water droplets of the same volumeresting on top of them.

Evaporation experiments. All evaporation experimentswere performed at 25 ± 1 ◦C, maintaining a R.H. of either30% or 50%. For the diffusion-limited evaporation regime,droplets were allowed to evaporate inside the control chamber.For the experiments corresponding to the transfer rate-limitedregime, a fan was placed inside the chamber, 10 cm away fromthe pillar. The evaporating substances used were milli-Q water,10% polyethylene glycol 400 (No. 202398, Sigma-Aldrich,USA), and blood plasma on different pillars. The dropletsof blood plasma were placed using a metal tip; 10 ml ofwhole blood were centrifuged for 5 min at 1600 rpm in alab centrifuge (5804-R, Eppendorf). Plasma was obtained bycollecting the top 4 ml of the 10 ml tube; 100 µl aliquots ofblood plasma were frozen at −20 °C and thawed before use.

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FIG. 5. Photographs of water droplets of the same volume resting on acrylic pillars of different diameters. The pillars’ diameters, d = 2a,are displayed at the bottom of each pillar.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

A. Pure substances: Interface transfer-rate-limited evaporation

We now compare the predictions of the model developedin Sec. II B against experimental data. Figures 6(a) and 6(c)display the measured drop’s height versus time for milli-Qwater droplets pinned on pillars. In these experiments, a fanpoints in the direction of the drop. Thus, we expect that theevaporation rate be proportional to the drop’s area A, so thatξA(τA) = (1 − τA)2, Eq. (16). The environmental conditions

FIG. 6. Transfer-rate-limited evaporation results: water. (Left)Measured height of droplets vs time at 25 ◦C and R.H. = 30% pinnedon pillars of different diameters for initial volumes of 2 μl (a) and3 μl (c). (Right) Square of the normalized height ξ (τ ) vs normalizedtime τ given by Eq. (16). The collapse of all data onto the theoreticalcurves (solid line) indicates that the liquid-vapor transfer-rate-limitedmodel correctly captures the physical mechanism for evaporation onpillars for this system. Solid lines depict the best fit of all the data toEq. (8) (φA = 4 × 10−4 ± 5.2 × 10−6 kg m−2 s−1 for V0 = 3 μl andφA = 3.44 × 10−4 ± 2.9 × 10−6 kg m−2 s−1 for V0 = 2 μl) assumingthe density for water ρ = 1000 kg m−3, temperature T = 298 K,and saturation vapor pressure Ps = 3169 Pa. Insets display thecorresponding data in linear scale. Data were acquired every 2 min.

under which evaporation take place are R.H. = 30% and T =25 ◦C. Two different initial volumes are tested on all pillars,V0 = 3 μl [Fig. 6(a)] and V0 = 2 μl [Fig. 6(c)]. Figures 6(b)and 6(d) show ξ (τ ) versus τA. The theoretical prediction forξA(τA) is depicted by the solid line on both figures. In calculat-ing the variable τA for each pillar, the density for water is takenas ρ = 1000 kg m−3 and φA is chosen as to minimize the sumof the square differences of the combined data for all pillardiameters for each initial volume. The constants that best fitthe data are φA = 3.44 × 10−4 ± 2.9 × 10−6 kg s−1 m−2 andφA = 4 × 10−4 ± 5.2 × 10−6 kg s−1 m−2 for initial volumesof 3 μl and 2 μl, respectively. We find an excellent agreementbetween experiment and theory, i.e., all experimental datacollapses onto the theoretical curve, in which ξA spans almostthree orders of magnitude. We now briefly give a possibleinterpretation of the values of φA found. Approximatingσe ≈ σc = σ , we obtain an expression for the evaporation rate,

σ =√

2πkbT

mm

φA

Ps(1 − R.H.), (25)

which, after substituting m = 2.99153 × 10−26 kg, T =298 ◦C, Ps = 3169 Pa (at 298 ◦C), and kb = 1.38 ×10−23 m2 kg s−2K−1, allows us to find σ given the R.H. leveland φA found above. The sticking coefficients calculated fromEq. (25) are 1.44 × 10−4 and 1.68 × 10−4 for V0 = 3 μl andV0 = 2 μl. The values of σ found are on the lower endof those commonly found [25]. Furthermore, both valuesshould coincide if indeed both the temperature and R.H.

are kept constant. However, as mentioned in the introductionand noted in Ref. [25] and more recently in Ref. [35],there exist important limitations to the theory leading tothe KHE, especially with respect to the interpretation of thesticking coefficients. In particular, the values found here differnotably from those found by Rizzuto et al. [36] using Ramanthermometry, most likely due to the fact that their experimentswere carried out in vacuum. A brief review of this work hasbeen added to the Appendix (Sec. A 3). Further experimentsof evaporation process on micropillars similar to the oneswe have conducted here could help validate new or presentevaporation theories. Nevertheless, the collapse of the datafor each group of experiments indicates that the scaling lawproposed in Eq. (16) is correct for the evaporation of pinneddroplets on pillars in this experimental setup in which theevaporation rate is proportional to the surface area of thedrop.

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FIG. 7. Diffusion-limited evaporation experimental results forwater, T = 25 ◦C. Panels (a) and (c) show the drops’ height vstime at R.H. = 50% and R.H. = 30%, respectively, and panels (b)and (d) show the corresponding squares of the normalized drop’sheight vs dimensionless time. Solid lines depict the best fits to thetheoretical prediction, Eq. (21), given the initial condition, x foreach pillar size (diameter ≡ d = 2a). Insets evidence, in log-linearscale, that the theoretical prediction describes the experimental datareasonably well for the initial 90% of the evaporation process, but,as expected, consistently underestimates the mass-loss rate for theremaining portion of the process. While the mass-loss rate found isnot constant for all pillars at the same R.H., the ratio (φ30%

h /φ50%h )

for each pillar size is close to the theoretical prediction of 1.4. Datawere acquired every 2 min.

B. Pure substances: Diffusion-limited evaporation

We now present the results for the evaporation of waterdroplets pinned on pillars in the diffusion-limited regime.Figures 7(a) and 7(c) show the evolution of the drops´ heightfor R.H. = 50% and 30%, respectively, on pillars of differentdiameters. Figures 7(b) and 7(d) display plots of the square ofthe normalized height versus scaled time for the correspondingR.H. levels. Solid lines in these plots are the best fits toEq. (21), obtained by fixing the initial condition for theevaporation through the variable x given by the initial height ofthe droplets. Insets show ξ in log-lineal scale, evidencing thatwhile the model describes well the evaporation for the initial90% of the entire process, it underestimates the evaporationrate for the remaining 10%. This same result was found forboth sets of experiments at different R.H. levels. Table Ishows the values for φh from the best fits to Eq. (16) forthe different pillars and relative humidity levels. On theother hand, as pointed out in Sec. II C, we expected φh tobe constant for all pillars, but as shown in Table I, thereis a considerable dispersion in the values obtained for thedifferent pillars’ sizes. For the data taken at R.H. = 50%,the standard deviation is 15% of the average, φ50%

h = 1.07,

TABLE I. Evaporation rates obtained from the best fits to the dataat two different R.H. levels as plotted in Figs. 7(b) and 7(d). The ratioof the mass-loss rates at the two R.H. levels tested on each pillar size(Fig. 8) is close to the theoretical value of 1.4.

R.H. = 50% R.H. = 30%

a/mm x φh/10−6 kg s−1m−1 x φh/10−6kg s−1m−1 φh30%/φh

50%

0.5 12.6 0.942 ± 1.0% 10.5 1.263 ± 1.3% 1.340.6 7.2 0.936 ± 1.7% 6.7 1.353 ± 2.2% 1.450.7 4.7 1.143 ± 1.8% 4.8 1.679 ± 1.6% 1.470.8 3.2 1.254 ± 2.2% 2.7 1.520 ± 4.0% 1.210.9 2.2 1.529 ± 2.6% 2.2 2.260 ± 1.2% 1.48

while for R.H. = 30%, the average φ30%h is 1.45 with a

standard deviation of 13%. Nevertheless, if the ratio of(φ30%

h /φ50%h ) for each pillar size is calculated separately (right

column, Table I and Fig. 8) and averaged, the expected ratiofrom the theoretical expression is well approximated. Explic-itly, assuming D and cs are constant at a temperature T for agiven pillar size, the theoretical ratio of the evaporation ratesyields (φ30%

h /φ50%h ) = (1−0.3)/(1−0.5) = 1.40, while the av-

erage of the experimental values is 1.38 (dashed line Fig. 8).The fact that the theoretical solution found for this system

fits well only to the first 90% of the evaporation process was tobe expected. After all, in this model the concentration gradientwas approximated as that of a free-standing spherical drop. Butas the drop evaporates, the spherical cap resembles less and lessa complete sphere, making this approximation progressivelyworse. It is thus remarkable that despite this shortcoming ofthe model, the ratio of the experimental evaporation ratesyields, for each pillar size, a reasonably close value to thetheoretical prediction. This implies that it may be possible thatan effective diffusion constant D′(a) be assigned to each pillarsize, while keeping the spherical droplet approximation. Notethat, in principle, introducing D′(a) would not be equivalentto introducing a similar “evaporation reduction factor,” f (θ ),used for the evaporation of droplets on relatively large surfaces,because the effective diffusion constant would hold for allangles for a given pillar radius. On the other hand, and perhapsmore importantly, Figs. 7(b) and 7(d) further shed light on

FIG. 8. Ratio of the evaporation rates, (φ30%h /φ50%

h ) (Table I) vspillar radius a. Dashed line is the average of these values (1.39),which is very close to the theoretical prediction of 1.4.

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the physics of the evaporation in this experimental condition,as they strongly suggest that the square of the normalizedheight decreases linearly with time for the most part of theprocess. Thus, it is possible that an evaporation rate that leadsto an expression of the form ξh(τh) ≡ [h(τh)/h0]2 = (1 − τh),actually collapses the experimental data. As shown in theAppendix, an evaporation rate proportional to the liquid-vaporinterface radius, dm/dt = −φr(t) [Eq. (2) and Fig. 1(b)],precisely leads to such an expression for ξ . But doing thiswould seem physically incorrect, since it is the result ofassuming the evaporation of a perfectly spherical suspendeddrop, a system that is very far from the conditions of the actualpinned drop. Nevertheless, note that the expression for themass-loss rate,

dm/dt = −φhh(t)

{[a

h(t)

]2

+ 1

}= −2φhr(t), (26)

is mathematically identical to Eq. (2) (except for a factor of2 that is absorbed by the constant φh), due to the particulargeometry of the spherical cap [r = (a2 + h2)/2h].

Also, note that the expression to which for the mass-lossrate derived by Saenz et al. reduces for axisymmetric dropscan be interpreted as consisting of a term proportional tothe drop’s height h, times a geometric correction of theform g[(a/h)2 + 1], which in their case is a power law withexponent 0.47. This geometrical factor serves the purpose ofaccelerating the evaporation rate as the drop’s height decreasesand has the same form as Eq. (29). Based on these twoarguments, we propose that for diffusion-limited evaporationthe mass-loss rate can be approximated by Eq. (29) in whichthe geometric factor g is just a linear function of its argument.It is straightforward to show that this expression leads to adimensionless area equal to (see Sec. A 1 for details):

α(τh) = α0 − τh(α0 − 1/4), (27)

which, in turn, yields

ξh(τh) ≡ [h(τh)/h0]2 = (1 − τh), (28)

where the dimensionless time is now given by

τh ≡ t(4φh/h0

2πρ). (29)

In Fig. 9 we plot again ξh(τh) for the same data presented inFig. 7, but with the normalized time of Eq. (29). Now the datacollapse reasonably well onto the prediction curve, Eq. (28).The evaporation rates from the theoretical prediction that bestfit the data are φ50%

h = 0.790 × 10−6 ± 3.7 × 10−9 kg s−1 m−1

and φ30%h = 1.07 × 10−6 ± 1.41 × 10−8 kg s−1 m−1. The ratio

φ30%h /φ50%

h = 1.35, is in fair agreement with the theoreticalprediction of 1.4, after assuming that φ ∝ Dcs(1 − R.H.),where both D and cs are constants. Furthermore, takingcs = 0.023 kg m−3 [37] and D = 2.55 × 10−5 m2 s−1 at T =25 ◦C [38], the product is Dcs = 5.88 × 10−7 kg s−1 m−1,that along with the values found for φ50%

h and φ30%h yield

values of 2φ/[πDcs(1 − R.H.)] equal to 1.711 and 1.655,respectively. These values are very close to 12/7 = 4(3/7) =1.714, which implies that the mass-loss rate of Eq. (26) is wellapproximated by

dm/dt = −12

7πr(t)Dcs(1 − R.H.), (30)

FIG. 9. Data collapse for pure substances. Square of the nor-malized height for the data presented in Fig. 7, vs normalized timeτh ≡ 4φht/h0

2πρ for (a) R.H. = 50% and (b) R.H. = 30%. Theevaporation rates from the theoretical prediction [Eq. (28)] thatbest fit the data are φh = 0.790 × 10−6 ± 3.7 × 10−9 kg s−1 m−1 andφh = 1.07 × 10−6 ± 1.41 × 10−8 kg s−1 m−1 for R.H. = 50% andR.H. = 30%, respectively. The ratio φ30%

h /φ50%h = 1.35 is in fair

agreement with the theoretical prediction of 1.4. Insets display thecorresponding data in log-linear scale.

leading to the scaling law ξh(τh) = (1 − τh) of Eq. (28). Ourresults suggest that Eq. (30) brings a good balance betweensimplicity and physical correctness. Moreover, it implies thatthe pillars effectively hinder evaporation by a factor of 7/3 ascompared to the case of a complete spherical drop evaporatingin free space [Eq. (2)].

C. Evaporation of multicomponent droplets pinned on pillars:Diffusion-limited evaporation

As stated in Sec. II D, to describe the evaporation of binarymixtures, we make use of the same model we developed fordiffusion-limited evaporation by considering the square of thenormalized height for partial evaporation, ξh(τ )∗ [Eq. (23)].To do this, it is necessary to take as final height, hf, the height ofthe remaining solute, i.e., the solute left once evaporation hasstopped. For both blood plasma and the solution of PEG, theshape of the remaining solute is found to be well approximatedby a spherical cap constrained by the pillar’s diameter.

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FIG. 10. Evaporation of a mixture of water and PEG (10%),T = 25 ◦C. Height vs time for 3 μl droplets on pillars of dif-ferent diameters for R.H. = 50% (a) and for R.H. = 30% (c).The corresponding squares of the normalized available heights,ξ ∗ [Eq. (24)], vs normalized time, τh = t/[x − log(

√e/2x)] are

plotted in panels (b) and (d). The best fit of the model yieldsφh = 1.043 × 10−6 ± 6.28 × 10−9 kg s−1m−1 for R.H. = 50% andφh = 1.437 × 10−6 ± 1.13 × 10−8 kg s−1m−1 for R.H. = 30%. Datawere acquired every 2 min. Insets display the data in linear-linearscale.

1. Polyethylene glycol (PEG) and water

We now present the experimental results for the evaporationof droplets composed of a 10% volume concentration of PEG400 (No. 202398, Sigma-Aldrich, USA) in water. Figures 10(a)and 10(c) show the drops’ height versus time at R.H. =50% and R.H. = 30%, respectively. When the new variablesξh(τ )∗ and τ are plotted, the curves for all pillars diameterscollapse onto the theoretical curve, Eq. (24) [Figs. 10(b)and 10(d)]. Furthermore, combining the data for all pillarsizes on each of these figures (and setting ρ = 1000 kg/m3),the values of the evaporation rates found to best fit thedata are φh = 1.0427 × 10−6 ± 6.28 × 10−9 kg s−1 m−1 andφh = 1.4376 × 10−6 ± 1.13 × 10−8 kg s−1 m−1, respectivelyfor R.H. = 50% and R.H. = 30%. The ratio of these mass-loss rates is 1.38, once again in very good agreement with thetheoretical prediction of 1.4.

2. Blood plasma

The evaporation and spreading of blood has been studiedextensively and reviewed by Brutin and coworkers [1,39]. Forexample, studying the patterns left after evaporation can giveinformation about the molecular and supramolecular structureof proteins [40] and has even been used for the detection ofdiseases [41]. Plasma makes up 55% of the total blood content,which in turn is composed of 90% water, with the remaining10% being ions, electrolytes, and proteins [39]. As a final

FIG. 11. Evaporation of a mixture of blood plasma, T = 25 ◦C.Height vs time for 3 μl droplets on pillars of different diameters forR.H. levels of (a) 50% and (c) 30%. (b–d) Corresponding squaresof the normalized available heights, ξ ∗, vs normalized time, τh =t/[x − log(

√e/2x)]. The best fits of the model to the data yield φh =

0.958 × 10−6 ± 4 × 10−9 kg s−1m−1 and φh = 1.438 × 10−6 ± 8 ×10−9 kg s−1m−1 for R.H. = 50% and R.H. = 30%, respectively.Data were acquired every 30 s. Inset displays collapsed data inlinear-linear scale.

test of this model, we repeat the exact same experiments thatwere carried out for PEG, but this time for blood plasma.Figures 11(a) and 11(c) show the height of the droplets versustime. As with the evaporation of PEG and water droplets, whenξh(τ )∗ is plotted against τ , the curves for all pillars diameterscollapse onto the theoretical curve, Eq. (24) [Figs. 11(b)and 11(d)].

However, the log-linear scale plots show that in thiscase the fit is not as good as in the case of the PEGand water droplets. The evaporation rates found in thiscase is φh = 0.958 × 10−6 ± 4 × 10−9 kg s−1 m−1 and φh =1.438 × 10−6 ± 8 × 10−9 kg s−1 m−1 for R.H. = 50% andR.H. = 30%, respectively. The ratio of these values is 1.5,in fair agreement with the theoretical ratio of 1.4.

Note that, despite the approximations made in its derivation,the theoretical prediction for both multicomponent system fitswell the experimental prediction.

V. CONCLUSIONS

We have developed models for the evaporation of axisym-metric droplets pinned on pillars for the diffusion-limitedand transfer-rate-limited regimes, for both pure and mixedsubstances. After physically sound approximations, we havefound simple closed-form expressions for the square of thenormalized drop’s height, ξ ≡ (h/h0)2, as a function of thenormalized time, τ . For single-substance droplets in thediffusion-limited regime, our experimental results have led us

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SIMPLE SCALING LAWS FOR THE EVAPORATION OF . . . PHYSICAL REVIEW E 96, 062803 (2017)

to propose an approximate mass-loss rate equal to dm/dt =−(12/7)πDcs(1 − R.H.)r(t). This rate greatly simplifies thescaling law found, ξ = (1 − τ ), while comparing very wellwith experimental data. A systematic study of this systemat different R.H. levels is necessary to further validate theuniversality of the rate proposed here.

For binary mixtures in this same regime, the scaling law isgiven by the square of the normalized drop’s height for par-tial evaporation, ξ ∗ = (1 − τ )2, while for single-componentdroplets in the transfer-rate-limited regime, the scaling lawfound is ξ = (1 − τ )2. These laws where successfully testedagainst experimental data from water, PEG and water, andblood plasma droplets. For the evaporation of binary mixturesin the diffusion-limited regime, a mass-loss rate proportionalto the drop’s height h (Rowan’s model) yields a model thatcompares very well with experimental data. We attribute thissuccess to the fact that the solute keeps the shape of thedroplet closer to a complete sphere for the entire durationof the evaporation process as compared to the case of puresubstances. In this respect, it would be very interesting to havesingle component droplets evaporate on pillars with sphericalcap-shaped surfaces, which would serve the same purposeas the solute. While the manufacturing of these pillars doesnot seem straightforward it does not seem impossible either.Carrying out bioassays on such pillars may lead to differentand possibly enhanced detection levels.

The scaling laws presented here can help validate the timeevolution of the drop’s height as a first step towards theunderstanding of more complicated physical and chemicalprocesses that may occur inside the drop, as in the case ofbioassays.

ACKNOWLEDGMENTS

J.V.E. gratefully acknowledges funding from DGAPA-UNAM No. IA101216 and No. IA103018 and CONACYTNo. CB 2013/221235. J.L.G.C. acknowledges funding fromCONACYT Grants No. CB-256097 and No. FC-1132.

APPENDIX

1. Diffusion-limited evaporation: Mass-loss rate proportional toliquid-vapor surface radius

For the sake of completeness, we briefly incorporate themodel corresponding to an evaporation rate proportional tothe radius of the liquid-vapor interface, dN(t)/dt = −r(t)φr .A model in which the mass-loss rate is proportional to R

was proposed by Birdi and Vu [42] for diffusion-limitedevaporation. Note that this model can be interpreted as theresult of considering the sessile droplets as free standingspherical ones in which the pillars play no role, neitherwith respect to limiting evaporation nor for constraining thegeometry of the drop. Necessarily, this is then just a zerothorder approximation to the problem at hand, but it does offer aclosed-form analytical solution as we now show. Substitutingthe radius on the right-hand side of Eq. (1) in place of the area,the equivalent differential equation leading to α(t) becomes

FIG. 12. Rescaling of the normalized available area. Normalizeddroplet height (a), original normalized available area (b), and rescaledone (c) as a function of the scaled time for different initial conditions.The corresponding insets are linear scales. Solid line in panel (c)shows the approximation that serves as a scaling law.

[using Eq. (8) for the dimensionless radius],

d

dt

[(α(t) + 1

2

)√α(t) − 1

4

]= − 2α(τ )√

4α(τ ) − 1ϕr, (A1)

an equation that reduces to

d

dtα(t) = −2

3ϕr, (A2)

whose solution is simply

α(t) = α0 − (2tϕR/3), (A3)

where now ϕr ≡ (3φr/4πa2ρ). The square of the normalizedheight in this case is

ξr (τr ) ≡ [h(τr )/h0]2 = (1 − τr ), (A4)

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where τr ≡ t/tmax and tmax = (3/8φr )(4α0 − 1) = h02πρ/2φr .

Equation (A3) predicts that the drop’s height will decrease asthe square root of time.

2. Approximation for the normalized available area for partialevaporation, ξh(τh)∗

A natural time scale for this system is that needed forthe evaporation rate to decelerate, or, in mathematical terms,the point in time at which the first derivative of ξ (τ ) has aninflection point.

This time is τdec ≡ x − log(√

e/2x), and has the interest-ing property that η(τdec) = (1/2)

√W(xexe−τdec ) = 1/(2

√2),

regardless of the value of x as shown in Fig. 12(a). We thuspropose that τf = τdec, which allows us to write the modifiedavailable normalized area as

ξh(τh)∗ =[√

W(xexe−τdecτh) − 1/√

2√x − 1/

√2

]2

, (A5)

where τh ≡ (τ/τdec) ∈ [0,1]. In writing the last equation, wehave assumed that evaporation stops at τdec. Noting that boththe function itself as well as its first derivative evaluated atτh = 1 are zero, it is natural to attempt to find an approximatesolution of the form ξh(τh)∗ ≈ μ(x)(1 − τh)2, where μ(x) isthe second order Taylor expansion coefficient which is oforder 1 within the experimentally relevant range of initialconditions. Actually, μ(x = 2) ≈ 0.92 and μ(x = 12) ≈ 1.57,which roughly correspond to the initial volumes of a 3 μldroplet pinned on pillars of diameters a = 0.5 mm and a =0.9 mm, respectively. Given the above considerations as well asthe fact that it is required that ξh(τh = 0)∗ = 1, we approximatethe desired function by setting μ = 1 for all x, so that

ξh(τh)∗ ≈ (1 − τh)2 = (1 − τ/τdec)2. (A6)

Figure 12(c) shows that, indeed, the approximate solution[Eq. (A6)] is in reasonable agreement with the exact expressionfor ξh(τ )∗ [Eq. (A5)]. Furthermore, this figure shows thatξh(τ )∗ depends only weakly on x. This contrasts with the

behavior of the original square of the available normalizedheight [Eq. (21)], ξh(τ ), which is strongly dependent on x

[Fig. 12(b)].

3. Recent studies on the microscopic mechanisms of evaporation

While this work does not concern itself with the mi-croscopic (molecular) mechanisms behind the process ofevaporation, as we have mentioned on the main text, saidmechanisms are a matter of present and vivid research. Inparticular, as noted in Ref. [36], it is of great interest to measurethe evaporation and condensation coefficients for differentsystems, since their deviation from unity signal the presenceof kinetic and thermodynamic barriers. In this respect, Rizzutoet al. performed the first experimental studies of ballisticevaporation using a jet Raman thermometry method [36].Using this technique, it is possible to infer the temperatureof the evaporating droplets from the position and line shape ofthe –OH stretching vibration signal. In turn, using a suitablecooling model, the evaporation coefficient can be calculated byfitting the experimental data. Interestingly, the authors find thatthe evaporation coefficients of aerosol droplets evaporatingin vacuum are strongly affected by the interfacial pH levelof the droplets. Specifically, the authors find a value forthe evaporation coefficient of 0.25 for 1.0 M HCl and 0.91for 0.1 M HCl, while for pure water they find a value of0.62. The authors attribute this effect to the presence ofhydronium perturbing the interfacial hydrogen bond structureof the droplet. Consistent with their experimental resultsis the molecular simulation study carried out by Nagataet al. [43]. The authors of the latter find that the ultrafast(order femtosecond) hydrogen-bond rearrangement dynamicsis crucial for evaporation to take place in the first place.Remarkably, the they show that the energy needed for a singlewater molecule to evaporate is provided by the collision witha second molecule that in turn is hydrogen bonded to a thirdone. In other words, they show that evaporation of a watermolecule is in fact a three-body effect.

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