Macroscopic Model for Sessile Droplet Evaporation on a Flat Surface.

Evaporation of sessile droplets on a flat surface involves a complex interplay between phase change, diffusion, advection, and surface forces. In an attempt to significantly reduce the complexity of the problem and to make it manageable, we propose a simple model hinged on a surface free-energy-based relaxation dynamics of the droplet shape, a diffusive evaporation model, and a contact line pinning mechanism governed by a yield stress. Our model reproduces the known dynamics of droplet shape relaxation and of droplet evaporation, both in the absence and in the presence of contact line pinning. We show that shape relaxation during evaporation significantly affects the lifetime of a drop. We find that the dependence of the evaporation time on the initial contact angle is a function of the competition between the shape relaxation and evaporation and is strongly affected by any contact line pinning.

Introduction

Understanding the dynamics of spreading and drying of droplets deposited on a substrate is of importance to many practices, such as inkjet printing,[1−3] pesticide spraying,[4] and semiconductor device manufacturing.[5,6] In the semiconductor industry, photolithographic methods are employed to define patterns for integrated circuits on wafers, coated with photosensitive polymer layers.[5,7] Often, water immersion is used to increase the resolution of the lithography process.[7,8] However, if any droplets are left behind on a wafer, they may induce the so-called watermark defects in the photoresist layer.[5,9]

Due to the importance of understanding the drying processes, the drying of droplets on surfaces has been intensely studied experimentally,[6,10−15] theoretically,[15−19] and numerically.[20−23] Nevertheless, the understanding of this multifaceted problem remains incomplete due to the multitude of coupled processes that determine the evaporation dynamics. Apart from the evaporation itself, processes such as convection and heat transport in the droplet, shape relaxation, and contact line pinning play a role.

Associated with the complex physics of the problem at hand are a large number of physical parameters, the relative importance of which depends on the initial and boundary conditions as well as the time and length scales of interest. Therefore, we aim to develop a macroscopic model that does not resolve the details of, for example, the velocity field inside the droplet or the vapor concentration field around it. Rather, we consider three constituents to make up our model: (1) interfacial free-energy-based relaxation for the droplet shape, (2) diffusion-limited evaporation, and (3) contact line pinning.

In the literature, various authors studied the evaporation of droplets, focusing on two limiting modes of evaporation: a droplet evaporates with either a constant contact area or a constant contact angle, allowing transitions between these limits.[13,16,24,25] Others have investigated the shape relaxation of droplets by measuring the contact angle of nonevaporating droplets in time.[26−28] To combine the both aspects, which are described in the literature separately, we propose a model that is not restricted to the two evaporation modes, but contains the shape relaxation of the droplet during the evaporation process. It captures and extends the evaluation of Stauber et al.[24] by taking into account the contact line dynamics, i.e., incorporating both advancing and receding contact lines and considering cases without contact line pinning. Moreover, to describe the transition between mobile and pinned contact lines, our model includes a yield stress that governs contact line pinning: contact line motion is inhibited for capillary driving forces below a critical stress.

The remainder of this paper is organized as follows. In the Theory section, we present the main ingredients of our phenomenological model. The Results and Discussion section compares experimental data and existing theories, as well as presents an overview of representative cases of evaporation with and without contact line pinning. We also discuss in detail the implications of choices made for certain parameters during the calculations. In the Summary and Conclusions section, we summarize our results and present our main conclusions.

Theory

The focus of this work is on droplets of sizes smaller than the capillary length , which allows us to model the droplet as a spherical cap.[29] Here, γLG denotes the surface tension of the liquid–gas interface, ρ is the mass density of the fluid, and g is the gravitational acceleration. For water in air at room temperature, lc ≃ 3 mm.[29] We presume the liquid to be incompressible. If the shape of the droplet is described as a spherical cap, it is uniquely defined by only two parameters. We choose for these the radius a of the contact area and the contact angle θ of the drop with the solid surface that we assume to be rigid (see Figure ).[30] Geometrically, they are related to the droplet volume V according toEquation implies that, for a given volume V, a prescribed value for the contact area radius a defines the contact angle θ and vice versa. Within a macroscopic description of the droplet shape, the equilibrium values of a and θ are determined by γLG, as well as by the solid–liquid and solid–gas interfacial tensions, γSL and γSG, respectively. We associate the droplet shape with an interfacial free energy F(a, θ), given by the sum of the interfacial tensions multiplied by the respective surface areasFor a given volume V, eq can be expressed as a function of θ only, using eq , that is, F(a, θ) → F(θ).[31] Minimizing this free energy F with respect to θ produces the following equation for its optimal valuewhich is the well-known Young’s equation for the equilibrium contact angle θeq.[32,33] We note that θeq denotes the contact angle corresponding to the free-energy minimum; however, this angle is not necessarily straightforwardly accessible experimentally.[34]

Figure 1

Schematic of an axisymmetric droplet on a planar surface, with contact angle θ and radius a of the contact area.

Out of equilibrium, eq does not hold. To describe how an out-of-equilibrium droplet shape relaxes toward equilibrium, we construct a kinetic equation for the contact angle using a relaxational dynamics approach based on our free-energy landscape.[35,36] It describes how the droplet adjusts its contact angle θ with the surface to move toward equilibrium. Together with the volume V, this defines a new radius of the contact area a. We refer to this process as “shape relaxation”, since both θ and a change simultaneously to accommodate a lower free-energy state. We note, however, that the overall shape of the droplet remains a spherical cap. This allows us to quite naturally include the effects of steady evaporation and of a potential pinning of the contact line. In the next subsections, we discuss separately and in detail, the three main components of our phenomenological model: the relaxation dynamics of the droplet shape, the description for diffusive evaporation, and the contact line pinning mechanism.

Relaxation Dynamics of the Droplet Shape

On the basis of the free energy F(θ) obtained from eqs and 2, we construct a relaxation equation for the contact angle θwhere Γ is a phenomenological relaxation rate that we specify in more detail below. Equation describes a rate of change that is proportional to the generalized force dF/dθ. This is in analogy to the so-called model A dynamics commonly applied in the kinetics of phase transitions of nonconserved order parameters.[35−37] We note that it is also possible to derive kinetic equations by equating capillary and viscous forces.[38,39] In our model, this balance is implicit in the parameter Γ.

Several experimental and theoretical works have identified the difference between the cosines of the instantaneous, time-dependent contact angle θ and its equilibrium value θeq, given by eq , to be the driving force for the motion of the contact line.[27,28,33,40,41] For small values of the difference cos θ – cos θeq, this relaxation can be described by a simple exponential function.[27,42] The exponential decay allows us to identify a characteristic time scale τrlx asWe discuss the functional expression for τrlx below.

To relate to eq , we transform eq into a kinetic equation for cos θ and expand it around the equilibrium cos θeq. A linearization produces an exponentially decaying cos θ, from which we determine Γ. This yieldswithEquations and 6 reproduce eq for small deviations from equilibrium. In eq , we introduce a time-dependent, dimensionless factor α(t) to account for changes in the dynamics of the droplet shape relaxation due to, e.g., a change in droplet size over time. We return to this below. Equations and 6 describe how a droplet deposited with a nonequilibrium initial contact angle θ0 relaxes to the equilibrium value θeq in a relaxation process characterized by a fundamental time scale τrlx, provided that the contact line is not pinned. We discuss our model for contact line pinning below.

The characteristic time scale τrlx for the shape relaxation has been identified in various experimental and theoretical works to be dependent on the fluid viscosity η, the liquid–gas interfacial tension γLG, and a length scale L(19,26,27,39,41−44) asIn experiments on spreading of polymeric fluids, this length scale L has been described as a measure of the slip or friction length of the interaction between a polymeric liquid and the solid,[26,27,42] which seems to be independent of droplet dimensions[27] and has been estimated to be of the order of micrometers.[42] In works on the coalescence of droplets, however, the length scale L has been shown to be proportional to the droplet size R,[19,39,41] which seems in agreement with the experimental and theoretical works on the spreading of polymer melts[43] and spherical droplets of simple liquids.[44] For this reason, we take the length scale L to be kV01/3, with V0 the initial volume of the droplet, making it proportional to the droplet size, and k a dimensionless proportionality constant that can be related to an Arrhenius factor.[41] HenceAs the droplet size decreases during evaporation, the length scale L related to the shape relaxation may (1) remain constant (in the case that L is related to a slip or friction length) or (2) decrease with the droplet size. The scale factor α(t) can be employed to incorporate either behavior into the dynamics described by eqs and 6. If L remains constant, we may choose α = 1, whereas for a size-dependent length scale, α(t) = (V(t)/V0)1/3. As we shall see, it turns out that the two expressions for α give rise to small differences in the droplet dynamics albeit that the lifetime of an evaporating droplet is not sensitive to whether α is proportional to the droplet size or not. For simplicity, we set α = 1 for the evaluation of our results. We discuss the implications of choosing the alternative α(t) in more detail in the Results and Discussion section.

The structure of eq allows for the implementation of different models for droplet shape relaxation, as long as it progresses exponentially in the limit of small deviations from equilibrium, as in eq . For example, from a microscopic perspective, the motion of the contact line is often described by the so-called molecular kinetic theory (MKT).[28] This theory describes the motion of the contact line in terms of small jumps over the intrinsically microscopically inhomogeneous surface, driven by thermal fluctuations. It has been shown to predict contact line dynamics in agreement with experiments and molecular simulation.[28,45−48] MKT relates the velocity da/dt of the contact line to the driving force via the expressionwhere ξ denotes the distance between adsorption sites on the surface; kBT is the usual thermal energy, with kB the Boltzmann constant and T the absolute temperature; vL is the molecular volume of the liquid; and G* is the surface contribution to the activation free energy of wetting.[28,46,49,50]

If we translate eq in terms of the time evolution of the cosine of the contact angle, i.e., make use of eq , and expand this to linear order for small deformations cos θ – cos θeq, we find that the characteristic time scale τrlx according to molecular kinetic theory must be given byWe see that the functional form of τrlx of eq is analogous to that of eq . This suggests that the characteristic shape relaxation time as predicted by MKT, which is a microscopic theory in origin, to linear order also is a function of macroscopic parameters such as droplet size, viscosity, and surface tension. Parenthetically, we find that a hydrodynamic theory for contact angle dynamics, as described by Voinov and de Ruijter et al., yields an analogous result.[45,51] If we expand the theory for small deformations cos θ – cos θeq, we again find a characteristic time scale τrlx, which is proportional to the fraction ηV1/3/γLG. This indicates that the scaling of the characteristic relaxation time scale with viscosity, droplet size, and interfacial tension, as described in eq , is universal.

This concludes our analysis of the relaxation dynamics of small drops. We next describe how quasi-steady evaporation affects the dynamics of a deposited droplet, presuming that an instantaneous free energy can be defined, in effect presuming a separation of time scales.

Evaporation of the Droplet

We take quasi-stationary, isothermal vapor diffusion into the surrounding gas phase to be the governing mechanism for evaporation, assuming the droplet to be in contact with an infinite volume of gas. We neglect thermal effects caused by the evaporation of the fluid, effectively assuming that heat transport occurs at much shorter times than the time scales associated with the evaporation process. For water in air, the evaporative cooling at the droplet surface has a negligible effect on the evaporation rate[52] and we consider an isothermal substrate, which is reasonable for surfaces with high thermal conductivity.[53−55] Picknett and Bexon[16] derived an expression for the rate of change in mass of a droplet as a function of contact angle θ. The rate of change of the volume V of a droplet can then be written aswhere D denotes the diffusion coefficient of vapor molecules in the gas phase and ρ is the mass density of the liquid.[17] Furthermore, Δc ≡ cs – c∞ denotes the difference between the vapor mass concentration cs near the liquid–gas interface (in units of mass per volume), presumed to be the saturation value of the fluid molecules in the gas phase, and the vapor mass concentration c∞ at infinity, i.e., that of the ambient atmosphere. Finally, f(θ) denotes a geometric factor for which an exact analytical expression is not available in closed form.[16,56] For our purposes, a polynomial representation for f(θ)is sufficiently accurate. Indeed, the error of the approximant is less than 0.2% for all values of θ.[16]

For a constant contact angle θ, eq can be expressed entirely in the contact area radius a(t) using eq and solved exactly. This giveswhere a0 denotes the initial value of the contact area radius a0 = a(0). It shows that the contact area πa2 decreases linearly in time, a known experimental result.[17] From eq , we deduce that the time tevap it takes to evaporate a droplet is the longest for θ = π/2. For this contact angle, the evaporation time τevap is given by the simple expressionIn the remainder of this work, we shall scale all evaporation times to τevap. Note that we have identified the two fundamental time scales that describe our problem: the fundamental relaxation time τrlx (eq ) and the fundamental evaporation time τevap (eq ). The actual evaporation time depends not only on the initial contact angle and the relaxation dynamics of the droplet shape, but also on whether or not contact line pinning takes place.

Contact Line Pinning

Contact line pinning is the phenomenon where the contact line of the droplet becomes stuck, permanently or temporarily, on structural or chemical inhomogeneities of the supporting surface.[40,57−63] In general, a droplet in the pinned state exhibits a contact angle different from the equilibrium angle θeq, as it cannot relax to its equilibrium shape. We model the influence of surface heterogeneities by introducing a net macroscopic threshold force per unit length, fp, exerted in the plane of the surface along the radial direction of the circular contact line. It has a direction opposite to the capillary driving force per unit length, fc. As both fp and fc are exerted on the perimeter of the contact area, we for simplicity refer to both as a force.

If the magnitude of the capillary driving force is smaller than the threshold fp, then the contact line remains pinned. On the other hand, if it is greater, we allow the contact line to move: the relaxation of the contact angle θ is calculated using eqs and 6 and the contact line moves accordingly. In the presence of contact line pinning, the droplet shape relaxes to the point where the capillary forces and pinning forces are balanced. The contact line motion is quasi-steady and hence the associated friction does not depend on the velocity of the contact line. For simplicity, we presume that the yield force fp does not depend on the position on the surface. We define the capillary force aswhere we have used Young’s law (eq ). In our prescription, we allow motion of the contact line as long as |fc| > fp. Equation is also referred to as the unbalanced Young’s force or unbalanced capillary forces.[25,33,50]

The magnitude of the pinning force fp defines a contact angle range in which the capillary force fc is too weak to overcome pinning. As long as the contact angle θ resides within this range, the contact area remains constant. We refer to this range as the “fixed-area” regime, and it turns out to be bounded by the receding and advancing contact angles, θr and θa, which are the contact angles for which fc and fp are balanced. Within our model, the values of these quantities depend on the pinning force fp,[40] according toThe receding and advancing contact angles indicate the points at which the pinning–depinning transitions occur. If the droplet evaporates while initially being in the pinned (fixed-area) state, the contact angle decreases until the droplet depins at a value equal to θr, after which the evaporation continues with a constant contact angle θr and a receding contact line. In contrast to a constant θr, a constant advancing angle θa is not encountered for droplets with decreasing volume, but it can only be observed as the point at which the droplet becomes pinned after initial spreading.

Results and Discussion

We now compare predictions of our phenomenological model with the full nonlinear response presented by molecular kinetic theory (MKT) and with experiments on droplet evaporation in the presence of contact line pinning. We quantify the competition between evaporation and relaxation using the ratio of the two time scales τevap and τrlx. It determines, together with the initial and equilibrium contact angles as well as the magnitude of the pinning force, the lifetime of an evaporating droplet. Both fundamental time scales depend only on the properties of the fluid and the surrounding vapor phase. Their ratio scales linearly with the droplet size V01/3. Presuming that the Arrhenius factor k in eq is of the order 108,[41] typical values of τevap/τrlx for water droplets of micrometer to millimeter sizes range from 10–4 to 100; however, the latter value may increase further under conditions of slow evaporation (i.e., high humidity). For fluids with higher viscosity η, presuming the other parameters remain unchanged, τevap/τrlx decreases. The ratio of the two time scales has also been addressed by Man and Doi[19] to be important in the context of evaporation problems. Directly connecting the parameter kev presented in ref (19) to our τevap/τrlx, however, is not straightforward due to the factor k. We choose the droplets to be hemispherical in equilibrium, i.e., θeq = π/2, which is typical for a water droplet on a polymer substrate. The implications of choosing a different equilibrium contact angle are discussed at the end of this section.

Because sessile droplet shape relaxation and evaporation have been described separately in the literature before, we feel it instructive to first investigate how our model compares to those works and to known experimental data. After the validation of the model with the literature, we discuss the predictions given by our more complete model that unites shape relaxation, droplet evaporation, and contact line pinning. Finally, we discuss the impact of the assumptions we make during our calculations.

Shape Relaxation and Pinning–Depinning Transition

To illustrate the relaxation dynamics predicted by our free-energy-based model and to compare the predictions to an existing model for contact line dynamics, we first compare our theory with the relaxation dynamics of a droplet deposited on a surface according to molecular kinetic theory (MKT). This theory, which has a microscopic basis, is shown to describe experimentally measured contact line dynamics rather well.[28,45−47] As discussed in the Theory section, for small values of cos θ – cos θeq, MKT predicts an exponential relaxation with a time scale τrlx given by eq . For greater values, however, the dynamics deviates from a simple single exponential description. To compare the nonlinear contact angle dynamics predicted by our model to that described by MKT, we solve eq numerically. For convenience, we set the equilibrium contact angle to θeq = π/2 and choose four initial angles θ0 symmetrically around this angle. In Figure , we compare the time dependence of the contact angle θ and the absolute value of the difference between the cosines of θ(t) and θeq. Indicated in the figures are the results of our model (blue triangles), the MKT result (green pluses), and a simple single exponential relaxation as given in eq (red crosses). Note that, as in all cases τrlx and θeq are fixed, the results indicated in the figures as “our model” are independent of the choice of the characteristic time scale and hence also describe the result for, e.g., the linearized version of the MKT model.

Figure 2

Comparison between predictions for the relaxation dynamics of the contact angle θ to its equilibrium value θeq = π/2 from our model (blue triangles), molecular kinetic theory (green pluses), and a simple exponential relaxation for comparison (red crosses). Left: contact angle θ as a function of dimensionless time t/τrlx; right: the corresponding |cos θ – cos θeq|.

Figure informs us of the following:

For deviations of ±π/6 from the equilibrium value of π/2 (see Figure c,e), the agreement between the evolution of the contact angle as a function of scaled time predicted by all three descriptions is excellent. For greater initial deviations from the equilibrium angle (Figure a,g), the agreement remains remarkably good, in particular for the larger initial angle.

Figure d,f highlights any inconsistencies for small deviations from the equilibrium by focusing on the difference of the cosines on a logarithmic scale. These figures show that well within one characteristic time scale simple single exponential decay is reached. Any small late-stage deviations between the curves is caused by the early-stage nonlinear behavior. Figure b,h shows that even for greater initial deviations from the equilibrium contact angle, simple single exponential decay occurs within one characteristic time scale.

The process of droplet evaporation in the presence of contact line pinning has been studied theoretically by Stauber et al.,[24] who describe the dependence of the evaporation time tevap on the initial contact angle θ0, where they fix the receding contact angle θr to several values. They consider two separate modes of evaporation, a constant contact radius (CCR, pinned) and a constant contact angle (CCA, receding) mode, allow for pinning–depinning transitions and model the evaporation dynamics accordingly using an evaporation description analogous to eq . Their results can be reproduced quantitatively by our model. However, our model also includes the relaxation of the droplet shape toward its equilibrium angle, after it is deposited on the surface with an angle different from the equilibrium value. We discuss in more detail the similarities and differences between their work and the results from our model in the next subsection.

We now relate results from our model to the experimental data of Belmiloud et al.[6] on the evaporation of a water droplet on a flat silicon surface (see Figure ). Figure shows the squared contact diameter (2a)2 (blue triangles) and contact angle θ (red crosses) as a function of time t. The results of Belmiloud et al., represented by the solid lines, can be readily reproduced by our phenomenological model (dashed lines). Initially, the contact line of the droplet is pinned, as is seen from the squared contact diameter remaining constant, while the contact angle decreases. As the receding contact angle θr is reached, a pinning–depinning transition occurs, after which the angle remains constant at the receding value and the diameter squared decreases linearly, as discussed in the Theory section.

Figure 3

Comparison between the results of experiments on the evaporation of sessile water droplets on a silica wafer[6] (solid lines) and the numerical evaluation of the droplet model (dashed lines). The squared contact diameter (2a)2 (left vertical axis, blue triangles) and contact angle θ (right vertical axis, red crosses) are shown as a function of time t. The two modes of evaporation, pinned and receding, are indicated. The model parameters are as follows: pinning force, fp ≃ 0.034 N m–1; vapor concentration difference, Δc = 11.6 × 10–3 kg m–3; and vapor diffusion coefficient, D = 3.15 × 10–5 m2 s–1.

To model the evaporation process of the initially pinned droplet, we choose our model parameters to correspond to the experimental values. The pinning force per unit length fp was set to a value of fp ≃ 0.034 N m–1 to obtain a receding contact angle of θr = 34°. The value of the pinning force per unit length is of the same order of magnitude as the liquid–air interfacial tension (γLG = 0.07 N m–1). The values reported by Belmiloud et al.[6] for the surface vapor concentration cs and the relative humidity were used to determine Δc = 11.6 × 10–3 kg m–3. The best correspondence between the measurement and our model is obtained for a vapor diffusion coefficient D = 3.15 × 10–5 m2 s–1, as opposed to the reported D = 2.60 × 10–5 m2 s–1. However, Belmiloud et al. also report on an underestimation of the evaporation rate: the droplet evaporates faster than predicted by eq .[6] This is arguably due to inaccuracies in measuring the properties of the ambient vapor.

Predictions by Full Model

We now consider the effect of the interplay between the three components of our model to predict the evaporation dynamics of a droplet. To that end, we first discuss two limiting cases. We report our findings on (1) the effect of contact line pinning on a nonevaporating, relaxing droplet and (2) the effect of shape relaxation on the lifetimes of droplets with an unpinned contact line. Subsequently, we present our results on simultaneous shape relaxation and evaporation of a droplet subject to contact line pinning.

If the shape relaxation of a droplet is affected by contact line pinning, the contact angle relaxation in the absence of evaporation studied in the previous section (Figure ) changes drastically, as is illustrated in Figure . If droplets start out within the fixed-area region, i.e., have an initial angle θr < θ0 < θa, indicated by the shaded region in Figure , then the contact line is not able to move. In other words, the droplets are not able to relax their shape to accommodate the equilibrium contact angle θeq. For initial angles outside of this regime, shape relaxation does occur, albeit only until the fixed-area region is reached, after which the motion of the contact line is halted. This phenomenon has strong implications for the lifetime of an evaporating droplet. The asymmetry in the time it takes for the droplet to become pinned for θ0 = π/6 and for θ0 = 5π/6 has its origin again, as is the case for the shape relaxation shown in Figure , in the nonlinearity of eqs and 6. We note that the curves shown in Figure depict the relaxation of the contact angle θ. The exponential relaxation of the cosine in eq is therefore not immediately evident from the figure.

Figure 4

Relaxation of the contact angle θ of a deposited drop on a surface toward the equilibrium value θeq = π/2. There is no evaporation and the pinning force fp is set such that the receding and advancing contact angles are θr = π/4 and θa = 3π/4. This results in the fixed-area range between θr and θa, where the capillary force fc cannot overcome fp and the contact line becomes or remains pinned.

If we allow for evaporation, the shape relaxation of a droplet from an initial contact angle θ0 toward its equilibrium angle θeq may have a strong impact on the evaporation dynamics of a droplet, also without any contact line pinning occurring. As discussed in the Theory section, the evaporation rate depends on the contact angle θ, and is at its minimum for θ = π/2. If a droplet with a certain θeq is deposited onto a surface with an initial angle θ0 ≠ θeq, the relative speeds at which the droplet relaxes to its equilibrium angle and at which it evaporates, characterized by the ratio τevap/τrlx, will influence the lifetime of such a droplet. In the remainder of our manuscript, we adopt the representation style of Stauber et al.[24] when discussing lifetimes of droplets, where we depict the scaled evaporation time tevap/τevap as a function of the initial contact angle θ0.

First, we consider the evaporation of droplets in the absence of contact line pinning. In Figure , we present the droplet lifetimes tevap/τevap as a function of the initial contact angle θ0, for τevap/τrlx = 10–4, 10–2, 100, and 102. In the limit of slow shape relaxation (τevap/τrlx ≪ 1), we exactly recover the result of Stauber et al.[24] for evaporation with a constant contact angle (see the blue triangles in Figure ). The droplet lifetime decreases rapidly for θ0 → 0, as the area-to-volume ratio increases. As already discussed in the Theory section, the lifetime is longest for θ = π/2, resulting in a maximum in the graph. For θ > π/2, the lifetimes slightly decrease again due to the increase in the area-to-volume ratio.

Figure 5

Scaled evaporation times tevap/τevap for sessile droplets as a function of the initial contact angle θ0, for ratios τevap/τrlx = 10–4, 10–2, 100, and 102. The equilibrium contact angle is θeq = π/2.

For increasing τevap/τrlx, we see a decreasing effect of the initial angle on the lifetimes, as the contact angles relax more quickly to the equilibrium value θeq = π/2. The increase in droplet lifetime due to faster shape relaxation is most notable for small contact angles, as the relaxation is the fastest in that regime and small changes to the contact angle induce large changes in the evaporation rate, according to eq . Close to θ0 = π/2, however, both shape relaxation and evaporation are slow and hence the changes in droplet lifetime upon changing τevap/τrlx are small. For τevap/τrlx ≫ 1, the droplet lifetime is effectively independent of θ0. We conclude that the evaporation dynamics of a droplet on a flat surface is strongly affected by the ratio between the rates of evaporation and shape relaxation, in the absence of contact line pinning.

If the contact line can become stuck on the surface due to pinning, however, the response of a drop to deposition on a surface becomes more complex. We can identify three regimes in the droplet dynamics, distinguished by the value of the initial contact angle θ0 relative to the receding and advancing contact angles θr and θa, respectively

θ0 < θr, where θ0 lies below the fixed-area range;

θr ≤ θ0 ≤ θa, where θ0 lies within the fixed-area range;

θ0 > θa, where θ0 lies above the fixed-area range.

Figure shows the droplet shape in terms of the scaled squared radius (a/a0)2 (blue triangles) and θ (red crosses) as a function of nondimensional time t/τevap. The pinning force fp is set to fp ≈ 0. 924 γLG, leading to θr = π/4 and θa= 3π/4. The contact angle range between the two (the fixed-area domain) is indicated by the shaded area. The ratio between evaporation and relaxation time scales is τevap/τrlx = 1. Three different graphs are shown to illustrate the three regimes. If θ0 < θr, the droplet starts out in the depinned state and thus the contact line moves to accommodate the contact angle relaxation toward the equilibrium value θeq (see Figure a). After some time, the receding contact angle θr is reached and the contact angle relaxation halts, resulting in the remainder of the evaporation process occurring with a constant contact angle θr. In the second case, shown in Figure b, the droplet starts out with a pinned contact line, since θr < θ0 < θa. Due to evaporation, the contact angle decreases until it reaches θr. At that point, a depinning transition occurs, the contact line is allowed to move again and evaporation continues with a constant contact angle θr. Parenthetically, in Figure b, we present the results for an initial contact angle θ0 < π/2, but the general behavior of the contact angle as a function of time is the same for π/2 ≤ θ0 < θa. Finally, for θ0 > θa, the contact line can initially move freely, causing the droplet to spread and increase its contact area (see Figure c). After the contact angle reaches the advancing value θa, however, the contact line becomes pinned. Once more, from this point on, the contact angle decreases due to evaporation. And again, as the contact angle reaches the receding value θr, a depinning transition occurs and evaporation continues with a constant contact angle.

Figure 6

Scaled contact area (a/a0)2 and contact angle θ as a function of nondimensional time t/τevap, for the evaporation of a droplet with equilibrium angle θeq = π/2. The shaded areas indicate the fixed-area domains between the receding and advancing contact angles, θr = π/4 and θa = 3π/4, respectively. The ratio between evaporation and relaxation time scales is set to τevap/τrlx = 1, and we show the time evolution of the droplet shape for three values of the initial contact angle θ0: (a) θ0 = π/6 < θr, (b) θr < θ0 = π/3 < θa, and (c) θ0 = 5π/6 > θa.

The emergence of the three regimes due to the presence of contact line pinning has a significant impact on the droplet lifetimes. The extent of the effect depends on the ratio of evaporation to relaxation time scales τevap/τrlx, as illustrated in Figure . Figure shows the scaled lifetime of an evaporating droplet tevap/τevap as a function of the initial contact angle θ0, for τevap/τrlx = 10–2, 100, and 102, covering the entire range from fast to slow evaporation. We can identify two limiting cases for the evaporation in all three graphs, shown in black, being evaporation with a constant contact radius (CCR, solid lines) and evaporation with a constant contact angle (CCA, dashed lines). These two limits are not dependent on the ratio τevap/τrlx, as we impose that either the contact area or the constant angle remains fixed.

Figure 7

Scaled evaporation times tevap/τevap of sessile droplets as a function of the initial contact angle θ0, for various values for fp/γLG ≈ 0.924, 0.707, 0.383, and 0, and three ratios τevap/τrlx of (a) 10–2, (b) 100, and (c) 102, covering the range from fast to slow evaporation. The values of the respective receding and advancing contact angles are indicated by arrows in the same color as the corresponding lines in the graph. The black lines represent the two limiting cases of evaporation at constant radius of the contact area (CCR, solid) and evaporation at a constant contact angle (CCA, dashed). The equilibrium contact angle is θeq = π/2.

The time it takes for a droplet to evaporate is shorter at a constant radius compared to a constant angle for the majority of the initial angle range 0 ≤ θ0 ≤ π. This is because the constant contact radius mode causes the contact angle to decrease during evaporation. Decreasing θ generally speeds up the evaporation process due to an increase of the surface-to-volume ratio, especially at late times. For large initial angles (θ0 → π), however, evaporation in the constant-angle mode becomes faster than evaporation in the constant-radius mode. As the latter causes a continuous decrease in the contact angle, it initially slows down the evaporation rate before speeding it up again. As discussed before, the maximum lifetime of a droplet evaporating in the constant-angle mode is tevap = τevap for θ0 = π/2, resulting in a maximum in the graph. For the constant-radius mode, the maximum lifetime is shorter than the maximum in the constant-angle mode, being tevap ≈ 0.9354τevap for θ0 ≈ 0.822π.[24]

The differently colored arrows at the top of Figure represent the receding contact angle θr and the advancing contact angle θa. These define the domain in which the contact area remains constant. The arrow colors correspond to the colors of the curves shown in the figure, which depict the droplet lifetimes tevap/τevap as a function of the initial contact angle θ0, for values of the pinning force fp/γLG of approximately 0.924 (blue triangles), 0.707 (green pluses), 0.383 (red crosses), and 0 (purple dots). The values for θr and θa remain constant in all three graphs, and as θeq = π/2, they take symmetric values around the equilibrium.

For all three graphs in Figure a–c, the curve segments between the bounding receding and advancing contact angles are identical. The reason is that if the contact line of a drop is initially pinned due to the choice of initial angle, this angle cannot relax toward its equilibrium value as is also shown in Figure b. Therefore, the magnitude of the shape relaxation rate does not affect the evaporation process. During evaporation, the contact angle only decreases from the initial to the receding angle and remains at that value until the drop has fully evaporated.

The relaxation of the contact angle toward its equilibrium value is only possible for initial angles outside of the fixed-area domain, as shown in Figure a,c, where shape relaxation occurs until the contact angle reaches either boundary. In other words, for values of θ0 outside of the fixed-area region, the relative shape relaxation rate does have an impact on the droplet lifetime. For values of τevap/τrlx ≳ 1, as shown in Figure b,c, evaporation is relatively slow and relaxation, in essence, is instantaneous. This leads to an evaporation time that is essentially an invariant of the initial angle, outside of the fixed-area domain, where the lifetime takes on the value at the nearest boundary (at θr or θa). If evaporation is very fast, i.e., τevap/τrlx ≪ 1, as shown in Figure a, relaxation cannot keep up and the evaporation time is dictated by a virtually constant contact angle. For sufficiently small θ0, however, relaxation can keep up with evaporation and the evaporation time deviates from the lifetimes for the constant contact angle mode. This deviation vanishes for τevap/τrlx → 0.

For initial angles above the advancing angle, the lifetime curves start to deviate from both limiting cases and from the curves reported by Stauber et al.,[24] when τevap/τrlx increases. This is caused by the evaporation dynamics predicted by our model being more complicated than a simple imposed transition from a pinned into a depinned state. As shown in Figure c, the contact line can initially move freely, implying that the contact angle starts to move toward its equilibrium value θeq. Upon reaching θa, the contact line becomes pinned and the contact angle decreases until it reaches θr. Subsequently, a pinning–depinning transition occurs and the droplet evaporates with a fixed contact angle. In other words, the droplet experiences two transitions, rather than one, by subsequently going through depinned, pinned, and depinned modes.

Impact of Assumptions

We developed a phenomenological model for the shape relaxation of an evaporating droplet. The characteristic time scale associated with this relaxation is found to be proportional to a length scale L. This length scale L has been connected to (1) a slip or friction length or (2) the size of the droplet. To incorporate the effects of either length scale on the evaporation dynamics, we have equipped eq , which describes the droplet relaxation, with a scale factor α. In the discussion of the results above, we chose α = 1 for simplicity.

Now we discuss in more detail the implications of considering an alternative α = α(t), which is proportional to the droplet sizewhere V(t) and V0 denote the instantaneous and initial droplet volumes, respectively. As a consequence, the relaxation process speeds up as the droplet size decreases. We find, however, that explicitly taking this effect into account hardly affects the droplet lifetime. This is caused by the circumstance that the capillary driving force is the strongest at short times, as the difference cos θ – cos θeq is then the greatest. In other words, the majority of the relaxation process occurs at short times. However, at short times, the droplet has hardly lost any volume by evaporation, which means that the scale factor α(t) ≈ 1, causing the relaxation processes for both expressions for α to occur in virtually the same manner. In Figure , we depict the squared scaled radius (a/a0)2 (blue triangles) and the contact angle θ (red crosses) of evaporating droplets as a function of scaled time t/τevap. We present the results for the scale factor α(t) = [V(t)/V0]1/3 (solid lines) compared to α = 1 (dashed lines), for three values of θ0. For these calculations, we do not incorporate a contact line pinning force, which means the droplet is allowed to relax its shape toward θeq = π/2, and τevap/τrlx = 1.

Figure 8

Comparison of the squared scaled radius (a/a0)2 and the contact angle θ between the alternative scale factor α(t) = [V(t)/V0]1/3 (solid lines) and the original α = 1 (dashed lines), as a function of scaled time t/τevap. τevap/τrlx = 1 and figures are shown for initial angles: (a) θ0 = π/6, (b) θ0 = π/3, (c) θ0 = 5π/6.

In all three graphs, we can clearly see that the dynamics described by the two expressions for α are identical at early times. Only after the droplet has partly evaporated, we see a slight deviation in the dynamics, due to the decrease of α(t). This effect only arises after approximately 30% of the evaporation time has passed. For large initial contact angles θ0, which we show in Figure c, the deviations between the graphs for the two expressions for α are slightly larger than those for smaller θ0 (Figure a,b). However, the time at which the droplet is fully evaporated is hardly affected. Note that the presence of contact line pinning would only decrease the effect shape relaxation has on the evaporation time, as it inhibits contact line motion for a certain range of the contact angle θ. We conclude from this that the lifetime of an evaporating droplet is not sensitive to our choice of α, hence explicitly taking into account the size dependence of the relaxation process has a negligible effect on the total evaporation time. We note that this phenomenon is valid regardless of the value of τevap/τrlx: a reproduction of Figure with the alternative scale factor α yields the same graph.

A second point left for discussion is the influence of the value of the equilibrium angle θeq on the evaporation dynamics of a droplet. We recall that an equilibrium contact angle of π/2 has two implications for the evaporation dynamics of a droplet. First, evaporation is slowest at θ = π/2, as described by eq . Second, in the presence of contact line pinning, the fixed-area domain is located symmetrically around the equilibrium angle: as cos θeq = 0, the advancing contact angle is given by θa = π – θr, as described by eqs and 18. Both properties are not valid for droplets with an equilibrium contact angle θeq ≠ π/2. A droplet taking on a nonhemispherical equilibrium results, on the one hand, in the fact that faster shape relaxation, or increasing τevap/τrlx, not necessarily implies slower evaporation, as we have shown in Figure , but that this depends on the initial and equilibrium contact angles. On the other hand, the receding and advancing contact angles are now located asymmetrically around the equilibrium angle.

As we have seen above, if contact line pinning occurs, the time it takes for a droplet to evaporate depends strongly on the values of the receding and advancing contact angles (see Figure ). For contact angles in between the two, contact line motion is inhibited. For θeq ≠ π/2, this principle is still valid, only θr and θa are located asymmetrically around θeq. For initial angles within the fixed-area domain, the lifetimes remain unchanged with respect to θeq = π/2. As contact line motion is inhibited there, shape relaxation is blocked, so the value of θeq is irrelevant. For initial angles outside of the domain, the lifetime as a function of initial angle behaves similar to what we show in Figure : for increasing shape relaxation rates, i.e., increasing τevap/τrlx, the evaporation time of a droplet converges to the values at the boundaries and it becomes increasingly less dependent on the initial contact angle θ0.

We conclude from the above that the specific value of the equilibrium contact angle θeq has little effect on the general behavior of the droplet lifetime as a function of the initial contact angle. It does affect the evaporation dynamics, but to an extent that is limited to two factors. On the one hand, it determines the evaporation time at the equilibrium angle, so it affects the droplet lifetime most in the absence of contact line pinning and in the limit of fast shape relaxation. On the other hand, it determines, together with the magnitude of the pinning force fp, the locations of the receding and advancing contact angles, which in turn define the region in which contact line motion is inhibited.

Summary and Conclusions

In conclusion, we propose a model for diffusive evaporation of a droplet on a flat surface, which accounts for the relaxation of the contact angle toward its equilibrium value. This shape relaxation is driven by the tendency of the droplet to reach its minimum free-energy state. We also model pinning of the contact line onto the surface by introducing a pinning force, insisting that the contact line remains pinned as long as the capillary forces are not able to overcome this threshold force.

Within our model description, the time it takes for a droplet to evaporate turns out to depend on five parameters: the initial and equilibrium contact angles, the characteristic time scales associated with shape relaxation and evaporation, and the magnitude of the contact line pinning force. The ratio between the two characteristic time scales describes the competition between shape relaxation and evaporation, which has a significant effect on the droplets’ lifetime. In the limit of slow relaxation (or fast evaporation), the total evaporation time of a droplet strongly depends on the initial contact angle, whereas for fast relaxation, the lifetime is virtually unaffected by the value of the initial contact angle.

The presence of a pinning force results in a contact angle range for which the contact line is fixed, as the capillary forces are not capable of overcoming the pinning force. This regime is bounded by the receding and advancing contact angles and as long as the contact angle resides within this range, the contact area remains constant. The magnitude of the pinning force determines the values of the receding and advancing contact angles and therefore has an impact on the lifetime of an evaporating droplet: the shape relaxation of a droplet becomes partly suppressed because the droplet cannot relax its shape for contact angles within this fixed-area regime.

We show that shape relaxation has a significant impact on the evaporation time of a droplet, both in the absence and in the presence of contact line pinning. Explicitly taking into account the size dependence of the relaxation process turns out to have virtually no effect on the droplet’s lifetime, since the majority of the relaxation occurs at short times for which the droplet size has hardly decreased. The value of θeq does also not affect the general dynamical behavior; however, it does define the lifetime for a droplet at its equilibrium angle and the location of the receding and advancing contact angles.

Finally, the simplicity of our model allows for relatively straightforward evaluation of the dynamics of an evaporating droplet. This means that it can also be readily extended to, e.g., take into account compound exchange between the solid phase and the liquid phase, or investigate an evaporation process wherein the droplet properties do not remain constant in time.