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- Droplet vaporization
- Macroscopic Model for Sessile Droplet Evaporation on a Flat Surface
- We apologize for the inconvenience...
- Evaporation of inclined water droplets

*During liquid evaporation, the equations for the vapor concentration in the atmosphere and for the temperature in the liquid are coupled and must be solved in an iterative manner. In the present paper, a combined field approach which unifies the coupled fields into one single hybrid field and thus makes the iteration unnecessary is proposed.*

## Droplet vaporization

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. 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.

It captures and extends the evaluation of Stauber et al. 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. 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.

If the shape of the droplet is described as a spherical cap, it is uniquely defined by only two parameters. Out of equilibrium, eq 3 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.

Together with the volume V , this defines a new radius of the contact area a. 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.

This is in analogy to the so-called model A dynamics commonly applied in the kinetics of phase transitions of nonconserved order parameters. This yields. Equations 4 and 6 reproduce eq 5 for small deviations from equilibrium. We return to this below. We discuss our model for contact line pinning below. In 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.

As 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 structure of eq 6 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 5. For example, from a microscopic perspective, the motion of the contact line is often described by the so-called molecular kinetic theory MKT.

It has been shown to predict contact line dynamics in agreement with experiments and molecular simulation. If we translate eq 10 in terms of the time evolution of the cosine of the contact angle, i. 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. This indicates that the scaling of the characteristic relaxation time scale with viscosity, droplet size, and interfacial tension, as described in eq 9 , 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.

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.

The rate of change of the volume V of a droplet can then be written as. Indeed, the error of the approximant is less than 0. This gives. 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 is the phenomenon where the contact line of the droplet becomes stuck, permanently or temporarily, on structural or chemical inhomogeneities of the supporting surface.

We model the influence of surface heterogeneities by introducing a net macroscopic threshold force per unit length, f p , 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, f c. As both f p and f c 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 f p , then the contact line remains pinned. 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 f p does not depend on the position on the surface. We define the capillary force as. The magnitude of the pinning force f p defines a contact angle range in which the capillary force f c is too weak to overcome pinning. Within our model, the values of these quantities depend on the pinning force f p , 40 according to.

The receding and advancing contact angles indicate the points at which the pinning—depinning transitions occur. 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.

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. 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.

We choose the droplets to be hemispherical in equilibrium, i. 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. 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. 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 10 numerically. 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 5 red crosses.

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.

The process of droplet evaporation in the presence of contact line pinning has been studied theoretically by Stauber et al. 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. The results of Belmiloud et al. Initially, the contact line of the droplet is pinned, as is seen from the squared contact diameter remaining constant, while the contact angle decreases. 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 two modes of evaporation, pinned and receding, are indicated.

To model the evaporation process of the initially pinned droplet, we choose our model parameters to correspond to the experimental values.

The values reported by Belmiloud et al. However, Belmiloud et al. 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 droplets start out within the fixed-area region, i. 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 exponential relaxation of the cosine in eq 5 is therefore not immediately evident from the figure.

In the remainder of our manuscript, we adopt the representation style of Stauber et al. First, we consider the evaporation of droplets in the absence of contact line pinning.

## Macroscopic Model for Sessile Droplet Evaporation on a Flat Surface

Vapor mass flux density has been calculated on the surface of spherical droplets under almost isothermal conditions. The calculations have been performed at droplet radii corresponding to the free-molecular, intermediate, and continuum flow regimes. The study has been performed by the direct numerical solution of the Boltzmann kinetic equation. The method of characteristics has been adapted for describing transfer processes in spherically symmetric systems. The applicability of different simplified approaches to the calculation of mass flux density on droplet surface has been estimated.

A new evaporation rate equation is developed valid for all K,. tion of a spherical droplet of radius R, whch is motionless relative to In the derivation the vapor.

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When a drop is placed on a flat substrate tilted at an inclined angle, it can be deformed by gravity and its initial contact angle divides into front and rear contact angles by inclination. Here we study on evaporation dynamics of a pure water droplet on a flat solid substrate by controlling substrate inclination and measuring mass and volume changes of an evaporating droplet with time. We find that complete evaporation time of an inclined droplet becomes longer as gravitational influence by inclination becomes stronger. The gravity itself does not change the evaporation dynamics directly, whereas the gravity-induced droplet deformation increases the difference between front and rear angles, which quickens the onset of depinning and consequently reduces the contact radius. This result makes the evaporation rate of an inclined droplet to be slow.

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### Evaporation of inclined water droplets

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.

The evolving composition of evaporating ethanol — water droplets initially The dependence of the evaporation rate on the relative humidity of the surrounding gas phase is also reported. The measured data are compared with both a quasi-steady state model and with numerical simulations of the evaporation process. Results from the numerical simulations are shown to agree closely with the measurements when the stimulated signal is assumed to arise from an outer shell with a probe depth of 2. Further, the time-dependent measurements are shown to be sensitive to the development of concentration gradients within evaporating droplets.

The wetting of solid surfaces using liquid droplets has been studied since the early s. Thomas Young and Pierre-Simon Laplace investigated the wetting properties, as well as the role of the contact angle and the coupling of a liquid and solid, on the contact angle formation. The geometry of a sessile droplet is relatively simple. However, it is sufficiently complex to be applied for solving and prediction of real-life situations for example, metallic inks for inkjet printing, the spreading of pesticides on leaves, the dropping of whole blood, the spreading of blood serum, and drying for medical applications. Moreover, when taking into account both wetting and evaporation, a simple droplet becomes a very complex problem, and has been investigated by a number of researchers worldwide.

The vaporizing droplet droplet vaporization problem is a challenging issue in fluid dynamics. It is part of many engineering situations involving the transport and computation of sprays: fuel injection , spray painting , aerosol spray , flashing releases… In most of these engineering situations there is a relative motion between the droplet and the surrounding gas. The gas flow over the droplet has many features of the gas flow over a rigid sphere: pressure gradient , viscous boundary layer , wake.

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