Dynamic Roughness Ratio-Based Framework for Modeling Mixed

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Dynamic Roughness Ratio Based Framework for Modeling Mixed Mode of Droplet Evaporation Madhu Ranjan Gunjan, and Rishi Raj Langmuir, Just Accepted Manuscript • Publication Date (Web): 22 Jun 2017 Downloaded from http://pubs.acs.org on June 23, 2017

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Dynamic Roughness Ratio Based Framework for

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Modeling Mixed Mode of Droplet Evaporation

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Madhu Ranjan Gunjan and Rishi Raj1

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Thermal and Fluid Transport Laboratory, Department of Mechanical Engineering, Indian Institute

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of Technology Patna, Bihar 801103, India

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KEYWORDS Droplet Evaporation, Mixed Mode, Contaminants, Contact line

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ABSTRACT The spatio-temporal evolution of an evaporating sessile droplet and its effect on

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lifetime is crucial to various disciplines of science and technology. While experimental

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investigations suggest three distinct modes through which a droplet evaporates, namely, the

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constant contact radius (CCR), the constant contact angle (CCA), and the mixed, only the CCR

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and the CCA modes have been modeled reasonably. Here we use experiments with water droplets

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on flat and micropillared silicon substrates to characterize the mixed mode. We visualize that a

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perfect CCA mode after the initial CCR mode is an idealization on a flat silicon substrate, and the

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receding contact line undergoes intermittent but recurring pinning (CCR mode) as it encounters

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fresh contaminants on the surface. Resulting increase in roughness lowers the contact angle of the

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droplet during these intermittent CCR modes until next de-pinning event followed by the CCA

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mode of evaporation. The airborne contaminants in our experiments are mostly loosely adhered to

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Corresponding Author: R113, Block III, IIT Patna, Bihta, Bihar 801103, India. Email: [email protected], Ph: +91-612302-8166

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the surface and travel along with the receding contact line. Resulting gradual increase in the

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apparent roughness and hence the extent of CCR mode over the CCA mode forces appreciable

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decrease in the contact angle observed during the mixed mode of evaporation. Unlike loosely

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adhered airborne contaminants on flat samples, micropillars act as fixed roughness features. The

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apparent roughness fluctuates about the mean value as the contact line recedes between pillars.

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Evaporation on these surfaces exhibits stick-jump motion with a short-duration mixed mode

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towards the end when the droplet size becomes comparable to the pillar spacing. We incorporate

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this dynamic roughness in a classical evaporation model to accurately predict the droplet evolution

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throughout the three modes, both for flat and micropillared silicon surfaces. We believe that this

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framework can also be extended to model evaporation of nanofluids and the coffee-ring effect,

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among others.

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Introduction

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Evaporation of small sessile droplets has been studied extensively for a wide range of applications

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such as spray cooling1, 2 and microlenses3. Similarly, the technique of controlled deposition of

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solute particles using droplet evaporation has been employed in DNA/RNA stretching4 and inkjet

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printing for fabricating miniaturized electronic devices5. Much of the thrust of the scientific

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research in this field is hence focused on the development of mathematical models for an accurate

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prediction of droplet lifetime.

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One of the striking features as illustrated through Figure 1 is the different modes in which

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a droplet evaporates on a surface. Since the contact angle hysteresis (CAH: the difference between

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the advancing/initial contact angle 𝜃𝐼 and receding contact angle 𝜃𝑅 ) is zero for a hypothetical

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ideal surface (uncontaminated, atomically smooth and homogenous), loss of droplet mass due to

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evaporation decreases the contact radius keeping the contact angle constant, i.e., a constant contact


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angle (CCA) mode (Figure 1a). On the other hand of the spectrum lie heavily contaminated high

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CAH surfaces (rough and heterogeneous) where a droplet evaporates with pinned contact line

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throughout its lifetime, i.e., a constant contact radius (CCR) mode (Figure 1b). Most practical

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engineering surfaces fall somewhere in between these two extremes wherein contaminants

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introduce a finite difference between the initial angle and the angle at which the contact line

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recedes. Irrespective of the extra care taken to make the surface free from any optically perceivable

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imperfections, a perfect CCA mode (Figure 1a) is an idealization and molecular reordering at the

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solid-solid or solid-liquid interface6 may still result in finite CAH as shown in Figure 1c.

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Figure 1. Modes of droplet evaporation: (a) Droplets on an ideal surface with zero CAH evaporates

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in the CCA mode; (b) Contact line on heavily contaminated and/or rough surfaces does not recede

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at all and the droplet evaporates in the CCR mode; (c) Droplets on surfaces with finite CAH

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evaporate first in the CCR mode followed by the CCA mode until complete evaporation; (d)

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Evaporation behavior observed in a typical experiment wherein CCR and CCA modes are followed

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by a mixed mode towards the end of droplet lifetime.


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Evaporating droplet in such a case accommodates the mass loss via a decrease in contact

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angle first through the pinning/constant contact radius (CCR) mode. Once the contact angle

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reduces from the initial value of 𝜃𝐼 to the receding contact angle of 𝜃𝑅 (Figure 1c), the contact line

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de-pins followed by evaporation in the CCA mode. Interestingly, most experimental reports

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suggest that the scenario in Figure 1c is also an idealization and the evaporation through the CCR

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and the CCA modes is usually followed by a third mode wherein both the contact angle and the

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contact radius decrease simultaneously (mixed mode in Figure 1d) towards the end of droplet

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lifetime7-14, 25-30.

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Following the pioneering work of Picknett and Bexon7, the science of droplet evaporation

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has been relentlessly updated with the emergence of newer models and their subsequent validation

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using well controlled experiments. Rigorous studies have been attempted to quantify the contact

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line dynamics during evaporation in terms of observable parameters and derive analytical

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expressions for droplet lifetime8, 15-21. Much of the earlier work8, 15, 16 was focused on modeling the

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CCR mode for the large CAH surfaces (Figure 1b) where the droplet evaporated with a pinned

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contact line throughout its lifetime. Assuming spherical cap geometry and a diffusion driven

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evaporation model, an approximate analytical expression was developed by Bourgès-Monnier and

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Shanahan8. Hu and Larson15 developed a similar expression for droplet lifetime in the CCR mode,

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but with a different function to account for the effect of the presence of a substrate. They reported

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that the evaporation rate was proportional to the contact radius, which corroborated with the

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qualitative findings of Picknett and Bexon7. Similarly, Erbil et al.22 explored droplet evaporation

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of organic fluids in pure CCA mode.

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Recent studies

13-14, 17, 20-21, 23-24

have attempted to develop unified model such that both the

CCR and the CCA modes can be captured together using a single model. For example, Raj et al.24


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used the classical model of Bourgès-Monnier and Shanahan8 along with the knowledge of contact

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line evolution to develop a unified model for droplet evaporation in the CCR mode followed by

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the CCA mode (the stick-slide (SS) mode). Despite the simplifying assumptions relating to the

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presence of modes of evaporation, experimental observations in most of these studies 13-14,

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suggest the presence of a third mode (Figure 1d) where the contact angle and the contact radius

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decrease simultaneously.

24

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While droplet evaporation models have been subjected to continuous improvements over the

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years and subsequent development of new models has provided us with a better understanding of

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the contact line dynamics during the CCR and CCA modes, studies elucidating reasons for the

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onset of mixed mode and its effect on droplet lifetime are limited. Mixed mode of droplet

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evaporation has been reported in the experimental studies of Bourgès-Monnier and Shanahan8,

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Kim et al.9, Nguyen et al.10, Pittoni et al.11, 12, Dash and Garimella13, Hu et al.14, Park et al.25, Lin

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et al.26 Askounis et al.27, Nguyen et al.28, Furuta et al.29 and Li et al.30. However, the discussion in

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these studies are mostly limited to a qualitative description of the mixed mode without any

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modeling attempts. Bourgès-Monnier and Shanahan8 attributed the onset of mixed mode to

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anchoring of the three-phase contact line at the local heterogeneities towards the end of the droplet

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lifetime. Conversely, Kim et al.9 attributed mixed mode to Marangoni instability towards the end

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of the droplet lifetime. More recently, Park et al.25 carried out an experimental study of droplet

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evaporation over a Teflon coated hydrophobic glass slide followed by a quantitative description

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of the mixed mode. They argued that the contact line deposits (contamination) and the associated

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adsorbed liquid films induces Cassie-Baxter heterogeneity which cause modulations in the contact

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line leading to a sharp decrease in contact angle during the mixed mode.


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Finally, most of the experimental studies9, 10, 25, 26, 28-30 report a slight but gradual decrease in

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contact angle even during the CCA mode. The decrease in contact angle becomes significant

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towards the end of droplet lifetime. Little understanding of the underlying contact line dynamics

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and the absence of quantitative modeling tools has forced the community to rely on rather

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qualitative visual interpretation of contact angle results to demarcate the mixed mode from the

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CCA mode.

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The preceding discussion highlights a very important aspect of solid-liquid interactions which

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has not been studied at a deeper level, especially in the context of droplet evaporation. Taking cue

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from the fact that a dynamic surface morphology can greatly influence the evolution of geometric

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parameters of an evaporating droplet, we perform systematic experiments with water droplets on

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flat silicon substrates with varying level of airborne contaminants. We analyze the experimental

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contact angle results to propose a unified framework for modeling droplet lifetime across all the

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three modes of evaporation, i.e. the CCR, the CCA, and the mixed modes. Our framework relies

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on experimental observations wherein the Wenzel surface roughness ratio31 at the receding contact

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line of an evaporating droplet was observed to increase continuously due to the accumulation of

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contaminants. We incorporate a classical evaporation model of Bourgès-Monnier and Shanahan8

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in this dynamic surface roughness based modeling framework to accurately predict the overall

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droplet lifetime on surfaces with various levels of contaminants. We next validate our model with

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experiments on micropillared silicon surfaces. Finally, we use the modeling framework to develop

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a quantitative criteria for defining the onset of mixed mode during evaporation on practical

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engineering surfaces.


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Experiments

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Well controlled dynamic contact angle measurements were performed using an automatic contact

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angle meter (MCA-3, Kyowa Interface Science). All experiments were carried out under normal

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laboratory conditions (temperature 𝑇: 23 − 28℃, relative humidity 𝑅𝐻: 43 − 60%). A piezo

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inkjet (PIJ) head was used to discharge water droplets at ~1.2 nL/s. Liquid addition was stopped

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when the overall droplet volume (~14 − 23 nL) was sufficient to allow enough time for

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monitoring the various modes of evaporation. Once the liquid addition was stopped, the evolution

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of contact angle, contact radius, and the droplet height were monitored using an in-built image

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analysis software (FAMAS) which employs half-angle method to measure the contact angle. Ultra-

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clean Millipore water was used as the test fluid. Presence of little residue after evaporation over a

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freshly prepared sample ruled out any appreciable concentration of solute particle in the test fluid.

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Ambient temperature and relative humidity were monitored by placing the respective sensors

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besides the substrate holder. The contact angle meter was positioned far from any convective

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device to minimize air flow near the substrate holder.

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Thermally oxidized undoped silicon test samples used for the experiments were cleaned using

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the methodology discussed in Raj et al.24. Freshly prepared samples were first sonicated for 10

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minutes in water and then rinsed with acetone and isopropyl alcohol. After the initial solvent

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cleaning, the samples were rinsed in water and blow dried using nitrogen. To understand the effect

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of airborne contaminants on the mixed mode of evaporation, we prepared two identical samples

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wherein no further cleaning was performed for the next four days. However, after the experiments

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on each day, the first sample was stored inside a vacuum desiccator (Sample A) while the second

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sample was left open under the laboratory environment (Sample B). Experiments in an interval of

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approximately twenty four hours were conducted for four consecutive days on each of these two


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surfaces. Top and side view images of the droplet were captured using the in-built microscopes of

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the contact angle meter. The top view images of the surface before the liquid addition and after the

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droplet evaporation were then processed using an in-house MATLAB® code to calculate dynamic

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surface roughness arising out of loosely adhered airborne contaminants.

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Results

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The evolution of the contact angle (𝜃), the contact radius (𝑎), and the droplet height (ℎ) for the

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two fresh samples, A and B, on day 1, are shown in Figure 2. The two fresh and nearly identical

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samples did not show any appreciable difference in the evolution profile wherein evaporation

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started with the CCR mode followed by the CCA mode. The contact angle was primarily constant

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in the CCA mode except for the very end of the droplet lifetime wherein a short duration mixed

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mode was observed in both cases. The tiny greenish spots formed after complete droplet

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evaporation (𝑡 = 44 s, Figure 2) may be attributed to the adsorbed liquid film and deposited ultra-

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small contaminants (either in the test fluid or on the surface) just before complete disappearance

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of the droplet as suggested by Park et al.25.


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Figure 2. Comparison of droplet evaporation on the first day. Evolution of the contact angle, the

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contact radius, and the height for (a) sample A, and (b) sample B. Time-lapse images of the

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evaporating droplet are shown below the respective profile evolution plots. Test conditions: 𝑅𝐻 =

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48 − 49 %, and 𝑇 = 26 − 27 ℃, initial droplet volume ~ 15nL. Scale bars correspond

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to 200 μm.

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A qualitative comparison of droplet evaporation on the two samples and on different days of experiments, essentially implying varying degrees of contamination, is shown in Figure 3.


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Figure 3. Comparison of droplet profile evolution for different degrees of contamination. Column

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(a) elucidates the geometric evolution of the droplet, column (b) is the top view image of the

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surface before liquid addition and column (c) is the image of the same surface after droplet

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evaporation. The figure illustrates a more pronounced mixed mode with increase in airborne

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contaminants on the surface. Test conditions for the experimental results of column (a): 𝑅𝐻 =

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47 − 49 %, and 𝑇 = 24 − 27 ℃, Droplet volume 𝑉 is as indicated in the plots of column (a). Scale

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bar corresponds to 200 μm.


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The first row illustrates the evaporation profile for sample A on the second day whereas

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the second and the third rows present the profiles for sample B (open to ambient) on the second

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and the third days, respectively. Visual inspection of the images clearly corroborate the increase

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in the extent of mixed mode to the increased contamination (indicated by black arrows in Figure

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3b) upon prolonged exposure to ambient. Comparison of the surface images before (Figure 3b)

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and after droplet evaporation (Figure 3c) suggests that most of the visible airborne contaminants

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are loosely adhered to the surface and travel along with the receding contact line.

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The effect of contaminants on droplet dynamics is further illustrated through Figure 4. The

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time-lapse images of the top view on the second day experiment on sample B elucidates the

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underlying dynamic interaction between the airborne contaminants deposited on the surface and

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the contact line that causes recurring pinning. Dragging of a loosely adhered contaminant and the

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subsequent pinning at different contact radii can be observed between 19 and 26 seconds.

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Simultaneous pinning and shifting of droplet footprint center can be observed between 35 and 48

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seconds. Time-lapse images at 35 and 36 seconds also illustrate that the contaminants which are

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either large or strongly adhered to the surface may not always get dragged along with the contact

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line. Nonetheless, these images confirm the fact that the majority of randomly deposited

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contaminants aggregate at the contact line causing a gradual rise in surface roughness.


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Figure 4. Temporal snapshots of the top view of an evaporating droplet on sample B for the

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experiments performed on the second day (corresponding to the results on the second row in Figure

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3). The locations of local pinning and de-pinning due to contaminants are marked by dashed red

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circles. The plus ‘+’ and the star ‘*’ symbols correspond to the location of the initial droplet

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footprint center and the shifted droplet footprint center, respectively. Scale bar corresponds to

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200 μm.


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Modeling

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Following the new insights through the contact angle experiments on evaporating water droplets,

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we now discuss the development of a unified modeling framework for droplet profile prediction

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through the three stages of evaporation. We propose that the pinning action of the contaminants

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present at the contact line alters the receding contact angle continuously until complete droplet

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evaporation. Most of the airborne contaminants in our experiments are loosely adhered to the

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surface and travel along with the receding contact line. Accordingly, the apparent roughness ratio

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increases gradually to force recurring pinning (CCR) and de-pinning (CCA) of the contact line.

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As the droplet recedes to small sizes, contact line pinning events become frequent and an

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appreciable decrease in contact angle may be observed even with the receding contact line. This is

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due to the fact that while the contact line may be pinned along the line-of-sight (line-of-

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visualization) in a particular experiment, the contact line may recede along other directions (Please

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see supplementary video SV1 which visualizes the line-of-sight dependence during a typical

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experiment). The resulting anisotropy forces the droplet contact line shape to deviate from the

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idealized circle (Figure 4). Accordingly, the apparent contact angle along any line-of-sight is

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affected by the actual contact line shape and the droplet experiences a simultaneous decrease in

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both the contact angle and the contact radius. The effect of pinning forces become dominant when

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the droplet becomes small and is usually manifested as the mixed mode of droplet evaporation.

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We use the classical Wenzel31 equation to represent the contaminant induced roughness as

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heterogeneities on the surface. We then use the modified Cassie-Baxter24 equation to account for

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these contaminant induced heterogeneities and estimate the effective receding contact angle. The

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concentric circles in the top view (Figure 5a) represent the contact line at various instants during


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droplet evaporation. The circle in red corresponds to the CCR mode and the green ones represent

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the receding contact line during the CCA and mixed modes. The small black spots represent

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contaminants. Let us consider a particular instant of the receding contact line shown by the dashed

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green circle in Figure 5a. Let there be 𝑛 number of randomly positioned contaminants/particles of

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varying shapes and sizes at this contact line. Assuming the contaminants as heterogeneities and

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using the modified Cassie-Baxter equation24, we define the modified receding contact angle 𝜃𝑅∗

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(Figure 5b) at this contact line by:

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cos 𝜃𝑅∗ = ∑𝑛𝑗=1 𝑓𝑗 cos 𝜃𝑗 + (1 − ∑𝑛𝑗=1 𝑓𝑗 ) cos 𝜃𝐹

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where 𝜃𝐹 is the intrinsic receding contact angle on the flat surface without contaminants and 𝑓𝑗 is

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the contact line fraction for the 𝑗𝑡ℎ contaminant.

(1)

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Figure 5. Schematic of the top (a) and the side (b) view of a typical contaminated substrate. The

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circle shown in red in (a) is the initial contact line with 𝑚 number of contaminants represented

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here by black spots. The intermediate contact lines during the receding phase are shown in green.

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The dashed green circle represents one of the instants of the receding contact line with 𝑛 number

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of contaminants randomly distributed over it. 𝜃𝑅 and 𝜃𝑅∗ represent the receding contact angle at the


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onset of the receding mode at the maximum contact radius and at an intermediate contact radius

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respectively. The contact angle profile in (c) illustrates the effect of recurring pinning. Local

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pinning during the CCA mode causes slight deviation from an idealized CCA mode which

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eventually turns into a mixed mode. The zoomed in portion of the mixed mode is illustrated with

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a staircase feature to characterize the recurring CCR-CCA framework along one line-of-sight.

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We define the contact line fraction as the ratio of the average diameter of the contaminant

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to the perimeter24 of the contact line on which it lies (please see section S1 in the supporting

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information for a detailed discussion on contaminant to heterogeneity transformation). To define

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the intrinsic contact angle 𝜃𝑗 of the 𝑗𝑡ℎ contaminant when modeled as a heterogeneity, we use a

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Wenzel formulation in terms of 𝜃𝐹 as follows:

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cos 𝜃𝑗 = 𝑟𝑗 cos 𝜃𝐹 , 𝑗 = 1, 2, 3, … , 𝑛

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where, 𝑟𝑗 is the Wenzel roughness ratio (defined as the ratio of the total surface area to the projected

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area) for the 𝑗𝑡ℎ contaminant. Please note that our model makes a simplifying assumption wherein

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the intrinsic wettability of the airborne contaminant is assumed to be equal to the intrinsic

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wettability 𝜃𝐹 of the flat surface without any contaminants. Using Equation 2 in Equation 1, we

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get the modified receding contact angle 𝜃𝑅∗ as follows:

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cos 𝜃𝑅∗ = {∑𝑛𝑗=1 𝑟𝑗 𝑓𝑗 + (1 − ∑𝑛𝑗=1 𝑓𝑗 )} cos 𝜃𝐹 .

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We next use the knowledge of number and size of contaminants at the maximum contact radius

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(which is the contact radius for CCR mode, red circle in Figure 5a) and the corresponding de-

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pinning/receding contact angle 𝜃𝑅 (for the onset of CCA mode) to estimate 𝜃𝐹 , i.e., the receding

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contact angle of the hypothetical uncontaminated sample as follows:

(2)

(3)


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cos 𝜃𝑅 + (1 − ∑𝑚 𝑘=1 𝑓𝑘 )]

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cos 𝜃𝐹 =

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Using this expression for cos 𝜃𝐹 in Equation 3 and rearranging the terms, we get the modified

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receding contact angle 𝜃𝑅∗ at every contact radius due to the presence of random contaminants as

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follows:

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cos 𝜃𝑅∗

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where 𝑟 ∗ = (1+∑𝑚𝑗=1( 𝑟

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Please note that the knowledge of Wenzel roughness ratio 𝑟𝑗/𝑘 and the contact line fraction 𝑓𝑗/𝑘

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for all contaminants lying between initial (red solid circle in Figure 5a, at time 𝑡 = 0) contact line

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and the current contact line (green dashed circle in Figure 5a) is required to estimate the modified

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receding contact angle 𝜃𝑅∗ at any instant. It can be readily observed that Equation 5 essentially

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transforms roughness into heterogeneity which suggests that the modified receding contact angle

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𝜃𝑅∗ depends on the effective roughness ratio 𝑟 ∗ and can be suitably modeled using the location of

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contaminants underneath the droplet. We use a fixed value of 4 as the Wenzel roughness ratio 𝑟𝑗/𝑘

271

for all the contaminants throughout our calculations. The rationale behind the assumed value of 4

272

is explained through Figure S2 and the following discussion in section S1 of the supporting

273

information. We next employ an image processing technique on the image of the surface before

274

liquid addition to identify individual contaminants and then calculate the effective contact line

275

fraction 𝑓𝑗/𝑘 . Further details on the image processing and the estimation of 𝑟𝑗/𝑘 and 𝑓𝑗/𝑘 for the

276

estimation of effective roughness ratio 𝑟 ∗ can be found in section S2 of the supporting information.

[∑𝑚 𝑘=1 𝑟𝑘 𝑓𝑘

1 + ∑𝑛𝑗=1( 𝑟𝑗 𝑓𝑗 − 𝑓𝑗 ) =( ) cos 𝜃𝑅 , or cos 𝜃𝑅∗ = 𝑟 ∗ cos 𝜃𝑅 ( ) 1 + ∑𝑚 𝑟 𝑓 − 𝑓 𝑘 𝑘=1 𝑘 𝑘 1+∑𝑛 ( 𝑟𝑗 𝑓𝑗 −𝑓𝑗 ) ) 𝑘=1 𝑘 𝑓𝑘 −𝑓𝑘 )

(4)

(5)

is the effective roughness ratio at any instant.


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We use a modified form of the lifetime equations23 for the CCR and CCA modes derived

278

from the classical diffusion controlled evaporation model of Bourgès-Monnier and Shanahan8 to

279

estimate the total droplet lifetime. Unlike Raj and Wang.23, the receding contact angle 𝜃𝑅∗ in our

280

model is not fixed and changes with the change in effective roughness ratio 𝑟 ∗ . The droplet

281

evaporates in very short duration CCR modes and the receding contact angle decreases repeatedly

282

due to the recurring pinning of the contact line at contaminants (Figure 5c). However, until the

283

very end of the life, the droplet spends significant portion of its lifetime in CCA mode (𝑡𝐶𝐶𝐴 ≫

284

𝑡𝐶𝐶𝑅 , Figure 5c). Hence, the overall decrease in the contact angle is not very significant and this

285

mode is generally referred to as the CCA mode in literature. Once the droplet recedes to very small

286

size, the effect of contaminants is more pronounced causing rapid switching between CCR and

287

CCA modes such that the time spent by the droplet in CCR and CCA modes are comparable

288

(𝑡𝐶𝐶𝐴 ~ 𝑡𝐶𝐶𝑅 ), i.e., a simultaneous decrease in the contact radius and the contact angle suggesting

289

the onset of the mixed mode of droplet evaporation.

290

The proposed framework captures the continuous switch between the CCR and CCA

291

modes by comparing the modified receding contact angle 𝜃𝑅∗ at a particular instant with the

292

instantaneous contact angle 𝜃 of the droplet. We have divided the total evaporation time as follows:

293

1 𝑡𝑡𝑜𝑡𝑎𝑙 = 𝑡𝐶𝐶𝑅 + ∑ 𝑡𝐶𝐶𝑅 + ∑ 𝑡𝐶𝐶𝐴

294

1 where 𝑡𝐶𝐶𝑅 is the time spent in the initial CCR mode which commences after liquid addition is

295

stopped. Conversely, ∑ 𝑡𝐶𝐶𝑅 and ∑ 𝑡𝐶𝐶𝐴 represent summation of times spent in the secondary CCR

296

and CCA modes (after the onset of CCA mode). The detailed expression and the solution algorithm

297

for the evaporation model is provided in section S3 of the supporting information. The framework

298

described above provides a numerical methodology to predict droplet profile evolution across all

(6)


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299

the three modes by coupling the dynamic roughness model (Equation 5) with the classical diffusion

300

-controlled evaporation model of Bourgès-Monnier and Shanahan8 which was originally derived

301

for the CCR mode only.

302

The implications of the dynamic roughness model (Equation 5) is illustrated through the

303

comparison of experiments and model prediction results (for results presented in Figure 3) in

304

Figure 6. Since we have assumed that the contact line drags away contaminants it encounters, the

305

effective roughness ratio 𝑟 ∗ increases with time (due to increasing contamination as well as

306

decreasing contact radius) and shoots up sharply towards the end of droplet lifetime (Figures 6b

307

and 6c, bottom row). This increase in effective roughness ratio 𝑟 ∗ continuously decreases the

308

modified receding contact angle 𝜃𝑅∗ . The model plots in the top row and the roughness ratio plots

309

in the bottom row mirror the effect on each other such that whenever there is an increase in the

310

effective roughness ratio 𝑟 ∗ , pinning of the contact line marked by a CCR mode is observed. When

311

the decreasing contact angle in the CCR mode attains the resulting modified receding contact

312

angle 𝜃𝑅∗ , the contact line de-pins marked by a CCA mode. However, unlike Figures 6b and 6c, the

313

model predicts almost a constant receding angle during the CCA mode for sample A with a sharp

314

dip towards the end as shown in Figure 6a. This is expected since sample A is much less

315

contaminated in comparison to sample B. Please note that our model not only captures the increase

316

in the extent of mixed mode due to increase in contaminants, but also models the gradual

317

slope/decrease in the contact angle, even during the CCA mode9, 10, 25, 26, 28-30 (Figure 6c and

318

supplementary video SV2). We rely on optical visualization to identify contaminants in our study,

319

and, hence, the effect of molecular or nanoscale inhomogeneity/roughness could not be

320

characterized in the current work. Nonetheless, we believe that the good agreement between the


18

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321

experiments and the prediction results suggests that the contribution of such factors are not

322

significant in our well controlled experiments.

323 324

Figure 6. Comparison between the model prediction and the experimental results presented in

325

Figure 3. The plots in top row in (a-c) compare the model predictions with the experimental results

326

corresponding to column (a) in Figure 3. The figures in bottom row are the corresponding plots of

327

effective roughness ratio 𝑟 ∗ . The inset images are the processed binary images of the surface before

328

liquid addition wherein the white spots correspond to visible airborne contaminants. Initial (𝜃𝐼 )

329

and receding contact angles (𝜃𝑅 ) corresponding to (a), (b) and (c) are: 48°, 34°; 50°, 40°; and

330

47°, 39°, respectively.

331

Our framework is based on prior (before the start of experiments) information of

332

contaminants and hence it cannot predict subtle deviations arising out of real time reordering of

333

contaminant due to dragging and separation from the three-phase contact line in space and time


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334

(supplementary video SV3). Actual information pertaining to the reordering of contaminants is

335

available only after the commencement of experiment beyond complete evaporation. Hence, it is

336

quite natural that if we use the information on time and location of separation of a particular

337

contaminant, we can account for the sporadic decrease in effective roughness ratio 𝑟 ∗ (and the

338

resulting increase/jump in contact angle) to predict experiments more accurately. This is shown

339

through Figure 7 where contact angle jump/increase observed at 𝑡 = 29 seconds and 𝑡 = 34

340

seconds can be related to the decrease in effective roughness ratio 𝑟 ∗ due to the separation of

341

strongly adhered contaminants from the three-phase contact line. Visual comparison of Figure 6c

342

and Figure 7a suggests that incorporation of real time information results in a more accurate

343

prediction of droplet profile evolution, which is reflected clearly in terms of the improved

344

agreement between the experimental and model predictions for contact radius (Figure 7a and

345

supplementary video SV4). Please note that a perfect agreement between the experiments (3-D)

346

and

347

contaminants/roughness are distributed in an axisymmetric manner such that a circular contact line

348

shape is maintained throughout evaporation.

the

model

(2-D

axisymmetric

predictions)

cannot

be

expected

unless

the


20

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349 350

Figure 7. Intermittent increase/jump in contact angle due to separation of contaminants from the

351

three-phase contact line. (a) Comparison of droplet profile evolution with the model prediction

352

based on the effective roughness ratio 𝑟 ∗ accounting for contaminant separation from the contact

353

line. (b) Effective roughness ratio 𝑟 ∗ as a function of time based on the information of contaminant

354

separation from the top view snapshots. Top and side view images of the evaporating droplet

355

corresponding to the instants of contact angle increase are shown below the plots. Locations of

356

pinning and subsequent separation of contaminants are indicated by arrowheads in the top view

357

snapshots. Scale bars correspond to 200 μm.


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358

We discuss next the overall effect of exposure to ambient on the duration of mixed mode

359

on sample A and sample B through Figure 8 which shows the contact angle profile for two sets of

360

experiments on the first and third day. The results for sample A are shown in Figure 8a and 8b

361

whereas those for sample B are shown in Figure 8c and 8d. An ideal CCA mode is described by

362

the violet dashed-dot horizontal straight line in Figure 8a and 8d. It can be readily observed that

363

increased contamination due to prolonged exposure to ambient has a significant effect wherein in

364

addition to a stronger mixed mode, the contact angle profile for sample B (third day, Figure 8d)

365

also shows an appreciable deviation from an idealized CCA mode. We attribute this to the

366

increased contaminants due to prolonged exposure to ambient for three days.

367

We next use the trends in Figure 8d to define the geometric criteria for the onset of mixed

368

mode. We recall the experimental results in Figure 3 and the schematic of the contact angle profile

369

in Figure 5c to note that the mixed mode is manifested when the evaporating droplet switches

370

frequently between the CCR and CCA modes. Frequent switching between the modes

371

implies 𝑡𝐶𝐶𝑅 ~ 𝑡𝐶𝐶𝐴 , suggesting that if we fit a straight line on to the mixed mode, it will intersect

372

the ideal (horizontal) CCA mode at ~ 45° (Figure 8d). Accordingly, a deviation of ~ 45 ° for the

373

contact angle profile from the ideal CCA mode will imply sharp reduction in contact angle and

374

can be assumed to mark the onset of mixed mode. Unlike Figure 8e, the general profile for a

375

contaminated surface would appear as shown in Figure 8f with a gradually decreasing contact

376

angle even during the CCA mode followed by a mixed mode towards the end. Please recall that

377

the gradual decrease in contact angle, even during the CCA mode, has also been reported in the

378

published literature9, 10, 25, 26, 28-30. However, lack of suitable modeling framework has forced the

379

community to determine the onset of mixed mode on a purely qualitative visual interpretation of


22

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380

the contact angle plots. We believe that the criteria for onset of mixed mode proposed above can

381

now onwards be used to standardize the demarcation of the mixed mode from the CCA mode.

382 383

Figure 8. Effect of prolonged exposure to ambient on the contact angle profile of an evaporating

384

droplet. A more pronounced mixed mode is observed for sample B on the third day along with a

385

gradual deviation from an idealized CCA mode (d). Unlike sample B, the contact angle profile

386

remains unaltered for sample A with a very little mixed mode even on the third day (b). Since in

387

mixed mode 𝑡𝐶𝐶𝑅 ~ 𝑡𝐶𝐶𝐴 , the contact angle would deviate from the idealized CCA line by ~ 45°

388

as shown in the inset image in (d). The images in (e) and (f) demonstrate the expected contact

389

angle behavior for an uncontaminated and a contaminated surface respectively wherein a more

390

pronounced mixed mode is predicted for the contaminated surface.


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391

The effect of the degree of atmospheric contamination on the model predictions in terms

392

of 𝑡̃𝐶𝐶𝐴 and mixed 𝑡̃𝑚𝑖𝑥𝑒𝑑 for both the samples is summarized in Table 1. The fraction of times

393

spent by the droplet in apparent CCA mode and mixed mode are denoted by 𝜒𝐶𝐶𝐴 and

394

𝜒𝑚𝑖𝑥𝑒𝑑 , where 𝜒𝐶𝐶𝐴 = 𝑡̃

395

we are only considering the model data points beyond the commencement of initial CCR mode.

396

The variation in 𝜒𝐶𝐶𝐴 and 𝜒𝑚𝑖𝑥𝑒𝑑 is presented in terms of the area fraction 𝑎𝑓 and the total number

397

𝑛̃ of the visible contaminants lying underneath the droplet at the start of the evaporation. The area

398

fraction 𝑎𝑓 here is defined as the projected area of the visible contaminants to the contact area of

399

the droplet at time 𝑡 = 0. The values of 𝑎𝑓, 𝑛̃, , 𝜒𝐶𝐶𝐴 , and 𝜒𝑚𝑖𝑥𝑒𝑑 presented in Table 1 are the

400

average of the values for two different tests on each days. We observe that unlike sample A which

401

does not show any marked difference between the durations of mixed mode on the first, second

402

and the third days, the duration of mixed mode on sample B increased significantly from 4 % to

403

30 %. This trend is also reflected in the value of area fraction 𝑎𝑓 for sample B which increased

404

from a negligible value of 8 × 10−6 on the first day to 2.7 × 10−3 on the third day. These results

405

emphasize that though the mixed mode is momentary on very clean surfaces and can be ignored

406

while modeling droplet evaporation, its effect may be significant on contaminated surfaces and

407

should be included in evaporation models.

𝑡̃𝐶𝐶𝐴 𝐶𝐶𝐴 +𝑡̃𝑚𝑖𝑥𝑒𝑑

𝑡̃𝑚𝑖𝑥𝑒𝑑 𝐶𝐶𝐴 +𝑡̃𝑚𝑖𝑥𝑒𝑑

, and 𝜒𝑚𝑖𝑥𝑒𝑑 = 𝑡̃

(Figure 8f). It is to be noted here that


24

Page 25 of 36

408

Table 1. . Model predictions for fractional time spent by the droplet in the CCA and the mixed

409

mode after the initial CCR mode. Contamination Sample A

Sample B

𝑎𝑓

𝑛̃

𝜒𝐶𝐶𝐴

𝜒𝑚𝑖𝑥𝑒𝑑

𝑎𝑓

𝑛̃

𝜒𝐶𝐶𝐴

𝜒𝑚𝑖𝑥𝑒𝑑

Day 1

0

0

1

0

8 × 10−6

1

0.96

0.04

Day 2

4 × 10−4

4

0.86

0.14

1.3 × 10−3

7

0.82

0.18

Day 3

5.3 × 10−4

5

0.87

0.13

2.7 × 10−3

12

0.7

0.3

Contamination

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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410 411

The results discussed so far deal with randomly distributed airborne contaminants wherein

412

the modeling framework required inputs from experiments to estimate the Wenzel roughness ratio

413

of 4. We believe that the dependence of our modeling framework on experimental results is

414

primarily due to the uncharacterized and uncontrolled and random nature of roughness features

415

(airborne contaminants) in the experiments. We next perform experiments on a well-defined

416

micropillared surface to show that our modeling framework, as such, does not need experimental

417

evidence

418

roughness/heterogeneities are well-defined.

to

calculate

the

Wenzel

roughness

ratio

for

surfaces

wherein

the

Droplet evaporation over textured surfaces also exhibit CCR, CCA and mixed modes13, 24,

419 420

32-38

421

between adjacent pillars/heterogeneities and is also referred to as the stick-slip35, 38 or stick-jump34

422

motion of the contact line. The mixed mode over textured surfaces are usually of short duration

423

and marked by a sharp dip in contact angle towards the end13, 24, 32, 34-36. Recently, Wang and Wu38

424

studied stick-slip motion on nanopillared surface during droplet evaporation using molecular

425

dynamics simulations. They found that the contact angle decreased in every stick stage to

. The CCA mode is marked by discrete jumps in contact angle as the contact line recedes


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426

overcome the pinning force due to pillars and then abruptly increased as the contact line de-pinned

427

and moved to the adjacent pillars resulting in stick-slip motion. We propose that the receding

428

contact line over textured surfaces experiences a fluctuating roughness as it moves between the

429

adjacent pillars/heterogeneities causing stick-jump motion during the CCA mode, and, as the

430

droplet size becomes comparable to that of the texture, the pinning mode (CCR) starts to dominate

431

leading to the onset of mixed mode.

432

We now validate our model with evaporation experiments on a textured silicon sample

433

with pillar diameter 𝐷 = 3 μm, pillar height 𝐻 = 2 μm, and pitch length 𝐿 = 12 μm. The surface

434

preparation and cleaning methodology are same as Raj et al.39 The plot of effective roughness ratio

435

𝑟 ∗ is shown in Figure 9a and the comparison between experimental results and model prediction

436

for contact angles are shown in Figure 9b. Unlike airborne contaminants on flat silicon samples,

437

the roughness features (pillars in this case) are not dragged away by the receding contact line. The

438

Wenzel roughness ratio 𝑟𝑗/𝑘 in this case is hence based only on the micropillar geometry (diameter

439

𝐷 and height 𝐻) and does not require inputs from experiments.

440

𝜋𝐷2 + 𝜋𝐷𝐻 4𝐻 4×2 𝑟= 4 =1+ =1+ = 3.67 2 𝜋𝐷 𝐷 3 4

(7)


26

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441 442

Figure 9. Comparison between experimental results and model predictions for droplet evaporation

443

over pillared surface: (a) Effective roughness ratio 𝑟 ∗ as a function of time, (b) Comparison of

444

droplet profile evolution with the model predictions based on the effective roughness ratio 𝑟 ∗ .

445

𝜃𝐼 and 𝜃𝑅 are 72.9° and 35° respectively. Relative humidity (𝑅𝐻) and temperature (𝑇) are 50 %

446

and 29℃, respectively. Scale bar corresponds to 200 μm.

447

The plot of effective roughness ratio 𝑟 ∗ reveals a predominantly constant value throughout

448

the droplet evaporation. However, the zoomed in image in the inset highlights minor but

449

continuous fluctuations as the contact line travels between the pillars (Figure 9a). These

450

fluctuations are relatively amplified as the droplet becomes smaller towards the end. Accordingly,

451

a stick-jump (since the roughness features are fixed) behavior is observed experimentally and also

452

captured through the model predictions (please see supplementary video SV5). Please note that a

453

similar stick-jump behavior was also observed for the case of fixed contaminant in Figure 7.

454

However, unlike Figure 7, our model did not need any inputs from experiment except for the

455

microstructure geometry. Moreover, these fluctuations and the drip towards the end of droplet


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456

lifetime (mixed mode) are insignificant in comparison to those with airborne contaminants

457

discussed earlier.

458

The results presented above highlight the significance of dynamic roughness in modeling

459

contact angles during droplet evaporation. The proposed framework accurately captures the

460

dynamic interaction of an evaporating fluid and the surface underneath, both for a randomly

461

contaminated sample and a well-defined textured sample, and throughout the three modes of

462

evaporation. The prediction of time and location of modes can be used in designing surfaces for

463

controlled evaporative deposition and can also be extended to develop predictive models for

464

nanofluids evaporation10, 27, 36 and coffee ring effect.10, 27

465

Conclusions

466

Systematic experiments were performed over flat and micropillared silicon surfaces to demonstrate

467

the effects of dynamic roughness on droplet evaporation. We show that the receding contact line

468

of an evaporating droplet is highly susceptible to contaminants deposited on the surface. Contact

469

angle measurement experiments suggest that a perfect CCA mode is an idealization and continuous

470

decrease in the receding contact angle can be observed even beyond the initial CCR mode of

471

evaporation. Time-lapse images of the top view of a receding droplet suggest that the majority of

472

the airborne contaminants are loosely adhered to the surface and gradually accumulate as they

473

travel along with the receding contact line. Receding contact line is then observed to pin and the

474

droplet evaporates in an intermittent CCR mode until next de-pinning event followed by the CCA

475

mode of evaporation. CCR mode starts dominating and the switch between the CCA and the CCR

476

modes along the line-of-sight become frequent towards the end of the droplet lifetime. The

477

resulting simultaneous decrease in the contact radius as well as the contact angle is manifested as


28

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478

the mixed mode of droplet evaporation. We next developed a unified modeling framework to

479

predict droplet evaporation throughout the three modes, namely CCR, CCA and mixed. We use

480

the Wenzel equation along with the modified Cassie-Baxter equation to transform the contaminant

481

induced roughness as heterogeneities on the surface and estimate the continuously decreasing

482

receding contact of an evaporating droplet. This model is then coupled with a classical diffusion-

483

controlled evaporation model to make accurate prediction for droplet lifetimes on surfaces with

484

varying degrees of contaminants. The framework is next applied to model droplet evaporation over

485

micropillared surface wherein we capture the stick-jump behavior of contact line observed in

486

experiments. We believe that the fundamental insights on the origin and nature of mixed mode

487

provided in this work has implications for analyzing evaporation of nanofluids and designing

488

microfluidics applications.

489

ASSOCIATED CONTENT

490

Supporting Information

491

Methodology adopted to compute effective roughness ratio 𝑟 ∗ for the estimation of modified

492

receding contact angle 𝜃𝑅∗ (Equation 5) is discussed in section S1. Image processing technique for

493

identification of contaminants is discussed in section S2. Estimation of droplet lifetime is described

494

in section S3. Comparison of experiments and model predictions results for day 4 are presented in

495

Section S4. Supplementary video SV1 illustrates line-of-sight dependence during experiments.

496

Supplementary videos SV2-SV4 visualize recurring contact line pinning due to contamination in

497

addition to comparison of experiments and model prediction results for flat silicon samples.

498

Supplementary video SV5 shows the result of model predictions and experiments for the


29

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499

micropillared sample. This material is available free of charge via the internet at http://

500

pubs.acs.org.

501

AUTHOR INFORMATION

502

Corresponding author

503

1

504

III, IIT Patna, Bihta, Bihar 801103, India.

505

Notes

506

The authors declare no competing financial interest.

507

ACKNOWLEDGEMENTS

508

The authors would like to acknowledge Prof. Evelyn N. Wang, Device Research Laboratory,

509

Department of Mechanical Engineering, Massachusetts Institute of Technology for providing

510

thermally oxidized silicon test samples.

Phone: +91-612-302-8166. Email: [email protected]. Corresponding author address: R113, Block


30

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REFERENCES

512

1.

Briones, A. M.; Ervin, J. S.; Putnam, S. A; Byrd, L. W.; Gschwender, L. Micrometer-Sized

513

Water Droplet Impingment Dynamics and Evaporation on a Flat Dry Surface. Langmuir

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2010, 26, 13272-13286.

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2.

516 517

Deng, W.; Gomez. A. Electroscopy Cooling for Microelectronics. Int. J. Heat Mass Transfer 2011, 54, 2270-2275.

3.

Bonaccurso, E.; Butt, H. J.; Hankeln, B.; Niesenhaus, B.; Graf, K. Fabrication of

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microvessels and microlenses from polymers by solvent droplets. Appl. Phys. Lett. 2005,

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86, 124101

520

4.

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Dynamic Roughness Ratio-Based Framework for Modeling Mixed

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