Evaporation and Marangoni driven convection in small heated water droplets

External Stefan and Internal Marangoni Thermo-Fluid Dynamics for Evaporating Capillary Bridges

We probe the evaporation mechanisms of wettability-moderated, confined capillary bridges and bulges. For the first time, we explore the internal Marangoni hydrodynamics and external Stefan advection dynamics in the surrounding gaseous domain due to evaporative effects. A transient simulation approach based on the level set (LS) method and the Arbitrary Lagrangian-Eulerian (ALE) framework was adopted to computationally model the capillary bridge profiles and evaporation phenomenon with generic contact line dynamics (both CCR and CCA modes). The governing equations corresponding to the transport processes in both the liquid and gaseous domains are simulated in a fully coupled manner with appropriate boundary conditions to precisely trace the liquid-vapor interface and the three-phase contact point during evaporation. The effect of the bridge confinement phenomenon, i.e., the extent of confined ambient surrounding the liquid-vapor interface between the solid surfaces, is explored. Also, the role of wetting state and contact line dynamics during CCR and CCA modes of evaporation were probed, and good agreement with experimental observations was noted. Results show that the evaporation rate is primarily dictated by the confinement phenomenon, and wettability effects play a marginal role. A higher confinement curtails the evaporation rate due to an increased local vapor concentration around the liquid bridges. However, the wetting state substantially affects the internal Marangoni effect dynamics and the Stefan advection dynamics due to its explicit influence on the nonuniform evaporative flux along the liquid-vapor interface. Between superhydrophobic confinements, the contact lines are confined in the wedge-shaped region, thereby locally augmenting the vapor concentration. As a result, the large evaporative flux near the bulge region develops a higher temperature gradient, thereby inducing upscaled thermal Marangoni flow compared to hydrophilic confinements. These findings may have significant implications for the efficient designing and development of thermofluidic systems involving thermal transport, mixing, and deposition of dissolved particles in liquid bridges.

Soluto-thermal Marangoni convection in stationary micro-bioreactors on heated substrates: Tool for in vitro diagnosis of PSA

The investigation of antigen-laden droplet deposition patterns on antibody-immobilized substrates has potential for disease detection. Stationary droplets that contain antigens on surfaces immobilized with antibodies can function as microreactors. Temperature modulation enhances reaction efficiency and reduces detection time in droplet-based systems. Thus, the aim of this study is to explore the impact of substrate heating on the structures of protein deposits and the influence of substrate temperature on thermo-solutal Marangoni convection within the droplets. Previous research has explored deposition patterns as diagnostic tools, but limited investigations have focused on the effects of substrate heating on protein deposit structures and the influence of substrate temperature on thermo-solutal Marangoni convection within droplets, creating a knowledge gap. In this study, we conducted experiments to explore how heating the substrate affects the deposition patterns of droplets containing prostate-specific antigen (PSA) on a substrate immobilized with anti-PSA IgG. Additionally, we investigated the thermo-solutal Marangoni convection within these droplets. Our findings reveal distinct deposition patterns classified into dendritic structures (heterogeneous), transitional patterns, and needle-like (homogeneous) structures. The presence of prominent coffee rings and the variation in crystal size across different groups highlight the interplay between thermal and solutal Marangoni advection. Entropy analysis provides insights into structural differences within and between patterns. This work optimizes substrate temperatures for reduced evaporation and detection times while preserving protein integrity, advancing diagnostic tool development, and improving understanding of droplet-based systems.

Active Role of Vapor Clouds around Evaporating Sessile Droplets in Microgravity: Marangoni Jets and Electroconvection

A benchmark microgravity experiment (dubbed "ARLES") is analyzed. It concerns evaporation of several-μL sessile droplets with a pinned millimetric circular contact line on a flat substrate into a vast calm (here nitrogen) atmosphere at nearly normal conditions. Hydrofluoroether (HFE-7100) is used as a working liquid whose appreciable volatility and heavy vapor accentuate the contrast between the micro- and normal gravity. A possibility of switching on a DC electric field (EF) of several kV/mm orthogonally to the substrate is envisaged. We here focus on the findings intimately associated with the visualization of the vapor cloud by means of interferometry and rationalized by means of extensive simulations. In particular, with different degrees of unexpectedness, we discover and explore a Marangoni jet (without EF) and electroconvection (with EF) in the gas, which would otherwise be masked by buoyancy convection. Using the same tools, we examine some malfunctions of the space experiment.

Mathematical Modeling of the Evaporation of a Water Drop from a Heated Surface

This paper presents the results of mathematical modeling of the evaporation of a single water drop from the surface of a copper substrate using a new model, which does not require special experiments to close the system of equations and the corresponding boundary conditions with empirical constants. On the basis of the results of mathematical modeling, it was found that convective currents that occur in a small water drop (≤1 mm in diameter) do not significantly affect the characteristics or conditions of heat and mass transfer processes occurring in a liquid drop heated on a copper substrate. The results of numerical simulation showed that during the initial period of droplet heating, the latter undergoes a rapid transformation of the flow field. Five seconds after the beginning of the thermal action, a quasi-stationary regime of flows in the drop sets in. The model is tested on known experimental data. The theoretical analysis of temperatures at the characteristic points of a water drop and the surface on which the drop is located is carried out in ranges of thermal loads quite typical for practice, conditions for transferring heat and water vapor to the environment. According to the results of mathematical modeling, the possibility of using the developed model in the analysis of the state of cooling of surfaces heated to high temperatures, in cases typically used, is substantiated.

Circulating Marangoni flows within droplets in smectic films

We present a theoretical study and numerical simulation of Marangoni convection within ellipsoidal isotropic droplets embedded in free-standing smectic films (FSSFs). The thermocapillary flows are analyzed for both isotropic droplets spontaneously formed in FSSF overheated above the bulk smectic-isotropic transition and oil lenses deposited on the surface of the smectic film. The realistic model for which the upper drop interface is free from the smectic layers, while at the lower drop surface the smectic layering persists is considered in detail. For isotropic droplets and oil lenses this leads effectively to a sticking of fluid motion at the border with a smectic shell. The above mentioned asymmetric configuration is realized experimentally when the temperature of the upper side of the film is higher than at the lower one. The full set of stationary solutions for Stokes stream functions describing the Marangoni convection flows within the ellipsoidal drops are derived analytically. The temperature distribution in the ellipsoidal drop and the surrounding air is determined in the frame of the perturbation theory. As a result, the analytical solutions for the stationary thermocapillary convection are obtained for different droplet ellipticity ratios and the heat conductivity of the liquid crystal and air. In parallel, the numerical hydrodynamic calculations of the thermocapillary motion in drops are made. Both analytical and numerical simulations predict the axially symmetric circulatory convection motion determined by the Marangoni effect at the droplet-free surface. Due to a curvature of the drop interface a temperature gradient along its free surface always exists. Thus, the thermocapillary convection within the ellipsoidal droplets in overheated FSSF is possible for the arbitrarily small Marangoni numbers. Possible experimental observations enabling the checking of our predictions are proposed.

A scaling law of particle transport in inkjet-printed particle-laden polymeric drops

Hydrogels with embedded functional particulates are widely used to create soft materials with innovative functionalities. In order to advance these soft materials to functional devices and machines, critical technical challenges are the precise positioning of particulates within the hydrogels and the construction of the hydrogels into a complex geometry. Inkjet printing is a promising method for addressing these challenges and ultimately achieving hydrogels with voxelized functionalities, so-called digital hydrogels. However, the development of the inkjet printing process primarily relies on empirical optimization of its printing and curing protocol. In this study, a general scaling law is proposed to predict the transport of particulates within the hydrogel during inkjet printing. This scaling law is based on a hypothesis that water-matrix interaction during the curing of inkjet-printed particle-laden polymeric drops determines the intra-drop particle distribution. Based on the hypothesis, a dimensionless similarity parameter of the water-matrix interaction is proposed, determined by the hydrogel's water evaporation coefficient, particle size, and mechanical properties. The hypothesis was tested by correlating the intra-drop particle distribution to the similarity parameter. The results confirmed the scaling law capable of guiding ink formulation and printing and curing protocol.

Numerical Simulation of Evaporation of Ethanol-Water Mixture Droplets on Isothermal and Heated Substrates

In many printing technologies involving multicomponent liquids, the deposition and printing quality depend on the small-scale transport processes present. For liquids with dispersed particles, the internal flow within the droplet and the evaporation process control the structure of the deposition pattern on the substrate. In many situations, the velocity field inside microdroplets is often subject to either thermal or solutal Marangoni convection. Therefore, to achieve more uniform material deposition, the surface tension-driven flow should be controlled and the effect of different fluid and chemical parameters should be identified. Here, we employ an axisymmetric numerical model to study droplet spreading and evaporation on isothermal and heated substrates. For ethanol-water droplets, the effects of the initial contact angle and initial ethanol concentration inside the droplet (solutal Marangoni number) have been studied. We explore the role of the initial ethanol concentration on the magnitude and structure of the internal flows for binary mixture droplets. In addition, we show that certain combinations of initial contact angle and initial ethanol concentration can lead to a more uniform deposition of dispersed particles after all of the liquid has been evaporated.

Evaporation of Sessile Droplets of Polyelectrolyte/Surfactant Mixtures on Silicon Wafers

The wetting and evaporation behavior of droplets of aqueous solutions of mixtures of poly(diallyldimethylammonium chloride) solution, PDADMAC, with two different anionic surfactants, sodium laureth sulfate, SLES, and sodium N-lauroyl N-methyl taurate, SLMT, were studied in terms of the changes of the contact angle θ and contact length L of sessile droplets of the mixtures on silicon wafers at a temperature of 25 °C and different relative humidities in the range of 30–90%. The advancing contact angle θa was found to depend on the surfactant concentration, independent of the relative humidity, with the mixtures containing SLES presenting improved wetting behaviors. Furthermore, a constant droplet contact angle was not observed during evaporation due to pinning of the droplet at the coffee-ring that was formed. The kinetics for the first evaporation stage of the mixture were independent of the relative humidity, with the evaporation behavior being well described in terms of the universal law for evaporation.

Modulation of Central Depletion Zone in Evaporated Sessile Drops via Substrate Heating

In this article, we report the influence of substrate temperature (T_sub) on the evaporation driven patterning of colloids on solid substrates. When the drops are dried in an environment maintained at temperature, T_env, lower than T_sub, the temperature difference between the drop apex and the three-phase contact line leads to thermal Marangoni flow. We show that the interplay between the radial capillary flow, the thermal Marangoni flow, and the descending rate of the drop surface can be tuned to modulate the spatial distribution of colloids in the dried deposits. At ΔT (=T_sub - T_env) ≥ 45 °C, the distribution of particles in the interior region of the pattern is nearly uniform with a significant decrease in concentration of particles in the ring-like deposit at the edge. The deposits formed at 15 °C ≤ ΔT ≤ 40 °C are accompanied by a particle depleted zone in the center, which has not been reported to date. The formation of the central depletion zone arises from the suppression of the thermal Marangoni flow at the penultimate stage of drying and the interplay between the radial capillary flow and the descending rate of the drop surface. At ΔT < 15 °C, the dried deposits are found to exhibit coffee-ring-like stains.

Microdrop Deposition Technique: Preparation and Characterization of Diluted Suspended Particulate Samples

The analysis of particulate matter (PM) in dilute solutions is an important target for environmental, geochemical, and biochemical research. Here, we show how microdrop technology may allow the control, through the evaporation of small droplets, of the deposition of insoluble materials dispersed in a solution on a well-defined area with a specific spatial pattern. Using this technology, the superficial density of the deposited solute can be accurately controlled. In particular, it becomes possible to deposit an extremely reduced amount of insoluble material, in the order of few μg on a confined area, thus allowing a relatively high superficial density to be reached within a limited time. In this work, we quantitatively compare the microdrop technique for the preparation of particulate matter samples with the classical filtering technique. After having been optimized, the microdrop technique allows obtaining a more homogeneous deposition and may limit the sample amount up to a factor 25. This method is potentially suitable for many novel applications in different scientific fields such as demanding spectroscopic studies looking at the mineral fraction contained in ice cores or to pollution investigations looking at the detection of heavy metals present in ultra-trace in water.

Reversing Coffee-Ring Effect by Laser-Induced Differential Evaporation

The coffee-ring effect, ubiquitously present in the drying process of aqueous droplets, impedes the performance of a myriad of applications involving precipitation of particle suspensions in evaporating liquids on solid surfaces, such as liquid biopsy combinational analysis, microarray fabrication, and ink-jet printing, to name a few. We invented the methodology of laser-induced differential evaporation to remove the coffee-ring effect. Without any additives to the liquid or any morphology modifications of the solid surface the liquid rests on, we have eliminated the coffee-ring effect by engineering the liquid evaporation profile with a CO₂ laser irradiating the apex of the droplets. The method of laser-induced differential evaporation transitions particle deposition patterns from coffee-ring patterns to central-peak patterns, bringing all particles (e.g. fluorescent double strand DNAs) in the droplet to a designated area of 100 μm diameter without leaving any stains outside. The technique also moves the drying process from the constant contact radius (CCR) mode to the constant contact angle (CCA) mode. Physical mechanisms of this method were experimentally studied by internal flow tracking and surface evaporation flux mapping, and theoretically investigated by development of an analytical model.

Molecular investigation of evaporation of biodroplets containing single-strand DNA on graphene surface

In this study, the water droplet behaviour of four different types of single-strand DNA with homogeneous base sequence on a graphene substrate during evaporation of the droplet was investigated using molecular dynamics (MD) simulation. The simulation results indicated that the evaporation depended on the DNA sequence. The observed changes can be divided into four parts: (i) vaporization mode, (ii) evaporation flux, (iii) mechanism of single-strand placement on the surface, and (iv) consideration of remaining single strands after evaporation. Our simulation observations indicated different evaporation modes for thymine biodroplets as compared to those for other biodroplets. The evaporation of the thymine biodroplets occurred with an increase in the contact angle, while that of the other biodroplets occur in a constant contact angle mode. Moreover, thymine biodroplets generate the lowest contact line compared to other single strands, and it is always placed far away from the centre of the droplets during evaporation. Investigating variations in the evaporation flux shows that thymine has the highest evaporation flux and guanine has the lowest. Moreover, during initial evaporation, the flux of evaporation increases at the triple point of the biodroplets containing thymine single strands, while it decreases in the other biodroplets. The following observation was obtained from the study of the placement of single strands on the substrate: guanine and thymine interacted slower than other single strands during evaporation with graphene, adenine single strand had a higher folding during evaporation, and guanine single strand showed the lowest end-to-end distance. The investigation of single-strand DNA after evaporation shows that adenine produces the most stable structure at the end of evaporation. In addition, cytosine is the most stretched single-strand DNA due to its lack of internal π-π stacking and hydrogen bonding. Therefore, cytosine single strand is more accessible for use in microarrays to detect target single strands.

Recent advances in droplet wetting and evaporation

Wetting and evaporation of a simple sessile droplet is a very complex problem involving strongly coupled physics.

Sessile droplets for chemical and biological assays

Sessile droplets are non-movable droplets spanning volumes in the nL-to-μL range. The sessile-droplet-based platform provides a paradigm shift from the conventional, flow-based lab-on-a-chip philosophy, yet offering similar benefits: low reagent/sample consumption, high throughput, automation, and most importantly flexibility and versatility. Moreover, the platform relies less heavily on sophisticated fabrication techniques, often sufficient with a hydrophobic substrate, and no pump is required for operation. In addition, exploiting the physical phenomena that naturally arise when a droplet evaporates, such as the coffee-ring effect or Marangoni flow, can lead to fascinating applications. In this review, we introduce the physics of droplets, and then focus on the different types of chemical and biological assays that have been implemented in sessile droplets, including analyte concentration, particle separation and sorting, cell-based assays, and nucleic acid amplification. Finally, we provide our perspectives on this unique micro-scale platform.

Influence of Local Heating on Marangoni Flows and Evaporation Kinetics of Pure Water Drops

The effect of localized heating on the evaporation of pure sessile water drops was probed experimentally by a combination of infrared thermography and optical imaging. In particular, we studied the effect of three different heating powers and two different locations, directly below the center and edge of the drop. In all cases, four distinct stages were identified according to the emerging thermal patterns. In particular, depending on heating location, recirculating vortices emerge that either remain pinned or move azimuthally within the drop. Eventually, these vortices oscillate in different modes depending on heating location. Infrared data allowed extraction of temperature distribution on each drop surface. In turn, the flow velocity in each case was calculated and was found to be higher for edge heating, due to the one-directional nature of the heating. Additionally, calculation of the dimensionless Marangoni and Rayleigh numbers yielded the prevalence of Marangoni convection. Heating the water drops also affected the evaporation kinetics by promoting the "stick-slip" regime. Moreover, both the total number of depinning events and the pinning strength were found to be highly dependent on heating location. Lastly, we report a higher than predicted relationship between evaporation rate and heating temperature, due to the added influence of the recirculating flows on temperature distribution and hence evaporation flux.

Insight into the Evaporation Dynamics of a Pair of Sessile Droplets on a Hydrophobic Substrate

In this work, we have demonstrated three unique regimes in the evaporation lifecycle of a pair of sessile droplets placed in variable proximity on a hydrophobic substrate. For small separation distance, the droplets undergo asymmetric spatiotemporal evaporation leading to contact angle hysteresis and suppressed vaporization. The reduced evaporation has been attributed quantitatively to the existence of a constrained vapor-rich dome between the two droplets. However, a dynamic decrease in the droplet radius due to solvent removal marks a return to symmetry in terms of evaporation and contact angle. We have described the variation in evaporation flux using a universal correction factor. We have also demonstrated the existence of a critical separation distance beyond which the droplets in the droplet pair do not affect each other. The results are crucial to a plethora of applications ranging from surface patterning to lab-on-a-chip devices.

Thermocapillarity in Microfluidics-A Review

This paper reviews the past and recent studies on thermocapillarity in relation to microfluidics. The role of thermocapillarity as the change of surface tension due to temperature gradient in developing Marangoni flow in liquid films and conclusively bubble and drop actuation is discussed. The thermocapillary-driven mass transfer (the so-called Benard-Marangoni effect) can be observed in liquid films, reservoirs, bubbles and droplets that are subject to the temperature gradient. Since the contribution of a surface tension-driven flow becomes more prominent when the scale becomes smaller as compared to a pressure-driven flow, microfluidic applications based on thermocapillary effect are gaining attentions recently. The effect of thermocapillarity on the flow pattern inside liquid films is the initial focus of this review. Analysis of the relation between evaporation and thermocapillary instability approves the effect of Marangoni flow on flow field inside the drop and its evaporation rate. The effect of thermocapillary on producing Marangoni flow inside drops and liquid films, leads to actuation of drops and bubbles due to the drag at the interface, mass conservation, and also gravity and buoyancy in vertical motion. This motion can happen inside microchannels with a closed multiphase medium, on the solid substrate as in solid/liquid interaction, or on top of a carrier liquid film in open microfluidic systems. Various thermocapillary-based microfluidic devices have been proposed and developed for different purposes such as actuation, sensing, trapping, sorting, mixing, chemical reaction, and biological assays throughout the years. A list of the thermocapillary based microfluidic devices along with their characteristics, configurations, limitations, and improvements are presented in this review.

Effects of hydrodynamic retardation and interparticle interactions on the self-assembly in a drying droplet containing suspended solid particles

Self-assembly of particles, suspended in a drying droplet, were studied by the Monte Carlo method. The Brownian diffusion of particles was simulated accounting for the effect of hydrodynamic retardation and interparticle interactions. The model allowed for explaining formation of the "coffee ring" patterns even without accounting for the radial flows towards the three-phase contact line. Morphologies of the drying patterns and their dependence on interparticle interactions and concentration of particles are discussed.

Electrostatic suppression of the "coffee stain effect"

The dynamics of a slender, evaporating, particle-laden droplet under the effect of electric fields are examined. Lubrication theory is used to reduce the governing equations to a coupled system of evolution equations for the interfacial position and the local, depth-averaged particle concentration. The model incorporates the effects of capillarity, viscous stress, Marangoni stress, elecrostatically induced Maxwell stress, van der Waals forces, concentration-dependent rheology, and evaporation. Via a parametric numerical study, the one-dimensional model is shown to recover the expected inhomogeneous ring-like structures in appropriate parameter ranges due to a combination of enhanced evaporation close to the contact line, and resultant capillarity-induced flow. It is then demonstrated that this effect can be significantly suppressed via the use of carefully chosen electric fields. Finally, the three-dimensional behavior of the film and the particle concentration field is briefly examined.

Simultaneous spreading and evaporation: recent developments

The recent progress in theoretical and experimental studies of simultaneous spreading and evaporation of liquid droplets on solid substrates is discussed for pure liquids including nanodroplets, nanosuspensions of inorganic particles (nanofluids) and surfactant solutions. Evaporation of both complete wetting and partial wetting liquids into a nonsaturated vapour atmosphere are considered. However, the main attention is paid to the case of partial wetting when the hysteresis of static contact angle takes place. In the case of complete wetting the spreading/evaporation process proceeds in two stages. A theory was suggested for this case and a good agreement with available experimental data was achieved. In the case of partial wetting the spreading/evaporation of a sessile droplet of pure liquid goes through four subsequent stages: (i) the initial stage, spreading, is relatively short (1-2 min) and therefore evaporation can be neglected during this stage; during the initial stage the contact angle reaches the value of advancing contact angle and the radius of the droplet base reaches its maximum value, (ii) the first stage of evaporation is characterised by the constant value of the radius of the droplet base; the value of the contact angle during the first stage decreases from static advancing to static receding contact angle; (iii) during the second stage of evaporation the contact angle remains constant and equal to its receding value, while the radius of the droplet base decreases; and (iv) at the third stage of evaporation both the contact angle and the radius of the droplet base decrease until the drop completely disappears. It has been shown theoretically and confirmed experimentally that during the first and second stages of evaporation the volume of droplet to power 2/3 decreases linearly with time. The universal dependence of the contact angle during the first stage and of the radius of the droplet base during the second stage on the reduced time has been derived theoretically and confirmed experimentally. The theory developed for pure liquids is applicable also to nanofluids, where a good agreement with the available experimental data has been found. However, in the case of evaporation of surfactant solutions the process deviates from the theoretical predictions for pure liquids at concentration below critical wetting concentration and is in agreement with the theoretical predictions at concentrations above it.

Convective flows in evaporating sessile droplets

The evaporation rate and internal convective flows of a sessile droplet with a pinned contact line were formulated and investigated numerically. We developed and analyzed a unified numerical model that includes the effects of temperature, droplet volume, and contact angle on evaporation rate and internal flows. The temperature gradient on the air/liquid interface causes an internal flow due to Marangoni stress, which provides good convective mixing within the droplet, depending upon Marangoni number. As the droplet volume decreases, the thermal gradient becomes smaller and the Marangoni flow becomes negligible. Simultaneously, as the droplet height decreases, evaporation-induced flow creates a large jet-like flow radially toward the contact line. For a droplet containing suspended particles, this jet-like convective flow carries particles toward the contact line and deposits them on the surface, forming the so-called "coffee ring stain". In addition, we reported a simple polynomial correlation for dimensionless evaporation time as a function of initial contact angle of the pinned sessile droplet which agrees well with the previous experimental and numerical results.

Aggregation kinetics of SERS-active nanoparticles in thermally stirred sessile droplets

The aggregation kinetics of silver nanoparticles in sessile droplets were investigated both experimentally and through numerical simulations as a function of temperature gradient and evaporation rate, in order to determine the hydrodynamic and aggregation parameters that lead to optimal surface-enhanced Raman spectroscopic (SERS) detection. Thermal gradients promote effective stirring within the droplet. The aggregation reaction ceases when the solvent evaporates forming a circular stain consisting of a high concentration of silver nanoparticle aggregates, which can be interrogated by SERS leading to analyte detection and identification. We introduce the aggregation parameter, Γa ≡ τ(evap)/τ(a), which is the ratio of the evaporation to the aggregation time scales. For a well-stirred droplet, the optimal condition for SERS detection was found to be Γ(a,opt) = kc(NP)τ(evap) ≈ 0.3, which is a product of the dimerization rate constant (k), the concentration of nanoparticles (cNP), and the droplet evaporation time (τ(evap)). Near maximal signal (over 50% of maximum value) is observed over a wide range of aggregation parameters 0.05 < Γa < 1.25, which also defines the time window during which trace analytes can be easily measured. The results of the simulation were in very good agreement with experimentally acquired SERS spectra using gas-phase 1,4-benzenedithiol as a model analyte.

Thermal patterns and hydrothermal waves (HTWs) in volatile drops

Experimental measurements of temperature and heat flux at the liquid-wall interface during the evaporation of sessile FC-72 droplets have been reported for the first time using infrared (IR) thermography. Simultaneous high-speed imaging of the evaporating drop was carried out to monitor the drop profile. The study demonstrates that recently evidenced hydrothermal waves are actually bulk waves that extend across the entire droplet volume. More importantly, thermal patterns occurring in the bulk of the drop affect the temperature and heat-flux distributions on the solid substrate and ultimately influence the droplet evaporation rate. These effects were found to be increasingly pronounced as the substrate temperature was raised. The implications for heat-transfer mechanisms and energy transport are discussed.

Evaporation of nanodroplets on heated substrates: a molecular dynamics simulation study

Molecular dynamics simulations of Lennard-Jones particles have been performed to study the evaporation behavior of nanodroplets on heated substrates. The influence of the liquid-substrate interaction strength on the evaporation properties was addressed. Our results show that, during the temperature-raising evaporation, the gas is always hotter than the droplet. In contrast to the usual experimental conditions, the droplet sizes in our simulations are in the nanometer scale range and the substrates are ideally smooth and chemically homogeneous. As a result, no pinning was observed in our simulations for substrates denoted either hydrophilic (contact angle θ < 90°) or hydrophobic (contact angle θ > 90°). The evaporative mass flux is stronger with increasing hydrophilicity of the substrate because the heat transfer from the substrate to the droplet is more efficient for stronger attraction between the solid and the droplet. Evaporation and heat transfer to the gas phase occur preferentially in the vicinity of the three-phase contact line in the hydrophilic system. However, in the case of a hydrophobic substrate, there is no preferential location for mass and heat fluxes. During the whole evaporation process, no pure behavior according to either the constant-angle or the constant-radius model was found; both the contact angle and contact radius decrease for the droplets on hydrophilic and hydrophobic substrates alike.

Thermal effects of the substrate on water droplet evaporation

We experimentally investigate the behavior of a pinned water droplet evaporating into air. The influence of the substrate temperature and substrate thermal properties on the evaporation process are studied in both hydrophilic and hydrophobic conditions. Our objective is to understand the effect of thermal mechanisms on the droplet evaporation process. The experimental results are compared with the quasisteady, diffusion-driven evaporation model, which is implemented under the influence of the temperature; the model assumes the isothermia of the droplet at the substrate temperature. The results highlight a favorable correlation between the model and the experimental data at ambient temperatures for most situations considered here. The model works to qualitatively describe the influence of the substrate temperature on the evaporation process. However, with an increase in the substrate temperature, the role of the thermal-linked mechanisms becomes increasingly important; this experiment highlights the need for more accurate models to account for the buoyant convection in vapor transport and the evaporative cooling and heat conduction between the droplet and the substrate. Finally, the experimental data reveal the modification of contact angle evolution as the temperature increases and the crucial role played by the nature of the substrate in the evaporation of a sessile droplet. The influence of the substrate thermal properties on the global evaporation rate is explained by the parallel thermal effusivity of the liquid and solid phases.

Evaporation of pure liquid sessile and spherical suspended drops: a review

A sessile drop is an isolated drop which has been deposited on a solid substrate where the wetted area is limited by a contact line and characterized by contact angle, contact radius and drop height. Diffusion-controlled evaporation of a sessile drop in an ambient gas is an important topic of interest because it plays a crucial role in many scientific applications such as controlling the deposition of particles on solid surfaces, in ink-jet printing, spraying of pesticides, micro/nano material fabrication, thin film coatings, biochemical assays, drop wise cooling, deposition of DNA/RNA micro-arrays, and manufacture of novel optical and electronic materials in the last decades. This paper presents a review of the published articles for a period of approximately 120 years related to the evaporation of both sessile drops and nearly spherical droplets suspended from thin fibers. After presenting a brief history of the subject, we discuss the basic theory comprising evaporation of micrometer and millimeter sized spherical drops, self cooling on the drop surface and evaporation rate of sessile drops on solids. The effects of drop cooling, resultant lateral evaporative flux and Marangoni flows on evaporation rate are also discussed. This review also has some special topics such as drop evaporation on superhydrophobic surfaces, determination of the receding contact angle from drop evaporation, substrate thermal conductivity effect on drop evaporation and the rate evaporation of water in liquid marbles.

Temperature distribution inside an evaporating two-dimensional droplet lying on curved or flat substrates

The temperature field inside an evaporating two-dimensional droplet resting on curved or flat isothermal substrates is studied under various evaporation conditions. An analytical solution for the temperature is derived, which can be directly used in the cases of pinned and depinned contact lines as well as in stick-slip evaporation modes. It is found that the temperature drop at the free surface of a droplet on a convex, hydrophobic substrate is far greater than that for flat or concave substrates of the same hydrophobicity. The analytical solution for the temperature field allows the direct estimate of the local temperature gradient and, hence, of the local surface tension gradient that gives rise to Marangoni flow directed from the contact lines to the top of the droplet.

Use of IR thermography to investigate heated droplet evaporation and contact line dynamics

In this paper we present the results of an experimental study investigating interfacial properties during the evaporation of sessile water droplets on a heated substrate. This study uses infrared thermography to map the droplet interfacial temperature. The measurements evidence nonuniform temperature and gradients that evolve in time during the evaporation process. A general scaling law for the interfacial temperature is deduced from the experimental observations. A theoretical analysis is performed to predict the local evaporation rates and their evolution in time. The use of energy conservation laws enabled us to deduce a general expression for the interfacial temperature. The comparison between the theory and experiments shows good agreement and allows us to rationalize the experimental observations. The thermography analysis also enabled the detection of the three-phase contact line location and its dynamics. To our knowledge, such measurements are performed for the first time using thermography.

Structure of circulation flows in polymer solution droplets receding on flat surfaces

In a previous report where internal flows were experimentally visualized in polymer solution droplets receding on a lyophobic surface [Kaneda et al., Langmuir 2008, 24, 9102-9109], the direction of the circulation flow was found to depend on solvent and solute concentration. To identify the reason for this finding, the internal flow in the droplet is investigated numerically. A mathematical model predicts that double circulation flows initiate after a single flow develops at high Marangoni numbers, while only a single circulation flow develops at low Marangoni numbers. The dependencies of the calculated velocities on the solvent and the initial solute concentration agree qualitatively with experiment. It is concluded that the difference of the flow directions that were investigated experimentally is due to such a change in the flow structures. The effects of the contact angle and dimensions on transport phenomena in a droplet are also discussed.

Stability of evaporating water when heated through the vapor and the liquid phases

The stability of a water layer of uniform thickness held in a two-dimensional container of finite or semi-infinite extent is examined using linear stability theory. The liquid-vapor interface can be heated both through the liquid and through the vapor, as previously experimentally reported. The need to introduce a heat transfer coefficient is eliminated by introducing statistical rate theory (SRT) to predict the evaporation flux. There are no fitting or undefined parameters in the expression for the evaporation flux. The energy transport is parametrized in terms of the evaporation number, Eev, that for a given experimental circumstance can be predicted. The critical Marangoni number for the finite, Macf, and for the semi-infinite system, Mac(infinity), can be quantitatively predicted. Experiments in which water evaporated from a stainless-steel funnel and from a polymethyl methacrylate (PMMA) funnel into its vapor have been previously reported. Marangoni convection was observed in the experiments that used the stainless-steel funnel but not with the PMMA funnel even though the Marangoni number for the PMMA funnel was more than 27,000. The SRT-based stability theory indicates that the critical value of the Marangoni number for the experiments with the PMMA funnel was greater than the experimental value of the Marangoni number in each case; thus, no Marangoni convection was predicted to result from an instability. The observed convection with the stainless-steel funnel resulted from a temperature gradient imposed along the interface.

Criterion for reversal of thermal Marangoni flow in drying drops

The thermal Marangoni flow induced by nonuniform surface temperature has been widely invoked to interpret the deposition pattern from drying drops. The surface temperature distribution of a drying droplet, although being crucial to the Marangoni flow, is still controversial. In this paper, the surface temperature in the drop central region is analyzed theoretically based on an asymptotic analysis on the heat transfer in such region, and a quantitative criterion is established for the direction of the surface temperature gradient and the direction of the induced Marangoni flow of drying drops. The asymptotic analysis indicates that these two directions will reverse at a critical contact angle, which depends not only on the relative thermal conductivities of the substrate and liquid, but also on the ratio of the substrate thickness to the contact-line radius of the droplet. The theory is corroborated experimentally and numerically, and may provide a potential means to control deposition patterns from drying droplets.