Interfacial conditions during evaporation or condensation of water

Evaporation Dynamics of Macro- and Nanodroplets on Heated Hydrophilic Rough Substrates: The Effect of Roughness and Scale

Droplet evaporation on rough substrates plays an essential role in cooling and micro/nanoparticle assembly. Currently, there are numerous macroscopic experiments and theoretical models to investigate the droplet evaporation behavior on rough substrates. However, due to the complexity of this phenomenon, understanding its mechanisms solely through macroscale studies is difficult. To this end, molecular dynamics simulations of the models with distinct roughness factors are performed, and the obtained results are compared with those of relevant experiments of droplet evaporation on three hydrophilic substrates with different roughness average of 0.1, 0.15, and 0.2 μm, respectively. In this way, we assess the evaporation on these rough systems and the effect of scale on macro- and nanodroplets, which allows us to explore deeper the mechanism of droplet evaporation on rough hydrophilic substrates. In particular, we find that in the case of macroscale droplets, the evaporation mode remains the same with increasing roughness, pointing to a combined mixed and constant-contact-radius (CCR) mode. In the case of nanoscale droplets, the evaporation model is the constant-contact-angle mode when the roughness factor r = 1, while the mixed and CCR modes are found for r = 1.5 and 2, respectively. The scale effect has significant influence on the evaporation pattern of droplets on rough hydrophilic substrates. Moreover, it is also found that increasing the roughness of substrates expands the substrate-droplet contact area on both the macro- and nanoscale, which in turn enhances the heat transfer from the substrate toward the droplet. We anticipate that this first systematic analysis of scale effects provides further insights into the evaporation dynamics of droplets on rough hydrophilic substrates and has significant implications for the advancement of nanotechnology.

Nonisothermal evaporation

Evaporation of a liquid layer on a substrate is examined without the often-used isothermality assumption, i.e., temperature variations are accounted for. Qualitative estimates show that nonisothermality makes the evaporation rate depend on the conditions at which the substrate is maintained. If it is thermally insulated, evaporative cooling dramatically slows evaporation down; the evaporation rate tends to zero with time and cannot be determined by measuring the external parameters only. If, however, the substrate is maintained at a fixed temperature, the heat flux coming from below sustains evaporation at a finite rate, deducible from the fluid's characteristics, relative humidity, and the layer's depth. The qualitative predictions are quantified using the diffuse-interface model applied to a liquid evaporating into its own vapor.

Ultrahigh evaporative heat transfer measured locally in submicron water films

Thin film evaporation is a widely-used thermal management solution for micro/nano-devices with high energy densities. Local measurements of the evaporation rate at a liquid-vapor interface, however, are limited. We present a continuous profile of the evaporation heat transfer coefficient ([Formula: see text]) in the submicron thin film region of a water meniscus obtained through local measurements interpreted by a machine learned surrogate of the physical system. Frequency domain thermoreflectance (FDTR), a non-contact laser-based method with micrometer lateral resolution, is used to induce and measure the meniscus evaporation. A neural network is then trained using finite element simulations to extract the [Formula: see text] profile from the FDTR data. For a substrate superheat of 20 K, the maximum [Formula: see text] is [Formula: see text] MW/[Formula: see text]-K at a film thickness of [Formula: see text] nm. This ultrahigh [Formula: see text] value is two orders of magnitude larger than the heat transfer coefficient for single-phase forced convection or evaporation from a bulk liquid. Under the assumption of constant wall temperature, our profiles of [Formula: see text] and meniscus thickness suggest that 62% of the heat transfer comes from the region lying 0.1-1 μm from the meniscus edge, whereas just 29% comes from the next 100 μm.

Competition between thermal and surfactant-induced Marangoni flow in evaporating sessile droplets

HYPOTHESIS

Thermal Marangoni flow in evaporating sessile water droplets is much weaker in experiments than predicted theoretically. Often this is attributed to surfactant contamination, but there have not been any in-depth analyses that consider the full fluid and surfactant dynamics. It is expected that more insight into this problem can be gained by using numerical models to analyze the interplay between thermal Marangoni flow and surfactant dynamics in terms of dimensionless parameters.

SIMULATIONS

Two numerical models are implemented: one dynamic model based on lubrication theory and one quasi-stationary model, that allows for arbitrary contact angles.

FINDINGS

It is found that insoluble surfactants can suppress the thermal Marangoni flow if their concentration is sufficiently large and evaporation and diffusion are sufficiently slow. Soluble surfactants, however, either reduce or increase the interfacial velocity, depending on their sorption kinetics. Furthermore, insoluble surfactant concentrations that cause an order 0.1% surface tension reduction are sufficient to reduce the spatially averaged tangential flow velocity at the interface by a factor 100. For larger contact angles and smaller droplets this required concentration is larger (typically <1% surface tension reduction). The numerical models are mutually validated by comparing their results in cases where both are valid.

Evaporation in nano/molecular materials

Evaporation is a physical phenomenon with fundamental significance to both nature and technology ranging from plant transpiration to DNA engineering. Various analytical and empirical relationships have been proposed to characterize evaporation kinetics at macroscopic scales. However, theoretical models to describe the kinetics of evaporation from nano and sub-nanometer (molecular) confinements are absent. On the other hand, the fast advancements in technology concentrated on development of nano/molecular-scale devices demand appropriate models that can accurately predict physics of phase-change in these systems. A thorough understanding of the physics of evaporation in nano/molecular materials is, thus, of critical importance to develop the required models. This understanding is also crucial to explain the intriguing evaporation-related phenomena that only take place when the characteristic length of the system drops to several nanometers. Here, we comprehensively review the underlying physics of evaporation phenomenon and discuss the effects of nano/molecular confinement on evaporation. The role of liquid-wall interface-related phenomena including the effects of disjoining pressure and flow slippage on evaporation from nano/molecular confinements are discussed. Different driving forces that can induce evaporation in small confinements, such as heat transfer, pressure drop, cavitation and density fluctuations are elaborated. Hydrophobic confinement induced evaporation and its potential application for synthetic ion channels are discussed in detail. Evaporation of water as molecular clusters rather than isolated molecules is discussed. Despite the lack of experimental investigations on evaporation at nanoscale, there exist an extensive body of literature that have applied different simulation techniques to predict the phase change behavior of liquids in nanoconfinements. We infer that exploring the effect of electrostatic interactions and flow slippage to enhance evaporation from nanoconduits is an interesting topic for future endeavors. Further future studies could be devoted to developing nano/molecular channels with evaporation-based gating mechanism and utilization of 2D materials to tune energy barrier for evaporation leading to enhanced evaporation.

Study on Evaporation Characteristics of Water in Annular Liquid Pool at Low Pressures

In order to investigate the energy transfer mechanism and the nonequilibrium effect during water evaporation in its own pure vapor at low pressures, a series of precise measurements are conducted to obtain the temperature profile near the liquid-vapor interface and the evaporation rates in an annular pool in a closed chamber. The results show that the interface temperature of the vapor side is higher than that of the liquid side when water evaporates in its own pure vapor at low pressures (ranging from 394 to 1467 Pa), the temperature discontinuity across the interface exists in all experimental conditions. The magnitude of the temperature discontinuity is strongly affected by the vapor pressure. A uniform temperature layer with a thickness of about 2 mm is found below the evaporating interface because of the coupling effect of evaporation cooling and thermocapillary convection. The energy required for evaporation is mainly transferred by thermocapillary convection in the uniform temperature layer. Furthermore, the numerical simulation results confirm that the evaporation flux near the cylinders is much larger than that at the middle region, which implies that most of the latent heat required for evaporation is transferred to the interface near the cylinders.

Evaporation Mass Flux: A Predictive Model and Experiments

Evaporation is a fundamental and core phenomenon in a broad range of disciplines including power generation and refrigeration systems, desalination, electronic/photonic cooling, aviation systems, and even biosciences. Despite its importance, the current theories on evaporation suffer from fitting coefficients with reported values varying in a few orders of magnitude. Lack of a sound model impedes simulation and prediction of characteristics of many systems in these disciplines. Here, we studied evaporation at a planar liquid-vapor interface through a custom-designed, controlled, and automated experimental setup. This experimental setup provides the ability to accurately probe thermodynamic properties in vapor, liquid, and close to the liquid-vapor interface. Through analysis of these thermodynamic properties in a wide range of evaporation mass fluxes, we cast a predictive model of evaporation based on nonequilibrium thermodynamics with no fitting parameters. In this model, only the interfacial temperatures of liquid and vapor phases along with the vapor pressure are needed to predict evaporation mass flux. The model was validated by the reported study of an independent research group. The developed model provides a foundation for all liquid-vapor phase change studies including energy, water, and biological systems.

An Ultrathin Nanoporous Membrane Evaporator

Evaporation is a ubiquitous phenomenon found in nature and widely used in industry. Yet a fundamental understanding of interfacial transport during evaporation remains limited to date owing to the difficulty of characterizing the heat and mass transfer at the interface, especially at high heat fluxes (>100 W/cm²). In this work, we elucidated evaporation into an air ambient with an ultrathin (≈200 nm thick) nanoporous (≈130 nm pore diameter) membrane. With our evaporator design, we accurately monitored the temperature of the liquid-vapor interface, reduced the thermal-fluidic transport resistance, and mitigated the clogging risk associated with contamination. At a steady state, we demonstrated heat fluxes of ≈500 W/cm² across the interface over a total evaporation area of 0.20 mm². In the high flux regime, we showed the importance of convective transport caused by evaporation itself and that Fick's first law of diffusion no longer applies. This work improves our fundamental understanding of evaporation and paves the way for high flux phase-change devices.

Effect of the Thermocouple on Measuring the Temperature Discontinuity at a Liquid-Vapor Interface

The coupled heat and mass transfer that occurs in evaporation is of interest in a large number of fields such as evaporative cooling, distillation, drying, coating, printing, crystallization, welding, atmospheric processes, and pool fires. The temperature jump that occurs at an evaporating interface is of central importance to understanding this complex process. Over the past three decades, thermocouples have been widely used to measure the interfacial temperature jumps at a liquid-vapor interface during evaporation. However, the reliability of these measurements has not been investigated so far. In this study, a numerical simulation of a thermocouple when it measures the interfacial temperatures at a liquid-vapor interface is conducted to understand the possible effects of the thermocouple on the measured temperature and features in the temperature profile. The differential equations of heat transfer in the solid and fluids as well as the momentum transfer in the fluids are coupled together and solved numerically subject to appropriate boundary conditions between the solid and fluids. The results of the numerical simulation showed that while thermocouples can measure the interfacial temperatures in the liquid correctly, they fail to read the actual interfacial temperatures in the vapor. As the results of our numerical study suggest, the temperature jumps at a liquid-vapor interface measured experimentally by using a thermocouple are larger than what really exists at the interface. For a typical experimental study of evaporation of water at low pressure, it was found that the temperature jumps measured by a thermocouple are overestimated by almost 50%. However, the revised temperature jumps are still in agreement with the statistical rate theory of interfacial transport. As well as addressing the specific application of the liquid-vapor temperature jump, this paper provides significant insight into the role that heat transfer plays in the operation of thermocouples in general.

Experimental and Numerical Study of the Evaporation of Water at Low Pressures

Although evaporation is considered to be a surface phenomenon, the rate of molecular transport across a liquid-vapor boundary is strongly dependent on the coupled fluid dynamics and heat transfer in the bulk fluids. Recent experimental thermocouple measurements of the temperature field near the interface of evaporating water into its vapor have begun to show the role of heat transfer in evaporation. However, the role of fluid dynamics has not been explored sufficiently. Here, we have developed a mathematical model to describe the coupling of the heat, mass, and momentum transfer in the fluids with the transport phenomena at the interface. The model was used to understand the experimentally obtained velocity field in the liquid and temperature profiles in the liquid and vapor, in evaporation from a concave meniscus for various vacuum pressures. By using the model, we have shown that an opposing buoyancy flow suppressed the thermocapillary flow in the liquid during evaporation at low pressures in our experiments. As such, in the absence of thermocapillary convection, the evaporation is controlled by heat transfer to the interface, and the predicted behavior of the system is independent of choosing between the existing theoretical expressions for evaporation flux. Furthermore, we investigated the temperature discontinuity at the interface and confirmed that the discontinuity strongly depends on the heat flux from the vapor side, which depends on the geometrical shape of the interface.

Coherent description of transport across the water interface: From nanodroplets to climate models

Transport of mass and energy across the vapor-liquid interface of water is of central importance in a variety of contexts such as climate models, weather forecasts, and power plants. We provide a complete description of the transport properties of the vapor-liquid interface of water with the framework of nonequilibrium thermodynamics. Transport across the planar interface is then described by 3 interface transfer coefficients where 9 more coefficients extend the description to curved interfaces. We obtain all coefficients in the range 260-560 K by taking advantage of water evaporation experiments at low temperatures, nonequilibrium molecular dynamics with the TIP4P/2005 rigid-water-molecule model at high temperatures, and square gradient theory to represent the whole range. Square gradient theory is used to link the region where experiments are possible (low vapor pressures) to the region where nonequilibrium molecular dynamics can be done (high vapor pressures). This enables a description of transport across the planar water interface, interfaces of bubbles, and droplets, as well as interfaces of water structures with complex geometries. The results are likely to improve the description of evaporation and condensation of water at widely different scales; they open a route to improve the understanding of nanodroplets on a small scale and the precision of climate models on a large scale.

From adsorption to condensation: the role of adsorbed molecular clusters

Phase transition from an adsorbed vapour to an adsorbed liquid at a subcooling temperature of 2.7 ± 0.4 K.

Humidity distribution affected by freely exposed water surfaces: simulations and experimental verification

Accurate models for the water vapor flux at a water-air interface are required in various scientific, reliability and civil engineering aspects. Here, a study of humidity distribution in a container with air and freely exposed water is presented. A model predicting a spatial distribution and time evolution of relative humidity based on statistical rate theory and computational fluid dynamics is developed. In our approach we use short-term steady-state steps to simulate the slowly evolving evaporation in the system. Experiments demonstrate considerably good agreement with the computer modeling and allow one to distinguish the most important parameters for the model.

Heat and mass transfer through interfaces of nanosized bubbles/droplets: the influence of interface curvature

Heat and mass transfer through interfaces is central in nucleation theory, nanotechnology and many other fields of research.

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.

Evaporation of freely suspended single droplets: experimental, theoretical and computational simulations

Evaporation is ubiquitous in nature. This process influences the climate, the formation of clouds, transpiration in plants, the survival of arctic organisms, the efficiency of car engines, the structure of dried materials and many other phenomena. Recent experiments discovered two novel mechanisms accompanying evaporation: temperature discontinuity at the liquid-vapour interface during evaporation and equilibration of pressures in the whole system during evaporation. None of these effects has been predicted previously by existing theories despite the fact that after 130 years of investigation the theory of evaporation was believed to be mature. These two effects call for reanalysis of existing experimental data and such is the goal of this review. In this article we analyse the experimental and the computational simulation data on the droplet evaporation of several different systems: water into its own vapour, water into the air, diethylene glycol into nitrogen and argon into its own vapour. We show that the temperature discontinuity at the liquid-vapour interface discovered by Fang and Ward (1999 Phys. Rev. E 59 417-28) is a rule rather than an exception. We show in computer simulations for a single-component system (argon) that this discontinuity is due to the constraint of momentum/pressure equilibrium during evaporation. For high vapour pressure the temperature is continuous across the liquid-vapour interface, while for small vapour pressures the temperature is discontinuous. The temperature jump at the interface is inversely proportional to the vapour density close to the interface. We have also found that all analysed data are described by the following equation: da/dt = P(1)/(a + P(2)), where a is the radius of the evaporating droplet, t is time and P(1) and P(2) are two parameters. P(1) = -λΔT/(q(eff)ρ(L)), where λ is the thermal conductivity coefficient in the vapour at the interface, ΔT is the temperature difference between the liquid droplet and the vapour far from the interface, q(eff) is the enthalpy of evaporation per unit mass and ρ(L) is the liquid density. The P(2) parameter is the kinetic correction proportional to the evaporation coefficient. P(2) = 0 only in the absence of temperature discontinuity at the interface. We discuss various models and problems in the determination of the evaporation coefficient and discuss evaporation scenarios in the case of single- and multi-component systems.

Self-assembly of colloidal particles from evaporating droplets: role of DLVO interactions and proposition of a phase diagram

The shape of deposits obtained from drying drops containing colloidal particles matters for technologies such as inkjet printing, microelectronics, and bioassay manufacturing. In this work, the formation of deposits during the drying of nanoliter drops containing colloidal particles is investigated experimentally with microscopy and profilometry, and theoretically with an in-house finite-element code. The system studied involves aqueous drops containing titania nanoparticles evaporating on a glass substrate. Deposit shapes from spotted drops at different pH values are measured using a laser profilometer. Our results show that the pH of the solution influences the dried deposit pattern, which can be ring-like or more uniform. The transition between these patterns is explained by considering how DLVO interactions such as the electrostatic and van der Waals forces modify the particle deposition process. Also, a phase diagram is proposed to describe how the shape of a colloidal deposit results from the competition among three flow patterns: a radial flow driven by evaporation at the wetting line, a Marangoni recirculating flow driven by surface tension gradients, and the transport of particles toward the substrate driven by DLVO interactions. This phase diagram explains three types of deposits commonly observed experimentally, such as a peripheral ring, a small central bump, or a uniform layer. Simulations and experiments are found in very good agreement.

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.

Absence of Marangoni convection at Marangoni numbers above 27,000 during water evaporation

Two mechanisms by which Marangoni convection can be produced at the interface of water with its vapor are: (1) by imposing a temperature gradient parallel to the water-vapor interface, and (2) by imposing a temperature gradient perpendicular to the interface that results in the liquid becoming unstable. A series of evaporation experiments conducted with H2O and with D2O maintained at the mouth of a stainless-steel funnel indicated the presence of Marangoni convection, but the mechanism producing the convection was unclear. We have investigated the mechanism using a funnel constructed with a polymethyl methacrylate that has a small thermal conductivity relative to that of water and repeating the evaporation experiments. Marangoni convection was eliminated with this funnel even though the Marangoni number, Ma, was in the range 8277< or =Ma< or =27 847 . A comparison of the assumptions made in the theories available to predict the onset of Marangoni convection with the observations made in this study indicates some of the assumptions are invalid: although generally neglected, energy transport through the vapor to the interface of evaporating water is significant; there is an interfacial temperature discontinuity, but it is in the opposite direction of that assumed in the existing theories: the interfacial-vapor temperature is greater than that of the liquid during evaporation; and the prediction of the critical Marangoni number is based on an arbitrarily chosen value of the heat-transfer coefficient. When the temperature gradient is perpendicular to the water-vapor interface, these invalid assumptions indicate present theories do not apply to volatile liquids.

Investigation of local evaporation flux and vapor-phase pressure at an evaporative droplet interface

In the steady-state experiments of water droplet evaporation, when the throat was heating at a stainless steel conical funnel, the interfacial liquid temperature was found to increase parabolically from the center line to the rim of the funnel with the global vapor-phase pressure at around 600 Pa. The energy conservation analysis at the interface indicates that the energy required for evaporation is maintained by thermal conduction to the interface from the liquid and vapor phases, thermocapillary convection at interface, and the viscous dissipation globally and locally. The local evaporation flux increases from the center line to the periphery as a result of multiple effects of energy transport at the interface. The local vapor-phase pressure predicted from statistical rate theory (SRT) is also found to increase monotonically toward the interface edge from the center line. However, the average value of the local vapor-phase pressures is in agreement with the measured global vapor-phase pressure within the measured error bar.

Role of molecular phonons and interfacial-temperature discontinuities in water evaporation

During steady-state water evaporation, when the vapor phase is heated electrically, the temperature on the vapor side of the interface has been reported to be as much as 27.83 degrees C greater than that on the liquid side. The reported interfacial temperatures were measured with thermocouple beads that were less than 50 microm in diameter and centered 35 microm from the interface in each phase. We examine the reliability of these measurements by using them with a theory of kinetics to predict the interfacial-liquid temperature. The predicted temperature discontinuities are found to be in agreement with those measured up to a temperature discontinuity of 15.69 degrees C , but larger discontinuities cannot be confirmed because of uncertainties in the vapor-phase pressure measurements. The theory of kinetics used in the analysis includes molecular phonons in the expression for the evaporation flux. We show it is essential to include these terms if the theory is to be used to predict the temperature discontinuities.

Heat transfer at the nanoscale: evaporation of nanodroplets

We demonstrate using molecular dynamics simulations of the Lennard-Jones fluid that the evaporation process of nanodroplets at the nanoscale is limited by the heat transfer. The temperature is continuous at the liquid-vapor interface if the liquid/vapor density ratio is small (of the order of 10) and discontinuous otherwise. The temperature in the vapor has a scaling form T(r,t)=T[r/R(t)], where R(t) is the radius of an evaporating droplet at time t and r is the distance from its center. Mechanical equilibrium establishes very quickly, and the pressure difference obeys the Laplace law during evaporation.

Recent advances on thermocapillary flows and interfacial conditions during the evaporation of liquids

Thermocapillary convection has a very different history for water than for other liquids. For water, several studies have pointed to the lack of evidence supporting the existence of thermocapillary (or Marangoni) convection. Other studies have given clear evidence of its existence and of the role it plays during steady-state water evaporation. We examine both sets of data and suggest a reason for the difference in the interpretation of the experimental data. For organic liquids, the evidence of thermocapillary convection has been clearly documented, but the issues are the type of flow that it generates during steady-state evaporation. We review the measurements and show that the flow field of the evaporating liquid is strongly affected by the presence of the thermocapillary convection. When the results obtained from both water and organic liquids are compared, they give further insight into the nature of thermocapillary convection.

Thermocapillary transport of energy during water evaporation

When evaporation occurs at a spherical water-vapor interface maintained at the circular mouth of a small funnel, studies of the energy transport have indicated that thermal conduction alone does not provide enough energy to evaporate the liquid at the observed rate. If the Gibbs model of the interface is adopted and the "surface-thermal capacity" is assigned a value of 30.6+/-0.8 kJ/(m2 K), then for evaporation experiments with the interfacial temperature in the range -10 degrees C< or =TLV< or =3.5 degrees C and Marangoni number (Ma) in the range 100<Ma<22,000, it was found that if energy transport by both thermocapillary convection and thermal conduction were taken into account, conservation of energy was fully satisfied. The question addressed herein is whether the assigned value of the surface-thermal capacity is an ad hoc empirical parameter or a property of the water-vapor interface that can be used in other circumstances. Accordingly, a series of experiments has been conducted in which water evaporated at cylindrical interfaces that were, on average, 4.4 times larger in area than that of the spherical interfaces used to measure the surface-thermal capacity initially. It is shown that using the value of the surface-thermal capacity determined at a spherical interface, the energy transported by thermocapillary convection and thermal conduction at a cylindrical interface is sufficient to evaporate the liquid at the observed rate. Knowing the value of the surface-thermal capacity also allows the local evaporation flux to be calculated from the measured temperature profiles in the liquid and vapor phases. The calculated local evaporation flux can then be used with statistical rate theory to calculate the vapor-phase pressure along the interface. The predicted mean vapor-phase pressure is in close agreement with that measured, and the predicted pressure gradient is consistent with that expected when thermocapillary convection is present.

Surface excess properties from energy transport measurements during water evaporation

When water evaporates at high rates, recent studies indicate thermal conduction to the interface does not provide enough energy to evaporate water at the observed rate and that it is perhaps thermocapillary convection that transports the remaining energy. This possibility is examined by applying the Gibbs dividing-surface approximation to develop an expression for the energy transported along the interface. When this energy transport rate is compared with that required to evaporate the liquid at the observed rate, it is found that a Gibbs excess property, the "surface-thermal capacity," can be evaluated. A series of 19 evaporation experiments has been conducted under conditions for which there was no buoyancy-driven convection and for which the evaporation rate was progressively increased. For Marangoni numbers, (Ma) less than approximately 100, the interface was quiescent and thermal conduction (the Stefan condition) correctly predicted the energy transport rate to the surface. For experiments with 100<Ma<22,000, thermocapillary convection was present and the thermal conduction did not fully account for the energy transport. However, if the surface-thermal capacity is assigned a value of 30.6+/-0.8 kJ/(m2K), then energy transport by thermocapillary convection and conduction provides the energy transport required to evaporate the liquid at the observed rate. For experiments with Ma>22,000, the interfacial flow was turbulent and viscous dissipation became important.

Surface-thermal capacity of from measurements made during steady-state evaporation

When D2O(l) evaporates into its vapor under steady-state conditions with the temperature field in the liquid arranged so that there is no buoyancy-driven convection and the Marangoni number is less than approximately 100, it is found that the interface is quiescent and thermal conduction to the interface supplies energy at a sufficient rate to evaporate the liquid. However, if the evaporation rate is raised so that the Marangoni number goes above approximately 100, the interface is transformed: a fluctuating thermocapillary flow occurs, and thermal conduction no longer supplies energy at a sufficient rate to evaporate the liquid. An energy analysis indicates conservation of energy can be satisfied only if thermocapillary convection is taken into account, and the surface-thermal capacity csigma is assigned a value of 32.5+/-0.8 kJ/(m2 K) when the temperature is in the range -10 degrees C< or =TLV< or =3.7 degrees C. This value is consistent with that found previously for H2O, and application of the Gibbs model gives a qualitative explanation for the value. Once the value of the surface-thermal capacity is known, the local heat flux along the interface can be calculated and statistical rate theory can be used to predict the local vapor-phase pressure on the interface. Since this theory introduces no adjustable parameters, the predicted pressure can be compared directly with that measured: this comparison indicates the mean of the pressures predicted to exist on the interface is in close agreement with those measured approximately 20 cm above the interface, and the small pressure gradient along the interface is consistent with the thermocapillary convection predicted from the interfacial temperature gradient.

Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet

We study the effects of Marangoni stresses on the flow in an evaporating sessile droplet, by extending a lubrication analysis and a finite element solution of the flow field in a drying droplet, developed earlier. The temperature distribution within the droplet is obtained from a solution of Laplace's equation, where quasi-steadiness and neglect of convection terms in the heat equation can be justified for small, slowly evaporating droplets. The evaporation flux and temperature profiles along the droplet surface are approximated by simple analytical forms and used as boundary conditions to obtain an axisymmetric analytical flow field from the lubrication theory for relatively flat droplets. A finite element algorithm is also developed to solve simultaneously the vapor concentration, and the thermal and flow fields in the droplet, which shows that the lubrication solution with the Marangoni stress is accurate for contact angles as high as 40 degrees. From our analysis, we find that surfactant contamination, at a surface concentration as small as 300 molecules/microm(2), can almost entirely suppress the Marangoni flow in the evaporating droplet.

Mean evaporation and condensation coefficients based on energy dependent condensation probability

A generalization of the classical Hertz-Knudsen and Schrage laws for the evaporation mass and energy fluxes at a liquid-vapor interface is derived from kinetic theory and a simple model for a velocity dependent condensation coefficient. These expressions, as well as the classical laws and simple phenomenological expressions, are then considered for the simulation of recent experiments [Phys. Rev. E 59, 419 (1999)]]. It is shown that mean condensation and evaporation coefficients in the mass flow influence the results only if they are small compared to unity and that the expression for evaporation mass flow determines the temperature of the liquid. Moreover, it is shown that the expression for evaporation energy flow plays the leading role in determining the interface temperature jump, which can be obtained in good agreement with the experiment from the generalized kinetic theory model and phenomenological approaches, but not from the classical kinetic-theory-based Hertz-Knudsen and Schrage laws. Analytical estimates show that the interface temperature jump depends strongly on the temperature gradient of the vapor just in front of the interface, which explains why much larger temperature jumps are observed in spherical geometry and the experiments as compared to planar settings.

Turbulent transition of thermocapillary flow induced by water evaporation

Water has been examined for thermocapillary convection while maintained just outside the mouth of a stainless-steel, conical funnel where it evaporated at different but steady rates. Evaporation at a series of controlled rates was produced by reducing the pressure in the vapor-phase to different but constant values while maintaining the temperature of the water a few millimeters below the interface at 3.56+/-0.03 degrees C in each case. Since water has its maximum density at 4 degrees C, these conditions ensured there would be no buoyancy-driven convection. The measured temperature profile along the liquid-vapor interface was found to be approximately axisymmetric and parabolic with its minimum on the center line and maximum at the periphery. The thermocapillary flow rate was determined in two ways: (1) It was calculated from the interfacial temperature gradient measured along the interface. (2) The deflection of a 12.7-microm-diameter, cantilevered probe inserted into the flow was measured and the liquid velocity required to give that deflection determined. The values determined by the two methods agree reasonably. As the vapor-phase pressure was reduced, the thermocapillary flow rate increased until a limiting value was reached. When the pressure was reduced further, certain of the variable relations underwent a bifurcation and the power spectrum of the probe displacement indicated it was a periodic function with frequency locking. These results suggest that thermocapillary flow plays an important role in the energy transport near the interface of evaporating water. In particular, it appears that the subinterface, uniform-temperature layer, reported in earlier studies, results from the mixing produced by the thermocapillary flow. The Stefan boundary condition is often applied to determine the energy flux to an interface where phase change is occurring; however, when there is strong convective flow parallel to the interface, the normal Stefan condition does not give an adequate description of the energy transport.