Analytical Model for Diffusive Evaporation of Sessile Droplets Coupled with Interfacial Cooling Effect

Unraveling the role of vaporization momentum in self-jumping dynamics of freezing supercooled droplets at reduced pressures

Supercooling of water complicates phase change dynamics, the understanding of which remains limited yet vital to energy-related and aerospace processes. Here, we investigate the freezing and jumping dynamics of supercooled water droplets on superhydrophobic surfaces, induced by a remarkable vaporization momentum, in a low-pressure environment. The vaporization momentum arises from the vaporization at droplet's free surface, progressed and intensified by recalescence, subsequently inducing droplet compression and finally self-jumping. By incorporating liquid-gas-solid phase changes involving vaporization, freezing recalescence, and liquid-solid interactions, we resolve the vaporization momentum and droplet dynamics, revealing a size-scaled jumping velocity and a nucleation-governed jumping direction. A droplet-size-defined regime map is established, distinguishing the vaporization-momentum-dominated self-jumping from evaporative drying and overpressure-initiated levitation, all induced by depressurization and vaporization. Our findings illuminate the role of supercooling and low-pressure mediated phase change in shaping fluid transport dynamics, with implications for passive anti-icing, advanced cooling, and climate physics.

Analysis of Evaporation of Droplet Pairs by a Quasi-Steady-State Diffusion Model Coupled with the Evaporative Cooling Effect

Multidroplet evaporation is a common phase-change phenomenon not only in nature but also in many industrial applications, including inkjet printing and spray cooling. The evaporation behavior of these droplets is strongly affected by the distance between neighboring droplets, and in particular, evaporation suppression occurs as the distance decreases. However, further quantitative information, such as the temperature and local evaporation flux, is limited because the analytical models of multidroplet evaporation only treat vapor diffusion, and the effect of the latent heat transfer through the liquid-vapor phase change is ignored. Here, we perform a numerical analysis of evaporating droplet pairs that linked vapor diffusion from the droplet surface and evaporative cooling. Heat transfer through the liquid and gas phases is also considered because the saturation pressure depends on the temperature. The results show an increase in the vapor concentration in the region between the two droplets. Consequently, the local evaporation flux in the proximate region significantly decreases with decreasing separation distance. This means that the latent heat transfer through the phase change is diminished, and an asymmetrical temperature distribution occurs in the liquid and gas phases. These numerical results provide quantitative information about the temperature and local evaporation flux of evaporating droplet pairs, and they will guide further investigation of multiple droplet evaporation.

Evaporation-driven liquid flow in sessile droplets

The evaporation of a sessile droplet spontaneously induces an internal capillary liquid flow. The surface-tension driven minimisation of surface area and/or surface-tension differences at the liquid-gas interface caused by evaporation-induced temperature or chemical gradients set the liquid into motion. This flow drags along suspended material and is one of the keys to control the material deposition in the stain that is left behind by a drying droplet. Applications of this principle range from the control of stain formation in the printing and coating industry, to the analysis of DNA, to forensic and medical research on blood stains, and to the use of evaporation-driven self-assembly for nanotechnology. Therefore, the evaporation of sessile droplets attracts an enormous interest from not only the fluid dynamics, but also the soft matter, chemistry, biology, engineering, nanotechnology and mathematics communities. As a consequence of this broad interest, knowledge on evaporation-driven flows in drying droplets has remained scattered among the different fields, leading to various misconceptions and misinterpretations. In this review we aim to unify these views, and reflect on the current understanding of evaporation-driven liquid flows in sessile droplets in the light of the most recent experimental and theoretical advances. In addition, we outline open questions and indicate promising directions for future research.

Lens Evaporation on Immiscible Liquid Surface with an Interfacial Cooling Effect

A theoretical heat and mass transfer model of volatile liquid lens evaporation on the surface of an immiscible liquid substrate is established in toroidal coordinates. According to the coupled boundary conditions of heat and mass transfer at the lens surface as well as the interfacial cooling effect, the analytical solutions of the temperature field inside the lens and the vapor concentration field around the lens are derived for the first time. Compared with the isothermal model, the change of contact radius calculated by the present model agrees well with the experimental data, especially when the liquid substrate reaches a relatively high temperature. It also reveals that the temperature distribution inside the lens is not uniform, which is similar to the sessile droplet evaporation on a solid substrate surface. In addition, the excess temperature, heat flux, and evaporation flux of the lens-air interface increase monotonically from the lens center to the contact line. Finally, the influences of density ratio and evaporative cooling number E ₀ on lens mass evaporation rate are analyzed, which shows that the lens mass evaporation rate decreases with increasing density ratio and evaporative cooling number.

Gas-Phase Temperature Mapping of Evaporating Microdroplets

Evaporation is a ubiquitous and complex phenomenon of importance to many natural and industrial systems. Evaporation occurs when molecules near the free interface overcome intermolecular attractions with the bulk liquid. As molecules escape the liquid phase, heat is removed, causing evaporative cooling. The influence of evaporative cooling on inducing a temperature difference with the surrounding atmosphere as well as within the liquid is poorly understood. Here, we develop a technique to overcome past difficulties encountered during the study of heterogeneous droplet evaporation by coupling a piezo-driven droplet generation mechanism to a controlled micro-thermocouple to probe microdroplet evaporation. The technique allowed us to probe the gas-phase temperature distribution using a micro-thermocouple (50 μm) in the vicinity of the liquid-vapor interface with high spatial (±10 μm) and temporal (±100 ms) resolution. We experimentally map the temperature gradient formed surrounding sessile water droplets having varying curvature dictated by the apparent advancing contact angle (100° ≲ θ ≲ 165°). The experiments were carried out at temperatures below and above ambient for a range of fixed droplet radii (130 μm ≲ R ≲ 330 μm). Our results provide a primary validation of the centuries-old theoretical framework underpinning heterogeneous droplet evaporation mediated by the working fluid, substrate, and gas thermophysical properties, droplet apparent contact angle, and droplet size. We show that microscale droplets residing on low-thermal-conductivity substrates such as glass absorb up to 8× more heat from the surrounding gas compared to droplets residing on high-thermal-conductivity substrates such as copper. Our work not only develops an experimental understanding of the heat transfer mechanisms governing droplet evaporation but also presents a powerful platform for the study and characterization of liquid-vapor transport at curved interfaces wetting and nonwetting advanced functional surfaces.

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.

Tuning the Splitting Behavior of Droplet in a Bifurcating Channel through Wettability-Capillarity Interaction

We present a comprehensive computational physics-based study of the influence of surface wettability on the displacement behavior of a droplet in a three-dimensional bifurcating channel. Various surface wettability configurations for the daughter branches are considered to gain insight into the wettability-capillarity interaction. Also, the influence of initial droplet size on the splitting dynamics for different wettability configurations is investigated. Time evolution of the droplet displacement behavior in the bifurcating channel is discussed for different physicochemical parameters including capillary number and wettability. Three distinct flow regimes are identified as the droplet interacts with the bifurcating tip of the channel, namely, splitting, nonsplitting, and oscillating regimes. Furthermore, the occurrence of Rayleigh-Plateau instability in different wettability scenarios is discussed. Additionally, the intricacies associated with the droplet dynamics are elucidated through the temporal evolution of the droplet surface area and mass outflow of the continuous phase. A flow regime map based on the capillary number and wettability contrast of the daughter branches is proposed for a comprehensive description of the droplet dynamics.

Evaporation of Initially Heated Sessile Droplets and the Resultant Dried Colloidal Deposits on Substrates Held at Ambient Temperature

The present study experimentally and numerically investigates the evaporation and resultant patterns of dried deposits of aqueous colloidal sessile droplets when the droplets are initially elevated to a high temperature before being placed on a substrate held at ambient temperature. The system is then released for natural evaporation without applying any external perturbation. Infrared thermography and optical profilometry are used as essential tools for interfacial temperature measurements and quantification of coffee-ring dimensions, respectively. Initially, a significant temperature gradient exists along the liquid-gas interface as soon as the droplet is deposited on the substrate, which triggers a Marangoni stress-induced recirculation flow directed from the top of the droplet toward the contact line along the liquid-gas interface. Thus, the flow is in the reverse direction to that seen in the conventional substrate heating case. Interestingly, this temperature gradient decays rapidly within the first 10% of the total evaporation time and the droplet-substrate system reaches thermal equilibrium with ambient thereafter. Despite the fast decay of the temperature gradient, the coffee-ring dimensions significantly diminish, leading to an inner deposit. A reduction of 50-70% in the coffee-ring dimensions is recorded by elevating the initial droplet temperature from 25 to 75 °C for suspended particle concentration varying between 0.05 and 1.0% v/v. This suppression of the coffee-ring effect is attributed to the fact that the initial Marangoni stress-induced recirculation flow continues until the last stage of evaporation, even after the interfacial temperature gradient vanishes. This is essentially a consequence of liquid inertia. Finally, a finite-element-based two-dimensional modeling in axisymmetric geometry is found to capture the measurements with reasonable fidelity and the hypothesis considered in the present study corroborates well with a first approximation qualitative scaling analysis. Overall, together with a new experimental condition, the present investigation discloses a distinct nature of Marangoni stress-induced flow in a drying droplet and its role in influencing the associated colloidal deposits, which was not explored previously. The insights gained from this study are useful to advance technical applications such as spray cooling, inkjet printing, bioassays, etc.

Theoretical Analysis of a Sessile Evaporating Droplet on a Curved Substrate with an Interfacial Cooling Effect

Sessile droplet evaporation is widely encountered in nature, and it has numerous applications in industrial and scientific communities; therefore, the accurate prediction of droplet evaporation has great significance in practical applications. In this paper, for the first time, a comprehensive theoretical model is built up for diffusion-controlled heat and mass transfer for sessile droplet evaporation on a curved substrate in toroidal coordinates. The evaporative mass transfer is coupled with the heat transfer across the gas-liquid droplet interface, as well as the heat transfer across the solid-liquid interface of the curved substrate. The effects of interfacial cooling and thermal conductivity of the droplet and substrate as well as their initial shapes on the droplet evaporation are provided in details. It is found that the evaporative flux usually increases sharply near the droplet edge due to the short distance for heat conduction from the substrate to the droplet; however, it can be reversed from sharp increasing to decreasing at a low thermal conductivity ratio k_R < 0.3 of the substrate over droplet or large initial droplet contact angle θ > 30°. The interfacial evaporative cooling effect can always suppress the droplet evaporation. The lifetime of evaporative droplet can be prolonged with the decreasing thermal conductivity ratio, increasing evaporative cooling number, and increasing initial droplet contact angle or tangential angle of a curved substrate. These findings may be of great significance in the applications of droplet evaporation on the curved substrate.

Evaporation-Induced Wetting Transition of Nanodroplets on Nanopatterned Surfaces with Concentric Rings: Surface Geometry and Wettability Effects

Droplet evaporation is widespread in natural and industrial application, and the rapid and efficient evaporation can significantly improve energy efficiency. However, the fundamental mechanism of contact line dynamics and the microscopic characteristics of evaporating nanodroplets are not well understood. Moreover, how to design a nanostructure surface to enhance nanodroplet evaporation remains unclear. Here, through molecular dynamics simulation, we investigated the evaporation dynamics of nanodroplets on various nanoring surfaces with different geometric parameters and wettability. By measuring the changes of contact radius and contact angle, the results showed that nanodroplets successively exhibit constant contact angle (CCA), constant contact radius (CCR), and mix mode during evaporation, and the evaporation-induced CCA-CCR transition, in essence, is a Cassie-Wenzel wetting transition, whose onset time is remarkably dependent on the surface roughness and wettability. We found that this evaporation-induced wetting transition is postponed on the surface with small nanostructure spacing and weak hydrophilicity, and the evaporation rate of nanodroplets improves accordingly. The dense and hydrophobic nanostructures can not only restrain the Cassie-Wenzel transition, but also enhance the evaporation rate of nanodroplets. Last, through the potential energy field analysis of nanoring substrates, we revealed that the Cassie-Wenzel wetting transition of nanodroplets is a process of molecule migration to low potential energy regions. Our work provides guidance for designing nanostructure surfaces to effectively control the droplet wetting state and enhance its mass transfer performance of phase change.

Spontaneous water adsorption-desorption oscillations in mesoporous thin films

Understanding fluid transport and phase changes in nanopore structures is of great interest to many application fields, from energy conversion to water harvesting. This work discusses the spontaneous oscillations of the water saturation of mesoporous thin films, in the zone adjacent to a sessile water drop, at ambient conditions. The wetting-front dynamics onto the film is described by considering three coexisting phenomena: infiltration from the water drop, condensation from air vapor, and evaporation to the ambient. It was found that the oscillations follow spontaneous condensation-evaporation imbalances, which are governed by the hysteretic character of the adsorption-desorption behavior of the mesoporous material. The outcomes of this work provide insights on the complex interplay between water and nanopore structures, which has practical implications for the handling of humid microenvironments in lab-on-a-chip technology, as well as for many processes that take part of the cycle of water in nature.

Liquid Leaks: Dripping Versus Evaporation

Liquid leaks often reveal themselves as pendant drops or drips emanating from a low point on a fluid handling component. For volatile liquids, understanding the contributions of interfacial properties, such as diffusivity of the liquid and wettability of the solid, is crucial to determining leak rates. To estimate the resolution of hydrostatic leak testing, the competing factors of leak and evaporation rates were analyzed. We used drop volumes and contact angles along with intrinsic fluid properties to calculate the detection limit of hydrostatic leak tests. For water and ethanol, we reckon that it is approximately 10^-4 to 10^-5 cm³/s in dry air.

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.