Wetting and Contact-Angle Hysteresis: Density Asymmetry and van der Waals Force.

A droplet depositing on a solid substrate leads to the wetting phenomenon, such as dew on plant leaves. On an ideally smooth substrate, the classic Young's law has been employed to describe the wetting effect. However, no real substrate is ideally smooth at the microscale. Given this fact, we introduce a surface composition concept to scrutinize the wetting mechanism via considering the liquid-gas density asymmetry and the fluid-solid van der Waals interaction. The current concept enables one to comprehend counterintuitive phenomenon of contact-angle hysteresis on a smooth substrate and increase of contact angle with temperature as well as gas bubble wetting.

Wetting Effect Induced Depletion and Adsorption Layers: Diffuse Interface Perspective.

When a multi-component fluid contacts arigid solid substrate, the van der Waals interaction between fluids and substrate induces a depletion/adsorption layer depending on the intrinsic wettability of the system. In this study, we investigate the depletion/adsorption behaviors of A-B fluid system. We derive analytical expressions for the equilibrium layer thickness and the equilibrium composition distribution near the solid wall, based on the theories of de Gennes and Cahn. Our derivation is verified through phase-field simulations, wherein the substrate wettability, A-B interfacial tension, and temperature are systematically varied. Our findings underscore two pivotal mechanisms governing the equilibrium layer thickness. With an increase in the wall free energy, the substrate wettability dominates the layer formation, aligning with de Gennes' theory. When the interfacial tension increases, or temperature rises, the layer formation is determined by the A-B interactions, obeying Cahn's theory. Additionally, we extend our study to non-equilibrium systems where the initial composition deviates from the binodal line. Notably, macroscopic depletion/adsorption layers form on the substrate, which are significantly thicker than the equilibrium microscopic layers. This macroscopic layer formation is attributed to the interplay of phase separation and Ostwald ripening. We anticipate that the present finding could deepen our knowledge on the depletion/adsorption behaviors of immiscible fluids.

Wetting Behavior of Inkjet-Printed Electronic Inks on Patterned Substrates.

In inkjet printing technology, one important factor influencing the printing quality and reliability of printed films is the interaction of the jetted ink with the substrate surface. This short-range interaction determines the wettability and the adhesion of the ink to the solid surface and is hence responsible for the final shape of the deposited ink. Here, we investigate wetting morphologies of inkjet-printed inks on patterned substrates by carefully designed experimental test structures and simulations. The contact angles, the surface properties, and drop shapes, as well as their influence on the device variability, are experimentally and theoretically analyzed. For the simulations, we employ the phase-field method, which is based on the free energy minimization of the two-phase system with the given wetting boundary conditions. Through a systematic investigation of printed drops on patterned substrates consisting of hydrophilic and hydrophobic areas, we report that the printed morphology is related not only to the designed layout and the drop volume but also to the printing strategy and the wettability. Furthermore, we show how one can modify the intrinsic wettability of the patterned substrates to enhance the printing quality and reliability. Based on the present findings, we cast light on the improvement of the fabrication quality of thin film transistors.

Evolution dynamics of thin liquid structures investigated using a phase-field model.

Liquid structures of thin-films and torus droplets are omnipresent in daily lives. The morphological evolution of liquid structures suspending in another immiscible fluid and sitting on a solid substrate is investigated by using three-dimensional (3D) phase-field (PF) simulations. Here, we address the evolution dynamics by scrutinizing the interplay of surface energy, kinetic energy, and viscous dissipation, which is characterized by Reynolds number Re and Weber number We. We observe special droplet breakup phenomena by varying Re and We. In addition, we gain the essential physical insights into controlling the droplet formation resulting from the morphological evolution of the liquid structures by characterizing the top and side profiles under different circumstances. We find that the shape evolution of the liquid structures is intimately related to the initial shape, Re, We as well as the intrinsic wettability of the substrate. Furthermore, it is revealed that the evolution dynamics are determined by the competition between the coalescence phenomenology and the hydrodynamic instability of the liquid structures. For the coalescence phenomenology, the liquid structure merges onto itself, while the hydrodynamic instability leads to the breakup of the liquid structure. Last but not least, we investigate the influence of wall relaxation on the breakup outcome of torus droplets on substrates with different contact angles. We shed light on how the key parameters including the initial shape, Re, We, wettability, and wall relaxation influence the droplet dynamics and droplet formation. These findings are anticipated to contribute insights into droplet-based systems, potentially impacting areas like ink-jet printing, drug delivery systems, and microfluidic devices, where the interplay of surface energy, kinetic energy, and viscous dissipation plays a crucial role.

Nanoliter Scale Parallel Liquid-Liquid Extraction for High-Throughput Purification on a Droplet Microarray.

In the current drug discovery process, the synthesis of compound libraries is separated from biological screenings both conceptually and technologically. One of the reasons is that parallel on-chip high-throughput purification of synthesized compounds is still a major challenge. Here, on-chip miniaturized high-throughput liquid-liquid extraction in volumes down to 150 nL with efficiency comparable to or better than large-scale extraction utilizing separation funnels is demonstrated. The method is based on automated and programmable merging of arrays of aqueous nanoliter droplets with organic droplets. Multi-step extraction performed simultaneously or with changing conditions as well as handling of femtomoles of compounds are demonstrated. In addition, the extraction efficiency is analyzed with a fast optical readout as well as matrix-assisted laser desorption ionization-mass spectrometry on-chip detection. The new massively parallel and miniaturized purification method adds another important tool to the chemBIOS concept combining chemical combinatorial synthesis with biological screenings on the same miniaturized droplet microarray platform, which will be essential to accelerate drug discovery.

Effect of wall free energy formulation on the wetting phenomenon: Conservative Allen-Cahn model.

Cahn introduced the concept of wall energy to describe the interaction between two immiscible fluids and a solid wall [J. W. Cahn, J. Chem. Phys. 66, 3667-3672 (1977)]. This quintessential concept has been successfully applied to describe various wetting phenomena of a droplet in contact with a solid surface. The usually formulated wall free energy results in the so-called surface composition that is not equal to the bulk composition. This composition difference leads to a limited range of contact angles which can be achieved by the linear/high-order polynomial wall free energy. To address this issue and to improve the adaptability of the model, we symmetrically discuss the formulation of the wall free energy on the Young's contact angle via Allen-Cahn model. In our model, we modify the calculation of the fluid-solid interfacial tensions according to the Cahn's theory by considering the excess free energy contributed by the distorted composition profile induced by the surface effect. Additionally, we propose a semi-obstacle wall free energy which enforces the surface composition to be the bulk composition within the framework of bulk obstacle potential. By this way, the accuracy of the contact angle close to 0° and 180° is significantly improved in the phase-field simulations. We further reveal that the volume preservation term in the conservative Allen-Cahn model has a more significant impact on the wetting behavior on superhydrophobic surfaces than on hydrophilic surfaces, which is attributed to the curvature effect. Our findings provide alternative insights into wetting behavior on superhydrophilic and superhydrophobic surfaces.

Janus Droplet Formation via Thermally Induced Phase Separation: A Numerical Model with Diffusion and Convection.

Microscale Janus particles have versatile potential applications in many physical and biomedical fields, such as microsensor, micromotor, and drug delivery. Here, we present a phase-field approach of multicomponent and multiphase to investigate the Janus droplet formation via thermally induced phase separation. The crucial kinetics for the formation of Janus droplets consisting of two polymer species and a solvent component via an interplay of both diffusion and convection is considered in the Cahn-Hilliard-Navier-Stokes equation. The simulation results of the phase-field model show that unequal interfacial tensions between the two polymer species and the solvent result in asymmetric phase separation in the formation process of Janus droplets. This asymmetric phase separation plays a vital role in the establishment of the so-called core-shell structure that has been observed in previous experiments. By varying the droplet size, the surface tension, and the molecular interaction between the polymer species, several novel droplet morphologies are predicted in the development process of Janus droplets. Moreover, we stress that the hydrodynamics should be reckoned as a non-negligible mechanism that not only accelerates the Janus droplet evolution but also has great impacts on the coarsening and coalescence of the Janus droplets.

Phase-Field Modeling of Multiple Emulsions Via Spinodal Decomposition.

Currently, multiple emulsions via liquid-liquid phase separation in ternary polymer solutions have sparked considerable interest because of its remarkable potential in physical, medical, and biological applications. The transient "onion-like" multilayers are highly dependent on the evolution kinetics, which is challenging to be scrutinized in experiments and has not yet been fully understood. Here, we report a thermodynamically consistent multicomponent Cahn-Hilliard model to investigate the kinetics of multiple emulsions by tracing the temporal evolution of the local compositions inside the emulsion droplets. We reveal that the mechanism governing the kinetics is attributed to the competition between surface energy minimization and phase separation. Based on this concept, a generalized morphology diagram for different emulsion patterns is achieved, showing a good accordance with previous experiments. Moreover, combining the analysis for the kinetics and the morphology diagram, we predict new emulsion structures that provide general guidelines to discovery, design, and manipulation of complex multiphase emulsions.

Wetting transition and phase separation on flat substrates and in porous structures.

In contrast to the wetting phenomenon of pure substance phase, we here concentrate on the wetting behavior of immiscible fluids with two components via numerical simulations. We investigate the energetic contribution, the wall energy and the surface excess energy, to the wetting behavior of liquid solutions varying with temperature. This investigation is in accordance with Cahn's wetting transition theory, where the surface composition plays a vital role. By analyzing the energetic contributions, we reveal two different physical mechanisms of complete wetting: (i) surface tension driven complete wetting, where the wetting microstructure is achieved via the outward spreading of the triple junction, and (ii) diffusion induced complete wetting, where the wetting film is achieved through a direct deposition of the solute on the substrate. The former one is consistent with the classic theory of Young's law, and the latter one is an alternative mechanism. To indicate the broad multiplicity of the microstructural arrangements, we take porous structures to exemplarily elucidate the formation of alternative perfect wetting structures. Differing from the wetting on a flat substrate, we show that the surface composition varying with temperature leads to a distinct wetting phenomenon in porous structures. The present findings provide an alternative interpretation for complete wetting and are expected to be exploited for designing more effectively and efficiently superhydrophilic structures.

How do chemical patterns affect equilibrium droplet shapes?

By utilizing a proposed analytical model in combination with the phase-field method, we present a comprehensive study on the effect of chemical patterns on equilibrium droplet morphologies. Here, three influencing factors, the droplet sizes, contact angles, and the ratios of the hydrophilic area to the hydrophobic area, are contemplated. In the analytical model, chemical heterogeneities are described by different non-linear functions. By tuning these functions and the related parameters, the analytical model is capable of calculating the energy landscapes of the system. The chemically patterned surfaces display complex energy landscapes with chemical-heterogeneity-induced local minima, which correspond to the equilibrium morphologies of the droplets. Phase-field (PF) simulations are accordingly conducted and compared with the predicted equilibrium morphologies. In addition, we propose a modified Cassie-Baxter (CB) model to delineate the equilibrium droplet shapes. In contrast to the classic CB model, our extension is not only restricted to the shape with a spherical cap. Both the energy landscape method and the modified CB model are demonstrated to have a good agreement with the PF simulations.

Investigation of Equilibrium Droplet Shapes on Chemically Striped Patterned Surfaces Using Phase-Field Method.

We systematically investigate the equilibrium shapes of droplets deposited on a set of chemically striped patterned surfaces by using an Allen?Cahn-type phase-field model. Varying the widths of the stripes d, the volume V, as well as the initial positions of the droplets, we release the droplets on the top of the surfaces and observe the final droplet shapes. It is found that there are either one or two equilibrium shapes for a fixed ratio of d/ V1/3 and each equilibrium shape corresponds to an energy minimum state. The aspect ratio of the droplets ? shows a periodic oscillation behavior with a decreasing amplitude as d/ V1/3 decreases, similar to the stick?slip?jump movement of a slowly condensing droplet on a chemically striped patterned surface. Additionally, by comparing the movements of slowly evaporating and condensing droplets, we have observed a hysteresis phenomenon, which reveals that the final shapes of droplets also rely on the moving paths. Through modifying the dynamic contact angle boundary condition, the contact line movements of droplets under condensation and evaporation, which are far from equilibrium, are addressed.

Electric-field-induced lamellar to hexagonally perforated lamellar transition in diblock copolymer thin films: kinetic pathways.

Symmetric block-copolymers, hitherto, are well known to evolve into parallel, perpendicular and mixed lamellar morphologies under the concomitant influence of an electric field and substrate affinity. In the present work, we show that an additional imposed confinement can effectuate a novel parallel lamellar to hexagonally perforated lamellar (HPL) transition in monolayer and bilayer films. Three dimensional numerical studies are performed using the Ohta-Kawasaki functional, complemented with an exact solution of Maxwell's equation. HPL is shown to stabilize at large substrate affinity in a narrow region of the phase diagram between parallel and perpendicular lamellar transitions in ultra-thin films. Additionally, we also identify perforated lamellae as intermediate structures during parallel-to-perpendicular lamellar transition. A systematic analysis using Minkowski functionals yields deeper insights into the associated kinetic pathways.

Compound Droplets on Fibers.

Droplets on fibers have been extensively studied in the recent years. Although the equilibrium shapes of simple droplets on fibers are well established, the situation becomes more complex for compound fluidic systems. Through experimental and numerical investigations, we show herein that compound droplets can be formed on fibers and that they adopt specific geometries. We focus on the various contact lines formed at the meeting of the different phases and we study their equilibrium state. It appears that, depending on the surface tensions, the triple contact lines can remain separate or merge together and form quadruple lines. The nature of the contact lines influences the behavior of the compound droplets on fibers. Indeed, both experimental and numerical results show that, during the detachment process, depending on whether the contact lines are triple or quadruple, the characteristic length is the inner droplet radius or the fiber radius.

A phase-field approach for wetting phenomena of multiphase droplets on solid surfaces.

We study the equilibrium wetting behavior of immiscible multiphase systems on a flat, solid substrate. We present numerical computations which are based on a vector-valued multiphase-field model of Allen-Cahn type, with a new boundary condition, based on appropriately designed surface energy contributions in order to ensure the right contact angles at multiphase junctions. Experimental investigations are carried out to validate the method and to support the numerical results.

Revealing process and material parameter effects on densification via phase-field studies.

Sintering is an important processing step in both ceramics and metals processing. The microstructure resulting from this process determines many materials properties of interest. Hence the accurate prediction of the microstructure, depending on processing and materials parameters, is of great importance. The phase-field method offers a way of predicting this microstructural evolution on a mesoscopic scale. The present paper employs this method to investigate concurrent densification and grain growth and the influence of stress on densification. Furthermore, the method is applied to simulate the entire freeze-casting process chain for the first time ever by simulating the freezing and sintering processes separately and passing the frozen microstructure to the present sintering model.

Brownian motion of droplets induced by thermal noise.

Brownian motion (BM) is pivotal in natural science for the stochastic motion of microscopic droplets. In this study, we investigate BM driven by thermal composition noise at submicro scales, where intermolecular diffusion and surface tension both are significant. To address BM of microscopic droplets, we develop two stochastic multiphase-field models coupled with the full Navier-Stokes equation, namely, Allen-Cahn-Navier-Stokes and Cahn-Hilliard-Navier-Stokes. Both models are validated against capillary-wave theory; the Einstein's relation for the Brownian coefficient D^{*}∼k_{B}T/r at thermodynamic equilibrium is recovered. Moreover, by adjusting the co-action of the diffusion, Marangoni effect, and viscous friction, two nonequilibrium phenomena are observed. (I) The droplet motion transits from the Brownian to Ballistic with increasing Marangoni effect which is emanated from the energy dissipation mechanism distinct from the conventional fluctuation-dissipation theorem. (II) The deterministic droplet motion is triggered by the noise induced nonuniform velocity field which leads to a novel droplet coalescence mechanism associated with the thermal noise.

Phase-field simulations at the atomic scale in comparison to molecular dynamics.

Early solidification is investigated using two different simulation techniques: the molecular dynamics (MD) and the phase-field (PF) methods. While the first describes the evolution of a system on the basis of motion equations of particles, the second grounds on the evolution of continuous local order parameter field. The aim of this study is to probe the ability of the mesoscopic phase-field method to make predictions of growth velocity at the nanoscopic length scale. For this purpose the isothermal growth of a spherical crystalline cluster embedded in a melt is considered. The system in study is Ni modeled with the embedded atom method (EAM). The bulk and interfacial properties required in the PF method are obtained from MD simulations. Also the initial configuration obtained from MD data is used in the PF as input. Results for the evolution of the cluster volume at high and moderate undercooling are presented.

Wetting Effect on Patterned Substrates.

A droplet deposited on a solid substrate leads to the wetting phenomenon. A natural observation is the lotus effect, known for its superhydrophobicity. This special feature is engendered by the structured microstructure of the lotus leaf, namely, surface heterogeneity, as explained by the quintessential Cassie-Wenzel theory (CWT). In this work, recent designs of functional substrates are overviewed based on the CWT via manipulating the contact area between the liquid and the solid substrate as well as the intrinsic Young's contact angle. Moreover, the limitation of the CWT is discussed. When the droplet size is comparable to the surface heterogeneity, anisotropic wetting morphology often appears, which is beyond the scope of the Cassie-Wenzel work. In this case, several recent studies addressing the anisotropic wetting effect on chemically and mechanically patterned substrates are elucidated. Surface designs for anisotropic wetting morphologies are summarized with respect to the shape and the arrangement of the surface heterogeneity, the droplet volume, the deposition position of the droplet, as well as the mean curvature of the surface heterogeneity. A thermodynamic interpretation for the wetting effect and the corresponding open questions are presented at the end.

Asymptotic analysis of multi-phase-field models: A thorough consideration of junctions.

The solutions of multi-phase-field models exhibit boundary layer behavior not only along the binary interfaces but also at the common contacts of three or more phases, i.e., junctions. Hence, to completely determine the asymptotic behavior of a multi-phase-field model, the inner analysis of both types of layers has to be carried out, whereas, traditionally, the junctions part is ignored. This is remedied in the current work for a phase-field model of simple grain growth in two spatial dimensions. Since the junction neighbourhoods are fundamentally different from those of the binary interfaces, pertinent matching conditions had to be derived from scratch, which is also accomplished in a detailed manner. The leading-order matching analysis of the junctions exposed the restrictions present on the interfacial arrangement at the common meeting point, while the next-to-the-leading one uncovered the law governing the instantaneous motion of the latter. In particular, it is predicted for the considered model that the Young's law is always satisfied at a triple point, whether or not it is at rest. Surprisingly, the mobilities and the curvatures of the involving interfaces as well as the driving forces on the them do not affect this result. However, they do play a significant role in determining the instantaneous velocity of the junction point. The study has opened up many new directions for future research.

Line tension of sessile droplets: Thermodynamic considerations.

We deduce a thermodynamically consistent diffuse interface model to study the line tension phenomenon of sessile droplets. By extending the standard Cahn-Hilliard model via modifying the free energy functional due to the spatial reflection asymmetry at the substrate, we provide an alternative interpretation for the wall energy. In particular, we find the connection of the line tension effect with the droplet-matrix-substrate triple interactions. This finding reveals that the apparent contact angle deviating from Young's law is contributed by the wall energy reduction as well as the line energy minimization. Besides, the intrinsic negative line tension resulting from the curvature effect is observed in our simulations and shows good accordance with recent experiments [Tan et al. Phys. Rev. Lett. 130, 064003 (2023)0031-900710.1103/PhysRevLett.130.064003]. Moreover, our model sheds light upon the understanding of the wetting edge formation which results from the vying effect of wall energy and line tension.