Surface patterning via evaporation of ultrathin films containing nanoparticles

Electric field-modulated evaporative thin film deposition of bio-particles for piezoelectric applications

Bio-based functional materials can be used to replace or limit the use of synthetic materials sourced from unsustainable sources. However, the potential of such materials remains largely unexplored. In this study, we demonstrate the use of weak AC electric fields to deposit ultra-thin piezoelectric films from cellulose nanocrystals (CNC). This is the first time electric fields are used to realize <50 nm thick uniform bio-based piezoelectric films wherein the bioparticles exhibit unidirectional arrangement. Interestingly, we found that the use of weak AC electric fields of suitable frequencies completely mitigates the coffee ring effect (CRE), which results in defect-free uniform ultra-thin films. Additionally, the electric fields appear to help in realizing unidirectional alignment of particles in the films, which enhances their piezoelectric properties. The method was also tested for chitin nanocrystals (ChNC), which have a similar aspect ratio but bear opposite polarity surface charges, and the influence of the field on coffee ring formation and particle orientation in CNC thin film deposition was validated. The phenomena can be attributed to the constant spatio-temporal curvature of the evaporating liquid film, the transient state between the three-phase contact (TPC) line, the electric field-dependent contact angle, and the permanent and field-induced dipole moments. These factors lead to particle polarization and alignment. The films have an optimum electrical frequency of deposition at which they are continuous and uniformly thin, have unidirectional alignment of particles, and function as a single dipole.

Drops on Polymer Brushes: Advances in Thin-Film Modeling of Adaptive Substrates

We briefly review recent advances in the hydrodynamic modeling of the dynamics of droplets on adaptive substrates, in particular, solids that are covered by polymer brushes. Thereby, the focus is on long-wave and full-curvature variants of mesoscopic hydrodynamic models in gradient dynamics form. After introducing the approach for films/drops of nonvolatile simple liquids on a rigid smooth solid substrate, it is first expanded to an arbitrary number of coupled degrees of freedom before considering the specific case of drops of volatile liquids on brush-covered solids. After presenting the model, its usage is illustrated by briefly considering the natural and forced spreading of drops of nonvolatile liquids on a horizontal brush-covered substrate, stick-slip motion of advancing contact lines as well as drops sliding down a brush-covered incline. Finally, volatile liquids are also considered.

Droplet evaporation on soft solid substrates

Droplet evaporation on soft solid substrates is relevant to applications such as fabrication of microlenses and controlled particle deposition. Here, we develop a lubrication-theory-based model to advance fundamental understanding of the important limiting case of a planar droplet evaporating on a linear viscoelastic solid. A set of partial differential equations describing the time evolution of the liquid-air and liquid-solid interfaces is derived and solved with a finite-difference method. A disjoining-pressure/precursor-film approach is used to describe contact-line motion, and the one sided model is used to describe solvent evaporation. Parametric studies are conducted to investigate the effect of solid properties (thickness, viscosity, shear modulus, wettability) and evaporation rate on droplet dynamics. Our results indicate that softer substrates speed up droplet evaporation due to prolonged pinning of the contact line. Results from our model are able to qualitatively reproduce some key trends observed in experiments. Due to its systematic formulation, our model can readily be extended to more complex situations of interest such as evaporation of particle-laden droplets on soft solid substrates.

Applying droplets and films in evaporative lithography

This review covers experimental results of evaporative lithography and analyzes existing mathematical models of this method. Evaporating droplets and films are used in different fields, such as cooling of heated surfaces of electronic devices, diagnostics in health care, creation of transparent conductive coatings on flexible substrates, and surface patterning. A method called evaporative lithography emerged after the connection between the coffee ring effect taking place in drying colloidal droplets and naturally occurring inhomogeneous vapor flux densities from liquid-vapor interfaces was established. Essential control of the colloidal particle deposit patterns is achieved in this method by producing ambient conditions that induce a nonuniform evaporation profile from the colloidal liquid surface. Evaporative lithography is part of a wider field known as "evaporative-induced self-assembly" (EISA). EISA involves methods based on contact line processes, methods employing particle interaction effects, and evaporative lithography. As a rule, evaporative lithography is a flexible and single-stage process with such advantages as simplicity, low price, and the possibility of application to almost any substrate without pretreatment. Since there is no mechanical impact on the template in evaporative lithography, the template integrity is preserved in the process. The method is also useful for creating materials with localized functions, such as slipperiness and self-healing. For these reasons, evaporative lithography attracts increasing attention and has a number of noticeable achievements at present. We also analyze limitations of the approach and ways of its further development.

Evaporation and morphological patterns of bi-dispersed colloidal droplets on hydrophilic and hydrophobic surfaces

Understanding the formation of different morphological patterns depending on the particle size and surface wettability has great relevance in the separation, mixing and concentration of micro/nano particles and biological entities. We report the evaporation and morphological patterns of evaporating bi-dispersed colloidal droplets on hydrophilic and hydrophobic surfaces. To explain the underlying mechanisms of various particle distribution patterns, we propose a phenomenological model that accounts for the drag force, van der Waals and electrostatic interaction forces, and surface tension force acting on the particles. In the case of the hydrophilic surface (θ ∼ 27°), there is a competition between the frictional force arising due to the van der Waals (∼10-8 N) and electrostatic interaction forces (∼10-10 N) and the surface tension force (∼10-7 N) that depends on the particle size. Consequently, the smaller particles (0.2 and 1.0 μm in diameter) are found to be pinned and form an outer ring at the contact line whereas the larger particles (3.0 and 6.0 μm in diameter) move inward, either forming an inner ring or flocculating depending on the particle size. Interestingly, a completely different morphological pattern of the micro/nano particles is observed on a hydrophobic substrate (θ ∼ 110°): contact line pinning is no longer observed and particles form a centralized deposition pattern. The order of the magnitude of the surface tension force is higher as compared to the frictional force (∼10-8 N); thus the particles are driven radially inward and accumulate at the center of the droplet. Owing to the mixed mode of evaporation toward the end of evaporation, only a fraction of smaller particles travel radially outward due to the coffee-ring effect. Scanning electron microscopy images reveal that smaller particles are present mostly at the center with a small fraction of smaller particles at the edge of the pattern, whereas larger particles are uniformly distributed throughout.

Connecting Monotonic and Oscillatory Motions of the Meniscus of a Volatile Polymer Solution to the Transport of Polymer Coils and Deposit Morphology

We study the deposition mechanisms of polymer from a confined  meniscus of volatile liquid. In particular, we investigate the physical processes that are responsible for qualitative changes in the pattern deposition of polymer and the underlying interplay of the state of pattern deposition, motion of the meniscus, and the transport of polymer within the meniscus. As a model system we evaporate a solution of poly(methyl methacrylate) (PMMA) in toluene. Different deposition patterns are observed when varying the molecular mass,  the initial concentration of the solute, and temperature; these  are systematically presented in the form of morphological phase diagrams. The modi of deposition and meniscus motion are correlated. They vary with the ratio between the evaporation-driven convective flux and the diffusive flux of the polymer coils in the solution. In the case of a diffusion-dominated solute transport, the solution monotonically dewets the solid substrate by evaporation, supporting continuous contact line motion and continuous polymer deposition. However, a convection-dominated transport results in an oscillatory ratcheting dewetting-wetting motion of the contact line with more pronounced dewetting phases. The deposition process is then periodic and produces a stripe pattern. The oscillatory motion of the meniscus differs from the well documented stick-slip motion of the meniscus, observed as well, and is attributed to the opposing influences of evaporation and Marangoni stresses, which alternately dominate the deposition process.

Drying of Droplets of Colloidal Suspensions on Rough Substrates

In many technological applications, excess solvent must be removed from liquid droplets to deposit solutes onto substrates. Often, the substrates on which the droplets rest may possess some roughness, either intended or unintended. Motivated by these observations, we present a lubrication-theory-based model to study the drying of droplets of colloidal suspensions on a substrate containing a topographical defect. The model consists of a system of one-dimensional partial differential equations accounting for the shape of the droplet and depth-averaged concentration of colloidal particles. A precursor film and disjoining pressure are used to describe the contact-line region, and evaporation is included using the well-known one-sided model. Finite-difference solutions reveal that when colloidal particles are absent, the droplet contact line can pin to a defect for a significant portion of the drying time due to a balance between capillary-pressure gradients and disjoining-pressure gradients. The time-evolution of the droplet radius and contact angle exhibits the constant-radius and constant-contact-angle stages that have been observed in prior experiments. When colloidal particles are present and the defect is absent, the model predicts that particles will be deposited near the center of the droplet in a cone-like pattern. However, when a defect is present, pinning of the contact-line accelerates droplet solidification, leading to particle deposition near the droplet edge in a coffee-ring pattern. These predictions are consistent with prior experimental observations, and illustrate the critical role contact-line pinning plays in controlling the dynamics of drying droplets.

Influence of a Propagating Megahertz Surface Acoustic Wave on the Pattern Deposition of Solute Mass off an Evaporating Solution

We study the influence of a megahertz Rayleigh surface acoustic wave (SAW), propagating in a solid substrate, on the pattern deposition of a solute mass off an evaporating solution. An experimental procedure, where a film of a solution undergoes a controlled evaporation in a chamber, shows that the SAW alters the state of the pattern deposition. Increasing the power of the SAW supports an increase in the density of the deposited patterns. Beyond threshold conditions, the deposited patterns merge and we observe the deposition of a solid film. A simplified theory suggests that the SAW deforms the geometry of the film, which is predominantly governed by the capillary stress. The deformation of the film taking place alongside with the evaporation of the solution increases the concentration near the pinned three phase contact line at the front of the film, which is closer to the source of the SAW, on the expense of the concentration at the rear. The increased concentration translates to the deposition of solute mass over an increased area near the front of the film, which explains the experimental observation.

Evaporation of Sessile Droplets Laden with Particles and Insoluble Surfactants

We consider the flow dynamics of a thin evaporating droplet in the presence of an insoluble surfactant and noninteracting particles in the bulk. On the basis of lubrication theory, we derive a set of evolution equations for the film height, the interfacial surfactant, and bulk particle concentrations, taking into account the dependence of liquid viscosity on the local particle concentration. An important ingredient of our model is that it takes into account the fact that the surfactant adsorbed at the interface hinders evaporation. We perform a parametric study to investigate how the presence of surfactants affects the evaporation process as well as the flow dynamics with and without the presence of particles in the bulk. Our numerical calculations show that the droplet lifetime is affected significantly by the balance between the ability of the surfactant to enhance spreading, suppressing the effect of thermal Marangoni stresses-induced motion, and to hinder the evaporation flux through the reduction of the effective interfacial area of evaporation, which tend to accelerate and decelerate the evaporation process, respectively. For particle-laden droplets and in the case of dilute solutions, the droplet lifetime is found to be weakly dependent on the initial particle concentration. We also show that the particle deposition patterns are influenced strongly by the direct effect of the surfactant on the evaporative flux; in certain cases, the "coffee-stain" effect is enhanced significantly. A discussion of the delicate interplay between the effects of capillary pressure and solutal and thermal Marangoni stresses, which drive the liquid flow inside of the evaporating droplet giving rise to the observed results, is provided herein.

A model for pattern deposition from an evaporating solution subject to contact angle hysteresis and finite solubility

We propose a model for the pattern deposition of the solute from an evaporating drop of a dilute solution on a horizontal substrate. In the model we take into account the three-phase contact angle hysteresis and the deposition of the solute whenever its concentration exceeds the solubility limit. The evaporating drop is governed by a film equation. We show that unless for a very small three-phase contact angle or a very rapid evaporation rate the film adopts a quasi-steady geometry, satisfying the Young-Laplace equation to leading order. The concentration profile is assumed to satisfy an advection diffusion equation subject to the standard Fick's law for the diffusive flux. We further use an integral boundary condition to describe the dynamics of the concentration in the vicinity of the three-phase contact line; we replace an exact geometric description of the vicinity of the contact line, which is usually assumed such that mathematical singularities are avoided, with general insights about the concentration and its flux. We use our model to explore the relationships between a variety of deposition patterns and the governing parameters, show that the model repeats previous findings, and suggest further insights.

Numerical simulation of dip-coating in the evaporative regime

A hydrodynamic model is used for numerical simulations of a polymer solution in a dip-coating-like experiment. We focus on the regime of small capillary numbers where the liquid flow is driven by evaporation, in contrast to the well-known Landau-Levich regime dominated by viscous forces. Lubrication approximation is used to describe the flow in the liquid phase. Evaporation in stagnant air is considered (diffusion-limited evaporation), which results in a coupling between liquid and gas phases. Self-patterning due to the solutal Marangoni effect is observed for some ranges of the control parameters. We first investigate the effect of evaporation rate on the deposit morphology. Then the role of the spatial variations in the evaporative flux on the wavelength and mean thickness of the dried deposit is ascertained, by comparing the 2D and 1D diffusion models for the gas phase. Finally, for the very low substrate velocities, we discuss the relative importance of diffusive and advective components of the polymer flux, and consequences on the choice of the boundary conditions.

Pattern formation in thin films of polymer solutions: Theory and simulations

Self-assembly has been recognized as an efficient tool for generating a wide range of functional, chemically, or physically textured surfaces for applications in small scale devices. In this work, we investigate the stability of thin films of polymer solutions. For low concentrations of polymer in the solution, long length scale dewetting patterns are obtained with wavelength approximately few microns. Whereas, for concentrations above a critical value, bimodal dispersion curves are obtained with the dominant wavelength being up to two orders smaller than the usual dewetting length scale. We further show that the short wavelength corresponds to the phase separation in the film resulting in uniformly distributed high and low concentration regions. Interestingly, due to the solvent entropy, at very high concentration values of polymer, a re-entrant behaviour is observed with the dominant length scale now again corresponding to the dewetting wavelength. Thus, we show that the binary films of polymer solutions provide additional control parameters that can be utilized for generating functional textured surfaces for various applications.

Sagging of evaporating droplets of colloidal suspensions on inclined substrates

A droplet of a colloidal suspension placed on an inclined substrate may sag under the action of gravity. Solvent evaporation raises the concentration of the colloidal particles, and the resulting viscosity changes may influence the sag of the droplet. To investigate this phenomenon, we have developed a mathematical model for perfectly wetting droplets based on lubrication theory and the rapid-vertical-diffusion approximation. Precursor films are assumed to be present, the colloidal particles are taken to be hard spheres, and particle and liquid dynamics are coupled through a concentration-dependent viscosity and diffusivity. Evaporation is assumed to be limited by how rapidly solvent molecules can transfer from the liquid to the vapor phase. The resulting one-dimensional system of nonlinear partial differential equations describing the evolution of the droplet height and particle concentration is solved numerically for a range of initial particle concentrations and substrate temperatures. The solutions reveal that the interaction between evaporation and non-Newtonian suspension rheology gives rise to several distinct regimes of droplet shapes and particle concentration distributions. The results provide insight into how evaporation and suspension rheology can be tuned to minimize sagging as well as the well-known coffee-ring effect, an outcome which is important for industrial coating processes.

Patterned deposition at moving contact lines

When a simple or complex liquid recedes from a smooth solid substrate it often leaves a homogeneous or structured deposit behind. In the case of a receding non-volatile pure liquid the deposit might be a liquid film or an arrangement of droplets depending on the receding speed of the meniscus and the wetting properties of the system. For complex liquids with volatile components as, e.g., polymer solutions and particle or surfactant suspensions, the deposit might be a homogeneous or structured layer of solute--with structures ranging from line patterns that can be orthogonal or parallel to the receding contact line via hexagonal or square arrangements of drops to complicated hierarchical structures. We review a number of recent experiments and modelling approaches with a particular focus on mesoscopic hydrodynamic long-wave models. The conclusion highlights open question and speculates about future developments.

Gradient dynamics description for films of mixtures and suspensions: dewetting triggered by coupled film height and concentration fluctuations

A thermodynamically consistent gradient dynamics model for the evolution of thin layers of liquid mixtures, solutions, and suspensions on solid substrates is presented which is based on a film-height- and mean-concentration-dependent free energy functional. It is able to describe a large variety of structuring processes, including coupled dewetting and decomposition processes. As an example, the model is employed to investigate the dewetting of thin films of liquid mixtures and suspensions under the influence of effective long-range van der Waals forces that depend on solute concentration. The occurring fluxes are discussed, and it is shown that spinodal dewetting may be triggered through the coupling of film height and concentration fluctuations. Fully nonlinear calculations provide the time evolution and resulting steady film height and concentration profiles.

Nonlinear dynamics of thin liquid films consisting of two miscible components

Recently, we systematically derived a system of two coupled conservation equations governing a thin liquid layer with a deformable surface composed of two completely miscible components [Phys. Fluids 22, 104102 (2010)]. One equation describes the location of the free surface and the second one the evolution of the mean concentration. This lubrication model was investigated previously in linearized form. The study is now extended to the fully nonlinear case of thin liquid films of a binary mixture (in one and two horizontal spatial dimensions) with and without heat transport. For an initially flat and motionless film heated from below, we analyze the component separation induced by the Soret effect. Nonlinear simulations show that the Soret effect can cause a multitude of interesting behaviors, such as oscillatory patterns and solitonlike structures (localized traveling drops or holes). A stronger component separation induced by stronger Soret effects favors faster-moving localized structures. For isothermal systems, we study the fusion and the mixing of two thin liquid films of different but perfectly miscible liquids. Marangoni-driven forces can cause delayed coalescence, ripple formation, and fingering patterns at the borderline between the two liquid layers. A systematic analysis for ripple pattern formation and finger instabilities at different diffusion constants shows that these phenomena appear more pronounced for lower diffusion in the system.

Modified spin-coating technique to achieve directional colloidal crystallization

Fabricating large single crystals with colloidal spheres as building blocks is challenging and of competitive interest. Spin-coating of colloids offers a robust technique, which is highly reproducible in obtaining colloidal crystals even at fast dynamical regimes; however, these crystals are intrinsically polycrystalline due to the axial symmetry of spin-coating. We report a new method that applies a nonuniform electric field during the spin-coating process. By arranging the field direction to be stationary in the rotating frame, we are able to break the axial symmetry and to orient the colloids along one predefined direction. By regulating the applied field strength, we demonstrate local control over the orientation of the crystallites, and thus, the orientation is determined by the applied field strength.

Dynamical model for the formation of patterned deposits at receding contact lines

We describe the formation of deposition patterns that are observed in many different experiments where a three-phase contact line of a volatile nanoparticle suspension or polymer solution recedes. A dynamical model based on a long-wave approximation predicts the deposition of irregular and regular line patterns due to self-organized pinning-depinning cycles corresponding to a stick-slip motion of the contact line. We analyze how the line pattern properties depend on the evaporation rate and solute concentration.

Dynamical density functional theory for the dewetting of evaporating thin films of nanoparticle suspensions exhibiting pattern formation

Recent experiments have shown that the striking structure formation in dewetting films of evaporating colloidal nanoparticle suspensions occurs in an ultrathin "postcursor" layer that is left behind by a mesoscopic dewetting front. Various phase change and transport processes occur in the postcursor layer that may lead to nanoparticle deposits in the form of labyrinthine, network, or strongly branched "finger" structures. We develop a versatile dynamical density functional theory to model this system which captures all these structures and may be employed to investigate the influence of evaporation or condensation, nanoparticle transport, and solute transport in a differentiated way. We highlight, in particular, the influence of the subtle interplay of decomposition in the layer and contact line motion on the observed particle-induced transverse instability of the dewetting front.

Modelling approaches to the dewetting of evaporating thin films of nanoparticle suspensions

We review recent experiments on dewetting thin films of evaporating colloidal nanoparticle suspensions (nanofluids) and discuss several theoretical approaches to describe the ongoing processes including coupled transport and phase changes. These approaches range from microscopic discrete stochastic theories to mesoscopic continuous deterministic descriptions. In particular, we describe (i) a microscopic kinetic Monte Carlo model, (ii) a dynamical density functional theory and (iii) a hydrodynamic thin film model. Models (i) and (ii) are employed to discuss the formation of polygonal networks, spinodal and branched structures resulting from the dewetting of an ultrathin 'postcursor film' that remains behind a mesoscopic dewetting front. We highlight, in particular, the presence of a transverse instability in the evaporative dewetting front, which results in highly branched fingering structures. The subtle interplay of decomposition in the film and contact line motion is discussed. Finally, we discuss a simple thin film model (iii) of the hydrodynamics on the mesoscale. We employ coupled evolution equations for the film thickness profile and mean particle concentration. The model is used to discuss the self-pinning and depinning of a contact line related to the 'coffee-stain' effect. In the course of the review we discuss the advantages and limitations of the different theories, as well as possible future developments and extensions.

Pinning, retraction, and terracing of evaporating droplets containing nanoparticles

We consider the dynamics of a slender, evaporating droplet containing nanoparticles. We use lubrication theory to derive a coupled system of equations that govern the film thickness and the concentration of nanoparticles. These equations account for capillarity, Marangoni stresses, evaporation, and disjoining pressure; the nanoparticle-induced structural component of the disjoining pressure is also considered. Contact line singularities are avoided through the adsorption of ultrathin films wherein evaporation is suppressed by the disjoining pressure; a similar approach has recently been used by Ajaev [J. Fluid Mech. 2005, 528, 279-296] who has built on the previous work of Moosman and Homsy [J. Colloid Interface Sci. 1980, 73, 212-223]. The results of our numerical simulations indicate that, depending on the value of system parameters, the droplet exhibits a variety of different behaviours, which include spreading, evaporation-driven retraction, contact line pinning, and "terrace" formation.

Thinning of drying latex films due to surfactant

Lateral non-uniformities in surfactant distribution in drying latex films induce surface tension gradients at the film surface and lead to film thinning through surfactant spreading. Here we investigate the influence of the surfactant driven to the air-water interface, during the early stages of latex film drying, on the film thinning process which could possibly lead to film rupture. A film height evolution equation is coupled with conservation equations for particles and surfactant, within the lubrication approximation, and solved numerically, to obtain the film height, particle volume fraction, and surfactant concentration profiles. Parametric analysis identifies the effect of drying rate, dispersion viscosity and initial particle volume fraction on film thinning and reveals the conditions under which films could rupture. The results from surface profilometry conform qualitatively to the model predictions.

Ethane-Bridged Zinc Porphyrin Dimers in Langmuir−Shäfer Thin Films: Structural and Spectroscopic Properties

This work reports on the structural and spectroscopic properties of ethane-bridged Zn porphyrin dimers (1) in Langmuir-Schäfer (LS) thin films by combining scanning force microscopy (SFM) with film balance, UV-vis absorption, fluorescence, and nanosecond laser flash photolysis measurements. Results show that depending on the surface pressure the Langmuir films of pure 1 can be arranged in two different condensed phases, whereas SFM of the LS films shows characteristic fractal networks constituted by nanoscopic aggregates. The spectral findings agree with a picture in which 1 is apparently present in the anti conformation but aggregated in a sort of H-type structure whose optical features resemble those of the syn conformer. This type of structure is not responsive to light stimuli. By diluting 1 in arachidic acid the porphyrin aggregation is significantly minimized with 1 exhibiting almost exclusively the anti conformation. As a result the LS films become photoresponsive, showing fluorescence emission and triplet-triplet transient absorption.

Simultaneous thermal and surfactant-induced Marangoni effects in thin liquid films

The deformation of a thin liquid film in the presence of a surfactant monolayer, varying temperature distributions, and limited mass flux is considered. Use of lubrication theory yields a coupled pair of partial differential equations for the film height and surfactant surface monolayer concentration. The long-wave stability of the isothermal film is examined over a wide range of parameter values. It is shown that droplet patterns are obtained under certain thermal conditions for both an isothermal and nonisothermal underlying substrate. For the case of a localized thermal gradient initially imposed at the air-liquid interface, severe film thinning beneath the heat source was observed, which was not accompanied by droplet formation; pseudo steady states are observed in this case. In all situations the surfactant is found to rigidify the air-liquid interface, retarding thermally driven flow, while evaporation (condensation) acts to destabilize (stabilize) the film.

Polymer patterns in evaporating droplets on dissolving substrates

Self-organized polymer patterns resulting from the evaporation of an organic solvent drop on a soluble layer of polymer are investigated. The patterns can be modulated by changing the rate of evaporation and also the rate of substrate dissolution controlled by its solubility. Both of these affect the contact zone motion and its instabilities, leading to spatially variable rates of substrate etching and redeposition that result from a complex interplay of several factors such as Rayleigh-Benard cells, thermocapillary flow, solutal Marangoni flow, flow due to differential evaporation, osmotic-pressure-induced flow, and contact-line pinning-depinning events. The most complex novel pattern, observed at relatively low rates of evaporation, medium solubility, and without macroscopic contact-line stick-slip, consists of a regularly undulating ring made up of a bundle of parallel spaghetti-like threads or striations and radially oriented fingerlike ridges. Increased rate of evaporation obliterates the polymer threads, producing more densely packed fingers and widely separated multiple rings due to a frequent macroscopic pinning-depinning of the contact line. Near-equilibrium conditions such as slow evaporation or increased solubility of the substrate engender a wider and less undulating single ring.