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.     

Molecular Transport within Polymer Brushes: A FRET View at Aqueous Interfaces

Molecular permeability through polymer brush chains is implicated in surface lubrication, wettability, and solute capture and release. Probing molecular transport through polymer brushes can reveal information on the polymer nanostructure, with a permeability that is dependent on chain conformation and grafting density. Herein, we introduce a brush system to study the molecular transport of fluorophores from an aqueous droplet into the external “dry” polymer brush with the vapour phase above. The brushes consist of a random copolymer of N-isopropylacrylamide and a Förster resonance energy transfer (FRET) donor-labelled monomer, forming ultrathin brush architectures of about 35 nm in solvated height. Aqueous droplets containing a separate FRET acceptor are placed onto the surfaces, with FRET monitored spatially around the 3-phase contact line. FRET is used to monitor the transport from the droplet to the outside brush, and the changing internal distributions with time as the droplets prepare to recede. This reveals information on the dynamics and distances involved in the molecular transport of the FRET acceptor towards and away from the droplet contact line, which are strongly dependent on the relative humidity of the system. We anticipate our system to be extremely useful for studying lubrication dynamics and surface droplet wettability processes.     

Marangoni flow alters wetting: Coffee ring and superspreading

Colloidal nano altered wetting is an everyday phenomenon with various applications. It occurs when an aqueous colloidal droplet is placed on a wet solid and evaporates with a pinned contact line (to form a shape similar to a coffee ring) and when a ‘superspreader’ is placed on a partially wet solid surface with an unpinned contact line. This article reviews recent advances on the mechanisms behind these two phenomena. For the coffee ring phenomenon, besides the evaporation flow, the thermal and solutal Marangoni flow are critical to the radial flow as they carry solutes from the droplet's interior to the contact line. The particle self-assembly into a layered structure at the droplet edge involves particle structuring under confinement and the stagnation flow. While superspreading is not fully understood, it is proposed that the Marangoni flow plays an essential role in superspreading.     

A micropump controlled by EWOD: wetting line energy and velocity effects

A Laplace pressure gradient between a droplet and a liquid meniscus was utilized to create an on-demand constant flow rate capillary pump. Electrowetting on dielectric was implemented to induce the pressure gradient in the microchannel. For an initial droplet volume of 0.3 μL and a power of 12 nW a constant flow rate of 0.02 μL s(-1) was demonstrated. The effects of the wetting line energy on the static contact angle and the wetting line velocity on the dynamic contact angle in the pump operation were studied. Sample loading on-demand could be achieved by regulating an electric potential.     

Self-similar solution to the problem of vapor diffusion toward the droplet nucleated and growing in a vapor-gas medium

The problem of vapor diffusion toward a droplet nucleated and growing in the diffusion regime is exactly solved using the similarity theory. The surface motion of droplets is taken into account in the solution. The constructed nonstationary concentration field of vapor satisfies the diffusion equation, the boundary condition of equilibrium on the surface of growing droplet, and the initial homogeneous condition. According to the found solution, the radius of a droplet is proportional to the square root of the time of its growth. Far from the critical point, at a low ratio between the densities of excess vapor and a liquid droplet, the proportionality coefficient coincides with that resulting from an approximate solution. The balance between the numbers of molecules removed from vapor and those composing a growing droplet exactly corresponds to the obtained solution.     

Optimisation of a microfluidic analysis chamber for the placement of microelectrodes

The behaviour of droplets entering a microfluidic chamber designed to house microelectrode detectors for real time analysis of clinical microdialysate is described. We have designed an analysis chamber to collect the droplets produced by multiphase flows of oil and artificial cerebral spinal fluid. The coalescence chamber creates a constant aqueous environment ideal for the placement of microelectrodes avoiding the contamination of the microelectrode surface by oil. A stream of alternating light and dark coloured droplets were filmed as they passed through the chamber using a high speed camera. Image analysis of these videos shows the colour change evolution at each point along the chamber length. The flow in the chamber was simulated using the general solution for Poiseuille flow in a rectangular chamber. It is shown that on the centre line the velocity profile is very close to parabolic, and an expression is presented for the ratio between this centre line velocity and the mean flow velocity as a function of channel aspect ratio. If this aspect ratio of width/height is 2, the ratio of flow velocities closely matches that of Poiseuille flow in a circular tube, with implications for connections between microfluidic channels and connection tubing. The droplets are well mixed as the surface tension at the interface with the oil dominates the viscous forces. However once the droplet coalesces with the solution held in the chamber, the no-slip condition at the walls allows Poiseuille flow to take over. The meniscus at the back of the droplet continues to mix the droplet and acts as a piston until the meniscus stops moving. We have found that the no-slip conditions at the walls of the chamber, create a banding effect which records the history of previous drops. The optimal position for sensors is to be placed at the plane of droplet coalescence ideally at the centre of the channel, where there is an abrupt concentration change leading to a response time ?16 ms, the compressed frame rate of the video. Further away from this point the response time and sensitivity decrease due to convective dispersion.     

Unconventional multiple ring structure formation from evaporation-induced self-assembly of polymers

The formation of multiring deposits of poly(2-vinylpyridine) (P2VP) from the evaporation of a P2VP-(2,6-lutidine + water) drop on a glass substrate does not conform to the conventional pinning-depinning mechanism. Instead, ringlike deposits are formed when the droplet undergoes several cycles of spreading and receding where, for each spreading event, a P2VP ridge is formed at the contact line when the polymer flows toward the outward advancing edge. The complex interplay between an outward solutal-Marangoni flow due to a higher concentration of the polymer at the contact line and an inward solvent-Marangoni flow arising from the differences in volatilities and surface tensions of the pure solvent components plays an important role in enhancing the droplet spreading rate. The newly discovered surface patterning mechanism has important implications in the development of novel techniques for inducing self-assembly of functional materials from evaporating drops.     

Internal flow in polymer solution droplets deposited on a lyophobic surface during a receding process

When a polymer solution droplet is deposited on a lyophobic surface, the contact line is moved back to some degree and subsequently pinned. An experimental setup is constructed to investigate not only the receding process but also an internal flow of polystyrene-acetophenone and -anisole solutions. As a result, the time variation of the evaporation rate per unit area during receding does not strongly depend on the initial solute concentration. The average solute concentration at the pinning of the contact line increases as the initial solute concentration increases. A convective circulation flow that is upward at the axis of symmetry is observed. This flow pattern is different from those of pure liquids such as water, acetone, benzene, and so forth, which have been previously reported. Furthermore, the observed flow is enhanced as the initial solute concentration increases, contrary to an increase in the fluid viscosity. To resolve these discrepancies, the mechanism of the flow is numerically investigated using a hemispherical droplet model considering the density and surface tension distributions. The numerical results demonstrate that the circulation flow that is experimentally observed is actually caused. It is also found that the solutal Rayleigh effect initially induces the internal flow, and subsequently the solutal Marangoni effect dominates the flow. Both effects are enhanced as the initial concentration increases because of the evaporative mass balance at the free surface.     

Analytical and Numerical Study of Equilibrium Characteristics of a Droplet with Charged Condensation Nucleus in the External Electric Field

The equilibrium parameters of small dielectric droplet with charged condensation nucleus in the external uniform electric field are studied. Two typical cases are considered: (i) the droplet with charged nucleus suspended by external uniform electric field in the gravitational field and (ii) the droplet moves steadily under the action of external electric field with allowance for the resistance of surrounding vapor–gas medium. It is taken into account that the charged condensation nucleus can be displaced from the mass center of the droplet to new equilibrium position inside the droplet under the action of external electric field and response field. The scheme of the numerical solution of a nonlinear system of differential equations for the droplet equilibrium profile and electric potentials inside the droplet and in the vapor–gas medium at the arbitrary values of droplet size, strength of external field, and the charge of condensation nucleus is formulated and realized. Dependences of an equilibrium profile and the thermodynamic characteristics of a droplet such as the chemical potential of condensate and formation work on the droplet size, mass, and charge of condensation nucleus, the strength of external field and ratio of permittivities of droplet and the vapor–gas medium are plotted. Results of numerical calculations are supplemented by the analytical relations for equilibrium droplet characteristics in the first orders of the perturbation theory for a weak external field.     

Evaporative micro-particle self assembly influenced by capillary evacuation

As evaporation does not incur energy introduction, the droplet coffee-stain patterning approach is attractive for biochemical tests conducted in the field or in third world environments. A practical strategy uses chemically functionalized microbeads for the coffee stain deposition process. From an application perspective, it will be necessary to minimize the coffee stain deposition time, as evaporation, depending on the volume of the droplet, can be a slow process. The introduction of a porous media will generate a capillary flow (or wicking) that removes any remnant liquid in the droplet, thus permitting it to be done inexpensively and in the field. Using optical profilometry, we were able to establish that polystyrene microspheres developed more copious and defined single ring coffee depositions than silica of the same size and concentration in a suspension. In analyzing the droplet capillary evacuation process with a porous media, we found the liquid bridge formed during the later stages to rupture and leave behind some liquid material for a second stage evaporation process. This was responsible for a two ring structure that was more visible with silica microspheres. A high degree of hysteresis of the contact angle was found to develop at the contact line in which values below 5° could be achieved. Dynamic observations showed the copious and dense packing of polystyrene particles to be more resistant to ring break up from the evacuation flow. Nevertheless, erosion of the back array portions of the ring was evident notwithstanding either type of microsphere used.     

Dynamics of droplet motion under electrowetting actuation

The static shape of droplets under electrowetting actuation is well understood. The steady-state shape of the droplet is obtained on the basis of the balance of surface tension and electrowetting forces, and the change in the apparent contact angle is well characterized by the Young-Lippmann equation. However, the transient droplet shape behavior when a voltage is suddenly applied across a droplet has received less attention. Additional dynamic frictional forces are at play during this transient process. We present a model to predict this transient behavior of the droplet shape under electrowetting actuation. The droplet shape is modeled using the volume of fluid method. The electrowetting and dynamic frictional forces are included as an effective dynamic contact angle through a force balance at the contact line. The model is used to predict the transient behavior of water droplets on smooth hydrophobic surfaces under electrowetting actuation. The predictions of the transient behavior of droplet shape and contact radius are in excellent agreement with our experimental measurements. The internal fluid motion is explained, and the droplet motion is shown to initiate from the contact line. An approximate mathematical model is also developed to understand the physics of the droplet motion and to describe the overall droplet motion and the contact line velocities.     

Evaporation dynamics of a binary sessile droplet: Theory and comparison with experimental data on a droplet of a sulfuric-acid solution

Diffusion evaporation of a sessile binary droplet in an atmosphere of a noncondensable carrier gas has been considered. For a droplet consisting of two infinitely miscible liquids, a relation between the current values of solution concentration and volume of the droplet has been derived in an explicit form under the ideal solution approximation. It has been shown that the volume of a sessile binary droplet may, as well as the volume of a free binary droplet, vary nonmonotonically with time. The evaporation of a droplet of an aqueous sulfuric-acid solution has been considered in detail taking into account the nonideality of the solution. Time variations in the volume, base area, and contact angle have been experimentally measured for the sessile droplet of an aqueous sulfuric-acid solution on a hydrophobized substrate. The experimental data obtained at different initial humidities of water-vapor and droplet-solution concentrations have been analyzed within the theory of the stationary isothermal diffusion evaporation of a sessile binary droplet.     

3-D transient electrophoretic motion of a spherical particle in a T-shaped rectangular microchannel

This paper considers the electrophoretic motion of a spherical particle in an aqueous electrolyte solution in a T-shaped rectangular microchannel, where the size of the channel is close to that of the particle. This is a complicated transient process where the electric field, the flow field, and the particle motion are coupled together. A theoretical model was developed to investigate the influences of the applied electric potentials, the zeta potentials of the channel and the particle, and the size of the particle on the particle motion. A direct numerical simulation method using the finite element method is employed. This method employs a generalized Galerkin finite element formulation that incorporates both equations of the fluid flow and equations of the particle motion into a single variational equation where the hydrodynamic interactions are eliminated. The ALE method is used to track the surface of the particle at each time step. The numerical results show that the electric field in the T-shaped microchannel is influenced by the presence of the particle, and that the particle motion is influenced by the applied electric potentials and the zeta potentials of the channel and the particle. The path of the particle motion is dominated by the local electric field and the ratio of the zeta potential of the channel to that of the particle. The particle's velocity is also dependent on its size in a small channel.     

A self-consistent field study of a hydrocarbon droplet at the air-water interface

A molecularly detailed self-consistent field (SCF) approach is applied to describe a sessile hydrocarbon droplet placed at the air-water interface. Predictions of the contact angle for macroscopic droplets follow from using Neumann's equation, wherein the macroscopic interfacial tensions are computed from one-gradient calculations for flat interfaces. A two-gradient cylindrical coordinate system with mirror-like boundary conditions is used to analyse the three dimensional shape of the nano-scale oil droplet at the air-water interface. These small droplets have a finite value of the Laplace pressure and concomitant line tension. It has been calculated that the oil-water and oil-vapour interfacial tensions are curvature dependent and increase slightly with increasing interfacial curvature. In contrast, the line tension tends to decrease with curvature. In all cases there is only a weak influence of the line tension on the droplet shape. We therefore argue that the nano-scale droplets, which are described in the SCF approach, are representative for macroscopic droplets and that the method can be used to efficiently generate accurate information on the spreading of oil droplets at the air-water interface in molecularly more complex situations. As an example, non-ionic surfactants have been included in the system to illustrate how a molecularly more complex situation will change the wetting properties of the sessile drop. This short forecast is aimed to outline and to stress the potential of the method.     

Enhanced mixing of droplets during coalescence on a surface with a wettability gradient

We investigated the dynamics of head-on collisions between a moving droplet and a stationary droplet on a surface with a wettability gradient. The mixing of fluids is achieved passively through convective mass transfer caused by the release of surface energy during coalescence, and also through diffusive mass transfer. The coalescence dynamics were visualized with a high-speed camera; the internal flow patterns were resolved with measurement of micro-PIV (particle image velocimetry). The results show that the released surface energy creates a pair recirculation flow inside the merged droplet when the stationary droplet is placed near the gradient, whereas most released surface energy is converted into oscillation when the stationary droplet is far from the gradient. This distinction is attributed to the motion of the contact line during coalescence. The mixing of fluorescently labeled oligonucleotides in these two modes is revealed with confocal micro-laser induced fluorescence technique. The results of 3D scans demonstrate that the motion of the contact line during coalescence distributes the fluids in a complicated manner, thus beneficial for mixing. This mechanism of enhanced mixing is applicable also for platforms other than a surface with a wettability gradient; prospective applications include improving the mixing of biochemical fluids.     

Continuous-flow production of polymeric micelles in microreactors: experimental and computational analysis

The dynamics and stability of a thin, viscous film of volatile liquid flowing under the influence of gravity over a non-uniformly heated substrate are investigated using lubrication theory. Attention is focused on the regime in which evaporation balances the flow due to gravity. The film terminates above the heater at an apparent contact line, with a microscopically thin precursor film adsorbed due to the disjoining pressure. The film develops a weak thermocapillary ridge due to the Marangoni stress at the upstream edge of the heated region. As for spreading films, a more significant ridge is formed near the apparent contact line. For weak Marangoni effects, the film evolves to a steady profile. For stronger Marangoni effects, the film evolves to a time-periodic state. Results of a linear stability analysis reveal that the steady film is unstable to transverse perturbations above a critical value of the Marangoni parameter, leading to finger formation at the contact line. The streamwise extent of the fingers is limited by evaporation. The time-periodic profiles are always unstable, leading to the formation of periodically-oscillating fingers. For rectangular heaters, the film profiles after instability onset are consistent with images from published experimental studies.     

Motion of a drop on a solid surface due to a wettability gradient

The hydrodynamic force experienced by a spherical-cap drop moving on a solid surface is obtained from two approximate analytical solutions and used to predict the quasi-steady speed of the drop in a wettability gradient. One solution is based on approximation of the shape of the drop as a collection of wedges, and the other is based on lubrication theory. Also, asymptotic results from both approximations for small contact angles, as well as an asymptotic result from lubrication theory that is good when the length scale of the drop is large compared with the slip length, are given. The results for the hydrodynamic force also can be used to predict the quasi-steady speed of a drop sliding down an incline.     

Surfactant and electric field strength effects on surface tension at liquid/liquid/solid interfaces

We performed a series of experiments designed to elucidate the effects of the presence of sodium dodecyl sulfate (SDS) surfactant and an applied electrical field on the wetting behavior in a system containing a sessile droplet of phenylmethyl polysiloxane (PMPS) oil on a polished stainless steel surface submersed in aqueous solution. The voltage difference ranged from -3 to +3 V, which is at least 3 orders of magnitude smaller than from comparable recent work. We report the measured equilibrium contact angle of the droplet as a function of surfactant concentration and field strength. We then modeled the system. We solved the Laplace equation to obtain the 3D field within our system. We expanded the three surface tensions (oil droplet-aqueous solution (oa), oil droplet-metal surface (os), and aqueous solution-metal surface (as)) in a Taylor series with respect to surfactant concentration and local field strength. We use these three surface tensions in Young's equation to obtain the theoretical contact angle of the organic droplet. We demonstrate that the large changes in contact angle due to the simultaneous presence of small concentrations of surfactant and small voltage differences can be accounted for by changes in the oa and as surface tensions.     

Influence of substrate heating on the evaporation dynamics of pinned water droplets

The dynamics of evaporating water droplets deposited on a heated substrate is investigated numerically. Droplets are pinned with a contact line radius of R = 1 mm. Evaporative mass flow and convection occurring inside the droplets are studied for different heating substrate sizes L S and heating temperatures T S. A simplified model neglecting hydrodynamics in air and evaporative cooling and assuming droplets to be spherical caps is simulated with a finite element method. A toruslike convective cell appears inside the droplets as evaporation takes place. For L S/ R > 1, the contact line is warmer than the apex of the droplets, and convection generates a downstream flow in the vicinity of the symmetry axis of the droplets. For L S/ R < 1, it is the apex that is warmer. Convection then generates an upstream flow. The overall evaporation time is described. It slows when L S/ R > 1.