Thermocapillary Migration Of Droplets Research Articles

A reduction-consistent phase field model for non-isothermal multiphase flows of N immiscible incompressible fluids

In this study, we developed a reduction-consistent phase field model for non-isothermal incompressible N-phase flows. The model is based on the high-order spectral element method. To account for thermocapillary effects, we reformulated the continuum surface force model specifically for N-phase flows. In order to enhance computational efficiency, time-independent coefficient matrices for all variables involved were reconstructed, and an unsymmetrized multifrontal LU factorization was employed to solve the linear algebraic equations, which is derived by discretization. To verify the model effectiveness in calculating surface tension and describing three-phase interfacial dynamics, we conducted several experiments, including the stationary two-droplet example, the three-phase droplet spreading, and the equilibrium morphology of double emulsion droplet. Through these experiments, both the reduction-consistency and robustness of our model were demonstrated. Moreover, we validated the proposed model's applicability to non-isothermal three-phase flows by investigating the thermocapillary migration of single droplet and two droplets of different phases. Notably, we explored the thermocapillary migration of three-phase double droplets, focusing particularly on how the encapsulation process affects overall thermocapillary motion. Our findings indicate that the interaction between the two vortices near the interfaces of the two droplets strongly influences their migration behavior.

On the migration of a droplet driven by thermocapillary action in a wettability-confined track

We report a study of the thermocapillary migration of droplets under a radial thermal gradient and in a wettability-confined track. A three-dimensional mathematical model is established based on the lubrication approximation. By considering the contact-line dynamics, a method for determining the velocity of the contact line in different directions is proposed for a three-dimensional droplet. Numerical simulations are performed to investigate the variations in the droplet profile, contact angle, and contact line. Three substrate-wettability cases are considered: uniform, temperature-dependent, and track-dependent wettability. The results show that when the substrate wettability is uniform, the droplet height initially decreases rapidly, and its center becomes concave and then gradually evolves into a ring-like morphology. Reducing the temperature sensitivity of the liquid–gas interfacial tension or increasing the temperature sensitivity of the liquid–solid interfacial tension decreases the equilibrium contact angle and accelerates thermocapillary migration. When a droplet spreads in a wettability-confined track, a wave-like peak is formed on each side of the droplet along the track direction until it finally separates into two distinct parts. As the track width is decreased, the time taken for a droplet to split into two smaller droplets advances, and the separation time presents a linear relationship with the track width.

Thermocapillary migration of thin droplet on wettability-confined track

The thermocapillary migration of droplets on a solid surface is widely used in daily life and industrial fields. Regulating droplet thermocapillary migration by changing surface wettability has received extensive attention. According to the lubrication theory and slip boundary conditions, we establish a mathematical model of the thermocapillary migration dynamics of a droplet on wettability-confined tracks subjected to a uniform temperature gradient. Combined with the contact line dynamics, a method of determining the velocity of the contact line in a different direction of the three-dimensional droplet is proposed, the simulation is carried out with FreeFEM++. The evolution of droplet migration is examined, and the effects of track width and wettability on the droplet migration dynamics are emphatically investigated. The results show that the main part of the droplet moves from the high-temperature region to the low-temperature region, the trailing edge of the droplet forms a small bulge during the movement, and a thin liquid film is formed between the bulge and the main part of the droplet. The droplet spreading perpendicular to the track direction is inhibited and remains pinned after shrinking to the track edge. Negative correlation between the velocity of the advancing contact line and the track width is observed. The velocity of the advancing line first rapidly and then slowly decrease to a steady state. The squeezing effect caused by the wettability confined perpendicular to the track direction accelerates the thermocapillary migration of the droplet on the track in the initial short time. The enhanced track wettability increases the initial velocity of the receding contact line but has little effect on its stable value. The velocity of the advancing contact line is positively correlated with track wettability. Changing the track width is possibly easier to regulate the thermocapillary migration of a droplet than varying the track wettability.

A numerical study on the thermocapillary migration of a droplet under microgravity with periodic thermal boundaries using the front-tracking method

This paper mainly studied the thermocapillary migration of deformable droplets induced by periodic temperature boundary under microgravity conditions. The finite-difference/front-tracking (FD/FT) method was used to solve the Navier–Stokes equation coupled with the energy equation, and the continuum surface force (CSF) model was used to simplify the surface tension of the phase interface. The results showed that the maximum droplet migration velocity increased with the increase of temperature amplitude, and the droplet cycle period became shorter with the increase of temperature angular frequency. In the 1/4 cycle, the initial movement time of droplet decreased with the increase of temperature phase. If the phase was reversed, the initial movement direction of the droplet changed. With the increase of Reynolds number (Re), the droplet tended to maintain its motion inertia.

Thermocapillary migration of droplets and bubbles in a viscous liquid (review)

Investigation of the process of accumulation of gas bubbles in the aria of a heat source is, from a physical point of view, quite interesting problem that leads to important conclusions for practical applications. The peculiarity of the process under consideration is that the surface tension of the bubble changes in an alternating temperature field, which, in turn, leads to the appearance of a flow in the boundary layer of the liquid. In the world scientific literature, the discovery and description of the effect of gas bubble migration in the direction of the temperature gradient is usually associated with the experimental work of Yang, Goldstein and Block (1959). Without diminishing its significance, we note that the effect was first predicted in the theoretical work of Fedosov (1956) as a result of solving the problem of the onset of a microflow of a liquid near plane and spherical interphase boundaries in the presence of a temperature gradient. In both works, a significant factor in explaining the described phenomenon was the dependence of surface tension on temperature. After some time, after which it was realized the need to take into account the migration of not only bubbles, but also droplets, in inhomogeneous temperature fields in space technologies, biomedical and other applications, there was a significant number of publications on this subject, and this phenomenon was called thermocapillary migration. This review is devoted to the analysis of the main, in the opinion of the authors of the article, results of experimental, theoretical and applied research to establish the mechanism of migration bubbles and drops in temperature gradient fields. In most works, it is assumed that there is no dependence of the physical properties of a liquid, except for surface tension, on temperature. There are only a few studies where the influence of the temperature dependence of the viscosity coefficient was considered, which gives a new impetus to the continuation of research and the development of the theory of the effect, taking into account the thermorheological properties of working media.

Thermocapillary migration and interaction dynamics of droplets in a constricted domain

Migration of confined droplets in a stationary fluid medium due to thermocapillary forces is considered. An in-house solver with isosurface based interface reconstruction developed in OpenFOAM has been employed to carry out numerical simulations. Thermocapillary migration of a single droplet in a constricted domain with constriction comparable to the droplet size shows that the migration velocity has non-monotonic dependence on the droplet radius. In the case of two droplets migrating in a constricted domain, the relative slowdown of a larger droplet when a smaller droplet is trailing behind reveals the possibility of interaction which is not observed in larger domains. The effects of the constricted domain size, the initial distance of separation, the radius of the trailing droplet, and the Marangoni number are analysed for this configuration. It is observed that the constriction size and Marangoni number have more influence on the interaction and dictate whether the droplets may coalesce or move with a constant separation distance. The final steady state separation distance between the droplets does not depend on the initial separation distance, but it varies with the radius of the trailing droplet. The results from the present study reveal the physical mechanisms influencing the thermocapillary migration of droplets in constricted domains and interactions between the migrating droplets.

Thermocapillary migration of droplets under molecular and gravitational forces

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniformly heated surface. The effect of a non-zero contact angle is imposed through a disjoining–conjoining pressure term. The numerical results for two different molecular interactions are compared: on the one hand, London–van der Waals and ionic–electrostatics molecular interactions that account for polar liquids; on the other hand, long- and short-range molecular forces that model molecular interactions of non-polar fluids. In addition, the effect of gravity on the velocity of the drop is analysed. We find that for small contact angles, the long-time dynamics is independent of the molecular potential, and the footprint of the droplet increases with the square root of time. For intermediate contact angles we observe that polar droplets are more likely to break up into smaller volumes than non-polar ones. A linear stability analysis allows us to predict the number of droplets after breakup occurs. In this regime, the effect of gravity is stabilizing: it reduces the growth rates of the unstable modes and increases the shortest unstable wavelength. When breakup is not observed, the droplet moves steadily with a profile that consists in a capillary ridge followed by a film of constant thickness, for which we find power law dependencies with the cross-sectional area of the droplet, the contact angle and the temperature gradients. For large contact angles, non-polar liquids move faster than polar ones, and the velocity is proportional to the Marangoni stress. We find power law dependencies for the velocity for the different regimes of flow. The numerical results allow us to shed light on experimental facts such as the origin of the elongation of droplets and the existence of saturation velocity.

Contact angle hysteresis effect on the thermocapillary migration of liquid droplets

Thermocapillary migration describes a phenomenon where a liquid droplet spreads from warm to cold regions due to the interfacial tension gradients. Since the contact angle hysteresis effect is involved during the migration process, we consider the hysteresis effect and rectify the theoretical model to predict the migration velocity on solid surfaces. By conducting migration experiments on surfaces with different magnitudes of the hysteresis effect, we verify the validity of the theoretical derivation. This study advances the understanding of the interfacial phenomenon of thermocapillary migration, moreover, offers an insight into the migration capacity of different materials and guides the design of key components associated with the thermal gradients.

Modeling thermocapillary migration of interfacial droplets by a hybrid lattice Boltzmann finite difference scheme

To understand the motion behavior of interfacial droplets subjected to a temperature gradient, we built a simulation strategy integrated the phase field theory and the continuum surface force in a three-fluid system. In practice, a hybrid numerical scheme, coupling lattice Boltzmann (LB) for flows and finite-difference (FD) for fluids and heat transfer, was developed and then validated by droplet spreading and thermocapillary flows in a heated microchannel. We employed a two-dimensional finite deep model to study the thermocapillary motion of interfacial droplets. The results reveal the relation between the intensity of ambient thermocapillary flow and the direction of droplet motion, and demonstrate the effect of both droplets and interface deformations on droplet transfer. In particular, when the droplet largely deviates from the interface between ambient fluids, the deformation of the interface is obvious. These results are useful for the systematic investigation on the thermocapillary migration of interfacial droplets.

Thermocapillary migration of an isolated droplet and interaction of two droplets in zero gravity

Fluid transfer within a stagnant liquid presents a significant challenge in zero-gravity conditions due to the lack of buoyancy effects. This challenge can be overcome by the utilisation of the Marangoni effect, or more specifically thermocapillary migration. The thermocapillary migration of droplets is driven by temperature gradients within the multiphase system which bring about a surface tension gradient driving the flow from the cold to the hot region. The migration speed of the droplet is significantly impacted by the heat transfer both inside the droplet and in its surroundings.This paper presents the analysis of drop movement in a stagnant liquid using computational fluid dynamics (CFD). The commercial software package Ansys-Fluent v.13 [1] is used to solve the governing continuum conservation equations for two-phase flow using the Volume of Fluid (VOF) method to track the liquid/liquid interface in 2D domain. This approach has been shown to be a valuable tool for studying the phenomena of liquid–liquid interaction. A strong agreement has been found with experimental observations conducted in microgravity.The inherent velocity of drops has been found to decrease with increasing Marangoni number. This finding is in line with the previous space experiments of Xie et al. (2005) [2] and in contrast to the numerical results of Ma (1999) [3] using the same liquid for the droplet and the host liquid.Data obtained in the present numerical study has been used to derive an expression predicting the scaled droplet velocity as a function of Marangoni number. A numerical study of the interaction of two spherical droplets undergoing thermocapillary migration in microgravity is also presented. The temperature thrust from the leading droplet towards the trailing droplet was found to disturb its migration velocity, but the trailing droplet was found to have no influence on the migration of the leading droplet.

Droplet actuation on a liquid layer due to thermocapillary motion: Shape effect

In the thermocapillary migration of droplets on the free surface of immiscible liquids, we observe that the lens-shaped drops move from warm toward cooler regions while spherical drops move in the opposite direction. We explain this dual behavior using an analysis of surface deformation and velocity profiles of thin liquid layers subject to a lateral thermal gradient. Liquid platforms allow thermocapillary transport of drops with higher migration speeds than solid substrates and lower internal temperature fluctuation. Such conditions may be exploited in biochemical microsystems where droplet evaporation, contamination, and surface pinning need to be avoided.

Regimes of thermocapillary migration of droplets under partial wetting conditions

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniform heated substrate. An equation for the thickness profile of the droplet is derived by employing lubrication approximations. The model includes the effect of a non-zero contact angle introduced through a disjoining–conjoining pressure term. Instead of assuming a fixed shape for the droplet, as in previous works, here we allow the droplet to change its profile with time. We identify and describe three different regimes of behaviour. For small contact angles, the droplet spreads into a long film profile with a capillary ridge near the leading edge, a behaviour that resembles the experiments on Marangoni films reported by Ludviksson & Lightfoot (Am. Inst. Chem. Eng. J., vol. 17, 1971, pp. 1166). For large contact angles, the droplet moves as a single entity, weakly distorted from its static shape. This regime is the usual one reported in experiments on thermocapillary migration of droplets. We also show some intriguing morphologies that appear in the transition between these two regimes. The occurrence of these three regimes and their dependence on various parameters is analysed.

Dynamics for Droplets in Normal Gravity and Microgravity

A numerical study has been carried out to investigate the deformation dynamics of droplets under normal gravity and the thermocapillary migration of droplets under microgravity. The Navier–Stokes equations coupled with the energy conservation equation are solved on a staggered grid by the method of lines, and the mass conserving level set method is used to predict the surface deformation of the droplet. The simulation for the falling droplet in the air under normal gravity shows that the value of Weber number affects mainly the deformation of the droplet, while the value of Reynolds number has direct impact on the falling velocity of the droplet. From the simulation for the droplet thermocapillary migration and lateral oscillation under microgravity, it is found that the value of Marangoni number has obvious effects on the moving velocity and temperature distribution of the droplet.

Effect of contact angle hysteresis on thermocapillary droplet actuation

Open microfluidic devices based on actuation techniques such as electrowetting, dielectrophoresis, or thermocapillary stresses require controlled motion of small liquid droplets on the surface of glass or silicon substrates. In this article we explore the physical mechanisms affecting thermocapillary migration of droplets generated by surface temperature gradients on the supporting substrate. Using a combination of experiment and modeling, we investigate the behavior of the threshold force required for droplet mobilization and the speed after depinning as a function of the droplet size, the applied thermal gradient and the liquid material parameters. The experimental results are well described by a hydrodynamic model based on earlier work by Ford and Nadim. The model describes the steady motion of a two-dimensional droplet driven by thermocapillary stresses including contact angle hysteresis. The results of this study highlight the critical role of chemical or mechanical hysteresis and the need to reduce this retentive force for minimizing power requirements in microfluidic devices.

Thermocapillary migration of liquid droplets in a temperature gradient in a density matched system

An experimental investigation of thermocapillary flow in droplets of a vegetable oil (partially hydrogenated soybean oil) immersed in silicone oil was conducted in a test cell with a heated top wall and a cooled bottom wall. The liquids are nearly immiscible and have equal densities at a temperature below the room temperature, thus providing a simulation of low-gravity conditions by reducing the buoyancy forces. The interfacial tension between the two oils was measured in the temperature range 20 to 50 C using a capillary tube and (d sigma)/(d T) was determined to be negative. Droplets ranging in sizes from 3 mm to 1 cm diameter were injected into the silicone oil. The vertical temperature profile in the bulk liquid (silicone oil) produces temperature variations along the interface which induce variations in the interfacial tension. The flow inside the droplet driven by the resulting interfacial shear stresses was observed using a laser light-sheet flow visualization technique. The flow direction is consistent with the sign of (d sigma)/(d T). The observed maximum surface velocities are compared to the theoretical predictions of Young et al. (1959).

The motion of droplets in a vertical temperature gradient

The thermocapillary migration of single water droplets in butyl benzoate was investigated using the Plateau configuration. The droplets were injected below the level of equal density in the stably stratified butyl benzoate. Because of buoyancy and Marangoni force, the droplets rise until these forces balance each other. For seven different temperature gradients the final position of four to eight single droplets was determined. The reciprocal diameter was found to be a linear function of the deflection from the height of equal density. A systematic deviation from the theory of Young, Goldstein, and Block [J. Fluid Mech. 6, 350 (1959)] was observed. This deviation was attributed to the convective heat transport inside and outside the droplet not covered by the theory. The experimental data could be fitted by introducing an effective temperature gradient that decreases with the square of the applied gradient.

Effect of direct current on the thermocapillary migration of droplets in microgravity

The migration of droplets under the effect of electromagnetic and thermocapillary forces is studied in the case where the electrical conductivity of droplets differs from that of host fluid. The bulk electromagnetic forces in a diluted suspension filling the non-electrical-conducting tube arise due to the interaction between the axial electric current and its magnetic self-field. The temperature gradient in a fluid is caused by Joule heating. The migration of a bubble and a solid particle is also considered. The modified Lamb solution for the Stokes equations is a theoretical basis for the analysis performed.

Thermocapillary migration of droplets: An exact solution for small marangoni numbers

The thermocapillary motion of immiscible spherical droplets placed in an unbounded fluid with a linear temperature distribution is analyzed. For small Marangoni numbers, an exact solution to the Navier-Stokes equations has been found. The velocity field and the terminal velocity of the droplet are the same as those calculated by previous workers in the small Reynolds number limit. Hence the range of applicability of their results has been extended to arbitrary Reynolds numbers, so long as Ma = PrRe remains small (i.e., the convection of energy can be neglected). The shape of the droplet has also been calculated, when the deformations from the spherical shape are small. The results are qualitatively in agreement with the previous results, in the limit Pr → 0, that droplets of the same density as the host fluid do not deform at all, that bubbles tend to deform oblately, and that droplets with the denser fluid inside tend to elongate in the flow direction.

Thermocapillary migration of bubbles and droplets

The migration of a droplet in a large liquid body possessing a uniform temperature gradient is analyzed in the creeping flow limit for small values of the Marangoni number (ε). The method of matched asymptotic expansions is used for the solution of the field equations. The migration velocity is calculated to 0(ε 2) and the result reduces to that of Young, Goldstein and Block [10] in the limit of zero Marangoni number.