The Marangoni Effect: Principles, Applications in Propulsion and Printing, and Beyond#

1. Fundamentals of the Marangoni Effect:

The Marangoni effect, also termed the Gibbs–Marangoni effect, describes the phenomenon of mass transfer occurring along the interface between two fluid phases driven by a gradient in surface tension.1 This surface tension gradient can arise from spatial variations in temperature, leading to what is known as thermocapillary convection or Bénard–Marangoni convection, or from differences in the concentration of chemical species present at the interface, referred to as the solutocapillary effect.1 The initial scientific observation of this phenomenon dates back to 1855 when James Thomson, brother of Lord Kelvin, noted it in the context of the intriguing “tears of wine” effect.2 Subsequently, in 1865, Italian physicist Carlo Marangoni conducted extensive research on this effect as part of his doctoral dissertation, which ultimately led to the phenomenon being named in his honor.2

Surface tension is an inherent property of liquid surfaces that manifests as a tendency for the surface to contract and behave like an elastic membrane under tension, striving to minimize its surface area.11 This behavior is a direct consequence of the cohesive forces that exist between the molecules within the liquid.11 The magnitude of surface tension, denoted by σ, is generally observed to decrease as the temperature, T, of the liquid increases. This inverse relationship, mathematically expressed as dTdσ​<0 12, occurs because higher temperatures increase the kinetic energy of the molecules, thereby weakening the intermolecular forces responsible for surface cohesion. Furthermore, the presence of surface-active agents, commonly known as surfactants, typically results in a reduction of the surface tension of a liquid.7 Consequently, spatial variations in the concentration of surfactants, represented by Γ, lead to corresponding gradients in surface tension, where ∇σ is proportional to ∇Γ.7

The fundamental mechanism underlying the Marangoni effect is the generation of a net force at a liquid-fluid interface due to a gradient in surface tension, ∇σ.2 This force propels the fluid from regions characterized by lower surface tension towards areas where the surface tension is higher.2 In simplified scenarios, the resulting flow speed, U, can be approximated by the relationship U∼μΔσ​, where Δσ represents the difference in surface tension across the interface and μ is the dynamic viscosity of the liquid.2 This indicates that even subtle variations in surface tension can induce noticeable fluid motion, particularly in liquids with low viscosity.

To quantitatively characterize the relative importance of Marangoni-driven transport compared to diffusive transport, the dimensionless Marangoni number (Ma) is employed.9 This number serves as an analogue to the Péclet number.9 In the context of thermal gradients, the Marangoni number is defined as Ma=μαΔTLdTdσ​​, where ΔT is the characteristic temperature difference, L is a representative length scale, dTdσ​ is the temperature coefficient of surface tension, μ is the dynamic viscosity, and α is the thermal diffusivity of the fluid.9 A similar expression can be formulated for concentration gradients, utilizing the concentration difference, the concentration coefficient of surface tension, and the mass diffusivity of the relevant species.9 When the Marangoni number is small, diffusion processes dominate the transport phenomena. Conversely, a large Marangoni number signifies that Marangoni flow, or convection, is the primary mode of transport, potentially leading to instabilities and complex flow patterns such as Bénard–Marangoni convection.9

Several readily observable phenomena illustrate the Marangoni effect in action. The “tears of wine” are a direct consequence of the differential surface tension between water and alcohol, exacerbated by the higher volatility of alcohol.2 When alcohol evaporates from a thin film of wine on a glass, the remaining liquid becomes enriched in water, which has a higher surface tension, leading to an upward flow and the formation of droplets. Another common demonstration involves sprinkling pepper on the surface of water; the subsequent addition of soap, a surfactant, lowers the surface tension of the water locally, causing the surrounding water with higher surface tension to pull the pepper flakes radially outwards.2 The simple “soap boat” toy operates on the same principle: a piece of soap placed at the rear of a floating boat reduces the surface tension behind it, allowing the higher surface tension at the front to propel the boat forward.7 Finally, Marangoni bursting describes the rapid spreading and subsequent fragmentation of a droplet composed of a binary mixture, such as alcohol and water, when deposited on an oil bath, driven by surface tension gradients arising from the differing evaporation rates of the components.11

2. Mathematical Description of Marangoni Flow:

A fundamental aspect of mathematically describing Marangoni flow lies in the tangential stress balance at a free surface. The tangential component of the hydrodynamic stress acting on the surface must be balanced by the tangential stress resulting from gradients in surface tension.7 This condition can be expressed as n⋅T⋅t=−t⋅∇s​σ, where n is the unit outward normal to the surface, T is the stress tensor of the fluid, t is any unit vector tangent to the surface, and ∇s​σ represents the surface gradient of the surface tension.7 In a static equilibrium scenario, where there is no fluid motion, this balance simplifies to ∇s​σ=0, indicating that the surface tension must be uniform along the interface.7

The Marangoni stress, denoted as τM​, can be conceptualized as a shear stress acting at the fluid interface, directly proportional to the gradient of the surface tension along that interface.2 When considering a temperature-dependent surface tension, σ(T), the Marangoni stress is specifically related to the temperature gradient existing along the surface. The force per unit area resulting from the Marangoni effect, fM​, can be expressed as fM​=∇s​σ, which can be further expanded to include contributions from both temperature and concentration gradients: fM​=dTdσ​∇s​T+dCdσ​∇s​C, where T represents temperature and C represents concentration.21 This formulation clearly illustrates that surface tension gradients, whether induced by thermal variations or by differences in chemical composition, contribute to the overall Marangoni force acting on the fluid interface.

In certain simplified scenarios, an approximate expression for the speed of the Marangoni flow, U, can be derived as U∼LμΔσ​H, where Δσ is the difference in surface tension observed over a characteristic length scale L, μ is the fluid’s viscosity, and H represents the depth of the flow.12 For a specific case involving a small droplet of surfactant spreading on a water surface, the speed of expansion, u, of the surfactant-covered patch, which has a radius r, can be approximated by u≈μwater​σwater​−σsurfactant​​(rνwater​​)2/3, where σ denotes the surface tension of the respective fluids and ν is the kinematic viscosity of water.2 These simplified equations offer valuable estimates of the flow velocity under specific conditions and highlight the key parameters influencing the Marangoni effect.

For a more comprehensive and accurate description of Marangoni flows, the Navier-Stokes equations, which govern the conservation of momentum and mass in fluid flow, must be employed. In these models, the Marangoni stress condition is applied as a crucial boundary condition at the free surface of the fluid.5 This boundary condition effectively incorporates the influence of surface tension gradients into the fundamental equations of fluid dynamics, allowing for the simulation and prediction of complex Marangoni-driven phenomena, especially in situations involving non-ideal fluids or intricate geometries.

3. Marangoni Effect in Propulsion:

The Marangoni effect has found significant utility in the realm of propulsion, particularly at small scales relevant to microfluidics and self-propelled droplets.

In microfluidic systems, the solutal Marangoni effect is often exploited for propulsion. This involves the localized release of surfactants onto an air-water interface, which reduces the surface tension in that area, creating a gradient. This gradient then propels floating objects, often referred to as Marangoni surfers, towards regions of higher surface tension.13 This mechanism draws inspiration from the locomotion strategies observed in water-walking insects that secrete lipids to generate propulsive forces.15 For these micro-robots to be effectively maneuvered, precise control over the timing and location of surfactant release is essential.14

The thermocapillary Marangoni effect, induced by temperature gradients along a liquid-air interface, can also be harnessed for propulsion in microchannels.26 These temperature gradients create surface tension gradients that drive fluid flow, which can then be used to actuate and propel bubbles, droplets, or other objects immersed within the microchannel.26 The efficiency of these thermocapillary-driven flows can be enhanced by utilizing superhydrophobic surfaces, which help to maintain a liquid-air interface in close proximity to the solid boundary of the microchannel.29

Another approach involves the integration of microfluidic pumps to deliver a “fuel,” such as alcohol droplets, onto a water surface to achieve controlled Marangoni propulsion for micro-robots.23 The release of this fuel can be triggered using magnetic actuation.14 The overall movement and behavior of these robots can be finely tuned by adjusting various design parameters of the microfluidic pump, including the diameter of the nozzle and the characteristics of the porous media used within the pump.24

Several examples illustrate the application of the Marangoni effect in microfluidic propulsion. These include “Marangoni surfers” propelled by the release of surfactants 13, insect-inspired robots that use controlled alcohol release for both propulsion and steering 15, micro-robots powered by flow-imbibition-driven microfluidic pumps that dispense alcohol droplets 23, and swimming robots fabricated using microfluidic techniques and superhydrophobic surfaces that enable controllable Marangoni propulsion.30

While specific, universally applicable equations for the propulsion velocity and force in all these microfluidic scenarios are not consistently provided in the available information, the fundamental principle remains that the Marangoni force (FM​) is proportional to the surface tension gradient (∇s​σ) and acts along the interface.2 The resulting velocity of the propelled object is then determined by the balance between this Marangoni force and the viscous drag it experiences as it moves through the fluid. One study did develop characteristic radius and time scales based on various fluid and droplet properties for a droplet instability phenomenon 31, suggesting that the relationships governing these dynamics can be complex. Furthermore, theoretical modeling is often employed to predict and analyze the interplay between Marangoni and other forces, such as buoyancy, in these systems.29

Self-propelled droplets represent another fascinating area where the Marangoni effect drives motion. In some cases, this self-propulsion is achieved through chemical reactions occurring at the droplet interface. For instance, droplets loaded with surfactants can move due to changes in interfacial tension resulting from chemical reactions at the oil/water interface.32 The deprotonation of a surfactant in response to changes in pH can induce Marangoni convection within the droplet, leading to its movement.32 The speed and direction of this motion can be influenced by factors such as the pH of the surrounding solution, the temperature, and the relative humidity.32

Evaporation can also induce Marangoni propulsion in droplets. The evaporation of volatile components from a droplet can create concentration gradients within the droplet, which in turn lead to surface tension gradients that drive self-propulsion.19 A dramatic example of this is Marangoni bursting, where rapid evaporation causes a droplet to spread and subsequently fragment into numerous smaller droplets.11 The familiar “tears of wine” phenomenon is also a manifestation of evaporation-driven Marangoni flow.2

Surfactants play a critical role in influencing the Marangoni-driven motion of self-propelled droplets by affecting the surface tension and stability of the droplets.7 Furthermore, the pH of the surrounding medium can be used to control the surface activity of certain surfactants, effectively allowing for the initiation or cessation of droplet mobility.32

The equations that describe the velocity and trajectory of self-propelled droplets often involve a consideration of the balance between the Marangoni stress and the viscous drag exerted by the surrounding fluid.2 For a simplified case of a spherical droplet experiencing a surface tension gradient, the velocity (V) can be related to the Marangoni stress (Δσ/a) and the viscosity (μ) through a relationship such as V∼(Δσ/a)/μ, where a is the droplet’s radius. More sophisticated models take into account the internal flow patterns within the droplet and the specific mechanisms responsible for generating the surface tension gradient, such as the kinetics of chemical reactions or the rates of evaporation.35 One study provided analytical formulas for the flow fields both inside and outside a self-propelled spherical emulsion droplet based on a non-uniform surface tension profile at its interface.35

4. Marangoni Effect in Printing Technologies:

The Marangoni effect plays a significant role in various printing technologies, influencing droplet formation, spreading, and the overall quality of the printed output.

In inkjet printing, Marangoni instability is a key factor in the formation of ink droplets, particularly in continuous inkjet printing systems.43 Thermal modulation applied at the nozzle orifice induces variations in surface tension along the continuous jet of ink. This pulsed heating and the resulting surface tension gradients lead to Marangoni instability, which ultimately causes the jet to break up into discrete droplets.43 The temperature dependence of the ink’s surface tension, often described by the linear relationship σ(T)=σ0​−β(T−T0​), where σ0​ is the surface tension at a reference temperature T0​ and β is the temperature coefficient, is crucial in this process.43

The Marangoni effect also significantly influences the spreading and wetting behavior of ink droplets once they land on the printing substrate.44 Surface tension gradients, whether induced by temperature differences between the droplet and the substrate or by concentration variations arising from differential evaporation of solvents within the ink, can drive Marangoni flows within the droplet. Inward Marangoni flows are particularly desirable as they can transport solutes towards the center of the droplet, leading to the formation of more uniform printed films.44 Achieving uniform films, especially in applications like printed electronics and organic light-emitting diode (OLED) panels, often hinges on careful control of the Marangoni flow and the movement of the droplet’s contact line during drying.45

A common challenge in inkjet printing is the coffee-ring effect, where the solute material in the ink tends to accumulate at the edges of the drying droplet, resulting in a non-uniform deposit. This effect can be effectively suppressed by inducing an inward Marangoni flow within the droplet. This is often achieved through the use of dual-solvent ink systems or by incorporating surfactants into the ink formulation.48 The inward Marangoni flow counteracts the outward capillary flow that drives the coffee-ring effect, leading to a more uniform distribution of the deposited material.48

However, uncontrolled Marangoni flows can also lead to various print defects, including absorption-driven flow, haloing, and micro-haloing.50 These defects typically arise from surface tension gradients caused by the preferential absorption of certain ink components by the print media or by differences in surface tension between adjacent ink droplets. Strategies to mitigate these defects include pretreating the print media with solutions that induce precipitation of the ink’s colorant upon contact, effectively halting the Marangoni flow and preventing the formation of defects.50

Several equations are relevant to understanding the role of the Marangoni effect in inkjet printing. As mentioned earlier, the temperature dependence of surface tension can be described by σ(T)=σ0​−β(T−T0​).43 The relative magnitude of the Marangoni flow can be estimated by the parameter Δγ/η, where Δγ represents the difference in surface tension within the droplet and η is the viscosity of the ink.45 Additionally, the morphology of the final printed film can be analyzed using equations from thin film hydrodynamics, such as a generalized fourth-order equation for the normal displacement field of a curved thin liquid film.51

Beyond inkjet printing, the Marangoni effect is utilized in other printing and coating methods. In bar coating, heating the substrate can induce Marangoni flows within the deposited liquid film, which can significantly affect the uniformity of the resulting thin film, particularly in semiconductor manufacturing.52 The direction and strength of this Marangoni flow depend on the temperature gradient and the properties of the solvent used.52 Photoinitiated Marangoni flow offers a novel approach for surface patterning. By using UV light to induce phase transitions in polymer films, surface energy patterns are created that drive Marangoni flow, leading to the formation of topographic structures. Inkjet-printed patterns can be used to guide this mass migration, providing a method for developer-free surface inscription.46 Furthermore, 3D printing techniques have been employed to create self-propelling particles that utilize the Marangoni effect. These particles, often designed with internal fuel tanks containing alcohol, can move across the surface of a fluid by releasing the alcohol, which creates a local surface tension gradient.18

5. Other Applications of the Marangoni Effect:

The Marangoni effect extends its influence to a wide array of other applications beyond propulsion and printing, demonstrating its versatility in manipulating fluids and materials.

In microfluidic devices, Marangoni flows are employed for various purposes. Surface tension gradients induced by the evaporation of volatile liquids or by ultrasound-induced heating can significantly enhance mixing within sessile droplets, providing efficient mixing at the microscale where laminar flow regimes typically limit convective transport.56 Thermocapillary forces are also utilized for actuation in lab-on-a-chip devices, enabling the manipulation of droplets and fluids for sensing, trapping, sorting, and chemical reactions.5 Localized and rapid temperature gradients, often achieved through laser heating of gold nanoparticles, allow for precise control over these Marangoni flows.58 Furthermore, Marangoni flows generated around microbubbles by temperature gradients can be used to control the movement and sorting of microparticles within microfluidic channels, with the ability to dynamically switch flow direction by modulating the laser power.58

The Marangoni effect plays a crucial role in heat transfer enhancement in several systems. In heat pipes that utilize self-rewetting fluids, Marangoni flow aids in the return of condensed liquid to the evaporator by driving the liquid towards hotter regions due to the anomalous temperature dependence of surface tension in these fluids.2 Marangoni convection also influences heat transfer in evaporating droplets and thin films by affecting internal flow patterns and evaporation rates.6 The effect can also impact boiling and condensation processes by influencing the dynamics of bubble formation and movement during boiling and the formation of microdroplets during condensation.2 Non-uniform evaporation from thin films creates temperature and concentration gradients that drive Marangoni flows, which in turn affect the overall heat and mass transfer within the film.63

The Marangoni effect is also a significant driving force in various self-assembly processes. It can induce the assembly of nanoparticles into ordered patterns, such as hexagonal rings and stripes, through the controlled evaporation of solvents and condensation of water from a nanoparticle suspension.64 The characteristics of the assembled patterns can be tuned by adjusting parameters like relative humidity and the wettability of the substrate.64 At liquid interfaces, Marangoni flows can drive the self-assembly of complex structures like biomimetic jellyfish-like hydrogels 66 and influence the vertical self-assembly of cellulose nanocrystals into specific domains.67 Moreover, Marangoni and elastocapillary effects are involved in the self-organization of floating objects and the formation of interconnected networks at air-water interfaces.68 Coupling chemical reactions with Marangoni flow can also lead to self-sustained flows that facilitate reagent delivery and drive self-assembly processes.71 Even the uniform coating of polymers onto substrates can be achieved by leveraging Marangoni flow during the evaporation of a droplet containing the polymer.72

6. Conclusion and Outlook:

The Marangoni effect stands as a remarkably versatile phenomenon, capable of inducing fluid flow through subtle variations in surface tension arising from temperature or concentration gradients. This fundamental principle underpins a diverse range of applications across various scientific and engineering disciplines. From propelling micro-robots and controlling ink droplet behavior in printing technologies to enhancing heat transfer in thermal management systems and directing the self-assembly of nanomaterials, the Marangoni effect offers a powerful means of manipulating fluids and materials, particularly at small scales.

Despite its broad applicability, utilizing Marangoni flows effectively presents certain challenges. Precise control over the surface tension gradients is often required to achieve the desired outcome, and these flows can sometimes be susceptible to instabilities. Furthermore, at larger scales, the influence of other forces, such as gravity, can become significant and may need to be carefully considered.

Looking ahead, research in this area is likely to focus on developing more sophisticated and reliable methods for generating and controlling Marangoni flows. Emerging trends include exploring novel applications in fields such as advanced materials fabrication, biomedical engineering for drug delivery and diagnostics, and energy harvesting through interfacial phenomena. A deeper understanding of the fundamental physics governing Marangoni effects, especially in complex systems and under non-equilibrium conditions, will continue to drive innovation and expand the potential of this fascinating phenomenon.