1 Introduction
The earliest observations of capillary action, at least within occidental records, are traced to Leonardo da Vinci (Reference da Vinci1452--1519), Maxwell & Strutt (Reference Maxwell and Strutt1911) and Galileo’s student Niccoló Aggiunti Nelli (Reference Nelli1759). The first measurements by Francis Hauksbee (Reference Hauksbee1709) were expanded upon by James Jurin, resulting in Jurin’s law (Jurin Reference Jurin1718, Reference Jurin1719). The formal introduction of surface tension by Johann Andreas von Segner (Reference Segner1751) paved the way for establishing the theoretical foundations of interfacial science with the Young–Laplace equation (Young Reference Young1805; Laplace Reference Laplace1806). The three century delay from the first observations by Leonardo (1452–1519) to the Young–Laplace equation was neither for lack of effort nor ingenuity, but for the fact that in spite of the simplicity of its basis, interfacial science is notorious for its misunderstandings and interpretational difficulties. Even Young and Laplace succumbed to such obscurities (Maxwell Reference Maxwell1878) until Carl Friedrich Gauss unified their approaches (Gauss Reference Gauss1830); such difficulties were not limited to theory alone but plagued experiments alike. For instance, easy contamination of the water surface rendered reliable experimental determination of surface tension nearly impossible until the development of the Pöckels scale (Reference Pöckels1891). All these advances concerned static interfaces where the equilibrium shape or geometry of the surface took centre stage.
An understanding of the Marangoni stress arising from gradients of surface tension arose along a similarly convoluted path. Although interfacial dynamics of self-propelled camphor particles was reported by the late 17th century (Heyde Reference Heyde1686) and investigated by some brilliant minds (Tomlinson (Reference Tomlinson1869) provides a historical review), the phenomenon was explained as arising from surface tension gradients only two centuries hence (Van der Mensbrugghe Reference Van der Mensbrugghe1869). Mistaken hypotheses percolating along the way had to be painstakingly disproven before the correct explanation could emerge; yet another example of the difficulties that beset this field. The Marangoni effect – mass transfer along a fluid interface due to a surface tension gradient – was only identified in 1855 by James Thomson, brother of Lord Kelvin (Thomson Reference Thomson1855), explained 14 years later by Carlo Marangoni (Marangoni Reference Marangoni1869) and eventually received complete theoretical treatment with the Gibbs adsorption isotherm (Gibbs Reference Gibbs1878).
Soap films hold special significance in interfacial science owing to them being the prototypical instance of surface tension dominated effects. Much like the development of interfacial science, static soap film behaviour, e.g. minimal surfaces and related geometric properties (Plateau Reference Plateau1873; Dewar, Ross & Dickson Reference Dewar, Ross and Dickson1927) formed the focus of early studies and soap film dynamics was rarely studied (Mysels Reference Mysels1959). Soap film hydrodynamics (Couder, Chomaz & Rabaud Reference Couder, Chomaz and Rabaud1989; Chomaz Reference Chomaz2001) as a study in and of itself is relatively recent and has rapidly gained a footing as a standard experimental tool in the investigation of two-dimensional hydrodynamics, including turbulence (Kellay & Goldburg Reference Kellay and Goldburg2002), vortex dynamics (Roushan & Wu Reference Roushan and Wu2005) and fluid–structure interactions (Zhang et al.
Reference Zhang, Childress, Libchaber and Shelley2000). Unlike early works, both the three-dimensional character of soap films and the effect of surface tension have become prominent in recent studies (Gilet & Bush Reference Gilet and Bush2009; Tran et al.
Reference Tran, Chakraborty, Gioia, Steers and Goldburg2009), thus prompting the question, ‘What is the surface tension of a flowing soap film?’
2 Overview
The article by Sane, Mandre & Kim (Reference Sane, Mandre and Kim2018) addresses precisely this question, both theoretically and experimentally. The question assumes significance in the first place because it is the surfactant that imparts stability to the soap film. The surfactant molecules populate the interface and the bulk fluid. In equilibrium, the molecules are distributed in the two phases according to the adsorption isotherm, and the surface tension of the soap film is reduced below that of pure water by the adsorbed surfactant. Upon stretching the film, bulk molecules are transferred to the freshly created surface, a process whose duration is dictated by surfactant transport and sorption kinetics. For a soap film constantly flowing under external pumping of a solution, it is not clear how the time scale of the hydrodynamics that deforms the interface compares with the surfactant equilibration time scales. The film’s surface tension could vary with the flow regime, thereby necessitating its experimental determination. Such a measurement probes the extent to which the surfactant dynamics is out of equilibrium.
Unfortunately, standard surface tension measurement techniques fail here. For instance, the pendant drop method requires bulk fluid and is therefore disqualified by default whereas other techniques such as the du Noüy and Wilhelmy plate methods are invasive and become especially problematic if the film flow speed exceeds the Marangoni wave speed (Kim & Mandre Reference Kim and Mandre2017). Sane et al. (Reference Sane, Mandre and Kim2018) circumvent all these difficulties by imaginatively exploiting the Euler–Bernoulli equation to relate the bending curvature of wires that bound the flowing film to its surface tension. Standard gravity-driven soap films employ two nylon wires held in tension by a hanging mass of known weight, and the soap solution is drained down the wires under gravity. When the nylon wires are pulled apart, a soap film is created between the wires and the solution now flows down the film. But the film’s surface tension, which tries to minimize its surface area, exerts a force that tries to pull the nylon wires back together and imparts curvature to them while in tension (see title image). Measuring the bending curvature of the nylon wire against the applied tension yields the surface tension of the film in flow.
Sane et al. (Reference Sane, Mandre and Kim2018) have measured the surface tension of flowing soap films under different film thicknesses and soap concentrations and found an apparent equivalence between surfactant dilution and thinning of the soap film. Since the film’s surface tension depends upon surfactant concentration at the interface, this translates to the fact that thinner films possess higher surface tension. Relating the film thickness and surfactant concentration through surfactant conservation, they develop a quantitative model based on Langmuir’s adsorption isotherm. This model finds confirmation when surface tension, measured at different values of the solution flux and at different concentrations, collapses onto a master curve. This observation also explains that the soap film’s Marangoni elasticity measured using the angle of oblique Marangoni shocks (see title figure) does not vary with the film width, flow rate or soap concentration because the interface is saturated with soap (see Kim & Mandre Reference Kim and Mandre2017).
3 Outlook
In the process of determining the surface tension of flowing soap films, Sane et al. (Reference Sane, Mandre and Kim2018) have also discovered a heretofore unknown dependence of the soap film’s terminal velocity on the film thickness and its flux. Standard two-dimensional hydrodynamics assumes the terminal velocity arises from air drag, which is expected to be substantial on account of the large exposed area of the film to surrounding air, and is usually modelled as a linear drag term in the two-dimensional Navier–Stokes equation. The additional dependence on film thickness and flux points to a gaping hole in our understanding and one hazards to proffer the informed opinion that there is most probably new physics in the two-dimensional hydrodynamics.
Furthermore, commercial detergents employed in soap film experiments are complex formulations comprised of different surfactants. The proof of principle presented by Sane et al. (Reference Sane, Mandre and Kim2018) begs careful characterization against pure surfactants not just for surface tension measurement, but more importantly to gain a better understanding of stationary state sorption kinetics. This point applies beyond soap films to the general problem of surfactant transport. Surfactants dissolve to varying degrees in bulk fluid and can be transported either in the adsorbed layer (at the interface) or in bulk phase. Simultaneous measurement of surface and bulk surfactant concentrations, Marangoni stresses and surface rheology, sorption kinetics and how these quantities are correlated is an experimentally formidable task (Langevin & Monroy Reference Langevin and Monroy2014; Elfring, Leal & Squires Reference Elfring, Leal and Squires2016; Bandi et al.
Reference Bandi, Akella, Singh, Singh and Mandre2017; Mandre Reference Mandre2017) and contains all of the hallmarks of the obscurities faced by interfacial science since its inception. It is in this context the work of Sane et al. (Reference Sane, Mandre and Kim2018) bears hope that perhaps a renaissance is afoot in interfacial science, prompting one to close with the famous quote by Leonardo da Vinci, the discoverer of capillary action, ‘che non terminano in nota experientia’.
1 Introduction
The earliest observations of capillary action, at least within occidental records, are traced to Leonardo da Vinci (Reference da Vinci1452--1519), Maxwell & Strutt (Reference Maxwell and Strutt1911) and Galileo’s student Niccoló Aggiunti Nelli (Reference Nelli1759). The first measurements by Francis Hauksbee (Reference Hauksbee1709) were expanded upon by James Jurin, resulting in Jurin’s law (Jurin Reference Jurin1718, Reference Jurin1719). The formal introduction of surface tension by Johann Andreas von Segner (Reference Segner1751) paved the way for establishing the theoretical foundations of interfacial science with the Young–Laplace equation (Young Reference Young1805; Laplace Reference Laplace1806). The three century delay from the first observations by Leonardo (1452–1519) to the Young–Laplace equation was neither for lack of effort nor ingenuity, but for the fact that in spite of the simplicity of its basis, interfacial science is notorious for its misunderstandings and interpretational difficulties. Even Young and Laplace succumbed to such obscurities (Maxwell Reference Maxwell1878) until Carl Friedrich Gauss unified their approaches (Gauss Reference Gauss1830); such difficulties were not limited to theory alone but plagued experiments alike. For instance, easy contamination of the water surface rendered reliable experimental determination of surface tension nearly impossible until the development of the Pöckels scale (Reference Pöckels1891). All these advances concerned static interfaces where the equilibrium shape or geometry of the surface took centre stage.
An understanding of the Marangoni stress arising from gradients of surface tension arose along a similarly convoluted path. Although interfacial dynamics of self-propelled camphor particles was reported by the late 17th century (Heyde Reference Heyde1686) and investigated by some brilliant minds (Tomlinson (Reference Tomlinson1869) provides a historical review), the phenomenon was explained as arising from surface tension gradients only two centuries hence (Van der Mensbrugghe Reference Van der Mensbrugghe1869). Mistaken hypotheses percolating along the way had to be painstakingly disproven before the correct explanation could emerge; yet another example of the difficulties that beset this field. The Marangoni effect – mass transfer along a fluid interface due to a surface tension gradient – was only identified in 1855 by James Thomson, brother of Lord Kelvin (Thomson Reference Thomson1855), explained 14 years later by Carlo Marangoni (Marangoni Reference Marangoni1869) and eventually received complete theoretical treatment with the Gibbs adsorption isotherm (Gibbs Reference Gibbs1878).
Soap films hold special significance in interfacial science owing to them being the prototypical instance of surface tension dominated effects. Much like the development of interfacial science, static soap film behaviour, e.g. minimal surfaces and related geometric properties (Plateau Reference Plateau1873; Dewar, Ross & Dickson Reference Dewar, Ross and Dickson1927) formed the focus of early studies and soap film dynamics was rarely studied (Mysels Reference Mysels1959). Soap film hydrodynamics (Couder, Chomaz & Rabaud Reference Couder, Chomaz and Rabaud1989; Chomaz Reference Chomaz2001) as a study in and of itself is relatively recent and has rapidly gained a footing as a standard experimental tool in the investigation of two-dimensional hydrodynamics, including turbulence (Kellay & Goldburg Reference Kellay and Goldburg2002), vortex dynamics (Roushan & Wu Reference Roushan and Wu2005) and fluid–structure interactions (Zhang et al. Reference Zhang, Childress, Libchaber and Shelley2000). Unlike early works, both the three-dimensional character of soap films and the effect of surface tension have become prominent in recent studies (Gilet & Bush Reference Gilet and Bush2009; Tran et al. Reference Tran, Chakraborty, Gioia, Steers and Goldburg2009), thus prompting the question, ‘What is the surface tension of a flowing soap film?’
2 Overview
The article by Sane, Mandre & Kim (Reference Sane, Mandre and Kim2018) addresses precisely this question, both theoretically and experimentally. The question assumes significance in the first place because it is the surfactant that imparts stability to the soap film. The surfactant molecules populate the interface and the bulk fluid. In equilibrium, the molecules are distributed in the two phases according to the adsorption isotherm, and the surface tension of the soap film is reduced below that of pure water by the adsorbed surfactant. Upon stretching the film, bulk molecules are transferred to the freshly created surface, a process whose duration is dictated by surfactant transport and sorption kinetics. For a soap film constantly flowing under external pumping of a solution, it is not clear how the time scale of the hydrodynamics that deforms the interface compares with the surfactant equilibration time scales. The film’s surface tension could vary with the flow regime, thereby necessitating its experimental determination. Such a measurement probes the extent to which the surfactant dynamics is out of equilibrium.
Unfortunately, standard surface tension measurement techniques fail here. For instance, the pendant drop method requires bulk fluid and is therefore disqualified by default whereas other techniques such as the du Noüy and Wilhelmy plate methods are invasive and become especially problematic if the film flow speed exceeds the Marangoni wave speed (Kim & Mandre Reference Kim and Mandre2017). Sane et al. (Reference Sane, Mandre and Kim2018) circumvent all these difficulties by imaginatively exploiting the Euler–Bernoulli equation to relate the bending curvature of wires that bound the flowing film to its surface tension. Standard gravity-driven soap films employ two nylon wires held in tension by a hanging mass of known weight, and the soap solution is drained down the wires under gravity. When the nylon wires are pulled apart, a soap film is created between the wires and the solution now flows down the film. But the film’s surface tension, which tries to minimize its surface area, exerts a force that tries to pull the nylon wires back together and imparts curvature to them while in tension (see title image). Measuring the bending curvature of the nylon wire against the applied tension yields the surface tension of the film in flow.
Sane et al. (Reference Sane, Mandre and Kim2018) have measured the surface tension of flowing soap films under different film thicknesses and soap concentrations and found an apparent equivalence between surfactant dilution and thinning of the soap film. Since the film’s surface tension depends upon surfactant concentration at the interface, this translates to the fact that thinner films possess higher surface tension. Relating the film thickness and surfactant concentration through surfactant conservation, they develop a quantitative model based on Langmuir’s adsorption isotherm. This model finds confirmation when surface tension, measured at different values of the solution flux and at different concentrations, collapses onto a master curve. This observation also explains that the soap film’s Marangoni elasticity measured using the angle of oblique Marangoni shocks (see title figure) does not vary with the film width, flow rate or soap concentration because the interface is saturated with soap (see Kim & Mandre Reference Kim and Mandre2017).
3 Outlook
In the process of determining the surface tension of flowing soap films, Sane et al. (Reference Sane, Mandre and Kim2018) have also discovered a heretofore unknown dependence of the soap film’s terminal velocity on the film thickness and its flux. Standard two-dimensional hydrodynamics assumes the terminal velocity arises from air drag, which is expected to be substantial on account of the large exposed area of the film to surrounding air, and is usually modelled as a linear drag term in the two-dimensional Navier–Stokes equation. The additional dependence on film thickness and flux points to a gaping hole in our understanding and one hazards to proffer the informed opinion that there is most probably new physics in the two-dimensional hydrodynamics.
Furthermore, commercial detergents employed in soap film experiments are complex formulations comprised of different surfactants. The proof of principle presented by Sane et al. (Reference Sane, Mandre and Kim2018) begs careful characterization against pure surfactants not just for surface tension measurement, but more importantly to gain a better understanding of stationary state sorption kinetics. This point applies beyond soap films to the general problem of surfactant transport. Surfactants dissolve to varying degrees in bulk fluid and can be transported either in the adsorbed layer (at the interface) or in bulk phase. Simultaneous measurement of surface and bulk surfactant concentrations, Marangoni stresses and surface rheology, sorption kinetics and how these quantities are correlated is an experimentally formidable task (Langevin & Monroy Reference Langevin and Monroy2014; Elfring, Leal & Squires Reference Elfring, Leal and Squires2016; Bandi et al. Reference Bandi, Akella, Singh, Singh and Mandre2017; Mandre Reference Mandre2017) and contains all of the hallmarks of the obscurities faced by interfacial science since its inception. It is in this context the work of Sane et al. (Reference Sane, Mandre and Kim2018) bears hope that perhaps a renaissance is afoot in interfacial science, prompting one to close with the famous quote by Leonardo da Vinci, the discoverer of capillary action, ‘che non terminano in nota experientia’.