2.3.1 Compressibility: Gibbs ( $E$ ) and Marangoni ( $E_{0}$ ) moduli

The surface compressional (or dilatational) modulus,

measures the resistance of the surface to compression, completely analogous to 3-D materials. For insoluble surfactants, the number of surfactants in a monolayer does not change during compression, meaning $A$ can be replaced with $N/\unicode[STIX]{x1D6E4}$ in (2.21) to give the insoluble dilatational modulus

The nomenclature of (2.21) and (2.22) varies across the surfactant literature, with the names of Gibbs, Marangoni, Gibbs–Marangoni or simply dilatational modulus used for both $E$ and $E_{0}$ . For the purposes of this review, we will consistently call $E$ the Gibbs modulus and $E_{0}$ the Marangoni modulus.

The Marangoni modulus $E_{0}$ measures the work done in squeezing surfactant molecules together, and generally increases with surfactant concentration. For example, ideal gas monolayers have Marangoni modulus

with expressions for other isotherms given in table 1. Notably, the Marangoni modulus for the van der Waals isotherm,

becomes negative for a range of $\unicode[STIX]{x1D6E4}$ whenever $\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6E4}_{\infty }/k_{B}T>27/4$ . Just like in three dimensions, a monolayer with negative compressibility is mechanically unstable, and undergoes phase separation to a two-phase coexistence between a high- $\unicode[STIX]{x1D6E4}$ condensed phase, and a low- $\unicode[STIX]{x1D6E4}$ expanded phase.

In what follows (§§ 2.4.2 and 3.2), we will find that monolayers (soluble or insoluble) do not always react instantaneously following compression – finite time scales are required for phase transitions to occur, for surfactants to adsorb or desorb to equilibrate with the surrounding bulk fluid, or for surfactant gradients in the surroundings to diffusively relax. In this regard, the Marangoni modulus $E_{0}$ reflects an intrinsic material property, whereas the Gibbs modulus $E$ describes the dynamic response of a macroscopic interface, which additionally depends on the shape, size and time scales of the forcing.

2.3.2 The chemical potential

In preparation for the upcoming transition to soluble surfactants, we discuss one final thermodynamic property of Langmuir monolayers. The chemical potential represents the free energy cost of adding one additional molecule to the monolayer, holding temperature and area constant

The chemical potential of a monolayer of ideal surfactants – as described by (2.8) – is

or, equivalently,

where $\unicode[STIX]{x1D707}_{s}^{0}$ is a reference chemical potential, valid at a particular concentration $\unicode[STIX]{x1D6E4}_{0}$ . In what follows, we will frequently use the Langmuir (lattice) isotherm as a model; its chemical potential is

where $\unicode[STIX]{x1D707}_{s}^{0}$ is the chemical potential at $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{\infty }/2$ .

2.4.1 Gibbs isotherm

It is frequently difficult to measure the surface concentration $\unicode[STIX]{x1D6E4}$ of soluble surfactants; more common is to measure surface pressure $\unicode[STIX]{x1D6F1}$ (or, equivalently, surface tension $\unicode[STIX]{x1D6FE}$ ) as a function of subphase concentration $C$ . In such cases, the Gibbs adsorption isotherm allows $\unicode[STIX]{x1D6E4}$ to be derived from measured $\unicode[STIX]{x1D6F1}(C)$ relations (Martínez-Balbuena et al. Reference Martínez-Balbuena, Arteaga-Jiménez, Hernández-Zapata and Márquez-Beltrán2017). The Gibbs relation connects changes in surface pressure $\unicode[STIX]{x1D6F1}$ to changes in the chemical potential $\unicode[STIX]{x1D707}_{s}$ of adsorbed surfactants at concentration $\unicode[STIX]{x1D6E4}$ , via (Adamson Reference Adamson1990; Myers Reference Myers2006; Kralchevsky et al. Reference Kralchevsky, Danov and Denkov2008)

The chemical potential of adsorbed surfactants $\unicode[STIX]{x1D707}_{s}$ is difficult to determine directly since $\unicode[STIX]{x1D6E4}$ is unknown for soluble surfactants. However, when bulk and adsorbed surfactants are equilibrated, $\unicode[STIX]{x1D707}_{s}$ must be equal to $\unicode[STIX]{x1D707}_{b}$ for the dissolved surfactants. In cases where the surfactant solution is dilute enough to behave as ideal, $\unicode[STIX]{x1D707}_{b}(C)$ is given by (2.30), in which case

The Gibbs relation (2.38) then gives

In other words, the adsorbed surfactant concentration $\unicode[STIX]{x1D6E4}$ can be determined from measurements of surface pressure $\unicode[STIX]{x1D6F1}$ as a function of dissolved surfactant concentration $C$ , so long as the system has equilibrated, and the concentration is well below the CMC (Martínez-Balbuena et al. Reference Martínez-Balbuena, Arteaga-Jiménez, Hernández-Zapata and Márquez-Beltrán2017).

2.4.2 Compressibility: $E$ and $E_{0}$ for soluble surfactants

The distinction between the Gibbs and Marangoni moduli, $E$ and $E_{0}$ , defined by (2.21) and (2.22) respectively, becomes significant for soluble monolayers. Recall that $E$ tracks surface pressure changes when the monolayer area $A$ is changed, whereas $E_{0}$ additionally holds the number $N$ of adsorbed surfactant molecules fixed – meaning $\unicode[STIX]{x1D6E4}$ changes when $A$ does. Compressing monolayers of soluble surfactants raises the chemical potential $\unicode[STIX]{x1D707}_{s}$ of the adsorbed surfactants, without changing the concentration or chemical potential $\unicode[STIX]{x1D707}_{b}(C)$ of the dissolved surfactant. A thermodynamic force drives adsorbed surfactants to desorb, until $\unicode[STIX]{x1D6E4}$ returns to the value predicted by (2.29). Once equilibrium is re-established, $\unicode[STIX]{x1D6E4}$ returns to its initial value, so that

How quickly the interface re-equilibrates cannot be determined from thermodynamic properties alone, as discussed in § 3.2.

Soluble surfactant monolayers do have a non-zero Marangoni modulus $E_{0}$ , however. After all, $\unicode[STIX]{x1D6E4}$ (and therefore $\unicode[STIX]{x1D6F1}$ ) must increase during a rapid compression of a Gibbs monolayer, before the surfactants have had the chance to desorb. The Gibbs adsorption equation (2.40), however, offers a route to $E_{0}$ for soluble surfactant monolayers from measured $\unicode[STIX]{x1D6F1}(C)$ isotherms

Substituting $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x2202}\ln C$ using (2.40) gives

where primes denote differentiation with respect to $\ln C$ .

While the Gibbs modulus $E$ describes the monolayer’s mechanical response to macroscopic compression or expansion $(dA)$ , the Marangoni modulus $E_{0}$ reflects sensitivity to intrinsic molecular concentrations $\unicode[STIX]{x1D6E4}$ , and will play an important role in establishing an effective surface dilatational viscosity for surfactant monolayers, as shown in § 4.2.

2.4.3 Soluble isotherms via dynamic equilibrium

Equilibrium between surfactants adsorbed at an interface and dissolved in the subphase can also be determined by explicitly balancing adsorption and desorption fluxes. This approach holds particular value to the fluid mechanics community, as it connects the equilibrium arguments and measurements made above to Marangoni stresses and dynamical processes in surfactant systems.

The simplest expressions for adsorption and desorption fluxes – which one should expect to hold for ideal mixtures in both the monolayer and in the bulk – is to take the adsorption flux, $j_{a}$ , to be proportional to the bulk concentration $C$ , and the desorption flux $j_{d}$ to be proportional to the adsorbed concentration $\unicode[STIX]{x1D6E4}$ :

These fluxes balance at equilibrium, $j_{a}=j_{d}$ , giving

reproducing the Henry isotherm (2.32). Previously, the adsorption constant $K^{ideal}$ was shown to depend upon the free energy of adsorption $\unicode[STIX]{x0394}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{b}^{0}-\unicode[STIX]{x1D707}_{s}^{0}$ via (2.33). Equation (2.46) additionally relates $K^{ideal}$ to the ratio of adsorptive to desorptive rate constants $k_{a}/k_{d}$ . Each individual rate constant $k_{a}$ and $k_{d}$ cannot be determined from an equilibrium quantity like $K$ ; however, the ratio of the two is set by thermodynamics.

Other isotherms require different adsorption and/or desorption rate kinetics. For example, the Langmuir isotherm (2.35) is formed when dissolved surfactants adsorb to vacant lattice sites (Levich Reference Levich1962; Adamson Reference Adamson1990; Kralchevsky et al. Reference Kralchevsky, Danov and Denkov2008), modifying the adsorption flux (2.44) to

At equilibrium, $j_{a}^{L}=j_{d}$ , so that

recovering (2.35) with $K=k_{a}/k_{d}$ . Adsorption and desorption fluxes for other common isotherms are listed in table 2.

This dynamic equilibrium approach forms a natural transition into § 3, which addresses the dynamic response of surfactant-laden interfaces when they are driven out of equilibrium. Specifically, we will describe how fluid flow transports surfactant molecules in the bulk and on the interface, and how the tendency of surfactant molecules to re-equilibrate in turn impacts interfacial fluid dynamics.

3.1.1 Surface advection

The surface advective flux can be decomposed into components normal to the interface, and components along the interface

and therefore has surface divergence

as in (3.10). The first term captures the compression or dilation of surfactant that occurs when curved interfaces ( $\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{n}\neq 0$ ) themselves deform (due to non-zero normal interfacial velocities), and the second term captures the compression or dilation arising from non-uniform convective flux within the plane of the interface.

3.1.2 Surface diffusion

The standard expression for the diffusive flux of adsorbed surfactant along a surface, used to derive (3.10), has a Fickian form,

A diffusive flux of this form, however, is built on assumptions that are rarely accurate in systems of practical interest.

A brief derivation of the diffusive flux highlights conditions under which the Fickian form holds. Because the chemical potential $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6E4},T)$ represents the free energy ‘cost’ of an adsorbed surfactant at concentration $\unicode[STIX]{x1D6E4}$ and temperature $T$ , any spatial gradients in $\unicode[STIX]{x1D707}$ point toward more ‘costly’ locations for surfactants to be placed. A chemical potential gradient thus represents a thermodynamic force ( $\boldsymbol{f}_{s}=-\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D707}$ ) on an adsorbed surfactant, which drives it to migrate along the interface, with relative velocity

where $b_{s}$ is the hydrodynamic mobility of the surfactant along the surface, and is related to its self-diffusivity by the Stokes–Einstein relation (Saffman & Delbrück Reference Saffman and Delbrück1975; Furst & Squires Reference Furst and Squires2017)

Each surfactant molecule moves with velocity (3.15) along the surface, so that a single-component monolayer of surface concentration $\unicode[STIX]{x1D6E4}$ establishes a flux

relative to the interface. In multi-component monolayers, the chemical potential of each species depends on the surface concentration of every other component, and the term in brackets in (3.17) is replaced by a generalized Maxwell–Stefan diffusivity tensor (Krishna Reference Krishna1990). In what follows, we restrict our discussion to single-component monolayers.

The Fickian form (3.14) holds only for the ideal gas monolayer, for which $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{s}^{ideal}(\unicode[STIX]{x1D6E4})$ from (2.27), and therefore

so that the diffusive flux (3.17) reduces to Fick’s law (3.14). As discussed in § 2.3, however, it is rare for any surfactant that reduces surface tension in any appreciable way to behave as an ideal gas. The Fickian form, then, rarely holds as explicitly derived. Of course, one can define an effective diffusivity

which differs from the true self-diffusivity of each surfactant molecule. If gradients are small enough that $D_{s}^{eff}$ is approximately constant, then the Fickian form would be appropriate, albeit with a modified diffusivity.

The effective surface diffusivity $D_{s}^{eff}$ can be shown using (2.22) and (2.23) to exceed the Fickian self-diffusivity by the Marangoni modulus $E_{0}(\unicode[STIX]{x1D6E4})$ relative to an ideal gas monolayer

The diffusive flux expressions for the Langmuir and Volmer isotherms (table 1), for example, become

Curiously, the diffusive flux of adsorbed surfactant within single-component monolayers can be expressed in terms of surface pressure gradients alone. The Gibbs adsorption relation (2.38) directly implies

which can be substituted for $\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D707}$ in (3.17) to give a diffusive flux

which appears to be independent of the surface concentration $\unicode[STIX]{x1D6E4}$ ! Although this result seems counter-intuitive at first, it can be understood physically as follows. The surface pressure gradient gives the force per unit area exerted on the surfactant monolayer, which is divided among $\unicode[STIX]{x1D6E4}$ molecules per unit area. The greater the concentration $\unicode[STIX]{x1D6E4}$ of adsorbed surfactant, the weaker the force on each ( $\boldsymbol{F}_{s}\sim \unicode[STIX]{x1D6FB}\unicode[STIX]{x1D6F1}/\unicode[STIX]{x1D6E4}$ ), and the slower each migrates: $\boldsymbol{V}\sim b_{s}(\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D6F1}/\unicode[STIX]{x1D6E4})$ . Ultimately, the concentration $\unicode[STIX]{x1D6E4}$ cancels out of the flux $\unicode[STIX]{x1D6E4}\boldsymbol{V}$ in (3.23).

Figure 6. (a) Geometry of a typical surfactant-laden interface. (b–e) Mass transport processes on the interface: (b) convection due to imposed (or Marangoni) velocity $u_{s}$ ; (c) diffusion due to a surface concentration gradient; (d) surface concentration evolution due to curvature modification; and (e) adsorption/desorption from the sublayer, showing the depletion length $L_{d}$ .

3.2.1 Adsorption/desorption to an oscillating bubble

Here we adapt the work of Johnson & Stebe (Reference Johnson and Stebe1994), who considered oscillations of a bubble with rest radius $R_{0}$ and equilibrium surface concentration $\unicode[STIX]{x1D6E4}_{0}$ in a liquid containing dissolved surfactant at concentration $C_{0}$ (figure 7). The bubble radius changes in response to a controlled oscillation of gas volume or pressure. Assuming departures from equilibrium to be small, the radius, surface concentration and bulk concentration are perturbed via

The surfactant conservation equation (3.10) for purely radial oscillations becomes

where $u_{r}=\text{d}R/\text{d}t$ is the radial velocity at the interface. Perturbing (3.30) via (3.29) gives

In systems where the surfactant is insoluble (for which $j_{n}=0$ ), the change in surface concentration is

We define $C_{s}$ and $\unicode[STIX]{x1D6FF}C_{s}$ as the bulk concentration and the amplitude of its oscillatory perturbation at the interface

Because the convective term in the bulk transport equation (3.26) is quadratic in perturbed quantities, the bulk concentration $C$ obeys the diffusion equation to leading order, with solution

where $\unicode[STIX]{x1D6FF}C_{s}$ is as yet unknown. The diffusive flux (3.27) onto the interface,

must equal the kinetic flux $j_{kin}$ (3.24), which is given to leading order by

Here

are effective rate constants associated with the equilibrium exchange fluxes (e.g. table 2). The ratio of $k_{C}$ and $k_{\unicode[STIX]{x1D6E4}}$ has units of length and is defined as the depletion depth

For example, the (linear) Henry isotherm (table 1) has $k_{C}=k_{a}$ and $k_{\unicode[STIX]{x1D6E4}}=k_{d}$ , so that the depletion depth is a constant equal to the equilibrium adsorption constant: $L_{d}=K^{ideal}$ .

Equating the diffusive (3.35) and kinetic fluxes (3.36) relates $\unicode[STIX]{x1D6FF}C_{s}$ to $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$ :

where the mass transfer Womersley number $Wo$ and Damköhler number $Da$ are

Then, using (3.39) in (3.35) or (3.36) to eliminate $\unicode[STIX]{x1D6FF}C_{s}$ and determine $j_{n}$ in terms of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$ , and substituting into the surface conservation equation (3.31) reveals $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$ to be

where the Stanton number $St$ is defined by

The Damköhler number controls the transition from kinetically limited ( $Da\ll 1$ ) to diffusion-limited ( $Da\gg 1$ ) surfactant exchange. In what follows (§§ 3.2.2–3.2.3), we consider kinetically and diffusion-controlled regimes separately. For later use, we rewrite the surfactant exchange flux, from (3.35) or (3.36), as

3.2.2 Kinetically limited mass transfer ( $Da\ll 1$ )

Adsorption is kinetically limited ( $Da\ll 1$ ) when molecular exchange between the interface and the subsurface layer is significantly slower than surfactant diffusion through the bulk. In this limit, the subphase concentration is approximately uniform: (3.39) gives $\unicode[STIX]{x1D6FF}C_{s}\rightarrow 0$ as $Da\rightarrow 0$ , implying $C_{s}(t)\approx C_{0}$ (figure 7 e).

Figure 8. (a) Real (solid line) and imaginary (broken line) components of the perturbed surface concentration for kinetically limited adsorption ( $Da\rightarrow 0$ ) to an oscillating bubble, from (3.44). (b) Surface concentration in diffusion-limited mass transfer ( $Da\rightarrow \infty$ ), from (3.50).  $\unicode[STIX]{x1D6EC}_{d}=0$ represents a planar interface following (3.48). The vertical dashed lines show locations where $2\unicode[STIX]{x1D6EC}_{d}\unicode[STIX]{x1D701}_{d}^{2}=1$ when $\unicode[STIX]{x1D6EC}_{d}\gtrsim 1$ , indicating modified diffusion times for small bubbles via (3.53).

Setting $Da=0$ in (3.41) gives the perturbed surface concentration

The $St\rightarrow 0$ limit corresponds to oscillations so rapid that surfactants do not have the times to adsorb or desorb, so that the monolayer behaves as if it were insoluble: $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{kin}\rightarrow \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{insol}$ . By contrast, the $St\rightarrow \infty$ limit occurs when surfactants adsorb/desorb much faster than bubble oscillations. In that case, $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{kin}\rightarrow 0$ and the interface maintains its equilibrium concentration $\unicode[STIX]{x1D6E4}(t)\approx \unicode[STIX]{x1D6E4}_{0}$ .

Time scales for bubble oscillation and adsorption/desorption are comparable when $St\sim O(1)$ , which defines the characteristic time scale for kinetically limited adsorption:

For finite $\unicode[STIX]{x1D70F}_{k}$ (or finite $St$ ), $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{kin}$ is always smaller than $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{insol}$ (figure 8 a). Additionally, the surface concentration lags the bubble radius by a phase shift $\tan ^{-1}(St)$ .

When kinetically limited ( $Da\rightarrow 0$ ), the adsorption flux (3.43) becomes

Throughout this work, we will use (3.46) as the kinetically limited sorptive flux for small departures from equilibrium.

3.2.3 Diffusion-limited mass transfer ( $Da\gg 1$ )

In diffusion-controlled surfactant exchange, adsorption kinetics are so fast that $\unicode[STIX]{x1D6E4}$ equilibrates with the subsurface concentration $C_{s}$ effectively instantaneously, via the appropriate isotherm $\unicode[STIX]{x1D6E4}(C_{s})$ (figure 7 d). In this limit, $j_{kin}$ can be ignored, so that $j_{n}=j_{diff}$ . The depletion depth $L_{d}$ follows by expanding $\unicode[STIX]{x1D6E4}(C_{s})$ around $\unicode[STIX]{x1D6E4}_{0}(C_{0})$ , and taking the $Da\rightarrow \infty$ limit of (3.39) to give

which we non-dimensionalize by the bubble radius $R_{0}$ to give

The perturbed surface concentration (3.41) in the diffusion-controlled limit ( $Da\rightarrow \infty$ ) is then

which can be expressed as

where

is a dimensionless ratio of the diffusive oscillatory boundary layer thickness $\unicode[STIX]{x1D6FF}_{BL}=\sqrt{D/2\unicode[STIX]{x1D714}}$ to the depletion depth $L_{d}$ .

Bubbles with radii much larger than the depletion depth ( $\unicode[STIX]{x1D6EC}_{d}\ll 1$ ) behave like planar interfaces. Indeed, (3.50) recovers the celebrated results of Lucassen & van den Tempel (Reference Lucassen and van den Tempel1972) in the $\unicode[STIX]{x1D6EC}_{d}\rightarrow 0$ limit, which we examine in detail in § 4.2.1. Adsorbed surfactants on a planar interface ( $\unicode[STIX]{x1D6EC}_{d}\rightarrow 0$ ) act as effectively insoluble if molecules in the subphase diffuse far less than the depletion depth in one oscillation ( $\unicode[STIX]{x1D701}_{d}\ll 1$ ). Conversely, the surface concentration remains close to its equilibrium value ( $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{diff}\rightarrow 0$ ) if molecules diffusively escape $L_{d}$ during an oscillation ( $\unicode[STIX]{x1D701}_{d}\gg 1$ ). Interfacial oscillations and diffusive mass transfer are comparable when $\unicode[STIX]{x1D701}_{d}=O(1)$ , which reveals the characteristic time scale for diffusion-limited mass transfer in the planar limit ( $\unicode[STIX]{x1D6EC}_{d}\rightarrow 0$ ) to be

Diffusive mass transfer is sensitive to interfacial curvature when $\unicode[STIX]{x1D6EC}_{d}\gtrsim 1$ , as shown in figure 8(b). The perturbed surface concentration $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{diff}$ still vanishes if diffusion is fast ( $\unicode[STIX]{x1D701}_{d}\gg 1$ ) and approaches the insoluble limit if diffusion is slow ( $\unicode[STIX]{x1D701}_{d}\ll 1$ ). However, the transition occurs around $\unicode[STIX]{x1D6EC}_{d}\unicode[STIX]{x1D701}_{d}^{2}=O(1)$ , where

which reveals the characteristic time scale of diffusive-controlled surfactant exchange for small bubbles ( $\unicode[STIX]{x1D6EC}_{d}\gtrsim 1$ )

Indeed, experiments and simulations show a smooth transition from $\unicode[STIX]{x1D70F}_{d,p}$ to $\unicode[STIX]{x1D70F}_{d,s}$ with decreasing bubble radius (Alvarez, Walker & Anna Reference Alvarez, Walker and Anna2010a ,Reference Alvarez, Walker and Anna b ).

Finally, the diffusion-limited surfactant exchange flux follows from the $Da\rightarrow \infty$ limit of (3.43). For small bubbles ( $R_{0}^{2}\ll D/\unicode[STIX]{x1D714}$ , or $Wo\,\ll 1$ ), the net adsorption flux is

which has the same form as (3.46), but with the diffusion-controlled time scale $\unicode[STIX]{x1D70F}_{d,s}$ in place of the kinetically controlled time scale $\unicode[STIX]{x1D70F}_{k}$ . For the same reasons, $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{diff}(\unicode[STIX]{x1D6EC}_{d}\gg 1)$ from (3.50) is identical to $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{kin}$ from (3.44) with $2\unicode[STIX]{x1D6EC}_{d}\unicode[STIX]{x1D701}_{d}^{2}$ in place of $St$ . In other words, diffusion-limited mass transfer to small bubbles ‘looks like’ it is kinetically limited, albeit with a characteristic time scale $\unicode[STIX]{x1D70F}_{d,s}$ that depends on the bubble radius via (3.54).

To summarize, the characteristic sorption time $\unicode[STIX]{x1D70F}_{s}$ is

In systems with dynamic interfaces, the mechanical response of a surface to deformation depends not only on the equilibrium properties of the surfactant (such as $E_{0}$ , § 2.3.1), but also on the adsorbed concentration profile $\unicode[STIX]{x1D6E4}(\boldsymbol{r}_{s},t)$ at any particular position and time. In § 4.2, we will again use the oscillating gas bubble example to quantify the apparent viscoelasticity of soluble monolayers, and its dependence on surfactant properties such as $k_{\unicode[STIX]{x1D6E4}}$ , $k_{C}$ , $D$ and $E_{0}$ .

3.2.4 Adsorption to a clean interface

So far, we have discussed surfactant transport on interfaces that are perturbed only slightly from equilibrium. Ward & Tordai (Reference Ward and Tordai1946) pursued a complementary problem: the diffusion-limited mass transfer to an initially clean planar interface. More recently, Jin, Balasubramaniam & Stebe (Reference Jin, Balasubramaniam and Stebe2004) and Alvarez et al. (Reference Alvarez, Walker and Anna2010a ,Reference Alvarez, Walker and Anna b ) established the critical role of interfacial curvature on surfactant transport. In what follows, we explore their calculations of surfactant exchange to an initially clean static bubble, in both diffusion- and kinetically limited regimes.

The system is the same as in § 3.2.1, except that the bubble interface is stationary, $R(t)=R_{0}$ , and is initially clean: $\unicode[STIX]{x1D6E4}(0)=0$ . Surfactant is dissolved in the bulk at concentration $C_{0}$ , and the subsurface concentration is $C(R,t)=C_{s}(t)$ . The surface conservation equation (3.10) with the diffusive flux $j_{diff}$ from the bulk is

where $C(R,t)$ follows from the solution of the bulk diffusion equation. Laplace transforming (3.57), denoted by tildes, gives

where $s$ is the Laplace transform variable. Laplace transforming the bulk diffusion equation gives

with solution

where $C_{1}$ is as yet unknown. Evaluating (3.58) with (3.60), then eliminating $C_{1}$ in (3.60) in favour of $C_{s}$ gives

Finally, inverting the Laplace transform of (3.61) gives

The first term on the right-hand side reflects the solution of Ward & Tordai (Reference Ward and Tordai1946) for adsorption onto a clean planar interface, whereas the last two terms reflect interfacial curvature (Jin et al. Reference Jin, Balasubramaniam and Stebe2004; Alvarez et al. Reference Alvarez, Walker and Anna2010a ).

The generalized Ward–Tordai result (3.62) is an implicit integral relation between $\unicode[STIX]{x1D6E4}(t)$ and the yet undetermined subsurface concentration $C_{s}(t)$ . Solving for the surface concentration requires another relationship between $\unicode[STIX]{x1D6E4}(t)$ and $C_{s}(t)$ , which follows from the kinetic flux condition (3.24). However, inverting this relation is not straightforward except for the simplest kinetic flux expressions, and is typically solved numerically (Jin et al. Reference Jin, Balasubramaniam and Stebe2004; Alvarez et al. Reference Alvarez, Walker and Anna2010a ). For example, the 2-D ideal gas assumption (table 2) gives

with Laplace transform

In principle, eliminating $\tilde{C}_{s}$ between (3.64) and (3.61), and inverse Laplace transforming $\tilde{\unicode[STIX]{x1D6E4}}(s)$ gives an explicit expression for $\unicode[STIX]{x1D6E4}(t)$ . In fact, Hansen (Reference Hansen1961) employs this strategy to examine adsorption to a planar interface ( $R\rightarrow \infty$ ).

Even for the simplest kinetic flux (3.63), however, adsorption to a spherical interface is intractable at arbitrary Damköhler number. Instead, we highlight the kinetically controlled ( $Da\ll 1$ ) and diffusion-controlled ( $Da\gg 1$ ) limits individually, by ignoring $j_{diff}$ and $j_{kin}$ , respectively. These limits are easier to calculate, and are illustrative in light of the discussion around oscillating bubbles in the previous section. For later use, we note that the equilibrium surface concentration following (3.63) is

with depletion depth

When surfactant transport is kinetically limited, diffusion in the bulk is assumed to be instantaneous, so that $C_{s}(t)=C_{0}$ . Laplace transforming (3.63) with $C_{s}=C_{0}$ and using (3.65) gives

with inverse

which recovers the kinetically limited sorption time $\unicode[STIX]{x1D70F}_{k}=1/k_{d}$ (3.45). At short times, $\unicode[STIX]{x1D6E4}_{kin}$ grows linearly with time, whereas $\unicode[STIX]{x1D6E4}_{kin}$ approaches $\unicode[STIX]{x1D6E4}_{eq}$ exponentially for $t\gg \unicode[STIX]{x1D70F}_{k}$ (figure 9 a). As with adsorption to an oscillating bubble, kinetically limited mass transfer is independent of bubble size.

Figure 9. (a) Kinetically controlled adsorption (3.68) to an initially clean spherical surface. (b) Diffusion-controlled adsorption (3.71) to a spherical surface. Adsorption occurs over a faster time scale $\unicode[STIX]{x1D70F}_{d,s}$ in smaller bubbles ( $\unicode[STIX]{x1D6EC}_{d}=L_{d}/R\gtrsim 1$ ), and approaches equilibrium exponentially, rather than as $1/\sqrt{t}$ in the case of large bubbles ( $\unicode[STIX]{x1D6EC}_{d}\rightarrow 0$ ).

By contrast, contact equilibrium is assumed between $\unicode[STIX]{x1D6E4}(t)$ and $C_{s}(t)$ when adsorption is diffusion controlled, so that

Using the Laplace transform of (3.69) to eliminate $\tilde{C}_{s}$ in (3.61) gives

Inverting (3.70) is laborious but straightforward, and yields

where

and $\unicode[STIX]{x1D6EC}_{d}=L_{d}/R$ (3.48).

The large- and small-bubble limits of (3.70) are particularly illustrative. Bubbles with radii much larger than the depletion depth ( $\unicode[STIX]{x1D6EC}_{d}\rightarrow 0$ ) have

recovering the result of Hansen (Reference Hansen1961) for a planar interface. Indeed, the characteristic diffusion time in (3.73) is $\unicode[STIX]{x1D70F}_{d,p}=L_{d}^{2}/D$ , like in diffusion-limited adsorption to large oscillating bubbles (3.52). $\unicode[STIX]{x1D6E4}_{diff}$ approaches $\unicode[STIX]{x1D6E4}_{eq}$ algebraically at long times (figure 9 b), much more slowly than the exponential approach during kinetically controlled adsorption (3.68).

When the bubble is much smaller than the depletion depth ( $\unicode[STIX]{x1D6EC}_{d}\gg 1$ ), however, surface concentration approaches equilibrium exponentially, via

The characteristic diffusion time $\unicode[STIX]{x1D70F}_{d,s}=L_{d}R/D$ that emerges is the same (3.54) that controls small, oscillatory bubbles. As with oscillating bubbles, diffusion-limited adsorption to small bubbles ( $R\ll L_{d}$ ) has the same form as kinetically limited adsorption (3.68), except with $\unicode[STIX]{x1D70F}_{d,s}$ replacing $\unicode[STIX]{x1D70F}_{k}$ . As shown in figure 9(b), diffusion-limited adsorption is faster for smaller bubbles, and $\unicode[STIX]{x1D6E4}_{diff}$ approaches $\unicode[STIX]{x1D6E4}_{eq}$ exponentially rather than algebraically over long times.

Finally, convection in the bulk fluid further enhances the rate of diffusive adsorption by ‘screening’ the characteristic diffusion length by the thickness of the diffusion boundary layer $\unicode[STIX]{x1D6FF}_{BL}$ . Alvarez et al. (Reference Alvarez, Vogus, Walker and Anna2012) demonstrated that the diffusion time is indeed $\unicode[STIX]{x1D70F}_{d,p}^{conv}\propto \unicode[STIX]{x1D6FF}_{BL}^{2}$ for large bubbles, and $\unicode[STIX]{x1D70F}_{d,s}^{conv}\propto \unicode[STIX]{x1D6FF}_{BL}$ for smaller bubbles. This scaling suggests strategies to further speed up diffusive surfactant transport using flow, as the boundary layer thickness decreases with increasing bulk convection, thereby increasing the range of measurable kinetic-limited adsorption (Alvarez et al. Reference Alvarez, Vogus, Walker and Anna2012).

3.3.1 Surface concentration gradients and hydrodynamic coupling

Recall from § 3.1.2 that spatial gradients in the surface chemical potential $\unicode[STIX]{x1D707}_{s}$ point to energetically unfavourable locations to place adsorbed surfactants. A thermodynamic force $\boldsymbol{f}_{s}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D707}_{s}$ drives surfactants down the gradient with a velocity given by the molecule’s hydrodynamic mobility (3.15). However, a molecule moves not only because it is forced, but also because its neighbours are forced, and drive fluid flows that entrain the molecule. Readers familiar with suspension dynamics will recognize hydrodynamic coupling in a 3-D fluid with background velocity $\boldsymbol{V}_{\infty }$

where the tensor $\unicode[STIX]{x1D642}(\boldsymbol{r}_{i}-\boldsymbol{r}_{j})$ gives the velocity at $\boldsymbol{r}_{i}$ in response to a force $\boldsymbol{F}_{j}$ on a particle centred at $\boldsymbol{r}_{j}$ (Happel & Brenner Reference Happel and Brenner1965; Guazzelli & Morris Reference Guazzelli and Morris2012). In many cases (e.g. sedimentation), the hydrodynamic coupling sum may overwhelm the ‘self-mobility’ term.

The precise analogue occurs at surfactant interfaces: the velocity of a surfactant molecule at $\boldsymbol{r}_{i}$ is a combination of the background velocity $\boldsymbol{u}_{s}$ , the ‘self’-mobility and the hydrodynamic coupling, so that (3.15) is modified to include

where the final term reflects the surface velocity at $\boldsymbol{r}_{i}$ , established by all neighbouring surfactant molecules – with concentration $\unicode[STIX]{x1D6E4}$ , each forced by $-\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D707}$ . The Green’s function $\unicode[STIX]{x1D642}(\boldsymbol{r}_{i}-\boldsymbol{r}_{j})$ in this case gives the fluid velocity on the interface at $\boldsymbol{r}_{i}$ , driven by a force at $\boldsymbol{r}_{j}$ on the fluid interface. The precise form of the Green’s function depends on the geometry of the interface, the subphase depth, etc. (§ 3.4.2) The added velocity due to hydrodynamic interactions, however, corresponds to a boundary integral solution (Pozrikidis Reference Pozrikidis1992) to the Stokes equations, where the fluid interface is driven by a specified traction:

which, using the Gibbs adsorption relation (3.22), becomes

The hydrodynamic coupling (3.76) between surfactant molecules is precisely equal to the net convective velocity driven by surface tension gradients. In other words, hydrodynamic coupling between surfactants is equivalent to the Marangoni flow driven on the interface by the surfactant monolayer.

Surface pressure gradients can drive or balance viscous shear stress following (3.78). Figure 10 illustrates two examples of the conjugate effects that usually go by the name of Marangoni, both of which involve surfactant gradients. Gradients in surface pressure may arise due to non-uniform surface concentrations $\unicode[STIX]{x1D6E4}(\boldsymbol{r}_{s})$ , or due to surface convective transport $\boldsymbol{u}_{s}\unicode[STIX]{x1D6E4}$ that establishes a concentration gradient. Depositing surfactant on an initially clean interface (figure 10 a) introduces a surface concentration gradient $\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D6E4}$ , and therefore a surface pressure gradient that exerts a traction

Flows are therefore driven down surface pressure (or surface concentration) gradients.

The functional form of $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6E4})$ depends on the particular surfactant isotherm (§ 2.4). It is common practice to assume an ideal gas monolayer, for which the Marangoni traction is

As discussed in § 2.3, however, the ideal gas assumption rarely holds in practice, and more accurate models or measured values of $\unicode[STIX]{x2202}\unicode[STIX]{x1D6F1}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}$ would be more appropriate. For example, the Langmuir and Volmer isotherms give tractions

both of which recover the ideal gas limit when $\unicode[STIX]{x1D6E4}\ll \unicode[STIX]{x1D6E4}_{\infty }$ .

Figure 10. Two conjugate effects commonly termed ‘Marangoni’ effects: (a) A local increase in surfactant concentration, shown here by the addition of a surfactant-rich drop, establishes a surface concentration gradient (and, therefore, a surface pressure gradient) that drives an outward surface flow (red arrows). (b) Surface compression due to flow (in this case, towards an interfacial barrier) establishes a surface tension gradient due to non-uniform surface concentration. This introduces a reverse Marangoni flow that ‘immobilizes’ the surface.

The second example, shown in figure 10(b), resembles the so-called ‘Reynolds ridge’ (Scott Reference Scott1982) and involves flows that compress (or dilate) a surfactant-laden fluid interface against a floating barrier. Such a flow creates surface concentration gradients that act against the interfacial compression (or dilatation). Marangoni stresses therefore act like surface-excess elasticity (Langevin Reference Langevin2014), working to lessen surface compression or dilatation. In what follows, we quantify the degree to which reverse Marangoni flows resist inhomogeneous surface compression.

3.3.2 Marangoni numbers and surface incompressibility

Surfactant monolayers are far more compressible than 3-D fluids. Compressing an insoluble surfactant increases the surface concentration $\unicode[STIX]{x1D6E4}$ and thus the surface pressure $\unicode[STIX]{x1D6F1}$ . Additionally, many surfactants exhibit phase transitions (Kaganer, Möhwald & Dutta Reference Kaganer, Möhwald and Dutta1999) and surface-pressure-dependent surface viscosity (Kurtz, Lange & Fuller Reference Kurtz, Lange and Fuller2006; Kim et al. Reference Kim, Choi, Zasadzinski and Squires2011) even under facile compression, as discussed in § 5. However, inhomogeneous compression of a surfactant monolayer drives reverse Marangoni flows that resist such deformations.

Figure 11. Illustration of surfactant-induced incompressibility. (a) Motion of a probe establishes a surface concentration gradient, shown in top and side views. In a soluble monolayer with instantaneous adsorption/desorption, surface concentration gradients are rapidly eliminated. (b) In this limit, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F1}\approx 0$ and reverse Marangoni flows are absent. The surface flow has a non-zero divergence ahead and behind the disk. Bulk fluid flow is indicated by the dashed arrows and is indistinguishable from that corresponding to a stress-free clean interface. (c) By contrast, if the surfactant is insoluble, a surface concentration gradient is sustained, and (d) the surface pressure difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F1}$ generates a reverse Marangoni flow that resists interfacial compression/dilatation. The modified surface flow is divergence free, which changes the bulk flow by constraining it to flow in planes parallel to the interface (see discussion).

For example, a disk of radius $R$ translating at velocity $U$ within a planar surfactant-laden interface compresses the monolayer ahead of the disk, and dilates the monolayer in its rear (figure 11). The surfactant conservation equation for insoluble surfactants (3.10) at steady state, in the absence of surface diffusion, gives the surface divergence

where $E_{0}$ is the Marangoni modulus (2.22). Balancing the surface pressure gradient $\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D6F1}$ in (3.82) with traction on the subphase via the Marangoni boundary condition (3.79) gives

Non-dimensionalizing gives a dimensionless surface divergence

where the Marangoni number

balances surface compressibility $E_{0}/R$ against viscous traction $\unicode[STIX]{x1D702}U/R$ .

In the large Marangoni number limit ( $Ma\gg 1$ ), surfactant molecules resist compression so strongly that the surface flow is effectively divergence free. In other words, the surface is 2-D incompressible when the time scale to establish reverse Marangoni flows

is much faster than the surface convection time scale $\unicode[STIX]{x1D70F}_{flow}=R/U$ , so that $Ma=\unicode[STIX]{x1D70F}_{flow}/\unicode[STIX]{x1D70F}_{m}\gg 1$ . Surface pressure then acts as a Lagrange multiplier to maintain surface incompressibility, much like bulk pressure in a 3-D fluid (figure 11). In unidirectional surface flows, such as in figure 10(b), surface incompressibility requires $\boldsymbol{u}_{s}$ to be constant. Marangoni flows then ‘immobilize’ the interface, effectively modifying the interfacial boundary condition from a free surface ( $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}z=0$ ) when $Ma\ll 1$ to a rigid surface $(u_{s}=v(0)=\text{const.})$ when $Ma\gg 1$ .

Insoluble surfactants are almost always surface incompressible around translating disks or particles. The Marangoni modulus of incredibly dilute monolayers with $\unicode[STIX]{x1D6E4}\sim 1/(100~\text{nm}^{2})$ in the ideal gas limit is $E_{0}^{ideal}=k_{B}T\unicode[STIX]{x1D6E4}\sim 0.04~\text{mN}~\text{m}^{-1}$ . Even for such dilute monolayers with immeasurably small surface pressures ( $\unicode[STIX]{x1D6F1}\sim 40~\unicode[STIX]{x03BC}\text{N}~\text{m}^{-1}$ ), a disk or particle must translate faster than $4~\text{cm}~\text{s}^{-1}$ before $Ma\lesssim 1$ and the interface compresses. The Marangoni modulus of real surfactants is much larger than $E_{0}^{ideal}$ (table 1), with typical values of $E_{0}\gtrsim 1~\text{mN}~\text{m}^{-1}$ (Arriaga et al. Reference Arriaga, López-Montero, Ignés-Mullol and Monroy2010; Kotula & Anna Reference Kotula and Anna2016), so that insoluble monolayers are effectively surface incompressible unless $U\gtrsim 1~\text{m}~\text{s}^{-1}$ .

Even the slightest amount of insoluble surfactant, therefore, fundamentally changes the interfacial boundary condition on the bulk fluid flow. Stone & Masoud (Reference Stone and Masoud2015) illustrated the change in subphase flow by considering the continuity equation for the bulk fluid at the interface,

where $w$ is the $z$ -component of subphase fluid velocity $\boldsymbol{v}$ . At the interface, the tangential velocity is $\boldsymbol{v}=\boldsymbol{u}_{s}$ , and thus surface incompressibility implies $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z=0$ . The vertical velocity $w$ vanishes at $z=0$ for an interface that does not deform out of plane, and along with $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z=0$ requires that $w$ be zero throughout the subphase. Surface incompressibility therefore constrains subphase fluid to flow in planes parallel to the interface (figure 11 d). Such flows set up stronger velocity gradients, dissipating more energy than in surfactant-free systems, where flow in the bulk is three-dimensional (figure 11 b).

Insoluble surfactants therefore substantially modify subphase flow relative to surfactant-free systems, increasing the translational resistance of particles within monolayers. For example, the drag on a circular disk translating in an incompressible monolayer exceeds the drag in a clean interface by a factor of $3/2$ (§ 4.6). In some cases, this increase in translational resistance has been misattributed to surface rheology, going back to Plateau’s experiments with oscillating needles on interfaces (Scriven & Sternling Reference Scriven and Sternling1960), while in fact it arises due to Marangoni flows. Indeed, surface incompressibility increases the hydrodynamic drag on translating probes even for a completely inviscid surfactant, causing significant confusion in the measurement and interpretation of surface rheology of insoluble surfactants (Sickert & Rondelez (Reference Sickert and Rondelez2003), Fischer (Reference Fischer2004a ), and § 3.4).

Soluble surfactants, on the other hand, may desorb (or adsorb) as the interface is compressed (or dilated) to restore equilibrium coverage (figure 11 a). The strength of the Marangoni stress then depends on the balance between surface advective and surfactant exchange fluxes. For small perturbations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$ in surface concentration around an equilibrium value $\unicode[STIX]{x1D6E4}_{0}$ , the surfactant balance (3.10) becomes

where $\unicode[STIX]{x1D70F}_{s}$ is the longer among kinetic and diffusive surfactant exchange times (3.56). Using (3.88) in the Marangoni boundary condition (3.79) gives

where $E_{0}$ is the Marangoni modulus (2.22). Significantly, any finite adsorption time scale $\unicode[STIX]{x1D70F}_{s}$ gives rise to a viscous-like force with apparent dilatational surface viscosity $E_{0}\unicode[STIX]{x1D70F}_{s}$ in (3.89) – a feature we will explore in detail in § 3.4.3. Non-dimensionalizing (3.89) gives

where

is a modified Marangoni number, that now compares the time scale $\unicode[STIX]{x1D70F}_{m}=\unicode[STIX]{x1D702}R/E_{0}$ to establish Marangoni flow with the time $\unicode[STIX]{x1D70F}_{s}$ for the surface to equilibrate via adsorption/desorption.

If a soluble monolayer equilibrates before Marangoni flows can be established ( $Ma_{K}\ll 1$ ), then the surface behaves as if it is compressible. Equation (3.90) recovers the stress-free boundary condition as $Ma_{K}\rightarrow 0$ , and fluid flows behave approximately as though the interface were clean (figure 11 b). In this limit, $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6F1}$ are largely unperturbed from the surfactant’s equilibrium isotherm. By contrast, if Marangoni flows are established before surfactants adsorb and desorb to equilibrate with the subphase concentration ( $Ma_{K}\gg 1$ ), then the surface divergence is

where $\dot{\unicode[STIX]{x1D716}}$ is a uniform compression or dilatation rate. In fact, interfaces with fixed area require $\dot{\unicode[STIX]{x1D716}}=0$ , and the interface acts as incompressible.

Finally, Marangoni flows may also be weakened by other surface processes, leading to alternative definitions of the Marangoni number. Table 3 summarizes common definitions of Marangoni numbers, obtained by comparing the Marangoni time scale $\unicode[STIX]{x1D70F}_{m}$ against competing system-specific surfactant processes. In each case, a large Marangoni number implies that the surfactant monolayer resists inhomogeneous surface compression or dilatation, and the interface can be approximated as 2-D incompressible. When Marangoni flows are weak, a fully compressible description becomes necessary (Barentin et al. Reference Barentin, Ybert, Di Meglio and Joanny1999; Elfring, Leal & Squires Reference Elfring, Leal and Squires2016).

3.4.1 The Boussinesq–Scriven model

Boussinesq (Reference Boussinesq1913) was the first to explicitly account for a viscous-like resistance to surface dilatation, using it to explain the anomalous settling velocity of drops (§ 4.1). Scriven (Reference Scriven1960) generalized Boussinesq’s model by treating the interfacial layer as a 2-D Newtonian fluid with intrinsic surface shear ( $\unicode[STIX]{x1D702}_{s}$ ) and surface dilatational ( $\unicode[STIX]{x1D705}_{s}$ ) viscosities, both with dimensions of 3-D viscosity $\times$ length, so that the extra rheological stress $\unicode[STIX]{x1D749}_{rheo}$ in (3.8) is Newtonian

where $\unicode[STIX]{x1D644}_{s}=\unicode[STIX]{x1D644}-\boldsymbol{n}\boldsymbol{n}$ is the surface identity tensor. Using (3.93) in the surface stress balance (3.8) for a planar liquid–air interface gives

which is the 2-D analogue of the Stokes equation for compressible fluids, with viscous traction from the subphase entering as a body force. More generally, out-of-plane deformations and 2-D viscous flow along curved interfaces introduce additional forces that have no analogue in 3-D Newtonian fluids (Scriven Reference Scriven1960; Aris Reference Aris1962; Edwards, Brenner & Wasan Reference Edwards, Brenner and Wasan1991; Slattery et al. Reference Slattery, Sagis and Oh2007).

The Boussinesq–Scriven model simplifies in many typical surfactant systems. As discussed in § 3.3.2, insoluble surfactants almost always behave as surface incompressible, in which case (3.94) reduces to the incompressible 2-D Stokes equation, forced by viscous traction from the subphase

Surface pressure then acts to enforce 2-D incompressibility, analogous to bulk pressure in a 3-D incompressible fluid.

Solving such problems is generally difficult, because two- and three-dimensional Stokes flows must be solved separately, but coupled via boundary conditions enforcing surface incompressibility and subphase traction. In systems where the subphase is very shallow, however, this coupling simplifies significantly, because the lubrication approximation relates the subphase velocity gradient to interfacial velocity and the subphase depth $H$ ,

in which case the subphase flow need not be solved explicitly (Evans & Sackmann Reference Evans and Sackmann1988). The lubrication approximation is particularly useful for compressible monolayers, where eliminating $\boldsymbol{u}$ permits analytical solutions (Barentin et al. Reference Barentin, Ybert, Di Meglio and Joanny1999; Elfring et al. Reference Elfring, Leal and Squires2016).

Figure 12. (a) A disk of radius $R$ rotating at a constant angular velocity $\unicode[STIX]{x1D6FA}$ within a monolayer of surface shear viscosity $\unicode[STIX]{x1D702}_{s}$ . (b) Surface velocity $u_{s}$ follows (3.100) when subphase dominant ( $Bq\ll 1$ , blue squares) and (3.101) when interface dominant ( $Bq\gg 1$ , green circles); adapted from Zell et al. (Reference Zell, Nowbahar, Mansard, Leal, Deshmukh, Mecca, Tucker and Squires2014).

3.4.2 Two-dimensional versus three-dimensional hydrodynamics and the Boussinesq number

Momentum propagation through a viscous interfacial layer fundamentally modifies familiar 3-D fluid dynamics, even when no other surfactant process is excited. For example, pure shear deformations like the swirling flow driven by rotating a circular disk at an interface (figure 12) do not generate surface concentration gradients, and thus do not give rise to Marangoni stresses. The planar Boussinesq–Scriven equation (3.94) then becomes

which couples to the Stokes equations for $\boldsymbol{u}$ in the subphase. Flow is driven by the rotation of a disk of radius $R$ at constant angular velocity $\unicode[STIX]{x1D6FA}$ : $\boldsymbol{u}_{s}(r\leqslant R)=\unicode[STIX]{x1D6FA}r\hat{\unicode[STIX]{x1D73D}}$ . Non-dimensionalizing (3.97) gives

where the Boussinesq number,

compares surface and subphase viscous stresses.

Subphase-dominant flows ( $Bq\ll 1$ ) in pure shear recover the stress-free condition at the interface. Solving the Stokes equations in the subphase with a stress-free interface driven by a rotating disk gives (Goodrich Reference Goodrich1969)

The velocity field due to rotation decays as $1/r^{2}$ in the absence of surface excess rheological stresses, as expected in Stokes flow (Guazzelli & Morris Reference Guazzelli and Morris2012).

By contrast, (3.98) implies that $\unicode[STIX]{x1D6FB}_{s}^{2}\boldsymbol{u}_{s}\approx \mathbf{0}$ when the flow is interface dominant ( $Bq\gg 1$ ), and

A viscous interfacial layer propagates momentum more extensively within the interface, resulting in a slower decay ( ${\sim}1/r$ ) of the surface velocity field relative to (3.100), as shown in figure 12(b). The torque required to sustain constant rotation transitions from ${\sim}R^{3}$ for an inviscid interface ( $Bq=0$ ) to ${\sim}R^{2}$ for a viscous monolayer ( $Bq\rightarrow \infty$ ; see § 4.6). This striking difference arises in systems where no other surfactant transport process – e.g. Marangoni flow, adsorption/desorption, surface dilatation – is active. Measurements of the rotational torque and flow field around a rotating disk can thus unambiguously and quantitatively detect surface shear viscosity (Choi et al. Reference Choi, Steltenkamp, Zasadzinski and Squires2011; Zell et al. Reference Zell, Nowbahar, Mansard, Leal, Deshmukh, Mecca, Tucker and Squires2014).

The translation of a particle embedded within a monolayer is more complex, as it deforms the surface via compression, dilation and extension, in addition to shear. Even in the seemingly simpler case of insoluble (and, therefore, 2-D incompressible from § 3.3.2) monolayers, translation introduces subtleties due to momentum transfer between a surface-shear-viscous interface and the underlying 3-D fluid.

Saffman & Delbrück (Reference Saffman and Delbrück1975) noticed this transition from 2-D to 3-D hydrodynamics in their seminal work on particle diffusion in biological membranes. The incompressible Boussinesq–Scriven equation (3.95) decouples from the subphase in the interface-dominant ( $Bq\rightarrow \infty$ ) limit. However, there is no solution for steady translation of a cylinder in 2-D creeping flow (Leal Reference Leal2007, the ‘Stokes paradox’). Saffman (Reference Saffman1976) recognized that subphase viscous stresses ( ${\sim}\unicode[STIX]{x1D702}U\ell _{SD}$ ) eventually catch up with surface-viscous stresses ( ${\sim}\unicode[STIX]{x1D702}_{s}U$ ) beyond a distance

ultimately regularizing the divergence inherent to 2-D Stokes flow. Surface viscous stresses dominate within the Saffman–Delbrück length $\ell _{SD}$ , and viscous traction from the subphase prevail beyond $\ell _{SD}$ .

The cross-over from 2-D to 3-D hydrodynamics is evident in the flow driven by a tangential point force on the interface (Lubensky & Goldstein Reference Lubensky and Goldstein1996; Levine & MacKintosh Reference Levine and MacKintosh2002; Fischer Reference Fischer2004b ; Oppenheimer & Diamant Reference Oppenheimer and Diamant2009). Velocity fields driven by arbitrarily shaped particles moving on an interface can be constructed using appropriate boundary integrals of the point-force solution (§ 4.6).

An incompressible Newtonian monolayer along the $x$ – $y$ plane acted upon by an in-plane point force $\boldsymbol{F}$ at $\boldsymbol{x}_{0}$ is governed by

where $\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{x}_{0}$ . The two-dimensional Fourier transform,

is defined so that $\boldsymbol{k}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{z}=0$ , and transforms (3.103) to give

The (3-D) hydrodynamic pressure field $p$ associated with flow on an incompressible surface is a constant everywhere (Stone & Ajdari Reference Stone and Ajdari1998), such that momentum balance for Stokes flow in the bulk reduces to Laplace’s equation: $\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}=\mathbf{0}$ . If the subphase extends to a finite depth $H$ such that $\boldsymbol{v}(-H)=\mathbf{0}$ , then

Using (3.106) in (3.105) and eliminating $\hat{\unicode[STIX]{x1D6F1}}$ by premultiplying by $\unicode[STIX]{x1D644}-\boldsymbol{k}\boldsymbol{k}/k^{2}$ gives

The Green’s function $\hat{\unicode[STIX]{x1D642}}(\boldsymbol{k})$ depends on the ratio $k\unicode[STIX]{x1D702}_{s}/\unicode[STIX]{x1D702}$ , which is the Boussinesq number (3.99) for a length scale $\unicode[STIX]{x1D706}=2\unicode[STIX]{x03C0}/k$ . In real space, the second-order tensor $\unicode[STIX]{x1D642}(\boldsymbol{r})$ is the surface analogue of the Oseen tensor in classical hydrodynamics. However, inverting $\hat{\unicode[STIX]{x1D642}}$ is not straightforward except in specific limits. We will focus on the deep subphase limit ( $H\rightarrow \infty$ ), where the inverse transform gives (Lubensky & Goldstein Reference Lubensky and Goldstein1996; Fischer Reference Fischer2004b ; Levine & MacKintosh Reference Levine and MacKintosh2002)

where $r=|\boldsymbol{r}|$ . The mobility coefficients $\unicode[STIX]{x1D707}_{\Vert }$ and $\unicode[STIX]{x1D707}_{\bot }$ are

where $H_{\unicode[STIX]{x1D708}}$ and $Y_{\unicode[STIX]{x1D708}}$ are, respectively, Struve functions and Bessel functions of the second kind of order $\unicode[STIX]{x1D708}$ , and

is distance scaled by the Saffman–Delbrück length $\ell _{SD}$ (3.102). Alternatively, non-dimensionalizing $r$ by the probe size $R$ makes $d$ in (3.110) equivalent to

where $Bq=\unicode[STIX]{x1D702}_{s}/\unicode[STIX]{x1D702}R$ is the Boussinesq number (3.99).

Momentum transport is interface-dominated over length scales smaller than $\ell _{SD}$ ( $d\ll 1$ or $Bq\gg 1$ ), and the mobility coefficients (3.109) become

To within an additive constant, the interfacial velocity driven by a point force in the $Bq\rightarrow \infty$ (or $d\rightarrow 0$ ) limit is then

which is, indeed, the 2-D Stokeslet.

The logarithmic divergence in (3.113) as $r\rightarrow \infty$ reflects the Stokes paradox, which is here resolved by viscous traction from the subphase. Subphase viscous stresses become dominant over length scales larger than $\ell _{SD}$ ( $d\gg 1$ or $Bq\ll 1$ ), in which case

The surface velocity profile becomes

in the surface-inviscid limit ( $Bq\rightarrow 0$ ). Notably, (3.115) is not a 2-D slice of the flow due to a 3-D Stokeslet – surface incompressibility (§ 3.3.2) modifies fluid streamlines to ensure that the resultant velocity profile is surface-divergence free (figure 11).

Prasad, Koehler & Weeks (Reference Prasad, Koehler and Weeks2006) experimentally mapped out surface velocities on surface-viscous protein monolayers via passive (colloid-tracking) two-particle microrheology. The displacement correlations between two points on the interface along and perpendicular to the line joining their centres are proportional to mobility coefficients $\unicode[STIX]{x1D707}_{\Vert }$ and $\unicode[STIX]{x1D707}_{\bot }$ , respectively. Experiments over a wide range of surface viscosities (from $O(1)\,\text{nN}~\text{s}~\text{m}^{-1}$ to $O(1)~\unicode[STIX]{x03BC}\text{N}~\text{s}~\text{m}^{-1}$ ) clearly show this transition from a logarithmic decay in the interface-dominated regime to $\unicode[STIX]{x1D707}_{\Vert }\sim 1/r$ and $\unicode[STIX]{x1D707}_{\bot }\sim 1/r^{2}$ when subphase stresses dominate (figure 13). Surface streamlines following (3.108) also transition distinctly between subphase-dominant and interface-dominant flows (figure 14).

Figure 13. Mobility coefficients in the radial ( $\unicode[STIX]{x1D707}_{\Vert }$ ) and azimuthal ( $\unicode[STIX]{x1D707}_{\bot }$ ) directions (in arbitrary units) extracted from two-particle microrheology. In the surface-dominated regime ( $Bq\gg 1$ or $d\ll 1$ ), both coefficients decay logarithmically with distance, following (3.112). In subphase-dominated cases ( $Bq\ll 1$ or $d\gg 1$ ), the coefficients decay as $1/r$ and $1/r^{2}$ . The symbols correspond to two-particle displacement correlations along the line of centres (filled symbols) and perpendicular to it (empty symbols), and the solid lines are fits to (3.109). The shapes of the symbols correspond to different surface viscosities. Adapted from Prasad et al. (Reference Prasad, Koehler and Weeks2006).

Figure 14. Streamlines of surface velocity as a result of a point force applied at the origin, following (3.108). Length is in units of $R$ such that $Bq=\unicode[STIX]{x1D702}_{s}/\unicode[STIX]{x1D702}R$ .

3.4.3 Intrinsic and apparent surface viscosity

As discussed in § 3.3, surface flows that compress or dilate the interface establish surfactant concentration gradients, generating Marangoni stresses. Surfactant exchange with the subphase returns the system to equilibrium. For small departures from equilibrium, the Marangoni stress due to a soluble surfactant (3.89) is

where $E_{0}$ is the Marangoni modulus (2.22), and $\unicode[STIX]{x1D70F}_{s}$ is the characteristic sorption time (3.56). Using (3.116) in the Boussinesq–Scriven equation (3.94) gives

As noted previously in § 3.3.2, adsorption/desorption introduces an apparent dilatational surface viscosity $\unicode[STIX]{x1D705}_{s}^{ads}=E_{0}\unicode[STIX]{x1D70F}_{s}$ . Equation (3.117) highlights the pitfalls of inferring a ‘true’ dilatational surface viscosity of a soluble surfactant. Any measurement that is sensitive to surface-viscous dissipation due to pure compression/dilatation would at best report $\unicode[STIX]{x1D705}_{s}+E_{0}\unicode[STIX]{x1D70F}_{s}$ . An intrinsic dilatational surface viscosity $\unicode[STIX]{x1D705}_{s}$ , should it exist, can only be established with complementary measurements of $E_{0}$ and $\unicode[STIX]{x1D70F}_{s}$ .

Furthermore, (3.117) reveals the coupling between surface shear and dilatation in one-dimensional (1-D) stretching or compression, as occurs in plate coating or thin-film drainage (§ 4.4). The 1-D version of (3.117) corresponding to deformation along the $x$ -direction is

Surface-viscous dissipation in 1-D derives from a combination of surface shear viscosity, surface dilatational viscosity and the apparent viscous-like term $E_{0}\unicode[STIX]{x1D70F}_{s}$ . Surface shear viscosity inferred from such systems, therefore, is also prone to mischaracterization when $E_{0}\unicode[STIX]{x1D70F}_{s}\neq 0$ . This rate- and system-dependent viscous-like contribution arising from adsorption/desorption is responsible, at least in part, for widely dissimilar values (spread over four orders of magnitude) reported for the surface viscosity of the same soluble surfactant (Stevenson Reference Stevenson2005; Zell et al. Reference Zell, Nowbahar, Mansard, Leal, Deshmukh, Mecca, Tucker and Squires2014).

Non-dimensionalizing (3.118) gives

wherein $\unicode[STIX]{x1D6F9}$ reflects a general ‘degree of immobilization’ in 1-D systems:

Here, $Bq_{\unicode[STIX]{x1D702},\unicode[STIX]{x1D705}}$ are Boussinesq numbers defined separately for the intrinsic surface shear and surface dilatational viscosities. Written this way, the exchange Marangoni number $Ma_{K}=E_{0}\unicode[STIX]{x1D70F}_{s}/\unicode[STIX]{x1D702}L$ (table 3) can be interpreted as a modified Boussinesq number defined with the apparent surface dilatational viscosity, $\unicode[STIX]{x1D705}_{s}^{ads}=E_{0}\unicode[STIX]{x1D70F}_{s}$ , in place of an intrinsic surface viscosity.

The immobilization parameter $\unicode[STIX]{x1D6F9}$ controls the transition from a stress-free interface ( $\unicode[STIX]{x1D6F9}\rightarrow 0$ ) to a no-slip surface ( $\unicode[STIX]{x1D6F9}\rightarrow \infty$ ) in 1-D compression/dilatation. In the following sections, we will see this combination appear in the contexts of settling drops, coating flows, and foams. In each of these applications, a macroscopically measurable quantity (such as the velocity of a settling drop) depends on $\unicode[STIX]{x1D6F9}$ in a manner that does not differentiate between $\unicode[STIX]{x1D702}_{s}$ , $\unicode[STIX]{x1D705}_{s}$ and $E_{0}\unicode[STIX]{x1D70F}_{s}$ .

4.1.1 Surface immobilization due to surface viscosity

One of the first quantitative attempts to address the inconsistency between (4.2) and measurements came from Boussinesq (Reference Boussinesq1913), who hypothesized that a thin interfacial layer provides its own ‘surface viscous’ resistance. In writing the tangential surface stress balance, Boussinesq neglected surface tension gradients and instead introduced a surface viscosity

Here, $\boldsymbol{n}$ and $\boldsymbol{t}$ are the normal and tangent, respectively, to the drop surface, $[\![\unicode[STIX]{x1D748}]\!]$ is the stress jump across the interface and $\unicode[STIX]{x1D705}_{s}$ is an interfacial dilatational viscosity. Solving the boundary-value problem with (4.4) as the boundary condition, the terminal settling velocity of a drop is (Boussinesq Reference Boussinesq1913; Levich Reference Levich1962; Agrawal & Wasan Reference Agrawal and Wasan1979)

where

is a retardation coefficient due to surface dilatational viscosity. Thus, surface viscosity has an effect analogous to increasing the effective viscosity of the drop from $\unicode[STIX]{x1D702}^{\prime }$ to $\unicode[STIX]{x1D702}^{\prime }+2\unicode[STIX]{x1D717}/3$ .

The retardation coefficient $\unicode[STIX]{x1D717}$ has units of bulk shear viscosity, and can be argued based on excess energy dissipation. The energy dissipated in the drop via conventional shear viscosity scales like

where $\dot{\unicode[STIX]{x1D716}}$ is the strain rate within the fluid; the dissipation per unit volume then becomes

By extension, the retardation coefficient $\unicode[STIX]{x1D717}$ scales like the surface-excess energy $P_{s}$ dissipated by the surfactant, normalized by the volume of the drop

The Boussinesq number quantifies the relative strength of surface and bulk viscous stresses

and the drop settling speed (4.5) can be rewritten as

Surface viscosity is negligible when $Bq_{\unicode[STIX]{x1D705}}\ll 1$ , whereas the interface is effectively immobilized and the drop behaves like a rigid sphere when surface-viscous stresses dominate ( $Bq_{\unicode[STIX]{x1D705}}\gg 1$ ). Because $Bq_{\unicode[STIX]{x1D705}}$ decreases as $R$ increases, Boussinesq’s calculation reveals that bulk viscous stresses dominate over surface-viscous stresses for sufficiently large drops. For fixed $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D705}_{s}$ , therefore, the Hadamard–Rybczynski prediction improves with an increase in drop size.

4.1.2 Marangoni stress and adsorption/desorption

The Boussinesq correction due to surface viscosity does not explicitly account for the Marangoni flow associated with surface tension gradients. Levich (Reference Levich1962) accounted for convection, adsorption/desorption and diffusion in both the surface and the bulk. When drops or bubbles translate, the interfacial fluid motion advects adsorbed molecules to the rear (figure 15 c), thereby establishing a gradient in $\unicode[STIX]{x1D6E4}$ . This sets up reverse Marangoni flows, and the tangential stress balance at the interface becomes

The strength of Marangoni stresses depends on the speed with which gradients are established, compared with how quickly various processes can cause them to relax. For small departures from equilibrium surface coverage, the terminal velocity takes the same form as (4.5), but with a retardation coefficient $\unicode[STIX]{x1D717}$ that depends on the dominant surfactant transport process (Levich Reference Levich1962; Agrawal & Wasan Reference Agrawal and Wasan1979). In what follows, we outline the cases of adsorption and surface diffusion as the rate-limiting steps to illustrate the relative strength of Marangoni flows in immobilizing the interface.

The adsorption/desorption of soluble surfactants contributes an apparent dilatational surface viscosity (§ 3.4.3),

where $E_{0}=\unicode[STIX]{x1D6E4}_{0}|\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}|$ is the Marangoni modulus and $\unicode[STIX]{x1D70F}_{s}$ is the sorption time (3.56). The retardation coefficient in this case is identical to (4.6) but with $\unicode[STIX]{x1D705}_{s}^{app,ads}$ in place of an intrinsic dilatational surface viscosity $\unicode[STIX]{x1D705}_{s}$ (Levich Reference Levich1962)

The drop settles at a velocity

where the strength of Marangoni-induced retardation relative to bulk viscous drag defines the exchange Marangoni number (table 3)

Marangoni flows immobilize the interface ( $U\rightarrow U_{rigid}$ ) when surfactant exchange with the bulk fluid is slow ( $\unicode[STIX]{x1D70F}_{s}\gg \unicode[STIX]{x1D70F}_{m}=\unicode[STIX]{x1D702}R/E_{0}$ ), so that $Ma_{K}\gg 1$ . For the same fluid and surfactant properties, smaller drops (with $R\ll E_{0}\unicode[STIX]{x1D70F}_{s}/\unicode[STIX]{x1D702}$ ) settle like rigid spheres, whereas larger drops follow the Hadamard–Rybczynski prediction.

As discussed in § 3.2, the time scale $\unicode[STIX]{x1D70F}_{s}$ depends on bulk diffusion gradients if exchange kinetics at the interface are sufficiently fast ( $\unicode[STIX]{x1D70F}_{s}\approx \unicode[STIX]{x1D70F}_{d}$ , $\unicode[STIX]{x1D70F}_{k}\approx 0$ ). Additionally, if the bulk concentration $C$ is above the critical micelle concentration (CMC), micelles act as surfactant reservoirs that dissociate to maintain a constant monomer concentration. As bulk concentration gradients vanish, $\unicode[STIX]{x1D70F}_{s}\approx \unicode[STIX]{x1D70F}_{d}\rightarrow 0$ , and surface concentration gradients also disappear (figure 15 d). Micelles thus act to diminish reverse Marangoni flows and reduce adsorption/desorption-based retardation by ‘remobilizing’ the interface (Stebe & Maldarelli Reference Stebe and Maldarelli1994).

4.1.3 Marangoni stress and surface diffusion

Diffusion of surfactant molecules on the interface can also relax surface concentration gradients, particularly when the surfactant is insoluble. Surface diffusion acts against gradients in $\unicode[STIX]{x1D6E4}$ established by surfactant advection, and the resultant profile $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D6E4})$ dictates the strength of the Marangoni reverse flow. Solving for a dilute system that is slightly perturbed from equilibrium, Levich (Reference Levich1962) obtained (4.5) again, but now with the retardation coefficient

where $D_{s}$ is the surface diffusivity of the surfactant.

This retardation coefficient captures the work dissipated as surfactant molecules are forced along the interface by chemical potential gradients. The rate of work done by a surfactant molecule of mobility $b_{s}$ forced to translate along the interface at velocity $U$ is $U^{2}/b_{s}\sim \dot{\unicode[STIX]{x1D716}}^{2}R^{2}/b_{s}$ , and the surface-excess power $P_{s}$ dissipated by ${\sim}\unicode[STIX]{x1D6E4}R^{2}$ surfactants is

Following (4.9), the retardation coefficient $\unicode[STIX]{x1D717}_{sd}$ is then related to the dissipation $\unicode[STIX]{x1D6F7}$ per unit volume of the drop,

The term within the brackets is the 3-D-viscosity-like coefficient $\unicode[STIX]{x1D717}_{sd}$ associated with dissipation in this process. In the dilute limit, the Marangoni modulus $E_{0}$ and surface concentration $\unicode[STIX]{x1D6E4}$ are related via $\unicode[STIX]{x1D6E4}=E_{0}/k_{B}T$ , and the surface mobility and diffusivity are related via the Stokes–Einstein relation $D_{s}=k_{B}Tb_{s}$ . Substituting $\unicode[STIX]{x1D6E4}$ and $b$ in (4.19) recovers Levich’s form (4.17) for the retardation coefficient $\unicode[STIX]{x1D717}_{sd}$ .

The surface diffusion of insoluble surfactants thus modifies the droplet settling velocity to

where the Marangoni number $Ma_{D}$ (table 3) is

The interface is immobilized by reverse Marangoni flow ( $U\rightarrow U_{rigid}$ ) when the surface diffusive time scale $R^{2}/D_{s}$ is much longer than the Marangoni time scale $\unicode[STIX]{x1D702}R/E_{0}$ (or $Ma_{D}\gg 1$ ). By contrast, the surface is mobile if $Ma_{D}\ll 1$ , in which case surface diffusion relaxes surface concentration gradients quickly enough that Marangoni flows do not develop.

It is tempting to express $\unicode[STIX]{x1D717}_{sd}$ from (4.17) in terms of an apparent surface viscosity, following (4.6), giving

where $\unicode[STIX]{x1D70F}_{d}=R^{2}/D_{s}$ is the surface diffusive time scale. In this case, however, the apparent surface viscosity $\unicode[STIX]{x1D705}_{s}^{app,sd}$ depends on the size of the drop, and therefore clearly does not represent an intrinsic material property. By extension, the $R^{2}$ dependence of $\unicode[STIX]{x1D705}_{s}^{app,sd}$ differs from both intrinsic surface viscosity $\unicode[STIX]{x1D705}_{s}$ and as well as the apparent $\unicode[STIX]{x1D705}_{s}^{app}$ due to other dynamic surfactant processes like adsorption/desorption (4.13), suggesting experimental strategies to differentiate between the possible mechanisms.

4.1.4 Inferring retardation mechanisms from measurements

The three different mechanisms of surface stress relaxation considered here – surface viscosity, Marangoni reverse flow with adsorption/desorption, and Marangoni reverse flow with surface diffusion – all lead to drop settling velocities of the form

The immobilization parameter $\unicode[STIX]{x1D6F9}$ (§ 3.4.3) acts in way that resembles an increase in the shear viscosity of the drop. The drop settles like a rigid sphere when $\unicode[STIX]{x1D6F9}\rightarrow \infty$ , and like a clean drop when $\unicode[STIX]{x1D6F9}\rightarrow 0$ . Equation (4.23) qualitatively maps out the illustrative plot in figure 1, where $\unicode[STIX]{x1D6F9}$ is determined by the ‘hidden’ surfactant variable(s). More general immobilization parameters $\unicode[STIX]{x1D6F9}=\unicode[STIX]{x1D6F9}(Ma,Pe,Da,Bq_{\unicode[STIX]{x1D705}})$ arise when surfactant transport in the bulk becomes comparable to surface processes (Levich Reference Levich1962; Agrawal & Wasan Reference Agrawal and Wasan1979), but the form of (4.23) remains unchanged. Here, the Péclet number $Pe=UR/D$ compares advection and diffusion in the bulk, and the Damköhler number $Da=k_{C}R/D$ compares adsorption to bulk diffusion.

Experimentally measuring drop settling velocity $U$ alone cannot reveal exactly what the surfactant does to the interface. Equation (4.23) links a measured velocity to the parameter $\unicode[STIX]{x1D6F9}$ , which could be $Bq_{\unicode[STIX]{x1D705}}$ , some form of $Ma$ , a function of $Pe$ and $Da$ or some non-trivial combination. How can we, if at all, demarcate the specific process responsible for slowing the drop?

Since one can at best measure the lumped parameter $\unicode[STIX]{x1D6F9}$ , one approach is to exploit the scaling of $\unicode[STIX]{x1D6F9}$ with experimentally controllable quantities such as the bubble radius or surfactant concentration. If $\unicode[STIX]{x1D6F9}$ , interpreted from measurements via (4.23), increases linearly with drop size, then Marangoni flow with surface diffusion would be consistent as the dominant mechanism ( $\unicode[STIX]{x1D6F9}=Ma_{D}$ ). Alternatively, if experiments suggest $\unicode[STIX]{x1D6F9}\propto 1/R$ , surface viscosity ( $\unicode[STIX]{x1D6F9}=Bq_{\unicode[STIX]{x1D705}}$ ), adsorption-based retardation ( $\unicode[STIX]{x1D6F9}=Ma_{K}$ ) or both would be consistent. However, one might expect these two mechanisms to scale differently with bulk surfactant concentration $C$ : an intrinsic surface viscosity $\unicode[STIX]{x1D705}_{s}$ likely increases with $C$ or stays constant, whereas $\unicode[STIX]{x1D705}_{s}^{app,ads}$ likely decreases with $C$ , since adsorptive equilibration speeds up as more surfactant is added to the bulk.

4.2.1 Apparent oscillatory surface rheology

Lucassen & van den Tempel (Reference Lucassen and van den Tempel1972) computed the surface pressure $\unicode[STIX]{x1D6F1}(t)$ on a planar surfactant monolayer whose area $A(t)$ is forced to oscillate, and connected the monolayer response to a rate-dependent surface viscoelasticity. Figure 16 depicts the model system, consisting of a soluble monolayer at $z=0$ subjected to an oscillatory dilatational deformation at frequency $\unicode[STIX]{x1D714}$ (e.g. by moving barriers in a Langmuir trough), while surfactants adsorb/desorb over a time scale $\unicode[STIX]{x1D70F}_{s}$ .

As discussed in § 3.2, the molecular exchange of surfactant between the surface and the subphase can be diffusion- or kinetically limited. Lucassen & van den Tempel (Reference Lucassen and van den Tempel1972) focused on the diffusion-limited regime ( $Da\gg 1$ ), for which the sorption time scale $\unicode[STIX]{x1D70F}_{s}\approx \unicode[STIX]{x1D70F}_{d}\sim L_{d}^{2}/D$ . The equilibrium surface area, surface concentration and bulk concentration fields are perturbed via

where the disturbance is homogeneous along the interface, and surfactant is dissolved in the bulk fluid below the interface ( $z<0$ ). The bulk concentration $C$ satisfies the diffusion equation (3.26), with solution

Both real and imaginary components of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$ resist interfacial dilatation, and a convenient measure of the net resistance is the complex dilatational modulus

where $E_{0}=\text{d}\unicode[STIX]{x1D6F1}/\text{d}\ln \unicode[STIX]{x1D6E4}$ is the Marangoni modulus (§ 2.4.2). We will assume that $\unicode[STIX]{x1D6F1}(t)$ is in phase with and uniquely determined by $\unicode[STIX]{x1D6E4}(t)$ , so that $E_{0}$ is independent of $\unicode[STIX]{x1D714}$ . Note, however, that $\unicode[STIX]{x1D6E4}$ and $A$ may not vary precisely in phase: even monolayers of insoluble surfactant monolayers may relax to equilibrium over surprisingly long times, as discussed in § 5. Surfactant transport thus affects $E^{\ast }$ via the dynamic behaviour of $\unicode[STIX]{x1D6E4}(A)$ in (4.28). The real part of $E^{\ast }$ is the dynamic Gibbs modulus, whereas the imaginary or out-of-phase component captures the viscous-like dissipation associated with surfactant exchange.

Surfactant conservation at the interface with diffusion-limited adsorption (§ 3.2) requires

Contact equilibrium is assumed between the surface concentration $\unicode[STIX]{x1D6E4}(t)$ and the bulk concentration $C_{s}(t)=C(0,t)$ when diffusion controlled, so that

where $L_{p}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x2202}C_{s}$ is the depletion depth (3.47). Using (4.30) in (4.29) gives

Substituting perturbed variables (4.26) in (4.31) then gives the complex modulus (4.28):

where

is a dimensionless ratio of the diffusive boundary layer thickness $\unicode[STIX]{x1D6FF}_{BL}=\sqrt{D/2\unicode[STIX]{x1D714}}$ to the depletion depth $L_{p}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x2202}C_{s}$ , first introduced as (3.51) in § 3.2.

The apparent surface-elastic and surface-viscous moduli are the real and imaginary parts of $E^{\ast }$ , respectively

During ‘fast’ oscillations ( $\unicode[STIX]{x1D701}_{d}\ll 1$ ), surfactant molecules do not diffusively escape the depletion depth before the barrier reverses direction, so the interface behaves as insoluble, with $E\rightarrow E_{0}$ . By contrast, during ‘slow’ oscillations ( $\unicode[STIX]{x1D701}_{d}\gg 1$ ), the oscillatory boundary layer is thicker than the depletion depth ( $\unicode[STIX]{x1D6FF}_{BL}\gg L_{p}$ ). In this limit, dissolved molecules diffuse across the depletion depth during each oscillation and restore the sublayer concentration to $C_{0}$ before the barrier reverses. Molecules then adsorb/desorb much more quickly than the interface is compressed and dilatated, so that $\unicode[STIX]{x1D6E4}(t)\approx \unicode[STIX]{x1D6E4}_{0}$ and the Gibbs modulus vanishes. The apparent dilatational surface viscosity $\unicode[STIX]{x1D705}_{s}^{ads}$ is highest when $\unicode[STIX]{x1D701}_{d}=O(1)$ , i.e. when $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}\sim \unicode[STIX]{x1D70F}_{d}$ .

More generally, the complex dilatational modulus $E^{\ast }$ depends on the Damköhler number $Da$ (3.40). We will generalize $E^{\ast }$ for finite $Da$ and non-planar interfaces in the next section, but simply note that the qualitative trends of $E^{\ast }$ resemble those in figure 16(b), regardless of system geometry: $|E^{\ast }|\rightarrow E_{0}$ for slow adsorption/desorption (or rapid deformation), and $|E|\rightarrow 0$ for slow deformations.

4.2.2 Oscillating bubble tensiometry

Oscillating bubbles make excellent experimental probes of dilatational surface rheology (Ravera et al. Reference Ravera, Loglio and Kovalchuk2010; Kotula & Anna Reference Kotula and Anna2016). Indeed, § 3.2.1 illustrated kinetically and diffusion-limited surfactant exchange via oscillating bubbles. Here, we will follow Johnson & Stebe (Reference Johnson and Stebe1994) in examining the surface stresses that arise in oscillating bubbles, how they depend on adsorption/desorption, surface rheology and the subphase viscosity.

Alvarez et al. (Reference Alvarez, Walker and Anna2010a ) developed a microbubble microtensiometer, which simultaneously measures the radius and gas pressure of $O(100~\unicode[STIX]{x03BC}\text{m})$ bubbles. For a static bubble with equilibrated surface coverage, the surface tension (or surface pressure) is then determined via the Young–Laplace equation (2.6). Indeed, the equilibrium isotherm of insoluble monolayers can be mapped out by quasi-static compression of a spherical bubble (Kotula & Anna Reference Kotula and Anna2016).

To quantify the impact of surfactant adsorption/desorption, mass transport and surface rheology on dynamic bubbles, we return to the work of Johnson & Stebe (Reference Johnson and Stebe1994), first introduced in § 3.2.1, involving an oscillating air bubble of radius $R(t)$ , with dynamic surface concentration $\unicode[STIX]{x1D6E4}(t)$ , in a liquid containing dissolved surfactant at concentration $C(t)$ . Fluid pressure and radial velocity in both gas and liquid phases are perturbed as before (3.29), via

Solving the unsteady Stokes equations gives

in the liquid phase.

Assuming the gas viscosity to be negligible, the gas pressure is uniform throughout the bubble ( $p_{g}=p_{g}(R)$ ) and is determined by the stress balance at the interface (3.8). Assuming also that the adsorbed surfactant forms a Newtonian monolayer and simplifying the Boussinesq–Scriven stress tensor (3.93) for radial deformation gives (Scriven Reference Scriven1960):

The perturbed surface tension is

so that the leading-order interfacial stress balance (4.37) is the Young–Laplace equation

At $O(\unicode[STIX]{x1D6FF}R)$ , however, the pressure jump at the interface has contributions due to Young–Laplace from the radial change, the resistance to dilatation due to changes in adsorbed surfactant concentration (and therefore $\unicode[STIX]{x1D6F1}$ ), the viscous resistance from the bulk liquid and dilatational surface viscous resistance

where

is the surface concentration change in the insoluble limit (3.32). Recall from § 3.2.1 that $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$ in (4.40) is complex, with components both in and out of phase with $\unicode[STIX]{x1D6FF}R$ .

Non-dimensionalizing length and pressure in (4.40) using $R_{0}$ and the Laplace pressure $2\unicode[STIX]{x1D6FE}_{0}/R_{0}$ respectively gives

where

and the capillary, Marangoni and Boussinesq numbers are

Evaluating (4.42)–(4.44) requires the solution for $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{insol}$ , described in § 3.2.1 and given by (3.41) and reproduced here

where the Stanton, Womersley and Damköhler numbers are

Surfactant-free bubbles lack surface-viscous and surface-elastic stresses ( $Ma=Bq_{\unicode[STIX]{x1D705}}=0$ ). The interfacial stress balance (4.42) then recovers the linear limit of the classic Rayleigh–Plesset equation (Marmottant et al. Reference Marmottant, van der Meer, Emmer, Versluis, de Jong, Hilgenfeldt and Lohse2005)

The normal stress jump balances the perturbed Laplace pressure due to change in radius and the viscous stress from the liquid phase.

Bubbles with insoluble surfactants have surface concentrations that change in phase with the oscillations, giving $\text{Re}[\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}]=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{insol}$ and $\text{Im}[\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}]=0$ , so that (4.42) becomes

The pressure jump across the interface then balances additional surface stresses: the terms proportional to $Ma_{\unicode[STIX]{x1D6FE}}$ and $Bq_{\unicode[STIX]{x1D705}}Ca$ quantify the elastic and intrinsic surface-viscous resistance to interfacial dilatation/compression, respectively.

Surface-elastic resistance to bubble expansion decreases for soluble surfactants, however, since adsorption/desorption diminishes perturbations to surface concentration, and $\text{Re}[\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}]<\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}_{insol}$ . Solubility also introduces an out-of-phase component of surface concentration, so that $\text{Im}[\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}]\neq 0$ , giving rise to an additional surface-viscous-like stress as in § 4.2.1.

The dynamic Gibbs modulus $E$ and the apparent dilatational surface viscosity $\unicode[STIX]{x1D705}_{s}^{ads}$ are the real and imaginary parts of the complex modulus $E^{\ast }$ (§ 4.2.1). For small oscillatory perturbations around a spherical bubble, (4.28) gives

so that using (4.45) for $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$ gives

Surfactant exchange between the interface and subphase is kinetically limited when the bulk diffusion time $\unicode[STIX]{x1D70F}_{d}$ is much smaller than the kinetic time $\unicode[STIX]{x1D70F}_{k}$ (see § 3.2.1). In the kinetically limited ( $Da=\unicode[STIX]{x1D70F}_{d}/\unicode[STIX]{x1D70F}_{k}\rightarrow 0$ ) case, (4.50) becomes

The monolayer is effectively insoluble, and the Gibbs modulus $E$ approaches the Marangoni modulus $E_{0}$ , in the limit that adsorption kinetics are so slow that $\unicode[STIX]{x1D6E4}(t)$ does not change appreciably before the oscillatory cycle reverses (or $St\rightarrow 0$ ). By contrast, $\unicode[STIX]{x1D6E4}(t)$ is approximately constant if adsorption/desorption is fast relative to bubble oscillations ( $St\rightarrow \infty$ ), so that the monolayer provides no resistance to compression/dilatation. Bubble size does not affect $E_{kin}^{\ast }$ when transport is kinetically limited: the oscillatory mechanical response of a soluble planar monolayer (§ 4.2.1) also follows (4.51) in the kinetically limited regime $Da\rightarrow 0$ .

Alternatively, adsorption is diffusion limited when $\unicode[STIX]{x1D70F}_{d}\gg \unicode[STIX]{x1D70F}_{k}$ (or $Da\gg 1$ ), and surface concentration rapidly equilibrates with the subsurface concentration $C_{s}(t)=C(R,t)$ via an isotherm $\unicode[STIX]{x1D6E4}(C_{s})$ . In the diffusion-controlled limit ( $Da\rightarrow \infty$ ), (4.50) becomes

where $\unicode[STIX]{x1D6EC}_{d}=(\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x2202}C_{s})/R_{0}$ is the dimensionless depletion depth (3.47). The monolayer behaves as if it were insoluble, so that $E_{diff}^{\ast }\rightarrow E_{0}$ when diffusive transport is so slow that surfactants desorbed during bubble compression do not diffuse away before they readsorb during bubble expansion ( $Wo\rightarrow \infty$ ). By contrast, $\unicode[STIX]{x1D6E4}(t)$ remains nearly in equilibrium if the bubble oscillates much more slowly than required for diffusion ( $Wo\rightarrow 0$ ), in which limit the monolayer offers no resistance.

The real and imaginary parts of (4.52) correspond to the excess elastic and dissipative terms in (4.42). Separating these contributions and simplifying gives

where

is the ratio of the diffusive oscillatory boundary layer thickness to the depletion depth (3.51),

Bubbles with radii significantly larger than the depletion depth ( $\unicode[STIX]{x1D6EC}_{d}=L_{d}/R\ll 1$ ) recover the classic results of Lucassen & van den Tempel (Reference Lucassen and van den Tempel1972) for oscillatory compression of planar interfaces, presented in § 4.2.1.

In the examples so far, the subphase was assumed to be infinitely deep so that the bulk fluid always contained enough surfactant to adsorb on to the interface given enough time. However, the bulk fluid may be entirely depleted of surfactant if the subphase is shallow, as may occur in thin films, coating flows, foams and concentrated emulsions. The film thickness $h$ then ‘cuts off’ the depletion depth $L_{d}$ , and the film is said to be confined (Quéré Reference Quéré1999; Delacotte et al. Reference Delacotte, Montel, Restagno, Scheid, Dollet, Stone, Langevin and Rio2012). Surface concentration perturbations are not diminished by adsorption/desorption if $h\ll L_{d}$ , and the effective dilatational modulus is maximal (equal to the Marangoni modulus $E_{0}$ ) for confined flows.

4.3.1 Waves on a clean liquid surface

The vertical displacement $\unicode[STIX]{x1D701}$ of a surface due to plane waves of wavelength $\unicode[STIX]{x1D706}$ propagating along the $x$ direction (figure 17) can be written

where $k=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D714}$ are, respectively, the wavenumber and frequency. For small wave amplitudes, the nonlinear term in the Navier–Stokes equation is negligible (Levich Reference Levich1962; Probstein Reference Probstein1994) and the hydrodynamic governing equations are the unsteady Stokes equations.

Figure 17. A planar wave of amplitude $\unicode[STIX]{x1D701}_{0}$ , length $\unicode[STIX]{x1D706}$ and frequency $\unicode[STIX]{x1D714}$ .

Waves are driven by the balance between fluid inertia and the restoring forces of gravity and/or surface tension. The excess hydrostatic pressure due to the disturbance is

and the excess capillary pressure, following the Young–Laplace equation and assuming small curvatures, is

The ratio of hydrostatic and capillary pressures is the Bond number,

Short wavelengths ( $Bo\ll 1$ ) correspond to capillary waves ( $\unicode[STIX]{x1D706}\lesssim O(1~\text{cm})$ for water), whereas large wavelengths ( $Bo\gg 1$ ) correspond to gravity waves.

Inviscid flows are simplest, as they are irrotational in this configuration, and can therefore be described by a velocity potential

Linearity and continuity require

so that the linearized inviscid Navier–Stokes equation gives

approximating the interface to be at $z=0$ , with $O(\unicode[STIX]{x1D701}^{2})$ corrections. Imposing the kinematic condition $D\unicode[STIX]{x1D701}/Dt=v_{z}$ at the interface then gives

where $Bo$ is the Bond number (4.59). From the dispersion relation (4.63), the wave velocity is given by

with a minimum $c_{min}\approx 20~\text{cm}~\text{s}^{-1}$ for water.

Waves described by (4.63) neither grow nor decay when the fluid is inviscid and the interface is clean. However, waves decay in fluids with finite viscosity. The relative strength of wave damping is captured by the ratio of viscous stresses to inertial stresses,

where $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70C}$ is the kinematic viscosity. Based on the minimum phase velocity in water, $m<1$ for wavelengths larger than $\unicode[STIX]{x1D706}\gtrsim 0.1~\text{mm}$ (Levich Reference Levich1962). Viscous stresses are therefore typically weak relative to inertia, and it is safe to assume that the wave frequency departs only slightly from its inviscid value $\unicode[STIX]{x1D714}_{0}$ , so that

where $|\unicode[STIX]{x1D6FD}|\ll \unicode[STIX]{x1D714}_{0}$ gives a weak viscous damping. The negative real part of $\unicode[STIX]{x1D6FD}$ is the decay rate, and its imaginary part gives a small correction to the wave frequency due to fluid viscosity.

The non-irrotational nature of viscous flows necessitates a streamfunction $\unicode[STIX]{x1D713}$ in addition to the potential function $\unicode[STIX]{x1D719}$ . Solving for the velocity, pressure and $\unicode[STIX]{x1D6FD}$ based on $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D713}$ is tedious – see Levich (Reference Levich1962) for details – here we simply present the resultant damping rate,

The damping time scale $\unicode[STIX]{x1D6FD}_{clean}^{-1}$ can be interpreted as the time taken for vorticity generated by shear stresses to diffuse a depth comparable to the wavelength $\unicode[STIX]{x1D706}$ .

In what follows, we examine how surfactants modify the damping rate $\unicode[STIX]{x1D6FD}$ . Levich (Reference Levich1962) shows the effect of surfactants to be significant only when $\unicode[STIX]{x1D706}\lesssim 10~\text{cm}$ , and therefore affect capillary waves more prominently than gravity waves. We will therefore restrict our attention to capillary waves ( $Bo\ll 1$ ) in the following sections, and approximate $\unicode[STIX]{x1D714}_{0}\approx \sqrt{\unicode[STIX]{x1D6FE}k^{3}/\unicode[STIX]{x1D70C}}$ . As Franklin et al. (Reference Franklin, Brownrigg and Farish1774) noted over two centuries ago, large waves on a stormy sea are unaffected, whereas smaller ripples are smoothed out resulting in ‘glassy’ surfaces.

4.3.2 Marangoni damping due to insoluble and surface inviscid surfactants

Surfactants strengthen wave damping through two main mechanisms – Marangoni stresses and surface rheology – and we start with the former. The simplest case is a surface-inviscid monolayer, for which $\unicode[STIX]{x1D702}_{s}=\unicode[STIX]{x1D705}_{s}=0$ . Additionally, surface concentration gradients do not relax by adsorption/desorption in insoluble monolayers, in which case $E$ is equal to the Marangoni modulus $E_{0}$ (§ 2.4.2). Surface diffusion may also weaken the Marangoni effect by smoothing out gradients in $\unicode[STIX]{x1D6E4}$ , but we will neglect diffusion by assuming the time to diffuse the wavelength $\unicode[STIX]{x1D706}^{2}/D_{s}$ is much longer than the oscillation period $\unicode[STIX]{x1D714}_{0}$ .

The surface concentration is perturbed away from its equilibrium value $\unicode[STIX]{x1D6E4}_{0}$ via

where $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}\ll \unicode[STIX]{x1D6E4}_{0}$ . As with the vertical displacement $\unicode[STIX]{x1D701}(x,t)$ (4.66), we assume that subphase viscous effects are weak, so that the frequency is well approximated by its inviscid value $\unicode[STIX]{x1D714}_{0}$ , and that the damping rate $|\unicode[STIX]{x1D6FD}|\ll \unicode[STIX]{x1D714}_{0}$ . For small surface velocities, the surfactant conservation equation (3.10) gives

to leading order, so that the normal and tangential stress boundary conditions (3.8) for a surface-inviscid monolayer are

The capillary force in (4.70a ) drives the wave, whereas the Marangoni stress in (4.70b ) resists it. We therefore anticipate the damping rate $\unicode[STIX]{x1D6FD}$ to depend on the relative magnitudes of $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D6E4}_{0})$ and $E_{0}(\unicode[STIX]{x1D6E4}_{0})$ .

Solving the unsteady Stokes equations with boundary conditions (4.70) gives the fluid velocity and the damping rate $\unicode[STIX]{x1D6FD}$ . The calculation is tedious, but analytically tractable, in the limits of weak viscosity ( $m=\unicode[STIX]{x1D708}k^{2}/\unicode[STIX]{x1D714}_{0}\ll 1$ ) and small damping rate ( $|\unicode[STIX]{x1D6FD}|\ll \unicode[STIX]{x1D714}_{0}$ ), giving (Levich Reference Levich1962; Alpers & Hühnerfuss Reference Alpers and Hühnerfuss1989)

Here, the modified Marangoni number $Ma_{\unicode[STIX]{x1D6FE}}$ (table 3) quantifies the competition between interfacial area creation due to capillary forces and resistance to interfacial stretching due to Marangoni forces

Reverse Marangoni flows immobilize the interface when $Ma_{\unicode[STIX]{x1D6FE}}\gg 1$ or $E_{0}\gg \unicode[STIX]{x1D6FE}$ , in which case the interface behaves like an incompressible sheet with

By contrast, weak $E_{0}$ gives $Ma_{\unicode[STIX]{x1D6FE}}\ll 1$ and recovers the clean interface limit $\unicode[STIX]{x1D6FD}_{clean}$ (4.67). The ratio of the damping coefficient in the two limits is

Because wave speeds are greater than $c_{min}\approx 0.2~\text{m}~\text{s}^{-1}$ on water–air interfaces, $\unicode[STIX]{x1D6FD}_{stiff}>\unicode[STIX]{x1D6FD}_{clean}$ for waves with millimetre (or larger) wavelength. Typical capillary waves therefore decay more rapidly when surfactants are present.

Notably, the damping coefficient depends non-monotonically on $Ma_{\unicode[STIX]{x1D6FE}}$ (figure 18), with a maximum value $\unicode[STIX]{x1D6FD}_{max}\approx 2\unicode[STIX]{x1D6FD}_{stiff}$ that occurs at an intermediate Marangoni number, as reported in both experiments and simulations (Davies & Vose Reference Davies and Vose1965; Alpers & Hühnerfuss Reference Alpers and Hühnerfuss1989). Lucassen (Reference Lucassen1968) ascribes this maximum to a resonance-like mechanism between transverse capillary waves and longitudinal Marangoni waves. Damping is maximum when the wavelengths of these transverse and longitudinal waves are equal.

Figure 18. Capillary wave damping rate $\unicode[STIX]{x1D6FD}$  (4.71) as a function of the Marangoni number $Ma_{\unicode[STIX]{x1D6FE}}=E_{0}/\unicode[STIX]{x1D6FE}$ and the normalized fluid viscosity $m=\unicode[STIX]{x1D708}k^{2}/\unicode[STIX]{x1D714}_{0}$ . The damping rate asymptotes to $\unicode[STIX]{x1D6FD}_{clean}=-2m\unicode[STIX]{x1D714}_{0}$ when Marangoni flows are weak ( $Ma_{\unicode[STIX]{x1D6FE}}\ll 1$ ) and to $\unicode[STIX]{x1D6FD}_{stiff}=-(\unicode[STIX]{x1D714}_{0}/2)\sqrt{m/2}$ when the interface is immobilized ( $Ma_{\unicode[STIX]{x1D6FE}}\gg 1$ ). Waves decay faster in a more viscous fluid (larger $m$ ) for all $Ma_{\unicode[STIX]{x1D6FE}}$ .

Probstein (Reference Probstein1994) notes that $\unicode[STIX]{x1D6FD}_{stiff}$ can be estimated by treating the interface as an incompressible plate. The damping coefficient in general takes the form $\unicode[STIX]{x1D6FD}\sim -\unicode[STIX]{x1D708}k/d$ , where $d$ is viscous gradient length scale in the bulk. Without surfactants, the dissipation extends to a depth $d\sim k^{-1}$ , and $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{clean}\sim -\unicode[STIX]{x1D708}k^{2}$ in agreement with (4.67). When surfactants immobilize the surface via Marangoni flows, the viscous boundary layer under the surface is akin to that of an incompressible plate oscillating at frequency $\unicode[STIX]{x1D714}$ . This is Stokes’ second problem, revealing a viscous boundary layer that extends to a depth $d\sim (\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714})^{1/2}$ , with damping rate $\unicode[STIX]{x1D6FD}_{stiff}\sim -(\unicode[STIX]{x1D708}\unicode[STIX]{x1D714})^{1/2}k$ .

The enhanced damping mechanism is illustrated in figure 19. Fluid elements on and near a surfactant-free interface travel in circular trajectories. Surfactants drive Marangoni flows that act to immobilize the interface, distorting these circular trajectories. The change from near-vertical motion at the interface to circular trajectories in the subphase increases velocity gradients and viscous dissipation. Notably, damping still occurs due to bulk viscous dissipation: surface incompressibility due to reverse Marangoni flow modifies the bulk velocity field in a manner that amplifies viscous dissipation in the bulk (§ 3.3.2).

Figure 19. (a) Circular motion of interfacial fluid particles on the clean surface of a wave moving to the right. The solid blue arrows along the interface depict compression and expansion of the surface. (b) Surfactants distort circular trajectories via Marangoni flows (red dashed arrows) that oppose the compression and expansion of the interface. The trajectories of fluid elements are then distorted, becoming straight lines in the $Ma_{\unicode[STIX]{x1D6FE}}\rightarrow \infty$ limit.