2.1. Hydrodynamics

We assume that the film is thin enough so that gravitational forces are negligible but thick enough so that disjoining pressure arising from van der Waals interactions can be neglected. The balance of viscous and surface-tension stresses provides a characteristic horizontal velocity scale $U = \sigma _0 H^3/ \mu L^3$ , where $H$ is the initial mean film thickness, $L$ is a characteristic horizontal length scale and $\sigma _0$ is the solvent surface tension. The characteristic horizontal length scale $L$ can be obtained from a balance of viscous and solute Marangoni stresses

(2.1) \begin{align} \frac {\mu U}{H} \sim \frac {w_{\textit{solute}} \Delta \sigma }{L} \Rightarrow L \sim \frac {H}{\sqrt {\textit{Ma}}}. \end{align}

Here, $w_{\textit{solute}}$ is the initial solute mass fraction and $\Delta \sigma = \sigma _{0} - \sigma _{\textit{solute}}$ is the difference between the solvent and solute surface tensions. The solute Marangoni number $Ma = (w_{\textit{solute}} \Delta \sigma )/\sigma _0$ provides a measure of the strength of solute-induced surface-tension gradients relative to the solvent surface tension.

We assume that the ratio between the vertical and horizontal characteristic lengths $\epsilon = H/L \ll 1$ , allowing the use of the lubrication approximation (Oron, Davis & Bankoff Reference Oron, Davis and Bankoff1997). We non-dimensionalise variables using the following scalings:

(2.2) \begin{gather} x^* = x L, \ z^* = z \epsilon L, \ h^* = h \epsilon L, \ \sigma ^* =\sigma \sigma _0, \ u^* = u \frac {\sigma _0 \epsilon ^3}{\mu }, \nonumber \\ w^* = w \epsilon \frac {\sigma _0 \epsilon ^3}{\mu }, \ t^* = t \frac {L \mu }{\sigma _0 \epsilon ^3}, \ p^* = p \frac {\sigma _0 \epsilon }{L}, \ \phi ^* = \phi \ \phi _0, \ \varGamma ^* = \varGamma \ \varGamma _0 , \end{gather}

where $(u^*,w^*)$ is the velocity and $p^*$ is the pressure. The solute concentration $\phi ^*$ is scaled by an initial concentration $\phi _0$ , the interface surfactant concentration $\varGamma ^*$ is scaled by the initial surface surfactant concentration $\varGamma _0$ and the pressure $p^*$ is scaled by a capillary pressure obtained from the normal stress balance at the interface. The kinematic condition gives the scale $ L/U$ for time. Note that $Ma = \epsilon ^2$ due to the choice of lateral length scale in (2.1).

The lubrication approximation of the Navier–Stokes equations in two dimensions leads to a nonlinear evolution equation for $h(x,t)$

(2.3) \begin{equation} \frac {\partial h}{\partial t} = -\frac {\partial }{\partial x} \left [\frac {\partial ^3 h }{\partial x^3} \frac {h^3}{3}\right ] - \frac {\partial }{\partial x} \left [\frac {h^2}{2} \left ( \frac {1}{\textit{Ma}} \frac {\partial \sigma }{\partial x} \right ) \right ]\!. \end{equation}

Here, $\sigma (x,t)$ is the surface tension of the liquid. The velocity field is given by

(2.4) \begin{align} u &= - \frac {\partial ^3 h}{\partial x^3} \frac {z^2}{2} + \left ( \frac {1}{\epsilon ^2} \frac {\partial \sigma }{\partial x} + \frac {\partial ^3 h}{\partial x^3} h \right ) z , \end{align}

(2.5) \begin{align} w &= \frac {\partial ^4 h}{\partial x^4} \frac {z^3}{6} - \left [ \frac {1}{\epsilon ^2} \frac {\partial ^2 \sigma }{\partial x^2} + \frac {\partial }{\partial x} \left ( \frac {\partial ^3 h}{\partial x^3} h\right ) \right ] \frac {z^2}{2}. \end{align}

A detailed derivation of (2.3), (2.4) and (2.5) is provided in § S.1 of the supplementary material.

When surfactants are absent and the solute is sufficiently dilute, the surface-tension equation of state is linear in the solute concentration (Serpetsi & Yiantsios Reference Serpetsi and Yiantsios2012; Köllner et al. Reference Köllner, Schwarzenberger, Eckert and Boeck2013; Larsson & Kumar Reference Larsson and Kumar2021)

(2.6) \begin{align} \sigma ^* = \sigma _0 - \left .\left ( \frac {\partial \sigma ^*}{\partial \phi ^*} \right )\right |_{\phi ^* = 0} \ \phi ^*|_{z = h}. \end{align}

In non-dimensional form, (2.6) becomes

(2.7) \begin{align} \sigma = 1 - \textit{Ma} \ \phi |_{z = h} . \end{align}

In the experiments of Horiuchi et al. (Reference Horiuchi, Suszynski and Carvalho2015), ethanol was the solute of interest. For ethanol mass fractions $w_{\textit{solute}} \lt 0.1$ , a linear dependence of the surface tension on $\phi$ was previously found to be reasonably accurate (Vazquez, Alvarez & Navaza Reference Vazquez, Alvarez and Navaza1995) and is the simplest relationship for this dependence to probe physical mechanisms. We take $Ma \gt 0$ because most alcohols are weakly surface active and lower the surface tension of the liquid (Kahlweit, Strey & Busse Reference Kahlweit, Strey and Busse1991; Horiuchi et al. Reference Horiuchi, Suszynski and Carvalho2015), and this was the case studied by Larsson and Kumar in the absence of surfactants (Larsson & Kumar Reference Larsson and Kumar2021). The case $Ma \lt 0$ can occur in practice, for instance in the case of salts (Benouaguef et al. Reference Benouaguef, Musunuri, Amah, Blackmore, Fischer and Singh2021) and acrylic resins (Wilson Reference Wilson1993), but is not explored in this work.

When surfactants are present in dilute concentrations, the surface tension can be written as a linear function of both the solute and surfactant concentrations (Pham & Kumar Reference Pham and Kumar2019)

(2.8) \begin{equation} \sigma = 1 - \textit{Ma} \ \phi |_{z = h} - {\textit{Ma}}_{s} \varGamma . \end{equation}

Here, ${\textit{Ma}}_{s} = (w_{\textit{surfactant}} \Delta \sigma _s )/\sigma _0$ is a surfactant Marangoni number defined with respect to the surfactant mass fraction $w_{\textit{surfactant}}$ and the solvent–surfactant surface-tension difference $\Delta \sigma _s = \sigma _0 - \sigma _{\textit{surfactant}}$ .

Inserting (2.8) into (2.3) yields

(2.9) \begin{equation} \frac {\partial h}{\partial t} = -\underbrace {\frac {\partial }{\partial x} \left [\frac {\partial ^3 h }{\partial x^3} \frac {h^3}{3}\right ] }_{\text{capillary levelling}} + \underbrace {\frac {\partial }{\partial x} \left [\frac {h^2}{2} \frac {\partial \phi |_{z = h}}{\partial x}\right ] }_{\text{solute Marangoni flow}} + \underbrace {\frac {\partial }{\partial x} \left [ \frac {{\textit{Ma}}_{s}}{\textit{Ma}} \frac {h^2}{2} \frac {\partial \varGamma }{\partial x} \right ] }_{\text{surfactant Marangoni flow}} . \end{equation}

The first term on the right-hand side describes a contribution due to pressure-driven flows produced by interface curvature, or capillary levelling. The second and third terms, respectively, contain contributions of surface-tension gradients arising from spatial variations of solute and surfactant concentrations at the interface.

2.2. Solute transport

The transport of solute is governed by the convection–diffusion equation

(2.10) \begin{align} \frac {\partial \phi }{\partial t} + u \frac {\partial \phi }{\partial x} + w \frac {\partial \phi }{\partial z} = \frac {1}{\textit{Pe}} \frac {\partial ^2 \phi }{\partial x^2} + \frac {1}{Pe \: \epsilon ^2} \frac {\partial ^2 \phi }{\partial z^2} . \end{align}

Here, ${\textit{Pe}}= L U/D$ is the solute Péclet number, which gives a ratio of horizontal diffusive and convective time scales. Equation (2.10) is subject to no-flux boundary conditions at the substrate and a mass-conservation condition at the liquid–air interface

(2.11) \begin{align} \left .\frac {\partial \phi }{\partial z}\right |_{z = 0} = 0, \end{align}

(2.12) \begin{align} \left .\frac {\partial \phi }{\partial z}\right |_{z = h} - \epsilon ^2 \frac {\partial h}{\partial x} \left .\frac {\partial \phi }{\partial x}\right |_{z = h} = 0 . \end{align}

We note that the lubrication approximation has not been invoked in (2.10) and (2.12) in order to retain vertical solute concentration gradients, which are expected to play a key role in vertically stratified multilayer systems (Larsson & Kumar Reference Larsson and Kumar2021).

2.3. Insoluble surfactant

The surfactant is assumed to be insoluble in § 4 and its transport is governed in the lubrication limit by (Pereira & Kalliadasis Reference Pereira and Kalliadasis2008)

(2.13) \begin{equation} \frac {\partial \varGamma }{\partial t} + \frac {\partial \left ( u|_{z = h} \varGamma \right ) }{\partial x} = \frac {1}{{\textit{Pe}}_{s}} \frac {\partial ^2 \varGamma }{\partial x^2} , \end{equation}

where ${\textit{Pe}}_{s}= L U/D_s$ is the surfactant Péclet number at the interface. The first term of (2.13) describes the evolution of surfactant concentration with time, the second term describes convective transport of surfactant at the interface and the last term accounts for surfactant diffusion along the interface.

2.4. Soluble surfactant

We assume in this work that equilibrium relationships between the surface tension and species concentration hold at all times, and thus the surface tension at a given time $\sigma$ is the surface tension $\sigma _{eq}$ achieved under equilibrium conditions with respect to the surface concentration $\varGamma$ . It is also assumed that the surfactant concentration is sufficiently dilute so that micelles do not form.

For dilute surfactant concentrations, a Henry’s law isotherm relates surfactant concentration in the bulk and at the interface (Prosser & Franses Reference Prosser and Franses2001)

(2.14) \begin{equation} \varGamma ^* = K_H c^*|_{z = h} . \end{equation}

The surfactant’s efficiency of adsorption is described by $K_H$ , an equilibrium constant measuring the surface activity of a given surfactant in a solvent of interest. A flux $J^*$ describing deviation from equilibrium is obtained using the relation

(2.15) \begin{equation} J^* = k_{\textit{ads}}^H c^*|_{z = h} - k_{\textit{des}}^H \varGamma ^* , \end{equation}

where $k_{\textit{ads}}^H$ and $k_{\textit{des}}^H$ are first-order rate constants of adsorption and desorption, respectively. We note that $K_H = k_{\textit{ads}}^H/k_{\textit{des}}^H$ , consistent with the equilibrium condition $J^* = 0$ .

We assume equilibrium at $t = 0$ between the surfactant-laden liquid layer and the interface. The scale for the bulk concentration is determined by

(2.16) \begin{align} c^* = c \ c_{s,0} \text{ with } c_{s,0} = \frac {k_{\textit{des}}^H \varGamma _{0}}{k_{\textit{ads}}^H } , \end{align}

where $c_{s,0}$ denotes the initial subsurface surfactant concentration in the bulk (i.e. at $t = 0$ and at $z = h$ ). This scale assumes fast surfactant-exchange kinetics between the bulk and the interface relative to surfactant diffusion in the bulk, and an initially uniform surfactant concentration $c_{s,0}$ at the subsurface in equilibrium with surfactant at the interface.

With this scaling, the dimensionless form of (2.15) is

(2.17) \begin{align} J = Bi \left ( c|_{z = h} - \varGamma \right ) , \end{align}

where $Bi = k_{\textit{des}}^H L/U$ acts as a Biot number and provides a ratio between the time scale of the flow and the time scale of desorption.

Surfactant interface transport is governed by the equation

(2.18) \begin{equation} \frac {\partial \varGamma }{\partial t} + \frac {\partial \left ( u|_{z = h} \varGamma \right ) }{\partial x} = \frac {1}{{\textit{Pe}}_{s}} \frac {\partial ^2 \varGamma }{\partial x^2} + J , \end{equation}

where the added term $J$ , given by (2.17), accounts for the exchange flux due to adsorption–desorption kinetics. Note that (2.13) is recovered when $J = 0$ . The bulk surfactant concentration $c(x,z,t)$ is governed by the convection–diffusion equation

(2.19) \begin{equation} \frac {\partial c}{\partial t} + u \frac {\partial c}{\partial x} + w \frac {\partial c}{\partial z} = \frac {1}{{\textit{Pe}}_b} \frac {\partial ^2 c}{\partial x^2} + \frac {1}{{\textit{Pe}}_b \: \epsilon ^2} \frac {\partial ^2 c}{\partial z^2} , \end{equation}

where ${\textit{Pe}}_{b}= L U/D_b$ is the bulk surfactant Péclet number. The lubrication approximation was not invoked here in order to retain the effects of vertical surfactant concentration gradients in the bulk.

Boundary conditions enforce no flux at the substrate and a balance at the interface relating the diffusive flux of surfactant from the bulk to the interface through $J$

(2.20) \begin{align} \left .\frac {\partial c}{\partial z}\right |_{z = 0} &= 0 , \end{align}

(2.21) \begin{align} \left .\frac {\partial c}{\partial z}\right |_{z = h} - \epsilon ^2 \frac {\partial h}{\partial x} \left .\frac {\partial c}{\partial x}\right |_{z = h} &= -\epsilon ^2 {\textit{Pe}}_b \textit{Bi} \beta \left ( c|_{z = h} - \varGamma \right ) . \end{align}

Here, (2.21) is determined from Fick’s law with $J = D_{b}\ \boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{\nabla }c|_{z = h}$ (Kalogirou & Blyth Reference Kalogirou and Blyth2020), where $\boldsymbol{n} = (-\partial h/ \partial x, 1 )$ defines the outward unit vector normal to the interface under the lubrication approximation. The parameter $\beta = k_{\textit{ads}}^{H}/(k_{\textit{des}}^{H} H)$ is a solubility coefficient describing the strength of adsorption relative to desorption. Larger $\beta$ values correspond to less soluble surfactants, with the surfactant behaving as insoluble for $\beta \gg 1$ .

The insoluble-surfactant case is described by (2.9), (2.10) and (2.13), while the soluble-surfactant case is described by (2.9), (2.10), (2.18) and (2.19). Note that (2.9) and (2.13) are equivalent to some of the equations derived in Edmonstone, Craster & Matar (Reference Edmonstone, Craster and Matar2006) when the capillary velocity scale is chosen (i.e. $Ca = 1$ , where $Ca$ is the capillary number). Typical values for variables and parameters are presented in tables 1 and 2.

Table 1.

Order-of-magnitude estimates of important dimensional parameters.

Table 2.

Typical range of dimensionless parameters explored in this work using the dimensional parameter values from table 1.

2.5. Initial conditions

Motivated by prior work (Larsson & Kumar Reference Larsson and Kumar2021), we approximate the initial condition of the two-layer coating (figure 1) through two transition functions $T_{\textit{solute}}$ and $T_{\textit{surfactant}}$ , achieving the desired stratification between the component-rich and component-depleted layers

(2.22) \begin{align} T_{\textit{solute}}(z; h_b, s) &=\!\begin{cases}1, & \text{if } w \leqslant 0 \\ \tfrac {\exp (-1/(1-w))}{\exp (-1/(1-w)) + \exp (-1/w)}, & \text{if } 0 \lt w\lt 1 \ \ \text{where } w = \tfrac {z - (h_b - 1/s)}{2/s}, \\ 0, & \text{if } 1 \leqslant w \end{cases} \end{align}

(2.23) \begin{align} T_{\textit{surfactant}}(z; h_b, s) &=\!\begin{cases} 0, &\! \text{if } w \leqslant 0 \\ 1 - \tfrac {\exp (-1/(1-w))}{\exp (-1/(1-w)) + \exp (-1/w)}, & \!\text{if } 0 \lt w\lt 1 \ \ \text{where } w = \tfrac {z - (h_b - 1/s)}{2/s}. \\ 1, &\! \text{if } 1 \leqslant w \end{cases} \end{align}

Here, $s$ is the slope of the transition between layers of distinct composition and $h_b$ is the dimensionless bottom-layer thickness. We take $s= 3$ and $h_b = 0.5$ to match previously chosen values for the surfactant-free case (Larsson & Kumar Reference Larsson and Kumar2021). The stratification of solute and surfactant respectively achieved by (2.22) and (2.23) can be visualised in figures 2(a) and 2(b).

Figure 2.

Transition functions for the (a) solute and (b) surfactant approximating each component’s vertical stratification. The solute concentration field at $t = 0$ is shown in (c), and its perturbation amplitude is amplified in (d) from $\phi _p = 0.01$ to $\phi _p = 0.3$ so it can be seen more easily. The $x$ -coordinate is rescaled from $[0, 2\pi /\alpha ]$ to $[0,1]$ .

The solute transition function (2.22) models a film with a solute-depleted top layer and a solute-rich bottom layer. Consistent with prior work (Larsson & Kumar Reference Larsson and Kumar2021), we refer to this as a two-layer film even though solute diffusion homogenises the two-layer structure as time progresses. In the absence of a lateral perturbation, diffusion would occur evenly across the film height. To better understand the onset and growth of non-uniformities driven by solute Marangoni stresses, we follow prior work (Larsson & Kumar Reference Larsson and Kumar2021) and introduce a lateral perturbation in the solute concentration localised at $z = h_b$

(2.24) \begin{equation} \phi (x, z, t = 0) = T_{\textit{solute}} \left [ 1 + \phi _p \cos (\alpha x) \exp \left (- \frac {(z - h_b)^2}{2 \nu ^2} \right ) \right ] . \end{equation}

Here, $\phi _p = 0.01$ is the perturbation amplitude (assumed small), $\alpha = 0.3$ is its wavenumber and $\nu = 0.1$ is the standard deviation of the Gaussian localising the initial perturbation at a height $h_b$ from the substrate. This perturbation satisfies the periodic boundary conditions and corresponds to a depletion of solute at the centre of the film localised at the diffuse region between the two layers. Its localisation at $z = h_b$ allows us to more easily probe the influence of solute vertical diffusion on the system’s response to the lateral disturbance, which is shown in figures 2(c) and 2(d).

Initially, the fluid is at rest, the film is flat and the height is subject to the initial condition $h(x,t = 0) = 1$ . Surfactant in the insoluble case is uniformly deposited as a monolayer such that $\varGamma (x,t = 0) = 1$ . To qualitatively compare our model with the experiments of Horiuchi et al. (Reference Horiuchi, Suszynski and Carvalho2015), we also investigated a configuration which involves soluble surfactant through the initial conditions $\varGamma (x,t = 0) = 1$ and $c(x,z,t = 0) = T_{\textit{surfactant}}$ . This corresponds to a surfactant-depleted bottom layer, a surfactant-rich top layer and equilibrium conditions at $t = 0$ between the surfactant-rich top layer and the liquid–air interface.

2.6. Numerical method

We implement a coordinate transformation to numerically handle the moving-boundary problem by mapping $(x,z,t) \to (\chi , \eta , \tau )$ , where

(2.25) \begin{align} \frac {\partial \chi }{\partial x} = 1, \ \ \eta = \frac {z}{h}, \ \ \frac {\partial \tau }{\partial t} = 1. \end{align}

Scaling the $z$ -coordinate using the height leads to a fixed-coordinate domain where the interface remains at $\eta = 1$ . Further details are given in § S.2 of supplementary material. The transformed system of equations is solved on the $[0, 2\pi /\alpha ] \times [0, 1]$ domain via a pseudospectral method (Trefethen Reference Trefethen1996). Periodicity in $\chi$ warrants the use of Fourier modes for the $\chi$ domain, while a finite $\eta$ domain on $[ 0, 1]$ warrants the use of Chebyshev polynomials for that domain (Boyd Reference Boyd2001). Mesh independence was achieved for the examined conditions using 50 Fourier modes or less and 150 Chebyshev polynomials or less. We used ode15s for time integration, a variable-step, variable-order implicit solver built into MATLAB. Results were validated through comparisons with prior work in the absence of surfactants (Larsson & Kumar Reference Larsson and Kumar2021).

3.1. Vertical-averaging approximation

When $\epsilon ^2 \textit{Pe} \ll 1$ , vertical diffusion of the solute is rapid relative to convection. Vertical solute concentration gradients are averaged out on a much faster time scale compared with the time needed to achieve maximum deformation of the film. We employ a vertical-averaging approximation (VAA) exploiting this time-scale disparity through an asymptotic expansion in powers of $\epsilon ^2 Pe$

(3.1) \begin{align} \phi = \phi _b (x,z,t) + \epsilon ^2 \textit{Pe} \ \phi _p (x,z,t) + \mathcal{O}\left [ \left(\epsilon ^2 \textit{Pe}\right)^2 \right ]\!. \end{align}

After substituting (3.1) into the solute convection–diffusion equation, (2.10), and employing boundary conditions (2.11) and (2.12), we recognise that $\phi _b$ is independent of $z$ and obtain the governing equation

(3.2) \begin{equation} \frac {\partial \phi _b}{\partial t} + \overline {u} \frac {\partial \phi _b}{\partial x} = \frac {1}{h \ \textit{Pe}} \frac {\partial }{\partial x} \left ( h \frac {\partial \phi _b}{\partial x} \right )\!, \end{equation}

where

(3.3) \begin{equation} \overline {u} = \frac {1}{h} \int _0^h u {\rm d}z = \frac {1}{3} h^2 \frac {\partial ^3 h}{\partial x^3} - \frac {1}{2} h \left (\frac {\partial \phi _b}{\partial x} + \frac {{\textit{Ma}}_s}{\textit{Ma}} \frac {\partial \varGamma }{\partial x} \right )\!. \end{equation}

A detailed derivation is provided in § S.3 of supplementary material. Equations (2.9), (3.2) and (2.13) are solved using a one-dimensional version of the numerical method described in § 2.6 which employs a Fourier expansion of $\phi _b(x,t)$ , $h(x,t)$ and $\varGamma (x, t)$ for spatial discretisation and ode15s for time stepping. The solution is presented under the ‘VAA’ label in subsequent figures.

3.2. Linearised theory

When $\epsilon ^2 Pe \ll 1$ , vertical solute diffusion rapidly homogenises any initial vertical solute concentration gradients. By vertically averaging initial condition (2.24), we obtain

(3.4) \begin{align} \phi (x, t = 0) = \int _{0}^{1} \phi (x,z,t) {\rm d}z = 0.5 + \overline {\phi _p} \ \cos (\alpha x), \end{align}

where

(3.5) \begin{align} \overline {\phi _p} = \phi _p \int _0^1 T_{\textit{solute}} \exp \left [-\frac {(z - h_b)^2}{2 \nu ^2}\right ] {\rm d}z \approx 0.00125 \ll 1. \end{align}

When solute diffusion is sufficiently fast, convection can be neglected relative to lateral diffusion, reducing (2.10) to the diffusion equation

(3.6) \begin{align} \frac {\partial \phi }{\partial t} = \frac {1}{\textit{Pe}} \frac {\partial ^2 \phi }{\partial x^2}, \end{align}

where we have retained the transient term in the limit ${\textit{Pe}} \to 0$ by rescaling time using a diffusive time scale. Equation (3.6) is subject to the vertically averaged initial condition (3.4).

We use separation of variables to look for a solution of the form $\phi (x,t) = X(x) T(t)$ . Periodic boundary conditions in the lateral direction $x$ suggest a cosine-series expansion for the solute concentration

(3.7) \begin{equation} \phi (x,t) = \sum _{n = 0}^{\infty } a_n \exp \left (-\frac {n^2 \alpha ^2 t}{\textit{Pe}}\right ) \cos \left (n \alpha x \right )\!. \end{equation}

Initial condition (3.4) imposes $a_0 = 0.5$ and $a_1$ = $\overline {\phi _p}$ . Since higher-order terms decay exponentially fast, we substitute the leading-order solution $\phi (x,t) = 0.5 + \overline {\phi _p} \cos (\alpha x) \exp {(-\alpha ^2 t/Pe)}$ into (2.9) and (2.13).

Since $\overline {\phi _p} \ll 1$ , the remaining variables are expanded in powers of $\overline {\phi _p}$ as

(3.8) \begin{align} h = 1 + g(t) \overline {\phi _p} \cos (\alpha x) + \mathcal{O}[(\overline {\phi _p})^2], \end{align}

(3.9) \begin{align} \varGamma = 1 + q(t) \overline {\phi _p} \cos (\alpha x) + \mathcal{O}[(\overline {\phi _p})^2]. \end{align}

This expansion represents an initially uniform film thickness and surfactant concentration field with a small lateral perturbation.

Inserting (3.8) and (3.9) into (2.9) and (2.13) yields the $\mathcal{O}[\overline {\phi _p}]$ problem

(3.10) \begin{align} \frac {{\rm d}g(t)}{{\rm d}t} &= -\frac {1}{3} \alpha ^4 g(t) -\frac {\alpha ^2}{2} \left [ \exp \left ({-\frac {\alpha ^2 t}{\textit{Pe}}} \right ) + \frac {{\textit{Ma}}_{s}}{\textit{Ma}} q(t) \right ], \end{align}

(3.11) \begin{align} \frac {{\rm d} q(t)}{{\rm d}t} &= - q(t) \left [\alpha ^2 \frac {{\textit{Ma}}_{s}}{\textit{Ma}} + \frac {\alpha ^2 }{{\textit{Pe}}_{s}} \right ] -\frac {\alpha ^4}{2} g(t) - \alpha ^2 \exp \left (-\frac {\alpha ^2 t}{\textit{Pe}} \right )\!, \end{align}

which forms a linear system of constant-coefficient inhomogeneous ordinary differential equations. A detailed derivation along with some analytical results are provided in § S.4 of supplementary material. Solutions were obtained numerically using MATLAB’s ode15s with initial conditions $g(t) = q(t) = 0$ , and are presented under the ‘linearised theory’ label in subsequent figures.

4.1. Fast-diffusion regime

We first discuss the regime of relatively low Péclet numbers ( ${\textit{Pe}} \lt 10^4$ ), where solute diffusion is rapid enough to homogenise vertical solute concentration gradients on a much faster time scale relative to other phenomena. The initial perturbation imposed on the solute concentration profile (figure 2 c) diffuses to the solute-depleted liquid–air interface and causes a laterally inhomogeneous solute composition there. The centre of the film is depleted of solute relative to the edges and will correspond to a high-surface-tension zone (figures 3 a and 3 c). Surface-tension gradients set the liquid in motion at the initially flat interface (figure 3 b), exerting a pull toward the centre of the film. At early times, the surfactant concentration gradients at the interface are relatively weak (figure 3 d), and this solute Marangoni flow acts mostly uninhibited, easily deforming the film thickness and producing a prominent non-uniformity in the height (figure 3 b).

Figure 3.

Profiles of (a) surface tension $\sigma$ , (b) height $h$ , (c) interface solute concentration $\phi |_{z = h}$ and (d) interface surfactant concentration $\varGamma$ for the insoluble case at various times. Here, $Ma = 0.01$ , ${\textit{Pe}} = 5 \times 10^3$ , ${\textit{Pe}}_s = 10^3$ and ${\textit{Ma}}_{s} = 0.001$ . The surface tension and the interface solute concentration at $t = 0$ are horizontal lines at $0.999$ and $0$ , respectively, and were left out of (a) and (c) for clarity.

Solute Marangoni flow redistributes surfactant so that it is more concentrated at the centre of the film relative to the edges (figure 3 d). Since surfactant lowers the surface tension, surface-tension gradients arising from solute Marangoni effects are weakened by this surfactant redistribution. In addition, surfactant concentration gradients drive Marangoni stresses that oppose solute Marangoni stresses. Given that solute-driven convection is responsible for the deformation of the initially flat interface, surfactants resist the growth of film-height perturbations.

The competition between solute and surfactant Marangoni flows can be understood through the influence of each species on the change in height as described by (2.9). The two Marangoni contributions in (2.9) are of opposite signs given the opposite direction of their interface concentration gradients. We compare the relative magnitude of each Marangoni flow through the respective solute and surfactant contribution terms, $1/2 \ \partial _x [h^2 \ \partial _x(\phi |_{z = h})]$ and $1/2({\textit{Ma}}_{s}/Ma)\partial _x [h^2 \partial _x \varGamma ]$ , where $\partial _x = \partial /\partial x$ . We compute the maximum gradient in absolute value over all $x$ -coordinate nodes for each time $t$ using a fourth-order centred finite-difference scheme and plot the result versus time for different ${\textit{Ma}}_s$ in figure 4. As ${\textit{Ma}}_s$ increases, the maximum solute contribution decreases at any given time while the surfactant contribution increases.

Figure 5(a) shows the resulting monotonic decrease in $\Delta h$ , the maximum amplitude of the interface deformation, with increasing ${\textit{Ma}}_{s}$ . This behaviour is consistent with the suppression mechanisms just described. From figure 5(c), we observe that surfactant decreases $t_{\textit{max}}$ , the time at which the maximum perturbation is reached. At low ${\textit{Ma}}_s$ , increasing ${\textit{Pe}}$ leads to larger height deformations $\Delta h$ and higher $t_{\textit{max}}$ . This is because the lateral solute diffusive time scale increases with ${\textit{Pe}}$ and sustains solute concentration gradients at the interface for a longer time. This effect exacerbates solute Marangoni stresses and delays the erasure of solute concentration gradients by lateral diffusion, producing larger $\Delta h$ and leading to higher $t_{\textit{max}}$ . The film’s behaviour at high ${\textit{Ma}}_s$ , which appears independent of ${\textit{Pe}}$ , is discussed through a scaling argument in § 4.1.2.

Figure 4.

Time evolution of (a) solute and (b) surfactant contributions to the change in height (second and third terms in (2.9)) for various ${\textit{Ma}}_{s}$ with $Ma = 0.01$ , ${\textit{Pe}} = 5 \times 10^3$ and ${\textit{Pe}}_s = 10^3$ .

Figure 5.

Variation of (a) $\Delta h$ and (c) $t_{\textit{max}}$ with ${\textit{Ma}}_{s}$ for various Péclet numbers in the regime of fast solute diffusion. Comparison of (b) $\Delta h$ and (d) $t_{\textit{max}}$ between the linearised theory, VAA and 2-D model for ${\textit{Pe}} = 100$ . The other parameters are fixed at $Ma = 0.01$ and ${\textit{Pe}}_s = 10^4$ . Note that the range $0.1 \lt {\textit{Ma}}_{s} \lt 0.9$ was investigated to gain insight into asymptotic behaviour at very high surfactant mass fractions.

4.1.2. Asymptotic behaviour at high ${\textit{Ma}}_s$

From figures 5(a) and 5(b), we note $\Delta h \sim {\textit{Ma}}_{s}^{-1}$ when ${\textit{Ma}}_{s} \gt 0.01$ . This is consistent with the asymptotic behaviour of the function $g(t)$ in (3.10), which describes the time dependence of the height perturbation in the linearised theory. It is straightforward to show that in the limit $b = {\textit{Ma}}_s/Ma \gg 1$ (see § S.4 of supplementary material for details)

(4.2) \begin{align} \Delta h \sim b ^{-1} . \end{align}

The regime where the linear slope is observed in figure 5(a) corresponds to $b \geqslant 1$ . Scaling relation (4.2) reflects the fact that stronger surfactant Marangoni stresses at increasing ${\textit{Ma}}_{s}$ (larger $b$ ) lead to smaller height non-uniformities (smaller $g(t_{\textit{max}})$ and thus smaller $\Delta h$ ) that are more rapidly attained (smaller $t_{\textit{max}}$ , figures 5 c and 5 d).

Figure 5(a) also shows nearly overlapping $\Delta h$ plots at high ${\textit{Ma}}_s$ as ${\textit{Pe}}$ varies over two orders of magnitude. Scaling relation (S.73) in the supplementary material suggests that $\Delta h$ is relatively insensitive to ${\textit{Pe}}$ in the fast-diffusion regime when $b \gg 1$ , since

(4.3) \begin{align} \Delta h \sim \exp \left (-t_{\textit{max}} \frac {\alpha ^2}{\textit{Pe}}\right )\!. \end{align}

The argument of the exponential in (4.3) is responsible for the influence of ${\textit{Pe}}$ on $\Delta h$ . For typical values $\alpha = 0.3$ , $t_{\textit{max}} \leqslant 1000$ and $100 \leqslant Pe \leqslant 10^4$ , we obtain $ t_{\textit{max}} \alpha ^2/Pe \leqslant 0.9$ . In contrast, the effect of surfactant Marangoni stresses on $\Delta h$ is given by (S.74) of supplementary material as

(4.4) \begin{align} \Delta h \sim \exp \left (-b \alpha ^2 t_{\textit{max}}\right )\!. \end{align}

From figure 5(a), we note that surfactants exert a significant influence on $\Delta h$ when ${\textit{Ma}}_s \geqslant 0.001$ ( $b \geqslant 0.1$ ). In this regime, the exponential argument in (4.4) exceeds $9$ . This value is an order of magnitude larger than the exponential argument responsible for ${\textit{Pe}}$ effects ( $9/0.9 = 10$ ). We thus expect that variations in ${\textit{Pe}}$ will have negligible influence on the maximum height non-uniformity in this regime compared with ${\textit{Ma}}_s$ . We also note that (4.3) quickly approaches $0$ as ${\textit{Pe}} \to 0$ , revealing that $\Delta h$ is insensitive to ${\textit{Pe}}$ relative to other parameters when solute diffusion is sufficiently fast. This reasoning agrees with prior work on films devoid of surfactant (Larsson & Kumar Reference Larsson and Kumar2021) and generalises the dependence of $\Delta h$ on ${\textit{Pe}}$ to the surfactant-laden case.

Physically, scaling relation (4.3) can be interpreted as follows. In the low- ${\textit{Pe}}$ regime, vertical diffusion of solute is rapid. The initial lateral perturbation imposed on the solute and localised at $h_b = 0.5$ will diffuse to the interface on a much faster time scale compared with the time needed for Marangoni flows to develop laterally and drive a height non-uniformity. As a result, one-layer predictions are accurate despite treating vertical diffusion as instantaneous in the asymptotic limit of ${\textit{Pe}} \to 0$ and employing a vertically uniform initial condition that ignores the initial solute stratification altogether.

4.2. Slow-diffusion regime

4.2.1. Behaviour of $\Delta h$ and $t_{\textit{max}}$

Figure 6(a–d) shows the behaviour of $\Delta h$ and $t_{\textit{max}}$ with respect to ${\textit{Ma}}_s$ when ${\textit{Pe}} \gt 10^4$ . A monotonic decrease of $\Delta h$ with increasing ${\textit{Ma}}_{s}$ is observed (figures 6 a and 6 b), consistent with trends reported in the low- ${\textit{Pe}}$ regime (§ 4.1). In contrast, $t_{\textit{max}}$ tends to plateau at ${\textit{Ma}}_s \geqslant 0.01$ in the high- ${\textit{Pe}}$ regime (figures 6 c and 6 d).

Figure 6.

Variation of (a) $\Delta h$ and (c) $t_{\textit{max}}$ with ${\textit{Ma}}_{s}$ for various Péclet numbers in the regime of slow solute diffusion. Comparison of (b) $\Delta h$ and (d) $t_{\textit{max}}$ between the linearised theory, VAA and 2-D model for ${\textit{Pe}} = 8 \times 10^4$ . The other parameters are fixed at ${\textit{Pe}}_s = 10^4$ and $Ma = 0.01$ .

The behaviour at low ${\textit{Ma}}_s$ in the high- ${\textit{Pe}}$ regime can be understood from results obtained in previous work where surfactants were absent (corresponding to ${\textit{Ma}}_s=0$ ) (Larsson & Kumar Reference Larsson and Kumar2021). In this regime, solute diffusion is slow compared with the low- ${\textit{Pe}}$ regime, so vertical concentration gradients remain important for longer time and contribute to the developing height non-uniformity. As a consequence, the values of $\Delta h$ and $t_{\textit{max}}$ are more sensitive to the value of ${\textit{Pe}}$ at low ${\textit{Ma}}_s$ (figures 6 a and 6 c) compared with the low- ${\textit{Pe}}$ regime (figures 5 a and 5 c). Moreover, the values of $\Delta h$ at low ${\textit{Ma}}_s$ are generally larger in the high- ${\textit{Pe}}$ regime because weaker solute diffusion exacerbates lateral gradients in the solute concentration at the interface. This effect makes it easier for solute-induced Marangoni stresses to deform the interface as previously reported (Larsson & Kumar Reference Larsson and Kumar2021). Since the interface is initially solute depleted, weaker diffusion of solute at increasing ${\textit{Pe}}$ delays the onset of solute Marangoni stresses, which is reflected in the larger values of $t_{\textit{max}}$ as ${\textit{Pe}}$ increases (figures 6 b and 6 d). Additionally, since the interface deforms more as ${\textit{Pe}}$ increases, it takes longer to reach its maximum deformation, contributing to larger $t_{\textit{max}}$ at higher ${\textit{Pe}}$ . Note that the $t_{\textit{max}}$ values at low ${\textit{Ma}}_s$ and high ${\textit{Pe}}$ can actually be smaller than those in the low- ${\textit{Pe}}$ regime due to the convective steepening of lateral solute concentration gradients driven by solute Marangoni stresses (Larsson & Kumar Reference Larsson and Kumar2021).

As might be expected, predictions of the simplified models, which assume a vertically uniform solute concentration, deviate from the 2-D model predictions at low ${\textit{Ma}}_s$ in the high- ${\textit{Pe}}$ regime (figures 6 b and 6 d), in contrast to the excellent agreement observed in the low- ${\textit{Pe}}$ regime (figures 5 b and 5 d). However, the values of $\Delta h$ become relatively independent of ${\textit{Pe}}$ at high ${\textit{Ma}}_s$ (figure 6 a) and agree closely with predictions of simplified models in this regime (figure 6 c). This suggests that sufficiently strong surfactant Marangoni flows can help erase vertical solute concentration gradients and produce a quasi-one-layer structure, as discussed below.

4.2.2. Marangoni-induced recirculation

To better understand the high- ${\textit{Pe}}$ regime, we plot the velocity field and solute concentration contours in figures 7 and 8 for various parameter values. For reference, we first consider the low- ${\textit{Pe}}$ regime in figure 7. When $t \approx 0.5 \, t_{\textit{max}}$ (figure 7 a), rapid vertical diffusion has already eliminated steep vertical concentration gradients. The resulting horizontal concentration gradients drive Marangoni flow toward the film centre. When $ t \approx 1.5 \, t_{\textit{max}}$ (figure 7 b), the film has already achieved a maximum height deformation at $t_{\textit{max}}$ and is in the process of relaxing back to a flat state with flow directed away from the height peak at the film centre. Surfactant has already accumulated at the centre of the film by this time, so surfactant Marangoni flows and capillary levelling act to recover a flat film. The general behaviour shown here is representative of other values of ${\textit{Ma}}_s$ , with larger values of ${\textit{Ma}}_s$ leading to smaller values of $\Delta h$ and $t_{\textit{max}}$ .

Figure 7.

Velocity field plotted over solute concentration contours at (a) $t \approx 24 \approx 0.5 \, t_{\textit{max}}$ and (b) $t \approx 72 \approx 1.5 \, t_{\textit{max}}$ for ${\textit{Pe}} = 10^3$ . Other parameters are $Ma = 0.01$ , ${\textit{Ma}}_s = 0.01$ and ${\textit{Pe}}_s = 10^4$ . At $t \approx 0.8 \, t_{\textit{max}}$ , the velocity field looks very similar to (a) and is not shown for brevity.

Figure 8.

Velocity field plotted over solute concentration contours at ${\textit{Pe}} = 10^5$ , $Ma = 0.01$ and ${\textit{Pe}}_s = 10^4$ . Panels show (a) ${\textit{Ma}}_s = 0$ at $t \approx 374 \approx 0.5 \,t_{\textit{max}}$ ; (b) ${\textit{Ma}}_s = 0$ at $t \approx 599 \approx 0.8 \,t_{\textit{max}}$ ; (c) ${\textit{Ma}}_s = 0.01$ at $t \approx 101 \approx 0.5 \,t_{\textit{max}}$ ; (d) ${\textit{Ma}}_s = 0.01$ at $t \approx 160 \approx 0.8 \,t_{\textit{max}}$ .

Figure 9.

Velocity magnitude contours at ${\textit{Pe}} = 10^5$ and $Ma = 0.01$ when (a) ${\textit{Ma}}_s = 0$ and $t \approx 0.8 \, t_{\textit{max}}$ ( $t_{\textit{max}} \approx 751$ ), (b) ${\textit{Ma}}_s = 0$ and $t \approx 0.9 \, t_{\textit{max}}$ , (c) ${\textit{Ma}}_s = 0.01$ and $t \approx 0.8 \, t_{\textit{max}}$ ( $t_{\textit{max}} \approx 200$ ), (d) ${\textit{Ma}}_s = 0.01$ and $t \approx 0.9 \, t_{\textit{max}}$ .

We now consider the high- ${\textit{Pe}}$ regime in figure 8. In figures 8(a) and 8(b), we observe sharp vertical concentration gradients when ${\textit{Ma}}_s = 0$ . At the times shown, flow is directed toward the film centre, similar to the low- ${\textit{Pe}}$ regime (figure 7 a) but with a vertically stratified solute concentration. The picture is dramatically different for sufficiently large ${\textit{Ma}}_s$ , as seen in figures 8(c) and 8(d). Although there are significant vertical concentration gradients when $t \approx 0.5 \, t_{\textit{max}}$ , those vertical concentration gradients are rapidly smoothed out by the time $t \approx 0.8 \,t_{\textit{max}}$ . Additionally, while the flow is directed toward the film centre when $t \approx 0.5 \, t_{\textit{max}}$ , a recirculation region is seen when $t \approx 0.8 \, t_{\textit{max}}$ . This recirculation mixes the vertically stratified solute and erases its vertical concentration gradients. As a consequence, predictions of $\Delta h$ at sufficiently high ${\textit{Ma}}_s$ approach those obtained from simplified one-layer models (figure 6 b).

In figures 9(a) and 9(c), we show velocity magnitude contours corresponding to figures 8(b) and 8(d). When ${\textit{Ma}}_s = 0$ , the largest velocities are located at the liquid–air interface for $t \lt t_{\textit{max}}$ (figures 9 a and 9 b). As $t$ approaches $t_{\textit{max}}$ , the significant height deformation produced at high ${\textit{Pe}}$ results in large capillary stresses that resist further height deformation. Although this is not immediately apparent from figure 9(b), recirculation regions are present at $t \approx 0.9 \, t_{\textit{max}}$ , similar to what has been previously reported for surfactant-free systems at high ${\textit{Pe}}$ for times approaching $t_{\textit{max}}$ (Larsson & Kumar Reference Larsson and Kumar2021). However, when ${\textit{Ma}}_s = 0.01$ , the largest velocities are near the film centre, and flow at the liquid-air interface is nearly stagnant (figures 9 c and 9 d). This is due to surfactant-induced Marangoni stresses effectively resisting the solute-induced Marangoni stresses when the value of ${\textit{Ma}}_s$ is large, thereby rigidifying the liquid–air interface.

Figure 10.

Maximum magnitude of surface-tension gradients at the liquid–air interface over all $x$ and $t$ for various ${\textit{Ma}}_s$ at representative values of the low and high- ${\textit{Pe}}$ regimes. Here, ${\textit{Pe}}_s = 10^4$ , $Ma = 0.01$ and dashed lines represent the surfactant-free case ( ${\textit{Ma}}_s = 0$ ).

A rigid liquid–air interface corresponds to vanishing tangential motion. In figure 10, we plot the maximum magnitude of the surface-tension gradient over all time and space at low and high ${\textit{Pe}}$ . We observe that this maximum, which is proportional to the maximum tangential stress magnitude at the interface, decreases as ${\textit{Ma}}_s$ increases, but it does so more rapidly at high ${\textit{Pe}}$ . When ${\textit{Ma}}_s = 0.01$ and ${\textit{Pe}} = 10^5$ , the magnitude is very small and is given by $|\partial \sigma /\partial x | \approx 7 \times 10^{-5}$ (figure 10). Since this tangential stress is responsible for liquid flow along the surface but remains negligible at all times $t \lt t_{\textit{max}}$ , interfacial flows nearly halt. This restriction on interfacial motion at high ${\textit{Pe}}$ and high ${\textit{Ma}}_s$ prevents further lateral flow at the interface once surfactants have been sufficiently redistributed.

The mechanism for the formation of recirculatory flows at high ${\textit{Pe}}$ and high ${\textit{Ma}}_s$ can be understood through this rigidification of the interface. At early times, the surfactant concentration at the interface is nearly uniform, and surfactant Marangoni stresses are negligible. This means interfacial velocities develop freely due to solute-driven Marangoni convection. Flow at the free surface propagates into the bulk because a solute-induced Marangoni stress must be balanced by a shear stress in the liquid. When surfactant Marangoni flows become very strong at high ${\textit{Ma}}_s$ , a minor surfactant redistribution is sufficient to greatly suppress solute Marangoni stresses, thereby suppressing lateral motion at the interface. While the flow underneath, which is being dragged toward the film centre, is expected to deform the liquid-air interface, it fails to do so when the interface resists motion and is effectively rigid. As a result, lateral flow is redirected downward at the centre of the film and fluid at the edges will flow upward to conserve mass in the liquid film, giving rise to recirculation regions. Surfactants can thus precipitate the formation of recirculation regions at high ${\textit{Pe}}$ in films with initial vertical gradients in the solute concentration.

Predictions of $t_{\textit{max}}$ from simplified models in figure 6(d) exhibit similar behaviour to the low- ${\textit{Pe}}$ regime (figure 5 d), given that these models ignore vertical concentration gradients and model the system as a one-layer film. In contrast, the 2-D model predicts smaller values of $t_{\textit{max}}$ at low ${\textit{Ma}}_s$ compared with the simplified models because of the convective steepening effect discussed by Larsson & Kumar (Reference Larsson and Kumar2021). This effect is recovered in the limit of ${\textit{Ma}}_s \to 0$ where surfactant Marangoni stresses are non-existent. As ${\textit{Ma}}_s$ increases, surfactants decrease the maximum height non-uniformity via mechanisms similar to those described in § 4.1 and help achieve $t_{\textit{max}}$ more rapidly. Note that since the interface becomes effectively rigid at sufficiently high ${\textit{Ma}}_s$ , the values of $t_{\textit{max}}$ reach a plateau at ${\textit{Ma}}_s = 0.01$ , as seen in figure 6(c). While $t_{\textit{max}}$ becomes insensitive to ${\textit{Ma}}_s$ in this regime, it remains sensitive to ${\textit{Pe}}$ . Solute must first reach the interface before a height deformation takes place and before the surfactant gets redistributed by solute Marangoni flows. The rate-limiting step for these processes is the slow vertical diffusion of solute, which is controlled by ${\textit{Pe}}$ . Higher ${\textit{Pe}}$ values delay interface rigidification and lead to large $t_{\textit{max}}$ as a result (figure 6 c).

We have also examined the influence of the surfactant Péclet number ${\textit{Pe}}_s$ on $\Delta h$ and found its effect to be negligible. A reversal in the height deformation at $t \gt t_{\textit{max}}$ can take place when surfactant diffusion is slow relative to solute diffusion (high ${\textit{Pe}}_s$ , low ${\textit{Pe}}$ ) because surfactant concentration gradients at the interface persist and generate Marangoni stresses that overwhelm those of the solute. Further details are provided in § S.5 of the supplementary material.

4.3. Connection to experimental observations

Although there are several differences between the model developed in this work and the experiments conducted by Horiuchi et al. (Reference Horiuchi, Suszynski and Carvalho2015), including but not limited to their use of two distinct solvents (water and glycerol) for the individual layers and the relatively concentrated SDS solutions they deposited, qualitative comparisons can provide valuable insights into underlying physical mechanisms. They found decreasing film-height non-uniformities at increasing surfactant mass fractions. We have found in this work that increasing ${\textit{Ma}}_s$ can significantly improve the uniformity of multilayer coatings by suppressing solute-driven Marangoni flows, which is reflected in smaller $\Delta h$ at increasing ${\textit{Ma}}_s$ observed in our simulations. Since ${\textit{Ma}}_s \sim w_{\textit{surfactant}}$ in our model (table 2), our findings are qualitatively consistent with the observations of Horiuchi et al., The lack of a non-uniformity metric in their experiments renders a quantitative comparison with our model difficult. However, the significant decrease in $\Delta h$ with increasing ${\textit{Ma}}_s$ we have observed suggests that non-uniformities can decrease and cease to induce dewetting through the mechanisms we have delineated when sufficiently high surfactant mass fractions are used.

5.1. Effects of $Bi$ and $\beta$

Surfactant solubility is characterised by two parameters that we vary here: $Bi$ and $\beta$ . The Biot number $Bi = k_{\textit{des}}^{H} L / U$ measures the strength of surfactant desorption relative to convection. The solubility coefficient $\beta = k_{\textit{ads}}^{H}/ (k_{\textit{des}}^{H} H )$ measures the strength of surfactant adsorption relative to desorption.

Figure 11(a) shows the sensitivity of the maximum height deformation $\Delta h$ to $Bi$ . In the limit of low $Bi$ , $\Delta h$ approaches a plateau close to its predicted value for insoluble surfactants. Surfactant desorption is weak in this regime because $Bi \sim k_{\textit{des}}^H$ . As a result, surfactant concentration gradients can develop at the interface and produce strong surfactant Marangoni stresses. As $Bi$ increases, faster surfactant desorption weakens the associated Marangoni stresses, leading to an increase in $\Delta h$ . For high values of $Bi$ , $\Delta h$ approaches a plateau close to the value for the no-surfactant case.

Figure 11.

Solubility effects measured through variations in $\Delta h$ as a function of (a) $Bi$ and (b) $\beta$ . We fix $\beta = 1$ in (a) and $Bi = 1$ in (b). Remaining parameter values are $Ma = 0.01$ , ${\textit{Ma}}_s = 2 \times 10^{-3}$ , ${\textit{Pe}}_b = {\textit{Pe}}_s = 10^4$ and ${\textit{Pe}} = 1.5 \times 10^5$ .

Note that the plateaus seen in figure 11(a) are offset from the limiting cases shown by the dashed lines. The exchange flux $J$ is small at low values of $Bi$ but is not negligible. For $\beta = 1$ and ${\textit{Pe}}_b = 10^4$ , the contribution $ ( c|_{z = h} - \varGamma )$ in (2.17) grows with time, causing loss of surfactant mass from the interface into the bulk. This effect decreases the amount of surfactant available at the interface over time and weakens surfactant Marangoni effects compared with the insoluble case, leading to the offset as $Bi \to 0$ (figure 11 a). The variation of $\Delta h$ with respect to $Bi$ in figure 11(a) is observed for a wide range of $\beta$ and ${\textit{Pe}}_b$ values, with offsets from the asymptotic limits $Bi \to 0$ and $Bi \to \infty$ that depend on surfactant solubility ( $\beta$ ) and the bulk surfactant diffusion time scale ( ${\textit{Pe}}_b$ ). When $\beta \geqslant 100$ and $Bi \leqslant 0.01$ (not shown), the height non-uniformity is within $2\,\%$ of the insoluble-surfactant case so that the offset from the insoluble limit is smaller than in figure 11(a).

At $Bi = 1$ , surfactant Marangoni effects are strong enough to suppress the height deformation by nearly an order of magnitude (figure 11 a). We fix $Bi = 1$ in figure 11(b), and vary $\beta$ to better understand the role of surfactant solubility on the growth of the height non-uniformity. In the limit of low surfactant solubility (high $\beta$ ), surfactants exhibit high affinity for the interface and strongly suppress $\Delta h$ due to a reduction in the associated Marangoni stresses. The value of $\Delta h$ approaches a plateau close to the value for insoluble surfactants. Note that because $Bi= 1$ in these calculations, there is still significant surfactant desorption relative to convection, so the plateau is offset from the insoluble-surfactant value. As $\beta$ decreases, the surfactant becomes more soluble in the liquid bulk, which weakens surfactant Marangoni stresses and increases $\Delta h$ . In the limit of high surfactant solubility (low $\beta$ ), the height non-uniformity approaches the surfactant-free limit.

5.2. Effect of ${\textit{Pe}}_b$

Figure 12.

(a) Variation of $\Delta h$ with ${\textit{Pe}}_b$ for soluble surfactant. (b) Variation of the subsurface surfactant concentration (at $z = h$ ) with time at fixed $x = 0.5$ (centre of the film). (c) Surfactant concentration at the interface at $t_{\textit{max}}$ . All other parameters in (a–c) are fixed at $Ma = 0.01$ , ${\textit{Pe}} = {\textit{Pe}}_s = 10^4$ , $\beta = 1$ , $Bi = 0.01$ and ${\textit{Ma}}_s = 0.001$ .

To investigate the effect of surfactant diffusion in the liquid bulk on the height non-uniformity, we vary ${\textit{Pe}}_b$ . Figure 12(a) shows that $\Delta h$ decreases as the diffusion of surfactant in the bulk slows down (increasing ${\textit{Pe}}_b$ ). Slower bulk diffusion strengthens surfactant Marangoni stresses and leads to greater height suppression. We note that the calculations become increasingly stiff as ${\textit{Pe}}_b \to 0$ (due to fast surfactant diffusion) and ${\textit{Pe}}_b \to \infty$ (due to steep concentration gradients). However, results from calculations we have performed for other parameter regimes (not shown here) indicate that $\Delta h$ should reach plateaus as ${\textit{Pe}}_b \to 0$ and ${\textit{Pe}}_b \to \infty$ . These plateaus may be offset from the limiting cases shown by the dashed lines in figure 12 depending on the values of $Bi$ and $\beta$ (see § 5.1).

To better understand the impact of ${\textit{Pe}}_b$ on surfactant-induced surface-tension gradients, we plot in figure 12(b) the subsurface concentration of surfactant $c(x,z = h, t)$ over time at the centre of the film. At high ${\textit{Pe}}_b$ ( $10^5$ ), surfactant diffusion within the liquid bulk is relatively slow, so it can maintain a high surfactant concentration at the subsurface. Slow vertical diffusion of surfactant in the film will aim to replenish the surfactant-depleted bottom layer but will fail to rapidly deplete the initially surfactant-rich top layer at high ${\textit{Pe}}_b$ . As a consequence, the subsurface region remains sufficiently enriched in surfactant so that it can sustain a high surfactant concentration at the interface through the rapid surfactant-exchange equilibrium. As surfactant accumulates at the centre of the interface due to solute Marangoni flows, it remains concentrated there for extended time and develops substantial lateral concentration gradients. Surfactants become more effective at suppressing surface-tension gradients that arise from the solute and can better counteract solute Marangoni flows when ${\textit{Pe}}_b$ is large, which helps achieve smaller $\Delta h$ (figure 12 a).

In contrast, figure 12(b) shows the subsurface is rapidly depleted of surfactant at lower ${\textit{Pe}}_b$ ( $10^2$ ). The fast diffusion of surfactant in the liquid bulk strips surfactant away from the interface and produces a depletion region at the subsurface. Figure 12(b) shows that $c(x = 0.5,z = h, t)$ sharply drops and remains at a vertically averaged concentration of $c \approx 0.5$ in the low- ${\textit{Pe}}_b$ regime. Surfactant in the liquid diffuses to the surfactant-depleted bottom layer because of vertical concentration gradients imposed by the initial stratification of the surfactant. The surfactant-rich regions of the interface that develop over time will experience rapid surfactant desorption into the depletion zone to re-establish the surfactant-exchange equilibrium with the subsurface (since $c(x,z = h, t) \approx 0.5 \lt 1$ ). As a result, lateral gradients in the surfactant concentration at the interface are rapidly erased as they form and fail to become substantial. Figure 12(c) reflects this through the contrast between the nearly uniform surfactant concentration at the interface at $t_{\textit{max}}$ for sufficiently small ${\textit{Pe}}_b$ and the prominent lateral gradients that exist at sufficiently large ${\textit{Pe}}_b$ .