3.1 Particle tracking velocimetry

Figure 2.

Particle tracking velocimetry to obtain the mean velocity of the tracked particles $v$ at the centre of the film as a function of time $\unicode[STIX]{x1D70F}$ ($v$ is positive when along gravity). The three images show the tracked particles, while the fields of view are at the centre of the pre-wetted film and are cropped for display. The mean velocity $v$ is calculated by averaging the tracked particle velocities across the field of view, which is $30~\text{mm}$ below the top of the film. Water with $6~\unicode[STIX]{x03BC}\text{m}$ diameter PMMA particles is injected onto the glass slide to form a pre-wetted film. The fluid injection stops at $\unicode[STIX]{x1D70F}=0$. The pre-wetted film then drains by gravity ($\unicode[STIX]{x1D70F}\lesssim 11.5~\text{s}$), and the particles move downwards following the drainage, so that $v>0$ but decreases with time as the film thins. The mean velocity decreases dramatically while the climbing front passes the field of view ($11.5~\text{s}\lesssim \unicode[STIX]{x1D70F}\lesssim 13~\text{s}$), and the particles near the air–water interface reverse direction and move upwards against gravity, following the climbing film (the SDS concentration in the bath $c_{0}=1\times 10^{-3}~\text{M}$). The mean velocity then becomes negative, indicating that the mean fluid velocity along the film reverses and turns upwards, against the direction of gravity. After encountering the climbing front ($\unicode[STIX]{x1D70F}\gtrsim 13~\text{s}$), the particles near the interface continuously rise.

To quantify the flow details in both the pre-wetted film and the climbing film, PTV was performed, where the processed data are shown in figure 2. Spherical poly(methyl methacrylate) (PMMA) particles (from Microbeads AS), diameter $d=6~\unicode[STIX]{x03BC}\text{m}$, were added to deionized water. Note that the diameter of the PMMA particles, which were used to track the fluid velocity in thick films, is usually much smaller than the film thickness (up to around 20 times smaller). For the situation that the film thickness is of the same order of magnitude as the particle diameter, the particles would cease to trace the local flow and may even adhere to the substrate and remain stationary. Also, the sedimentation velocity of the PMMA particles in water, as estimated by $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}gd^{2}/18\unicode[STIX]{x1D707}$, where $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}\approx 2\times 10^{2}~\text{kg}~\text{m}^{-3}$ is the density difference between PMMA particles and water, $g$ is gravitational acceleration and $\unicode[STIX]{x1D707}$ is the dynamic viscosity of water, is approximately $4~\unicode[STIX]{x03BC}\text{m}~\text{s}^{-1}$ and thus is negligible relative to the draining and climbing speeds, which are of the order of a millimetre per second. The water with PMMA particles was injected onto a glass slide to form a pre-wetted film, and the top of the film was set $60~\text{mm}$ above the bottom of the glass slide, as sketched in figure 1(b). The glass slide with the film was placed between a high-speed camera and a light-emitting diode light panel. The centre of the pre-wetted film, which was $30~\text{mm}$ below the top of the film, was recorded at 100 frames per second with a field of view of $2.3~\text{mm}\times 2.3~\text{mm}$ (see the PTV movie in supplementary movie 2).

The PMMA particles in the field of view are tracked and velocities are thus measured, using a modified code, which was originally developed by Blair & Dufresne (Reference Blair and Dufresne2008). Note that the tracked particles in the film may be either near the air–liquid interface or near the glass substrate; thus the tracked particle velocity depends on the position in the film. Whether the particles are near the air–liquid interface or the substrate can be determined qualitatively by the degree to which they are in focus or out of focus in the image. Typically, hundreds of particles across all of the entire thickness of the film are tracked in the field of view, and the mean velocity of the particles, $v$, is calculated by averaging the velocities of the tracked particles. Hence, assuming that averaging the velocity of the particles is equivalent to the average of the velocity across the full thickness of the film, $v$ represents the mean flow in the film and changes with $\unicode[STIX]{x1D70F}$ during the film climbing (figure 2, $v$ is positive along the direction of gravity). In figure 2, the drainage time interval is $\unicode[STIX]{x1D70F}_{0}\approx 10~\text{s}$, which is the time interval from the end of the water injection ($\unicode[STIX]{x1D70F}=0$) to the contact of the glass slide with the bath ($t=0$).

Before the bath contacts the glass slide, the pre-wetted film on the glass slide drains freely. The particles move downwards following the gravitational film drainage (figure 2, $\unicode[STIX]{x1D70F}\lesssim 11.5~\text{s}$); $v$ decreases with $\unicode[STIX]{x1D70F}$, on account of the pre-wetted film thickness decreasing due to drainage. As the bath, which contains $1\times 10^{-3}~\text{M}$ SDS solution, contacts the glass slide, a liquid film rises upwards along the pre-wetted film. The climbing front then passes through the field of view (centre of the pre-wetted film) where the velocities of the particles are significantly decreased (figure 2, $11.5~\text{s}\lesssim \unicode[STIX]{x1D70F}\lesssim 13~\text{s}$). The particles near the air–liquid interface move upwards, following the climbing film, while the particles near the glass substrate move downwards, following the gravitational drainage. As more particles follow the climbing film, the mean velocity of the particles decreases and becomes negative, indicating that the mean fluid velocity reverses and opposes gravity. After encountering the climbing front and even after the climbing front reaches the top of the pre-wetted film, the particles near the air–liquid interface rise continuously, with a higher magnitude of velocity than the particles near the substrate, which move downwards following the drainage (figure 2, $\unicode[STIX]{x1D70F}\gtrsim 13~\text{s}$), since at this time the Marangoni stress is stronger than the gravitational drainage. The details of the rising and falling flows depend on parameters such as the film thickness and the surface tension difference between the pre-wetted film and the bath. The effects of the surfactant and the pre-wetted film on the climbing film will be the focus of the next sections.

3.2 Interferometry: drainage

Figure 3.

The pre-wetted film on the vertical substrate during the drainage. (a) A schematic of the pre-wetted film during drainage. The glass slide is fixed vertically as a substrate, and approximately $3~\text{ml}$ deionized water is injected onto the glass slide ($60~\text{mm}$ above the bottom of the slide) to form a pre-wetted film. The film thickness at the centre ($L_{0}=30~\text{mm}$) of the film is measured by interferometry. (b) The film thickness at the centre of the pre-wetted film $h_{0}$ during the drainage, as a function of the drainage time $\unicode[STIX]{x1D70F}$. The black solid line denotes our experimental measurements and the blue dashed line denotes Jeffreys’ similarity solution $h_{0}(\unicode[STIX]{x1D70F})=(\unicode[STIX]{x1D707}L_{0}/\unicode[STIX]{x1D70C}g\unicode[STIX]{x1D70F})^{1/2}$, equation (3.1). The value $\unicode[STIX]{x1D70F}=0$ denotes the time that the injection ends. The error bars correspond to standard deviations at $\unicode[STIX]{x1D70F}=5,~10,~20$ and $40~\text{s}$ (experiments repeated four times), which are typical values of drainage time interval $\unicode[STIX]{x1D70F}_{0}$ for the later controlled experiments. The inset shows a plot of $h_{0}(\unicode[STIX]{x1D70F})$ with logarithmic axes.

The shape of the pre-wetted film that contacts the bath is set by gravitational drainage. The draining of a thin liquid film on a vertical plate was first systematically analysed by Jeffreys (Reference Jeffreys1930), who considered a fixed contact line. Applying the lubrication approximation, and balancing the gravitational force on the liquid and the viscous force, a similarity solution for the wetted film thickness was found as

(3.1)$$\begin{eqnarray}h=(\unicode[STIX]{x1D707}\ell /\unicode[STIX]{x1D70C}g\unicode[STIX]{x1D70F})^{1/2},\end{eqnarray}$$

where $h$ denotes the wetted film thickness, $\ell$ the vertical distance to the top (i.e. to the pinned contact line) of the film (figure 3a) and $\unicode[STIX]{x1D70C}$ the density of the fluid.

In our experiments, interferometry is performed to measure the film thickness $h_{0}$ at the centre of the pre-wetted film; see figure 3(b). The injection terminates at $\unicode[STIX]{x1D70F}=0$, and then the interferometric pattern at the centre of the draining film is recorded by a high-speed camera (150 frames per second in this set of experiments). The thickness $h_{0}$ is then estimated by measuring the number of fringes that pass the centre, as a function of the drainage time $\unicode[STIX]{x1D70F}$, marked as the black solid line in figure 3(b). The inset in figure 3(b) shows a comparison (using a log–log plot) between Jeffreys’ similarity solution and our experimental measurements of $h_{0}$. The blue dashed line in figure 3(b) refers to $h_{0}=(\unicode[STIX]{x1D707}L_{0}/\unicode[STIX]{x1D70C}g\unicode[STIX]{x1D70F})^{1/2}$ predicted by (3.1), where we have used $\unicode[STIX]{x1D707}=1\times 10^{-3}~\text{Pa}~\text{s}$, $\unicode[STIX]{x1D70C}=1\times 10^{3}~\text{kg}~\text{m}^{-3}$, $g=9.8~\text{m}~\text{s}^{-2}$ and the distance from the centre to the top of the film $L_{0}=30~\text{mm}$.

For the drainage time $\unicode[STIX]{x1D70F}\lesssim 20~\text{s}$ in figure 3, our experiments agree with Jeffreys’ (Reference Jeffreys1930) similarity solution, indicating $h_{0}\propto \unicode[STIX]{x1D70F}^{-1/2}$ during early drainage. However, at later times ($\unicode[STIX]{x1D70F}\gtrsim 40~\text{s}$), our results deviate from the Jeffreys’ solution and we attribute this difference to the observed contact line depinning on the top of the film: the contact line on the top of the film slips downwards slowly with a speed of approximately $0.3~\text{mm}~\text{s}^{-1}$ (it takes around 100 s for the contact line to slip from the initial injection position ($\ell =0~\text{mm}$) to the field of view at the middle of the substrate ($\ell =30~\text{mm}$)). The effect of the dewetting appears when the drainage time $\unicode[STIX]{x1D70F}$ is comparable to the dewetting time (100 s). At later times ($\unicode[STIX]{x1D70F}\gtrsim 40~\text{s}$), the decrease of the measured film thickness is affected by both gravitational drainage as well as film dewetting, and therefore the experimental values for $h_{0}$ decrease faster than the prediction of Jeffreys’ solution. Note that most of the experiments throughout this article are in the regime in which the dewetting effect is less significant ($\unicode[STIX]{x1D70F}\lesssim 20~\text{s}$), and the film climbing speed is typically much faster than this downward dewetting speed.

One implication from our interferometric measurements is that the thickness of the pre-wetted film can be tuned by the drainage time $\unicode[STIX]{x1D70F}$. In our subsequent experiments, in order to study the effect of the pre-wetted film thickness on the climbing film, four thicknesses are formed by controlling the drainage time interval $\unicode[STIX]{x1D70F}_{0}=5,~10,~20$ and $40~\text{s}$, respectively, i.e. we tune the drainage time interval $\unicode[STIX]{x1D70F}_{0}$, the time interval from the termination of water injection to the contact of the pre-wetted film with the bath, to achieve different initial pre-wetted film profiles. The resulting centre film thicknesses are $h_{0}=26.2\pm 0.8,~18.2\pm 0.9,~11.9\pm 0.5$ and $6.4\pm 0.3~\unicode[STIX]{x03BC}\text{m}$, respectively (experiments are repeated four times). For results presented below, $h_{0}$ represents the initial pre-wetted film thickness.

3.3 Interferometry: climbing front position

Figure 4.

Interferometric measurements of the position of the climbing front with high SDS concentrations ($c_{0}\geqslant 1\times 10^{-3}~\text{M}$) in the bath. (a–d) A time series of the interferometric patterns (images on the right) and the processed image signal (plots on the left), as the climbing film front rises from a bath with $3\times 10^{-3}~\text{M}$ SDS. The intensity change $I^{\prime \prime }$ ($1/\text{pixel}^{2}$) is calculated by vertically taking the second forward difference of the normalized grey value (averaged horizontally) of the interferometric image, and is plotted as a function of vertical position $z$, which is the distance to the liquid level of the bath. The red dashed line denotes the climbing front position, labelled as the position that maximizes $I^{\prime \prime }$. (e) A schematic of a film climbing on the pre-wetted layer; $z_{f}$ denotes the distance from the climbing film front to the level of the bath. (f) The climbing front position $z_{f}$ as a function of time $t$, for baths of $1\times 10^{-2}$ (black circles), $3\times 10^{-3}$ (blue squares) and $1\times 10^{-3}~\text{M}$ (red triangles) SDS solution; $t=0$ denotes the contact of the bath with the pre-wetted film. The data points that refer to the interferometric patterns in (a–d) are marked. The error bars correspond to averaged standard deviations (the average values of the standard deviations among experiments with the same SDS concentration) from three experiments.

Typical results from the interferometric measurements for the climbing front position, in experiments with high SDS concentrations ($c_{0}\geqslant 1\times 10^{-3}~\text{M}$) in the bath, are shown in figure 4. At $t=0$, the bath contacts the pre-wetted film and a film front starts to rise. The appearance of the climbing film changes the total film thickness, and thus the interferometric pattern changes, as recorded by a high-speed camera. Figure 4(a–d) shows a characteristic interferometric image sequence of the climbing front (see the interferometric movie of the climbing front in supplementary movie 3). A schematic of the climbing film on the pre-wetted layer is displayed in figure 4(e). Above the climbing front (the area far above the red dashed lines in figure 4a–d) is the draining pre-wetted film, undisturbed by the climbing film. The fringes on the pre-wetted film are relatively wide, indicating that the slope of the film thickness is low, and is consistent with our measurements on the draining film thickness as well as the Jeffreys’ solution. Near the climbing front (close to the red dashed lines in figure 4a–d), the fringes become much narrower, indicating a more rapid change of film thickness. The climbing film induces a sharp gradient on the overall film thickness so that the narrow fringes exceed the resolution of the images in figure 4. Below the climbing front (below the red dashed lines in figure 4a–d), the film thickness still changes rapidly, while the fringes are much narrower than those in the freely draining film. The shape of the climbing film is flat at some points, indicating that the film thickness reaches a local maximum or minimum. Detailed measurements of the film thickness with higher resolution will be reported in § 3.4.

There is a meniscus that connects the thin film on the substrate and the liquid in the bath; see the dark region on the bottom of the glass slide in figures 1(a) and 10(b) in appendix A. The height and thickness of the meniscus are typically of the order of the capillary length $\ell _{c}=\sqrt{\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}g}\approx 3~\text{mm}$. Hence, the thickness of the meniscus is much larger than the thin film (order of $100~\unicode[STIX]{x03BC}\text{m}$) on the substrate and thus the meniscus contains much more fluid. Also, according to the experiments, the shape and the position of the meniscus remain approximately unchanged. Therefore, the meniscus can be considered as a part of the liquid in the bath, and we hereby denote the top of the meniscus as the bottom of the thin film, i.e. $z=0$ (figure 4e). Note that the schematic in figure 4(e) amplifies the film thickness on the substrate and in reality the thin-film thickness is much smaller than the meniscus thickness.

In order to robustly mark the climbing fronts in the interferometric images, the following image processing is performed. We first extract the image signal in an interferometric image by cropping a vertical stripe ($5~\text{pixel}\times 686~\text{pixel}$, where $1~\text{pixel}\approx 19~\unicode[STIX]{x03BC}\text{m}$) in the middle of the image. The normalized grey value $I$ (as a function of $z$, the distance to the level of the bath) is calculated by taking the average of the grey values horizontally along the strip (i.e. the mean value across the five horizontal pixels) and then normalizing (dividing by 255). The light intensity along the glass slide is indicated by $I(z)$, so that high light intensity $I$ refers to bright fringes and low intensity refers to dark fringes. Then $I^{\prime }\approx \text{d}I/\text{d}z$ ($1/\text{pixel}$) is calculated by taking the first forward difference of $I(z)$. The peaks in $I^{\prime }$ refer to the boundaries of the fringes on the image.

In order to estimate the position of the climbing front where the film thickness changes sharply and the fringes become narrower, $I^{\prime \prime }\approx \text{d}^{2}I/\text{d}z^{2}$ ($1/\text{pixel}^{2}$) is calculated by taking the second forward difference of $I(z)$; see the plots on the left in figure 4(a–d). The $I^{\prime \prime }$ value increases as the fringes get narrower, and we find that the peak of $I^{\prime \prime }$ robustly marks the front of the climbing film. The red dashed lines in figure 4(a–d) mark the peaks of $I^{\prime \prime }$, and also capture the position in the image where the fringes narrow rapidly, indicating that the film thickness increases sharply due to the climbing film.

In addition to the climbing film front (marked with red dashed lines in figure 4a–d), large values and local peaks of $I^{\prime \prime }$ are observed below the climbing film front, which indicate that, in addition to the front of the climbing film, there are other sharp slopes of the film thickness on the climbing film. Moreover, wide fringes and circles are also observed in the interferometric images (for example, in figure 4d), which indicate that the film thickness reaches a local maximum (peak) or minimum (valley). Local peaks of $I^{\prime \prime }$ are found near the wide fringes and circles, which means the film thickness increases or decreases sharply behind and after the local peak or valley of the film. The detailed thickness profile of the climbing film is complex (see the schematic in figure 4e) and is an interesting problem for further investigation. In this article, we focus on the position of the climbing film front rather than the detailed thickness profile during the Marangoni climbing.

At $t=0~\text{s}$, the bath with SDS solution ($c_{0}=3\times 10^{-3}~\text{M}$ in figure 4a–d) contacts the pre-wetted film and the liquid from the bath starts to rise. The film rises rapidly and the front position of the climbing film $z_{f}(t)$ approaches $9.6~\text{mm}$ within $40~\text{ms}$ (figure 4f). The data in figure 4(f) also indicate that the velocity of the climbing front continuously decreases. The climbing front reaches the top of the field of view ($z_{f}\approx 20~\text{mm}$) at $t=250~\text{ms}$.

To study the effect of the SDS concentration in the bath $c_{0}$, interferometry experiments are performed with different $c_{0}$ and the climbing front positions $z_{f}$ are measured. The drainage time interval is controlled uniformly as $\unicode[STIX]{x1D70F}_{0}=10~\text{s}$, so that the pre-wetted films in this set of experiments are approximately the same. The climbing front position $z_{f}$ as a function of time $t$ is shown in figure 4(f) for bath SDS concentrations $c_{0}=1\times 10^{-2}$ (black circles), $3\times 10^{-3}$ (blue squares) and $1\times 10^{-3}~\text{M}$ (red triangles), respectively. The data for $c_{0}=1\times 10^{-2}~\text{M}$ show a similar trend as $c_{0}=3\times 10^{-3}~\text{M}$, while the rise speed continuously decreases during the ascent. It is intuitive that the front rises faster with higher SDS concentration $c_{0}$ (the climbing front reaches the top of the field of view ($z_{f}\approx 20~\text{mm}$) by $0.1~\text{s}$ with $c_{0}=1\times 10^{-2}~\text{M}$ and by $0.25~\text{s}$ with $c_{0}=3\times 10^{-3}~\text{M}$, figure 4f), since the surface tension in the bath decreases more with higher surfactant concentration. As for $c_{0}=1\times 10^{-3}~\text{M}$, the climbing front rises more than 10 times slower than that with $c_{0}=3\times 10^{-3}~\text{M}$, taking $3~\text{s}$ to reach the top of the field of view. This slow rate (approximately $5~\text{mm}~\text{s}^{-1}$) of rise is due to the effect of gravitational drainage as the Marangoni stress weakens.

Figure 5.

Interferometric measurements of the position of the climbing front with low SDS concentrations ($c_{0}\leqslant 1\times 10^{-3}~\text{M}$) in the bath. (a–d) A time series of the interferometric patterns (images on the right) and the processed image signal (plots on the left), as the climbing film front rises from a bath with $3\times 10^{-4}~\text{M}$ SDS. The intensity change $I^{\prime \prime }$ ($1/\text{pixel}^{2}$) is calculated by vertically taking the second forward difference of the normalized grey value (averaged horizontally) of the interferometric image, and is plotted as a function of vertical position $z$, which is the distance to the liquid level of the bath. The red dashed line denotes the climbing front position, labelled as the position that maximizes $I^{\prime \prime }$. (e) The climbing front position $z_{f}$ as a function of time $t$, for baths of $1\times 10^{-3}$ (black circles), $3\times 10^{-4}$ (blue squares), $1\times 10^{-4}$ (red upward-pointing triangles) and $3\times 10^{-5}~\text{M}$ (magenta downward-pointing triangles) SDS solution; $t=0$ denotes the contact of the bath with the pre-wetted film. The data points that refer to the interferometric patterns in (a–d) are marked. The error bars correspond to averaged standard deviations (the average values of the standard deviations among experiments with the same SDS concentration) from three experiments.

The trend of the climbing front position $z_{f}(t)$ differs when the SDS concentration in the bath is low, e.g. less than $1\times 10^{-3}~\text{M}$. Figure 5 shows the interferometric measurements on the front position $z_{f}$ with low SDS concentrations in the bath, for $c_{0}=1\times 10^{-3}$ (black circles), $3\times 10^{-4}$ (blue squares), $1\times 10^{-4}$ (red upward-pointing triangles) and $3\times 10^{-5}~\text{M}$ (magenta downward-pointing triangles), respectively. The speed of the climbing film is much slower with low SDS concentrations in the bath (compare the data in figures 4 and 5). For example, the climbing front with $c_{0}=3\times 10^{-4}~\text{M}$ rises more than 10 times slower than that of $c_{0}=1\times 10^{-3}~\text{M}$, while the surface tension difference is only approximately three times smaller. Also, after early times ($t\gtrsim 2~\text{s}$ for $c_{0}=3\times 10^{-4},~1\times 10^{-4}$ and $3\times 10^{-5}~\text{M}$), the rise speed remains approximately constant, i.e. the climbing front position $z_{f}$ is linear with time $t$ for almost the entire experiment. It is also worth noting that for $c_{0}=3\times 10^{-4}~\text{M}$, the film front rises much faster at early times ($t\lesssim 2~\text{s}$, figure 5a,b) than at late times ($t\gtrsim 2~\text{s}$, figure 5c,d) (see the interferometric movie of the climbing front from the bath with $3\times 10^{-4}~\text{M}$ SDS solution (figure 5a–d) in supplementary movie 4). Gravitational drainage of the climbing film is also observed in figure 5(c,d), which indicates that, for low bath SDS concentration $c_{0}$, the climbing film is affected by gravitational drainage at late times ($t\gtrsim 2~\text{s}$). In general, the climbing film front at lower concentration ($c_{0}\leqslant 1\times 10^{-3}~\text{M}$) rises much slower due to the low surface tension difference and enhanced gravitational drainage. The effect of both the surface tension difference and the gravitational drainage will be discussed in more detail in the discussion, § 3.5. Note that the temperature variation (e.g. from evaporation) or random contamination (e.g. from dust in the ambient air) in the liquid may affect the dynamics of the film climbing in the experiments for low SDS concentration $c_{0}\lesssim 1\times 10^{-4}~\text{M}$. The surface tension of water decreases approximately $0.1~\text{mN}~\text{m}^{-1}$ by increasing the temperature by $1\,^{\circ }\text{C}$, but adding SDS surfactant with concentration $c_{0}=3\times 10^{-4}~\text{M}$ into water leads to a decrease of surface tension of $2.4~\text{mN}~\text{m}^{-1}\gg 0.1~\text{mN}~\text{m}^{-1}$. Therefore, the effect of the temperature variation is negligible for the majority of our experiments ($c_{0}\gtrsim 3\times 10^{-4}~\text{M}$).

Figure 6.

The effect of the pre-wetted film thickness $h_{0}$ on the film climbing front position $z_{f}(t)$, with (a) high ($c_{0}=3\times 10^{-3}~\text{M}$) and (b) low ($c_{0}=3\times 10^{-4}~\text{M}$) SDS concentrations in the bath. The pre-wetted film thicknesses are tuned by setting the drainage time interval $\unicode[STIX]{x1D70F}_{0}=5$, $10$, $20$ and $40~\text{s}$, and the resulting pre-wetted film centre thicknesses $h_{0}=26$ (black circles), $18$ (blue squares), $12$ (red upward-pointing triangles) and $6~\unicode[STIX]{x03BC}\text{m}$ (magenta downward-pointing triangles), respectively. The error bars correspond to averaged standard deviations (the average value of the standard deviations of experiments with the same $h_{0}$) with three experiments.

We postulate that the pre-wetted film thickness may affect the climbing film. To study this effect, we tune the drainage time and the resulting pre-wetted film centre thicknesses (the film thickness $30~\text{mm}$ below the top of the pre-wetted film) $h_{0}=26$, $18$, $12$ and $6~\unicode[STIX]{x03BC}\text{m}$, respectively. The effect of the pre-wetted film thickness on the film climbing front position $z_{f}(t)$ with high ($c_{0}=3\times 10^{-3}~\text{M}$) and low ($c_{0}=3\times 10^{-4}~\text{M}$) bath SDS concentrations is shown in figure 6(a) and (b), respectively.

For high concentration in the bath ($c_{0}=3\times 10^{-3}~\text{M}$, figure 6a), the film front position $z_{f}$ for different $h_{0}$ shows a similar trend, and the speed of the climbing front increases as $h_{0}$ increases. For example, the speed approximately doubles as a result of increasing $h_{0}$ from $6~\unicode[STIX]{x03BC}\text{m}$ to $26~\unicode[STIX]{x03BC}\text{m}$. On the other hand, for low concentration in the bath ($c_{0}=3\times 10^{-4}~\text{M}$, figure 6b), the speed of the climbing front decreases as $h_{0}$ increases, which is opposite to the trend for films with high surfactant concentration (figure 6a). In this case, as $h_{0}$ increases from $6~\unicode[STIX]{x03BC}\text{m}$ to $26~\unicode[STIX]{x03BC}\text{m}$, the rise speed is approximately halved. Also, the trend of the film front position $z_{f}$ as a function of time $t$ also differs for different film thickness (figure 6b): for example, for a thin-film thickness $h_{0}=6~\unicode[STIX]{x03BC}\text{m}$, at early times ($t\lesssim 5~\text{s}$), the front first rises rapidly and the rise speed decreases continuously until the velocity remains approximately constant at $t\gtrsim 5~\text{s}$; for a larger film thickness $h_{0}=26~\unicode[STIX]{x03BC}\text{m}$, the climbing front rises with an approximately constant velocity throughout the measurement, and no ‘early-time’ speed decrease is observed.

For high surfactant concentration in the bath, the climbing speed decreases as the pre-wetted film thickness decreases. In contrast, at low surfactant concentration in the bath, the climbing speed increases as the pre-wetted film thickness decreases. We rationalize this effect as a consequence of the competition among the surface tension difference (Marangoni driving stress), viscous resistance and gravitational drainage. For high SDS concentration, Marangoni stress is strong and gravitational drainage is negligible. Then the Marangoni stress balances the viscous resistance, and hence a thicker pre-wetted film thickness results in less viscous resistance and therefore a higher rise speed. In contrast, for low SDS concentration, Marangoni stress is weak and balances gravitational drainage. As a result, a thicker pre-wetted film thickness results in more drainage, and therefore a slower rise speed. A detailed discussion and modeling will be presented in the discussion in § 3.5.

3.4 Interferometry: film thickness

Figure 7.

Interferometric measurements on the film thickness profile while the film climbs, for SDS concentrations (a) $c_{0}=3\times 10^{-4}~\text{M}$, (b) $1\times 10^{-3}~\text{M}$, (c) $3\times 10^{-3}~\text{M}$ and (d) $1\times 10^{-2}~\text{M}$. The interferometric pattern at the centre of the film is recorded and $\unicode[STIX]{x0394}t$ denotes the time interval from when the climbing front reaches the centre of the field of view (images on the left) to the top of the field of view (images on the right). The pre-wetted film profiles are expected to be approximately the same by setting the same drainage time interval $\unicode[STIX]{x1D70F}_{0}=10~\text{s}$; $h_{0}=18~\unicode[STIX]{x03BC}\text{m}$. (e) Schematic of the climbing film thickness profile. Here $z^{\ast }$ denotes the vertical distance (positive downwards) to the climbing front and $\unicode[STIX]{x0394}h$ the film thickness difference to the climbing front. (f) The film thickness difference $\unicode[STIX]{x0394}h$ as a function of $z^{\ast }$ when the climbing front reaches the top of the field of view (images on the right of (a–d)). (g) The film thickness difference $\unicode[STIX]{x0394}h$ at the fixed centre point of the film during the climbing of the film, as a function of time $t^{\ast }$. Here $t^{\ast }=0$ denotes the time when the climbing front passes the fixed point. The inset shows a zoomed-in view of early times $t^{\ast }\leqslant 0.2~\text{s}$. The SDS concentration in the bath is (a) $c_{0}=3\times 10^{-4}$ (magenta downward-pointing triangles), (b) $1\times 10^{-3}$ (red upward-pointing triangles), (c) $3\times 10^{-3}$ (blue squares) and (d) $1\times 10^{-2}~\text{M}$ (black circles), respectively.

In this section, we focus on measuring the local film thickness profile (near the climbing front) with increased resolution (smaller pixel size) in the interferometric set-up. Figure 7 shows the film thickness profile deduced using the interferometric technique for different SDS concentrations $c_{0}=3\times 10^{-4}$, $1\times 10^{-3}$, $3\times 10^{-3}$ and $1\times 10^{-2}~\text{M}$ (figure 7a–d, respectively). Note that for $c\lesssim 1\times 10^{-4}~\text{M}$, the climbing film is slow and fails to rise into the field of view of this set-up. The interferometric images at the centre of the film ($30~\text{mm}$ below the top and also $30~\text{mm}$ above the bottom of the pre-wetted film) are recorded by a high-speed camera. The frame rate is 1000 frames per second and the exposure time is $250~\unicode[STIX]{x03BC}\text{s}$. The field of view is $96~\text{pixel}\times 1200~\text{pixel}$ where $1~\text{pixel}\approx 1.7~\unicode[STIX]{x03BC}\text{m}$.

The interferometric images when the climbing front reaches the centre and the top of the field of view are displayed in figure 7(a–d), on the left and the right, respectively. Similar to the interferometric images measuring the climbing front position in § 3.3, the film above the climbing front is the draining pre-wetted film and the fringes are wide while the change of film thickness is slow. The fringes that are near and below the climbing front are narrower, since the film thickness changes sharply as the film rises. When the SDS concentration in the bath $c_{0}$ increases, the Marangoni stress increases and the change of film thickness becomes more rapid. The film thickness difference $\unicode[STIX]{x0394}h$ along the film is estimated by measuring the number of fringes in the interferometric images, and is plotted as a function of vertical position $z^{\ast }$ (the distance to the climbing front) (figure 7e). The fringes become narrower and the number of fringes in the field of view increases at higher $c_{0}$ (figure 7a–d). Also, as $c_{0}$ increases, the climbing front velocity increases, while $\unicode[STIX]{x0394}t$, which denotes the time interval for the climbing front to reach from the centre to the top of the field of view, decreases (figure 7a–d), and is consistent with our observations measuring the climbing front position in § 3.3 (see the movies showing the climbing film rising for figure 7a–d in supplementary movies 5–8).

We also measure the film thickness change locally at a fixed position corresponding to the centre point of the film, during the film ascent, by analysing the fringes that pass this fixed point; see figure 7(g). Encountering the climbing film, the film thickness at this fixed point first slightly decreases (${\sim}1~\unicode[STIX]{x03BC}\text{m}$) and then increases (inset in figure 7g). A later secondary increase of the film thickness is also observed at high surfactant concentration (for example, see figure 7g, $c=1\times 10^{-2}~\text{M}$ (black line and circles), $t^{\ast }>1~\text{s}$). As a result, increasing the SDS concentration in the bath, $c_{0}$, leads to a more rapid change in film thickness and also a larger film thickness, which is a consequence of the enhanced Marangoni stress.

Using interferometry, the effect of the pre-wetted film thickness on the climbing film thickness is also studied. In figure 8 we show the results of the film thickness profile, with different pre-wetted film centre thicknesses $h_{0}=26$ (black circles), $18$ (blue squares), $12$ (red upward-pointing triangles) and $6~\unicode[STIX]{x03BC}\text{m}$ (magenta downward-pointing triangles), respectively. The SDS concentration $c_{0}=3\times 10^{-3}~\text{M}$ is in the high-surfactant-concentration region where we have shown that the resistance from drainage can be neglected. As a result, a thicker pre-wetted film leads to thicker films near the climbing fronts (figure 8a), but a thinner pre-wetted film also leads to thicker secondary films; for example, a strong secondary climbing film is observed at $t^{\ast }>2~\text{s}$ with $h_{0}=6~\unicode[STIX]{x03BC}\text{m}$ (figure 8b). We also note that the film thickness differences $\unicode[STIX]{x0394}h(t^{\ast })$, for different $h_{0}$, coincide at early times (inset in figure 8b) after encountering the climbing front, which is to say, for this experimental set-up, that the film thickness variation with time is unaffected by the pre-wetted film thickness.

Figure 8.

The effect of the pre-wetted film thickness on the shape of the climbing film, as measured by interferometry. The SDS concentration in the bath $c_{0}=3\times 10^{-3}~\text{M}$. (a) The film thickness difference $\unicode[STIX]{x0394}h$ as a function of $z^{\ast }$, when the climbing front reaches the top of the field of view. (b) The film thickness difference $\unicode[STIX]{x0394}h$ at the centre point of the film as the film rises, as a function of time $t^{\ast }$. Here $t^{\ast }=0$ denotes the time that the climbing front passes the fixed point. The inset shows a zoomed-in view of the early rising $t^{\ast }\leqslant 0.2~\text{s}$. The pre-wetted film centre thickness is $h_{0}=26$ (black circles), $18$ (blue squares), $12$ (red upward-pointing triangles) and $6~\unicode[STIX]{x03BC}\text{m}$ (magenta downward-pointing triangles), respectively.