Hydrodynamic signatures of stationary Marangoni-driven surfactant transport

TL;DR

This study experimentally investigates steady Marangoni-driven surfactant transport on water interfaces, revealing how different surfactants influence flow dynamics based on their adsorption and dissolution properties, without prior chemical knowledge.

Contribution

It introduces a hydrodynamic measurement method to distinguish surfactant transport regimes without needing detailed chemical properties.

Findings

Low concentration surfactants cause adsorption-driven flow.

High concentration surfactants lead to dissolved surfactant-driven flow.

The method effectively characterizes unknown surfactants like camphoric acid.

Abstract

We experimentally study steady Marangoni-driven surfactant transport on the interface of a deep water layer. Using hydrodynamic measurements, and without using any knowledge of the surfactant physico-chemical properties, we show that sodium dodecyl sulphate and Tergitol 15-S-9 introduced in low concentrations result in a flow driven by adsorbed surfactant. At higher surfactant concentration, the flow is dominated by the dissolved surfactant. Using Camphoric acid, whose properties are {\it a priori} unknown, we demonstrate this method's efficacy by showing its spreading is adsorption dominated.

Equations

$$u(r,z=0)=f^{\prime }(0)C_a{r}^{-3/5},{\delta }_a(r)=r^{4/5}\sqrt{\nu /C_a},$$

$$u(r,z=0)=C_d{r}^{-1},{\delta }_d(r)=r\sqrt{\nu /C_d}$$

$$u(r,z)$$

$$f^{\prime \prime \prime }(\xi )+\frac{3}{5}{f}^{\prime }(\xi {)}^2+\frac{6}{5}f(\xi )f^{\prime \prime }(\xi )=0,$$

$$u(r,z)=C_d{r}^{-1}{\text{sech}}^2\left(\frac{z}{{\delta }_d(r)\sqrt{2}}\right)+O\left(\frac{{\delta }_d(r)}{r}\right).$$

$$u_z(r,z=0)\approx \frac{u(r,0)-u(r,-h)}{h}+\frac{|h|u_{zz}(r,z=0)}{2}+\frac{ϵ}{|h|}+\dots ,$$

$$u(r,0)=\frac{1}{r^{3/5}}\left(a+\frac{b}{r}+\frac{c}{r^2}\dots \right),$$

$$u(r,0)=\frac{1}{(r-\Delta r{)}^{3/5}}\left(a+\frac{c}{(r-r_0{)}^2}\dots \right),,$$

Full text

marangoni, boundary layer, self-similar, surfactant transport

Surfactants introduced at liquid interfaces give rise to Marangoni stresses that drive a flow Levich and Krylov (1969). The fundamental process of surfactant spreading, governed by its diffusion and transport via self-induced flow has many applications from materials chemistry to biomechanics Jobe (1993); Grotberg (1994); Harshey (2003); Eisner et al. (2005); Bush and Hu (2006); Cantat et al. (2013); Ivanov and Kralchevsky (1997); Barnes (2008); Herzig et al. (2011). Many surfactants are soluble in the fluid and could be transported in a phase dissolved in the bulk or adsorbed at the interface Roché et al. (2014); Le Roux et al. (2016). A complete description of the resulting flow is hindered by the complexity of surfactant dynamics, which includes characterizing the equilibrium adsorption characteristics, the adsorption-desorption kinetics, and the transport by the flow Chang and Franses (1995); Eastoe and Dalton (2000). Whereas methods based on molecular Lu et al. (2000) or radiometric Salley et al. (1950); Matuura et al. (1958); Tajima et al. (1970) markers can measure surface excess during a flow Manning-Benson et al. (1998), low surfactant diffusivity into bulk fluid renders bulk concentration measurements at the interface difficult during flow. Direct Marangoni stress measurements via in situ surface tension gradient measurements are equally challenging. Simultaneous access to bulk and surface concentrations, Marangoni stress, sorption kinetics, and their subsequent correlation with one another to deduce the surfactant dynamics remains a formidable task.

In a recent study Roché et al. (2014), for example, surfactant was introduced on the air-water interface through a steady point source. Simple scaling laws for surfactant spreading were derived by assuming the sorption kinetics to be much faster than the hydrodynamics so that the dynamics were dominated by the dissolved phase. Verification of this assumption was not possible owing to the aforementioned difficulties. A possible alternative is that the sorption kinetics are too slow compared to the hydrodynamics, so that the dynamics are governed by the adsorbed phase. Either of these assumptions reduce the complexity of the problem by enabling semi-analytical steady solutions to the governing equations Mandre (2017). Our objective in this setting is an experimental validation of these assumptions using hydrodynamic measurements alone.

Consider a surfactant released steadily on the interface through a source much smaller in radial extent than the container size (see Figure 1) such that a steady axisymmetric flow is established (see Supplemental Material - Movie M1 for visualization). In the region much larger than the source but much smaller than the container, the source may be idealized as a point and the container assumed infinite. Furthermore, consider the fluid viscosity and surfactant diffusivity to be small enough that most of the flow and surfactant concentration is established within a boundary layer near the surface. These approximations, along with the assumption of adsorption- or dissolution-dominated surfactant dynamics, render the governing physical description scale invariant. Consequently, the fluid radial $u(r,z)$ velocity components in cylindrical coordinates $(r,z)$, exhibit a self-similar structure Mandre (2017). Three experimentally measured invariant characteristics of this self-similar flow serve as hydrodynamic signatures of the simplified surfactant transport.

In this letter, we present experimental verification of these flow signatures using two generic surfactants in water – sodium dodecyl sulphate (SDS) and Tergitol 15-S-9 (Tergitol). Both these surfactants are water soluble (solubility 0.2 kg/l and 0.7 kg/l, respectively) and span a range of critical micellar concentration (CMC) from $8\times 10^{-5}$ mM for Tergitol to $8\times 10^{-3}$ mM for SDS. SDS is ionic in nature, while Tergitol is non-ionic. Without using any knowledge of the surfactant physico-chemical parameters, we show that for concentrations less than 15% CMC, both surfactants exhibit flows dominated by adsorbed surfactant. In the same manner, mixture concentrations between $24-50\%$ CMC exhibit flow dominated by the dissolved surfactant. Finally, we also determine which of the two processes dominate the dynamics of a third surfactant, camphoric acid (CA), released at the interface from a gel tablet at unknown rates and concentrations.

Solution of SDS or Tergitol was introduced on air-water interface via a borosilicate capillary (tip inner diameter of 3-5 $\mu$m) by Marangoni suction, a procedure empirically determined to minimize forcing a radial jet due to hydrodynamic pumping Squire (1951); Landau and Lifshitz (1959); Laohakunakorn et al. (2013). Four different concentrations for SDS and Tergitol ranging from about $0.05$ CMC to $0.5$ CMC (labeled C1-8 in Figure 2) were used to span the range of surfactant dynamics from adsorption-dominated to dissolution-dominated. CA was introduced on the interface through an agarose gel tablet (diameter 3 mm, thickness 1 mm) infused with CA (case C9 in Figure 2). The gel tablet was mounted on a vertical motion stage and brought in contact with the interface. In our experiments, the velocity boundary layer was minimally influenced by the dish bottom. The velocity profiles $u(r,0)$ and $u(r=r_{1},z)$, and the surface shear $u_{z}(r,z=0)$ of the axisymmetric flow that developed due to the Marangoni flow were measured using Laser Doppler Velocimetry (LDV).

The reproducibility required for the experiment and the measurement precision in velocity up to 4${}^{\text{th}}$ decimal place to ascertain the power laws and the boundary layer profile reported here require a tight protocol (for full experimental details, please see Supplemental Material).

Signature 1: The measured surface radial velocity $u(r,0)$ is shown on a logarithmic scale in Fig. 2(a). A correction to account for higher order effects due to finite size of the CA tablet is applied, as detailed in Supplemental Materials. For all the nine cases considered, $u(r,0)$ reaches a maximum $u_{\text{max}}$ at $r=r_{\text{max}}$ (about 1 mm), and decays approximately as a power law in a range of radii $1<r/r_{\text{max}}<20$. For $r/r_{\text{max}}\gtrsim 20$, $u(r,0)$ decreases much faster than the power-law decay.

The exponent of the power-law decay in the intermediate range of radii is the first hydrodynamic signature of the surfactant dynamics. In this range, five of the nine cases (C1, C2, C5, C6 and C9), those with surfactant concentrations $<0.15$ CMC and the one with CA, exhibit an approximate decay of $u(r,0)$ as $r^{-3/5}$. (For enhanced visibility, cool colors depict these cases in Figures 2 and 5.) The remaining four cases (C3, C4, C7 and C8), which include surfactant concentration $>0.24$ CMC (shown in warm colors), exhibited decay as $r^{-1}$.

To confirm the measured slopes, Fig. 2(b) plots the log derivative (Selke’s method Selke (1991)) $n=\text{d}~{}\log u/\text{d}~{}\log r$ as a function of $r$. The differentiation is performed using finite differences between neighboring experimentally measured data points. For the lowest concentrations of the SDS (0.049 CMC) and Tergitol (0.046 CMC), and in the range $1<r/r_{\text{max}}<10$, the value of $n$ lies between -0.565 and -0.618. For the next lowest concentration (0.146 CMC for SDS And 0.138 CMC for Tergitol), $n$ departs from this range at $r/r_{\text{max}}\gtrsim 8$. As the concentration is increased further (0.243 CMC for SDS and 0.25 CMC for Tergitol), $n$ lies in the range $-0.87$ to $-1.04$, with a systematic departure from $\approx-1$ occuring in the range $1<r/r_{\text{max}}\lesssim 4$. And finally, for the largest concentration (0.5 CMC for SDS and 0.46 CMC for Tergitol), $n$ lies in the range $-0.98$ to $-1.03$. For the flow driven by CA, $n$ lies in the range $-0.60$ to $-0.63$. Based on these oservations, we posit two values for the power-law exponents, $n\approx-0.6$ and $n\approx-1$, with the random variation attributed to measurement noise and the systematic deviations to departures from the asymptotic regimes of validity.

These power laws can be understood in terms of the competing fluid and surfactant-induced stresses as follows. Due to self-similar nature of the flow, the length scale in the radial and depth-wise directions are $r$ and the boundary layer thickness, $\delta(r)$, respectively. Fluid inertia scales as $\rho u^{2}/r$ ($\rho$ is fluid density) while viscous forces scale as $\mu u/\delta^{2}$ ($\mu$ is dynamic viscosity). A balance between the two is expected in the boundary layer, which furnishes one relation, $\delta\sim\sqrt{\mu r/\rho u}$. Imposing the Marangoni stress, which scales as $\Delta\sigma/r$ ($\Delta\sigma$ being the reduction in surface tension) to be equal to the scale of the fluid’s shear stress, $\mu u/\delta$, leads to $\delta=\mu ur/\Delta\sigma$. The two cases are distinguished by the relation between $\Delta\sigma$ and the surfactant concentration, and how the surfactant is transported.

When the surfactant dynamics are dominated by the adsorbed phase, surfactant conservation implies $2\pi urc_{2}=q_{2}$, where $c_{2}$ is the surface concentration of the surfactant and $q_{2}$ its surface flux. Here we neglect the diffusion of surfactant. The surface tension depends on surfactant concentration as $\Delta\sigma=-\Gamma_{2}c_{2}$, where $\Gamma_{2}$ is a proportionality constant. Eliminating $c_{2}$ and $\Delta\sigma$ leads to

[TABLE]

where $C_{a}=(\Gamma_{2}^{2}q_{2}^{2}\nu/(4\pi^{2}\mu^{2}))^{1/5}$, $f^{\prime}(0)$ is a dimensionless proportionality constant to be determined, and $\nu=\mu/\rho$.

When surfactant dynamics are dominated by the dissolved phase, surfactant bulk concentration $c_{3}(r,z)$ obeys an advection-diffusion equation with diffusivity $D$, and $\Delta\sigma=-\Gamma_{3}c_{3}$, where $\Gamma_{3}$ is a material-dependent constant. The surfactant diffuses in a boundary layer of thickness $\delta_{c}=\sqrt{Dr/u}$, and hence surfactant conservation implies $2\pi ruc_{3}\delta_{c}\propto q_{3}$, where $q_{3}$ is the volumetric surfactant release rate, which yields $c_{3}\propto q_{3}/\sqrt{ur^{3}D}$. The resulting Marangoni stress scales as $\Gamma_{3}c_{3}/r\propto\Gamma_{3}q_{3}/\sqrt{ur^{5}D}$, which balances the fluid viscous shear stress. The shear stress at the surface, due to a peculiarity in the boundary layer flow structure, does not scale as $\mu u/\delta$, but scales one order weaker in the small parameter $\delta/r$, as $\mu u/r$. Balancing the scales for Marangoni stress and shear stress yields

[TABLE]

where $C_{d}={(\Gamma_{3}^{2}q_{3}^{2}/(8\pi^{3}\mu^{2}D))^{1/3}}$. These scaling estimates and the appropriate dimensionless proportionality constant are determined from an exact similarity solution by Bratukhin and Maurin Bratukhin and Maurin (1967) (for details see Mandre, 2017).

Signature 2: To ensure that the power law exponents arise due to the fluid dynamics presented here, and not due to any unexpected coincidences, we compare the depth-wise profile $u(r_{1},z)$ with theoretical expectations. In the case of adsorption-dominated surfactant dynamics, the solution may be expressed as

[TABLE]

in terms of a similarity coordinate $\xi=z/\delta_{a}(r)$ and a self-similar profile $f(\xi)$. Here $f$ satisfies Mandre (2017)

[TABLE]

and $f(0)=0$, $f^{\prime\prime}(0)f^{\prime}(0)=\dfrac{2}{5}$, and $f^{\prime}(-\infty)=0$. This third order ordinary differential equation is solved using a shooting method to obtain $f$ and the $u(r,z)$ is re-constructed using (3). The proportionality constant, $f^{\prime}(0)\approx 0.9943$ in (1), is obtained as part of this solution.

Similarly, a leading order approximation to the boundary layer flow profile driven by the surfactant whose dynamics are dominated by the dissolved phase Mandre (2017) is

[TABLE]

Figure 3 and 4 show a comparison of experimentally measured depth-wise profiles $u(r=r_{1},z)$ for the cases exhibiting a power-law exponent of $-3/5$ and $-1$, respectively. The values of $C_{a}$ and $C_{d}$ are determined using the relation $u(r_{1},0)=C_{a}f^{\prime}(0)r_{1}^{-3/5}$ and $u(r_{1},0)=C_{d}/r_{1}$, which are subsequently used to determine the boundary layer thickness $\delta_{a,d}(r_{1})$ for that profile. When the profiles are rescaled according to (3) or (5), and plotted against the similarity coordinate, they collapse close to a universal curves. The theoretical profiles $f^{\prime}(\xi)$ and $\text{sech}^{2}(\xi/\sqrt{2})$, respectively, well-approximate these universal curves. Apart from random measurement noise, systematic departure of the data from these curves occurs due to two reasons: the return flow in the region outside the boundary layer and departures from the power-law behavior at the measurement location $r=r_{1}$. This collapse validates the thickness of the boundary layer arising from the adsorption- and dissolution-dominated regimes.

Signature 3: The combination of radial decay as $r^{-3/5}$ and depth-wise profiles shown in Figure 3 is only possible when driven by an adsorbed layer of surfactant spreading as $2\pi ru(r,0)c_{2}(r)=q_{2}$, or a small perturbation thereof. However, the agreement in Figure 4 of the measured velocity profile with the leading order of (5) is not conclusive proof of the flow being driven by a dissolved surfactant. It is so because, as explained in Ref. Mandre (2017), Squire’s radial jet Squire (1955) forced by a momentum source at the origin also exhibits $r^{-1}$ decay and the velocity profile (5) to leading order. Only higher order corrections to the flow in the small parameter $\delta_{d}/r$ distinguish between Squire’s radial jet and the complete solution (5). The shear rate $u_{z}(r,z=0)$ is such a quantity; $u_{z}=0$ for Squire’s radial jet, and $u_{z}=2u/r$ from the exact solution for dissolved surfactant driven flow by Bratukhin and Maurin Bratukhin and Maurin (1967). Based on this argument, we define the third hydrodynamic signature to be $\zeta=u_{z}l/u$ at $z=0$, where $l=\delta_{a}(r)$ if the surface velocity decays as $r^{-3/5}$, and $l=r$ if it decays as $r^{-1}$.

Figure 5 shows the experimentally measured values of $\zeta$ for all the nine cases. As expected, for cases C1 and C5 where adsorbed surfactant dominates the dynamics, $\zeta$ is scattered around the theoretically expected value $f^{\prime\prime}(0)/f^{\prime}(0)\approx 0.404$. For the cases C2 and C6, the reduction of $\zeta$ for $r\gtrsim 8r_{\text{max}}$ coincides with the departure of $n$ from $-3/5$. For the remaining cases, $\zeta$ is scattered around $2$ and not around zero, implying that the flow is driven by Bratukhin and Maurin’s surfactant mechanism and not a localized momentum source near the origin.

Conclusion: The agreement of the power-law exponent in Figure 2, the depth-wise profile in Figures 3-4, and the dimensionless shear rate with the theoretically expected ones prove that the flow is driven by a surface stress caused by an agent transported in a manner homologous to the restrictive assumptions underlying the theoretical derivation. Since our experimental protocol has carefully eliminated all other sources of surface stress, we are left with the unavoidable conclusion that the stress is caused by surfactant alone. Therefore, the surfactant dynamics within the power-law region in these cases must be as assumed in the theoretical model. In particular, for SDS and Tergitol released on the interface at concentrations $<0.14$ CMC, the adsorbed surfactant governs the resulting dynamics, while for concentrations $>0.25$ CMC, the dissolved surfactant dynamics dominates. A transition between the two behaviors is expected for intermediate concentrations, as suggested by the systematic deviations of $n$. For both surfactants, the deviation of $n$ from $-3/5$ towards $-1$ at $r/r_{\text{max}}\gtrsim 8$ for cases C2 and C6 suggests the beginning of transition, and in the cases C3 and C7 at $r/r_{\text{max}}\lesssim 4$ suggests the end of the transition. Given that the transition occurs within this range implies that the surfactant and hydrodynamic time-scales approximately overlap, rendering simple order-of-magnitude estimates unreliably to distinguish between the two regimes. Furthermore, there is no convenient independent way to measure a pivotal parameter in characterizing the dynamics – the fraction of the surfactant flux that is transported in an adsorbed phase. Therefore, using invariant hydrodynamic signatures to determine the validity of the assumptions about surfactant dynamics without a priori knowledge of the physico-chemical parameters represents a fundamental advance on the topic.

Our result is quite robust, as we demonstrated for two surfactants varying in their CMC values by factor 100, and can be used with other surfactants. We used these signatures to determine that CA released from a gel tablet spreads in an adsorbed phase, a result that bears upon Marangoni-driven self-assembly Soh et al. (2008); Suematsu et al. (2010a); Soh et al. (2011); Heisler et al. (2012); Kitahata et al. (2013); Grzybowski et al. (2017) and propulsion Nakata and Murakami (2010); Suematsu et al. (2010b); Liakos et al. (2016); Bandi et al. (2017). Assumptions about surfactant dynamics, such as made in Ref. Roché et al. (2014), can also be verified using the hydrodynamic signatures. A theoretical description of the transition between the two behaviors and its dependence on the physico-chemical parameters remain to be developed.

In closing, we note a vast majority of studies Fay (1969); Camp and Berg (1995); Jensen and Grotberg (1992); Jensen (1995); Dussaud and Troian (1998); Dussaud et al. (1998); Fallest et al. (2010); Bandi et al. (2011) to date have focused on transient Marangoni-driven surfactant spreading dynamics, where the flow ceases once the surfactant saturates the available interface area. Here, we have explored the much less studied class of statistically stationary Marangoni-driven flows Bratukhin and Maurin (1967, 1968); Zebib et al. (1985); Roché et al. (2014) which arise when a mechanism for surfactant outflux balances its influx rate onto the interface, thus achieving a steady-state balance.