Hydrodynamic signatures of stationary Marangoni-driven surfactant transport

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.

Surfactants introduced at liquid interfaces give rise to Marangoni stresses that drive a flow [1]. The fundamental process of surfactant spreading, governed by its diffusion and transport via self-induced flow has many applications from materials chemistry to biomechanics [2][3][4][5][6][7][8][9][10]. Many surfactants are soluble in the fluid and could be transported in a phase dissolved in the bulk or adsorbed at the interface [11,12]. 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 [13,14]. Whereas methods based on molecular [15] or radiometric [16][17][18] markers can measure surface excess during a flow [19], 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 [11], 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 [20]. 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 [20].
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 × 10 −5 mM for Tergitol to 8 × 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 µm) by Marangoni suction, a procedure empirically determined to minimize forcing a radial jet due to hydrodynamic pumping [21][22][23]. 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    (1) and (2) expected for the dissolution (solid black line) and adsorption (dashed black line) dominated cases.
The velocity is rescaled by its maximum value u max on the interface, and r is rescaled by r max , the location where the maximum velocity occurs. (b) Same data as (a), but presented in the form of power law exponent n = d(log u)/d(log r).
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 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 max at r = r max (about 1 mm), and decays approximately as a power law in a range of radii 1 < r/r max < 20. For r/r max 20, u(r, 0) decreases much faster than the power-law decay. To confirm the measured slopes, Fig. 2 Fluid inertia scales as ρu 2 /r (ρ is fluid density) while viscous forces scale as µu/δ 2 (µ is dynamic viscosity). A balance between the two is expected in the boundary layer, which furnishes one relation, δ ∼ µr/ρu. Imposing the Marangoni stress, which scales as ∆σ/r (∆σ being the reduction in surface tension) to be equal to the scale of the fluid's shear stress, µu/δ, leads to δ = µur/∆σ. The two cases are distinguished by the relation between ∆σ where C a = (Γ 2 2 q 2 2 ν/(4π 2 µ 2 )) 1/5 , f (0) is a dimensionless proportionality constant to be determined, and ν = µ/ρ. µu/r. Balancing the scales for Marangoni stress and shear stress yields where C d = (Γ 2 3 q 2 3 /(8π 3 µ 2 D)) 1/3 . These scaling estimates and the appropriate dimensionless proportionality constant are determined from an exact similarity solution by Bratukhin and Maurin [25] [for details see 20].
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 in terms of a similarity coordinate ξ = z/δ a (r) and a self-similar profile f (ξ). Here f satisfies [20] f (ξ) and , and f (−∞) = 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 (0) ≈ 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 [20] is which are subsequently used to determine the boundary layer thickness δ 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 (ξ) and sech 2 (ξ/ √ 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 2πru(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. [20], Squire's radial jet [26] 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 δ 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 [25]. Based on this argument, we define the third hydrodynamic signature to be ζ = u z l/u at z = 0, where l = δ a (r) if the surface velocity decays as r −3/5 , and l = r if it decays as r −1 . 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 [27][28][29][30][31][32] and propulsion [33][34][35][36]. Assumptions about surfactant dynamics, such as made in Ref. [11], 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 [37][38][39][40][41][42][43][44]  Clean room preparation: All experiments were performed within a static-dissipative vinyl coated (to reduce particulate matter) softwall cleanroom (5 m × 5 m) expressly converted to meet approximate class 1000 cleanroom conditions. Following thorough scrubbing of the room floor and ceiling, portable floor mount dehumidifiers and particle collectors (Terra Universal) were continuously run for 2 weeks to remove particulate matter up to 0.5 µm in size. Sticky floor mats were installed outside and inside the strip curtain entrance to the room. The room was constantly maintained at 25 ± 1 • C temperature.
Cleaning protocol: All glass components (glass syringe, petri dish, glass reservoir, and capillary) were washed in acetone followed by methanol three times and dried in an oven for 10 minutes at 100 • C. They were then soaked in sulfochromic acid bath for 10 minutes, followed by a thorough rinse with de-ionized water. The glass components were once again baked in the oven for 30 minutes at 100 • C, and irradiated in plasma to remove any residual organic impurities. PVC tubing used were washed in acetone followed by methanol three times, thoroughly rinsed with de-ionized water and dried in an oven for 20 minutes at 40 Experimental preparation: Following initial cleaning procedures, the setup was constructed within an enclosed space modified to meet approximate class 1000 clean room conditions. The setup (see Figure 1 2) The capillary was placed vertically above the air-water interface with its tapered outlet positioned 150 µm above the air-water interface.
3 before start of experiment. The second probe was then moved to a distance r = 0.0402 m.
We waited for 2 minutes from contact of tablet or capillary at the interface for transients to die out before data collection commenced. All data collection was automated through a LabView interface, with independent control for each probe. Each probe collected one second worth of data at a given radial position, then moved a 200 µm step along the radial direction at 300 µm/s speed, and repeated the measurement. In this manner, the first probe a high error for measurements made at least 90µm apart. We therefore resorted to a an unconventional method outlined below to measure shear stress: 1) Two LDV probes were employed, one mounted vertically above the petri dish looking down onto the interface, and the second mounted from below the dish, looking up into the dish.
2) Since surface radial velocities were high enough at radial positions where shear stress was measured, the Bragg cell was disconnected and replaced with the beam splitter to generate a static fringe pattern.
3) The pair of laser beams for each LDV probe were intentionally misaligned to reduce the intersecting spot size down to 3 static fringes with fringe spacing ∆s = 2.55µm.
4) The intentional misalignment resulted in an elliptical beam intersection spot of length 5.1µm along the radial direction and 2µm along vertical direction.
5) The two LDV probes were carefully positioned such that the vertical distance between the spots was 1µm, thus yielding a vertical measurement distance of 5µm. 8) The passage of a colloidal tracer registered as a scatter event with three peaks closely separated in time (∆t). The velocity was then directly obtained by dividing fringe spacing ∆s with ∆t, u = ∆s/∆t. 9) A total of 10 6 such measurements were made to obtain velocity values of 0.1% precision.
The finite difference ∆u = u r (r = r 1 , z = 0) − u r (r = r 1 , z = −5µm) divided by total vertical spacing of 5µm provided the shear stress for data presented in figure 5 of the main text.
Theoretical explanation for the scatter in u z : This measurement of u z is especially susceptible to inaccuracies because it is O(u/r) in magnitude and exists in the presence of a strong second derivative u zz which is O(u/δ 2 d ). The finite-difference procedure yields u z (r, z = 0) ≈ u(r, 0) − u(r, −h) h + |h|u zz (r, z = 0) 2 + |h| + . . . , where h = 5 µ m is the finite difference spacing and represents the measurement error in the velocity. The error is minimized for |h| ∝ /u zz ∝ δ d /u, and the minimum error scales as ∆u z = √ u/δ d , where we consider the dissolution-dominated cases. Given the weak magnitude of the shear u/r, the relative error ∆u z /u z in the measurements scales as /u × r/δ d . The four digits of accuracy in velocimetry implies /u = O(10 −3 ) and typically for our measurements r/δ ≈ 10, which implies that the minimum relative error is about 3×10 −1 . The optimal |h| according to this analysis is about 10 µm. The experimental |h| = 5 µm was determined through experimental trials. The above analysis rationalizes the need of an accurate velocimetry and a small finite-difference step |h| for measuring u z , thus explaining the need for the unconventional technique to measure u z and the increased scatter in those data.
Applying data correction to CA surface velocity: In general, the power-laws presented here and the underlying similarity solutions for the flow are only approximate representations of the exact solutions of the governing equations.
The comparison with the measurements for CA tablets require these corrections to be accounted. While theoretical expressions for the corrections are difficult owing to the sensitive nature of their dependence on the system details, their general structure may be deduced readily. The similarity solution is considered to merely be the first term in an asymptotic series solution, which in this case is u(r, 0) = 1 where the leading order a = K 2/5 2 ν 1/5 f (0) is derived from the similarity solution. The higher order constants b and c depend on the specific system details that cause these corrections, but the general structure of the series in powers of r is generic. The series may be alternatively written as u(r, 0) = 1 (r − ∆r) 3/5 a + c (r − r 0 ) 2 . . . , , where the first order correction is absorbed in the leading order by simply shifting the coordinate. We determine the constants a, ∆r, and c by fitting (2) with the experimental data; we find that the best fits to be a = 7.5 × 10 −4 m 8/5 /s, ∆r = 1.50 × 10 −3 m, with a maximum difference between the experimental data and the fit of 100 µm/s. The difference is too small to discern the next term in the series, i.e. c ≈ 0 from our data. Therefore, we plot u(r, 0) against r − ∆r for the flow driven by CA to account for the finite size of the source. We note that a similar attempt to fit the measured surface velocity to the corrected version of (2) does not yield such a good fit.