Curvature- and temperature-dependent transport of soluble surfactant mixtures to the air-aqueous surface with applications in fluorine-free firefighting foams

Abstract

Understanding surfactant adsorption to air-water interfaces is crucial to eliminating toxic fluorocarbon-based surfactants while retaining firefighting performance. The adsorption of commercial siloxane and Glucopon surfactants is investigated by measuring dynamic surface tension at different length scales using a pendant drop tensiometer and capillary pressure microtensiometer (CPM) for millimeter and micrometer sized bubbles at 23°C and 60°C. The effect of interfacial curvature can be rationalized by rescaling respective times scales for diffusion-limited adsorption. Higher surfactant concentration, higher curvature, and higher temperature favor surfactant adsorption. For constant area adsorption in the capillary pressure microtensiometer, surfactants relevant to firefighting foams show stepwise adsorption. Model mixtures of ethoxylated surfactants of different chain lengths also show this stepwise adsorption, suggesting heterogeneity in tail lengths in the commercial surfactants. Surfactant adsorption is modeled by treating the mixtures as a quasi-single component using the Ward-Tordai equation and the Langmuir adsorption isotherm to characterize the temperature-dependent surfactant properties. Higher temperature increases the adsorption of Dow 502W but decreases the adsorption of Triton X-100 and the Glucopon surfactants, A deeper understanding of how different head and tail group lengths and temperature affect surfactant adsorption will help optimize new surfactant replacements for enhanced firefighting performance.

Graphical Abstract

1. Introduction

Rapidly generating low surface tension at liquid-gas interfaces via surfactant adsorption is essential to optimizing firefighting foams (1, 2), sea spray aerosol generation (3, 4), detergent performance (5), and tablet coating in the pharmaceuticals industry (6, 7). The low surface tensions of fluorinated surfactants result in a positive spreading coefficient for aqueous film forming foams (AFFF) used for fighting liquid hydrocarbon fuel fires (1, 2, 8). Unfortunately, fluorocarbon surfactants are toxic and cause long-term damage to animals, humans and the ecosystem (9–11), which makes it imperative to find replacements that can maintain fire extinguishing performance while being environmentally friendly.

Commercial siloxane and alkyl polyglucoside surfactants have shown potential as replacements for fluorinated surfactants in firefighting foams and are capable of extinguishing bench-scale heptane pool fires within an acceptably short time (2, 12–14). Foams made from mixtures of these surfactants suppress fires more rapidly than foams containing either of the individual surfactants (2, 12–14). As firefighting foams are generated at ambient temperature, then applied to burning fuels, it is also essential to understand how surfactant molecules adsorb and desorb at the air-liquid interface both at ambient and elevated temperatures. Conroy et al. determined that fluorinated and fluorine-free foams lowered a heptane fuel pool temperature to 60°C within seconds of foam deposition onto the pool surface (15). Hence, we have measured the dynamic surface tension and used surfactant transport models to address siloxane and alkyl polyglucoside surfactant adsorption to air-water interfaces at 23°C and 60°C.

Surface and interfacial tension reduction by surfactants requires two processes – surfactant diffusion to the interface, followed by the adsorption of surfactant to the interface. If the characteristic time for diffusion is much greater than that for adsorption, the process is referred to as “diffusion-controlled” or “diffusion-limited” and the surfactant in the sublayer of solution is related to that on the interface by an equilibrium relation such as the Langmuir isotherm (See Eqns. 5–8) (16–25). However, if the characteristic time for diffusion is much less than that for adsorption, surfactant transport is “adsorption-controlled” or “kinetically controlled” (26) ; the concentration profile in the solution is uniform up to the interface and the rate of adsorption is determined by the relative on and off rates of surfactants from the interface. For most small molecule surfactant adsorption described in the literature, diffusion control predominates (17–25, 27).

To quantify the mechanisms of surfactant adsorption, we measure the dynamic surface tension. For a static drop or bubble suspended in a liquid, the capillary pressure difference across the interface is determined by the surface tension and the local mean curvature of the interface. For larger bubbles, gravity causes variations in the hydrostatic pressure across the bubble that modifies the capillary pressure as described by the modified Young-Laplace equation:

$$\begin{gathered} \Delta P_{\mathit{\text{cap}}} \\ =\left({\rho }_o-{\rho }_i\right)\mathit{\text{gz}}+\gamma \nabla \cdot n \end{gathered}$$ (1)

in which $\Delta P_{\mathit{\text{cap}}}$ is the capillary pressure or the difference in pressure between the inside and outside of the bubble at some height, z. ${\rho }_o-{\rho }_i=\Delta \rho$ is the density difference between the fluid outside the bubble and the fluid within the bubble, $\gamma$ is the interfacial tension, $g$ is the acceleration due to gravity, and $z$ is the vertical position along the interface. $\nabla \cdot n$ is the divergence of the normal vector, $n$, to the interface; $\nabla \cdot n=2H=\frac{1}{R_1}+\frac{1}{R_2}$ in which $H$ is the mean curvature of the interface, and $R_1$ and $R_2$ are the principal curvatures (23). In the absence of gravity, the mean curvature is constant, the bubble is spherical and $2H=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$R$}\right.$ in which $R$ is the sphere radius, leading to the Laplace equation: $\Delta P_{\mathit{\text{cap}}}=2\gamma /R$. Scaling the curvature, $\bar{H}=R_{c}H$, and the interface position, $\bar{z}=\raisebox{1ex}{$z$}\!\left/ \!\raisebox{-1ex}{$R_c$}\right.$, by the radius of the capillary tube in the CPM or pendant drop apparatus, $R_c$, leads to a characteristic capillary pressure, $\raisebox{1ex}{$\gamma $}\!\left/ \!\raisebox{-1ex}{$R_c$}\right.$, and a dimensionless expression for the shape of a spherical interface:

$$\begin{gathered} \frac{{\Delta P}_{\mathit{\text{cap}}}}{\raisebox{1ex}{$\gamma $}\!\left/ \!\raisebox{-1ex}{$R_c$}\right.} \\ =\frac{\Delta \rho g{R}_c^2}{\gamma }\bar{z}+2\bar{H} \end{gathered}$$ (2a)

$$\overline{\Delta P}=B_o\bar{z}+2\bar{H}$$ (2b)

The Bond number, $B_o=\frac{\Delta \rho g{R}_c^2}{\gamma }$ is a measure of the extent gravity alters the bubble away from the purely spherical shape dictated by surface tension. For the 30 – 100 μm bubbles in the capillary pressure microtensiometer (CPM), $B_o\ll 1$ and the bubble held at the end of a glass capillary tube is spherical with a constant mean curvature, or $\overline{\Delta P}\cong 2\bar{H}$, which results in the Laplace equation: $\Delta P_{\mathit{\text{cap}}}=\raisebox{1ex}{$2\gamma $}\!\left/ \!\raisebox{-1ex}{$R$}\right.$. As a result, even as the surface tension changes due to surfactant adsorption, the interface remains spherical in the CPM, requiring that the bubble radius and the capillary pressure be measured simultaneously to evaluate the surface tension, $\gamma$ (23, 27). As long as $B_o\ll 1$, the spherical shape of the bubble in the CPM does not change as surfactant adsorbs and the surface tension decreases (23, 27).

In the pendant drop tensiometer, $B_o~1$ and gravity distorts the shape of the bubble away from spherical. These shape changes can be imaged and compared to solutions to Eqn. 2 to directly determine the surface tension. The capillary pressure does not need to be measured, the changes in the bubble shape are sufficient to evaluate the surface tension (28). However, as the surface tension changes during surfactant adsorption, the effective radius and surface area of the bubble change, complicating comparison to models of adsorption (See Eqn. 12) that assume a spherical bubble of constant radius.

The pendant drop tensiometer and CPM methods have other advantages and disadvantages as well; in this work both are used to measure dynamic surface tension to determine the rates of surfactant adsorption. Pendant drop measurements are analyzed by comparing the axisymmetric image analysis of the drop profile to solutions of Eqn. 2; for typical pendant bubble experiments, $R\approx 1.5\text{mm}$ and $B_o\approx 0.3$ for an air-water interface. This value of the Bond number is sufficient to perturb the bubble shape away from spherical sufficiently that image analysis alone can be used to accurately determine the dynamic surface tension (28) .

In the CPM, the capillary radius, $R_c<100\mu \mathrm{m}$, which makes $B_o\approx 0.001$. Hence, the interface has a constant mean curvature and $\Delta P_{\mathit{\text{cap}}}$ is uniform over the bubble surface, which results in a spherical cap shape for the bubble or drop during dynamic adsorption (23, 27). This necessitates the simultaneous measurement of $\Delta P_{\mathit{\text{cap}}}$ and the bubble radius, $R$ to determine the surface tension, , $\gamma$. The capillary pressure across the bubble interface is controlled and measured via a commercial Fluigent Lineup Flow EZ pressure-based flow controller. To determine the bubble radius, a dedicated camera and lens system is used to obtain video images of the capillary and bubble as shown in Fig. S3(b). Edge detection software run by Labview determines the bubble outline in each video frame. A circle (which is the 2-D projection of a spherical cap) of the appropriate radius (red in image) is then fit to the projected bubble edge over the range of the arc shown by the green wedge. This fit determines the radius of curvature of the bubble, $R$. Deviations of the fit to the circle (yellow) must be below a cut-off value or the data is discarded. The inner capillary radius, ${\mathrm{R}}_{\mathrm{c}}$ is determined directly from the image. From the measure values of $\Delta P_{\mathit{\text{cap}}}$ and $R$, the dynamic surface tension is calculated via the Laplace equation, $\frac{R\Delta P_{\mathit{\text{cap}}}}{2}=\gamma$ (23, 27, 29–32). The surface area of the bubble, $A$ is determined from the capillary radius and the bubble radius, $A=2\pi R\left(R-\sqrt{R^2-R_c^2}\right)$ (27).

To simplify the analysis of dynamic adsorption, both the pendant drop and CPM experiments should begin with a pristine bubble with no surfactant adsorbed, or a surface excess concentration, $\Gamma =0$. In practice, forming and stabilizing a bubble takes time during which surfactant can adsorb to the bubble interface along with an additional time lag before accurate surface tension values can be recorded. This complicates comparing the adsorption to the Ward and Tordai equation (Eqn.12) as it requires extrapolating to the $\Gamma =0$ initial condition. This is less of a problem if the total adsorption time is much longer than the initial bubble formation time, which depends on the total surfactant concentration in solution and the bubble size. The CPM provides better time resolution as the smaller bubbles can be formed within ~100 msec compared to the ~ 1 sec for the pendant bubble. However, adsorption to the smaller CPM bubbles reaches equilibrium faster than the larger pendant bubble. The total adsorption time can be increased by decreasing the surfactant concentration in solution; slower adsorption results in more accurate measurements and modeling (25).

The characteristic time, $\tau$, for diffusion limited adsorption is set by a diffusive length scale, $h$, and the monomer surfactant diffusivity, $D_S:\tau =h^2/D_S$. For a flat interface, the diffusive length scale is the planar depletion length, ${\mathrm{h}}_{\mathrm{p}}=\raisebox{1ex}{$\Gamma $}\!\left/ \!\raisebox{-1ex}{${\mathrm{C}}_0$}\right.$, which is the distance from the interface required to saturate a flat interface to a surface concentration, $\Gamma$, for a given bulk concentration, ${\mathrm{C}}_0$. At equilibrium ${\mathrm{C}}_0$ and $\Gamma$ are related by the Langmuir adsorption isotherm (33):

$$\frac{\Gamma \left(\mathrm{t}\right)}{{\Gamma }_{\infty }}=\frac{C_0\left(t\right)}{\frac{1}{\kappa }+C_0(t)}$$ (3a)

$$C_0(t)=\frac{1}{\kappa }\frac{\Gamma \left(t\right)}{{\Gamma }_{\infty }-\Gamma \left(t\right)}$$ (3b)

In which $\kappa$ is the relative affinity or the ratio of adsorption to desorption to the interface of the surfactant molecules exhibiting Langmuir kinetics, and ${\Gamma }_{\infty }$ is the maximum or saturating surface excess. Hence for the Langmuir isotherm,

$${\mathrm{h}}_{\mathrm{p}}=\raisebox{1ex}{$\Gamma $}\!\left/ \!\raisebox{-1ex}{${\mathrm{C}}_0$}\right.=\frac{{\Gamma }_{\infty }}{\frac{1}{\kappa }+C_0}$$ (4)

Eqn. 4 shows that $h_p$ decreases with increasing surfactant concentration. For the surfactants and concentrations used here (See Tables 1, 2), $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\kappa $}\right.\ll C_0$ and ${\mathrm{h}}_{\mathrm{p}}\approx \raisebox{1ex}{${\Gamma }_{\infty }$}\!\left/ \!\raisebox{-1ex}{${\mathrm{C}}_0$}\right.$. The depletion depth, $h_s$, for a spherical cap of radius $R$ is reduced compared to a planar interface due to the curved geometry:

$$\frac{h_s}{h_p}=\left(\frac{R}{h_p}\right)\left[{\left(\frac{3h_p}{R}+1\right)}^{\frac{1}{3}}-1\right]$$ (5)

For $h_p\ll R,\frac{h_s}{h_p}\cong 1-\frac{h_p}{R}$, which is the case for high surfactant concentrations or “large” spheres as is the case for the pendant drop apparatus (23). The literature suggests several possible combinations of $h_s$ and $h_p$ as the appropriate diffusive length scales for a spherical interface (23, 26). One combination is the product of the spherical depletion length, $h_p$, and the dimensionless absorbing boundary condition, $\frac{d\Gamma \left(t\right)}{dt}={\left.D_S\frac{\partial c}{\partial r}\right|}_{r=R}$ that results in ${\tau }_1=\frac{h_s{h}_p}{D_S}$ (26). Another combination is based on scaling the spherical diffusion equation, ${\tau }_2=\frac{h_s^2}{D_S}$ (23). Here, we take the geometric mean of these two to give the time scale for diffusion (23, 24):

$${\tau }_d=\sqrt{{\tau }_1{\tau }_2}=\frac{{({h_s}^3{h}_p)}^{\frac{1}{2}}}{D_s}$$ (6)

For $h_p\ll R,{\tau }_d\approx \frac{h_p^2}{D_s}{\left(1-\frac{h_p}{R}\right)}^{\frac{3}{2}}$ and there is little difference between ${\tau }_1,{\tau }_2$ and ${\tau }_d$ (Table 2). We use Eqn. 6 to determine the ratio of diffusive time scales for the CPM relative to the pendant drop from values of ${\Gamma }_{\infty },{\mathrm{C}}_{\mathrm{s}}$, and $R$ for Triton X100, Dow 502W, and Glucopon surfactants. The timescales and ratios are listed in Table 2.

Table 1.

${\Gamma }_{\infty },\kappa$ and ${\mathrm{D}}_{\mathrm{s}}$ for Triton X100 and Dow 502W at 23°C from fits of the dynamic surface pressure from pendant drop experiments to the Ward-Tordai model. ${\Gamma }_{\infty }$ and $\kappa$ for Glucopon 215UP and Glucopon 225DK at 23°C fitting the equilibrium surface tension from the CPM to the Szyszkowski equation of state (Eqn. 8a). Average values obtained from 3 sets of reading are listed.

Surfactant	${\Gamma }_{\infty }\left(\text{mol}/{\text{m}}^2\right)\times {10}^6$	$\mathit{\kappa }\left(\frac{{\mathit{m}}^3}{\mathit{m}\mathit{o}\mathit{l}}\right)$	${\mathit{D}}_{\mathit{s}}\left({\text{m}}^2/\text{s}\right)\times {10}^{10}$
Triton X100	4.0 ± 0.24	334 ± 57	5.8 ± 1.2
Dow 502W	4.6 ± 0.03*	576 ± 9*	8.7 ± 0.57*
Glucopon 215UP	4.9 ± 0.4	60 ± 13	-
Glucopon 225DK	2.6 ± 0.4	1874 ± 1678	-

^*The values are for the first step of adsorption and should be considered as apparent parameters.

Table 2.

Planar (${\mathrm{h}}_{\mathrm{p}}$) and spherical depletion depths, ${\mathrm{h}}_{\text{SCPM}}$ for CPM and ${\mathrm{h}}_{\text{SPD}}$ for the pendant drop tensiometer with ${\tau }_{\mathit{\text{DCPM}}}/{\tau }_{DPD}$ from Eqns. 4, 5. ${\tau }_{\mathit{\text{DCPM}}}/{\tau }_{DPD}$ was used to rescale the adsorption time for the PD data to overlay the CPM data in Figure 6. The error in ${\tau }_{\mathit{\text{DCPM}}}/{\tau }_{DPD}$ is within ±0.01.

Surfactant	Concentration (ppm)	${\mathit{h}}_{\mathit{p}}\left(\mathit{\mu }\mathit{m}\right)$	${\mathit{h}}_{\mathit{S}\mathit{C}\mathit{P}\mathit{M}}\left(\mathit{\mu }\mathit{m}\right)$	${\mathit{h}}_{\mathit{S}\mathit{P}\mathit{D}}\left(\mathit{\mu }\mathit{m}\right)$	$\frac{{\mathit{\tau }}_{{\mathit{D}}_{\mathit{C}\mathit{P}\mathit{M}}}}{{\mathit{\tau }}_{\mathit{D}\mathit{P}\mathit{D}}}$
Triton X100	25	89	46	82	0.41
	50	46	29	44	0.55
	75	31	22	30	0.63
Dow 502 W	40	110	51	97	0.38
	80	55	33	51	0.51
	120	37	25	35	0.60
Glucopon 215 UP	40	26	19	25	0.65
	80	13	11	13	0.78
	120	8.8	7.8	8.8	0.84
Glucopon 225 DK	40	30	21	30	0.62
	80	15	12	15	0.75
	120	10	8.8	10	0.81

To relate the surface pressure, $\pi ={\gamma }_0-\gamma ,$ (${\gamma }_0$ is the surface tension of the clean water-air interface, 72.8 mN/m at 23°C) to the bulk concentration, $C_0$, Eqn. 3b is combined with the Gibbs adsorption isotherm, Eqn.7, and integrated:

$$\begin{gathered} \Gamma \\ =\frac{-1}{R_{G}T}\frac{\partial \gamma }{\partial \text{In(}C\text{)}} \end{gathered}$$ (7)

to give the Szyszkowski equation of state (EOS):

$$\pi ={\gamma }_0-\gamma =R_{G}T{\Gamma }_{\infty }\text{ln}\left(1+\kappa C_0\right)$$ (8a)

$R_G$ is the gas constant (8.3 J⋅mol⁻¹⋅K⁻¹) and $T$ is the absolute temperature (33). Eqn. 8a is valid below the critical micelle concentration (CMC). At concentrations near but below the CMC, $\raisebox{1ex}{$\partial \gamma $}\!\left/ \!\raisebox{-1ex}{$\partial \text{ln}\left(C\right)$}\right.$ is constant for small molecule surfactants, meaning the surface is close to saturated, and the slope of the surface tension vs $\text{ln}\left(C\right)$ gives $\Gamma \cong {\Gamma }_{\infty }$. Eqn. 8a has been used to fit equilibrium surface tension vs surfactant concentration data to obtain values for ${\Gamma }_{\infty }$ and $\kappa$ for a variety of different surfactants (17, 26, 27, 29–31). From Eqns. 3b and 8a, the surface pressure and the surface concentration are related as:

$$\pi =-R_{G}T{\Gamma }_{\infty }\mathit{\text{ln}}\left(1-\frac{\Gamma }{{\Gamma }_{\infty }}\right)$$ (8b)

The rate of diffusion of surfactant to the surface of a spherical bubble of constant radius $R$ (which is maintained in the CPM experiments by the feedback loop) is given by the transient spherical diffusion equation:

$$\frac{\partial c}{\partial t}=D\left[\frac{{\partial }^{2}c}{\partial r^2}+\frac{2}{r}\frac{\partial c}{\partial r}\right]$$ (9)

which describes the time dependent concentration profile, $c(r,t)$, in the domain $R<\mathrm{r}<\infty$. This assumes a spherically symmetric geometry in which the surfactant concentration is a function only of the radial coordinate, $r$, and time, $t$. The initial condition is that the concentration is uniform, $\mathrm{c}(r,t=0)=C_0$ for $r>R$. Far from the bubble interface, $c(r\to \infty ,t)=C_0$ for all t. An interfacial mass balance shows the rate of change of the surface concentration, $\Gamma \left(t\right)$, is equal to the diffusive flux of surfactant at the bubble surface:

$$\frac{d\Gamma \left(t\right)}{dt}={\left.D_S\frac{\partial c}{\partial r}\right|}_{r=R}$$ (10)

with an initial condition $\Gamma (t=0)=0$. For diffusion-limited adsorption, $\Gamma \left(t\right)$ is related to the surfactant concentration in solution at the bubble interface $C_s\left(t\right)=\mathrm{c}(r=R,t)$ by:

$$\Gamma \left(t\right)={\Gamma }^{*}\left(C_s\left(t\right)\right)$$ (11)

in which ${\Gamma }^{*}\left(C_s\right)$ denotes the equilibrium isotherm. Ward and Tordai derived an implicit solution to Eqn. 9 for $\Gamma \left(t\right)$ (16):

$$\Gamma (t)=\sqrt{\frac{D_s}{\pi }}{C}_0\left[2\sqrt{t}-{\int }_0^t \frac{\frac{C_s\left(\tau \right)}{C_0}d\tau }{\left(\sqrt{t-\tau }\right)}\right]+\frac{D_s{C}_0}{R}\left[t-{\int }_0^t \frac{C_s\left(\tau \right)}{C_0}d\tau \right]$$ (12a)

The first two terms describe adsorption to a planar surface and the second two terms describe how adsorption depends on the radius of curvature of the interface, $R$. Eqn. 3b is used to replace $C_s\left(t\right)$ by the surface excess concentration, $\Gamma \left(t\right)$, in Eqn. 12a assuming local equilibrium at the interface:

$$\left.\left.\Gamma (t)=\sqrt{\frac{D_s}{\pi }}{C}_0\left[2\sqrt{t}-\frac{1}{\kappa C_0}{\int }_0^t \frac{\frac{\Gamma \left(\tau \right)}{{\Gamma }_{\infty }-\Gamma \left(\tau \right)}d\tau }{\left(\sqrt{t-\tau }\right)}\right]+\frac{D_s{C}_0}{R}\right]t-\frac{1}{\kappa C_0}{\int }_0^t \frac{\Gamma \left(\tau \right)}{{\Gamma }_{\infty }-\Gamma \left(\tau \right)}d\tau \right]$$ (12b)

The integral terms make the equation implicit and numerical analysis is required to find $\Gamma \left(t\right)$. Details of the numerical scheme used here are provided in the SI Section 3. At early times of the adsorption, $\frac{C_s\left(t\right)}{C_0}\ll 1$, so the initial adsorption rate can be approximated as:

$$\Gamma (t)\cong 2\sqrt{\frac{D_{s}t}{\pi }}{C}_0+\frac{D_s{C}_{0}t}{R}$$ (12c)

Eqn. 12c is also the exact solution for diffusion to a spherical interface with a perfectly adsorbing boundary condition, $C(r=R,t)=0$ for all $t$. Eqn. 8b relates $\Gamma \left(t\right)$ to $\pi \left(t\right)$ to fit to experimental surface pressure vs time data. In Eqn. 12b, the bubble radius of curvature, $R$, is assumed to be constant during adsorption with a spherically symmetric geometry; the constant bubble radius control of the CPM makes the data easier to match to the model predictions compared to the PD in which the effective bubble radius and surface area change with time.

Eqns. 3–12 describe adsorption of a single surfactant rather than a surfactant mixture. As a single surfactant adsorbs, the Ward and Tordai model predicts that surface concentration and surface pressure increase monotonically until the interface is saturated and both $\Gamma \left(t\right)$ and $\pi \left(t\right)$ reach constant values (Figures 1–4) (18, 25)^23,34. This is true for many commercial surfactants over certain concentration ranges. However, for other concentrations, commercial surfactant adsorption occurs in two steps; a step of rapid change in surface pressure followed by a plateau indicating limiting adsorption by one surfactant in the mixture, then a second rapid change in surface pressure followed by another plateau corresponding to the limiting adsorption of the second surfactant in the mixture (Figures 2–4). These steps and plateaus depend on the concentration and surface activity of the individual species that make up a surfactant mixture (34–37) (See Figure 7). We use Eqns. 5–12 to model the first step and plateau of such surfactants.

Figure 1.

Dynamic surface pressure for 25 (red, circle), 50 (green, triangle), and 75 (blue, square) ppm Triton X 100 at 23°C using (a) the pendant drop tensiometer (PD) and (b) capillary pressure microtensiometer (CPM). A stable surface pressure is achieved roughly an order of magnitude faster in the CPM (10 s) than with the pendant drop (100 s). The data was interpolated to have constant log (time) intervals. The early time data is shown only after 1 s as it takes time to form a stable bubble. Adsorption begins before the bubble is stabilized as the surface pressure does not begin at 0. The long time (equilibrium) values of the surface pressure were the same for each concentration on either apparatus. Data shown represents 1 representative run out of 3 repeats. Results are consistent across runs with a reproducibility within ±1 mN/m.

Figure 4.

Dynamic surface pressure for a bubble in water with 40 (red, circle), 80 (green, triangle), and 120 (blue, square) ppm Glucopon 225DK at 23°C using (a) PD and (b) CPM. Like Glucopon 215DK, adsorption occurs in two stages in the CPM. At 40 ppm in the CPM, an initial rapid rise leads to a plateau from 10 −500 seconds. A second rapid rise in surface pressure occurs from 500 – 2000 seconds before leveling off at the equilibrium surface pressure, again suggesting chemical heterogeneity. Data shown represents 1 representative run out of 3 repeats. Results are consistent across runs with a reproducibility within ±1 mN/m.

Figure 2.

Dynamic surface pressure for 40 (red, circle), 80 (green, triangle), and 120 (blue, square) ppm Dow 502W at 23°C using (a) PD and (b) CPM. At 40 ppm, Dow 502W has a two-step adsorption in the CPM. The first step is less visible at higher concentrations. The pendant drop tensiometer is only able to capture the first step as the bubble goes unstable and detaches before the second step begins. Data shown represents 1 representative run out of 3 repeats. Results are consistent across runs with a reproducibility within ±1 mN/m.

Figure 7.

Dynamic surface pressure for a bubble in water at 23°C with 13.5 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$ (red, circle), 1.5 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$ (black, triangle), and their 1:4 (green, triangle-upward) and 4:1 by volume mixtures (blue, square) for (a) the CPM at constant area and (b) the pendant drop with no area control. The equilibrium surface pressure of the 13.5 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$ is ~ 17 mN/m which is reached in 5–6 seconds. The equilibrium surface pressure of the 1.5 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$ is ~ 37 mN/m but requires 200–300 seconds to reach. Both 1:4 and 4:1 ${\mathrm{C}}_{10}{\mathrm{E}}_6$ to ${\mathrm{C}}_{14}{\mathrm{E}}_6$ ratio mixtures show two-step adsorption in the CPM when measured at constant bubble area. The general features of the adsorption are similar to those observed for the commercial surfactants in Figures 2–4. Data shown represents 1 representative run out of 3 repeats. Results are consistent across runs with a reproducibility within ±1 mN/m.

In this study, the pendant drop tensiometer and the CPM were used to measure the dynamic surface tension of air bubbles in aqueous solutions of four non-ionic commercial surfactants, Triton X100, Dow 502W, Glucopon 215UP, Glucopon 225DK. In addition, the dynamic surface tension of two pure ethoxylated alcohol surfactants, decyl hexaethylene glycol ether $\left({\mathrm{C}}_{10}{\mathrm{E}}_6\right)$, and tetradecyl hexaethylene glycol ether $\left({\mathrm{C}}_{14}{\mathrm{E}}_6\right)$ was measured. Adsorption of 1:4 and 4:1 mixtures of ${\mathrm{C}}_{10}{\mathrm{E}}_6:{\mathrm{C}}_{14}{\mathrm{E}}_6$ were also measured to show how mixtures of surfactants with different concentrations and surface activity can lead to a two-step adsorption. Dow 502W and both Glucopon surfactants have been used in mixtures as replacements for perfluorinated surfactants (PFAS) in fire-fighting foams (13, 38). Adsorption rates and equilibrium surface tension as a function of surfactant concentration are used to characterize the surfactant molecular parameters ${\Gamma }_{\infty },\kappa$ and $D_s$ at different temperatures (23°C and 60°C) for Triton X100 and Dow 502W. ${\Gamma }_{\infty }$ and $\kappa$ are determined for Glucopon 215UP and Glucopon 225DK from equilibrium surface tension data. All the commercial surfactants exhibit stepwise adsorption over certain concentration ranges in the CPM on small bubbles of constant radius. Stepwise adsorption is also found for known mixtures of ${\mathrm{C}}_{10}{\mathrm{E}}_6$ and ${\mathrm{C}}_{14}{\mathrm{E}}_6$. The similarity of the model mixture stepwise adsorption confirms that the commercial surfactants are mixtures of surfactant species with different length hydrophobic groups.

The time evolution of surface tension in 100 μm bubbles in the CPM and millimeter sized bubbles in the pendant drop apparatus can be superimposed by scaling the adsorption time, ${\tau }_d=\frac{{\left({h_s}^3{h}_p\right)}^{\frac{1}{2}}}{D_s}$ using the values of ${\Gamma }_{\infty },{\mathrm{C}}_0$, and $R$ for each surfactant and bubble. The agreement of the scaled data between the CPM and PD confirms adsorption is diffusion-limited. This work provides the data necessary to understand dynamic surfactant transport and interface stabilization for fluorine-free firefighting foam alternatives. We also point out the dynamic and equilibrium effects of surfactant mixtures on adsorption that may provide benefits for environmentally friendly fire-fighting foams.

2. Materials and Methods

2.1. Materials

Triton X100 (Fisher Scientific), Glucopon 215UP (BASF Corporation), Glucopon 225DK (BASF Corporation), and Dow Corning 502W Additive (Dow Corning Co, referred to as Dow 502W) were used as received. These surfactants are liquids at room temperature, so all concentrations are measured in terms of parts per million by volume. All four commercial surfactants are advertised as mixtures of closely related species (see Section S1 in the SI for details). Two pure ethoxylated alcohol surfactants, decyl hexaethylene glycol ether (${\mathrm{C}}_{10}{\mathrm{E}}_6$), and tetradecyl hexaethylene glycol ether $\left({\mathrm{C}}_{14}{\mathrm{E}}_6\right)$ were purchased from Fisher Scientific and were stored at 4°C until used. To make the test solutions, the appropriate volume of surfactant was added to 200 mL HPLC grade water (Fisher Scientific, CAS No. 7732–18-5). Fresh solutions of Dow 502W were made and used the same day because Dow 502W hydrolyzes in aqueous solutions. The relevant literature values of properties and structures of these surfactants are given in Table S1 and Figure S1 of the SI. All concentrations used were less than the reported critical micelle concentrations (CMC) of each surfactant, as shown in Table S1.

2.2. Pendant Drop Tensiometer

A Krüss DSA-30S Pendant Drop Tensiometer was used to measure the dynamic surface tension of surfactant solutions at 23°C and 60°C. The instrument software program ADVANCE was used to control the drop volume and the drop shape analyzer. A bubble is formed at the tip of the J-shaped needle (outer diameter ~2 mm) at a given injection volume and rate, as shown in Figure S2 in the SI. The assigned time (t = 0) began at bubble injection to capture the initial surface tension change. The surface tension was evaluated from image analysis once per second and the total time for data collection was set. The magnification of the outer diameter of the needle tip was calibrated using pure water at 23°C. The surface tension was determined through fitting multiple bubble profiles for each freshly made solution (28) and was repeated with a second fresh solution to determine the reproducibility and minimize the standard error.

To maintain a temperature of 60°C, a water pump continually drove water at 70°C through the stage holding the sample cuvette and the syringe holder in the Krüss Pendant Drop Tensiometer. A thermocouple was connected to a Thermocouple Data Logger (Pico Technology) and the temperature was recorded by Picolog 6 via a Windows laptop. Simultaneously, surfactant solutions were preheated in a sealed glass bottle (to minimize evaporation) on a hot plate at 100°C. The entire data collection process for the reported results required less than 20 mins to minimize the effects of evaporation on adsorption. For Dow 502W, preheating was done no longer than 30 minutes as prolonged exposure to higher temperatures speeds up surfactant hydrolysis (39). Water evaporation was minimal at these temperatures and had negligible effects on the measured surface tensions.

2.3. Capillary pressure microtensiometer

In the capillary pressure microtensiometer (CPM), an air bubble in the shape of a hemispherical cap is pinned at the end of a 110 μm inside diameter capillary in a 1 mL volume liquid reservoir. The schematic is shown in Figure S3 in the SI. Capillaries of prescribed diameter were pulled from 1.5 mm OD, 1.1 mm ID, 10 cm long borosilicate glass capillaries in a Sutter Instrument P-1000 micropipette puller (Novato, CA), cleaned with Alnochromix and sulfuric acid (Millipore-Sigma), and rinsed with Millipore water. The capillaries were immersed in 5% Xiameter OFS-6124 Silane (Dow Chemical) in ethanol solution and cured by heating under house vacuum at 100°C for one hour to make the capillary hydrophobic. The capillary pressure across the bubble interface is controlled via a commercial Fluigent Lineup Flow EZ pressure-based flow controller with 0.02 mbar resolution with a Lineup Link that allows the pump to interface with the computer (27). The dynamic response of the bubble interface is visualized via an optical lens and camera system as the bubble expands and contracts. Edge detection software run by Labview determines the bubble contour in each video frame. A circle (which is the 2-D projection of a spherical cap) of the appropriate radius is then fit to the projected bubble contour over the range of the arc shown by the green wedge (See Fig. S3b). This fit determines the radius of curvature of the bubble, $R$. Deviations of the fit to the circle must be below a cut-off value or the data is discarded. The capillary radius, ${\mathrm{R}}_{\mathrm{c}}$ is determined directly from the images. The surface tension is determined from the Laplace equation, $\gamma =\frac{R\Delta P_{\mathit{\text{cap}}}}{2}$. To compensate for the variable depth of the bubble relative to the liquid interface in the reservoir (a few millimeters; See Figure S3); the effective capillary pressure is calibrated by comparing to the known surface tension of water at the temperature of interest.

Model-based bubble area control is implemented based on the Laplace equation using LabVIEW software to control the applied capillary pressure during adsorption to maintain a constant bubble radius (27). A new bubble can be generated in a second to provide a fresh interface and surface tension variations caused by surfactant adsorption can be determined with 10 msec time and ± 1 mN/m stress resolution. The interfacial area, $A$, of the bubble can vary from $A=\pi R_c^2$ (flat interface, bubble radius, $R\to \infty$) to $2\pi R_c^2$ (hemispherical interface, bubble radius, $R=R_c$); $R_c$ is the inside radius of the capillary (27). For the 60°C experiments, surfactant solutions were preheated to 80°C on a hot plate before being added to the CPM reservoir. For Dow 502W, the time for preheating was ≤ 15 mins to avoid hydrolysis. Due to the lack of temperature control for the CPM reservoir, the dynamic surface tension evolution was captured over a temperature range of 56.5°C to 63.5°C over ~ 1 min.

2.4. Parameter fitting based on numerical solution of the Ward-Tordai equation

${\Gamma }_{\infty },\kappa$, and $D_s$ are used as fitting parameters to match the experimental data to solutions of the Ward-Tordai equation, Eqn.12b. This work uses the full Ward-Tordai solution to fit the surface tension vs time data throughout the measurement process instead of relying on the early-stage approximation, Eqn. 12c (40). A variety of different numerical approaches have been proposed to discretize the integral terms in the Ward-Tordai equation (25, 41–43). Our approach starts with numerical solutions to the Ward-Tordai equation (Eqn. 12b) for a curved interface of radius, $R$, and then Eqn. 8 is used to relate the calculated surface excess concentration to the measured surface tension. Details are discussed in Section 4 of the SI.

3. Results and Discussion

3.1. Effect of interfacial curvature on surfactant adsorption

Figures 1–4 show the dynamic surface pressure, $\pi ={\gamma }_0-\gamma$, for Triton X100, Dow 502W, Glucopon 215UP, and Glucopon 225DK, respectively, using the pendant drop tensiometer (1a-4a) and the CPM (1b-4b). As the bulk surfactant concentration increases, the rate of adsorption increases, and the surfactants reach a higher surface pressure, indicative that these concentrations are below the surfactant CMC.

For a given bulk concentration, Eqn. 3 shows that the ratio of spherical to planar depletion lengths, $h_s/h_p$, decreases as the bubble radius decreases, and hence adsorption will be faster to the smaller bubbles in the CPM than the PD as shown in Figures 1–4. 25 ppm Triton X100 solution reaches a stable surface pressure in ~100 s for a 110 μm diameter bubble in the CPM compared to ~1000 s for a millimeter sized bubble in the pendant drop tensiometer. The faster kinetics in the CPM makes it more convenient to evaluate the equilibrium surface pressure for slower adsorbing surfactants such as the Glucopon surfactants (Figures 3 and 4).

Figure 3.

Dynamic surface pressure for a bubble in water with 40 (red, circle), 80 (green, triangle), and 120 (blue, square) ppm Glucopon 215UP at 23°C using (a) PD and (b) CPM. In the PD, the surface pressure increases almost linearly, but had not reached equilibrium by 1000 sec. In the CPM, Glucopon 215UP has a two-stage adsorption. The slope changes at ~ 10 seconds with a plateau that ends with a second rapid change in adsorption at ~ 200 seconds. Compare to Figure 7. The two-step adsorption is consistent with the main homologs being alkyl chains of 47% C8, 36% C10, and 10% C12, like the model mixtures of ethoxylated alcohols shown in Figure 7. Data shown represents 1 representative run out of 3 repeats. Results are consistent across runs with a reproducibility within ±1 mN/m.

The small Bond number in the CPM allows the bubble shape to remain a spherical cap during adsorption even as the surface pressure increases. By controlling the applied capillary pressure using the Laplace equation as model feedback control, adsorption in the CPM can be done at a constant radius and surface area, consistent with the assumptions on which the Ward and Tordai equation is based (Eqn. 12). The bubble shape, and hence the bubble radius and surface area change as the surface pressure changes in the pendant drop; this is a requirement of obtaining the surface pressure from analysis of the bubble shape. As a result of the shape changes, especially for large decreases in surface tension, the bubble remains stable for longer in the CPM than the PD. For instance, Dow 502W (Figure 2) bubbles do not remain stable and eject off the needle tip in the PD after ~200s when the surface pressure increases above 40 mN/m.

3.2. Estimation of ${\mathit{\Gamma }}_{\infty },\mathit{\kappa }$ and ${\mathit{D}}_{\mathit{s}}$

We used the fitting approach in Section 2.4 to model adsorption of Triton X100 and Dow 502W adsorption measured using the pendant drop apparatus with three free parameters, ${\Gamma }_{\infty },\kappa$, and ${\mathrm{D}}_{\mathrm{s}}$. For Glucopon 215UP and Glucopon 225DK, insufficient data was collected before the bubble went unstable in the PD to fit the Ward and Tordai equation, so the Szyszkowski equation of state (EOS) was used to obtain values for ${\Gamma }_{\infty }$ and $\kappa$ (Table 1).

Figure 5 shows the dynamic surface pressure of 25 ppm Triton X100 (5a) and 40 ppm Dow 502W (5b) at 23°C using the pendant drop tensiometer (open triangles and circles), as well as the fitted curves based on Eqns. 8 and 12 with three free parameters, ${\Gamma }_{\infty },\kappa$, and ${\mathrm{D}}_{\mathrm{s}}$. Data before 1 second was truncated as the bubbles were unstable, but the solutions to Eqn. 12 were initiated from $\Gamma (t=0)=\pi (t=0)=0$. The fits capture the main features of the data. For Dow 502W, the values of ${\Gamma }_{\infty },\kappa$, and ${\mathrm{D}}_{\mathrm{s}}$ likely correspond to the highest concentration species in Dow 502W, as the higher concentration species adsorbs first. From the fits to the dynamic data, Dow 502W has a larger ${\Gamma }_{\infty }$ than Triton X100 and a higher diffusivity, $D_s$. From the fits to the equilibrium data, Glucopon 215UP has a larger ${\Gamma }_{\infty }$ than Glucopon 225DK. In terms of molecular dimensions, $1/{\Gamma }_{\infty }=0.25\frac{{nm}^2}{\mathit{\text{molecule}}}$ for Triton X100, $0.22\frac{{nm}^2}{\mathit{\text{molecule}}}$ for Dow $502,0.20\frac{{nm}^2}{\mathit{\text{molecule}}}$ for Glucopon 215UP, and $0.38\frac{{nm}^2}{\mathit{\text{molecule}}}$ for Glucopon 225DK. An alkane chain in a crystalline packing has an area/molecule of $~0.2-0.25\frac{{nm}^2}{\mathit{\text{molecule}}}$ (44) suggesting these surfactants have a liquid-like organization on the interface. As discussed in the introduction, ${\Gamma }_{\infty }$ and $\kappa$ can also be obtained from fitting the equilibrium data based on Eqn. 8. As a result, multiple pairs of ${\Gamma }_{\infty }$ and $\kappa$ are possible that can result in similar fits to the Langmuir EOS. More discussion on possible multiple pairs of ${\Gamma }_{\infty }$ and $\kappa$, relatively higher uncertainty in $\kappa$, and comparison with literature data is provided in Section S4 of SI.

Figure 5.

Data (open triangles and circles) and fits to Eqn. 12b (solid lines) of dynamic surface tension for a bubble in water using PD at 23°C for (a) 25 ppm Triton X100 and (b) 40 ppm Dow 502W. The fitted curves are generated from Eqns. 8b and 12b with three free parameters, ${\Gamma }_{\infty },\kappa$ and ${\mathrm{D}}_s$. Data before 1 second was truncated as the bubbles were unstable, but the solutions to Eqn. 12b used the initial condition $\Gamma (t=0)=\pi (t=0)=0$. Data shown represents 1 representative run out of 3 repeats. Experimental data is consistent across runs with a reproducibility within ±1 mN/m while some variability in the estimated parameter values is observed and is listed in Table 1.

3.3. Effect of bubble size on diffusion-controlled surfactant adsorption

Figures 1–4 showed the differences in adsorption rates of surfactants to bubbles in the different devices with different radii. For diffusion-controlled adsorption, the adsorption rates measured with the two devices should agree once the effects of curvature are incorporated into ${\tau }_d$. From the equilibrium EOS parameters, ${\Gamma }_{\infty }$ and $\kappa$, the planar and spherical depletion depths $h_p$ and $h_s$ can be calculated once the bubble radius is known (Eqn. 4, 5). Table 2 lists the planar depletion depth and spherical depletion depths for CPM and pendant drop tensiometer (PD) based on Eqns. 4 and 5, for Triton X100, Dow 502W, Glucopon 215UP, and Glucopon 225DK at the tested concentrations. For CPM, the bubble radius is $60\mu m$. For PD, the bubble radius is an average of the volume equivalent radius of the bubble. This equivalent radius is around $1.2\mathit{\text{mm}}.h_p$ decreases with increasing bulk concentration, which decreases the diffusion time, ${\tau }_d$, for surfactant adsorption consistent with Figures 1–4. The spherical depletion depth for the CPM is smalle than in the PD, as the bubble radii in the CPM (~60 μm) are more than an order of magnitude smaller than those in the PD (~1.5 mm). To a good approximation, $h_{sPD}\approx h_p$, the planar depletion length for the pendant drop.

The ratio of spherical diffusion timescales, $\frac{{\tau }_{{\mathrm{D}}_{\mathit{\text{CPM}}}}}{{\tau }_{\mathrm{D}\text{PD}}}={\left(\frac{h_{s\mathit{\text{CPM}}}}{h_{s\mathit{\text{PD}}}}\right)}^{\frac{3}{2}}$ in the CPM relative to the PD was used to re-scale time for the PD adsorption to compare to the CPM. Figure 6 shows that the adsorption rates in the PD and CPM were similar when the adsorption time in the PD was multiplied by ${\tau }_{{\mathrm{D}}_{\text{CPM}}}/{\tau }_{\text{DPD}}$ (See Figures 1–4). The close correspondence confirms that adsorption was diffusion-limited in both devices.

Figure 6.

Rescaling the adsorption time in the PD by multiplying actual adsorption time (Figs. 1–4) by ${\tau }_{{\mathrm{D}}_{\mathit{\text{CPM}}}}/{\tau }_{\text{DPD}}$ to scale the adsorption time for the larger bubbles in the PD to the smaller bubbles in the CPM. (a) 25 ppm (red, circle), 50 ppm (green, triangle), and 75 ppm (blue, square) Triton X100, (b) 40 ppm (red, circle), 80 ppm (green, triangle), and 120 ppm (blue, square) Dow 502W, (c) 40 ppm (red, circle), 80 ppm (green, triangle), and 120 ppm (blue, square) Glucopon 215UP, and (d) 40 ppm (red, circle), 80 ppm (green, triangle), and 120 ppm (blue, square) Glucopon 225DK. The open symbols represent PD, and the closed symbols represent CPM data. The overlap between the PD and CPM data showed that adsorption was

3.4. Two-step adsorption is a mixture effect

The adsorption of Dow 502W, Glucopon 215UP, and Glucopon 225DK occurred in two distinct steps in the CPM as shown in the semi-log plots in Figures 2b–4b. An initial rapid rise in surface pressure was followed by a plateau in which the surface pressure was roughly constant, then a second rapid rise and plateau led to a final, equilibrium surface pressure. These commercial surfactants are mixtures with varying numbers of ethylene oxide or glucose head groups and different number of methylene or siloxyl units in the tail groups (See Section S1 and Figure S1 in SI). Traube’s rule states that for a homologous series of surfactants with the same headgroup and tailgroups made up of alkane chains, the concentration needed to reach a given surface pressure decreases by a factor of three for each methylene group in the chain (33). Hence, the shorter chain length species requires a much higher concentration to reach a given surface pressure than the longer chain length species. To demonstrate these differences can lead to the two-step adsorption we see in Figures 2b–4b, we mixed decyl hexaethylene glycol ether $\left({\mathrm{C}}_{10}{\mathrm{E}}_6\right)$, and tetradecyl hexaethylene glycol ether $\left({\mathrm{C}}_{14}{\mathrm{E}}_6\right)$ that had identical headgroups and a methylene group difference of 4 in alkane chain length. From Traube’s rule, it should require a $3^4=81$ higher concentration of ${\mathrm{C}}_{10}{\mathrm{E}}_6$ compared to ${\mathrm{C}}_{14}{\mathrm{E}}_6$ to reach a given surface pressure. This is consistent with the ratios of CMC’s as ${\mathrm{C}}_{10}{\mathrm{E}}_6$ has a reported CMC of ~ 63 × 10⁻⁵ mol/liter, while ${\mathrm{C}}_{14}{\mathrm{E}}_6$ is reported to have a CMC of 0.8 × 10⁻⁵ mol/liter, a factor of 79 (45). However, the planar depletion length, $h_p=\frac{\kappa {\Gamma }_{\infty }}{1+\kappa C_s}$, is inversely proportional to the bulk concentration, making the diffusion time, ${\tau }_d$, inversely proportional to the square of the concentration (Eqn. 4). Hence, a ten-fold higher concentration of ${\mathrm{C}}_{10}{\mathrm{E}}_6$ should adsorb to the interface faster and reach an intermediate surface pressure rapidly, while a lower concentration of ${\mathrm{C}}_{14}{\mathrm{E}}_6$ should adsorb more slowly but increase the surface pressure more.

Figure 7a shows the dynamic surface tension measured with the CPM for 13.5 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$ (red circles) and 1.5 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$ (black triangles) aqueous solutions and their mixtures to validate this model of step-wise adsorption. These concentrations were chosen so that the ~10-fold ratio of ${\mathrm{C}}_{10}{\mathrm{E}}_6$ to ${\mathrm{C}}_{14}{\mathrm{E}}_6$ should result in significantly different time scales for diffusion-limited adsorption of the two surfactants. However, as the 10-fold ratio is much less than the 81-fold ratio predicted by Traube’s rule for equal surface pressures, the equilibrium surface pressure of the 13.5 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$ is ~ 17 mN/m (red circles) while that of the 1.5 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$ is ~ 35 mN/m (black triangles). Figure 7a shows that the time required to reach these equilibrium surface pressures is about 5–6 seconds for the 13.5 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$ and about 200–300 seconds for the 1.5 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$, roughly consistent with the inverse square dependence of ${\tau }_d$ on bulk concentration for diffusion limited adsorption; the variation from the inverse square dependence on concentration is likely due to differences in $\kappa$ and ${\Gamma }_{\infty }$ between the two surfactants (see Eqn. 4).

When these two surfactant solutions are mixed in either a 1:4 and 4:1 ${\mathrm{C}}_{10}{\mathrm{E}}_6$ to ${\mathrm{C}}_{14}{\mathrm{E}}_6$ ratio by volume, both mixtures show two-step adsorption in the CPM when measured at constant bubble area. The general features of the two-step adsorption are similar to those observed for the commercial surfactants (Figures 2b–4b). For the 4:1 mixture (blue squares), with a net concentration of 10.8 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$ and 0.3 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$, the adsorption initially overlays the pure ${\mathrm{C}}_{10}{\mathrm{E}}_6$ adsorption with a rapid initial increase in surface pressure over the first 5–6 seconds to reach a plateau at ~ 17 mN/m. The surface pressure is then roughly constant until about 50 seconds, when a second rise occurs to a surface pressure of ~ 27 mN/m, which is approximately the surface pressure of an 0.3 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$ solution (45). The second rise occurs at a somewhat slower rate than the rise in the pure ${\mathrm{C}}_{14}{\mathrm{E}}_6$ solution, consistent with the lower concentration of ${\mathrm{C}}_{14}{\mathrm{E}}_6$.

For the 1:4 mixture with a net concentration of 2.7 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$ and 1.2 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$, the mixture overlays the long-time surface pressure behavior of the pure ${\mathrm{C}}_{14}{\mathrm{E}}_6$ solution (inverted green triangles) and reaches a similar equilibrium surface pressure. The initial adsorption is slowed relative to the pure ${\mathrm{C}}_{10}{\mathrm{E}}_6$ solution due to the lower concentration of ${\mathrm{C}}_{10}{\mathrm{E}}_6$, but the mixture adsorbs faster than the pure ${\mathrm{C}}_{14}{\mathrm{E}}_6$ solution, even though the ${\mathrm{C}}_{14}{\mathrm{E}}_6$ concentration is lower in the mixture as the total surfactant concentration is higher. The surface pressure undergoes an inflection point rather than a full plateau at about 17 mN/m; all four curves cross at about 17 mN/m and 50 seconds. In both mixtures, the adsorption curves are dominated by the surfactant in relative volume excess in the mixture.

The two-step adsorption is not as apparent in the pendant drop apparatus (Figure 7b). Adsorption is slower in the pendant drop apparatus, and it is difficult to maintain a stable bubble for sufficiently long to reach the final equilibrium surface pressures. The changing area of the bubble in the pendant drop likely smooths out the rapid rises and plateaus visible in the CPM data. Figure 8a shows the change in bubble surface area for the 4:1 mixture of ${\mathrm{C}}_{10}{\mathrm{E}}_6$ and ${\mathrm{C}}_{14}{\mathrm{E}}_6$. During the adsorption process, the bubble surface area increases from 21 to 26 mm², or about ~20%. This increase in surface area dilutes the surface concentration and requires additional adsorption during the plateau before the surface pressure can increase. This slows the rapid rise of surface pressure and extends the plateaus. Figure 8b shows that if the CPM operates at constant capillary pressure, which causes the surface area of the bubble to increase throughout the adsorption process, the two-step adsorption disappears just like in the pendant drop apparatus. The surface concentration is continuously diluted by the increasing bubble surface area which eliminates the second step in the adsorption. However, if the surface area is kept constant (except for the initial 5 seconds during the rapid rise in surface pressure during which the bubble is stabilizing and area control is difficult), the second step of the adsorption is clearly visible.

Figure 8.

(a) Surfactant adsorption for 4:1 mixture of 13.5 ppm ${\mathrm{C}}_{10}{\mathrm{E}}_6$:1.5 ppm ${\mathrm{C}}_{14}{\mathrm{E}}_6$ (by volume) in the pendant drop apparatus. The bubble surface area increases by ~20% throughout the adsorption process (blue stars, right axis). The change in surface pressure, $\pi$, (black squares, left axis) is monotonic with no plateaus. (b) Dynamic surface pressure (black, left axis) for the same 4:1 mixture using the CPM operating at constant applied capillary pressure, $\Delta P$, (black squares) or constant surface area (black circles). The y-axis on left (blue) shows the change in surface area of the bubble, A, with time, for the two modes. The two-step behavior only occurs when the bubble is held at constant area. The initial decrease in area is due to imperfect feedback control during the initial rapid change in surface tension. Data shown represents 1 representative run out of 3 repeats. Results are consistent across runs with a reproducibility within ±1 mN/m for surface pressure and ±0.5 % for surface area.

These results show that the likely origin of the two-step adsorption of the commercial surfactants is the variation in number of methylene groups in the alkyl chains of Glucopon215UP and Glucopon225DK (See Figure S1 in the SI). Rodriguez et al. used HPLC to determine the alkyl range and distribution in Glucopon 215UP to be 47% ${\mathrm{C}}_8$,36% ${\mathrm{C}}_{10}$, and 10% ${\mathrm{C}}_{12}$ chains (46). The small fraction of ${\mathrm{C}}_{12}$ and the larger fractions of ${\mathrm{C}}_8$ and ${\mathrm{C}}_{10}$ are consistent with the concentration ratios used in the two-step adsorption proposed for the model mixtures of ${\mathrm{C}}_{10}{\mathrm{E}}_6$ and ${\mathrm{C}}_{14}{\mathrm{E}}_6$ (Figure 4, 7). This should result in fast adsorption of the higher concentration ${\mathrm{C}}_8$ and ${\mathrm{C}}_{10}$ alkyl chain fractions to an intermediate surface tension value followed by slow adsorption of the more dilute but higher surface activity ${\mathrm{C}}_{12}$ fraction as we see in Figure 3. Values reported in the original patent claim the degree of glucoside polymerization ranges from 1.5 to 1.7, indicating one or two glucoside units per surfactant (47). This small variation in the headgroups likely would not lead to significant changes in surface tension at a given concentration or the CMC (46).

Similar to Glucopon 215UP, Glucopon225DK is reported to have a distribution of alkyl chain lengths resulting in a two-step adsorption. Dow 502W is reported by the manufacturer to have a distribution of trisiloxane tail groups which should have a similar effect to the distribution of alkyl chain lengths in the Glucopon surfactants (2, 13, 33, 45). The implications of a two-step adsorption on the generation and stability of foams are unknown, although the fast adsorption, followed by slower equilibration to a higher surface pressure may be advantageous.

3.5. Surfactant adsorption at elevated temperatures

Figure 9 shows the surface pressure of Triton X100, Dow 502W, Glucopon 215UP, and Glucopon 225 DK at 23°C (open blue symbols) and 60°C (filled red symbols). The surface tension of water decreases to 66 mN/m at 60°C compared to the 72.8 mN/m at 23°C so a surface pressure of $\pi =6\mathrm{m}\mathrm{N}/\mathrm{m}$ corresponds to a surface tension of $\gamma =60\mathrm{m}\mathrm{N}/\mathrm{m}$ at 60°C compared to a surface tension of $\gamma =66\mathrm{m}\mathrm{N}/\mathrm{m}$ at 23°C. The general features of the adsorption rates are similar to those at 23°C for each surfactant. Dow 502 adsorption and surface pressure are greater at 60°C compared to 23°C at all concentrations tested, while Triton X100 and the Glucopon surfactants have slower adsorption and lower surface pressures at 60°C compared to 23°C.

Figure 9.

Plots of surface pressure change for a bubble in water using the PD at 60°C (red, filled symbols) and 23°C (blue open symbols) for (a) Triton X100, (b) Dow 502W, (c) Glucopon 215UP and (d) Glucopon 225DK. Only Dow 502W adsorbs more and has a higher surface pressure at 60°C compared to 23°C at short and long times. Data shown represents 1 representative run out of 3 repeats. Results are consistent across runs with a reproducibility within ±1 mN/m.

Figure 10 shows the fit of the Ward and Tordai equations (Eqn. 12b and 8b) to the pendant drop measurements of the dynamic surface pressure of 25 ppm Triton X100 (10a) and 40 ppm Dow 502W (10b) at 60°C. The fits to Eqns. 8 and 12 at both 60 and 23°C (Figure 5) are equally good.

Figure 10.

Plots and fitted curves of dynamic surface tension for a bubble in water using PD at 60°C with (a) 25 ppm Triton X100 and (b) 40 ppm Dow 502W. The fitted curves are generated based on Eqns. 8b and 12b. Data before 1 second was truncated as the bubbles were unstable, but the solutions to Eqn. 12b used the initial condition $\Gamma (t=0)=\pi (t=0)=0$. Data shown represents 1 representative run out of 3 repeats. Experimental data is consistent across runs with a reproducibility within ±1 mN/m while some variability in the estimated parameter values is observed and is listed in Table 2.

Table 3 lists ${\Gamma }_{\infty },\kappa$, and ${\mathrm{D}}_{\mathrm{s}}$ generated from the fits to Eqns. 8 and 12 for Triton X100 and Dow 502W at 60°C compared to those at 23°C (in parentheses). ${\Gamma }_{\infty }$ for Triton X100 decreases with increasing temperature, while ${\Gamma }_{\infty }$ for Dow 502W increases with increasing temperature. From Eqn. 8a, increasing ${\Gamma }_{\infty }$ increases the equilibrium surface pressure, ${\pi }_{eq}$, while decreasing ${\Gamma }_{\infty }$ decreases ${\pi }_{eq}$, as we observe for Dow 502 and Triton X100, respectively (Figure 9). $\kappa$ has a similar magnitude and decreases with increasing temperature for both Triton X100 and Dow 502W.

Table 3.

${\Gamma }_{\infty },\mathrm{k}$ and ${\mathrm{D}}_{\mathrm{s}}$ for Triton X100 and Dow 502W at 60°C from fits of the dynamic surface pressure from pendant drop experiments to the Ward-Tordai model. The numbers in parentheses correspond to the values at 23°C. The differences in ${\Gamma }_{\infty }$ and k are not significant between 60°C and 23°C. The main difference is in the increase in diffusivity at the higher temperature. Average values obtained from 3 sets of reading are listed.

Surfactant	${\Gamma }_{\infty }\left(\text{mol}/{\text{m}}^2\right)\times {10}^6$	$\kappa \left(\frac{m^3}{mol}\right)$	$D_s\left({\text{m}}^2/\text{s}\right)\times {10}^{10}$
Triton X100	3.7 ± 0.3 (4.0)	195 ± 68 (334)	11.5 ± 1.1 (5.8)
Dow 502W	6.1 ± 0.1 (4.6)	207 ± 3.4 (576)	39.6 ± 3.9 (8.7)

The increased rate of adsorption may be an effect of the increased surfactant diffusivity, $D_s$ at the higher temperature. Table 3 shows that $D_s$ increases with temperature which is consistent with the Stokes-Einstein Law for diffusion, $D_s=\frac{k_{B}T}{6\pi \eta a}$, where $k_B$ is the Boltzmann constant, $\eta$ is the viscosity of the solvent (water in this work), and $a$ is the radius of the diffusing particle (surfactant molecules in this work) (48). The Stokes-Einstein Law predicts an increase in the diffusivity with temperature directly as well as through the temperature effects on the solution viscosity. For the temperatures studied in this work, temperature changes from 296 K (23°C) to 333 K (60°C), and the viscosity of water decreases from 0.94 mPa.s to 0.47 mPa.s (48). According to the Stokes-Einstein law, the surfactant diffusivity should increase by a factor of ~2.25 from 23°C to 60°C. From Table 3, the diffusivity increases by a factor of 1.97 for Triton X100 and 4.54 for Dow 502W, both of which are of the correct order of magnitude given the limitations of the model.

Conclusions

The dynamic surface tension of aqueous solutions of Triton X100, Dow 502W, Glucopon 215UP, and Glucopon 225DK was measured with both a commercial pendent drop apparatus and a newly designed constant area capillary pressure microtensiometer (CPM). The dynamic and equilibrium surface tensions as a function of surfactant concentration are necessary to evaluate these commercial surfactants as replacements for toxic fluorinated surfactants in new firefighting foam formulations. Scaling arguments show that adsorption measurements using the pendent drop and capillary pressure microtensiometer, in which the bubble radii differ by more than an order of magnitude, superimpose when adjusted for the different dimensionless time scales consistent with diffusion-controlled adsorption. The smaller bubbles used in the CPM show faster adsorption and reach equilibrium faster as expected for diffusion-limited adsorption.

The commercial surfactant adsorption was also measured at 60°C to model the temperature of a heptane fuel pool fire following application of an aqueous firefighting foam (15). Both the dynamic and equilibrium surface pressure for Triton X100, Glucopon 215UP, and Glucopon 225DK decrease with increasing temperature, while the dynamic and equilibrium surface pressure for Dow 502W increases with increasing temperature. For Triton X100 and Dow 502W, fits of adsorption rates to the Ward-Tordai equation (Eqn. 12) and Langmuir adsorption isotherm (Eqn. 8) were used to determine values of the pseudo single component Langmuir isotherm parameters ${\Gamma }_{\infty }$ and $\kappa$, and estimate the diffusivity, $D_S$, at 23 and 60°C$.\kappa$ did not change significantly with temperature, while $D_s$ increased with temperature. Fits of the equilibrium surface pressure vs concentration to the Szyszkowski equation of state were used to predict $\kappa$ and ${\Gamma }_{\infty }$ for the slower adsorbing Glucopon 215UP, and Glucopon 225DK that could not be fit to the Ward-Tordai equation.

As the Bond number, $B_o=\frac{\Delta \rho g{R}_c^2}{\gamma }\ll 1$, in the CPM, the spherical cap shape of the bubble is maintained during adsorption and the bubble radius can be kept constant by adjusting the applied capillary pressure via the microfluidic pressure controller to satisfy the Laplace equation, $\Delta P_{\mathit{\text{cap}}}=\raisebox{1ex}{$2\gamma $}\!\left/ \!\raisebox{-1ex}{$R$}\right.$. Maintaining a constant area and constant shape during adsorption in the CPM showed that the commercial surfactants adsorbed in two-steps; an initial rapid rise in surface pressure followed by a plateau, followed by a second rapid rise to an equilibrium surface pressure plateau. The two-step adsorption was not observed in the pendant drop apparatus in which the bubble area and shape changed continuously during adsorption. Model mixtures of ${\mathrm{C}}_{10}{\mathrm{E}}_6$ and ${\mathrm{C}}_{14}{\mathrm{E}}_6$ ethoxylated alcohol surfactants at various ratios showed similar step-wise adsorption. The two-step adsorption is likely a mixture effect; the first step is due to the rapid adsorption of the higher concentration, but less surface active, ${\mathrm{C}}_{10}{\mathrm{E}}_6$, which leads to the first plateau, followed by the slower adsorption of the lower concentration, higher surface active ${\mathrm{C}}_{14}{\mathrm{E}}_6$, which leads to the second, and equilibrium plateau. Glucopon 215UP has a larger fraction of the shorter alkyl chain length species, which have a lower surface activity, and a smaller fraction of the higher alkyl chain length species, with a higher surface activity (46) (46). This step-wise adsorption occurs when all species that make up the commercial surfactant mixture are below their respective CMCs. If the shorter chain surfactant is at higher concentration than the longer chain surfactant, the shorter chain surfactant adsorbs first as the characteristic time for diffusion-limited adsorption is inversely proportional to the square of the surfactant concentration. The lower concentration, longer chain surfactant adsorbs more slowly but increases the surface pressure more due to Traube’s rule that states the decrease in surface tension increases by a factor of three for every additional methylene group at a given concentration (33). This two-step behavior confirms the chemical heterogeneity of the commercial surfactants as well as suggesting the relative concentrations of the various species in Dow 502W and Glucopon 225DK where the actual fractions are not known. The pendant drop method does not capture this behavior because the surface area increase and shape change during the surfactant adsorption process dilutes the surface concentration and smooths out the stepwise surfactant adsorption.

These results can be used to understand the properties of surfactants under consideration to replace toxic fluorinated surfactants in foams for fire suppression at both 23°C, the temperature at which the foams are typically prepared, and 60°C, the temperature of the liquid surface during pool fires. The dynamic and equilibrium surface pressure for Glucopon 215UP, and Glucopon 225DK decrease with increasing temperature, while the dynamic and equilibrium surface pressure for Dow 502W increases with increasing temperature. This may be one reason why the Dow 502W has shown promise as a replacement surfactant for fire-fighting foams. The commercial surfactants tested are mixtures with different length head and tail groups at various mole fractions; the detailed composition of these mixtures can lead to significantly different rates of surfactant adsorption and minimum surface tension. Correlating the adsorption results for bubbles of significantly different sizes shows that adsorption is diffusion-limited over the range of concentrations typically used in firefighting foam applications. The adsorption behavior of these replacement surfactants may be responsible for foam dynamics and stability. During the initial stages of foam formation higher adsorption rates will result in faster bubble generation and at later times, lower surface tensions may result in more stable foams. Pure surfactants have a monotonic decrease in surface tension with time while the commercial surfactants have an intermediate step during which the surface tension is roughly constant before starting the second surface tension decrease, eventually reaching a final equilibrium value. This may produce more stable foams with optimized properties for fire-fighting.