Effect of impurities in description of surface nanobubbles

Surface nanobubbles emerging at solid-liquid interfaces of submerged hydrophobic surfaces show extreme stability and very small (gas-side) contact angles. In a recent paper Ducker [W. A. Ducker, Langmuir 25, 8907 (2009)]. conjectured that these effects may arise from the presence of impurities at the air-water interface of the nanobubbles. In this paper we present a quantitative analysis of this hypothesis by estimating the dependence of the contact angle and the Laplace pressure on the fraction of impurity coverage at the liquid-gas interface. We first develop a general analytical framework to estimate the effect of impurities (ionic or nonionic) in lowering the surface tension of a given air-water interface. We then employ this model to show that the (gas-side) contact angle and the Laplace pressure across the nanobubbles indeed decrease considerably with an increase in the fractional coverage of the impurities, though still not sufficiently small to account for the observed surface nanobubble stability. The proposed model also suggests the dependencies of the Laplace pressure and the contact angle on the type of impurity.

Figure 1

Schematic of Nanobubble, with $\theta$ being the gas-side contact angle and $R_b$ being the radius of the bubble.

Figure 2

(a) Variation in the surface pressure with bulk ionic concentration (here $c$ is in $\text{mol}/{\text{m}}^3$ and $c_0=1\text{mol}/{\text{m}}^3$) of nonionic surfactants ${\text{BHBC}}_{16}$. The continuous line is the result obtained from the present simulation [by iteratively solving Eqs. (16, 17), with ${\omega }_0=6.023\times {10}^4{\text{m}}^2/\text{mol}$, ${\omega }_1=2.5\times {10}^5{\text{m}}^2/\text{mol}$, $K$ $\left(\text{fitting}\text{variable}\right)=1.9\times {10}^7$, $R=8.314\text{J}/\text{mol}\text{K}$, and $T=300\text{K}$], whereas the squares are the data from experimental results [28, 29]. (b) Variation in the surface pressure with bulk ionic concentration (expressed in $\text{mol}/{\text{m}}^3$) of ionic surfactants SDS. The continuous line is the result obtained from the present simulation [by iteratively solving Eqs. (16, 25), with ${\omega }_0=6.023\times {10}^4{\text{m}}^2/\text{mol}$, ${\omega }_1=3.4\times {10}^5{\text{m}}^2/\text{mol}$ [50], $K$ $\left(\text{fitting}\text{variable}\right)=4\times {10}^3$, $R=8.314\text{J}/\text{mol}\text{K}$ and $T=300\text{K}$], whereas the squares are the data from experimental results [26, 27].

Figure 3

Variation in (a) the modified surface tension ${\sigma }_{\text{lg}}^{/}$ (with ${\sigma }_{\text{lg}}=0.072\text{N}/\text{m}$) (b) the gas side contact angle $\theta$ (with $\Delta \sigma ={\sigma }_{\text{sl}}-{\sigma }_{\text{sg}}=0.025\text{N}/\text{m}$) and (c) the Laplace pressure $\Delta p$ (with radius of the spherical cap $R_b=100\text{nm}$) with the total fractional coverage of impurities for different possible values of the partial molar area $\left({\omega }_0\right)$ of the solvent for the case of nonionic surface impurities. In these plots ${\omega }_{0,\text{th}}=6.023\times {10}^4{\text{m}}^2/\text{mol}$. Other constant parameters used for the plots are the gas constant $R=8.314\text{J}/\text{mol}\text{K}$ and $T=300\text{K}$.

Figure 4

Variation in (a) the modified surface tension ${\sigma }_{\text{lg}}^{/}$ (with ${\sigma }_{\text{lg}}=0.072\text{N}/\text{m}$) (b) the gas side contact angle $\theta$ (with $\Delta \sigma ={\sigma }_{\text{sl}}-{\sigma }_{\text{sg}}=0.025\text{N}/\text{m}$) and (c) the Laplace pressure $\Delta p$ (with radius of the spherical cap $R_b=100\text{nm}$) with the total fractional coverage of impurities for the case of ionic surface impurities. Results are shown both with and without the ionic contribution to the surface pressure. Here, we study the effects of variation partial molar area of the impurities $\left({\omega }_1\right)$ and consider that there is only one kind of impurity at the interface which is ionic in nature $\left(\text{valence}=1\right)$, with all the ions acting as surface active or potential-determining ions. Other parameters used for the plots are ${\omega }_0={\omega }_{0,\text{th}}=6.023\times {10}^4{\text{m}}^2/\text{mol}$, $R=8.314\text{J}/\text{mol}\text{K}$, and $T=300\text{K}$.