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The dynamic steady state of an electrochemically generated nanobubble Yuwen Liu, Martin A. Edwards, Sean R. German, Qianjin Chen, and Henry S. White Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b04607 • Publication Date (Web): 26 Jan 2017 Downloaded from http://pubs.acs.org on January 31, 2017
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The dynamic steady state of an electrochemically generated
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nanobubble
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Yuwen Liu†, ‡, Martin A. Edwards†, Sean R. German†, Qianjin Chen† and Henry S. White*†
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†
Department of Chemistry, University of Utah, Salt Lake City, UT 84112, United States
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‡
College of Chemistry and Molecular Sciences, Wuhan University, Wuhan, 430072, China
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Liu Page 2
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Abstract
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This article describes the dynamic steady state of individual H2 nanobubbles generated by H+
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reduction at inlaid and recessed Pt nanodisk electrodes. Electrochemical measurements coupled
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with finite element simulations allow analysis of the nanobubble geometry at dynamic
5
equilibrium. We demonstrate that a bubble is sustainable at Pt nanodisks due to the balance of
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nanobubble shrinkage due to H2 dissolution and growth due to H2 electrogeneration. Specifically,
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simulations are used to predict stable geometries of the H2/Pt/solution three-phase interface and
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the width of exposed Pt at the disk circumference required to sustain the nanobubble via steady-
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state H2 electrogeneration. Experimentally measured currents, i ss , corresponding to the
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electrogeneration of H2, at or near the 3-phase interface, needed to sustain the nanobubble are
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between 0.2 to 2 nA for Pt nanodisks electrodes with radii between 2.5 nm to 40 nm. However,
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simple theoretical analysis shows that the diffusion-limited currents required to sustain such a
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single nanobubble at an inlaid Pt nanodisk are 1 to 2 orders larger than the observed values. Finite
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element simulation of the dynamic steady state of a nanobubble at an inlaid disk also
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demonstrates that the expected steady-state currents are much larger than the experimental
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currents. Better agreement between the simulated and experimental values of i ss is obtained by
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considering recession of the Pt disk nanoelectrode below the plane of the insulating surface,
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which reduces the outward flux of H2 from the nanobubble and results in smaller values of i ss .
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Introduction
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Our group has recent reported the electrogeneration of individual H2 1 - 3 O23, N2 4
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nanobubbles at Pt nanodisk electrodes. Nanobubbles are electrogenerated by oxidation or
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reduction of an species. For instance, an individual N2 nanobubble can be formed at a Pt nanodisk
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electrode by oxidizing hydrazine, N2H4 → N2 + 4H+ + 4e-, creating a supersaturated solution of
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N2 adjacent to the electrode surface. A liquid-to-gas phase transformation associated with the
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formation of a single nanobubble at the electrode surface is readily indicated in the voltammetric
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reponse by a sudden drop in N2H4 oxidation current to a small but non-zero value. Similarly, an
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individual H2 nanobubble can be formed at a Pt nanodisk by electroreduction of H+, 2H+ + 2e- →
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H2, creating a supersaturated solution of H2 that nucleates a nanobubble13, while an individual O2
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nanobubble can be generated by oxidation of water3, 2H2O → O2 + 4H+ + 4e-.
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Due to their nanoscale dimensions, the Laplace pressure within these nanobubbles is on
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the order of ~100 atm, resulting in a large outward flux of the gas from the bubble towards the
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bulk solution. The nanobubble stability results from continous electrogeneration of dissolved gas
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at the circumference of the Pt nanodisk, resulting in a dynamic steady state. The rate of steady-
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state electrogeneration of dissolved gas is readily measured as an electrical current, i ss , in the
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experiment, providing insight into the nanobubble dynamics, as discussed below. If the potential
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of the electrode is shifted to a value where the electrochemical reaction no longer occurs, the
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nanobubble dissolves on the time scale of 10s of milliseconds.5
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A recent report of the dependence of the electrochemical current on external pressure
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demonstrates that the stable nanobubble has a radius that is equal, within error, to the radius of the
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Pt nanodisk. 6 Thus, the width of the exposed Pt at the disk circumference required to sustain the
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electrochemical reaction is expected to be exceedingly small, on the order of a nanometer (for
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calibration, the diameter of a Pt atom is ~0.25 nm). 2
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This article addresses the geometry of a H2 nanobubble formed at the Pt nanodisk of
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radius a, as schematically depicted in Figure 1. We focus on H2 nanobubbles because we have
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accumulated sufficient experimental data to compare and draw meaningful conclusions with
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theory and simulations; however, the results presented in this aricle are general and can be
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applied to other electrogenerated nanobubbles.
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Figure 1. Schematic of a spherical H2 nanobubble at a Pt nanodisk electrode of radius a. In this
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generalized drawing, the nanobubble has a contact angle equal to θ, and H+ reduction to H2 occurs
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at the exposed ring of Pt of width w. The radius of bubble footprint on the Pt nanoelectrode is
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thus rnb = a – w. The outward flux of H2 from the nanobubble (i.e., dissolution) is balanced by an
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inward flux of H2 (i.e., growth), resulting from the electrochemical reaction at the exposed Pt
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surface. The rate of this reaction is a function of w, decreasing as w → 0; thus, a dynamic
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equilibrium exists for specific values of a, w, and θ.
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As shown in Figure 1, the nanobubble exists at a dynamic equilibrium that is established
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by a balance of the outward flux of H2 from the nanobubble (i.e., bubble dissolution) with an
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inward flux of H2 (i.e., bubble growth), the latter supported by H+ reduction at a ring of exposed
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Pt of width w at the bubble edge. A portion of the Pt nanodisk must remain unobstructed by the
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nanobubble in order to support H2 electrogeneration, as a bubble would immediatedly begin to
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dissolve if the bubble covered the entire Pt electrode surface (i.e., if w = 0), which would prevent
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the electron-transfer reaction from occuring. Similarly, the nanobubble covers a finite portion of
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the surface, as evidenced from the decrease in electrochemical current upon nucleation and
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growth. Thus, w must have a value between 0 and a. As noted above, the lifetime of nanobubbles
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prepared in our laboratory is on the order of 10s of milliseconds when the electrochemical
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reaction is turned off by shifting the electrode potential. 5 The condition of dynamic equilibrium
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thus suggests that a unique geometrical condition(s) exists that is defined by the bubble contact
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angle, θ (the parameter largely defining the rate of H2 flux out of the bubble), and w (the
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parameter largely defining the rate of H2 flux into the bubble). Herein, we report finite element
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simulations that determine the states θ, w of dynamic equilibrium that allow for the existence of
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an electrogenerated steady-state bubble. Steady-state solutions are compared to experimental
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values of i ss from our previous studies to determine the most likely bubble geometry.
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Over the past decade there have been numerous reports regarding the anomalous behavior
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of very long-lived “nanobubbles” 7-22. In contrast, the nanobubbles observed in our laboratory
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behave classically1-6, 23 . Specifically, the measured surface tensions agree with macroscopic
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values,6 and the nanobubbles rapidly dissolve when the solution is not supersaturated with the
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corresponding gas.
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9 10
The Electrochemical Current Corresponding to H2 Generation at Dynamic Equilibrium
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Figure 2a shows the cyclic voltammetric response (100 mV/s) at a 27 nm Pt disk
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electrode in deoxygenated 0.5 M H2SO4 (reproduced from Q. Chen, et al 20141). The Pt disk was
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prepared by a previously reported method in which a sharpened Pt wire is embedded in glass, and
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the glass is polished to expose a Pt disk that is nominally flush with the glass surface. 24 However,
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scanning electron microscopy shows that in most cases (including our own work involving
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nanobubbles),6 the Pt electrode is recessed into the glass, a point that we address in the
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simulations and conclusions, vide infra. The radius of the Pt disk electrodes, assuming that they
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are coplanar with the insulating glass shroud, are determined by measuring the diffusion limited
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current for oxidation of a soluble redox species (e.g., ferrocene), as detailed in reference 1 and
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elsewhere.
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The voltammogram shows the characteristic behavior corresponding to electrogeneration
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of a single nanobubble. Starting from an initial positive value, the potential of the Pt nanodisk
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electrode is scanned towards negative values. Beginning at ~-0.25 V, a cathodic current is
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observed that corresponds to the reduction of H+ to H2. The current rapidly increases until
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p reaching a peak value, inb , where it suddenly drops to a low ‘steady-state current’, i ss . This drop
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in current is characteristic of the sudden nucleation and growth of a single stable nanobubble that
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occupies a significant portion of the electrode surface, blocking the H+ flux to the electrode. The
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nanobubble persists in a dynamic steady state, as illustrated in Figure 1, as long as the production
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of H2 is maintained. Dissolution of H2 away from the nanobubble-solution interface occurs by
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diffusion and is balanced by production of H2 at the exposed portion of the electrode that enters
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into the bubble. Additional details of nanobubble nucleation and growth, and their dependencies
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on reactant concentration, scan rate, and surfactant are found in our previous publications.1-6, 23 p
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We have previously reported how the observed peak currents ( inb ) are related to the
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critical surface concentration of H2 (or N2 and O2) necessary for the formation of a critical
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p nucleus that grows to a stable nanobubble. 123 We observe that inb increases linearly with the
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electrode radius, indicating that the critical nucleus size and pressure are independent of the
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electrode size.1, 4 The critical concentration of H2 at the instant of nucleation is 310 times greater
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than the saturation concentration at room temperature and atmospheric pressure. Analysis of the
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supersaturation data indicate that the critical nucleus has a radius of 3.6 nm and contains ~1700
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molecules of H2.23
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Once the nanobubble forms, the steady-state current ( i ss ) is a measure of the rate of H2
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electrogeneration required to balance the diffusive loss of H2 from the bubble. We note that while
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H2 electrogeneration required to sustain the nanobubble, not all of the electrogenerated H2 is
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required to enter into the bubble; a fraction of the H2 may simply diffuse away from the electrode
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into the bulk solution.
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Figure 2b shows a summary of measured values of i ss as a function of the electrode
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radius, a, for 20 different nanoelectrodes. The values of i ss are in the range of 0.2 to 2.4 nA for
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p electrodes with radii between 2.5 and 45 nm, and, unlike inb , these steady-state currents show no
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consistent dependence on the electrode radius. The magnitude and variability of i ss provide
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constraints on interpretation of possible dynamic equilibrium states determined by finite element
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simulations, as described in the following section.
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(a)
(b) 0
ss
-10
2
i / nA
a = 27 nm
i ss i / nA
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p inb
-0.5 0.0 0.5 Voltage (V) vs Ag/AgCl
1 0 0
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a / nm
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Figure 2. Voltammetric generation of individual H2 nanobubbles at Pt nanoelectrodes. (a) Cyclic
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voltammogram (100 mV/s) of a 27 nm Pt electrode in deoxygenated 500 mM H2SO4. (b) Plot of
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residual current vs. electrode radius (voltammetrically determined) for 20 different
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nanoelectrodes. Figure reproduced from Q. Chen, et al. 20141.
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Finite Element Simulations.
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Two electrode geometries were simulated in our investigation of the dynamic equilibrium of
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a nanobubble. In case 1 (Figure 3a), a single spherical-cap nanobubble resides on a nanoelectrode
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that is coplanar with the surrounding glass. In case 2 (Figure 3b) the bubble resides on a recessed
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disk nanoelectrode and the bubble can be described as a hemispherical cap atop a cylindrical
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column. A detailed description of the simulations, including the equations solved, geometry,
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mesh, and boundary conditions, is given in the Supporting Information.
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Figure 3. Schematic of the electrode and gas-transfer processes involved in maintaining an electrogenerated nanobubble in dynamic equilibrium at a nanodisk electrode: (a) the Pt disk is coplanar with the glass surface, and (b) the Pt disk is recessed relative to the glass. The color scale is from a specific finite element simulation, and represents the concentration of H+ in vicinity of the nanobubble and electrode. Red corresponds to a high H+ concentration (500 mM ) while dark blue corresponds to a very low H+ concentration (~0 mM). The color plot is presented here to provide a semi-quantitative representation of how the electrochemical reaction creates local H+ concentration gradients. In this and following figures, the simulation domain extends into the bulk solution far beyond the region shown.
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Briefly, in all of the simulations, it is assumed that the bubble dissolution is limited by diffusion and that the dissolved gas concentration at the bubble-solution interface, Cs, is always in equilibrium with the bubble’s internal pressure, pin, as described by Henry’s Law (Eq 1):
Cs = K H pin
(1)
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-1 -1 where KH is Henry’s constant (0.00078 mol ⋅ L ⋅ bar 25). The Young-Laplace equation is used to
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define the internal pressure of the nanobubble:
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pin =
2γ + pex rnb / sin θ
(2)
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where γ is the surface tension of gas/solution interface ( = 0.072 N/m26), rnb is the pinned radius
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of the nanobubble, θ is the inner contact angle, and pex is the external pressure (1 atm is assumed
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throughout). Application of these classical macroscopic thermodynamic expressions to nanoscale
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bubbles will be discussed in a later section.
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The reduction of H+ at the exposed Pt is assumed to occur at a diffusion-limited rate
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described by Fick’s 1st and 2nd laws. The concentration of H+ at the Pt surface was set equal to
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zero because iss is experimentally measured at a large negative overpotential relative to the
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standard redox potential of the H+/H2 couple. The total amount of H2 entering or leaving the
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bubble per unit time was calculated by an integral of the H2 flux over the bubble surface; a similar
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integration, performed over the exposed electrode surface was used to calculate the
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electrochemical current for H+ reduction.
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Results and Discussion
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1. Steady-state nanobubble geometries at an inlaid nanodisk electrode
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We first examine the results from simulations of a bubble on an electrode coplanar with its
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glass support, corresponding to the schematic illustration shown in Figure 3a. To find the
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possible stationary states of nanobubbles at a disk nanoelectrode, the fluxes at the bubble-solution
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interface for various combinations of w and θ were computed. For a nanobubble to maintain its
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size, the net integrated flux of H2 across the total nanobubble-solution surface should be zero,
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which means that any H2 outflow should be exactly compensated by the H2 entering the
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nanobubble resulting from the electrode reaction. A net positive H2 inflow corresponds to a
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growing nanobubble, while a net negative inflow corresponds to a shrinking nanobubble.
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Figure 4 shows the H2 concentration distributions (color) and the H2 flux distribution at the
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gas/solution interface for three nanobubbles at a 27 nm electrode where the inner contact angle
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was fixed at 150° and the exposed radius (w) was varied at values of 0.55 nm, 0.33 nm and 0.15
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nm as labeled.
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stationary (w = 0.33 nm), and shrinking bubbles (w = 0.15 nm). Arrows are used to indicate the
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direction and magnitude of the H2 flux, and scale bars for the flux magnitude are shown to the
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right of the figures.
As discussed below, these bubbles correspond to growing (w = 0.55 nm),
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Figure 4. Finite element simulations of nanobubble bubbles at inlaid disk nanoelectrodes as a
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function of the width, w, of exposed Pt needed to support H2 electrogeneration. Colors represent
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the concentration of dissolved H2, while arrows on the surface of the bubble represent the flux of
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H2 into/out of the bubble. Top panel: simulations over entire surface of the bubble. Bottom
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panel: expanded view of the simulation near the Pt/bubble/solution interface, that show the H2
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concentrations and fluxes near the three phase boundary. The inward H2 flux arrows in the upper
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panel are too small to be observed at the scale used, but are shown in the expanded views in the
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lower panel. NB: The scale of the flux arrows differs by a factor of 100 between the upper and
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lower (magnified) plots; see scale bars (right). Widths of exposed Pt (a) w = 0.55 nm, (b) 0.33 nm,
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and (c) 0.15 nm, and correspond to growing, stationary or shrinking bubbles, respectively. Other
3
= 500 mM, CHbulk = 0 mM). simulation parameters were held constant (θ = 150o, a = 27 nm, CHbulk + 2
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The three bubbles shown in Figure 4 have nearly identical radii of curvature (53.1 – 53.7 nm)
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and thus their internal pressures given by the Young-Laplace equation (eq 2) are also almost
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identical (28.0 – 27.7 Bar). This is again reflected by nearly equal surface concentrations (21.8 –
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21.6 mM) as governed by Henry’s law (Eq. 1). Because we assume equilibrium at the gas/liquid
8
interface, the concentration of dissolved H2, Cs, just outside the bubble is constant at any point on
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the bubble surface. Consequently, the outflux of H2 at nanobubbles/solution interface is
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approximately equal along the upper surface regions of the three bubbles, as shown by flux
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arrows in the upper part of Figure 4.
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Influx of H2 occurs at the bubble-solution interface near the three-phase boundary. The
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same interfacial concentration Cs again determined by Henry’s law, but the H2 flux is now into
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the bubble due to the high concentration of electrogenerated H2 at the exposed Pt. The total H2
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influx increases significantly as the width of exposed Pt is increased, as is shown in the lower part
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of Figure 4 where the scale of the flux arrows differs by a factor of 100 from those in the upper
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part. For the nanobubble with w = 0.33 nm of exposed Pt (center), the integrated flux of H2 across
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the total nanobubble surface equals zero, and this bubble geometry represents a true steady-state,
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as long as the electrode is held at a potential to generate H2.
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In essence, the stable state exists because there is an effective negative feedback between the
21
radius of exposed Pt and the amount of H2 entering into the nanobubble. An exposed Pt width (w)
22
larger than 0.33 nm will result in net inflow of H2 into the bubble and growth of the nanobubble
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(e.g., w = 0.55 nm, Figure 4). Conversely, a value of w less than 0.33 nm results in a shrinking
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nanobubble (e.g., w = 0.15 nm, Figure 4).
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Figure 5. Finite element simulations of bubbles on inlaid disk nanoelectrodes as a function of
4
contact angle, θ, at the three-phase contact line. Colors represent the concentration of dissolved
5
H2, while arrows on the surface of the bubble represent the flux of H2 into/out of the bubble. Top
6
panel: simulations over the surface of the bubble. Bottom panel: expanded view of the simulation
7
near the Pt/bubble/solution interface, that show the H2 concentrations and fluxes near the bubble.
8
The inward H2 flux arrows are not apparent in the upper panel, but are shown in the expanded
9
views in the lower panel. NB: The scale of the flux arrows differs by a factor of 100 between the
10
upper and lower (magnified) plots; see scale bars (right). Contact angles of (a) θ = 150°, (b) θ =
11
159°, and (c) θ = 170° represent growing, stationary or shrinking bubbles, respectively. Other
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bulk simulation parameters were held constant (w = 1 nm, a = 27 nm, CH+ = 500 mM, CHbulk = 0 mM ). 2
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Henry’s law (eq.1) combined with Young-Laplace equation (eq 2) dictate that the surface
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concentration is highest for the smallest bubble, which has the highest internal pressure. While
15
this is difficult to see directly in Figure 5 a-c, this relationship has the consequence of reducing
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the flux of H2 as the contact angle θ in increased, which is clearly visible in the upper plots from
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the size of the flux arrows. While the flux of H2 out of the nanobubble decreases with increasing
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θ, the integrated outflux of H2 from the nanobubble increases with θ, due to the increase in
2
surface area with increasing θ.
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The lower plots in Figure 5 show that the flux of H2 into the bubble near the three-phase
4
boundary decreases as θ is increased, and this variation can be attributed to a decrease in
5
production of H2 at the exposed Pt electrode due to diminished H+ transport through the
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narrowing ‘wedge’ of solution between the bubble and the electrode/glass surface. Analogous to
7
the interpretation of the results in Figure 4, where we fixed the θ and varied w, a stable
8
equilibrium configuration occurs when the integrated flux of H2 at the nanobubble surface equals
9
zero. This condition occurs for θ = 159°, which corresponds to the middle plot of Figure 5.
10
In above paragraphs, we described two specific examples where stable configurations of a
11
nanobubble exist at a 27 nm radius Pt electrode for specific values of the w,θ parameters (i.e., w =
12
0.33 nm, θ = 150o in Figure 4 and w = 1.0 nm, θ = 159o in Figure 5). We used this strategy of
13
determining other stationary states by identifying w,θ combinations where the integrated H2 flux
14
is zero. In Figure 6a, we plot the integrated H2 flux from simulations over a wider range of w and
15
θ; points along the zero flux intercept represent stable nanobubble configurations. NB: only
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representative choices of θ are shown to aid clarity. By integrating the flux of H+ on the exposed
17
Pt surface, the current iss for each of these configurations was also calculated and is presented in
18
Figure 6b. In particular, iss corresponding to the nanobubbles in stationary states are shown as the
19
dashed line.
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Figure 6. ). (a) Integrated flux of H2 as a function of w for a family of contact angles θ
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(increasing contact angles, 90°, 120°, 150° and 170°, indicated by curved arrows). (b)
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Corresponding plot of the current vs. w. Circles/dashed lines represent unique combinations of w
4
and θ giving rise to a stable bubble, i.e., unique pairs of the θ,w parameters where the integrated
5
H2 flux is equal to zero.
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Figure 7 shows the dependence of w and iss on θ. In general, the width of the exposed Pt
7
ring decreases with decreasing inner contact angles. For instance, w = 12 nm, 5 nm and 1.3 nm
8
for nanobubbles with contact angle of 175o, 170o and 160o, respectively. This dependence arises
9
because a larger current (and thus larger w) is required as the area of the bubble/solution interface increases with θ. The results are for a Pt disk radius, a, of 27 nm.
stationary width i ss i eq
10 9
8
8
4
7
0 100
11
12
120
140
160
w / nm
10
i / nA
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θ /ο
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Figure 7 Dependence of the width of exposed Pt ring, w, (red line) and residual current (blue
13
circles) on the contact angle, θ, for stable nanobubbles. Blue solid line represents the H2 flux out
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of a stable bubble (in units of equivalent electrochemical current) assuming no electrode
15
reaction27, see text for details. The results are for a Pt disk radius, a, of 27 nm.
16
Our simulations also indicate that stable nanobubbles can also exist when the exposed Pt
17
width is less than w = 5 pm and the inner contact angles is less than 90o. However, this is not
18
physically realistic, as the van der Waals radius of a hydrogen atom (106 pm28) is considerably
19
larger than this value of w, and the hydrogen evolution reaction (HER) mechanism (Tafel-
20
Volmer-Heyrovsky 29 ) involves the chemiadsorption of hydrogen atoms on exposed Pt atoms.
21
Thus, stationary nanobubble states with contact angles less than 90° were not further considered.
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Figure 7 also shows the equivalent electrode current ieq for a nanobubble with the same
2
geometry in the absence of electrode reaction (blue line). While this quantity monotonically
3
increases with the contact angle, the stationary state current reaches a minimum when the inner
4
contact angle is ~125° (blue circles in Figure 7). This difference between ieq and iss occurs
5
because only a fraction of H2 produced by the electrode reaction flows into the nanobubble when
6
the inner contact angle is less than 125°, the remaining fraction of H2 diffuses away into the bulk
7
solution. Conversely, essentially 100% of H2 produced by electrode reaction enters into the
8
nanobubble near the 3-phase contact line when θ is larger than 125o.
9
Figure 7 shows that the minimum value of iss for stationary nanobubbles formed at a 27 nm
10
Pt disk nanoelectrode is about ~8 nA, which occurs when contact angle θ = 125° and w = 0.013
11
nm. However, the experimental residual currents are in the range of 0.2 nA to 2.4 nA (see Figure
12
2b). The difference between the simulated steady-state and experimental values of iss suggest the
13
failure of at least one of the assumptions used in simulations and is discussed in the following
14
paragraphs.
15
The validation of Henry’s Law for the solubilities of H2, He, and N2 in water under pressures as
16
high as 300 atmospheres (which is the inner pressure of a 5 nm bubble in water) has been
17
confirmed by Gerth30 through experimental measurements. Although Henry’s Law is valid for
18
nanobubbles at thermodynamic equilibrium state, transfer of H2 at the gas-liquid interface may be
19
rate limiting, as we previously suggested5 in explaining an ~1000x longer life of electrogenerated
20
nanobubbles compared with predictions based on diffusion limited dissolution. Instead of using
21
the thermodynamic equation of Henry’s Law, we suggested that the flux of H2 out of the bubble is
22
described by the kinetic expression:
23
J = pin k f − kbCs
(3)
24
where J is the flux out of the bubble (in mol⋅m-2⋅s-1) at every point, pin again is the internal
25
pressure of nanobubble, Cs the surface concentration, kf and kb are rate constants of the forward
26
and backward directions, respectively. We previously estimated kf ~ 10-9 mol·N-1·s-1 based on
27
experimental electrochemical measurements5, which corresponds to the dissolution of a ~20 nm
28
radius H2 bubble on the timescale of 10 ms (directly measured by experiment 5). Similarly, we
29
have measured a lower limit of ~3 ms for the timescale of growth of a H2 nanobubble to steady
30
state.2 These timescales are several orders of magnitude longer than the voltammetric timescale
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for measuring iss, which in slow scan voltammetric experiments (e.g., the data in Figure 2a) is on
2
the order of 1 s. Thus, the use of Henry’s law to compute the surface concentration in the
3
simulations, and the subsequent comparison of the resulting iss values to experimental data,
4
appear valid.
5
We also considered the assumption that the surface tension of the nanoscale bubbles may
6
be different from macroscopic values used in our simulations. However, we recently reported
7
experimental measurements of for H2 bubbles with radius as small as 7 nm, and found no
8
dependence of on bubble radius6. Nagayama et al.31 suggested that Young-Laplace equation is
9
not applicable to nanobubbles based on molecular dynamics simulation. However, nanobubbles in
10
their simulations are single component, in contrast to nanobubbles filled with (non-vapor) gas.
11
Conversely, validation of the Young-Laplace equation to nanobubbles and the lack of any
12
dependence of on the radius for nanobubble radius were reported by Matsumoto32, also based
13
molecular dynamics simulation of the nucleation of stable nanobubbles. Matsumoto’s research
14
considered “ideal” bubbles, without adsorption at the surfaces. However, microbubbles have
15
often been shown to have negative charges.33,34 The reported charge surface charge density of
16
bubbles is not very high (about 5 × 10-5 electron per nm2 in pure water 35) would not expect to
17
change γ appreciably.
18
In addition to the above assumptions regarding the applicability of Henry’s law and the
19
Laplace-Young equation for nanoscale bubbles, we also have assumed above that the Pt
20
nanoelectrode is ideally flush with the surface of the glass shroud. Recent reports in the literature
21
suggest that the Pt surface can become recessed in the glass by physical or chemical processes
22
(e.g., chemical dissolution).
23
nanoelectrode on nanobubble steady states through finite element simulations. Qualitative, partial
24
confinement of the H2 nanobubble in a recession would reduce the dissolution rate of H2,
25
resulting in smaller values of iss. This occluded geometry is anticipated to yield lower simulated
26
values of iss that would be in better agreement with experimental values. We consider this
27
electrode geometry in the following section.
28
2. Steady-state nanobubble geometries at a recessed disk nanoelectrode.
36,37
In the next section we consider the consequences of recessed
29
A schematic illustration of the geometry used for simulations of a nanobubble at a recessed
30
nanodisk electrode is shown in Figure 8, with geometric quantities labeled. We make the
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assumption of a hemispherical cap, θ = 90o, atop a cylindrical gas pillar of height H. The
2
nanobubble is contained within a cylindrical recession of height L.
3 4 5
Figure 8. Schematic of the geometry used for simulating a bubble on a recessed Pt disk electrode.
6
Pertinent geometric parameters are labelled. Simulation domain (detailed in Supporting
7
Information Figure SI 3) extends into the bulk solution far beyond the region shown.
8
As in the previous sections, we simulate the concentration profile of the gas in solution for
9
various geometrical parameter (exposed Pt and pillar height), from which we calculate H2 fluxes
10
and iss, and deduce the conditions for a stable geometry by finding solutions when the integrated
11
H2 flux at the gas/solution interface is zero.
12
Figures 9 and 10 show the H2 concentration profiles (color) and fluxes at the liquid-gas
13
interface (arrows) as function of the exposed Pt (w) and gas pillar height (H) varied, respectively
14
at a 27 nm disk nanoelectrode position at the bottom of a 54 nm deep recession.
15
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Figure 9. Finite element simulations of bubbles on recessed disk nanoelectrodes (a = 27 nm, L =
3
54 nm) as a function of the width of exposed Pt, w. Colors represent the concentration of
4
dissolved H2, while arrows on the surface of the bubble represent, and are proportional to, the flux
5
of H2 into/out of the bubble. Note, the scale of the flux arrows differs by a factor of 40 between
6
the upper and lower (magnified) plots, see scale bars to the right of the figure. Exposed Pt widths
7
of (a) w = 1 nm, (b) w = 0.8 nm, and (c) w = 0.6 nm represent growing, stationary, and shrinking
8
bubbles, respectively. Other simulation parameters were held constant (θ = 90°, a = 27 nm, L =
9
bulk 2a = 54 nm, H = 0.3a, CH+ = 500 mM, CHbulk = 0 mM ). 2
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1 2
Figure 10. Finite element simulations of bubbles on recessed disk nanoelectrodes (a = 27 nm, L =
3
54 nm) as a function of the nanobubble height, H. Colors represent the concentration of dissolved
4
H2, while arrows on the surface of the bubble represent, and are proportional to, the flux of H2
5
into/out of the bubble. Note, the scale of the flux arrows differs by a factor of 40 between the
6
upper and lower (magnified) plots; see scale bars to the right of the figure. Bubble heights of (a)
7
H/L = 0.10, (b) H/L = 0.36, and (c) H/L = 0.60, represent growing, stationary and shrinking
8
bubbles, respectively. Other simulation parameters were held constant (θ = 90°, a = 27 nm, L =
9
bulk 2a = 54 nm, w = 1.0 nm, CH+ = 500 mM, CHbulk = 0 mM ). 2
10
It can be readily seen from Fig. 10 that the flux of H2 from a nanobubble at the recessed
11
electrode is mainly determined by the height of nanobubble, H. Nanobubbles with smaller H (i.e.,
12
deeper in the recessed electrode) will have a smaller H2 outflux, requiring a smaller
13
electrochemical current to maintain the stationary state. On the other hand, the flux of H2 into the
14
nanobubble is decreased when either H and w decrease. Negative feedback also exists in this
15
situation.
16
Pairs of geometrical parameters (H,w) corresponding to nanobubbles in stationary states,
17
corresponding as before to zero integrated H2 flux at the gas-solution interface) were determined.
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We have plotted these results in Figure 11, showing the steady-state bubble height normalized to
2
the recess depth (H/L) as a function of the recess depth normalized to the electrode radius, a (L/a).
3
Data are plotted for two different values of the Pt disk radius, a = 27 and 55 nm. Figure 11b also
4
shows the values of iss corresponding to these steady state configurations.
(a)
5
(b) w = 1.0 nm, a = 27 nm w = 1.0 nm, a = 55 nm w = 0.1 nm, a = 27 nm w = 0.1 nm, a = 55 nm
0.2
0.0
w = 1.0 nm, a = 27 nm w = 1.0 nm, a = 55 nm w = 0.1 nm, a = 27 nm w = 0.1 nm, a = 55 nm
4.0
0.4
i ssnb / nA
Recess depth H / L
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2.0
0.0
0
10
20
Recess ratio L / a
0
10
20
Recess ratio L / a
6
Figure 11. Stable bubbles on recessed electrodes. (a) The steady-state height ratio (bubble
7
height/electrode recess depth, H/L) as a function of recession depth (normalized by electrode
8
radius, L/a), for different electrode sizes and dimensions of exposed Pt (see legend). (b) Steady-
9
state current ( i ss ) calculated for the stable bubble configurations reported in part (a).
10
In Fig.11a, the dependence of the normalized stable nanobubble height (H/L) as a function of
11
the recess ratio is shown for exposed Pt widths (w) of 1 nm and 0.1 nm. It can be seen from Fig.
12
11a that for L/a larger than 5, and for a particular value of w, the normalized height of the stable
13
nanobubble is nearly constant (e.g., ~0.43 for w = 1 nm). This independence of H/L on L/a occurs
14
because H2 diffusion out of the bubble is limited by the diffusion of H2 in solution within the
15
recession, the rate of which is inversely proportional to the depth of the bubble in the recessed
16
electrode (L – H). Similarly the flux of H2 into the bubble is dependent on the electroreduction of
17
H+ at the exposed part of Pt electrode, which is limited by the diffusion of H+ from the bulk
18
solution down the recession between the bubble and recession cavity to the Pt electrode;
19
transport rate is inversely proportional to the height of the bubble (H). These two transport
20
processes create a negative feedback loop that defines the steady-state nanobubble geometry.
21
With increasing height H, diffusion of H2 out of the bubble increases while diffusion of H+ to the
22
electrode decreases (corresponding to a smaller H2 concentration near the electrode to
23
compensate for the loss of H2 by diffusion). As the flux of H+ decreases, the volume of the bubble
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will adjust by decreasing H. Conversely, with decreasing H, diffusion of H2 out of the bubble
2
decreases while the reduction of H+ increases (corresponding to a higher H2 concentration near
3
the electrode); in this case, the volume of the bubble will increase by increasing H. Thus the value
4
of (L – H)/H and H/L should be relatively independent of the ratio L/a, as seen in Figure 11a.
5
Figure 11b shows that the current for the nanobubble in stationary state (iss) decreases with
6
increasing L/a. That is because the H2 outflow from the bubble decreases with increasing L/a, as
7
H2 diffusion is constrained by the recessed geometry. The range of iss for nanobubbles in
8
stationary states, obtained by simulation is between 0.3 nA and 4 nA, which appear in reasonable
9
agreement with the experimentally measured values of 0.2 to 2.4 nA, Figure 2b. For example, for
10
a recess ratio (L/a) between 1 and 5, for electrodes with a = 55 nm, the corresponding value of iss
11
is between 1 and 4 nA (w = 1 nm) or 1 and 3 nA (w = 0.1 nm).
12
For the experimental data in Figure 2b, we do not know the precise depth of recession of the
13
Pt disks, and thus cannot correct these data for a quantitative comparison to the simulations.
14
However, the much better agreement between experimental data and simulate values for iss for the
15
recessed geometry (relative to the inlaid disk geometry) suggests that the electrodes have a
16
recessed geometry. The large scatter in the values of iss (Figure 2b) also indicate that depth of
17
recession is quite variable, and this is consistent with the currently poorly understand chemical
18
and/or physical mechanism that leads to recession of the electrode.
19
Conclusion
20
Simulations of single nanobubbles at nanodisk electrodes were used to determine the
21
dynamic stationary nanobubble geometries and to calculate the faradaic currents required to
22
sustain the nanobubble. Our results demonstrate that a stable gas filled nanobubble can exist at
23
dynamic equilibrium at both inlaid and recessed Pt nanodisk electrodes of radius between 10 and
24
50 nm. Better quantitative agreement between experimental and simulated values of the current
25
rquired to sustain the nanobubble are obtained assuming a recessed electrode geometry.
26
The geometry of the nanobubble is determined by a dynamic equilibrium that provides a
27
negative feedback in balancing the rate of H2 dissolution from the nanobubble (outflux) by the
28
rate of gas entry (influx) resulting from electrogeneration of the H2 near the three-phase interface.
29
The contact angle, θ, and width of the exposed Pt ring, w, describing a nanobubble at an inlaid
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disk determines the balance of these rates. Similarly, the height, H, and w for nanobubbles
2
generated at recessed Pt disk electrodes are governed by the same feedback mechanism.
3
Our conclusions regarding nanobubble geometries at a dynamic stationary state appear to be
4
entirely consistent with current experiment observations. However, confirmation of the exact
5
geometry of these bubbles requires further experimental studies.
6 7
Acknowledgements
8
This work was funded by the Office of Naval Research Award Number N00014-16-1-2541. YL
9
acknowledges support of the China Scholarship Council (201506275068).
10 11
Support Information. The Supporting Information is available free of charge on the ACS
12
Publications Website at DOI:
13
Part 1 describes the details of the finite element model and simulations. Part 2 provides a
14
COMSOL model report.
15 16
References
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TOC Figure:
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