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Voltammetric Determination of the Stochastic Formation Rate and Geometry of Individual H2, N2 and O2 Bubble Nuclei Martin A. Edwards, Henry S. White, and Hang Ren ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.9b01015 • Publication Date (Web): 22 Mar 2019 Downloaded from http://pubs.acs.org on March 22, 2019

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Voltammetric Determination of the Stochastic Formation Rate and Geometry of Individual H2, N2 and O2 Bubble Nuclei Martin A. Edwards*§, and Henry S. White§, and Hang Ren*† §Department

of Chemistry, University of Utah, Salt Lake City, UT 84112, United States

†Department

of Chemistry and Biochemistry, Miami University, Oxford, OH 45056, United States *[email protected] and [email protected]

Abstract Herein, we report a general voltammetric method to characterize the electrochemical nucleation rate and nuclei of single nanobubbles. Bubble nucleation is indicated by a sharp peak in the current in the voltammetry of gas evolving reactions. In contrast to expectations based on the stochastic nature of nucleation events, the peak current signifying a stable nucleus is extremely reproducible over hundreds of cycles (~3% deviation). By applying classical nucleation theory, this seemingly deterministic behavior can not only be understood, but used to quantify the nucleation rate and size of bubble nuclei. A statistical model is developed whereby properties of single critical nuclei (contact angle, the radius of curvature, activation energy, and Arrhenius pre-exponential factor) can be readily measured from the narrow distribution of peak currents (mean, standard deviation) from hundreds of voltammetric cycles at a nanoelectrode. Single nanobubbles formed from gas evolving reactions (H2 from H+ reduction, N2 from N2H4 oxidation, O2 from H2O2 oxidation) are analyzed, to find that their critical nuclei have contact angles of ~150°, ~160° and ~154° for H2, N2, and O2, respectively, corresponding to ~50, ~40 and ~90 gas molecules in each nucleus. The energy barriers for heterogeneous nucleation of H2, N2, and O2 bubbles are, respectively, 2%, 0.4%, and 0.7% of those required for homogeneous nucleation under the same supersaturation. Keywords: single-entity electrochemistry; nucleation; survival analysis; nanoelectrode; activation energy; contact angle

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Gas is the product of many important electrocatalytic processes, including Cl2 production from the chlor-alkali process, H2 and O2 production from water splitting, and CO2 generation in many hydrocarbon fuel cells.1-3 Yet the formation of gas bubbles on electrode surfaces hinders the transport of reactant species to electrocatalytic sites, creating an additional overpotential, and lowering efficiency.4-6 Thus a detailed quantitative description of the behavior of bubbles at electrode surfaces, especially a quantitative description of their formation, is of great interest in electrocatalysis.7,8 The first step in the formation of a new phase, e.g., a gas bubble in an aqueous solution, is typically the nucleation of a sufficiently sized cluster of molecules, such that growth of a new phase can proceed favorably from a supersaturated solution. Bubble formation serves as a model system to study nucleation, lacking complications reported to be involved in the nucleation of other phases, e.g., competition between polymorphs and the formation of pre-nuclei in the hydrous phase, reported during nucleation of macromolecular crystals.9-11 Classical nucleation theory describes the free energy barrier for bubble nucleation as the sum of two competing contributions: a volume energy is released from transferring a supersaturation of dissolved gas into the gas phase, and favors the formation of a new phase, whereas the surface energy, the cost of forming a new liquid/gas interface, opposes the formation. The sum of these competing terms, i.e., the energy to form a bubble as a function of the size, displays a maximum, an energy barrier, corresponding to a critical size cluster of molecules. The growth of gas clusters with sizes greater than the critical nucleus size is energetically favorable, whereas nuclei smaller than this critical size are favored to shrink. Thus the rate of nucleation is governed by the formation of a critical nucleus, a random process (see the derivation of classical nucleation theory in the SI), which can be expressed as an Arrhenius process. Quantitative analysis of the properties of the critical nucleus can provide critical information about nucleation. However, as a critical nucleus is typically nanoscopic (often containing only a handful of molecules) and kinetically unstable,12 such measurements demand highly sensitive and rapid detection schemes. Moreover, due to the stochasticity of nucleation and its strong dependence on conditions (vide infra), such measurements must be repeated under precisely controlled conditions. Complications arising from heterogeneity between nucleation sites and interactions between multiple nucleation events, have hindered the interpretation of experiments of bubble nucleation

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on a macro-sized surface, e.g., macro-sized electrodes.13 To address this challenge, we used the electrochemical generation of a supersaturation of gas at a nanoelectrode to induce the formation of a single nanobubble, during a voltammetric cycle (see Figure 1, detailed discussion below).14 The high temporal resolution of the nanoelectrode current response, repeatability of the experiment, and localization of nucleation to the electrode surface, make it an attractive approach to study nucleation. By using a nanoelectrode, the number of nucleation events and bubbles formed is reduced to unity, greatly simplifying the interpretation of the experimental results. Nanoelectrode-induced formation of bubbles of several technologically important gases (e.g., H2, O2, and N2), and on different electrode surfaces, has allowed determination of supersaturations necessary for nucleation and the Laplace pressure inside critical nuclei of bubbles.14-18 Most recently, we have measured the rates of nucleation of single H2 and O2 bubbles at a nanoelectrode, via galvanostatic control and a triggered switching algorithm using custom programmable hardware.19,20 Herein, we introduce a fast, convenient method to measure the rate of bubble nucleation and properties of the critical nucleus (radius of curvature, contact angle (𝜃), number of molecules) at nanoelectrodes. We use cyclic voltammetry, an electrochemical technique that is routinely performed using conventional electrochemical apparatus, and which has previously been used to study many electrochemical systems, including stochastic nucleation of metal particles and nanobubbles.16,17, 21-24 Our method uses only the variability and mean nucleation supersaturation, which are directly accessed from the voltammetric peak current. A model based on classical nucleation theory and statistical principles allows one to rapidly look up properties of the nuclei from a table, app, or graphical method. The method is demonstrated through a comparison of H2, N2 and O2 bubble nucleation at Pt nanoelectrodes.

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Figure 1. Nucleation of a H2 bubble induced by the reduction of H+ to H2. A) Schematic of bubble nucleation and growth. ① H+ reduction at Pt electrode, ② formation of a critical nucleus, ③ a steady state bubble at Pt electrode. The yellow spherical cap represents the gaseous region inside which no solvent exists. B) Cyclic voltammetry indicating bubble formation (10 consecutive cycles) at a 10 nm radius Pt disk electrode in 1.0 M H2SO4 at 2 V/s. Nucleation is indicated by a sharp drop in the current, which is shown in the zoomed-in inset. C) H2 concentration at the electrode surface, 𝐶surf H2 (left axis, black line, calculated using eq 1), and nucleation rate, J (right axis, blue line, calculated from classical nucleation theory - eq 3); both derived from the voltammetric current in B) before the formation of a bubble. Values of 𝐽0 = 1010 s-1 and 𝜃 = 150°, which are relevant to H2 bubble nucleation at Pt (vide infra), were used in the calculation. RESULTS AND DISCUSSION Nucleation and subsequent growth of a single H2 bubble at a Pt nanoelectrode can be induced by the electroreduction of H+ during a voltammetric scan (Figure 1). In a typical voltammogram for H2 bubble generation (Figure 1B), the potential is initially scanned negatively from a potential 4


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value where no H+ reduction occurs (-0.05 V). At this potential, the Pt electrode is bare (no bubble) and the local concentration of H2 is zero. As the potential is scanned negative of -0.2 V, the current increases exponentially due to the increasing rate of H+ reduction, and the dissolved H2 concentration at the electrode surface (𝐶surf H2 ) increases accordingly (① in Figure 1), yet no bubble has nucleated. Once 𝐶surf H2 is sufficiently high (~-0.4 V in this experiment), the stochastic nucleation of a bubble, i.e., the formation of a critical-size cluster of H2 molecules, becomes increasingly probable (② in Figure 1). Once nucleation occurs, the bubble grows favorably, to rapidly cover almost the entire electrode surface, which greatly blocks the transport of H+ to the electrode surface. This fast nucleation and growth results in a precipitous drop from the peak current, 𝑖pH2 nb, to a small residual current. Since the growth of the nanobubble is fast (100°), 𝐽0 has a negligible effect on nucleation supersaturation in the range investigated here. Figure 3B shows the relative standard deviation of the supersaturation (σs/S̅), which quantifies the relative spread of the distribution, and is only affected by 𝐽0/𝜈 and not θ (see Figures S4 and S5). Lower values of 𝐽0 result in larger σs/S̅ and vice versa. Qualitatively, this is because a lower attempt frequency (J0) will result in nucleation occurring over a wider range of supersaturations in order for the same nucleation rate. We can use S̅ and σs/S̅, determined from the distribution of 𝑖pnb in voltammetry, to rapidly determine 𝐽0 and θ as described below. Figure 3C shows a histogram of the 𝑖pH2 nb distribution from 400 voltammetric cycles where H2 nanobubbles were formed via H+ reduction at a 10 nm radius Pt electrode (10 cycles shown in Figure 1B). The mean and standard deviation in 𝑖pH2 nb (8.48 ± 0.22 nA) are converted to supersaturation (top axis) using eq 1 and Henry’s law to give S̅ = 312.1 and σs = 8.1. Taking σs/S̅ = 0.026, the value of log10[𝐽0/ν (V-1)] can be read from Figure 3B as ~12.5, as shown graphically by the dashed lines. These measurements were taken with a scan rate of 2 V/s, giving log10[𝐽0 (s-1)] =12.8 ± 0.5. Next, taking this calculated value of log10[𝐽0/ν (V-1)] and reading across from the vertical axis in Figure 3A, until the contour of the experimentally measured

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mean supersaturation is intersected (S̅ = 312), we obtain θ = 150°. These parameter values produce an excellent fit to the experimental distributions (solid line, Figure 3C). To provide a simple alternative to the graphical fitting process described above, we developed a MATLAB App where 𝐽0 and θ can be directly calculated from S̅ and σs/S̅ based on our model (see Figure S6 for the screenshot of the App and SI for the source code and a description of usage). Alternatively, a lookup table is compiled, which can be used to obtain 𝐽0 and θ from S̅ and σs/S̅ (Table S1). Figure 4, which shows distributions of 𝑖pH2 nb from voltammetry for H2 bubble nucleation performed at different scan rates on the same electrode, (100 cycles at each 𝜈), confirms the prediction (Figure 2) that an increase in 𝑖pH2 nb is predicted with increasing scan rate (𝜈). Importantly, distributions of 𝑖pH2 nb at all four different 𝜈 are well fitted with a single set of 𝐽0 and θ (parameters obtained from 𝜈 = 1 V/s data), suggesting that the voltammetrically induced nucleation of individual bubbles is well described by our model, 𝐽0 and θ are independent of scan rate, and thus measurements at a single scan rate are adequate to determine 𝐽0 and θ. Reassuringly, these values of J0 and θ for H2 bubble nucleation are in agreement with those obtained from nucleation rate measurements by our group using a galvanostatic (constant current) method.19

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S

280

260

240

N=100

20 0.2 V/s 10 0 20 1 V/s

N=100

10

Counts

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 20 2 V/s

N=100

10 0

N=100

20 10 V/s

Experiment Theory

10 0 -6.5

-6.0

i pH

2

-5.5

/ nA nb

-5.0

Figure 4. Distribution of 𝑖pH2 nb as a function of scan rate at a 8 nm radius Pt disk electrode. The histograms of experimentally measured distributions of 𝑖pH2 nb (bars) are well fit at all scan rates by theoretical curves (black lines) with a single pair of parameters (𝐽0 = 1014 s-1and θ = 151°, calculated from σs and S̅ for 𝜈 = 1 V/s). N = 100 measurement of 𝑖pH2 nb at each scan rate.

The voltammetric method presented above allows one to monitor whether any changes in 𝐽0 and θ occur during the experiment, which might occur, e.g., through the restructuring of the nucleation sites on the electrode surface and affect the energetics and kinetics for nucleation.33 A demonstration of monitoring nucleation is presented in Figure 5, which shows 𝑖pH2 nb in 500 consecutive voltammetric cycles of H+ reduction at a single electrode. The data are divided into 7 time segments, with a histogram for each segment shown in the middle portion of the figure.

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Variations in θ and 𝐽0 over time (cycle number), which are shown in the lower portion, were readily obtained by applying the previously described method to each data subset. A clear step change in 𝑖pH2 nbis apparent after ~40 cycles, which is caused by the changes in both J0 and θ. During the 500 cycles of voltammetry for H2 bubble nucleation, θ (pink line) was found to vary between 144° to 148°. Though small, this change in θ is beyond the uncertainty of the measurement (dashed lines, 1 standard deviation). On the other hand, 𝐽0 (blue line) varies between 1013 to 1017 s-1, but with much larger errors, which stem from the less sensitive dependence of the nucleation rates on 𝐽0 at larger θ (see discussion above and observe in Figure 3B that an uncertainty of 0.01 in σs/S̅ gives ~4 orders of magnitude of uncertainty in 𝐽0). Large errors in the measurement of the preexponential factor are quite common in nucleation studies, because θ usually exerts a much larger effect than 𝐽0 on the nucleation rate.33 We observed no noticeable change in electrode size before and after the 500 cycles via voltammetric measurement in ferrocene.

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Figure 5. Determination of H2 bubble nucleation parameters (𝐽0 and θ) vs time (cycle number) from cyclic voltammetry. A) Top: 𝑖pH2 nb (left axis) and corresponding supersaturation, S, (right axis) from 500 consecutive cycles of H2 bubble nucleation at a 10 nm radius Pt electrode: 𝜈 = 2 V/s. Bottom: Distributions of experimental 𝑖pH2 nb, each corresponding to the shaded region directly above. Theoretical distributions (blue lines) are plotted using eqs A7 and A2 with 𝐽0 and θ obtained from the mean and standard deviation of 𝑖pH2 nb in each segment. B): θ and 𝐽0 obtained from the distributions of 𝑖pH2 nb as a function of cycle number. Dashed lines represent 1 standard deviation.

The method applied above to quantify H2 bubble nucleation, is applicable to studying nucleation of bubbles of other gases, as shown in Figure 6A, which shows voltammograms of single N2

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nanobubble nucleation induced by N2H4 oxidation at a 24 nm radius Pt disk electrode. Qualitatively, the voltammograms are similar to those of a H2 nanobubble (Figure 2A), i.e., a precipitous drop in current results in a peaked current, 𝑖pN2 nb, indicating the nucleation of a N2 bubble. However, the sign of the current and voltage are now positive due to the electrode reaction being an oxidation. The distribution of 𝑖pN2 nb from 40 voltammetric cycles is shown in Figure 6B, from which we obtain S̅ = 178 and σs/S̅ = 0.03 Applying the graphical method, we first read off from σs/S̅ = 0.03 in Figure 3B, to give log10[𝐽0/ν (V-1)] =11.3 (therefore, log10[𝐽0 (s-1)] = 11.0). Next, we take this value and observe that it intersects the contour for S̅ =178 at θ = 158° in Figure 3A. The experimental distribution of 𝑖pN2 nb from N2 nanobubble nucleation is also well fit by the theoretical distribution using these parameters (solid line). To our knowledge, these values represent the first measurements of N2 nanobubble nucleation rates and critical nuclei.

Figure 6. Measurement of θ and 𝐽0 of N2 bubble nucleation from voltammetry of N2H4 oxidation at a 24 nm radius Pt disk electrode. A) 10 typical voltammetric cycles in 1.0 M N2H4 solution at 2

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V/s. Inset highlights the variation in 𝑖pN2 nb. B) distribution of 𝑖pN2 nb from 40 consecutive voltammetric cycles (bars). Theoretical nucleation likelihood distribution (black line) uses with log10[𝐽0(s-1)] = 11 ± 1 and θ = 158 ± 1°, which were obtained from σs and S̅.

Comparing H2 nanobubble nucleation at the same 24 nm radius electrode used in Figure 6B (data are shown in Figure S7), we observe θ of the critical nucleus for N2 (158 ± 1°) is larger than that for H2 (146 ± 1°), indicating a flatter bubble nucleus for N2 than H2. Confidence that these angles are significantly different (p < 0.01), is due to the high sensitivity of S̅ to θ when θ is large (Figure 3A) giving small errors (~1°). The pre-exponential factor for N2 bubble formation (log10(J0 /s) = 11 ± 1) is much smaller than that for H2 bubbles (log10(J0/s) = 18.6 ± 1.2). Smaller θ (160°) and J0 (log10[𝐽0(s-1)] = 8.5) for N2 bubble nucleation on a different electrode (Figure S6) suggests that the mechanism for nucleation of H2 and N2 bubbles may be different, e.g., it may occur at different sites, or the attempt frequency of desolvation, an important step in gas transfer from the aqueous phase to gas phase, may be different between H2 and N2. However, as the measurement of J0 innately involves larger errors and a variation of 3 orders of magnitude was obtained on the same electrode during the same measurement of H2 bubble nucleation (Figure 5), strong conclusions cannot be drawn without further investigation. The literature contains a number of reports of single nanobubble nucleation voltammetry.15-18, 34 The method described above offers an opportunity to rapidly gain insight, by re-analyzing such data. In the published voltammograms for O2 bubble nucleation (Figure S2 of ref 12), the mean and standard deviation in 𝑖pO2 nb at 2 V/s, 24.7 nA and 0.34 nA, respectively. Using a MATLAB App that automates the graphical method described above (see Supporting Information, section S5 for details), these were rapidly converted to θ = 153° and J0 = 4 × 1022 s-1, values of which are comparable reported by a galvanostatic method.20 The measured parameters of S̅ and J0 allow us to conclude additional geometric and thermodynamic parameters pertaining to bubble nucleation, as summarized in Figure 7 and Table 1. Specifically, assuming an equilibrium between the solution and the bubble nuclei, Henry’s law gives internal pressures of ~300, ~180, and ~170 atm for H2, N2 and O2 critical nuclei, respectively, consistent with our previous report.19,20 The Laplace pressure (𝑃Laplace) inside the critical nuclei

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can then be obtained from 𝑃gas = 𝑃Laplace + 𝑃0. The radii of the curvature of the critical nuclei, 𝑟crit = ~5 nm, ~ 8 nm, and 9 nm for H2, N2, and O2, respectively are then derived from the Young– Laplace equation, 𝑃Laplace = 2𝛾 𝑟crit.35 Knowing the contact angle and radius of curvature of the critical nuclei, we can apply geometric relations for a sphere cap (eq S20) to deduce the volume, V, and footprint radius, rfootprint, of the spherical cap. NB: herein, we implicitly assume a spherical-cap (energy minimizing surface) on a flat electrode, similar relations can be derived if nuclei form in pits or defects. The number of gas molecules in the critical nucleus (N) can then be obtained either from the ideal gas law or real gas data (see Table 1).36 From these analyses, on average, a bubble nucleus of N2, H2 and O2 contain ~40, ~50, and ~90 gas molecules, respectively. The geometry of critical nuclei for H2 and N2 bubbles are shown in Figure 7, both of which are flat (θ > 140°), with a width of ~3 nm (measured as footprint radius, rfootprint, see Figure 7). Interestingly, this width is very close to the lower limit of a stable meniscus (~1.9 nm) from a reported lattice gas Monte Carlo simulation.37 Note that the similar size of the critical bubble nucleus and the narrowest water meniscus are both affected by the same property of water (i.e., surface tension) at the nanoscale. Since thermal fluctuations govern the smallest stable meniscus,37 such fluctuations are also likely to play a major role in determining the properties of the critical nucleus. Note that the size of the critical nucleus obtained in this study is comparable to the size of the smallest bubbles observed from an in situ TEM experiment.38 Given the volume (V) and the number of molecules (N) in the critical nucleus, the average distance between the gas molecules inside can be estimated via

3

𝑉 𝑁,

which yields a

distances of ~5 Å between H2 molecules and ~6 Å between both N2 and O2 molecules. These intermolecular distances of gases inside the bubble nuclei are closer than the average distance of gas molecules in solutions that are equilibrated with the critical nuclei (~19 Å for average H2 distance in a 0.24 M H2 solution, ~24 Å for average N2 distance in a 0.11 M N2 solution, and ~20 Å for average O2 distance in a 0.21 M O2 solution). The energetics of the critical nuclei, i.e., the 4

activation energy for nucleation, are obtained via Δ𝐺 ‡ = 3𝜋𝑟2crit𝛾Φ(𝜃) (see eqs S28 and S29 and their derivation in the SI), which yield ~34, ~20, and ~ 40 kBT for H2, N2 and O2 bubble nucleation, respectively.

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Figure 7. Schematic of bubble nuclei for A) H2 and B) N2 on Pt (111) derived from voltammetry (Figures S7 and 6). The top and bottom panels are the cross-section and 3-D view of the bubble nuclei, respectively. The shaded area in the 3D scheme represents the time-averaged liquid/gas interface of the critical nuclei, determining a region in which the solvent molecules are excluded; the red and blue balls represent instantaneous snapshots of H2 and N2 molecules, respectively.

Table 1. Properties of the H2 and N2 nuclei measured from voltammetry in this work. Values are reported as mean ± standard deviation. 𝑃Laplace /atm

𝑟curv /nm

𝜃 /°

𝑟footprint /nm

V /nm3

# of molecules (ideal gas)

# of molecules (real gas36)

Δ𝐺 ‡

H2

305 ± 5

4.7 ± 0.1

146 ± 1

2.7 ± 0.1

9±1

61 ± 4

50 ± 3

~34 kBT

N2

177 ± 6

8.1 ± 0.3

158 ± 1

3.0 ± 0.1

9±2

39 ± 5

38 ± 5

~20 kBT

O2

167 ± 3

8.6 ± 0.2

154 ± 1

3.9 ± 0.2

22 ± 2

84 ± 6

87 ± 6

~40 kBT

The contact angle reported here represents the geometric angle of a spherical cap for a flat surface. However, even if the assumption of spherical geometry or flat substrate is not valid, θ still quantifies the ease of nucleation. Higher θ means a bubble easier to nucleate, with θ = 0° 18


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corresponding to homogeneous nucleation. The geometric factor here, Φ(𝜃), is the same as the potency factor, which quantifies the energetic ratio between heterogeneous and homogeneous nucleation (see the derivation of classical nucleation in SI).39 The energy barriers for heterogeneous nucleation of H2,N2 and O2 bubbles are 2%, 0.4%, and 0.7% of those required for homogeneous nucleation under the same supersaturation, respectively, which does not rely on any assumptions about the geometry of the critical nucleus or the surface. Our analysis use the bulk value of surface tension for the liquid/gas interface, and the effect of curvature on surface tension (e.g., Tolman’s correction) is not included.40 We have previously demonstrated experimentally that surface tension holds the macroscopic value down to at least ~7 nm.41 Computational studies have shown that cluster of molecules (water) of size sub-1 nm still manifest bulk properties.42,43 Another potentially important contribution to the energetics of nucleation is line tension, which is not explicitly included in the classical nucleation theory. The measurement of line tension, τ, for a three-phase interphase is challenging and is prone to errors in the measurement of contact angle.44 Nonetheless, the most consistent magnitude of line tension for a three-phase contact line is ~10-11 J/m.44-46 Using this value, the contribution of line tension to the energetics of nucleation can be estimated using a typical 2π𝑟footprintτ and, resulting in a line energy of ~ 0.9 kBT for a H2 bubble nucleus and ~2 kBT for a N2 bubble nucleus, both of which are small compared to the activation energy calculated without line tension (Table 1).

CONCLUSIONS A statistical model of single nanobubble nucleation during voltammetry was developed, which describes the distribution of supersaturations for nucleation and depends only on the contact angle, θ, of the critical nuclei and the pre-exponential factor in an Arrhenius rate expression, J0. Higher θ and a lower J0 result in lower mean supersaturations for nucleation, while lower J0 predicts higher variability. Experimentally measured distributions of nucleation supersaturation for H2, O2, and N2 were well fit by the model. Fits are obtained, from a graphical method, and a look-up table or app, each which uses only the mean and standard deviation of the supersaturation of nucleation (i.e., the peak current), to determine the properties of critical nuclei. Contact angles of ~150 °, ~154° 19


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and ~160° with spherical-cap bubble nuclei containing ~50, ~90 and ~40 molecules, were determined for H2, O2, and N2, respectively. Results for H2 and O2 compare favorably to those from a galvanostatic method that required custom programmable hardware, whereas those for N2 has not been reported in the galvanostatic method.19,20 The voltammetric method takes advantage of commonly available electrochemical instrumentation and offers superior temporal resolution, allowing monitoring changes in nucleation. As demonstrated herein, our model may be used to gain insight into nucleation data reported in the literature. The statistical analysis can easily be modified to reflect an alternative electrochemical technique, or a different (e.g., multi-step) nucleation mechanism, which might be more appropriate for the nucleation of a different phase (e.g., metal particles or crystals),23,47 by use of an appropriate hazard function. As such, this work provides a guide to those studying stochastic single entity events.48-51

EXPERIMENTAL SECTION Sulfuric acid (98%) and hydrazine solution (35 wt. % in H2O) (both Sigma Aldrich) were used as received. Ferrocene (Sigma Aldrich, 98%) was purified twice by sublimation. All aqueous solutions were prepared using deionized water (18.2 MΩ·cm, Barnstead Smart2Pure, ThermoFisher), with no additional electrolyte added. Pt nanodisk electrodes were prepared according to a procedure reported previously.52 Briefly, a 25-μm-diameter Pt wire (Goodfellow Corp., 99.99%) was electrochemically sharpened and sealed in a borosilicate glass capillary (Dagan Corp., OD = 1.65 mm, ID = 1.10 mm). The tip of the sealed capillary was then carefully polished until the tip of the Pt was exposed. The apparent radii of the disk electrodes, 𝑎, were measured from the steady-state diffusion-limited currents (ilim) for the electro-oxidation of 5 mM ferrocene in acetonitrile with 0.1 M tetrabutylammonium hexafluorophosphate (TBAPF6) (see Figure S7 for the voltammograms). Electrochemical measurements were performed using a HEKA EPC10 amplifier, with low-pass filtering at 10 kHz (3-pole Bessel) and sampling at 20 kHz. A two-electrode setup, employing a Ag/AgCl reference electrode (3 M NaCl, BASi Inc.) as a combined reference/counter electrode was used for all the electrochemical measurements, which were performed inside a homebuilt

20


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Faraday cage. All solutions were purged with Ar to prior to measurements, and were protected by flowing Ar in the solution headspace throughout the experiment. All the experiments were performed at room temperature (21±1 °C). Errors in the measurements of 𝐽0 and θ are reported as standard deviations, which were estimated using the Bootstrap method with a resampling number of 50,000.53 APPENDIX: Derivation of Probability Distribution of 𝒊𝐩𝐧𝐛 in CV For simplicity, we consider the case of the reduction of H+ to H2, the behavior for oxidation (N2 from N2H4) follows naturally, through appropriate changes of signs. In the forward scan of a voltammogram (prior to bubble formation) performed under typical bubble nucleation conditions (i.e., with high concentrations of the gas forming reactants, when the potential is in the Tafel region, far from the mass transport limit) the current at a nanoelectrode, 𝑖, is well described by the exponential function (A1)

𝑖 ≈ 𝑘exp (𝑓𝜈𝑡) where 𝜈 is the scan rate, 𝑓 =

𝛼𝑛𝐹 𝑅𝑇 ,

𝑅 is the molar gas constant, 𝑇 the thermodynamic temperature,

𝐹 Faraday’s constant, 𝛼 the electron transfer coefficient, 𝑛 the number of electrons transferred, and 𝑘 is a constant the takes into account combined effects of the formal electrode potential, initial potential, electrode geometry, diffusion coefficients and electron-transfer kinetics. Substituting this into eq 1 from the main text gives the concentration on the surface of the electrode as 𝐶surf H2 ≈ 𝐾exp (𝑓𝜈𝑡)

(A2)

where 𝐾 = 𝑘/4𝑛𝐹𝑎𝐷H2. Substituting eq A2 and Henry’s law into the rate expression for classical nucleation theory (eq 2, derivation shown in SI section 4) gives the rate of bubble formation during voltammetry, J, as a function of time to be 𝐽 = 𝐽0exp

(

―16𝜋𝛾3𝜙(𝜃)

)

3𝑘B𝑇(𝐾exp (𝑓𝜈𝑡)/𝐾H ― 𝑃0)2

21


(A3)

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Close to nucleation, the ambient pressure, 𝑃0, is negligible in comparison to the partial pressure of electrochemically generated gas, i.e., 𝐾exp (𝑓𝜈𝑡)/𝐾H ≫ 𝑃0, and so we can approximate the rate of nucleation as 𝐽 = 𝐽0exp

(

―16𝜋𝛾3𝜙(𝜃)

3𝑘B𝑇(𝐾exp (𝑓𝜈𝑡) 𝐾H)

)

2

(A4)

= 𝐽0exp ( ―𝛽exp ( ―2𝑓𝜈𝑡)) where 𝛽 =

16𝜋𝛾3𝜙(𝜃)𝐾2H 3𝑘B𝑇𝐾2

.

To obtain the distribution of 𝑖pnb, we use survival analysis, which is an application of probability theory to understand the distribution of lifetimes. To perform the analysis, we will use the eq A5, a result from survival theory. For derivations see an introductory text on probability.32 Let tnuc be a random variable representing the time required for an event (bubble nucleation), then the probability density function of the tnuc, PDF(𝑡), can be written as

(

)

𝑡

(A5)

PDF(𝑡nuc) = ℎ(𝑡)exp ― ∫0ℎ(𝑠)𝑑𝑠

where ℎ(𝑡) is the hazard function, that is, the instantaneous rate at which an event (bubble nucleation) occurs, at time t, assuming it has not previously occurred. In nucleation studies, ℎ(𝑡) = 𝐽(𝑡), and the probability density function of the time required for nucleation is:

(

𝑡

)

(A6)

PDF(𝑡) = 𝐽(𝑡)exp ― ∫0𝐽(𝑠)𝑑𝑠

A brief derivation of eq A6 is included in the Supporting Information, section 3. For cyclic voltammetry, the expression for 𝐽(𝑡) from eq A4 can be substituted in eq A6, which results in the following solution, which is expressed in terms of exponential integrals (defined as Ei(𝑥) = ∞

∫𝑥 𝑒 ―𝑡 𝑡𝑑𝑡) 54 PDF(𝑡) = 𝐽0exp ( ―𝛽exp ( ―2𝑓𝜈𝑡))exp

(

―𝐽0(Ei(𝛽exp ( ―2𝑓𝜈𝑡))) 2𝑓𝜈

)

(A7)

From PDF(t), the distribution of nucleation as a function of potential (𝐸) can be obtained by a change of variable, 32 using 𝑡 = 𝑔(𝐸) = (𝐸 ― 𝐸init) 𝑣 : PDF(𝐸) = PDF(𝑔(𝐸)) ⋅ 𝑔′(𝐸)

22


(A8)

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=

𝐽0 ―𝐽0(Ei(𝛽exp ( ―2𝑓(𝐸 ― 𝐸init))) exp ( ―𝛽exp ( ―2𝑓(𝐸 ― 𝐸init)))exp 𝜈 𝑣 2𝑓

(

)

In eq A8, 𝐽0 𝑣 can be separated out, suggesting that the effect of 𝑣 on the distribution of 𝐸 can be eliminated if a normalized pre-exponential factor 𝐽0 𝑣 is used. The corresponding distribution of 𝑖pnb, 𝐶nuc nb and S can be further expressed using the 𝑖- 𝐸 relation in voltammetry (eq A1), an 𝑖-𝐶 relation (eq 1) and equilibrium saturation concentration. All the probability distribution plots in the main text are obtained using eq A8, which were confirmed by comparison to those obtained using Monte Carlo simulation, as described in Supporting Information, section 2.

Associated Content Supporting Information. The Supporting Information is available free of charge on the ACS Publications website at DOI: xxx.xxxx/xxxx.xxxx Additional data showing the effect of current noise, validation of the analytical solution using Monte Carlo simulation, normalizing the effect of scan rate, procedures for obtaining J0 and θ using MATLAB app or look-up tables, H2 bubble nucleation at a 24-nm radius Pt electrode, measurement of electrode radii from ferrocene oxidation, and derivation of classical nucleation theory for bubble nucleation (pdf). Table for obtaining J0 and θ from the mean and standard deviation of the supersaturation (xlsx). The code for the MATLAB app to obtain J0 and θ from voltammetric data (zip) Author Information Corresponding Authors: *E-mail: [email protected] *E-mail: [email protected] 23


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Acknowledgments This work was funded by the Office of Naval Research Award Number N000141211021. HR acknowledges the start-up support and CFR Faculty Research Grant from Miami University. We thank Dr. Yuqing Qiu and Dr. Sean German for helpful discussion.

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