Water nanoelectrolysis: A simple model

# Water nanoelectrolysis: A simple model

Juan Olives    Zoubida Hammadi    Roger Morin    Laurent Lapena
July 26, 2019
###### Abstract

A simple model of water nanoelectrolysis—defined as the nanolocalization at a single point of any electrolysis phenomenon—is presented. It is based on the electron tunneling assisted by the electric field through the thin film of water molecules (0.3 nm thick) at the surface of a tip-shaped nanoelectrode (micrometric to nanometric curvature radius at the apex). By applying, e.g., an electric potential during a finite time , and then the potential during the same time , we show that there are three distinct regions in the plane : one for the nanolocalization (at the apex of the nanoelectrode) of the electrolysis oxidation reaction, the second one for the nanolocalization of the reduction reaction, and the third one for the nanolocalization of the production of bubbles. These parameters and completely control the time at which the electrolysis reaction (of oxidation or reduction) begins, the duration of this reaction, the electrolysis current intensity (i.e., the tunneling current), the number of produced or molecules, and the radius of the nanolocalized bubbles. The model is in good agreement with our experiments.

## I Introduction

Water electrolysis is used for hydrogen productionTurner (2004); Zeng and Zhang (2010); An et al. (2017) and more generally for the production of bubbles and the study of their formation, dissolution, stability, acoustic properties, etc.,Fernández et al. (2014); Luo and White (2013); Czarnecki et al. (2015) owing to the numerous applications, e.g., in medicine.Bloch et al. (2004); Qamar et al. (2010); Acconcia et al. (2013) Microbubbles produced by electrolysis are also used to manipulate a microobject.Li and Hu (2013) Very small electrodes, with diameter 1 m, were used for the study of microbubble/substrate forces or the formation of nanobubbles.Donose, Harnisch, and Taran (2012); Luo and White (2013); German et al. (2016) An effective control of the micro/nanobubbles, e.g., concerning their localization and size, is crucial for all these studies and applications. Nevertheless, if the electrode is not of micro/nanometric size, microbubbles will generally appear everywhere on the electrode surface. With our new method, called nanoelectrolysis, the production of microbubbles can be controlled and nanolocalized at a single point, namely, the apex of a tip-shaped electrode (with micrometric to nanometric curvature radius at the apex).Hammadi, Morin, and Olives (2013) By means of nanoelectrolysis, a strong control of the microbubbles is obtained: a single bubble can be immobilized (or moved to any point) in the liquid, at some distance from the apex of the electrode.Hammadi et al. (2016)

In addition to the known electrocatalytic effect of a high surface area on the electrode,Stojić et al. (2003); Nikolic et al. (2010); Wang et al. (2014) our approach shows the importance of the nanostructure/nanogeometry of the electrode surface: by applying a low electric potential during a finite time (e.g., using an alternating potential), the production of bubbles can be nanolocalized at a single reaction site with a nanometric/micrometric curvature radius on the electrode.Hammadi, Morin, and Olives (2013) In the same way, nanometric heterogeneities (optically invisible) are potential reaction sites for bubble production (see Fig. 1 and Note 16 in Ref. Hammadi, Morin, and Olives, 2013), and the activity of macroscopic electrodes is probably due to the presence of many such uncontrolled heterogeneities. According to our approach, nanostructured electrode surfaces—with arrays of nanotips or nanopillars, i.e., sites with nanometric curvature radii—could probably improve the electrode activity.

However, the main interest of nanoelectrolysis is to produce calibrated microbubbles at a single site, in a controlled way. Calibrated microbubbles are needed in various medical applications, e.g., as ultrasound contrast agents for capillary imaging, drug delivery, or blood clot lysis.Bloch et al. (2004); Lindner (2004); Hernot and Klibanov (2008); Acconcia et al. (2013) They are used to study ultrasounds–microbubbles interactions with applications to the detection and sizing of bubbles, e.g., for the prevention of decompression sickness (scuba diving and extra-vehicular astronaut activity) or the monitoring of liquid sodium coolant in nuclear reactors.Wu and Tsao (2003); Czarnecki et al. (2015); Buckey et al. (2005); Kim et al. (2000) The advantage of nanoelectrolysis over the microfluidics technique of production of bubblesWhitesides (2006) is that no surfactant (biologically harmful) is used and that arbitrary bubble production frequency (even a single bubble productionHammadi et al. (2016)) can be obtained. Calibrated microbubbles were recently produced from nanoelectrolysis combined with ultrasounds using tap water (non-chemically controlled) solution.Achaoui et al. (2017) In this paper, we show that calibrated microbubbles of any size can be obtained at a single site by nanoelectrolysis with a chemically controlled solution, by applying a suitable electric potential during a finite time.

Although electrolysis is classically described using the electric potential (which is constant on the whole surface of each electrode), nanoelectrolysis reveals the fundamental role of the electric field, which is higher at the apex of the electrode (where the curvature radius is very small). Nanoelectrolysis is caused by the electron tunneling through the thin film of water layers at the electrode/solution interface, assisted by this high electric field.Hammadi, Morin, and Olives (2013) In this paper, we present a general and simple model based on this tunneling and field effect, which is in agreement with the experiments and explains the various aspects of nanoelectrolysis, i.e., the nanolocalization of each electrolysis reaction (oxidation and reduction) and of the production of bubbles. It leads to a complete control of these reactions and the bubble production, at a single point, by means of the applied electric potential. The model applies to any type of electrolysis in aqueous solutions involving gas production (provided the electrode surface is not altered by solid deposition).

## Ii The model

In our experiments, one of the two electrodes—called the nanoelectrode—is tip-shaped (and made of Pt), with a curvature radius, at the apex of the electrode, ranging from 5 m to 1 nm.Hammadi, Morin, and Olives (2013); Hammadi et al. (2016) An aqueous solution of ( to mol/L) is generally used. The presence of a few water layers at the surface of the nanoelectrodeXia and Berkowitz (1995); Guidelli and Schmickler (2000); Rossmeisl et al. (2008); Osawa et al. (2008) will be modeled as a dielectric film of constant thickness ( 0.3 nm). Let us denote as the region occupied by the dielectric film, as that occupied by the solution, the surface of the nanoelectrode, that of the counter electrode, the interface between and , and the unit vector normal to , oriented from to (Fig. 1).

Maxwell’s equations give the discontinuities of the normal components of the electric field and the electric current at the interface

 ε(E2n −E1n)=ρs j2n −j1n=−∂ρs∂t (1)

(assuming the same permittivity in and ; is the surface charge density on ), hence

 γE2n=−ε∂∂t(E2n−E1n)+j1n, (2)

with the help of Ohm’s law ( being the conductivity of the solution). The term represents a “charging current”—more precisely, the current due to the discharge of the interface , according to Eq. (1)—and will be denoted . Owing to the low thickness of the dielectric film, electrons can cross this film by quantum tunneling if the electric field in the film is high enough, producing the electric current responsible for the electrolysis reactions. This tunneling current will thus be called the electrolysis current and denoted . Its non linear dependence on the electric field in the dielectric film—and then on the electric potential applied to the dielectric film, being the potential on a point of minus that on the neighbouring point of —will be simply modeled using a threshold value and a high slope for (). For the sake of simplicity in the notations, we will use the same threshold value and slope in the region , i.e.,

 jE =0if−E0E0 jE =γ0(E+E0)ifE<−E0 (3)

(Fig. 2).Asy () Denoting , the current in the solution (at ), Eq. (2) may thus be written as

 j=jC+jE. (4)

At a given time , all the points of (of , respectively) have practically the same potential—because the electrodes are made of metal—and we denote the potential on minus that on . On the contrary, the surface is not equipotential because the thickness of the dielectric film is constant but the electric field (in this film) varies, being higher at the apex of the nanoelectrode (where the curvature radius of is very small). Then, at a given time , the potential value varies with the position on (or on ). In the following, the nanoelectrode surface will be simply modeled as (i) a hemisphere of radius at the apex, denoted zone , and (ii) a cylinder (of the same radius) of length , denoted zone (Fig. 3).

As a first approximation, we assume that the electric potential field in zone (except near the junction with the cylinder of zone ) is that produced by a spherical electrode and that in zone (except near the two ends of the cylinder) is that produced by a (infinite) cylindrical electrode. With the help of Maxwell’s equations, we then obtain the values of and in zones and

 E1n =β1v E2n =β2(V−v), (5)

the values of , , and being different in zones and , and distinguished with the respective subscripts and , i.e.,

 β1a =r1(r1+d)d≈1d β2a =r1+d+l(r1+d)l≈1r1 (6)

in zone , and

 β1b =1(r1+d)logr1+dr1≈1d β2b =1(r1+d)logr1+d+lr1+d≈1r1loglr1 (7)

in zone (see Appendix A; refers to the natural logarithm; the approximations hold if ).

Equation (2) may then be written as

 γβ2(V−v)=−εddt(β2(V−v)−β1v)+jE,

i.e.,

 V−v=τ(dvdt−1αdVdt)+jEγβ2, (8)

where and , which, according to Eq. (3) (with and ), leads to

 v+τdvdt =V+ταdVdtif−v0v0, (10) v+τ′dvdt =V−+ταdV−dtifv<−v0, (11)

where , , , , and . Note that , , , and have different values in zones and , which will be distinguished with the respective subscripts and . For a given applied potential , Eq. (8) or Eqs. (9)–(11) determine as a function of time [i.e., using , , , and , and using , , , and ].

Since , the current intensity (in the solution, in zone or in zone ) is

 I=γβ2a(V−v)=γaα′d(V−v), (12)

being the corresponding area of , i.e., for zone and for zone . In fact, the total intensity in the electric circuit is the sum of the contributions of zone and zone :

 I=Ia+Ib. (13)

Similarly, the electrolysis current intensity and the charging current intensity (in zone or in zone ) are, respectively, and ; thus

 IE =0if−v0v0 IE =γaα′d~γα′(v+v0)ifv<−v0, (14)

and

 I=IC+IE (15)

(in zone or in zone ).

## Iii A simple example: potential of rectangular shape

Let us show the consequences of the preceding model with a simple example. The simplest case is the application of a constant electric potential (between the two electrodes) during a finite time and is treated below as phase I (Sec. III.1). The nanoelectrode is thus anode (cathode, respectively) during the time if (, respectively). In order to treat both cases (anode and cathode) and to study the possible occurrence of both oxidation and reduction reactions (in zone a and in zone b), we will consider the simple case of a potential of rectangular shape in which the preceding phase I is followed by a second phase (phase II) with a constant opposite potential during the same time , after which no potential is applied (phase III) (Fig. 4)

 V(t) =0ift<0 V(t) =V1if02t1(phaseIII).

We here consider , but the case is exactly similar. In Sec. III.1, we will see that the oxidation reaction may occur or not during phase I, in zone a and in zone b, depending on the values of and . This will explain the nanolocalization of the oxidation reaction (when this reaction only occurs at the apex of the nanoelectrode, i.e., in zone a but not in zone b). Similarly, Sec. III.2 concerns the possible occurrence of the reduction reaction during phase II and its nanolocalization at the apex of the electrode.

### iii.1 Phase I

#### iii.1.1 Before the electrolysis reaction

During phase I, let us consider the solution of Eq. (8) or (9) (in zone or in zone ), assuming that remains lower than . At , the discontinuity jump produces the term in ( being the Dirac measure at 0), then, according to Eq. (8), the term in , and then the discontinuity jump for at 0. If (which corresponds to our usual experimental conditionsVal ()), the solution is then

 v(t) =0ift<0 v(t) =V1−(V1−V1α)e−t/τift>0 (16)

represented in Fig. 5, with the corresponding current intensity given by Eq. (12).

Clearly, is always lower than and, if , will always remain lower than . In the following, we suppose , so that will reach the value at the time

 t′=τlog(1−1α)V1V1−v0 (17)

(if the duration of phase I is large enough; Fig. 6).

Note that if , then , so that in this case [and the solution of Eq. (10) shows that for ].

Consider now the relative position of and . If , will not reach the value during phase I (i.e., for ), and if , will reach the value during phase I, at the time . In this last case, according to Eq. (3), there will be an electrolysis current , i.e., the oxidation reaction of electrolysis will occur, for . Let us represent, in the plane , the two “oxidation” curves in zone a (given by Eq. (17) with and ) and in zone b [Eq. (17) with and ] (Fig. 7).

In the region , there is no oxidation reaction (neither in zone nor in zone ). In the region , the oxidation reaction occurs everywhere on the nanoelectrode (in zone , after , and in zone , after ). In the region , the oxidation reaction is nanolocalized at the apex of the nanoelectrode (it occurs in zone , after , but not in zone ).

#### iii.1.2 During the electrolysis reaction

If (in zone or in zone ), reaches the value at and, for , is the solution of the new Eq. (10), i.e.,

 v(t)=V1+−(V1+−v0)e−(t−t′)/τ′ift>t′, (18)

where , represented in Fig. 8. Note that, for any finite value of , is continuous at and .

Since , we may consider that tends to and the corresponding limit value of any quantity will be denoted . Thus, tends to for (since tends to ; see Fig. 8). According to Eq. (12), the current intensity and its limit value are then represented in Fig. 9.

Similarly, Eq. (14) gives the electrolysis current intensity and its limit value (Fig. 10).

Note that and have the same asymptotic value and that for . Thus, while before the electrolysis reaction (), the current intensity is equal to the charging current intensity (), during the electrolysis reaction (), owing to , the (limit) current intensity is equal to the (limit) electrolysis current intensity (). This is a consequence of constant for , which implies that and are constant [from Eq. (5)], and then .

From Eqs. (14) and (18), we then obtain the electric charge given by the nanoelectrode to the solution for the electrolysis reaction (in zone a or zone b)

 Q′ =∫t1t′IE(t)dt =γaα′d~γα′1+~γα′(V1−v0)(T′−τ′(1−e−T′/τ′)), (19)

where is the electrolysis duration, and its limit value (for )

 ¯Q′=γaα′d(V1−v0)T′. (20)

Note that this charge is due to the electrons —tunneling through the dielectric film, from the solution to the nanoelectrode— produced by the electrolysis oxidation reaction

 2H2O→O2+4H++4e−,

which gives the number of produced molecules

 nO2=Q′4qe (21)

( the elementary charge).

### iii.2 Phase II

#### iii.2.1 Before the electrolysis reaction

First note that the following results are based on the simple assumption of a unique threshold value (and slope ).Asy () As previously mentioned, the following study relates to either zone a or zone b. Because of Eq. (8), the discontinuity jump produces the discontinuity jump (as it occurred at ). Since ( because ; because is either or owing to the high value of ), the solution of Eq. (9) gives

 v(t)=−V1+(V1+v(t+1))e−(t−t1)/τift>t1 (22)

(as long as ). In the case , we know that, during phase I, and increases but does not reach the value for . If were equal to , we would have the same situation (with the opposite sign for ) during phase II, so that (now decreasing) would not reach the value for . Since (because ), this implies that will not reach the value during phase II. In other words, if there is no electrolysis reaction (oxidation) during phase I, there will be no electrolysis reaction (reduction) during phase II.

We then consider the case . If the duration of phase II is large enough, will reach the value at the time , with [from Eq. (22)]

 t′′ =τlogV1+v(t+1)V1−v0 =τlog(1−2α)V1+v0+V1−v01+~γα′(1−e−(t1−t′)/τ′)V1−v0 (23)

[using given by Eq. (18)] and its limit value (for )

 ¯t′′=τlog(1−2α)V1+v0V1−v0. (24)

Clearly, because . Thus, if , the electrolysis duration (reduction) during phase II is lower than the electrolysis duration (oxidation) during phase I.

Strictly speaking, is a function of [Eq. (23) with given by Eq. (17)], but its dependence on is very low and, for large values of , which only depends on . Let us represent, in the plane , the two “reduction” curves in zone a [given by Eq. (23) with , , , , and ] and in zone b [Eq. (23) with , , , , and ]. Thus, in the region , there is no reduction reaction of electrolysis during phase II (neither in zone nor in zone ). In the region , the reduction reaction occurs everywhere on the nanoelectrode (in zone , after , and in zone , after ). In the region , the reduction reaction is nanolocalized at the apex of the nanoelectrode (it occurs in zone , after , but not in zone ).

It may be shown that in general (e.g., if , , and ) for any such that and . We thus have five regions in the plane (see Fig. 11).

Below the curve , there is no oxidation reaction and no reduction reaction (neither in zone nor in zone ). Between the two curves and , the oxidation reaction is nanolocalized at the apex of the nanoelectrode (it occurs in zone but not in zone ) and there is no reduction reaction (neither in zone nor in zone ). Between the two curves and , the oxidation reaction and the reduction reaction are nanolocalized at the apex of the nanoelectrode (they occur in zone but not in zone ). Between the two curves and , the oxidation reaction occurs everywhere on the nanoelectrode (in zone and in zone ) and the reduction reaction is nanolocalized at the apex of the nanoelectrode (it occurs in zone but not in zone ). Above the curve , the oxidation reaction and the reduction reaction occur everywhere on the nanoelectrode (in zone and in zone ).

#### iii.2.2 During the electrolysis reaction

If (in zone or in zone ), reaches the value at and, for , is the solution of the new Eq. (11), i.e.,

 v(t)=−V1++(V1+−v0)e−(t−t1−t′′)/τ′ ift>t1+t′′, (25)

and being represented in Fig. 12. Note that, for , is continuous at and .

According to Eq. (12), the current intensity and its limit value are then represented in Fig. 13.

Similarly, Eq. (14) gives the electrolysis current intensity and its limit value (Fig. 14).

Note that and have the same asymptotic value and that for (these are the opposite values compared to those of phase I). Just as in phase I, while before the electrolysis reaction (), the current intensity is equal to the charging current intensity (), during the electrolysis reaction (), owing to , the (limit) current intensity is equal to the (limit) electrolysis current intensity ().

From Eqs. (14) and (25), we then obtain the electric charge given by the nanoelectrode to the solution for the electrolysis reaction (in zone a or in zone b)

 Q′′ =∫2t1t1+t′′IE(t)dt =−γaα′d~γα′1+~γα′(V1−v0)(T′′−τ′(1−e−T′′/τ′)), (26)

where is the electrolysis duration, and its limit value (for )

 ¯Q′′=−γaα′d(V1−v0)¯T′′ (27)

(). Note that this charge is due to the electrons tunneling through the dielectric film, from the nanoelectrode to the solution, and producing the electrolysis reduction reaction

 2H++2e−→H2

which gives the number of produced molecules

 nH2=|Q′′|2qe. (28)

As noted earlier, , which, according to Eqs. (19) and (26), implies that : the absolute value of the electric charge used for the electrolysis reaction during phase II (in our case, reduction) is lower than that used for the electrolysis reaction during phase I (in our case, oxidation). In our case, this means that the number of produced molecules is lower than twice the number of produced molecules.

### iii.3 Phase III

Because of Eq. (8), the discontinuity jump produces the discontinuity jump . Since ( because is either or owing to the high value of ; is proved in Appendix B), the solution of Eq. (9) gives

 v(t)=v(2t+1)e−(t−2t1)/τift>2t1, (29)

which has the form represented in phase III of Fig. 12. Thus, there is no electrolysis reaction during phase III.

## Iv Bubble production

If (in zone a or in zone b), the electrolysis oxidation reaction (during phase I, between and ) produces molecules which diffuse in the solution. If the volume mole density of these molecules in the solution exceeds the saturation value ( the Henry’s constant for in water, the atmospheric pressure) in some region of the solution, at some time, a bubble of will be produced. In the simple case , the flux of moles produced at the surface of the nanoelectrode (more precisely, at the interface ), in zone a or in zone b, is

 jm=¯jE4NAqe=γ4NAqeα′d(V1−v0) (30)

constant between and (see Sec. III.1.2 and bottom of Fig. 14; is the Avogadro constant). Close to the surface (corresponding to the high values of ), this surface may be assimilated to its tangent plane, and an approximate solution of the diffusion equation is

 ρ(x,t) =jm(πD)1/2∫t−t′0e−x2/(4Du)u1/2duift′

(see Appendix C; is the distance to , here assumed small with respect to , and the diffusion coefficient for in water). Note that we have to add to the preceding value the initial constant density of in the solution due to the equilibrium with the atmospheric ( being the partial pressure of in the atmosphere).

Clearly, at fixed , is a decreasing function of , with a maximum value at

 ρ(0,t) =jm(πD)1/2∫t−t′0duu1/2 =2jm(πD)1/2(t−t′)1/2ift′