## Abstract

The influence of electric fields on the velocity of the chemical reaction 4HF+SiO_{2}→SiF_{4}+2H_{2}O in aqueous solution is investigated experimentally. The field strengths used were high enough to measure nonlinear effects. The results permit a critical analysis of a theoretical model known in literature. The basic idea of dipole orientation changing the rate of the primary step of the chemical reaction can explain the experimental data, but several important details of the original model had to be changed. The primary step involves two hydrogen fluoride (HF) molecules rather than one, and field screening by mobile ions has a significant influence causing nonlinear effects. The fact that field screening plays an important role implies that electric field-assisted HF etching of silica may by used as an instrument for measuring ion concentrations in highly concentrated electrolytes. The data measured may be well described by a theoretical model based on mean field approximations. The results give an insight into the structure of highly concentrated hydrofluoric acid and also permit a critical analysis of applications of the effect in measuring electric fields written in glass samples by electrothermal poling. The effect may also be used for shaping glass surfaces.

## 1. Introduction

It is difficult to investigate electrochemical properties of highly concentrated electrolytes. Experimental data on conductivity at high concentration do not permit determining ion concentrations in a simple way. Electric field assisted HF etching of silica may be an alternative way of measuring ion concentrations. Though the original motivation for studying this effect was a different one.

Etching of silica glass (SiO_{2}) with hydrofluoric acid has been used to measure intense electric fields written in glass samples (Margulis & Laurell 1996; Triques *et al*. 2000, 2003). This measuring method relies on the fact that the HF-SiO_{2} etching velocity is changed by electric fields. The influence of electric fields on the etching velocity of hydrogen fluoride (HF) has been determined with very precise interferometric measurements and a linear relation between velocity change and electric field has been observed with fields up to 0.2×10^{8} V m^{−1} (Lesche *et al*. 1997). In that work, a theoretical model has been proposed that could explain the observed field dependence of etching velocity. According to this model, the HF molecules get partially orientated when they approach the glass surface and this alteration in angular distribution entails a change in the primary reaction rate. The model predicts a nonlinear dependence between velocity change and electric field. For fields within the range of −0.2×10^{8} to +0.2×10^{8} V m^{−1}, the nonlinearity is not perceptible with current experimental techniques. However, the application of the effect in measurements of fields written in silica samples by electrothermal poling uses the model of Lesche *et al*. (1997) in the nonlinear region, where the model has never been tested experimentally. Also, most of the practical measurements have been performed with hydrofluoric acid of 20 per cent, whereas the experimental calibration in Lesche *et al*. (1997) was performed with a concentration of 40 per cent. The present work investigates electric field assisted HF etching of silica with several concentrations of HF and with fields strong enough to observe nonlinear effects.

The theoretical model proposed in Lesche *et al*. (1997) was based on the assumption that the primary step of the HF–silica reaction was performed by one HF molecule, and that this primary reaction could only take place when the molecular axis formed an angle smaller than some critical angle, *θ*_{0}, with the normal vector of the glass–acid interface. A simple analysis of statistical mechanics gives the following expression for the etching rate as a function of the applied electric field1*E*_{ap}:
1.1
Equation (1.1) is valid as long as . In this equation, is a constant related to the dipole moment *μ* of the HF molecule, the absolute temperature *T*, and the electric permittivities *ϵ*_{glass}, *ϵ*_{acid}, *ϵ*_{0} of the glass sample, the acid and the vacuum, respectively:
1.2
where *c*_{ LF}=1+*χ*_{acid}/3 is the local field correction factor. The empirical value of that constant, which in the following shall be written as *α*, and which is defined in terms of experimental etching velocities :
1.3
has been found to be of the right order of magnitude but a little bit smaller than the theoretical prediction of equation (1.2) (prediction: ; experimental: *α*=1.26×10^{−9} m V^{−1}).

The experimental findings of the present work permit a critical analysis of the model, which reveals a complex scenario of effects contributing to electric field assisted HF etching. The findings may provide new insight into the properties of aqueous HF solutions. Hydrofluoric acid is a complicated electrolyte. At low concentration, it is a weak acid with a large variety of complex species in it. On the other hand, pure HF is a super acid. Even the answer to the most basic question, whether there exists an appreciable amount of the molecular species HF in the solution, is not exactly known. Infrared spectroscopic data seem to indicate that molecular HF is practically absent and instead ion pairs H_{3}O^{+}F^{−} are formed (Giguere & Turrell 1980). But conductivity and potentiometric data indicate that a large amount of HF does exist (McTigue *et al*. 1985). Ion concentrations and activity coefficients at high concentrations of that acid are unknown. Electric field assisted HF etching of silica may give new insight into these questions.

The etching mechanism of silica in HF solutions is not completely understood as the chemical species involved are unknown. Moreover, the reaction route and the involved species depend on the concentration and pH level (Knotter 2000). For the case of pure aqueous HF solutions with stoichiometric HF concentrations larger than 9.5 moll^{−1}, the present work gives unexpected evidence that the speed-limiting step involves two HF molecules.

## 2. Experimental methods

In the present work, five field-assisted etching experiments were performed with thin (approx. equal to 0.2 mm) Herasil silica plates in order to test the theory up to field strengths sufficiently high that the function *V* (*E*_{ap})−*V* (0) becomes notably nonlinear. The samples were etched in 150 ml of aqueous HF solution. The etched surface areas were 1.6±0.4 cm^{2}.

The initial thickness of the plates *d*_{0} was first measured in order to be able to determine the electric field strength from voltage values in the subsequent etching experiments. In order to reduce errors, thickness measurements were carried out with a Michelson interferometer, which permitted determining the thickness at the centre of the samples with precision better than 0.1 mm. Comparative measurements with a digital micrometer screw were also performed.

The measured samples were cleaned in an ultrasonic bath and glued with silicon rubber to the plain surface of a Teflon tube-like sample holder, as shown in figure 1. Two thin (0.2 mm) elastic wires were touching the glass surface inside the tube. These wires were electrically connected to plugs to facilitate the application of high voltage. An NTC thermistor (1.2 mm in diameter and 3.4 mm in length) with a tiny droplet of thermal interface paste at its end also touched the glass sample. Once the sample, wires and thermistor had been installed in the sample holder, a thin soot film was deposited on the sample surface inside the holder in order to provide a transparent electrode on the glass sample. The soot film had an optical density of and electric connectivity with the soot film was tested measuring the resistance between the two wires (approx. equal to 1 M2, although part of the conductance was due to soot on the Teflon walls). Then, the sample holder was screwed on to the lower end of a vertical Teflon tube hanging from an optical tower down into a box with an air exhaust that protected the laboratory from HF vapour during the etching process.

The thermistor was connected to a 12-bit battery-powered readout electronics unit mounted inside a metal box. During the experiments, this metal box, as well as the outer screen of the connecting coaxial cable, had the same electrical potential as the soot film. The acid, which etches the lower surface of the sample, is always kept at ground potential with the help of graphite electrodes. Because of this, the thermistor readout had to be transmitted to the data acquisition computer by means of an optical communication channel. The readout electronics had been constructed with resistor combinations selected so that the room temperature would not influence the temperature measurements. The whole temperature measuring system was carefully calibrated with the help of an Al isothermal block with temperature varying on a time scale of 7 days. Temperature measurements made with this thermometer have an estimated error of 0.05^{°}C.

During the etching experiment, an expanded and collimated He–Ne laser beam (*λ*=0.6328 mm) impinged vertically from above onto the sample. The reflected waves that come from the air–glass and glass–acid interfaces were sent to the matrix of a complementary metal oxide semiconductor camera that registers spatial interference patterns of the reflected waves. Every 1 or 2 s (depending on the concentration of the acid), an image was captured and saved in the data acquisition computer. The two glass surfaces were not perfectly parallel and therefore fringe images were formed. The analysis of fringe images consists of several steps.

Light intensities are averaged along straight lines (*y*-axis in figure 2) in a chosen rectangular region (white rectangle in figure 2). The line direction is manually chosen to be parallel to the fringes. The resulting time-dependent spatial intensity curves *I* are then shifted on the intensity scale so as to obtain oscillations around zero:
2.1
The sum is taken over all *x* values in the specified rectangle. Next, the shifted intensity curves are normalized,
2.2
and a trigonometric function is adjusted for each *t*. That means for each *t* one determines parameters *a*(*t*), *b*(*t*) and *c*(*t*) so that
2.3
The resulting phase constant *a*(*t*) permits calculation of the thickness *s*(*t*) of the etched layer:
2.4
A critical analysis of the resulting curves reveals that *s* contains a systematic error that oscillates in time. This error is caused by static defects in the images that do not correspond to fringes. This systematic error can be reduced considerably using the following correction. After a complete analysis of fringe images along the lines described above, a static intensity profile is defined by time averaging the deviation of normalized intensity curves from the trigonometric fittings
2.5
In this expression, *N*_{T} is the number of frames in a sufficiently large time interval (at least 10 oscillations or approx. 300 frames). This profile *S* represents static intensity patterns, which do not correspond to the moving fringes. With the help of this static function, one defines a corrected normalized intensity profile,
2.6
and repeats the phase analysis. The resulting corrected thickness, *s*_{corr}(*t*), no longer shows perceptible periodic errors. With this method the thickness of the removed glass layer can be determined, with a typical error of 2 nm. Etching velocities measured during one application of a given value of electric potential have a typical error of |*δ**V*_{error}/*V* |≈0.2%.

The measured etching velocity *V*_{measured}=d*s*_{corr}/d*t* is finally correlated with the values of the applied electric field,
2.7
where *φ*_{soot} and *φ*_{acid} are electric potential values of the soot film and the acid, respectively. During each experiment, several different potential values were applied in an almost random order. The zero value was continually applied in order to get information concerning systematic errors caused by temperature variation, as explained below.

The HF–SiO_{2} etching rate *V* depends on the temperature. In order to be able to separate the electric field dependence, an effort was made to keep the temperature constant during the experiments. Dilution of the acid was made 40–120 min before etching to dissipate temperature changes due to dilution. During the experiment, the windows of the laboratory were closed with thermal insulation and the experimenters, data acquisition computer and high-voltage power supply were in a neighbouring room. The effect of remaining temperature variations was subtracted to first order. To that end, the etching velocity with no voltage applied was plotted against temperature, and from this correlation a function *V*_{0}(*T*) was determined for each experiment. The measured velocities with applied electric field *E*_{ap} were then corrected by subtracting , where is the mean temperature during the experiment,
2.8
Typical values of temperature corrections were of the order of |*δ**V*_{T}/*V* |≈0.5%.

A smaller correction |*δ**V*_{c}/*V* |≈10^{−3}, due to the change of overall HF concentration caused by the chemical reaction during the experiments, was also considered. However, there may be an error due to a concentration change caused by HF evaporation during the experiments. The evaporation was kept low by closing the acid-containing recipient with a Teflon ring. Besides these global changes of concentration, one might expect to have local changes of concentration caused by the chemical reaction and low diffusion constants of the chemical species. However, experimental evidence exists showing that local changes of concentration play a minor role. Local HF depletion should lead to lower etching rates at the centre of the sample. But no difference between the centre and the border has ever been observed.

An additional etching experiment was also performed in order to study the dependence of etching velocity on HF concentration without an electric field. A 2 mm thick disc of silica (Herasil) was first shaped with selective HF etching in order to create an angle of 10^{−4} rad between the two glass surfaces so that the method of spatial fringe analysis could be applied in the subsequent etching experiment. During this experiment, the temperature of the laboratory was controlled actively (*T*=21.4±0.2^{°}C). The same sample was exposed to five different aqueous HF solutions, starting with the highest concentration (40%). In order to keep the acid consumption and production of contaminated waste low, the etching in this experiment was performed with only 30 ml, and another 10 ml of the solutions were used to rinse the sample and sample holder before using a new concentration. The rather small rinse volume causes some error, which was estimated using a repetition of the highest concentration.

In order to obtain necessary data for theoretical modelling of the effect, the conductivities of aqueous HF solutions were measured with 60 Hz and 1 kHz AC voltages using platinum electrodes. The usage of AC voltages is necessary in order to avoid depositing any kind of electrolytic material on the electrodes. In order to get rid of possible resistances of platinum–acid interfaces, the distance *l* of the electrodes was varied in a 10 cm long Teflon tube of 5 mm diameter, and the conductivity of the acid was determined from the slope of a linear fit of the resistance correlation of data.

## 3. Results

Figure 3 shows the measured etching velocities as a function of the applied electric field strength for a sample etched with hydrofluoric acid of 40 per cent. Figure 3*a* exhibits the uncorrected velocity values, whereas figure 3*b* includes temperature corrections. The normalized slope of the curve at *E*_{ap}=0 coincides perfectly well with the value measured in Lesche *et al*. (1997).

Current value (HF 40%, at 21.400^{°}C)
3.1
Value of Lesche *et al*. (1997)
3.2
However, for different concentrations, this value changes. Table 1 gives an overview of measured *α*-values for different concentrations and temperatures. As can be seen, lower concentrations result in larger *α*-values.

All five experiments showed a nonlinear relation between etching velocity and applied electric field. The curvature, as measured by quadratic coefficients of second-order polynomial fits, was found to be of the order of magnitude predicted by the model of Lesche *et al*. (1997), where the experimental *α*-values were used as an input parameter in the theoretical predictions. But systematically in all experiments, the experimental value of the second-order term was larger than the theoretical prediction. This indicates that the model needs corrections. There is other evidence that one basic assumption of the model of Lesche *et al*. (1997) is wrong. The etching rate at zero field strength is approximately proportional to the square of the stoichiometric concentration of HF
3.3
This linear dependence between *V* (*E*=0) and *c*^{2}_{ HF}, as illustrated in figure 4, is valid in the interval of concentrations used in the present work. At very low concentrations, other reaction mechanisms involving may become important and deviations of this simple relation may appear (Knotter 2000).

Equation (3.3) indicates that the primary step of the reaction 4HF+SiO_{2}→SiF_{4}+2H_{2}O is not performed by a single HF molecule, but requires simultaneous action of two HF molecules. A very simple-minded extension of the original model, that takes into account simultaneous action of two HF molecules in the primary step, would simply square the probabilities of the original model, assuming that the two molecules have statistically independent orientations. This leads to the following expression for the etching velocity:
3.4
and the theoretical prediction for *α* with this model would be
3.5Figure 5 shows a comparison of experimental data (20 October 2008) with theoretical curves. Curve A corresponds to the original model of Lesche *et al*. (1997), which uses only one HF molecule, and curve B corresponds to equation (3.5). In both models, *α* was used as an adjustable parameter. That means the experimental value was used rather than the predictions (1.2) or (3.5). Curve B adjusts itself to the experimental data slightly better than curve A, but the discrepancies are still larger than the experimental uncertainties. Moreover, the discrepancy of the experimental first-order coefficient *α* and theoretical prediction is much worse in the case of two molecules: m V^{−1}, m V^{−1}.

Improving the two-HF model was attempted by introducing a correlation between the molecular orientations, based on the electric dipole–dipole interaction. An extra term of dipole–dipole interaction was introduced into the energy of the two HF molecules. The probability of having two adjacent molecules both oriented within the critical angle *θ*_{0} is given by
3.6

where *E*_{L} is the local field that describes the effective interaction between the HF molecules and the applied field and polarized adjacent matter. And
3.7
is the dipole–dipole interaction energy of the two HF molecules, which are supposed to occupy two locations with the same distance from the glass surface and separated by a distance *d*.

The integrals were solved numerically and, in fact, the theoretical curve approaches the experimental data slightly more. However, the discrepancy between experimental data and theoretical prediction still exceeds experimental uncertainties and the problem of having to adjust *α* to a value much smaller than the theoretical prediction still persists.

In order to explain the remaining discrepancy another assumption of the model of Lesche *et al*. (1997) was investigated more carefully. So far, it has been assumed that the molecules are subject to a local field *E*_{L} that is strictly proportional to the applied electric field *E*_{ap},
3.8
This would be true if the electric field penetrated far into the acid. In order to determine the penetration depth, in the present work, measurements of conductivity were performed. The conductivity (at 23–25^{°}C) as a function of stoichiometric concentration *c*_{ HF} of HF is well described by the following relation:
3.9

From the measured conductivities, one may roughly estimate the ion concentration *n* of F^{−} ions with the help of the Fuoss–Onsager theory of molar conductivities (Fuoss & Onsager 1957; Miyoshi 1973). The estimated values are summarized in table 2.

The dielectric constant of the acid is also a parameter that influences the penetration depth of electric fields. The dielectric constants of the acid shown in table 2 were assumed to be a linear combination of the known dielectric constants of pure HF and H_{2}O at 0^{°}C with mol fractions *X* as weight and a temperature correction, which was assumed to be the same as that of water
3.10

With these values, the effective penetration of the electric field is much smaller that the one assumed in Lesche *et al*. (1997) (approx. 2.4–3.3 Å, rather than 130 Å). The field screening due to mobile ions in the solution reduces the theoretical *α*-values and it brings in two additional nonlinear effects. Using a mean field approach, one may describe the electric potential *φ* in the acid by means of the Poisson–Boltzmann equation:
3.11
where *z* is a spatial coordinate with origin in the glass–acid interface, *q* is the elementary charge, *n* is the density of F^{−} ions2 at and the electric potential *φ* is assumed to be zero at infinity. A one parameter family of exact solutions is known
3.12
with
3.13
which is the inverse of the Debye length. The electric field is given by
3.14
and the parameter *γ* is determined by the boundary condition
3.15
The logarithmic term is actually much smaller than the potential of the soot film. If one neglects this term, equation (3.15) can be solved analytically for *γ*,
3.16
For the fields applied in our experiments, |*γ*^{−1}| is much larger than *γ* and the field at position *z* can safely be approximated by
3.17
The local field *E*_{l}=(1+*χ*_{acid}/3)*E*(*z*) at the position of an HF molecule near enough to react with a silicon atom is now a slightly nonlinear function of the applied electric field. But the small penetration depth of the electric field brings yet another nonlinear effect into play. The polar HF molecules, which have a dipole moment slightly larger than water molecules (*μ*_{HF}=6.394×10^{−30} A s m, *μ*_{H2O}=6.161×10^{−30} A s m), are pulled into the regions of high electric field. This enhances the HF concentration near the glass surface. The probability of equation (3.6) has to be considered a conditional probability that two molecules have the right orientations to perform a chemical reaction, given that they are already at the right places. To get the probability of a chemical reaction, one has to multiply this conditional probability with the probability to find the molecules at the right spots. So far, this probability had been assumed to be independent of the applied electric field. But, as the field is capable of enhancing the HF concentration near the glass surface, this assumption has to be abandoned.

In order to evaluate this effect quantitatively, one must calculate the concentration changes induced by the heterogeneous electric field. A naive approach would be to take the Boltzmann e-factor to get an electrostatic version of barometric height formula. However, the acid is not at all an ideal gas: an HF molecule that gets pulled near to the glass surface has to expel other molecules that occupied the place. This means molecular interaction is an important factor. The Boltzmann factor describes the quotient of occupation probabilities of microstates *i*, *j* of a subsystem of a thermal bath, only as long as the interaction energy *W* of the subsystem and bath is small compared with the subsystem energies Δ*E*_{ij}. However, one may apply the Boltzmann e-factor if one includes the mean value 〈*W*〉 of the interaction energy in the energy of the subsystem so that the interaction part gets reduced to a much smaller quantity (*W*−〈*W*〉). This approach is completely within the spirit of theoretical equations used so far; equations (1.1), (3.4), (3.6) and (3.11) all rely on mean field approximations. Using such mean field approach, one may now construct a Boltzmann e-factor. The principal contribution to the subsystem energy is of course the energy of the dipole in the local electric field
3.18
where *α*_{P} is the mean (isotropic) polarizability of the HF molecule: *α*_{P}=9.23×10^{−41} A s m^{2} V^{−1} (Christiansen *et al*. 1998; Pecul & Rizzo 2003). The polarizability term actually gives only a small contribution so that a simplified isotropic treatment can be used. But this is not the only contribution. The interaction with neighbouring molecules that take care of the fact that two molecules do not occupy the same spatial region has to be considered too. A convenient way of estimating the mean value of this interaction energy is to extrapolate macroscopic hydrostatics into the microscopic realm. The averaging is automatically included in the macroscopic approach. The macroscopic electric force density on the liquid is (Becker & Sauter 1973)
3.19
where *ρ*_{el} is the free electric charge density and *ρ*_{M} is the mass density of the acid. The derivative d*ϵ*_{acid}/d*ρ*_{M} can be estimated using water as a similar liquid. For water, one has d*ϵ*_{H2O}/d*ρ*_{M}=1.231×*ϵ*_{0}*χ*_{H2O}/*ρ*_{M} (Marshall 2008).

One HF molecule in the 40 per cent acid occupies approximately 22.9×10^{−30} m^{3}, which can be estimated from the volume occupied by one molecule of H_{2}O in pure water (30.0×10^{−30} m^{3}), assuming that H_{2}O occupies the same volume in the acid. The volume 22.9×10^{−30} m^{3} corresponds to a sphere of radius *R*=1.76 Å. Integrating the force density (3.19) over such a sphere gives the average long-range electric force that acts on such a volume of acid. Using the Archimedes principle, the mean pressure force acting on the HF molecule can then be estimated as the negative of that integral. Integrating this buoyancy force over *z*, one gets an effective potential energy
3.20
where and is an abbreviation for the lengthy expression that multiplies the exponential factor. With the sum one can now form the Boltzmann factor and integrate over the angle *θ* to obtain the HF density at a position *z*,
3.21
where it has been supposed that . One may now assume that the chemical reaction initiates with the two involved HF molecules at some typical distance *z*, which is expected not to be larger than a few ångström. Then the etching velocity should be well described by the following expression:
3.22
where *P*(*θ*_{1}≤*θ*_{0},*θ*_{2}≤*θ*_{0}|*c*_{ LF}*E*(*z*)) is the numerically calculated probability of equation (3.6), but now with the local field *c*_{ LF}*E*(*z*) using equation (3.14) rather than equation (3.8). This theoretical prediction uses two adjustable parameters: the reaction distance *z* and the distance *d* between the reacting HF molecules. The parameter *α* should no longer be an adjustable one. The curve C in figure 5 shows a theoretical curve with parameter values
3.23
The theoretical curves A–C are shown beyond the experimental range in order to show that the application of HF etching in field measurements with poled glass, which result in relative etching rates of about 0.6, depend dramatically on the theoretical model.

The data of the experiment (20 October 2008) were chosen to determine the most appropriate values of *z* and *d* because this experiment had furnished the widest and cleanest dataset. The adjusted theoretical curve coincides perfectly well with the experimental data, and the deviations are just of the size of inherent experimental error. This fact alone would not be too impressive. But the same parameter values describe the experimental curves of the experiments (13 November 2007, 22 April 2008 and 13 May 2008) equally well, despite the fact that these experiments used a considerably different concentration. Figure 6 shows the comparison of theoretical curves and experimental data for the experiments (20 October 2008, 13 May 2008 and 9 October 2008). Data of the experiments (13 November 2007 and 22 April 2008) are not shown in order to maintain the graph’s readability. The quality of fit of the theoretical curves of these experiments is comparable to the one of the experiment (13 May 2008). Only the experiment (9 October 2008) shows deviations from the theoretical prediction that are still unsatisfactory. This discrepancy may be due to wrong estimates of ion concentrations and dielectric constant *ϵ*_{acid}. The Fuoss–Onsager theory is not reliable for hydrofluoric acid of such high concentration. Unfortunately, there is no reliable method to determine these concentrations for such high values. If one relies on the validity of the proposed theoretical model, one may inversely use the etching experiments to determine ion concentrations. Also the question, whether there exists the molecular species HF in the solution may be answered: if the reactive species were an ion pair H_{3}O^{+}F^{−} the observed *α*-values would be difficult to explain. H_{3}O^{+}F^{−} has a larger dipole moment than HF and one would need much stronger field screening and consequently unreasonably high ion concentrations to explain the observed values.

It is known that water catalyses the etching reaction (Ku Kang & Musgrave 2002). The present model considers only the HF molecules. One might argue that the changes of concentration induced by the heterogeneous electric field give rise to corresponding changes of water. But the effective change of the reaction rate due to changes of concentration can be described by the phenomenological relation (3.3), irrespective of the fact that water plays an important role. However, it might be that the catalytic action of water also depends on the orientation of the H_{2}O and this effect has not been considered in the present model.

## 4. Conclusion

Electric field induced changes of HF etching velocity was studied experimentally for different acid concentrations and with fields high enough to perceive nonlinear effects. The basic idea that dipole orientation causes the change in etching rate provides a reasonably good description of experimental data. Nevertheless, the original model proposed in Lesche *et al*. (1997) had to be modified: the primary step of the chemical reaction involves two HF molecules and field screening by mobile ions modifies the field effect. A theoretical description of the effect can be based on mean field approximations. The data of the present work can be used to measure with higher reliability fields written in silica samples. One may also use the known field dependence of etching velocity for shaping glass surfaces in a controlled way.

As the change of etching velocity induced by the electric field depends on the ion concentration, one may use the effect to determine ion concentration in highly concentrated electrolytes. This method may even be extended to other electrolytes using only small quantities of HF that act as a probe. To that end, the etching experiment has to be performed on a much longer time scale. Experiments for measuring ion concentrations this way are currently planned.

## Acknowledgements

We thank I. C. S. Carvalho and R. C. Matos for discussions and providing experimental materials. Financial support of the Fundação de Amparo à Pesquisa de Minas Gerais (FAPEMIG) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) is gratefully acknowledged. R.B.S.N. thanks the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for a grant. We especially thank one of the reviewers for extremely pertinent criticism, which stimulated an additional experiment (data of figure 4) and helped us to understand the relevance of the findings.

## Footnotes

- © 2009 The Royal Society