Short Communication
On the electrochemical applications of the bending beam method

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Abstract

Measuring the bending of a plate to determine the stress in thin films coated on one side of a substrate is a common technique. The surface stress changes can be estimated by using an appropriate form of Stoney's equation, if the thickness of the film is sufficiently less than that of the substrate. The changes in the bending of the substrate are usually calculated from the displacement of the reflected laser beam measured, e.g., with a position sensitive photo detector (PSD). Since the displacement of the light spot on the PSD is measured in air outside the liquid phase (where the mirroring surface is located), the measured values should be corrected according to Snell's law of refraction. A relation between the reciprocal of curvature radius of the substrate and the displacement of the laser beam on PSD was derived by taking into account the bending of the laser beam due to refraction at the optical window. It is shown that the neglect of the refraction at the optical window may cause an error of 25–30% in the estimation of the stress changes.

Introduction

The specific surface energy of solid electrodes provides important information on the solid  electrolyte interface. Most systems involving solids are, in fact, capillary systems, because any interaction between the bulk solid and the remainder of the system takes place via the surface region, convection effects being precluded, and therefore the understanding of the thermodynamics of solid surfaces is of importance to all surface scientists and electrochemists. During the past decades several attempts have been made to derive thermodynamic equations for the solid  liquid interface [1], [2], [3], and several methods were suggested for measurements of changes of the specific surface energy or surface stress of solid electrodes [4], [5].

Attempts to determine the surface energies of solid electrodes fall into two main categories: measurement of the potential dependence of the contact angle established by the liquid phase on the solid surface and measurement of the variation in surface stress experienced by the solid as a function of potential. Variation in the stress may either be measured directly [2], [6] with a piezoelectric element, or it can be obtained indirectly [1], [4], [5] by measuring the potential dependence of the strain (i.e. electrode deformation) and then obtaining the variation in stress from the appropriate form of Hooke's law. Usually, the methods yield only changes of surface energies as a function of various physicochemical parameters.

The principles of the so-called bending beam method were first stated by Stoney [7], who derived an equation relating the stress in a thin layer to the radius of curvature of the beam. Measuring the bending of a plate to determine the stress in thin films is a common technique [1], [4], [8], [9], [10], [11], [12], [13]. It has been used also for the investigation of the origin of electrochemical oscillations at silicon electrodes [14] and in the course of galvanostatic oxidation of formic acid on platinum [15], for the study of volume changes in polymers during redox processes [16], and for the investigation of the response kinetics of the bending of polyelectrolyte membrane platinum composites by electric stimuli [17].

According to several publications (e.g. [1], [8], [9], [10]), in the case of small bending the changes of the surface stress (Δγs) in a thin metal film coated on one side of a (e.g. glass) plate can be estimated from the changes in the bending of the substrate. If the thickness of the film, tf, is sufficiently less than that of the substrate, ts, the change of γs can be estimated by an expression based on Stoney's equation:Δγs=Ests261−νsΔ1r

In Eq. (1) Es, νs, and r are Young's modulus, Poisson's ratio and the radius of curvature of the bending of the substrate, respectively. In the derivation of Eq. (1) it was implicitly assumed that Δγs=tfΔσf, where Δσf is the change in the film stress. According to Eq. (1), for the calculation of Δγs the changes of the radius of curvature of the substrate Δ(1/r) should be also known.

The validity of Stoney's original equation has been questioned recently by Chu [18], who derived a more complicated expression. However, apart from a factor of 2 in the denominator (12 instead of 6) Chu's equation reduces to Stoney's formula in the special case of a very thin film on a substrate [12], [13], [18], and therefore both equations can be written as:Δγs≈kiΔ1rwhere k1=Ests261−νs in Stoney's equation, and k2=Ests2121−νs in Chu's equation.

Nevertheless, according to the above considerations it is clear that correct values of Δ(1/r) or r are always necessary for the calculations. In addition, the expressions derived for greater deformations also contain the difference of the reciprocal radii of curvature [16], [17].

It is very important, that although the numerical values of the stress may be rather uncertain due to the problems e.g. with Eq. (2) and due to other experimental and theoretical difficulties, the values from different experiments can be compared directly. Therefore the correct determination of the changes in the radius of curvature of the electrode is of great importance.

The values of Δ1r can be calculated if the changes Δα of the deflection angle (α) of a laser beam reflected by the metal layer are measured using an appropriate experimental setup. A typical arrangement for an electrochemical bending beam experiment is shown in Fig. 1. Such a setup can be used mainly for the investigation of small deflections, and several details may be different in special cases. However, it clearly demonstrates the principles and the possible pitfalls of the method. Fig. 2 shows the optical configuration of the experimental setup in Fig. 1.

According to Fig. 2 a laser beam coming from a direction normal to the plane of the optical window is reflected by the electrode surface (point R). The incident beam is perpendicular to the optical window, therefore no refraction occurs at C. The direction of the reflected beam is RC′, which coincides with the direction of C′M, and the angle between CR and RC′ is α. Without refraction at C′, the reflected beam would result in a light spot at M on the detector plane. (The displacement due to the thickness of the optical window has been neglected.) However, because of the refraction at C′ the direction of the beam changes so that the new direction is C′N, and a light spot can be detected at N. The angle between P′C′ (or PC) and C′ is denoted by β. The distance between P and M is a, and between P and N is b. Obviously, only b can be determined by direct measurement, because there is no light spot on the detector at M.

The distance between the electrode and the photo detector isw=d+lwhere d is the distance between the optical window and the reflection point (R) on the electrode, and l is the distance between the optical window and the detector plane (Fig. 2). If α is very small, the radius of curvature of the cantilever can be well approximated asr≈2sαwhere s is the length of the electrode in the solution. Sometimes half of the total deflection angle is denoted by θ, thenθ=α2andr≈sθIf y=PP′, from Fig. 2α≈tanα=aw=a−ylThe measurement of y is impossible in the usual experimental setup, but it is practically always possible to arrange the electrode and the detector in such a manner that ld. In these cases ya, and instead of Eq. (7) we arrive at the following expression:α≈tanα≈alFrom the practical point of view, Eq. (8) means that for the determination of α knowledge of a and w (or a and l) is necessary.

However, it is well known that in the case of non-normal incidence, if the deflection of a light beam is measured outside the phase where the mirroring surface is located, the deflection angle should usually be corrected according to Snell's law of refraction. According to this law, when light travels from one medium into another the incident and refracted rays lie in one plane where the normal to the surfaces are on opposite sides of the normal and make angles with the normal whose sines have a constant ratio to one another. This means that due to the refraction we cannot obtain a light spot on the detector plane at M (which could be used for the determination of a and α), and only the light spot at N can be detected (Fig. 2). As a consequence, only b is accessible experimentally.

According to the experimental setup shown in Fig. 1, the laser beam reflected from the lower end of the electrode is directed to a position sensitive detector. It means that for small deflectionsb−yl=b−yw−d=tanβ≈βIf ld and by, thenbwtanβ≈βConsequently, by measuring b and w the approximate value of β can be determined experimentally.

However, according to Eq. (4), the value of α (and not β) is necessary for the correct determination of r. The relationship between α and β can be obtained from Snell's law:sinαsinβ=na,s1nswhere na,s is the (relative) refractive index of the air with respect to the solution, and ns is the (absolute) refractive index of the solution. For small deflectionssinαsinβαβ1nsand thereforeα≈βnsThis means that the angle of deflection can be calculated asα≈bnswand the radius of curvature of the cantilever asr≈2nsswb

If the conditions ld and by are not fulfilled, e.g. because the detector is close to the electrode, Eq. (8) should be used. In this case the expressions for α and r are the following:α=b−ynsw−dandr=2nssw−db−yi.e. for the calculation of r the determination of y and d is also necessary.

On the other hand, according to Eq. (2) the value of Δ1r is necessary for the determination of the change in the film stress. Applying Eq. (15) for two different deflections of the electrode (see Fig. 2) and assuming that CRCR′, we arrive at the equationΔ1rΔb2nsswwhere Δb is the displacement of the light spot on the position sensitive detector due to the bending of the electrode.

Without the refractive index of the solution, it is impossible to determine correctly the values of the radius of curvature (or the change of the reciprocal value of the radius of curvature) of the electrode. Since refractive indices of aqueous solutions are in the range 1.33–1.48, neglect of the bending of the laser beam due to refraction at the optical window may cause an error of 25–30% in the determination of r−1 or Δr−1.

In the light of the above results it is clear that the effect of refraction at the optical window cannot be ignored in electrochemical bending beam experiments.

Unfortunately, in many papers reporting results based on such experiments no reference is made to the refractive index of the solution, the effect of refraction is ignored, or the value of the refractive index of the solution is not indicated (e.g. [8], [9], [10], [11], [12], [13], [14]). On the other hand, in figures showing the schemes of experimental arrangements the direction of the reflected beam before and after passing the optical window is indicated coincidentally (e.g. [10], [11], [13], [14]), which is misleading.

Section snippets

Conclusions

Summarizing the main points of the above considerations, the following conclusions can be drawn:

  • 1.

    Many figures in the literature showing experimental setups or optical configurations for electrochemical bending beam experiments are misleading, since in the figures the direction of the reflected beam in the two different media adjacent to the optical window is the same, which is impossible in the case of non-normal incidence.

  • 2.

    If the effects of refraction are ignored in a bending beam experiment,

Acknowledgements

Financial support provided by the Hungarian National Research Fund (OTKA T030150, F019444), the Ministry of Education (FKFP 0161/1997), and OMFB (Hungarian–Japanese Intergovernmental Cooperation in S&T, JAP-14/98) is acknowledged. G.L. thanks the Széchenyi Foundation and the Japan Society for the Promotion of Science for research fellowships.

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