Experimental Methods for the Measurement of Surface Stress Changes

  • Gyözö G. Láng
  • Cesar A. Barbero
Part of the Monographs in Electrochemistry book series (MOEC)


In Chap. 4, methods for the experimental determination of (interfacial) stress changes in electrochemical systems are presented. The piezoelectric method, the extensometer method and its variants (e.g., optical fiber interferometry with Mach–Zehnder interferometer), the “bending beam” (“bending cantilever,” “laser beam deflection,” “wafer curvature”) method and related techniques, and the method based on the measurement of contact angle are discussed in detail. Special attention has been paid to problems related to the use of the electrochemical bending beam and bending plate/disc methods. The theories of the methods are summarized, and typical experimental arrangements are presented. The kind and quality of information that can be achieved using these methods are discussed.


Contact Angle Electrolyte Solution Surface Stress Potential Dependence Zehnder Interferometer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

4.1 Introduction

As it has been pointed out in the previous chapters, the thermodynamic theory of the electrified solid/liquid interface and the thermodynamic interpretation of the results from various methods in terms of physicochemical properties of the system are not without problems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

In principle, the results of the theoretical work can be checked experimentally; however, the study of the specific surface energies of electrified solid/liquid interfaces is complicated by many factors. The value of the “absolute surface tension” of some relatively simple covalently bonded, ionic, rare-gas, and metallic crystals could be estimated theoretically [22], and the surface tensions of some solid surfaces have been determined experimentally in some special cases [22, 23]. For the calculation of the surface stress, σ, Vermaak and coworkers [24, 25, 26] measured the radial strain in small (solid) spheres of Au, Ag, and Pt by electron diffraction and determined an average surface stress using the following equation:
$$ \sigma = - \frac{3}{2}E\varepsilon r, $$
where E is the bulk modulus, ε is the radial strain, and r is the radius of the sphere, respectively. Their results are listed in Table 4.1 in order to allow easy comparison with other results in the literature.
Table 4.1

Experimental surface stress data from [24, 25, 26]


Surface stress, σ (J m−2)

Temperature, T (°C)


1.18 + 0.20



1.42 + 0.30



2.57 + 0.40


However, the methods used in these experiments are designed for the solid/gas interface and are mostly inappropriate for use in the presence of an electrolyte solution; consequently, they cannot be applied to study the surface energetics of solid electrodes. Theoretical calculations of surface stresses generally involve calculating the surface free energy and its derivative with respect to elastic strain. Both first principles and semiempirical atomic potential calculations involving computer simulations have been attempted; however, only first-principle approaches yield accurate values for surface stress [27]. Tabulated values of the surface stress and surface energy for a variety of metals, ionic solids, and semiconductors can be found in reviews [27, 28].

It is not surprising, therefore, that during the past decades, several methods were suggested for the measurements of changes of the interfacial stress (“interfacial tension,” “surface stress,” “specific surface energy,” etc.) of solid electrodes (e.g., [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, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70]).

According to the classification proposed by Morcos [71], attempts to determine the interface stress of solid electrodes fall into two main categories: measurement of the potential dependence of contact angle established by liquid phase on the solid surface [72, 73, 74] and the measurement of the variation in interface stress experienced by the solid as a function of potential. Changes in the interface stress may either be measured “directly” [39, 57], with a piezoelectric element, or be obtained indirectly [36, 75, 76, 77, 78], by measuring the potential dependence of the strain (i.e., the deformation of the electrode) and then calculating the variation in stress from the appropriate form of Hooke’s law.

It should be emphasized again that the above methods only yield changes of the interfacial stress as a function of various physicochemical parameters, e.g., as a function of electrode potential, and in principle, if there are both “plastic” and “elastic” contributions to the total strain, the changes of the “generalized surface parameter” [79] (i.e., the interfacial intensive parameter conjugate to the surface are) can be determined.

Unfortunately, most of the proposed methods have drawbacks, i.e., they are technically demanding, they cannot be used to monitor changes of the interfacial stress, they are semiempirical and depend on doubtful assumptions, etc.

In this chapter, the different techniques used for the determination of changes of interface stress of electrodes (the piezoelectric method (e.g., [39, 53, 75]), the extensometer method (e.g., [80, 81]) and its variants, and the electrochemical “bending beam” method (e.g., [30, 31, 32, 33, 34, 35, 36, 37, 38, 75, 78, 82, 83, 84, 85])) as well as the kind and quality of information that can be achieved using these methods are discussed. Special attention has been paid to problems related to the use of the “bending beam” (“bending cantilever,” “laser beam deflection,” “wafer curvature”) and “bending plate/disc” methods.

In bending beam measurements, a thin metal strip or a thin strip of glass or other substrate (on which the metal film is deposited) is rigidly clamped at one end in a fixed mount to form a cantilever. The deflection of the free end, as the strip becomes bent, is then measured by some means.

It should be noted that in most of the literature reviewed here, the intensive parameter conjugate to the surface area is usually called “surface tension” or “surface stress” and is denoted generally by γ s or σ s, respectively. In order to be consistent with the original literature, wherever and whenever possible, we will keep the “original” notation throughout the next chapters.

4.2 The Piezoelectric Method

According to our knowledge, Gokhshtein [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] was the first to measure changes ∂γ s/∂E of γ s with the electrode potential E at platinum electrodes in sulfuric acid using the “piezoelectric method.” This method, originally developed by Gokhshtein and further improved by other scientists [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70], especially by Seo et al. [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68], is a powerful in situ method for the rapid determination of surface energy changes.

The method is “direct” in the sense that it is the variation in the electrode deformation that is “registered” directly by a piezoelectric element. A metal plate is rigidly connected, in a special manner, to a highly sensitive piezoelectric element (Fig. 4.1).
Fig. 4.1

Schematic illustration of a device for the “piezoelectric method”

The applied potential consists of a mean (DC) component upon which is superimposed on a high-frequency component. Usually, a sinusoidal signal is superimposed on a linear potential sweep. Electrode potential oscillation with the amplitude ΔE results in oscillation with an amplitude Δγ s in the surface stress, generating mechanical oscillation in the entire electrode–piezoelement unit. The piezoelectric element converts the mechanical oscillations to alternating electric signals, which can be detected by using a lock-in amplifier, an oscilloscope, or a frequency response analyzer. In fact, this is some kind of electromechanical immittance measurement. The geometry of the electrode and its oscillation is of no fundamental importance. The metal in contact with the electrolyte solution may be made in the form of a tight filament which can make lengthwise oscillations, a straight rectangular rod which can make bending oscillations, etc. (Fig. 4.2). The alternating surface stress sets in motion not only the electrode, but the whole electrode and piezoelement system since the inertia forces are essential.

The oscillations of surface stress can also be excited by the application of high-frequency current. In this case, the amplitude of the surface charge density is kept constant by specifying the amplitude of the alternating current. According to [50], under such conditions, Δγ s will be proportional to the derivative ∂γ s/∂E, which is called the “ϕ-estance.” If the electric variable is the potential, and the measurement is carried out with a constant amplitude ΔE, the amplitude of the piezoelectric voltage |A| is proportional to the derivative of the surface stress with respect to the electrode potential ∂γ s/∂E, and the phase angle (ϕ) contains information about the change in the sign of ∂γ s/∂E. The ∂γ s/∂q value is related to ∂γ s/∂E (designated by Gokhshtein as q-estance and ϕ-estance, respectively [39, 49]) by
$$ \frac{{\partial {\gamma_{\rm{s}}}}}{{\partial q}} = \frac{{\partial {\gamma_{\rm{s}}}}}{{\partial E}}\frac{{\partial E}}{{\partial q}} = \frac{1}{C}\frac{{\partial {\gamma_{\rm{s}}}}}{{\partial E}}, $$
where C is the electrode capacitance.
Fig. 4.2

Piezoelement units designed by Gokhshtein: (a) with plunger and (b) with foot. (1a, 1b) metal plate, rod, or filament; (2) holder; (3) rib; (4) piezoelement; and (5) plug

If the contribution of components resulting from the measuring instruments and the mechanical properties of the system to ϕ is kept constant, the electrode potential corresponding to the change of the sign of ∂γ s/∂E can be evaluated from the relative change of ϕ [57]. According to [57], the relation between Δγ s and ∂γ s/∂E can be given as
$$ \Delta {\gamma_{\rm{s}}} = \int {\frac{{{\text{d}}\,{\gamma_{\rm{s}}}}}{{{\hbox{d}} E}}} {\hbox{d}} E \propto \int {\left| A \right|} {\hbox{d}} E. $$
By applying this method, ∂γ s/∂E is measured at high frequencies, and the quantitative determination of Δγ s requires a difficult calibration procedure (the transfer function characteristic for the mechanical coupling is rather complicated [39]). However, the potentials of extrema of the surface stress vs. potential function can be obtained directly. The relation between the piezoelectric signals (|A| and ϕ) and the Δγ s vs. E curve is shown schematically in Fig. 4.3.
Fig. 4.3

Measured and computed quantities in a typical piezoelectric experiment: The amplitude of the piezoelectrical signal |A|, the phase angle ϕ, and the change of surface stress Δγ s, as a function of the electrode potential E. Numerical integration of the experimentally determined |A| curve with respect to the electrode potential yields a quantity proportional to Δγ s

A series of piezoelectric surface stress change measurements has been performed to date in order to understand electrode processes such as electrosorption and initial oxidation. This technique was capable of detecting sensitively the shift in potential of zero charge (pzc) due to the adsorption of ions and the sign reversal of surface charge due to the formation and reduction of surface oxide phases. For example, in case of platinum in sulfuric acid solutions, Gokhshtein observed two extrema in the hydrogen adsorption region [39]. Similar results were obtained by Seo et al. [57], applying the same experimental technique to platinum in 0.5 M acid sulfate solutions. On the other hand, Malpas et al. [53] observed only one extremum at E ≈ 0.05 V for platinum in 0.1 M sulfuric acid. The electrode potential of the maximum was found to shift with pH to more negative values according to ∂E m/∂pH −40 mV [57].

Obviously, because of the dynamic features of the method, the recorded variation in surface stress does not always correspond to equilibrium conditions. For perfectly polarizable electrodes, e.g., Au in contact with aqueous sodium sulfate solution in a certain electrode potential range, equilibrium may be reached during the measurement because the time for charging/discharging of the electrochemical double layer is shorter than the period of oscillation of the interfacial tension. In case of platinum, the period of oscillation is shorter than the time necessary for adsorption of hydrogen or oxygen to reach equilibrium; thus, the results depend on the frequency of oscillation as well as on the rate on the scan rate of the linear sweep.

An important advantage of the piezoelectric method is the selective separation of surface energy contributions from other side effects, such as changes in bulk stress due to diffusion or Joule heating of the electrolyte solution [39].

4.3 The Extensometer Method and Related Techniques

An extensometer instrument that directly measures change in the length of a very thin metal ribbon or wire has been proposed in [80]. In this instrument, the ribbon in contact with an electrolyte solution served as a working electrode in an electrochemical cell. The ribbon was kept under an approximately constant force (a mechanically applied tensile stress) throughout the experiment by mounting it axially inside a glass tube.

The schematic design of the extensometer is shown in Fig. 4.4.
Fig. 4.4

Schematic design of extensometer

As depicted in Fig. 4.4, the upper end of the ribbon or wire is attached to a spring. The spring constant of the spring should be small compared with the stiffness of the ribbon. The lower end of the ribbon is attached to the glass tube through a Teflon plug. An aluminum bobbin mounted on the quartz spindle forms two capacitors with fixed plates mounted in the head. Small changes in the two capacitances due to length changes of the ribbon are measured with an electronic capacitance sensor. The output voltage VL) and the change in the length of the ribbon are described by the relationship:
$$ V(\Delta L) = {K_1}\left[ {\left( {\frac{1}{{{C_1}}}} \right) - \left( {\frac{1}{{{C_2}}}} \right)} \right] = {K_{{2}}}\Delta L, $$
where C 1 and C 2 are the two capacitances and K 1 and K 2 are proportionality constants.

In a series of experiments, Beck et al. [80, 81, 86, 87, 88] attempted to determine variations in surface stress as a function of potential by using this method. According to the authors’ opinion, the change in the surface stress with potential causes a change in the length of a ribbon, and the extensometer can be used to determine the potential of zero charge.

The method is based on the following considerations: Suppose an unstressed metal ribbon and a spring, one end of each is fixed. The ribbon of length L is immersed into the electrolyte solution in the extensometer. There is a gap L 0 between the other ends, as shown in Fig. 4.5, case 1. If the ribbon and spring are stretched to connect their free ends, the ribbon is stretched by Δx 1 (case 2 in Fig. 4.5), and the spring force F 1 is
Fig. 4.5

Schematic diagram of the spring and ribbon system: (1) nonstretched ribbon and spring, without potential applied; (2) stretched ribbon and spring, without potential applied; and (3) stretched ribbon and spring, with potential applied [81]

$$ {F_{{1 }}} = {k_{\rm{r}}}\Delta {x_{{1 }}} = {k_{\rm{s}}}(\Delta {L_0} - \Delta {x_1}), $$
where k s is the spring constant of the spring. With the bulk elastic modulus (Young’s modulus, E b) and the cross-sectional area (A) of the ribbon, respectively, the spring constant of the ribbon, k r, can be given as
$$ {k_{\rm{r}}} = \frac{{{E_{\rm{b}}}A}}{L}. $$
Eliminating Δx 1 gives
$$ {F_1} = \frac{{{k_{\rm{r}}}{k_{\rm{s}}}\Delta {L_0}}}{{{k_{\rm{r}}} + {k_{\rm{s}}}}}. $$

If the electrode potential is applied in the extensometer experiment (the ribbon in contact with the electrolyte solution serves as the working electrode), the ribbon is increased (decreased) in length by an amount ΔL.

Suppose that the free ends are again connected, and the ribbon is stretched by ΔL + Δx 2. Assuming, that since ΔL is very small, the spring constant of the ribbon remains the same after the change in length, the new force F 2 on the ribbon and spring is
$$ {F_{{2 }}} = {k_{\rm{r}}}\Delta {x_{{2 }}} = {k_{\rm{s}}}(\Delta {L_0} - \Delta L - \Delta {x_2}). $$
After eliminating Δx 2, we have
$$ {F_2} = \frac{{{k_{\rm{r}}}{k_{\rm{s}}}(\Delta {L_0} - \Delta L)}}{{{k_{\rm{r}}} + {k_{\rm{s}}}}}, $$
$$ \Delta F = {F_2} - {F_1} = \frac{{ - {k_{\rm{r}}}{k_{\rm{s}}}\Delta L}}{{{k_{\rm{r}}} + {k_{\rm{s}}}}}. $$
If k r ≫ k s, then
$$ \Delta F \approx - {k_{\rm{s}}}\Delta L. $$
In [81], it has been assumed that the force on the ribbon is resisted by separable bulk and surface stresses, σ b and σ s, respectively. The surface stress has been assumed to be in a region of atomic thickness on the surface of the metal ribbon. Thus,
$$ F = {\sigma_{\rm{b}}}A + {\sigma_{\rm{s}}}P $$
$$ \Delta F = {\sigma_{\rm{b}}}\Delta A + A\Delta {\sigma_{\rm{b}}} + {\sigma_{\rm{s}}}\Delta {S_{\rm{r}}} + {S_{\rm{r}}}\Delta {\sigma_{\rm{s}}}, $$
where S r is the periphery of the cross section of the ribbon.
According to Hooke’s law,
$$ \Delta {\sigma_{\rm{b}}} = {E_{\rm{b}}}\frac{{\Delta L}}{L}. $$
It has been assumed that E b remains unchanged if the electrode potential changes. If t is the thickness and w is the width of the metal ribbon, the change in area is
$$ \Delta A = (t + \Delta t)(w + \Delta w) - tw, $$
$$ \frac{{\Delta A}}{A} = \left( {1 + \frac{{\Delta t}}{t}} \right)\left( {1 + \frac{{\Delta w}}{w}} \right) - 1. $$
According to Poisson’s law,
$$ \frac{{\Delta t}}{t} = - \nu \frac{{\Delta L}}{L} $$
$$ \frac{{\Delta w}}{w} = - \nu \frac{{\Delta L}}{L}. $$
From Eq. (4.4), by neglecting the quadratic term,
$$ \frac{{\Delta A}}{A} = {\left( {1 - \nu \frac{{\Delta L}}{L}} \right)^2} - 1 \approx - 2\nu \frac{{\Delta L}}{L}. $$
By combining Eqs. (4.3) and (4.6),
$$ \left| {\frac{{{\sigma_{\rm{b}}}\Delta A}}{{A\Delta {\sigma_{\rm{b}}}}}} \right| = \frac{{{2}\nu {\sigma_{\rm{b}}}}}{{{E_{\rm{b}}}}}. $$
The bulk strain σ b/E b is expected to be very small; thus,
$$ A\Delta {\sigma_{\rm{b}}} \gg {\sigma_{\rm{b}}}\Delta A. $$
In [81] a constant elastic modulus E s has been defined for the atomic layer on the surface of the metal ribbon. This assumption is rather questionable since E s may be affected by adsorption or absorption. The change in the surface stress has been expressed as
$$ \Delta {\sigma_{\rm{s}}} = {E_{\rm{s}}}\frac{{\Delta L}}{L}, $$
and the periphery of the cross section of the ribbon has been approximated as S r ≈ 2w. This means that
$$ \frac{{\Delta {S_{\rm{r}}}}}{{{S_{\rm{r}}}}} \approx \frac{{\Delta w}}{w} = - \nu \frac{{\Delta L}}{L}. $$
Since the surface linear strain σ s/E s is very small, from Eqs. (4.9) and (4.10),
$$ {S_{\rm{r}}}\Delta {\sigma_{\rm{s}}} \gg {\sigma_{\rm{s}}}\Delta {S_{\rm{r}}}. $$
From Eqs. (4.1), (4.2), (4.8), and (4.11)
$$ \Delta F \approx A\Delta {\sigma_{\rm{b }}} + {S_{\rm{r}}}\Delta {\sigma_{\rm{s}}} \approx - {k_{\rm{s}}}\Delta L. $$
On the other hand,
$$ A\Delta {\sigma_{\rm{b}}} = A{E_{\rm{b}}}\frac{{\Delta L}}{L} = {k_{\rm{r}}}\Delta L, $$
$$ {S_{\rm{r}}}\Delta {\sigma_{\rm{s}}} \approx - ({k_{\rm{r}}} + {k_{\rm{s}}})\Delta L, $$
and since k r ≫ k s,
$$ \Delta {\sigma_{\rm{s}}} \approx - \frac{{A{E_{\rm{b}}}}}{{{S_{\rm{r}}}L}}\Delta L. $$

This means that the variation in surface stress can be obtained from the change in the ribbon length.

The most serious problem to overcome in the design of the extensometer is minimizing errors due to thermal expansion. Even very small temperature variations can cause changes in the length of the ribbon which are expected to be considerably larger than those produced by changes in the surface stress.

Several design considerations were made to minimize the effects of thermal expansion [80]:
  • Invar and quartz, each having low thermal expansion coefficients, were used for the measuring head above the ribbon out of the electrolyte solution. [Invar, also known generically as FeNi36 (64FeNi in the US), is a nickel steel alloy notable for its uniquely low coefficient of thermal expansion. The name, Invar, comes from the word invariable, referring to its lack of expansion or contraction with temperature changes.]

  • The whole instrument was placed in a thermally insulated box and measurements were made after the thermal drift has decreased to a low value.

  • The potential sweep rate was high to provide adequate resolution from residual thermal drift.

  • The measurements were limited to metals, electrolyte solutions, and potential ranges in which electrolysis current and consequently the heating effects are minimum.

However, none of these measures can completely eliminate thermal effects due to Joule heating or electrochemical polarization. It has been therefore suggested that the method be limited to systems with low electrochemical activity. Nevertheless, as pointed out by Morcos [71], unless the effect of electrochemical processes on thermal expansion can be quantitatively accounted for, the results of the extensometer method cannot be conclusively interpreted.

The temperature sensitivity of the extensometer can be estimated using the data presented in Table 4.2. The difference between the thermal expansion coefficients (Δα t ) of gold and crown glass is about 5 × 10−6 K−1. If l r = 50 cm length of the metal ribbon is immersed in the electrode solution, the differential thermal expansion (α t l r) is on the order of 2,500 nm K–1. This means that at a sensitivity of the order of 1 nm [86], a temperature stability of
Table 4.2

Coefficients of thermal expansion (α t , t = 20°C)

Au: 14.2 × 10−6 K−1

Pt: 8.9 × 10−6 K−1

Ag: 19.7 × 10−6 K−1

Quartz: 0.16–14.5 × 10−6 K−1

Optical glass: 7.6 × 10−6 K−1

Crown glass: 9.65 × 10−6 K−1

$$ \Delta t = \frac{{\Delta l}}{{{\alpha_t} \cdot {l_{\rm{t}}}}} = \frac{{1\,{\text{nm}}}}{{{2,500}\,{\hbox{nm}}\;{{\hbox{K}}^{{ - {1}}}}}} = 4 \times {10^{{ - 4}}}\,{\hbox{K,}} $$
is required.
In [89], the effect of mechanical stress on electrode potential, E, was studied under zero current conditions. The experimental setup was very similar to the extensometer described above. A tensile stress machine was used for mechanical characterization of the samples. A two-electrode geometry was used for the potential measurements; the potential of a thin Ag wire made taut vertically in an electrolyte solution (AgNO3) was measured against a reference electrode with a high input impedance voltmeter. (The total length of the wire was 50 mm; the portion in contact with the electrolyte solution was 30 mm.) One end of the wire was fixed to one arm of a balance; the other arm of which was loaded with standard weights. The applied force was low enough to maintain proportionality between stress and strain (elastic deformation conditions). A schematic diagram of the apparatus is shown in Fig. 4.6.
Fig. 4.6

Schematic drawing of the apparatus: (1) metal (Ag) wire; (2) reference electrode; and (3) force (F = mg, where m is the load mass and g is the local acceleration of free fall)

The numerically large response reported in [89] has been questioned in [90]. It has been pointed out that the dE/dσ values (where σ is a measure for tangential stress, which scales with the elastic strain ε) reported in the literature [89, 91, 92] differ by several orders of magnitude and even by sign. In [90], the measurement of the response, ΔE, of the electrode potential of a polarizable electrode (gold 10 mmol dm−3 aqueous perchloric acid solution) to elastic strain under open circuit has been described. A lock-in technique was used to measure the potential variation during cyclic elastic deformation of a thin-film electrode supported on a polymer substrate. The method allowed the potential–strain response to be accurately resolved for elastic strain amplitudes as small as 10−4. It has been shown that the approach similar to that originally suggested by Gokhshtein (see the piezoelectric method discussed above [39, 46]) can provide quantitative data for dE/dε. The results reported in [90] were commented in [93]. As a response to the criticism, the experiment with Ag was repeated [94] using the identical electrolyte as in [89] (aqueous 10 mmol dm−3 AgNO3 and 0.1 mmol dm−3 KNO3). According to the authors of [94], the results confirmed the prediction that ς = ∂E/∂e is small and ς at the potential of zero charge should be negative for Ag, similar to Au.

Figure 4.7 shows the experimental setup used in [90, 94]. The entire apparatus was housed in a stainless steel chamber. Before experiments, the chamber was flushed repeatedly with Ar and then sealed in Ar at atmospheric pressure.
Fig. 4.7

Schematic representation of the experimental setup in [90]: (1) sample with Au layer; (2) working electrode; (3) reference electrode with Luggin capillary; (4) auxiliary electrode; (5) mobile grip; (6) fixed grip; (7) lock-in amplifier; (8) potentiostat; (9) electrolyte solution; and (10) stainless steel chamber

Au layers for the experiments were prepared by dc magnetron sputtering onto the top of a thin Ti adhesion layer on much thicker polyimide substrates. The principle of operation can be understood by referring to Fig. 4.8. The entire gold surface was wetted during the measurements. The surrounding regions of wetted substrate are insulating. The electrode has been strained by applying a uniaxial stress at the open circuit potential (E ocp). When the axial strain imposed on the substrate is Δl/l (l is an axial linear dimension), then the area strain of the substrate is Δε = (1 − ν)·Δl/l, with ν is the substrate Poisson’s ratio. This strain is transferred to the electrode.
Fig. 4.8

The sample with gold film in Fig. 4.7: (1) gold layer; (2) substrate (Kapton®); (3) three-phase boundary between the surface of the sample, the electrolyte solution, and the gas phase; a: 10 mm; b: 25 mm

In the experiments, the substrate was strained by displacing one of two grips using a computer-controlled piezoactuator equipped with a calibrated displacement sensor that serves to record the time dependence of Δl. A gentle prestraining prevented the buckling of the substrate in the negative-going part of the strain cycle. The sample was mounted horizontally with the gold film facing down and was contacted from below by a meniscus of electrolyte solution. To assure constant wetted Lagrangian area (constant number of surface metal atoms in contact with the electrolyte solution) throughout the strain cycles, the wetted region was larger than the circular electrode section (see the boundary 3 in Fig. 4.8). The reference electrode was separated from the main body of the cell (made of Teflon®) by a Luggin capillary. Au wire in a compartment separated from the main reservoir by a channel served as the counter electrode. The frequency of the cyclic strain was limited to 100 Hz.

According to the authors, their results demonstrate that, when artifacts from Faraday currents and adsorbate coverage are avoided, the prediction that “electrical work–mechanical response is the same as mechanical work–electrical response” (as proposed almost 40 years ago by Gokhshtein) could be confirmed experimentally.

Cyclic voltammetric experiments combined with dilatometric detection of the length change have been reported in [95, 96]. Nanoporous Pt samples prepared by consolidating commercial Pt black having a grain size of 6 nm were immersed in different aqueous electrolyte solutions (H2SO4, HClO4, and KOH), and the strain upon varying the electrode potential E was measured in situ by dilatometry and diffractometry. According to the results, reversible strain amplitudes comparable to those of commercial piezoceramics could be induced in metals by introducing a continuous network of nanometer-sized pores with a high surface area and by controlling the surface charge density through an applied potential [95]. In [96], the experimental results for cuboids of porous gold of dimension 1.2 × 1.2 × 1 mm3 were reported.

In [97, 98], a laser technique, based on optical fiber interferometry, is described for in situ measurements of electrode strain during electrode reactions. The basic concept utilizes a metal coated optical fiber in contact with an electrolyte solution as the working electrode in an electrochemical cell, while simultaneously using the fiber as one arm of a Mach–Zehnder interferometer (see later). The optical path length of the metal coated fiber was monitored during the electrochemical process. Strain induced in the working electrode also strained the fiber and modified its effective optical path length.

The measurement system is shown schematically in Fig. 4.9.
Fig. 4.9

Schematic diagram of the system for in situ measurement of strain during electrode reaction (adapted from [97, 98])

Light from the He–Ne laser is split into two parts which are introduced into two single-mode optical fibers using microscope objectives. Both fibers pass through the electrolyte solution and are brought together on a microscope slide to which they are glued. The interference fringe pattern in the diode array is sent to the computer. The light beams coming from the two fiber ends interact to produce an interference pattern.

The Mach–Zehnder interferometer is a classical mirror interferometer [99, 100]. For a long time, it was the most common dual-beam interferometer used to measure continuously refractive index distributions or thickness variation of transparent objects. Figure 4.10 shows a Mach–Zehnder interferometer with a “phase object” in its measuring beam (“phase object” is a transparent object which influences the phase of light passing through it).
Fig. 4.10

Rectangle arrangement of a Mach–Zehnder interferometer. M1, M2: reflecting mirrors

It consists of two reflecting and two beam-splitting mirrors in a rectangle (can also be a parallelogram) arrangement. A light beam from a source, say a He–Ne laser, is first split into two parts by a beam splitter and then recombined by a second beam splitter. First, the interferometer has to be adjusted in such a way that the two beams have equal optical path lengths. When all reflecting surfaces are perfectly parallel, the recombined beams do not produce interference fringes. When the physical process of interest, e.g., strain, heat transfer, etc., is introduced into the measuring beam, an optical path difference between reference and measuring beam is produced. A superposition of the two beams then generates an interference pattern.

4.4 Contact Angle Measurements

The direct measurement of the potential dependence of contact angles in systems with a solid phase started with the work of Möller [101, 102]. Later, Frumkin observed [103] that the shape of a drop of water placed on a solid electrode could be altered by varying the electrode potential: The stronger the polarization, the flatter the drop. This phenomenon, commonly known as electrowetting, has numerous applications from liquid lenses to moving drops in microfluidic devices.

Contact angles can either be measured directly or obtained indirectly. In the latter case, the rise of a liquid meniscus at a partially immersed plate or the capillary rise inside a metal capillary is measured. The contact angle is then obtained by using a relationship connecting the measured parameter with the contact angle. The potential dependence of contact angles can be measured on metals, semiconductors, and semimetallic electrodes of both single-crystal and polycrystalline type, as well as on conductive polymers [71, 104].

An interesting example for the direct measurement of the potential dependence of contact angles can be found in [105]. In this study, the potential dependence of the contact angle between perfluorodecaline (C10F18) and copper, copper(I) sulfide, and copper telluride (Cu4Te3) in aqueous sodium acetate solution (concentration, 0.1 M) was investigated. The contact angles were measured in a plane–parallel quartz cuvette with a metallographic microscope. The contact angle was determined using a goniometer.

Contact angle measurements between water and a poly(vinylferrocene) film on a potential-controlled platinum electrode have been reported in [106].

In [107], an unwalled electrochemical cell formed by a sessile droplet of 1-butyl-3-methylimidazolium hexafluorophosphate resting on electroactive surfaces was used to investigate the voltammetry and surface energy of different electroactive film/ionic liquid interfaces. The following films were investigated: ferrocene films, Au140 nanoparticle films, and Au38 nanoparticle films. The substrate was a 200-nm Au film formed by evaporation of Au onto a glass substrate, with a 10-nm Cr undercoat. The contact angles were measured with a Rame-Hart optical goniometer.

Very little potential dependence of the contact angle between an electrolyte solution droplet and an etched germanium surface was reported in [108]. The variations of the advancing and receding contact angles during a potential scan have been measured by a gravimetric method for Si and Ge in different acid solutions [109]. For p-Si in dilute fluoride electrolyte solutions, the change from the reduced state to the oxidized state (electropolishing regime) is associated with a large decrease of the contact angles. In accordance with the results reported in [108], for Ge in H2SO4 or HClO4 solutions, the change of the contact angles between the reduced state and the oxidized state is much smaller than for silicon and is dependent upon crystal orientation.

The principles of the indirect measurement of contact angles by meniscus-rise techniques in electrochemical systems have been reported by Morcos [71, 104]. Experimental data on the potential dependence of contact angles on solid electrodes have been obtained by the method of meniscus rise at partially immersed plates. The equation relating the meniscus rise h at an infinitely wide vertical plate to the contact angle θ is [110, 111]
$$ \sin \theta = 1 - \frac{{g\rho }}{{{2}{\gamma_{\rm{L}}}}}{h^2}, $$
where γ L and ρ are the liquids surface tension and density and g is the local acceleration of free fall. Taking into account that h is the difference between the level of the meniscus at the solid surface and the level of the liquid surface in the cell, the heights of both levels must be experimentally determined. The measurement can be accurately performed, e.g., by a cathetometer (Fig. 4.11). According to [111], the advantage of this method compared to the direct measurement of contact angle lies in the fact that it is much easier to measure the height of a meniscus than to measure a contact angle between a sessile drop and a solid plate. It is important that the cell used for this purpose should be made of a material that is not wetted by the solvent.
Fig. 4.11

Schematic of the apparatus for measuring of meniscus rise. W: test electrode; R: reference electrode; C: counter electrode; B: box; M: adjustable stand; T: telescope

A method, based on the response of a thickness shear mode sensor to electrocapillary phenomena, for determination of the potential of zero charge and changes in the surface stress at solid metal/solution interfaces has been described in [112]. Essentially, the method involves measuring the change in the rise of the solution meniscus due to electrocapillary phenomena at the surface of a partially immersed vertical metal plate. This change has been determined by a thickness shear mode (TSM) bulk acoustic wave sensor.

The principle of the measurement can be summarized as follows: Changing the electrode potential E causes a change in the contact angle and meniscus rise and its associated height h (see Fig. 4.12). The change of h causes a change in the electrode mass and the mass of solution adhering to the piezoelectric quartz crystal sensor (PQC). As Δh is very small, the effect of the change of oscillating medium on the resonant frequency of the PQC–TSM sensor (such as viscosity, surface stress, and static pressure) can be neglected. A similar method, but with an electrochemical quartz crystal impedance system, was used in [113].
Fig. 4.12

The height of the solution meniscus at two-electrode potentials: (1) E 1; (2) E 2; a: wettable metal surface; b: nonwettable metal surface; δ: the characteristic attenuation length of the shear wave of the TSM sensor

However, contact-angle methods are not without drawbacks. The most important disadvantages of these methods are as follows:
  • In a three-phase system, the risk of surface contamination is high.

  • Hysteresis complicates the measurements and the interpretation of the data.

  • The evaluation of experimental data usually involves questionable assumptions and complicated correction procedures.

4.5 Bending Plate and Bending Beam Methods

The theory of the bending of a film–substrate system is an old problem dating back at least to 1909. The principles of the “bending beam” (“bending cantilever,” “laser beam deflection,” “wafer curvature,” etc.) method were first stated by Stoney [114, 115], who derived an equation relating the isotropic surface stress in the film (γ s) to the radius of curvature (R) of the beam
$$ {\gamma_{\rm{s}}} = \frac{{{E_{\rm{S}}}t_{\rm{S}}^2}}{{{6}R}}, $$
where t S is the substrate thickness and E is the modulus of elasticity of the substrate.

Early attempts to obtain the potential dependence of surface stress from measurement of the deformation of electrodes as a function of electrode potential have been reported in [30, 31]. The deflection of the electrode was measured with a laser optical lever. The measuring apparatus consisted of a He–Ne laser, optical components, and a detector.

Measuring the bending of a plate or strip to determine surface stress change or the stress in thin films is a common technique today, even in electrochemistry [71, 115, 116]. It has been used, for instance, for the investigation of the origin of electrochemical oscillations at silicon [117] or in the course of galvanostatic oxidation of organic compounds on platinum [78, 82], for the study of volume changes in polymers during redox processes [118], for the investigation of the response kinetics of the bending of polyelectrolyte membrane platinum composites by electric stimuli [119], for the experimental verification of the adequacy of the “brush model” of polymer modified electrodes [120], etc.

The generation of internal (residual) stress during the electrochemical deposition of films on substrates is illustrated in Fig. 4.13. Static equilibrium requires that no net forces and no body torques act of an infinitesimal volume element. In particular, the net force (F) and bending moment (M) vanish on the film–substrate cross section; thus,
Fig. 4.13

Events leading to: (a) internal tensile stress in film or (b) internal compressive stress in film. (1) Film and substrate are separated; (2) the film is firmly attached to the substrate; balanced external forces are applied on the film and the substrate to compensate for the difference in length; (3) the substrate– film system bends; S: substrate; f: film

$$ F = \int \sigma {\hbox{d}}A = 0, $$
$$ M = \int {\sigma z} {\hbox{d}}A = 0, $$
where z is the distance from the neutral axis, and A is the sectional area, respectively.

In case (a) shown in Fig. 4.13, the growing film initially shrinks relative to the substrate. There can be a multitude of reasons for this to happen, e.g., surface tension forces or lattice mismatch during epitaxial growth. However, compatibility requires that both the film and the substrate have the same length (and width); consequently, the film stretches while the substrate contracts to accommodate the constraints. The tensile forces developed within the film are balanced by the compressive forces in the substrate. The film–substrate system is still not in mechanical equilibrium because of the uncompensated end moments. If the substrate is not rigidly held, i.e., the film–substrate pair is not restrained from moving, it will elastically deform to counteract the unbalanced moments.

Thus, films containing internal (residual) tensile stresses bend the substrate concavely, as shown in Fig. 4.13a. Similarly, a film which develops residual compressive stresses (i.e., the film wants to be larger than the substrate) will expand relative to substrates and elastically bend the substrate, but in the opposite direction (Fig. 4.13b). By convention, the radius of curvature, R, of the substrate–film structure is positive for concave curvature (tensile stress) and negative for convex curvature (compressive stress). These results are perfectly general regardless of the specific mechanisms that cause the film to stretch or shrink relative to the substrate [121].

In the derivation of Eq. (4.13), Stoney considered a “thin steel rule” with a thin nickel layer of a thickness t f deposited on it. Assuming that the thickness of the rule (t S) is very small in comparison with the radius of curvature (R), the following equation can be written for the bending moments in the steel:
$$ \int\limits_{{{t_{\rm{s}}}}}^{{0}} {\frac{E}{R}} \left( {b - x} \right)x {\hbox{d}} x = 0, $$
where is the depth from the surface of the rule to the neutral axis (Fig. 4.14).
Fig. 4.14

The coordinate system for the derivation of Stoney’s equation

$$ \int\limits_{{{t_{\rm{S}}}}}^{{0}} {\frac{E}{R}} \left( {b - x} \right)x {\hbox{d}} x = \frac{E}{R}\left( { - \frac{{bt_{\rm{S}}^{{2}}}}{{2}} + \frac{{t_{\rm{S}}^{{3}}}}{{3}}} \right), $$
so that b = 2t S/3.
On the other hand,
$$ {\sigma_{\rm{f}}}{t_{\rm{f}}} = \int\limits_{{{t_{\rm{s}}}}}^{{0}} {\frac{E}{R}} \left( {b - x} \right) {\hbox{d}} x = \frac{E}{R}\left( {b{t_{\rm{S}}} - \frac{{t_{\rm{S}}^2}}{2}} \right) = \frac{{Et_{\rm{S}}^2}}{{6R}}, $$
which is identical with Eq. (4.13) since γ s = σ f· t f, where σ f is the film stress.

Equation (4.17) or its variants have been used in almost all experimental determinations of film stress or surface stress.

As has been pointed out in several publications (see, e.g., [122, 123]) that the effect of stress in two dimensions has been neglected in the original derivation. A modified equation can be obtained by referring to Fig. 4.15, which depicts a film–substrate system of width w. The thickness of the film and Young’s modulus are denoted by t f and E f, respectively, and the corresponding parameters of the substrate are t S and E S (see Fig. 4.15a). The interfacial forces can be replaced by a force acting on the entire cross section of the film (or substrate) and a corresponding moment as shown in the force diagram (“free body diagram”) in Fig. 4.15b. The statically equivalent combinations of forces and moments are (F f, M f) in the film and (F S, M S) in the substrate. The force F f must be equal to the force F S and can be imagined to act uniformly over the cross-sectional area, A f = t f·w, giving rise to the (tensile or compressive) film stress. The bending of the sample results from the moments M f and M S of the film and substrate, respectively. Since according to Eq. (4.13), the total moment in the system must be zero, the sum of the counterclockwise moments M f and M S must be equal to the clockwise moment acting on the entire system:
Fig. 4.15

Illustration of the relationship between film stress and substrate curvature: (a) The composite structure; (b) force diagram of film and substrate; (c) bending of the beam under an applied moment; and (d) an element of the substrate–film combination

$$ \frac{{{t_{\rm{f}}}}}{2}{F_{\rm{f}}} + \frac{{{t_{\rm{s}}}}}{2}{F_{\rm{S}}} = \frac{{{t_{\rm{f}}} + {t_{\rm{s}}}}}{2}{F_{\rm{f}}} = {M_{\rm{f}}} + {M_{\rm{S}}}. $$
Since the film–substrate system is not restrained from moving, it will bend to counteract the unbalanced moments, and it can be treated as a beam with the radius of curvature R defined as (Fig. 4.15c):
$$ \frac{1}{R} = \frac{1}{{{{{{\hbox{d}}i}} \left/ {{{\hbox{d}}\vartheta }} \right.}}} = {\hbox{C}}\frac{{{{\text{d}}^{{2}}}z}}{{{\hbox{d}}{x^2}}}, $$
$$ C = {\left[ {1 + {{\left( {\frac{{{\text{d}}z}}{{{\hbox{d}}x}}} \right)}^2}} \right]^{{ - {{3} \left/ {2} \right.}}}}. $$
For small deflections, dz/dx ≪ 1; therefore, C ≈ 1 and the radius of curvature can be approximated as
$$ R \approx \frac{1}{{{{{{{\hbox{d}}^{{2}}}z}} \left/ {{{\hbox{d}} {x^2}}} \right.}}}. $$
If the longitudinal strain varies linearly with the distance z from the neutral axis and is proportional with the curvature of the beam, then according to Hooke’s law:
$$ {\sigma_{\rm{S}}}(z) = \frac{{{E_{\rm{S}}}z}}{R}. $$
In case of an isolated beam (made of the substrate material) bent by the moment M s (Fig. 4.15c), the stress varies linearly across the section from maximum tension (+σ m) to maximum compression (−σ m). In terms of R and \(\vartheta \), Hooke’s law yields
$$ \pm {\sigma_{\rm{m}}} = \left[ {\frac{{\bigg( {R\pm \displaystyle\frac{{{t_{\rm{S}}}}}{{2}}} \bigg)\vartheta - R\vartheta }}{{R\vartheta }}} \right]{E_{\rm{S}}} = \pm \frac{{{E_{\rm{S}}}{t_{\rm{S}}}}}{{2R}}. $$
The bending moment corresponding to this stress distribution can be given as (Fig. 4.15d)
$$ {M_{\rm{S}}} = \int\limits_{{{{{ - {t_{\rm{S}}}}} \left/ {2} \right.}}}^{{{{{{t_{\rm{S}}}}} \left/ {2} \right.}}} {{\sigma_{\rm{S}}}} (z) z {\hbox{d}} A = \int\limits_{{{{{ - {t_{\rm{S}}}}} \left/ {2} \right.}}}^{{{{{{t_{\rm{S}}}}} \left/ {2} \right.}}} {\frac{{{E_{\rm{S}}}}}{R}} {z^2} {\hbox{d}} A = \int\limits_{{{{{ - {t_{\rm{S}}}}} \left/ {2} \right.}}}^{{{{{{t_{\rm{S}}}}} \left/ {2} \right.}}} {\frac{{{E_{\rm{S}}}}}{R}} {z^2} w {\hbox{d}} z = \frac{{{E_{\rm{S}}}t_{\rm{S}}^3w}}{{12R}}. $$
By analogy,
$$ {M_{\rm{f}}} = \frac{{{E_{\rm{f}}}t_{\rm{f}}^3w}}{{12R}}. $$
Substituting the expressions for M S and M f into Eq. (4.18) gives
$$ \frac{{{t_{\rm{f}}} + {t_{\rm{S}}}}}{2}{F_{\rm{f}}} = \frac{w}{{12R}}\left( {{E_{\rm{f}}}t_{\rm{f}}^3 + {E_{\rm{S}}}t_{\rm{S}}^{{3}}} \right). $$
In order to account for the biaxial nature of the stress, we have to replace E f by E f/(1 − ν f) and, similarly, E S by E S/(1 − ν S), where ν f and ν S are the film and substrate Poisson’s ratios, respectively. Thus,
$$ \frac{{{t_{\rm{f}}} + {t_{\rm{S}}}}}{2}{F_{\rm{f}}} = \frac{w}{{12R}}\left( {\frac{{{E_{\rm{f}}}}}{{1 - {\nu_{\rm{f}}}}}t_{\rm{f}}^3 + \frac{{{E_{\rm{S}}}}}{{1 - {\nu_{\rm{S}}}}}t_{\rm{S}}^{{3}}} \right). $$
Since t S is usually much larger than t f, the film stress can be given by
$$ {\sigma_{\rm{f}}} = \frac{{{F_{\rm{f}}}}}{{{t_{\rm{f}}}w}} = \frac{{{E_{\rm{S}}}t_{\rm{S}}^{{2}}}}{{6\left( {1 - {\nu_{\rm{S}}}} \right)\,{t_{\rm{f}}}}} \cdot \frac{1}{R} $$
$$ \Delta {\gamma_{\rm{s}}} = \frac{{{F_{\rm{f}}}}}{w} = \frac{{{E_{\rm{S}}}t_{\rm{S}}^{{2}}}}{{6\left( {1 - {\nu_{\rm{S}}}} \right)}} \cdot \frac{1}{R}. $$

Both Eqs. (4.22) and (4.23) are referred to as Stoney’s formula or Stoney’s equation and are valid under the following conditions: (a) The substrate is homogeneous, linearly elastic, and uniformly thick. (b) The stress is uniform throughout the film thickness. (c) The substrate thickness is much greater than the film thickness (according to [124]); the t f/t S ratio should be ≤10−3. (d) The radius of curvature of the substrate is much greater than the thickness of the composite structure, i.e., the bending displacement is small compared to the thickness of the substrate.

It should be noted here that an average stress can be defined by the relation [124]
$$ {\bar{\sigma }_{\rm{f}}} = \frac{1}{{{t_{\rm{f}}}}}\int\limits_0^{{{t_{\rm{f}}}}} {{\sigma_{\rm{f}}}(z) {\hbox{d}} z}, $$
where σ f(z) is the stress distribution through the film thickness t f. Experimentally, data are usually obtained for the product of the average stress and the film thickness.

Stoney’s equation has become the standard expression for the analysis of surface stress problems in materials physics. It has been pointed out [125, 126] that the equation holds for thickness ratios much larger than expected in the context of the thin-film approximation. This can be attributed to self-compensating errors in its derivation [127, 128].

The calculation of the bending of a sheet material subject to a change in the surface stress on one side has been carried out for two boundary conditions in [29]. One of them is that the sheet is allowed to bend only in one direction, the other one is that the sheet can bend freely in both principal directions. For simplicity in [29] a crystal plate is considered to be oriented such that the surfaces are (1 0 0) surfaces and that the sides of the rectangular shaped sheet coincide with the \( \langle 1\;0\;0\rangle \) direction. The solution satisfying the second boundary condition is formally identical with Eq. (4.23). For the case of a bending only in one direction the corresponding expression for the interface stress has been given as
$$ \tau_{{{11}}}^{\rm{S}} = \frac{{E{t^2}}}{{6\left( {\,1 - {\nu^{{2}}}} \right)}} \cdot \frac{1}{R}, $$
where E is Young’s modulus, ν is Poisson’s ratio, and t is the thickness of the crystal.
As it has already been mentioned earlier, the “bending beam” method can be effectively used in electrochemical experiments, since the changes of the surface stress (Δγ s) for a thin metal film on one side of an insulator (e.g., glass) strip (or a metal plate, one side of which is coated with an insulator layer) in contact with an electrolyte solution can be estimated from the changes of the radius of curvature of the strip. If the potential of the electrode changes, electrochemical processes resulting in the change of γ s can take place exclusively on the metal side of the sample. The change in the surface stress induces a bending moment and the strip bends. In case of a thin metal film on a substrate if the thickness of the film t f is sufficiently smaller than the thickness at of the plate, t S ≫ t f, the change of γ s can be obtained by an expression based on a generalized form of Stoney’s equation
$$ \Delta {\gamma_{\rm{s}}} = {k_{\rm{i}}}\Delta \left( {\frac{1}{R}} \right), $$
where k i depends on the design of the electrode. In the simplest case [see Eq. (4.23)]:
$$ {k_{\rm{i}}} = \frac{{{E_{\rm{S}}}t_{\rm{S}}^{{2}}}}{{6\left( {1 - {\nu_{\rm{s}}}} \right)}}, $$
where E S, ν S, and R are Young’s modulus, Poisson’s ratio, and radius of curvature of the plate, respectively. It should be noted that a number of authors have tried to modify Stoney’s approach over the years, and several modified equations have been derived; however, these equations can usually be written in a form equivalent to Eq. (4.24) [126, 127, 128, 129].
The values of Δ(1/R) = Δγ s/k i can be calculated (a) if the changes of the deflection angle of a laser beam mirrored by the cantilever are measured using an appropriate experimental setup (Fig. 4.16) (the optical detection techniques most used are optical beam deflection and interferometry, discussed in  Chap. 5) or (b) the deflection of the plate is determined directly, e.g., with a nanoindenter, an atomic force microscope (AFM), or a scanning tunneling microscope, etc. (discussed in  Chap. 6).
Fig. 4.16

Schematic of experimental setup for electrochemical bending beam measurements

The combination of the two methods was used in [130, 131, 132]. In order to improve the sensitivity and stability of the technique, microfabricated cantilevers, usually employed in atomic force microscopes, were used instead of thin, but still macroscopic plates [131], and the deflection of the cantilever was measured with an optical lever using the head of a commercially available atomic force microscope. The deflection detection has been calibrated before each experiment: The end of the cantilever was pushed a defined distance upward by a piezoelectric crystal, and the corresponding change in the photodiode signal was measured. However, as it is well known from elasticity theory, when pushing the end of the cantilever (a bar with rectangular cross section) downward or upward (“concentrated load mode”), its shape is not circular anymore, but it is described by a third-order polynomial. Using the notation in Fig. 4.15
$$ z(x) = - \frac{F}{{EI}}\left( {\frac{{l{x^2}}}{2} - \frac{{{x^3}}}{6}} \right), $$
where l is the length of the cantilever and I is the second moment of area (moment of inertia)
$$ I = \int\limits_A {{z^2}} {\hbox{d}} A. $$
The slope of the beam is
$$ \frac{{{\text{d}} z(x)}}{{{\hbox{d}} x}} = - \frac{F}{{EI}}\left( {lx - \frac{{{x^2}}}{2}} \right), $$
and in case of small deflections (l ≈ L) at the end of the cantilever,
$$ \frac{{{\text{d}} z(L)}}{{{\hbox{d}} x}} \approx - \frac{{F{L^2}}}{{2EI}} = \frac{3}{L}z(L). $$
For circular bending, the deflection can be calculated as (Fig. 4.17)
Fig. 4.17

The cylindrical bending of the cantilever. At small deflection, z(l) ≪ R and L ≈ l

$$ z(x) = {({R^2} - {x^2})^{{{{1} \left/ {2} \right.}}}} - R. $$
The slope of the beam is
$$ \frac{{{\text{d}} z(x)}}{{{\hbox{d}} x}} = - \frac{x}{{{{({R^2} - {x^2})}^{{{{1} \left/ {2} \right.}}}}}}. $$
At the end of the beam, at small deflections,
$$ \frac{{{\text{d}} z(L)}}{{{\hbox{d}} x}} \approx - \frac{L}{{{{({R^2} - {L^2})}^{{{{1} \left/ {2} \right.}}}}}}, $$
and so
$$ \frac{1}{R} \approx \frac{{2z(L)}}{{{L^2}}}. $$

It is clear that if the beam is pushed upward or downward at its end (i.e., during calibration), the inclination cannot be described by Eq. (4.31).

In [130, 131], the deflection, z, and inclination were related by
$$ \frac{{{\text{d}} z(L)}}{{{\hbox{d}} x}} = \frac{3}{{2L}}z(L), $$
since the shape of V-shaped cantilevers was taken to be equivalent to two rectangular cantilevers of the same length [133, 134]. During surface stress measurements, the signal of a photodiode was recorded. Since the instrument was calibrated, this signal corresponded to a “virtual” deflection, i.e., the deflection it would have if the cantilever were not bent circular, as described by Eq. (4.31), but if it was pushed by a force applied to its end, as described by Eq. (4.33). With Eq. (4.33), the “virtual” deflection has been converted into dz(L)/dx. The reciprocals radius of curvature was calculated by comparing Eqs. (4.31) and (4.33):
$$ \frac{1}{R} = \frac{{3z(L)}}{{L{{\left[ {4{L^2} + 9z{{(L)}^2}} \right]}^{{{{1} \left/ {2} \right.}}}}}} \approx \frac{{3z(L)}}{{2{L^2}}}, $$
where z(L) is the deflection signal of the calibrated instrument. The accuracy of the method has been reported to be better than 0.005 J m−2. A similar calibration of the cantilever setup was performed in other studies [135, 136]. A condition for the validity of the calibration procedure is that the laser is reflected from the same position along the length of the cantilever in each experiment, typically near the free end of the cantilever. Nevertheless, this is often not easy to accomplish. Measurements of the bending of commercial rectangular cantilevers under the concentrated load mode and the bending moment mode were reported in [137]. A detailed theoretical study of the effects of homogeneous surface stress on rectangular AFM cantilever plates has been presented in [138, 139].

It has been found in [140] that as long as the deflection is much smaller than the overall length of the cantilever, the radius of curvature of the cantilever is to a good approximation constant over its length when exposed to an isotropic surface stress.

As it has already been mentioned above, if the equation of the bending plate is known, the corresponding radius of curvature can be therefore calculated using Eq. (4.35), which is analogous to Eq. (4.17):
$$ \frac{1}{R} = - \frac{{z^{\prime}{\prime}(x)}}{{{{\left\{ {1 + {{\left[ {z^{\prime}(x)} \right]}^2}} \right\}}^{{{{3} \left/ {2} \right.}}}}}}. $$
According to the theory of elasticity, the deflected shape of a composite beam can be represented in terms of elementary functions. Since the deflection is smaller than the specimen length, the quadratic function,
$$ z(x) = a{x^2} + bx + c, $$
is a reasonable approximation of the bending profile [141]. It has been shown in [142] that in case of coated plates, the bending of the sample is accurately fitted by the quadratic function. The theory was also confirmed in [143, 144]. In [142], the equation for the bending plate could be written as y = ax 2, and so
$$ \frac{1}{R} = - \frac{{2a}}{{{{\left\{ {\,1 + {{(2ax)}^2}\,} \right\}}^{{{{{ 3}} \left/ {2} \right.}}}}}}. $$
Since 2ax ≪ 1, R is well approximated by
$$ R = - \frac{1}{{2a}}, $$
and a can be determined as a function of the geometrical parameters of the bending beam setup and then related to R.

Nevertheless, it can be shown that the fact that there is practically no difference between “parabolic” and “circular” approaches is a mathematical consequence of the small deflection approximation.

Let us consider a circle C of radius R centered at a point O (Fig. 4.18). The equation of the circle
Fig. 4.18

A circle centered at (0, −R) with radius R

$$ {x^2} + {(z + R)^2} = {R^2}, $$
can be written as
$$ {x^2} + z(z + 2R) = 0. $$
For z ≪ R,
$$ {x^2} + 2Rz \approx 0, $$
and so
$$ z \approx - \frac{1}{{2R}}{x^2}. $$
$$ z = \sqrt {{{R^2} - {x^2}}} - R = R \sqrt {{1 - \frac{{{x^2}}}{{{R^2}}}}} - R \approx R\left( {1 - \frac{{{x^2}}}{{2{R^2}}}} \right) -R =- \frac{1}{{2R}}{x^2}, $$
and the result is identical to Eq. (4.38).]
By comparing Eq. (4.38) with Eq. (4.36), we get Eq. (4.37). On the other hand Eq. (4.36), can be rewritten as
$$ {x^2} + \frac{b}{a}x + \frac{c}{a} = \frac{z}{a}. $$
With c = 0
$$ {x^2} + {z^2} - {z^2} - \frac{z}{a} + \frac{b}{a}x = 0, $$
$$ {x^2} + {\left( {z - \frac{1}{{2a}}} \right)^2} = \frac{{1 - 4abx +4a^2z^2}}{{4{a^2}}}. $$
If |4a(bx −az 2)| ≪ 1, then
$$ {x^2} + {\left( {z - \frac{1}{{2a}}} \right)^2} = \frac{1}{{4{a^2}}}. $$

The last being the equation of a circle centered at \(( 0,{\frac {1}{2a}})\) and radius \( R=\frac{1}{2a}\).

It should be noted that in the case of a cubic polynomial, the situation is much more complicated, and the shape of the cantilever cannot be characterized by a single radius of curvature.


  1. 1.
    Linford RG (1978) Chem Rev 78:81–95CrossRefGoogle Scholar
  2. 2.
    Láng G, Heusler KE (1994) J Electroanal Chem 377:1–7CrossRefGoogle Scholar
  3. 3.
    Heusler KE, Láng G (1997) Electrochim Acta 42:747–756CrossRefGoogle Scholar
  4. 4.
    Guidelli R (1998) J Electroanal Chem 453:69–77CrossRefGoogle Scholar
  5. 5.
    Láng G, Heusler KE (1999) J Electroanal Chem 472:168–173CrossRefGoogle Scholar
  6. 6.
    Guidelli R (1999) J Electroanal Chem 472:174–177CrossRefGoogle Scholar
  7. 7.
    Valincius G (1999) J Electroanal Chem 478:40–49CrossRefGoogle Scholar
  8. 8.
    Lipkowski J, Schmickler W, Kolb DM, Parsons R (1998) J Electroanal Chem 452:193–197CrossRefGoogle Scholar
  9. 9.
    Couchman PR, Jesser WA, Kuhlmann-Wilsdorf D (1972) Surf Sci 33:429–436CrossRefGoogle Scholar
  10. 10.
    Couchman PR, Jesser WA (1973) Surf Sci 34:212–224CrossRefGoogle Scholar
  11. 11.
    Couchman PR, Everett DW, Jesser WA (1975) J Colloid Interface Sci 52:410–411CrossRefGoogle Scholar
  12. 12.
    Rusanov AI (1978) J Colloid Interface Sci 63:330–345CrossRefGoogle Scholar
  13. 13.
    Rusanov AI (1989) Pure Appl Chem 61:1945–1948CrossRefGoogle Scholar
  14. 14.
    Rusanov AI (1996) Surf Sci Rep 23:173–247CrossRefGoogle Scholar
  15. 15.
    Everett DW, Couchman PR (1976) J Electroanal Chem 67:382–386CrossRefGoogle Scholar
  16. 16.
    Grafov BM, Paasch G, Plieth W, Bund A (2003) Electrochim Acta 48:581–587CrossRefGoogle Scholar
  17. 17.
    Marichev VA (2006) Surf Sci 600:4527–4536CrossRefGoogle Scholar
  18. 18.
    Marichev VA (2007) Chem Phys Lett 434:218–221CrossRefGoogle Scholar
  19. 19.
    Marichev VA (2008) Prot Met 44:99–104CrossRefGoogle Scholar
  20. 20.
    Marichev VA (2010) Adv Colloid Interface Sci 157:34–60CrossRefGoogle Scholar
  21. 21.
    Marichev VA (2010) Prot Met 46:383–402Google Scholar
  22. 22.
    Bikerman JJ (1978) Surface energy of solids. In: Dewar MJS, Hafner K, Heilbronner E, Ito S, Lehn J-M, Niedenzu K, Rees CW, Schaefer K, Wittig G, Boschke FL (eds) Topics in current chemistry, vol 77. Springer, BerlinGoogle Scholar
  23. 23.
    Adamson AW (1967) Physical chemistry of surfaces. Interscience, New York (Chap. 5)Google Scholar
  24. 24.
    Mays CW, Vermaak JS, Kuhlmann-Wilsdorf D (1968) Surf Sci 12:134–140CrossRefGoogle Scholar
  25. 25.
    Wassermann HJ, Vermaak JS (1970) Surf Sci 22:164–172CrossRefGoogle Scholar
  26. 26.
    Wassermann HJ, Vermaak JS (1972) Surf Sci 32:168–174CrossRefGoogle Scholar
  27. 27.
    Cammarata RC, Sieradzki K (1994) Annu Rev Mater Sci 24:215–234CrossRefGoogle Scholar
  28. 28.
    Cammarata RC (1994) Prog Surf Sci 46:1–38CrossRefGoogle Scholar
  29. 29.
    Ibach H (1997) Surf Sci Rep 29:193–263CrossRefGoogle Scholar
  30. 30.
    Fredlein RA, Damjanovic A, Bockris JO’M (1971) Surf Sci 25:261–264CrossRefGoogle Scholar
  31. 31.
    Fredlein RA, Bockris JO’M (1974) Surf Sci 46:641–652CrossRefGoogle Scholar
  32. 32.
    Sahu SN, Scarminio J, Decker F (1990) J Electrochem Soc 137:1150–1154CrossRefGoogle Scholar
  33. 33.
    Tian F, Pei JH, Hedden DL, Brown GM, Thundat T (2004) Ultramicroscopy 100:217–233CrossRefGoogle Scholar
  34. 34.
    Cattarin S, Pantano E, Decker F (1999) Electrochem Commun 1:483–487CrossRefGoogle Scholar
  35. 35.
    Cattarin S, Decker F, Dini D, Margesin B (1999) J Electroanal Chem 474:182–187CrossRefGoogle Scholar
  36. 36.
    Raiteri R, Butt HJ, Grattarola M (1998) Scanning Microsc 12:243–251Google Scholar
  37. 37.
    Kongstein OE, Bertocci U, Stafford GR (2005) J Electrochem Soc 152:C116–C123CrossRefGoogle Scholar
  38. 38.
    Stafford GR, Bertocci U (2006) J Phys Chem B 110:15493–15498CrossRefGoogle Scholar
  39. 39.
    Gokhshtein AYa (1976) Surface tension of solids and adsorption. Nauka, MoscowGoogle Scholar
  40. 40.
    Gokhshtein AYa (1966) Elektrokhimiya 2:1318–1326Google Scholar
  41. 41.
    Gokhshtein AYa (1968) Elektrokhimiya 4:619–624Google Scholar
  42. 42.
    Gokhshtein AYa (1968) Elektrokhimiya 4:665–670Google Scholar
  43. 43.
    Gokhshtein AYa (1968) Dokl Akad Nauk SSSR 181:385–388Google Scholar
  44. 44.
    Gokhshtein AYa (1969) Elektrokhimiya 5:637–638Google Scholar
  45. 45.
    Gokhshtein AYa (1969) Dokl Akad Nauk SSSR 187:601–604Google Scholar
  46. 46.
    Gokhshtein AYa (1970) Electrochim Acta 15:219–223CrossRefGoogle Scholar
  47. 47.
    Gokhshtein AYa (1970) Elektrokhimiya 6:979–985Google Scholar
  48. 48.
    Gokhshtein AYa (1971) Elektrokhimiya 7:3–17Google Scholar
  49. 49.
    Gokhshtein AYa (1971) Dokl Akad Nauk SSSR 200:620–623Google Scholar
  50. 50.
    Gokhshtein AYa (1972) Elektrokhimiya 8:1260–1260Google Scholar
  51. 51.
    Gokhshtein A (1975) Ya. Russ Chem Rev 44:921–932CrossRefGoogle Scholar
  52. 52.
    Gokhshtein AYa (2000) Physics-Uspekhi 43:725–750CrossRefGoogle Scholar
  53. 53.
    Malpas RE, Fredlein RA, Bard AJ (1979) J Electroanal Chem 98:171–180CrossRefGoogle Scholar
  54. 54.
    Malpas RE, Fredlein RA, Bard AJ (1979) J Electroanal Chem 98:339–343CrossRefGoogle Scholar
  55. 55.
    Handley LJ, Bard AJ (1980) J Electrochem Soc 127:338–343CrossRefGoogle Scholar
  56. 56.
    Dickinson KM, Hanson KE, Fredlein RA (1992) Electrochim Acta 37:139–141CrossRefGoogle Scholar
  57. 57.
    Seo M, Makino T, Sato N (1986) J Electrochem Soc 133:1138–1142CrossRefGoogle Scholar
  58. 58.
    Seo M, Jiang XC, Sato N (1987) J Electrochem Soc 134:3094–3098CrossRefGoogle Scholar
  59. 59.
    Seo M, Jiang XC, Sato N (1989) Electrochim Acta 34:1157–1158CrossRefGoogle Scholar
  60. 60.
    Jiang XC, Seo M, Sato N (1990) Corros Sci 31:319–324CrossRefGoogle Scholar
  61. 61.
    Jiang XC, Seo M, Sato N (1990) J Electrochem Soc 137:3804–3808CrossRefGoogle Scholar
  62. 62.
    Jiang XC, Seo M, Sato N (1991) J Electrochem Soc 138:137–140CrossRefGoogle Scholar
  63. 63.
    Seo M, Aomi M (1992) J Electrochem Soc 139:1087–1090CrossRefGoogle Scholar
  64. 64.
    Seo M, Aomi M (1993) J Electroanal Chem 347:185–194CrossRefGoogle Scholar
  65. 65.
    Seo M, Aomi M, Yoshida K (1994) Electrochim Acta 39:1039–1044CrossRefGoogle Scholar
  66. 66.
    Seo M, Ueno K (1996) J Electrochem Soc 143:899–904CrossRefGoogle Scholar
  67. 67.
    Ueno K, Seo M (1998) Denki Kagaku 66:713–719Google Scholar
  68. 68.
    Ueno K, Seo M (1999) J Electrochem Soc 146:1496–1499CrossRefGoogle Scholar
  69. 69.
    Valincius G (1998) Langmuir 14:6307–6319CrossRefGoogle Scholar
  70. 70.
    Valincius G, Reipa V (2000) J Electrochem Soc 147:1459–1466CrossRefGoogle Scholar
  71. 71.
    Morcos I (1978) The interfacial tension of solid electrodes. In: Thirsk HR (ed) Specialist periodical reports: electrochemistry, vol 6. The Chemical Society, Burlington House, LondonGoogle Scholar
  72. 72.
    Morcos I, Fischer H (1968) J Electroanal Chem 17:7–11CrossRefGoogle Scholar
  73. 73.
    Morcos I (1972) J Phys Chem 76:2750–2753CrossRefGoogle Scholar
  74. 74.
    Morcos I (1972) J Chem Phys 56:3996–4000CrossRefGoogle Scholar
  75. 75.
    Jaeckel L, Láng G, Heusler KE (1994) Electrochim Acta 39:1031–1038CrossRefGoogle Scholar
  76. 76.
    Ibach H, Bach CE, Giesen M, Grossmann A (1997) Surf Sci 375:107–119CrossRefGoogle Scholar
  77. 77.
    Haiss W (2001) Rep Prog Phys 64:591–648CrossRefGoogle Scholar
  78. 78.
    Láng G, Seo M, Heusler KE (2005) J Solid State Electrochem 9:347–353CrossRefGoogle Scholar
  79. 79.
    Trasatti S, Parsons R (1986) Pure Appl Chem 58:437–454CrossRefGoogle Scholar
  80. 80.
    Beck TR (1969) J Phys Chem 73:466–468CrossRefGoogle Scholar
  81. 81.
    Lin KF, Beck TR (1976) J Electrochem Soc 123:1145–1151CrossRefGoogle Scholar
  82. 82.
    Láng GG, Ueno K, Ujvári M, Seo M (2000) J Phys Chem B 104:2785–2789CrossRefGoogle Scholar
  83. 83.
    Láng G, Heusler KE (1997) J Chem Soc Faraday Trans 93:583–589CrossRefGoogle Scholar
  84. 84.
    Láng G, Heusler KE (1995) J Electroanal Chem 391:169–179CrossRefGoogle Scholar
  85. 85.
    Láng G, Heusler KE (1995) Russ J Electrochem 31:759–767Google Scholar
  86. 86.
    Beck TR, Beach KW (1974) Measurement of changes in surface stress of electrodes. In: Breiter MW (ed) Proceedings of the symposium on electrocatalysis. The Electrochemical Society, PrincetonGoogle Scholar
  87. 87.
    Lin KF (1978) J Electrochem Soc 125:1077–1078CrossRefGoogle Scholar
  88. 88.
    Beck TR, Lin KF (1979) J Electrochem Soc 126:252–256CrossRefGoogle Scholar
  89. 89.
    Horváth Á, Schiller R (2001) Phys Chem Chem Phys 3:2662–2667CrossRefGoogle Scholar
  90. 90.
    Smetanin M, Kramer D, Mohanan S, Herr U, Weissmüller J (2009) Phys Chem Chem Phys 11:9008–9012CrossRefGoogle Scholar
  91. 91.
    Craig PP (1969) Phys Rev Lett 22:700–703CrossRefGoogle Scholar
  92. 92.
    Unal K, Wickramasinghe HK (2007) Appl Phys Lett 90:113111CrossRefGoogle Scholar
  93. 93.
    Horváth Á, Nagy G, Schiller R (2010) Phys Chem Chem Phys 12:7290–7290CrossRefGoogle Scholar
  94. 94.
    Smetanin M, Deng Q, Kramer D, Mohanan S, Herr U, Weissmüller J (2010) Phys Chem Chem Phys 12:7291–7292CrossRefGoogle Scholar
  95. 95.
    Weissmüller J, Viswanath RN, Kramer D, Zimmer P, Würschum R, Gleiter H (2003) Science 300:312–315CrossRefGoogle Scholar
  96. 96.
    Kramer D, Viswanath RN, Weissmüller J (2004) Nano Lett 4:793–796CrossRefGoogle Scholar
  97. 97.
    Butler MA, Ginley DS (1987) J Electrochem Soc 134:510–511CrossRefGoogle Scholar
  98. 98.
    Butler MA, Ginley DS (1988) J Electrochem Soc 135:45–51CrossRefGoogle Scholar
  99. 99.
    Zehnder L (1891) Z Instrumentenk 11:275–285Google Scholar
  100. 100.
    Mach L (1892) Z Instrumentenk 12:89–93Google Scholar
  101. 101.
    Möller G (1908) Ann Physik 27:665–711CrossRefGoogle Scholar
  102. 102.
    Möller G (1908) Z Phys Chem 65:226–234Google Scholar
  103. 103.
    Froumkin AN (1936) Actualites Scientifiques et Industrielles 373:5–36Google Scholar
  104. 104.
    Morcos I (1975) J Electroanal Chem 62:313–340CrossRefGoogle Scholar
  105. 105.
    Batrakov VV, Makarov AG (2003) Russ J Electrochem 39:1351–1511CrossRefGoogle Scholar
  106. 106.
    Willman KW, Murray RW (1983) Anal Chem 55:1139–1142CrossRefGoogle Scholar
  107. 107.
    Wang W, Murray RW (2007) Anal Chem 79:1213–1220CrossRefGoogle Scholar
  108. 108.
    Sparnaay MJ (1964) Surf Sci 1:213–224CrossRefGoogle Scholar
  109. 109.
    Chazalviel JN, Maroun F, Ozanam F (2004) J Electrochem Soc 151:E51–E55CrossRefGoogle Scholar
  110. 110.
    Neumann AW (1964) Z Phys Chem Neue Folge 41:339–352CrossRefGoogle Scholar
  111. 111.
    Morcos I (1971) J Colloid Interface Sci 37:410–421CrossRefGoogle Scholar
  112. 112.
    Chen JH, Nie LH, Yao SZ (1996) J Electroanal Chem 414:53–59CrossRefGoogle Scholar
  113. 113.
    Xie Q, Zhang Y, Xiao X, Guo Y, Wang X, Yao S (2001) Anal Sci 17:265–275CrossRefGoogle Scholar
  114. 114.
    Stoney GG (1909) Proc Roy Soc London A 32:172–175Google Scholar
  115. 115.
    Láng GG (2008) Bending beam method. In: Bard AJ, Inzelt G, Scholz F (eds) Electrochemical dictionary. Springer, BerlinGoogle Scholar
  116. 116.
    Láng GG, Sas Ns, Vesztergom S (2009) Chem Biochem Eng Q 23:1–9Google Scholar
  117. 117.
    Lehmann V (1996) J Electrochem Soc 143:1313–1318CrossRefGoogle Scholar
  118. 118.
    Pei Q, Inganas O (1992) J Phys Chem 96:10507–10514CrossRefGoogle Scholar
  119. 119.
    Asaka K, Oguro K (2000) J Electroanal Chem 480:186–198CrossRefGoogle Scholar
  120. 120.
    Láng GG, Ujvári M, Rokob TA, Inzelt G (2006) Electrochim Acta 51:1680–1694CrossRefGoogle Scholar
  121. 121.
    Ohring M (1992) The materials science of thin films. Academic, San DiegoGoogle Scholar
  122. 122.
    Hoffman RW (1966) The mechanical properties of thin condensed films. In: Hass G, Thun RE (eds) Physics of thin films, vol 3. Academic, New YorkGoogle Scholar
  123. 123.
    Brenner A, Senderoff S (1949) J Res Nat Bur Stand 42:105–123CrossRefGoogle Scholar
  124. 124.
    Klokholm E (1969) Rev Sci Instrum 40:1054–1058CrossRefGoogle Scholar
  125. 125.
    Klein C, Miller RP (2000) J Appl Phys 87:2265–2272CrossRefGoogle Scholar
  126. 126.
    Klein C (2000) J Appl Phys 88:5487–5489CrossRefGoogle Scholar
  127. 127.
    Pureza JM, Lacerda MM, De Oliveira AL, Fragalli JF, Zanon RAS (2009) Appl Surf Sci 255:6426–6428CrossRefGoogle Scholar
  128. 128.
    Pureza JM, Neri F, Lacerda MM (2010) Appl Surf Sci 256:4408–4410CrossRefGoogle Scholar
  129. 129.
    Janssen GCAM, Abdalla MM, van Keulen F, Pujada BR, van Venrooy B (2009) Thin Solid Films 517:1858–1867CrossRefGoogle Scholar
  130. 130.
    Raiteri R, Butt HJ (1995) J Phys Chem 99:15728–15732CrossRefGoogle Scholar
  131. 131.
    Butt HJ (1996) J Coll Interf Sci 180:251–260CrossRefGoogle Scholar
  132. 132.
    Raiteri R, Butt HJ, Grattarola M (2000) Electrochim Acta 46:157–163CrossRefGoogle Scholar
  133. 133.
    Gould SAC, Drake B, Prater CB, Weisenhorn AL, Manne S, Kelderman GL, Butt HJ, Hansma H, Hansma PK, Magonov S, Cantow HJ (1990) Ultramicroscopy 33:93–98CrossRefGoogle Scholar
  134. 134.
    Butt HJ, Siedle P, Seifert K, Fendler K, Seeger T, Bamberg E, Weisenhorn AL, Goldie K, Engel A (1993) J Microsc 169:75–84CrossRefGoogle Scholar
  135. 135.
    Friesen C, Dimitrov N, Cammarata C, Sieradzki K (2001) Langmuir 17:807–815CrossRefGoogle Scholar
  136. 136.
    Vasiljevic N, Trimble T, Dimitrov N, Sieradzki K (2004) Langmuir 20:6639–6643CrossRefGoogle Scholar
  137. 137.
    Miyatani T, Fujihira M (1997) J Appl Phys 81:7099–7115CrossRefGoogle Scholar
  138. 138.
    Sader JE (2002) J Appl Phys 89:2911–2921CrossRefGoogle Scholar
  139. 139.
    Sader JE (2002) J Appl Phys 91:9354–9361CrossRefGoogle Scholar
  140. 140.
    Godin M, Tabard-Cossa V, Grütter P (2001) Appl Phys Lett 79:551–553CrossRefGoogle Scholar
  141. 141.
    Suhir E (1988) J Appl Mech 55:143–148CrossRefGoogle Scholar
  142. 142.
    Moulard G, Contoux G, Motyl G, Gardet G, Courbon M (1998) J Vac Sci Technol 16:736–742Google Scholar
  143. 143.
    Schwarzer N, Richter F, Hecht G (1993) Surf Coat Technol 60:396–400CrossRefGoogle Scholar
  144. 144.
    Ramsey PM, Chandler HW, Page TF (1990) Surf Coat Technol 43(44):223–233CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Inst. ChemistryEötvös Loránd UniversityBudapestHungary
  2. 2.Chemistry DepartmentUniversidad Nacional de Rio CuartoRio Cuarto CórdobaArgentina

Personalised recommendations