**2.3 Repassivation current transients**

Methods based on analysis of potential or current transients (Ponthiaux et al., 1995) are particularly well suited to study reciprocating sliding tests (Mischler et al., 1997). These methods are used to study between successive contact events, the rebuild of damaged surface layers (oxide, passive film ...). Under imposed polarization e.g. in the passivation range, at each stop-start event, a transient variation of current is noticed with time (see Figure 5). The charge corresponding to this transition can be attributed to the re-growth of a uniform film in the damaged area.

Fig. 5. Schematic evolution of the current transients obtained under reciprocating sliding.

Tribocorrosion: Material Behavior Under

the polarization curves, and

electron microscopy.

dissolution and passivation kinetics.

Combined Conditions of Corrosion and Mechanical Loading 91

reflecting the contribution of an elementary adsorption step in the reaction mechanism.



This can be interpreted by considering that in a non-rubbed condition the semi-circle linked to a possible second adsorbate does not appear because its concentration at the surface of the sample does not vary substantially with the potential. Under sliding conditions, the kinetics of some dissolution steps evolves, in particular the kinetics of steps in which the adsorbate

The in-depth interpretation of these impedance diagrams encounters however some difficulties as those already reported for open circuit potential and polarization

A first difficulty results from the fact that the impedance data reflect the overall state of the tested surface integrating the contributions of non-rubbed and rubbed surfaces. Such data must thus be de-convoluted in order to obtain the specific impedances of these two types of surface states. A first approach to this problem might be to use models similar to those describing the impedance of a sample undergoing a localized corrosion (Oltra & Keddam, 1990). In that specific case, the overall impedance can be considered as the result of two impedances in parallel, namely the impedance of the non-rubbed surface and the one of the rubbed surface. A strict interpretation requires further an evaluation of the areas of these surfaces, e.g. by using profilometry and surface observations by light optical or scanning

The second difficulty results from the non-uniform state of the rubbed surface. Behind the slider the sample surface can be laid bare for some time before some new surface layers are rebuilt. The restored surface increases gradually with the distance behind the slider along the sliding track. Even if the first difficulty was already solved and the overall impedance of the rubbed surface is obtained, it can not be used as such to characterize the non-uniform distribution of the electrochemical states behind the slider. However, it is expected that impedance measurement procedures already developed for analyzing non-uniform distributions of surface states and the electrochemical models developed for the interpretation of such measurements (Zhang et al., 1987) could be transferred to tribocorrrosion test conditions. Such a study could allow a localized characterization of

If theoretically this approach seems promising, experimental data analyses show that impedance measurements under sliding are often disturbed at low frequencies due to the random fluctuations of potential or current. This "electrochemical noise" limits unfortunately to some extent the application of impedance measurements. Limitations frequently originate from the sliding action itself and more specifically from the localized damages induced in the contact area by the mechanical interaction. Notwithstanding that, the in-depth analysis of the electrochemical noise will surely in the future be fruitful since it will contain useful information on the progress of the process at both spatial and time scales.

interacts. The variation of the concentration at the surface becomes then detectable.

measurements which are related to the heterogeneous state of the tested surfaces.

Under sliding, the shape and size of the plot are modified as follows:

a second adsorbate in the dissolution mechanism.

In order to study in more details the mechanism and kinetics of a re-growth of the passive film, the "potential jump" method can be used. However, such a method cannot be applied on metals such as aluminum, for which the passive film can not be reduced by a cathodic polarization, or for some steels for which the contamination of the surface by reduction products can affect the initial current increase at the potential jump.

#### **2.4 Electrochemical impedance measurements**

This method requires quasi-stationary electrochemical conditions of currents and potentials. The impedance measurements allow the study of the influence of sliding on the elementary processes involved in the corrosion mechanism. By performing a systematic analysis of the changes in the impedance diagrams with the sliding parameters like normal force and sliding speed or contact frequency, a model can be developed incorporating the sliding effects in the mechanism.

Impedance measurements recorded on a non-rubbed and a rubbed Fe-31% nickel alloy are shown in Figure 6. The measurements were done in 0.5 M sulfuric acid at a potential of -675 mV / SSE located in the active region (Boutard et al., 1985). The impedance diagrams were recorded in a limited range of measurement frequencies. In particular, under sliding, the impedance was not measured at frequencies below 0.01 Hz, to limit the duration of the measurements and to avoid in this way the influence of the long-term evolution of the electrochemical state of the surface due to wear - induced changes in the "rubbed area/unrubbed area" ratio. By taking these precautions, we can consider that the condition of average electrochemical steady-state required for impedance measurements is fulfilled.

Fig. 6. Electrochemical impedance plots recorded on Fe-31% Ni immersed in 0.5 M sulfuric acid polarized in the active region (-0,675 V/SSE): (a) in absence of any sliding, and (b) under continuous sliding against alumina at a mean contact pressure of 2,6 MPa and a sliding speed of 3,4 cm s-1 (Ponthiaux et al., 1999).

SSE: 'mercury/mercurous sulfate/saturated potassium sulfate' reference electrode (ESSE = +0.65 V/NHE).

In absence of friction, the plot consists of two successive semi-circles, a capacitive one reflecting the dielectric properties of the electrochemical double layer, and an inductive one

In order to study in more details the mechanism and kinetics of a re-growth of the passive film, the "potential jump" method can be used. However, such a method cannot be applied on metals such as aluminum, for which the passive film can not be reduced by a cathodic polarization, or for some steels for which the contamination of the surface by reduction

This method requires quasi-stationary electrochemical conditions of currents and potentials. The impedance measurements allow the study of the influence of sliding on the elementary processes involved in the corrosion mechanism. By performing a systematic analysis of the changes in the impedance diagrams with the sliding parameters like normal force and sliding speed or contact frequency, a model can be developed incorporating the sliding

Impedance measurements recorded on a non-rubbed and a rubbed Fe-31% nickel alloy are shown in Figure 6. The measurements were done in 0.5 M sulfuric acid at a potential of -675 mV / SSE located in the active region (Boutard et al., 1985). The impedance diagrams were recorded in a limited range of measurement frequencies. In particular, under sliding, the impedance was not measured at frequencies below 0.01 Hz, to limit the duration of the measurements and to avoid in this way the influence of the long-term evolution of the electrochemical state of the surface due to wear - induced changes in the "rubbed area/unrubbed area" ratio. By taking these precautions, we can consider that the condition of average electrochemical steady-state required for impedance measurements is fulfilled.

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Fig. 6. Electrochemical impedance plots recorded on Fe-31% Ni immersed in 0.5 M sulfuric acid polarized in the active region (-0,675 V/SSE): (a) in absence of any sliding, and (b) under continuous sliding against alumina at a mean contact pressure of 2,6 MPa and a

In absence of friction, the plot consists of two successive semi-circles, a capacitive one reflecting the dielectric properties of the electrochemical double layer, and an inductive one

SSE: 'mercury/mercurous sulfate/saturated potassium sulfate' reference electrode

Real part (ohms)

*10 Hz*

*0.1 Hz*

*10 Hz*

*1 Hz*

**a**

**b**

products can affect the initial current increase at the potential jump.

**2.4 Electrochemical impedance measurements** 

0


sliding speed of 3,4 cm s-1 (Ponthiaux et al., 1999).

(ESSE = +0.65 V/NHE).

1.0


2.0

*100 Hz*

*0.01 Hz*

3.0

effects in the mechanism.

reflecting the contribution of an elementary adsorption step in the reaction mechanism. Under sliding, the shape and size of the plot are modified as follows:


This can be interpreted by considering that in a non-rubbed condition the semi-circle linked to a possible second adsorbate does not appear because its concentration at the surface of the sample does not vary substantially with the potential. Under sliding conditions, the kinetics of some dissolution steps evolves, in particular the kinetics of steps in which the adsorbate interacts. The variation of the concentration at the surface becomes then detectable.

The in-depth interpretation of these impedance diagrams encounters however some difficulties as those already reported for open circuit potential and polarization measurements which are related to the heterogeneous state of the tested surfaces.

A first difficulty results from the fact that the impedance data reflect the overall state of the tested surface integrating the contributions of non-rubbed and rubbed surfaces. Such data must thus be de-convoluted in order to obtain the specific impedances of these two types of surface states. A first approach to this problem might be to use models similar to those describing the impedance of a sample undergoing a localized corrosion (Oltra & Keddam, 1990). In that specific case, the overall impedance can be considered as the result of two impedances in parallel, namely the impedance of the non-rubbed surface and the one of the rubbed surface. A strict interpretation requires further an evaluation of the areas of these surfaces, e.g. by using profilometry and surface observations by light optical or scanning electron microscopy.

The second difficulty results from the non-uniform state of the rubbed surface. Behind the slider the sample surface can be laid bare for some time before some new surface layers are rebuilt. The restored surface increases gradually with the distance behind the slider along the sliding track. Even if the first difficulty was already solved and the overall impedance of the rubbed surface is obtained, it can not be used as such to characterize the non-uniform distribution of the electrochemical states behind the slider. However, it is expected that impedance measurement procedures already developed for analyzing non-uniform distributions of surface states and the electrochemical models developed for the interpretation of such measurements (Zhang et al., 1987) could be transferred to tribocorrrosion test conditions. Such a study could allow a localized characterization of dissolution and passivation kinetics.

If theoretically this approach seems promising, experimental data analyses show that impedance measurements under sliding are often disturbed at low frequencies due to the random fluctuations of potential or current. This "electrochemical noise" limits unfortunately to some extent the application of impedance measurements. Limitations frequently originate from the sliding action itself and more specifically from the localized damages induced in the contact area by the mechanical interaction. Notwithstanding that, the in-depth analysis of the electrochemical noise will surely in the future be fruitful since it will contain useful information on the progress of the process at both spatial and time scales.

Tribocorrosion: Material Behavior Under

under sliding C has thus to be expressed as:

with CM the amount of corrosion induced by wear.

to the mass loss, M0, in a non-corrosive environment as:

processes in case of tribocorrosion as:

In which:

the electrochemical mass loss under sliding, *C*, from the equation:

were developed over the past decades are briefly reviewed hereafter.

**3.2.1 Quinn's model of mild oxidative wear** 

**3.2 Models of oxidative wear and application in aqueous environment** 

Combined Conditions of Corrosion and Mechanical Loading 93

with T the total duration of the test. However, the electrochemical mass loss due to corrosion under sliding, C, is generally not equal to the mass loss of the metal, C0, obtained under similar test conditions but in absence of any sliding. The electrochemical mass loss

The total mass loss, *W*, can be determined at the end of a sliding test by some *ex situ* technique like surface profilometry or gravimetry. That total mass loss can be compared to

in which M is the mechanical mass loss. This mechanical mass loss M can also be compared

in which MC represents the excess mechanical mass loss due to corrosion. This formalism now allows a general definition of the synergy, S, between corrosion and mechanical

The term *S* defined as such reflects the fact that the material mass loss in a corrosive environment cannot be predicted simply by the sum of the mass loss due to corrosion in the environment in absence of any mechanical interaction, and the material loss due to wear recorded under similar testing conditions but in an non-corrosive environment. There is a synergy between these two processes. The formalism of tribocorrosion originally proposed (Watson et al., 1995) has surely an educational value since it allows a diagnostic on the "origin of evil" under the given set of experimental conditions. The wear M0 is by some authors measured in a test under cathodic polarization in either dry air, de-ionized water or in the presence of corrosion inhibitors (Smith, 1985; Stemp et al, 2003; Takadoum, 1996). There is still some controversy about the validity of such procedures. In practice, the results often depend on the method used which limits the overall benefit of such a decoupling. Moreover, the concepts used do not have a physical meaning, and the synergies between wear and corrosion cannot really be simplified to a summing up. Other approaches that

A two-step model of mild oxidational wear for steel in air was developed (Quinn, 1992, 1994). The author observed that at sliding speeds below 5 ms-1 the wear debris consists only

C = C0 + CM (4)

W = C + M (5)

M = M0 +MC (6)

W = C0 +M0 + S (7)

S = CM +MC (8)

#### **2.5 Electrochemical noise analyses**

In the case of tribocorrosion as in cases involving stochastic processes of local surface damages like pitting (Uruchurtu & Dawson, 1987), stress corrosion (Cottis & Loto, 1990), and corrosion-abrasion (Oltra et al., 1986), sliding destroys locally to some extent surface layers. The bare areas created by the slider generate fluctuations of potential or current which consist of the superposition of elementary transients. The sudden increase in current occurs at the time that a bare surface is brought in contact with the electrolyte. The subsequent current decrease reveals the restoration of a protective surface film. The analysis of the transient characteristics (shape, amplitude, duration) provides information on the mechanisms and kinetics of the reactions involved (e.g. dissolution, passivation). The frequency of the transients depends in turn on the number of contacting points in the sliding track at a given time and on the sliding rate of the slider. It is therefore a quantity that provides useful information on the nature of the contact.

Sliding conditions affect the amplitude of the electrochemical noise, namely fluctuations of the open circuit potential as shown in Figure 2 (Ponthiaux et al., 1997). It is usually impossible to isolate the elementary transients. However, the spectral analysis of such a noise allows characteristic quantities to be derived such as the mean amplitude or duration and the average frequency at which transients occur. These characteristics are essential for getting a better understanding of the nature of the contact and the dissolution and passivation kinetics on the bare surface.

#### **3. Modeling approaches**

Given the complexity of the tribocorrosion phenomena there is currently no universal predictive model of the wear-corrosion process available. Such a problem solving is largely empirical, and designers rely on expert systems fed by experimental feedback to select the material couples for a given tribological system. In parallel to this technical approach, scientists are developing the modeling elements needed to unravel the phenomena. These models are discussed hereafter.

#### **3.1 Formalism of the wear-corrosion synergy**

It was shown here above that electrochemical methods allow under certain conditions to measure in real time the current I related to the corrosion reaction in a sliding track. At the end of a tribocorrosion experiment (e.g. sliding or erosion tests), the electrochemical mass loss, *C*, can be calculated using Faraday's relationship from the total charge consumed by the corrosion process:

$$\mathbf{C} = \frac{M}{nF} \mathbf{Q} \tag{3}$$

in which M is the atomic mass of the metal, n is the valency of the oxidized metal in the environment studied, F is the Faraday's constant and Q is the total charge related to the corrosion process, namely

$$Q = \int\_0^T I\left(t\right)\,dt$$

with T the total duration of the test. However, the electrochemical mass loss due to corrosion under sliding, C, is generally not equal to the mass loss of the metal, C0, obtained under similar test conditions but in absence of any sliding. The electrochemical mass loss under sliding C has thus to be expressed as:

$$\mathbf{C} = \mathbf{C}\_0 + \mathbf{C}\_M \tag{4}$$

with CM the amount of corrosion induced by wear.

The total mass loss, *W*, can be determined at the end of a sliding test by some *ex situ* technique like surface profilometry or gravimetry. That total mass loss can be compared to the electrochemical mass loss under sliding, *C*, from the equation:

$$\mathbf{W} = \mathbf{C} + \mathbf{M} \tag{5}$$

in which M is the mechanical mass loss. This mechanical mass loss M can also be compared to the mass loss, M0, in a non-corrosive environment as:

$$\mathbf{M} \equiv \mathbf{M}\_0 + \mathbf{M}\_\mathbf{C} \tag{6}$$

in which MC represents the excess mechanical mass loss due to corrosion. This formalism now allows a general definition of the synergy, S, between corrosion and mechanical processes in case of tribocorrosion as:

$$\mathbf{W} = \mathbf{C}\_0 + \mathbf{M}\_0 + \mathbf{S} \tag{7}$$

In which:

92 Corrosion Resistance

In the case of tribocorrosion as in cases involving stochastic processes of local surface damages like pitting (Uruchurtu & Dawson, 1987), stress corrosion (Cottis & Loto, 1990), and corrosion-abrasion (Oltra et al., 1986), sliding destroys locally to some extent surface layers. The bare areas created by the slider generate fluctuations of potential or current which consist of the superposition of elementary transients. The sudden increase in current occurs at the time that a bare surface is brought in contact with the electrolyte. The subsequent current decrease reveals the restoration of a protective surface film. The analysis of the transient characteristics (shape, amplitude, duration) provides information on the mechanisms and kinetics of the reactions involved (e.g. dissolution, passivation). The frequency of the transients depends in turn on the number of contacting points in the sliding track at a given time and on the sliding rate of the slider. It is therefore a quantity that

Sliding conditions affect the amplitude of the electrochemical noise, namely fluctuations of the open circuit potential as shown in Figure 2 (Ponthiaux et al., 1997). It is usually impossible to isolate the elementary transients. However, the spectral analysis of such a noise allows characteristic quantities to be derived such as the mean amplitude or duration and the average frequency at which transients occur. These characteristics are essential for getting a better understanding of the nature of the contact and the dissolution and

Given the complexity of the tribocorrosion phenomena there is currently no universal predictive model of the wear-corrosion process available. Such a problem solving is largely empirical, and designers rely on expert systems fed by experimental feedback to select the material couples for a given tribological system. In parallel to this technical approach, scientists are developing the modeling elements needed to unravel the phenomena. These

It was shown here above that electrochemical methods allow under certain conditions to measure in real time the current I related to the corrosion reaction in a sliding track. At the end of a tribocorrosion experiment (e.g. sliding or erosion tests), the electrochemical mass loss, *C*, can be calculated using Faraday's relationship from the total charge consumed by

in which M is the atomic mass of the metal, n is the valency of the oxidized metal in the environment studied, F is the Faraday's constant and Q is the total charge related to the

> 0 ( ) *T Q I t dt*

*<sup>M</sup> C Q nF* (3)

**2.5 Electrochemical noise analyses** 

passivation kinetics on the bare surface.

**3. Modeling approaches** 

models are discussed hereafter.

the corrosion process:

corrosion process, namely

**3.1 Formalism of the wear-corrosion synergy** 

provides useful information on the nature of the contact.

$$\mathbf{S} = \mathbf{C}\_{\text{M}} + \mathbf{M}\_{\text{C}} \tag{8}$$

The term *S* defined as such reflects the fact that the material mass loss in a corrosive environment cannot be predicted simply by the sum of the mass loss due to corrosion in the environment in absence of any mechanical interaction, and the material loss due to wear recorded under similar testing conditions but in an non-corrosive environment. There is a synergy between these two processes. The formalism of tribocorrosion originally proposed (Watson et al., 1995) has surely an educational value since it allows a diagnostic on the "origin of evil" under the given set of experimental conditions. The wear M0 is by some authors measured in a test under cathodic polarization in either dry air, de-ionized water or in the presence of corrosion inhibitors (Smith, 1985; Stemp et al, 2003; Takadoum, 1996). There is still some controversy about the validity of such procedures. In practice, the results often depend on the method used which limits the overall benefit of such a decoupling. Moreover, the concepts used do not have a physical meaning, and the synergies between wear and corrosion cannot really be simplified to a summing up. Other approaches that were developed over the past decades are briefly reviewed hereafter.

#### **3.2 Models of oxidative wear and application in aqueous environment**

#### **3.2.1 Quinn's model of mild oxidative wear**

A two-step model of mild oxidational wear for steel in air was developed (Quinn, 1992, 1994). The author observed that at sliding speeds below 5 ms-1 the wear debris consists only

Tribocorrosion: Material Behavior Under

the electrochemical conditions.

**3.2.3 Application to the synergy formalism** 

contact event depassivates the surface:

area at the applied potential.

Combined Conditions of Corrosion and Mechanical Loading 95

gripper latch arms of the control rods command mechanisms in PWR (Lemaire & Le Calvar,

Where W is the total worn volume, W0 a constant, and N the number of sliding steps applied to the alloy surface inducing removal of the passive film. t is the mean time interval between two successive sliding steps, and t0 is a characteristic repassivation time constant. n is a positive exponent whose value was found experimentally close to 0.65 for the cobaltbased alloy coating. The authors explain the wear law expressed by equation (12) by the evolution of the repassivation current Ip given by expression (2). This model implies that the growth of the oxide film between two sliding steps is proportional to t(1-n). The Quinn's law appears as a particular case of such a model for n = 0.5. In triborrosion studies, different values of n (between 0.6 and 0.9) where found depending on the metal, the environment and

Studies in corrosive aqueous solutions (Garcia et al., 2001; Jemmely et al., 2000) are suitable to follow *in situ* the growth of passive films by electrochemical methods, and allow thus the development of more sophisticated models. In these studies performed under continuous or reciprocating sliding conditions, a modeling of currents measured at an applied potential is done. It is then assumed that a unit area of depassivated material repassivates according to a simple repassivation transient, Ja (t), which is not affected by the electrochemical conditions on the areas surrounding the rubbed area. The measured total current is then the sum of the contributions of the different surface areas. A freshly depassivated area produces a large current while a area depassivated some time before produces lower currents. In the case of a reciprocating tribometer operated at a sliding frequency, f, the steady state current I can be expressed as follows assuming that each

1/ 1/

*f f*

0 0

with At the total area in contact with the solution, Aa the depassivated area on the sample during one cycle, f the frequency at which the surface is depassivated, jp the passive current density at the applied potential, and ja (t) the transient repassivation current density of a unit

Taking into consideration the synergy formalism developed above, the components of mass

*M aa p*

1/ 1/

0 0 ( () () ) *f f*

loss per cycle, CM and C0 in Equation (4) can now be written as follows:

() ( ) ()

*a a ta p I A f j t dt A A f j t dt* (13)

*<sup>M</sup> C A j t dt j t dt nF* (14)

0

*<sup>t</sup> W WN*

1

*n*

(12)

0

*t* 

2000). In this model, the wear law is given by the following expression:

of iron oxides and that the particle size is quite constant in the range of several micrometers. In a first step oxides grow on surface asperities as a result of local heating at contact points. When the oxide reaches a critical thickness, xc, the mechanical stresses generated in the material become too large and a detachment of the oxide layer which reaches the critical thickness takes place on the passage of the slider. The worn volume by unit of sliding distance, w, can be written as:

$$w = \frac{\mathcal{X}\_c}{Vt\_c} \cdot A\_r \tag{9}$$

with Ar the real contact area, V the sliding speed, and tc the time necessary to reach the critical oxide thickness. Quinn's model is thus a law that corresponds to the Archard's wear law with:

$$K\_{Archard} = \frac{\mathcal{X}\_c}{V \ t\_c} \tag{10}$$

It is possible to connect xc and tc through a thermally activated oxidation kinetics in air which can be considered in first instance as a parabolic function with time:

$$\alpha(t) = \alpha \sqrt{k\_0 \exp\left(-\frac{Q\_{act}}{R \ T\_f}\right)} t \tag{11}$$

where:


The critical thickness of the oxide, xc, can only be obtained experimentally by characterizing the debris. The model is thus not a predictive one. Moreover as pointed out (Smith, 1985) the oxide growth laws based on mass can hardly be used under the conditions of contact characterized by a low air supply, and a poor knowledge of the real contact temperature. In practice, the use of oxidation constants leads to wear rates that are several orders of magnitude different from the experimentally verified ones.

#### **3.2.2 Model of Lemaire and Le Calvar**

In conclusion, the Quinn's model can not be easily adapted to analyse the depassivation repassivation process determining the wear laws observed in tribocorrosion conditions. However, the main idea of this model, namely that wear proceeds by a succession of growth and delamination of an oxide layer, can be retained. It was at the origin of the tribocorrosion model presented to explain the the wear of a cobalt-based alloy coating applied on the

of iron oxides and that the particle size is quite constant in the range of several micrometers. In a first step oxides grow on surface asperities as a result of local heating at contact points. When the oxide reaches a critical thickness, xc, the mechanical stresses generated in the material become too large and a detachment of the oxide layer which reaches the critical thickness takes place on the passage of the slider. The worn volume by unit of sliding

> *<sup>c</sup> <sup>r</sup> c <sup>x</sup> w A Vt*

with Ar the real contact area, V the sliding speed, and tc the time necessary to reach the critical oxide thickness. Quinn's model is thus a law that corresponds to the Archard's wear

*<sup>c</sup> Archard*

It is possible to connect xc and tc through a thermally activated oxidation kinetics in air

<sup>0</sup> exp *act*


The critical thickness of the oxide, xc, can only be obtained experimentally by characterizing the debris. The model is thus not a predictive one. Moreover as pointed out (Smith, 1985) the oxide growth laws based on mass can hardly be used under the conditions of contact characterized by a low air supply, and a poor knowledge of the real contact temperature. In practice, the use of oxidation constants leads to wear rates that are several orders of

In conclusion, the Quinn's model can not be easily adapted to analyse the depassivation repassivation process determining the wear laws observed in tribocorrosion conditions. However, the main idea of this model, namely that wear proceeds by a succession of growth and delamination of an oxide layer, can be retained. It was at the origin of the tribocorrosion model presented to explain the the wear of a cobalt-based alloy coating applied on the

*<sup>Q</sup> xt k <sup>t</sup>*

which can be considered in first instance as a parabolic function with time:


magnitude different from the experimentally verified ones.

*<sup>x</sup> <sup>K</sup>*

*c*

*f*

*R T*

 

*V t*

(9)

(10)

(11)

distance, w, can be written as:


**3.2.2 Model of Lemaire and Le Calvar** 



law with:

where:

gripper latch arms of the control rods command mechanisms in PWR (Lemaire & Le Calvar, 2000). In this model, the wear law is given by the following expression:

$$\mathcal{W} = \mathcal{W}\_0 \text{ N}\left(\frac{t}{t\_0}\right)^{(1-n)}\tag{12}$$

Where W is the total worn volume, W0 a constant, and N the number of sliding steps applied to the alloy surface inducing removal of the passive film. t is the mean time interval between two successive sliding steps, and t0 is a characteristic repassivation time constant. n is a positive exponent whose value was found experimentally close to 0.65 for the cobaltbased alloy coating. The authors explain the wear law expressed by equation (12) by the evolution of the repassivation current Ip given by expression (2). This model implies that the growth of the oxide film between two sliding steps is proportional to t(1-n). The Quinn's law appears as a particular case of such a model for n = 0.5. In triborrosion studies, different values of n (between 0.6 and 0.9) where found depending on the metal, the environment and the electrochemical conditions.

#### **3.2.3 Application to the synergy formalism**

Studies in corrosive aqueous solutions (Garcia et al., 2001; Jemmely et al., 2000) are suitable to follow *in situ* the growth of passive films by electrochemical methods, and allow thus the development of more sophisticated models. In these studies performed under continuous or reciprocating sliding conditions, a modeling of currents measured at an applied potential is done. It is then assumed that a unit area of depassivated material repassivates according to a simple repassivation transient, Ja (t), which is not affected by the electrochemical conditions on the areas surrounding the rubbed area. The measured total current is then the sum of the contributions of the different surface areas. A freshly depassivated area produces a large current while a area depassivated some time before produces lower currents. In the case of a reciprocating tribometer operated at a sliding frequency, f, the steady state current I can be expressed as follows assuming that each contact event depassivates the surface:

$$I = A\_a \cdot f \cdot \int\_0^{1/f} j\_a(t)dt + (A\_t - A\_a)f \int\_0^{1/f} j\_p(t)dt\tag{13}$$

with At the total area in contact with the solution, Aa the depassivated area on the sample during one cycle, f the frequency at which the surface is depassivated, jp the passive current density at the applied potential, and ja (t) the transient repassivation current density of a unit area at the applied potential.

Taking into consideration the synergy formalism developed above, the components of mass loss per cycle, CM and C0 in Equation (4) can now be written as follows:

$$\mathbf{C}\_{M} = \frac{M}{nF} \cdot A\_{a} \left( \int\_{0}^{1/f} j\_{a}(t)dt - \int\_{0}^{1/f} j\_{p}(t)dt \right) \tag{14}$$

Tribocorrosion: Material Behavior Under

**4. Tribocorrosion testing**

during testing, and

future predictive approach.

measurements texture and internal stress analyses.

tools.

technological tests.

Combined Conditions of Corrosion and Mechanical Loading 97

Current measurements performed at different loads and sliding speeds for materials with

Similarly as in classical mechanical testing, tribocorrosion tests can be classified into two categories based on their different but complementary purposes, namely fundamental and

*Fundamental tests* are implemented in research laboratories and their objective is to clearly identify and to understand under well defined testing conditions, the basic mechanisms and their synergy that govern the phenomena of tribocorrosion. These tests require the development of experimental methodologies for both the test themselves and the techniques to be used for analyzing and measuring data and other experimental outcomes. Concerning



*Technological tests* are designed to reproduce at lab scale mechanical loading and/or environmental conditions corresponding to actual operating conditions, or to mimic particular conditions intending to accelerate material degradation processes. These tests are widely used to predict precisely the behaviour of mechanical devices in actual conditions of service and to improve their reliability and durability. In that respect, they are very useful

The full investigation of the tribocorrosion tests requires generally the use of *in situ* tools like open circuit measurements, polarization measurements, current transients, impedance spectroscopy, and noise measurements, and *ex situ* tools like elemental surface analysis techniques, optical or electron microscopy, micro-topography, micro and nanohardness

different hardness, allow the validation of the general form of this law.

**4.1 Specificity of laboratory and industrial tribocorrosion tests** 

friction in particular, two types of tests can be considered:

$$\mathbf{C}\_{0} = \frac{M}{nF} A\_{t} \int\_{0}^{1/\mathfrak{f}} \mathbf{j}\_{\mathbf{p}}(\mathbf{t}) d\mathbf{t} \tag{15}$$

When the component jp related to passive zones can be neglected, C0 becomes zero and equation (4) expressing the mass loss by corrosion under sliding becomes then:

$$\mathbf{C} = \mathbf{C}\_{M} = \frac{M}{nF} A\_{a} \int\_{0}^{1/f} j\_{a}(t)dt\tag{16}$$

The depassivated area during one cycle, Aa, can hardly be assessed. Some authors assume in first instance that it is equal to the apparent area of the sliding track. However, it is well known that the contact takes place only on a fraction of that area. An evaluation of the depassivated area from currents resulting from an electrochemical depassivation achieved by a potential jump was proposed (Garcia et al., 2001). This method also allowed them to evaluate the oxide thickness formed in between two successive depassivation events. They obtained oxide layer thicknesses in the range of a few nanometers.

Another approach was developed (Jemmely et al., 2000). The authors proposed to express the depassivated area in terms of a depassivation ratio per unit of time, Rdep :

$$\mathbf{R}\_{\rm dep} = \mathbf{f}.\mathbf{A}\_{\rm a} \tag{17}$$

with f the contact frequency. The currents can then be expressed as:

$$\mathbf{I} = \mathbf{R}\_{\text{dep}} \mathbf{Q}\_{\text{rep}} \tag{18}$$

with Qrep the charge density for repassivation. The rate at which an active area is generated per unit of time depends on the morphology and hardness of the surfaces in contact. A derivation of Rdep from the scratching of a ductile material by a hard abrasive one was proposed (Adler & Walters, 1996). That approach was taken over (Jemmely et al., 2000) and extended in more general terms (Mischler et al., 1998) in the following expression:

$$\mathbf{R}\_{\text{dep}} = \mathbf{K} \mathbf{V} \frac{F\_N^{\beta}}{H} \tag{19}$$

with K an empirical constant, V the sliding velocity, FN the applied load, H the hardness of the tested material, = 0.5 in the case of a contact between two counterparts with a similar roughness, = 0.5 in the case of a rough and hard body against a smooth and ductile counterpart, = 1 in the case of a hard and smooth body against a rough and ductile counterpart, and between 0.5 and 1 in the general case.

One empirical constant K remains in this model which approximates Archard's constant and which is related to the probability that a given contact becomes depassivated. The mass loss by corrosion under sliding, C, can thus finally be written as:

$$\mathbf{C} = \frac{M}{nF} \mathbf{K} V \frac{F\_N^{\beta}}{H} \cdot Q\_{np}(f) \tag{20}$$

Current measurements performed at different loads and sliding speeds for materials with different hardness, allow the validation of the general form of this law.
