The Lubricity of Oxides Revised Based on a Polarisability Approach

Abstract

The lubricity of a wide variety of solid oxides at high temperature is analysed mainly based on the existing polarisability approach. The starting point is that the interaction between ions in an oxide depends on how strongly electrons are interacting in overlapping orbitals. This is related to the polarisability of ions. From this polarisability, the interaction between cation and anion present in an oxide was calculated for binary/mixed oxides and for six new simple oxides (values for 17 oxides collected from literature) and correlated to the coefficient of friction of these oxides below their melting temperature. Based on that correlation, the mechanism of friction of these oxides is proposed.

Introduction to Chemical Approach and Polarisability

A chemical approach which relates the coefficient of friction with the ionic potential was worked out by Erdemir [1]. The ionic potential is the ratio of the charge state of cation to its radius. Erdemir’s approach was, however, not suitable to explain the friction of few oxides. In addition to his own work, Erdemir screened and collected values from so many authors, mainly from the work of Peterson et al. [2–10]. These authors carried out most of their experiments with Ni and Ni based alloys as counterbody. Since friction in sliding contact conditions is a system property, one must be aware that the counterbody material affects the friction value. Erdemir reported that with increasing ionic potential, the coefficient of friction steadily decreases (Fig. 1). The large spread noticed in the coefficient of friction for few oxides is most probably due to a variation in test conditions like counterbody material and test temperature. The following mechanism of friction was proposed by Erdemir [1]. The higher the ionic potential, the greater the screening of cations in an oxide by surrounding anions. The highly screened cations in an oxide will interact very little with other cations in their surroundings and this will allow them to shear more easily at elevated temperatures.

Fig. 1

Variation of the coefficient of friction with ionic potential for simple oxides as reported by Erdemir (reproduced from [1]), where z and r are the charge state and radius of cation

However, some limitations to Erdemir’s existing crystal-chemical approach can be formulated. At first, values of the coefficient of friction he used were collected at tests done at temperatures varying from above down to below the melting temperature of the oxides. Second, the frictional behavior of few very ionic oxides like PbO, Bi₂O₃, and BaO do not fit in the approach based on the ionic potential. Finally, the ionic potential does not clearly reflect the interactions between cations and anions in an oxide. These limitations become evident when considering the case of B₂O₃. A coefficient of friction ranging from 0.3 down to 0.15 was considered in the Erdemir’s approach (Fig. 1). B₂O₃ exist as a glassy or amorphous material and hence it does not have a sharp melting point. However these values are valid only at and above the melting temperature of B₂O₃, namely 500 °C. In the case of B₂O_3, the coefficient of friction increases sharply from 0.5 to 0.85 below the melting temperature of this oxide [2] as shown in Fig. 2. For the other oxides shown in Fig. 1, data reported were obtained at test temperatures below the respective melting temperatures. It seems that if one consider the values of the coefficient of friction obtained below the melting temperature of oxides, ionic potential fail to explain their frictional behavior (Fig. 3). Additionally, oxides having ionic potential of 1–3 are not considered in Fig. 1. The variation in the coefficient of friction with ionic potential after incorporating oxides with such an ionic potential like PbO, Bi₂O₃, and BaO is shown in Fig. 4. No clear correlation between coefficient of friction and ionic potential is appearing anymore. PbO is an oxide which give low friction till 700 °C and above. This is an oxide with layered structure as like CuO and WO₃. But the coefficient of friction of PbO (very ionic) is lower than the ionic oxides like CuO and WO₃. The low coefficient of friction noticed for PbO at room temperature could be linked to its layered structure, but the mechanism of friction at high temperature is not the same.

Fig. 2

Variation of the coefficient of friction with temperature for PbO, Re₂O₇, B₂O₃, and PbO–12% B₂O₃ as reported by Peterson et al. [2, 3]

Fig. 3

Variation of the coefficient of friction with ionic potential for simple oxides at test temperature below the melting point

Fig. 4

Variation of the coefficient of friction with ionic potential for simple oxides after incorporating data related to PbO, Bi₂O₃, and BaO

Finally, it should be mentioned that the ionic potentials of boron oxide and rhenium oxide are almost identical (Fig. 1). Notwithstanding that, these oxides behave in terms of friction totally differently below their melting temperature of 500 and 300 °C, respectively (Fig. 2). Indeed, in the case of rhenium oxide, the coefficient of friction remains constant at about 0.2 below and above its melting temperature, while for B₂O₃ the coefficient of friction increases largely below its melting temperature. This was verified with Ti–B coatings deposited by plasma immersion ion implantation and tested for their frictional behavior at temperatures below 500 °C [11]. The coefficient of friction increased to 0.85 against corundum in the sliding contact, when the test temperature is decreased from 500 to 100 °C. The coefficient of friction decreased with increase in the test temperature. Wear track analyses revealed the presence of B₂O₃. The reason for this different behavior could be linked to the fact B₂O₃ is a covalent oxide and Re₂O₇ is a more ionic one. Since the ionic potential does not consider the nature of interactions between anions and cations in oxides, it seems not to be the appropriate way to explain the frictional behavior of oxides at different temperatures.

These drawbacks in the chemical approach supports the need for finding a new approach that can explain the frictional behavior of oxides. At high temperature, during sliding in a tribo-contact, the mechanical and thermal energy may favor the movement cations and anions in an oxide by breaking up bonds. A parameter that accounts for the interaction between cation and anion in an oxide, is thus of interest to refine our understanding of friction of oxides.

Ionic covalent bonding in an oxide is closely related to the electronegativity and polarising power of cation/ionic potential. Main limitation with this is the disagreement observed between the optical basicity data reported by different researchers [12]. Polarisability or polarising power of cation is closely related to the binding energy of the electron in the cation. Ionic potential (z/r) physically mean the same thing. But the ionic potential/polarisability of cation does not give any idea of the degree of ionicity in an oxide. Dimitrov et al. did an extensive work on the polarisability approach and they have calculated the interaction parameter for many simple and mixed oxides [12–15]. They considered that the bond strength in an oxide depends on the electron bonding in overlapping orbitals of anions and cations. That can be related to the polarisability of ions [13], namely a tendency of charge centres to shift relatively. If the polarisability is large, the electron cloud in overlapping orbitals shifts and separates more, and hence a larger amount of free electrons becomes available. The cation and anion polarisability can be expressed by a single parameter, namely the interaction parameter [13]. Dimitrov et al. classified oxides into acidic, basic, and very basic materials on the basis of their polarisability and interaction parameter.

According to Dimitrov [13], the interaction parameter, A, of an oxide with a stoichiometry A_pO_q is given by:

$$ A = [(\alpha _{\rm{f}}^ - - \alpha _{02 - } )]/2(\alpha _{\rm{c}} + \alpha _{\rm{f}}^ - )(\alpha _{\rm{c}} + \alpha _{02 - } ) $$

(1)

with α ⁻_f the polarisability of an anion in the free state, α₀₂₋ the polarisability of an anion in the crystal, and α_c the polarisability of a cation. A Pauling constant with a value of 3.921 Å³ is commonly used for α ⁻_f [16].

The polarisability of an anion in an oxide can be calculated based on either the refractive index, n, or the band gap value, E _g, of that oxide. According to Dimitrov, the following equations are valid:

$$ \alpha _{02 - } (n) = [(V_{\rm{m}} /2.52)(n^2 - 1/n^2 + 2) - p\alpha _{\rm{c}} ]q^{ - 1} $$

(2)

$$ \alpha _{02 - } (E_{\rm{g}} ) = \left[ {(V_{\rm{m}} /2.52)\left( {1 - \sqrt {E_{\rm{g}} } /20} \right) - p\alpha _{\rm{c}} } \right]q^{ - 1} $$

(3)

with V _m the molar volume of the oxide.

The interaction parameter was calculated in this work based on the existing polarisability approach for oxides of interest as solid lubricant and/or wear resistant materials. For oxides like FeO and Cr₂O₃, the refractive index was used to calculate their polarisability, whereas for SnO₂, Co₂O₃, PdO, and Re₂O_7, band gap values were used [17, 18] (see Table 1).

Table 1 Bandgap and refractive index of few simple oxides [17, 18]

The polarisability of the cations in these oxides was calculated from the ionic refraction, R, given by Kordes [19]

$$ R = 2.52\alpha _{\rm{c}} . $$

(4)

Since values of ionic refraction are only available for a few ions, the polarisability of some cations of interest in this study had to be calculated from the classical theory of electronic polarisation [20]:

$$ \alpha _{\rm{c}} = {\hbox{ }}\mu */E = 4\pi \varepsilon _0 R^3 $$

(5)

with μ* the dipole moment, E the electric field strength, and ε₀ the permittivity of free space. In esu (electro static unit), the polarisability is equal to the cube of the atomic radius.

An attempt was then done to relate these calculated interaction parameters together with the Dimitrov data [13] to the coefficient of friction for simple and binary oxides. The coefficient of friction of these oxides were collected from literature [1–4, 21–24]. Unfortunately, we have not found any reliable data of oxides like Sb₂O₃, SrO, and Cs₂O–MoO₃. So these oxides were not considered in this manuscript.

Possible Link Between Interaction Parameter and Friction for Simple Oxides

The interaction parameter, the polarisability, the coefficient of friction, and the corresponding T/T _m ratio are given in Table 2. Most of the data shown in Table 2 were collected from literature, except for the six oxides marked with * for which the values of α_c, α_02−, and A, were calculated from Eqs. 1–5. Values for the coefficient of friction were collected from literature [1–3, 21–24].

Table 2 Polarisability of cations (α_c) and anions (α₀₂−), the interaction parameter (A), and the coefficient of friction (μ) at different ranges of T/T _m for various oxides

The calculated polarisability for the six oxides of interest to achieve low friction wear resistant coatings, namely Re₂O₇, PdO, FeO, Co₂O₃, Cr₂O₃, and SnO₂ decreases with increasing interaction parameter, and the data fit well with the values calculated by Dimitrov for other oxides (Fig. 5).

Fig. 5

Variation of the oxide ion polarisability with the interaction parameter in simple oxides

The relationship between the experimentally determined coefficient of friction of simple oxides in sliding contacts, and the calculated interaction parameter for these oxides, is shown in Fig. 6. The coefficient of friction of acidic (ionic covalent) oxides, like boron oxide characterized by a high interaction parameter of 0.244, is high in comparison to the coefficient of friction of basic (ionic) and very basic oxides (very ionic). Indeed, basic oxides, like WO₃ and MoO₃, show a coefficient of friction that is lower than the one of acidic oxides, but higher than the one of very basic oxides, like PbO, Bi₂O₃, and BaO. All these oxides have a positive interaction parameter, except Re₂O₇ that possesses a negative interaction parameter (Fig. 6a). This could explain the large difference in the coefficient of friction of Re₂O₇ and B₂O₃ noticed in Fig. 6. Indeed the two oxides differ markedly in the polarisability of their anions, which is very high in the case of Re₂O₇ but low in the case of B₂O₃ (see Table 2). The dependence of the coefficient of friction on the interaction parameter is shown in Fig. 6 for three different T/T _m ranges. Data were taken selectively from Table 2. It appears that the coefficient of friction decreases at decreasing interaction parameter independently of the T/T _m range considered.

Fig. 6

Variation of the coefficient of friction of simple oxides with the interaction parameter at T/T _m in the range of (a) 0.3–1, (b) 0.3–0.4, (c) 0.4–0.5, and (d) 0.9–0.95

At increasing polarisability of anions, the interaction parameter decreases (Fig. 5), and that corresponds to an increase of the free electron density in overlapping orbitals of the oxides. That increase in the free electron density weakens the bonding between cations and anions. In boron oxide (B₂O₃), the B³⁺ cations possess a high ionic potential of 12. This ionic potential is the ratio of the charge state, Z, to the ionic radius, r. This high ionic potential increases the cationic field strength being the force acting on a unit coulombic charge. This increases the polarising power, which is the tendency of a cationic field to keep the electrons in overlapping orbitals of this cation in the oxide. This increase in the polarising power decreases the polarisation of anions, and decreases the free electron density in the outer overlapping orbitals. In the case of very basic oxides like PbO, Bi₂O₃, and BaO, the orbitals involved in bonding are 5f 6s, and the charge state is also low. This decreases the ionic potential of these cations and hence increases the cation polarisability. The low field strength of these cations decreases the polarising power of these cations, and increases the anion polarisability as well as the free electron density in the overlapping orbitals. The case of Re₂O₇ is interesting. Re₂O₇ has the characteristics of very basic oxides, but shows a negative interaction parameter and a very high anion polarisability above the Pauling value for free oxide ions. The polarisability of the anion was calculated from the band gap of Re₂O₇. A negative band gap of −0.716 eV and a negative interaction parameter suggest that this oxide could be classified as a metallic material. The ionic potential of Re₂O₇ is 12.5. Even though, the cationic field strength is large in Re₂O₇, still the polarisability of the anion in this oxide is high in contrast to other oxides with a large field strength as B₂O₃. The reason for this high oxygen ion polarisability in Re₂O₇ could be linked to the increase in the number of oxygen ion contribution to the cation.

Interaction Parameter and Friction for Mixed/Binary Oxides

The interaction parameter for a number of binary and mixed oxides was calculated by using the polarisability approach. Data on the coefficient of friction of such oxides were collected from the published articles [1, 2, 4, 23, 24]. They prepared 50 atomic percentage compounds by mixing the appropriate quantities of the constituting simple oxides. The interaction parameter for these binary oxides was calculated as proposed by Dimitrov [14]:

$$ A = {\hbox{ }}X_1 A_1 + X_2 A_2 $$

(6)

with X _i and A _i the fraction and interaction parameter of the respective oxides.

The calculated interaction parameter and the coefficient of friction of a wide range of binary/mixed oxides are shown in Table 3. It seems that the coefficient of friction decreases with decreasing interaction parameter. For example, Cu(ReO₄)₂ that has an interaction parameter of 0.013, shows a coefficient of friction of 0.20 that is lower than the one for Ni(ReO₄)₂ and FeO–MoO₃ which possesses an interaction parameter of 0.041 and 0.0515. The coefficient of friction of 0.55 was noticed for Al₂O₃–TiO₂ with an interaction parameter of 0.143. The dependence of the coefficient of friction for these binary and mixed oxides on their interaction parameter at T/T _m = 0.7–0.8 is shown in Fig. 7. The coefficient of friction decreases with decreasing interaction parameter as noticed for simple oxides (see Fig. 6).

Table 3 Coefficient of friction (μ) and interaction parameter (A) for binary and mixed oxides

Fig. 7

Variation of the coefficient of friction with the interaction parameter in binary/mixed oxides at T/T _m = 0.7–0.8. Values for the coefficient of friction were taken from [2, 4, 23, 24]

The coefficient of friction of 88 PbO–12 B₂O₃ determined by Peterson was in the range of 0.2–0.9 [2]. The interaction parameter calculated for this binary oxide is low, namely 0.033, although a high coefficient of friction is noticed. So, this binary system 88 PbO–12 B₂O₃ do not fit into this polarisability approach to explain the frictional behavior. This can be understood when the evolution of the coefficient of friction of this binary oxide with temperature is compared to the one of B₂O₃ and PbO (Fig. 2). The coefficient of friction of this binary oxide follows the same trend as for B₂O₃. The presence of a low melting point oxide B₂O₃, as one of its constituent results in the presence of that oxide at the surface, when the test temperature is above or nearer to its softening/melting temperature. This could be the reason why 88 PbO–12 B₂O₃ system is showing the frictional behavior similar to that of its low melting point component B₂O₃ in a wide range of test temperatures.

In the case of binary/mixed oxides, if minor amount of lubricating oxide is present, its friction will dominate that for the alloy [4]. This is because the presence of a very basic oxide in a binary/mixed oxide reduces the interaction parameter of the whole system and hence a decrease in the coefficient of friction. So, the coefficient of friction is decided by the kind of oxide present at the surface. Hence, it is very essential to have an exact idea of the different phases, composition of the phases at the tribo-contact and their melting temperature. Such information will be really useful to explain the frictional behavior of a wide range of binary/mixed oxides based on this polarisability approach.

Friction: Role of Interaction Parameter and Activation Energy

In general, oxides with a low interaction parameter possess a high free electron density. These electrons do not contribute to bonding and thus an increase of the surface free energy is expected. Indeed, an explanation relating surface free energy to bond strength of transition metals was reported in literature by Buckley [25]. For instance, there is a decrease in the coefficient of friction in the case of transition elements with an increase in the d-bond character of the metal. That means that the greater the tendency of a metal to bond to itself, the lesser the surface free energy is, and hence the lesser the bonding across an interface is. An increase in the surface free energy could lead to a decrease of the activation energy for the formation of vacancies and for the hopping of ions causing surface diffusion. To prove this, available data on the activation energy for the formation of an anion vacancy were collected from literature [26] and it is shown in the Table 4. Indeed, it appears that the activation energy decreases to some extent with decreasing interaction parameter. Since there is a lack of data available for many oxides on the activation energy for anion vacancy, a strong correlation between the activation energy and interaction parameter is not noticeable.

Table 4 Interaction parameter and activation energy for simple oxides

In oxides, the mechanism of friction at different temperatures can be explained by considering the number of vacancies, N, at equilibrium given by [27]

$$ N = c\exp ( - Q/kT) $$

(7)

with Q the activation energy, T the temperature, k the boltzman constant, and c a constant. Indeed, a decrease of the activation energy increases the number of vacancies, and thus increases the mobility of ions which is reflected by a decreasing interaction parameter. This increased mobility of ions could be the reason for the low coefficient of friction in solid oxides with a small interaction parameter. That the coefficient of friction varies exponentially with the inverse of temperature (μ ∝ e^(1/T)) has been proven by Peterson et al. in the case of B₂O₃ [2]. Since the coefficient of friction and the number of vacancy formation varies exponentially with temperature, it seems that the number of vacancies formed at high temperature has a direct impact on the coefficient of friction. In the case of oxides with a high interaction parameter, the mobility of the ions decreases because a lower number of vacancies are available, and this may be at the origin of a higher sensitivity to crack formation. Indeed, Peterson reported that the coefficient of friction of B₂O₃ increases at decreasing temperature and that below a certain temperature cracks were noticed [2]. On the contrary, cracks or damage were not detected in very ionic oxides, like PbO, with a low interaction parameter [2]. The decrease in the coefficient of friction with decreasing interaction parameter at different T/T _m ratio (Fig. 6), suggests that the formation of vacancies and the hopping of ions at the surface of an oxide can play a prominent role on the coefficient of friction in sliding contacts. The exo-electron emission observed by Evdokimov et al. [28] in sliding contacts as a result of vacancies leaving the surface, confirms the formation of a large number of vacancies at the surface.

Among binary and mixed oxides, rhenates show a lower coefficient of friction than other oxides [2, 4]. Presence of Re₂O₇ (very ionic) in a binary/mixed oxide reduces the interaction parameter of the whole system and become more ionic. A very ionic oxide possesses more free electrons than an ionic and acidic one and has a larger surface free energy. This increase in surface free energy is expected to be accompanied by a decrease of the activation energy for the formation of vacancies. The number of vacancies increases and, as a consequence, a decrease in the coefficient of friction is expected.

Conclusions

A new approach based on the existing polarisability approach is proposed to explain the frictional behavior of a variety of oxides below their melting temperature. The interaction parameter tells how strongly ions in an oxide are bounded. This parameter is a function of the polarisability of cation and anion in an oxide. The interaction parameter of six simple oxides were calculated and data for other simple oxides were collected from the work reported by Dimitrov. In addition, the interaction parameter of few mixed/binary oxides were also calculated.

It was found that at a given T/T _m ratio, an oxide with a low interaction parameter shows a low friction. In the case of binary/mixed oxides, if minor amount of lubricating oxide is present, its friction dominate that for the alloy. This is because the presence of a very basic oxide in a binary/mixed oxide reduces the interaction parameter of the whole system and hence a decrease in the coefficient of friction. Oxides with a low interaction parameter possess high free electron density. This increase in surface free energy is expected to be accompanied by a decrease of the activation energy for the formation of vacancies. In such oxides, the number of vacancies at their surface must be higher at high temperature, and this increases the degree of freedom and mobility of ions. This increased mobility of ions at their surface, is assumed to be at the basis of the low coefficient of friction of such oxides in sliding contacts. The coefficient of friction is decided by the kind of oxide present at the sliding contact. Hence, it is very essential to have an exact idea of the different phases, composition of the phases at the tribo-contact and their melting temperature. Such information will be really useful to explain the unexplained frictional behavior of a wide range of binary/mixed oxides, based on this polarisability approach.

This approach is most helpful in selecting simple, binary, or mixed oxides for getting the desired coefficient of friction at high temperature. This study suggests that the formation of vacancies and the hopping of ions at the surface of oxides determine to a large extent the coefficient of friction at high temperature.