**2. Background and the present work goal**

### **2.1. Recent papers focused on the activation energy (Ea) new approach**

Paper [2] has proposed the new hypothesis concerning kinetics of chemical reactions and Ea. Shortly the hypothesis was confirmed indirectly [3] by using earlier published results on energy angular distribution of electrons emitted from MIM systems [7] and angular distribution of photoelectrons emitted from solids [8]. The next confirmation step was based on irreversible thermodynamics [4] and, it was demonstrated that the equation of the found reaction rate is also consistent with two different pathways for tribochemical reactions:

$$\mathbf{J}\_{\mathbb{C}} = \{ \mathbf{v} \, \exp(\mathbf{-} \mathbf{E}\_{\mathbb{A}} \mathbf{/} \mathbf{R} \mathbf{T}) \, (1 - \mathbf{e}^{\mathbf{A} \cdot \mathbf{R} \mathbf{T}}) + \mathbf{k} \, [1 - \exp(\mathbf{-} \mathbf{J}\_{\mathbb{A}} \mathbf{c} \mathbf{t} \mathbf{\tau} / \mathbf{T})] \} \, \|\, \Pi \mathbf{a}^{\mathbb{N}}\| \tag{2}$$

where: Jc – reaction rate, Ju – energy flow towards the reagents due to triboreactions, – shear stress due to friction, A – chemical affinity, c and k – constants, t – time, ai – activities of reagents, i – stoichiometric coefficient.

The first one is the thermal mechanism, typical for non-friction conditions and, the second concerns the direct transfer of energy from triboelectrons to molecules. The latter one generates special excited or activated molecules, such as radicals or ions which react rapidly to form the products, enhancing the global reaction rate.

#### **2.2. Aim of the paper**

The major goal of the present paper is to enhance the developed idea and provide it with the better understanding of Arrehnius equation and its activation energy. To make a progress in that field, i model developed by Kulczycki [9] was here applied along with its detailed thermodynamic interpretation. Detailed disscussion is focused on the basic Arrehnius equation.

#### **3. Thermodynamic interpretation of i model**

#### **3.1. Theoretical information**

48 Tribology in Engineering

thermodynamics.

catalytic surfaces.

reactions:

**2. Background and the present work goal** 

of reagents, i – stoichiometric coefficient.

**2.2. Aim of the paper** 

equation.

to form the products, enhancing the global reaction rate.

**2.1. Recent papers focused on the activation energy (Ea) new approach** 

Jc = {i exp(-Ea/RT) (1 – e-A/RT) + k [1 – exp(-Juct/T)] } ai

Paper [2] has proposed the new hypothesis concerning kinetics of chemical reactions and Ea. Shortly the hypothesis was confirmed indirectly [3] by using earlier published results on energy angular distribution of electrons emitted from MIM systems [7] and angular distribution of photoelectrons emitted from solids [8]. The next confirmation step was based on irreversible thermodynamics [4] and, it was demonstrated that the equation of the found reaction rate is also consistent with two different pathways for tribochemical

where: Jc – reaction rate, Ju – energy flow towards the reagents due to triboreactions, – shear stress due to friction, A – chemical affinity, c and k – constants, t – time, ai – activities

The first one is the thermal mechanism, typical for non-friction conditions and, the second concerns the direct transfer of energy from triboelectrons to molecules. The latter one generates special excited or activated molecules, such as radicals or ions which react rapidly

The major goal of the present paper is to enhance the developed idea and provide it with the better understanding of Arrehnius equation and its activation energy. To make a progress in that field, i model developed by Kulczycki [9] was here applied along with its detailed thermodynamic interpretation. Detailed disscussion is focused on the basic Arrehnius

i

(2)

supported the new approach to Ea by theoretical considerations based on irreversible

Heterogeneous catalysis provides the link between reactants and products on a reaction pathway which involves simultaneous motion of several to very many atoms [5]. Predictability of the outcome of catalytic reactions is controlled by their molecular mechanisms. Thus, the importance of the activation energy better understanding can not be overestimated. Some forty years ago, work [6] demonstrated that a "chemically stimulated" exo-electron emission (EEE) occurs simultaneously during the partial oxidation process of ethylene. It was also found that the emission rate was proportional to the rate of ethylene oxide formation. Therefore, discussing heterogeneous catalytic reactions, EEE process should also be taken into account, because they involve electro physical phenomena. Additionally, such electrons are of low-energy and are produced from the excited active

The main assumption of i model, described in detail in [9] introduced the **new measure – <sup>i</sup> coefficient** of reagent/lubricating oil or additive **properties / structure** influence on its **reactivity** related to **reaction conditions**. This model was worked out on the bases of the results of tribochemical investigations of different lubricating oils. In relation to tribological processes, **tribochemical reaction conditions** depend on the work done on tribological system - L. The work L is the function of applied load P, which can be treated as the only one variable y – L = f(y). The reagents **reactivity** is described by the second function of variable y - ( y ). Basing on Cauchy theorem the relation between two functions of the same one variable y is as follows:

$$\mathbf{u} = \left[ \left[ \mathbf{f} \left( \mathbf{b} \right) - \mathbf{f} \left( \mathbf{a} \right) \right] \right] / \left[ \boldsymbol{\phi} \left( \mathbf{b} \right) - \boldsymbol{\phi} \left( \mathbf{a} \right) \right] \right) / \left[ \mathbf{f}' \left( \mathbf{b} \right) / \boldsymbol{\phi}' \left( \mathbf{b} \right) \right] \tag{3}$$

where *a* and *b* are values of y parameter. For different reagents value *b* is the only one variable (assumption of Cauchy theorem) and *a* is constant. Variable *b* was related to tribological process conditions – for example applied load P. In i model P is critical value and it relates either to seizure load or weld load. Consequently work done on the system means the work needed to achieve the seizure or weld.

Work done on tribological system can be related to thermodynamic description of tribological process. In the consequence reactivity can be related to the internal energy change u, and Eqn. (4) can be written as follows:

$$
\alpha = \left[ \left( \text{L} - \text{L} \right) / \left( \Delta \text{u} - \Delta \text{u} \right) \right] \times \left( \text{d} \,\Delta \text{u} / \text{d} \,\text{L} \right) \tag{4}
$$

L = Pvt; - friction coefficient; v – speed; t – time; P – applied load (test result); where: Lo = f(a), u0 = (a)

Since it is difficult to define relationship u= (P) because u is not linear dependence of the applied load P, we can use the first low of thermodynamics L to express it as a function of u:

$$\mathbf{L} = \mathbf{Q} + \boldsymbol{\Lambda}\mathbf{u} \tag{5}$$

Q is energy dissipated by system during tribological process; mainly it is a dissipated heat, which in relation to tribological process can be described by the following dependence:

$$\mathbf{Q} = \mathbf{c}\_{\text{V}} \left( \mathbf{T} - \mathbf{T}\_{\text{ot}} \right) \tag{6}$$

$$\mathbf{T} = \mathbf{T}\_{\mathbb{B}} + \mathbf{A} \; \mathbf{P}^{0, \mathbb{B}} \tag{7}$$

where: ch - average specific heat capacity,

Tb – temperature of lubricant out of friction area, Tot – temperature of environment.

Assuming that both average specific heat capacity and Tb are constant for different oils, Q can be expressed as

$$\mathbf{Q} = \mathbf{A}\_1 \mathbf{P}^{\tag{8}} \tag{8}$$

A New Attempt to Better Understand Arrehnius Equation and Its Activation Energy 51

k = A exp (B – Ea/RT) (13)

Consequently for the given constant value of C only some of values P meet equation (10)

What is physical and chemical meaning of those dependences? Answers seem to be also related to tribocatalysis and heterogeneous catalysis. Exponential part of this dependence

where temperature T may be connected with load P applied in tribological process, as

T = Tb + P0,5 D D = 10-5VsPp0,5(k1 + k2)-1 where: Vs – sliding speed, Pp - unit pressure of the metal flow, k1 and k2 - coefficients of thermal conductivity of cooperated elements of tribological system [10].

The i model and the equation (12) combine the activation energy delivered to molecules from the energy stream. Considering the energy delivered to molecules for each reaction,

 Cauchy`s theorem requires the only one variable in functions f(y) and (y); application of i model to tribochemistry/ tribocatalysis and heterogeneous catalysis this variables can be temperature T or connected with them applied load P (tribochemistry/

 To use i model in tribochemical and chemical problems solving it was assumed that variable y describes critical stage of reaction system, which is equivalent to critical rate

 A critical rate of catalytic reactions or triboreactions is reached at a critical temperature (eg. temperature of catalytic reaction at which there is maximum rate of this reaction)

 Connecting critical rate of triboreaction / catalytic reaction with C in Eqn. 9 and 10 it was concluded that the value of Ea is the same for different reactants in the case of the same mechanism of triboreactions / catalytic reactions, because Ea is neither a function



On the basis of this mathematical model, physical model of tribocatalysis / catalysis was created. The model assumes that mechanical work done on the system (containing liquid

or, equivalently, under a critical load, at which seizure load takes place.

of *T* nor *P* and Cauchy`s theorem does not accept another variables than y.

additives / reactants and different values of critical load P.

can be connected with kinetics of chemical reactions by Arrhenius equation:

and (11). (red points in Figure1.)

earlier shown in equation (7)

the following should be noted:

of described reactions.

Concluding for tribocatalysis:

tribocatalysis).

**3.3. Cauchy`s theorem application to the model** 

Consequently

$$\mathbf{A} \cdot \mathbf{u} = (\mu \mathbf{P} \mathbf{v} \mathbf{t} - \mu \mathbf{P} \mathbf{v} \mathbf{t}) \left[ (\mu \mathbf{v} \mathbf{t} \, \mathbf{dP} - \mathbf{A} \mathbf{u} \, \mathbf{dP} \, \mathbb{S}) / \, \mu \mathbf{v} \mathbf{t} \, \mathbf{dP} \right] / \left( \mu \mathbf{v} \mathbf{t} \mathbf{P} - \mathbf{A} \mathbf{u} \, \mathbf{P}^{\otimes 5} - \mathbf{A} \mathbf{u} \mathbf{o} \right) \tag{9}$$

### **3.2. Description of the C value as the function of applied load P in terms of Figure 1**

It was experimentally found during tribological tests, described in [9], that i is linear function of applied load P. This linear dependence takes place in case the number of lubricants containing additives of different activity, but similar chemical structures were tested in the given tribological test. In this case

$$
\alpha = \text{(\mu Pvt - \mu Povt)} \text{ C} \tag{10}
$$

and for additives of similar chemical structure (for example zinc dithiophosphates) C value is constant. For these additives in the given tribological test

$$\mathbf{C} = \left[ \left( \mu \text{vt dP } - \text{A} \mathbf{\iota} \text{ dP}^{0.5} \right) / \left( \mu \text{vt dP } \right) / \left( \mu \text{vt} \mathbf{P} - \text{A} \mathbf{\iota} \text{ P}^{0.5} \right. \\ \left. - \text{A} \mathbf{\iota} \text{\iota} \text{\iota} \right) \tag{11}$$

The fact that for additives of similar chemical structure (for example zinc dithiophosphates) C value is constant and C is the function of P, which value is different for each additive, requires that C is a harmonic function of P, described by the following dependence:

$$\mathbf{C} = \mathbf{A} \exp(-a\mathbf{y}) / \cos \ (b\mathbf{y} + d) \tag{12}$$

where y = P or T and A, *a*, *b* and *d* are constant value.

**Figure 1.** Dependence between C and applied load P

Consequently for the given constant value of C only some of values P meet equation (10) and (11). (red points in Figure1.)

What is physical and chemical meaning of those dependences? Answers seem to be also related to tribocatalysis and heterogeneous catalysis. Exponential part of this dependence can be connected with kinetics of chemical reactions by Arrhenius equation:

$$\mathbf{k} = \mathbf{A} \exp\left(\mathbf{B} - \mathbf{E}\mathbf{\sqrt{R}T}\right) \tag{13}$$

where temperature T may be connected with load P applied in tribological process, as earlier shown in equation (7)

$$\mathbf{T} = \mathbf{T}^{\rhd} + \mu \mathbf{P}^{0, \mathbf{s}} \mathbf{D}$$

D = 10-5VsPp0,5(k1 + k2)-1 where: Vs – sliding speed, Pp - unit pressure of the metal flow, k1 and k2 - coefficients of thermal conductivity of cooperated elements of tribological system [10].

#### **3.3. Cauchy`s theorem application to the model**

50 Tribology in Engineering

Consequently

**Figure 1** 

tested in the given tribological test. In this case

is constant. For these additives in the given tribological test

where y = P or T and A, *a*, *b* and *d* are constant value.

**Figure 1.** Dependence between C and applied load P

Q = A1 P 0,5 (8)

i = (Pvt - P0vt) C (10)

C = A exp(-*a*y) / cos (*b*y + *d*) (12)

C = [(vt dP –A1 dP 0,5) / vt dP] / (vtP - A1 P 0,5 –uo) (11)

i = (Pvt - P0vt) [(vt dP –A1 dP 0,5) / vt dP] / (vtP - A1 P 0,5 –uo) (9)

**3.2. Description of the C value as the function of applied load P in terms of** 

It was experimentally found during tribological tests, described in [9], that i is linear function of applied load P. This linear dependence takes place in case the number of lubricants containing additives of different activity, but similar chemical structures were

and for additives of similar chemical structure (for example zinc dithiophosphates) C value

The fact that for additives of similar chemical structure (for example zinc dithiophosphates) C value is constant and C is the function of P, which value is different for each additive,

requires that C is a harmonic function of P, described by the following dependence:

The i model and the equation (12) combine the activation energy delivered to molecules from the energy stream. Considering the energy delivered to molecules for each reaction, the following should be noted:


Concluding for tribocatalysis:


On the basis of this mathematical model, physical model of tribocatalysis / catalysis was created. The model assumes that mechanical work done on the system (containing liquid

reagents = lubricants), is transformed to internal energy increase and dissipated energy. Internal energy is distributed in the system: one part is distributed to the liquid phase and is responsible for ambient temperature increase of the lubricant (Ta), the second part is cumulated in solid body (solid elements of tribological system) and is emitted as electrons or photons by its surface as impulses of high intensity. Energy cumulated in the liquid phase is not sufficient to reach value of Ea. Energy emitted by surface as impulses can reach value of Ea and reaction / triboreaction begins to proceed.

A New Attempt to Better Understand Arrehnius Equation and Its Activation Energy 53

2 vs vp-1 + 0,2 – 0,00073 40 (14)

boundary conditions is selected on AW and EP types. The AW and EP coefficients are assigned to each tested lubricants. The method used in calculation of AW and EP values on

The i values assigned to those lubricants were related to the results of tribological standard tests. Therefore, it was necessary to find out tribological experimental methods able to provide test results concerning lubricant's ability to create both AW and EP types of the boundary layer. In the already cited work [5], there were selected two different tests carried

The first test was used to determine seizure load Pt under the following operating

Following dependence for AW determined applying the above described procedure and Pt

This dependence points out that the mechanism of AW type of protective layer formation

The EP values determined using above procedure were related to the second 4-ball test, where welding load Pw was the test result. This test was performed under the following


It was found that Pw values depend on durability of EP type of the boundary layer and was

EP = 0,48 – 0,00013 Pw (15)

Using both these standard tests the value of AW and EP can be determined easily and than used to predict the results of another tribological tests (eg. FZG) or analyze the structure of

the bases of tribotests results for each lubricant was described in [9].

out using a four ball machine.



AW = ( 0,000086 40 – 0,01 )0,5 Pt

40 – kinematic viscosity of tested oil at 40 oC


experimentally found the linear dependence between EP and Pw:

vs - sliding velocity ( 0,18 m/s ) vp - speed of load increase ( 45 N/s )

depends on base oil viscosity.

μ- friction coefficient

operating conditions:

Ball Method)



conditions:

values is found:

where:

Eqn. 10 points that value of C depends not only on reaction rate constant but on intensity of energy emitted in angle from solid body to reaction space. Angle can not be another variable in Eqn. 8 and 10 – requirements of Cauchy`s theorem, so it should depend on T or P. Accordingly, the value of the angle �depends on the system energy flux. The critical state of a tribological system appears at conditions resulting in destruction of the protective film. It has been observed that for different reactants, the critical rate of reaction leading to protective layer destruction was achieved for different values of energy flux into the system (different values of applied load *P*). The same value of *C* obtained for different reactants and different densities of energy streams introduced into the system (characteristic of each reactant) leads to the conclusion that the same critical rate of destruction reaction was achieved and thus for each reactant a different angle is connected with the critical rate of reaction (different values of T or P).

Consequently, for each reactant there is a specific value of the energy flux density (*e*), where e = e0 cos e0 - intensity of energy stream emitted by the solid/catalyst in normal direction to solid surface ( = 0) and the value of activation energy *Ea* is constant. Accordingly, it is possible to emphasize that:


## **3.4. Empirical verification of i model and its derivative equation concerning heterogeneous catalysis**

#### *3.4.1. Verification of i model in tribological tests*

Basing on the above dependences and obtained test results the i values were calculated for a number of gear, hydraulic and transmission oils of viscosity at 40 0C in range between 32 to 220 mm2/s. Because of the structure of protective layer created by lubricant under boundary conditions is selected on AW and EP types. The AW and EP coefficients are assigned to each tested lubricants. The method used in calculation of AW and EP values on the bases of tribotests results for each lubricant was described in [9].

The i values assigned to those lubricants were related to the results of tribological standard tests. Therefore, it was necessary to find out tribological experimental methods able to provide test results concerning lubricant's ability to create both AW and EP types of the boundary layer. In the already cited work [5], there were selected two different tests carried out using a four ball machine.

The first test was used to determine seizure load Pt under the following operating conditions:


Following dependence for AW determined applying the above described procedure and Pt values is found:

$$\mathbf{q}\_{\text{A}\text{av}} = (0.000086 \,\text{v} \,\text{ov} - 0.01 \,\text{J}) 0.5 \,\mu \,\text{P}^2 \,\text{v} \,\text{v}\_{\text{P}} \,\text{v}\_{\text{P}}^{-1} + 0.2 - 0.00073 \,\text{v} \,\text{w} \tag{14}$$

where:

), where

52 Tribology in Engineering

Ea and reaction / triboreaction begins to proceed.

reaction (different values of T or P).

reaction activation energy *Eac*;

or its rate enhancement.

**heterogeneous catalysis** 

*3.4.1. Verification of* 

possible to emphasize that:

reagents = lubricants), is transformed to internal energy increase and dissipated energy. Internal energy is distributed in the system: one part is distributed to the liquid phase and is responsible for ambient temperature increase of the lubricant (Ta), the second part is cumulated in solid body (solid elements of tribological system) and is emitted as electrons or photons by its surface as impulses of high intensity. Energy cumulated in the liquid phase is not sufficient to reach value of Ea. Energy emitted by surface as impulses can reach value of

Eqn. 10 points that value of C depends not only on reaction rate constant but on intensity of energy emitted in angle from solid body to reaction space. Angle can not be another variable in Eqn. 8 and 10 – requirements of Cauchy`s theorem, so it should depend on T or P. Accordingly, the value of the angle �depends on the system energy flux. The critical state of a tribological system appears at conditions resulting in destruction of the protective film. It has been observed that for different reactants, the critical rate of reaction leading to protective layer destruction was achieved for different values of energy flux into the system (different values of applied load *P*). The same value of *C* obtained for different reactants and different densities of energy streams introduced into the system (characteristic of each reactant) leads to the conclusion that the same critical rate of destruction reaction was achieved and thus for each reactant a different angle is connected with the critical rate of

Consequently, for each reactant there is a specific value of the energy flux density (*e*

energy stream (time of the tribological process is constant for each load);

e = e0 cos e0 - intensity of energy stream emitted by the solid/catalyst in normal direction to solid surface ( = 0) and the value of activation energy *Ea* is constant. Accordingly, it is

i. *Ea* is constant for a given type of reaction and the critical rate of reaction depends not only on the energy quantity added to reactants but on the density of the introduced

ii. The catalyst emits pulses of energy flux of high density at an angle The value of emitted in short time energy is equal to the difference between activation energy calculated for reaction without catalyst *Ea* and activation energy calculated for catalytic

iii. The catalyst collects energy done as mechanical work and emits it as pulses of the high energy density flux, thereby decreasing temperature needed for the reaction initiation

**3.4. Empirical verification of i model and its derivative equation concerning** 

Basing on the above dependences and obtained test results the i values were calculated for a number of gear, hydraulic and transmission oils of viscosity at 40 0C in range between 32 to 220 mm2/s. Because of the structure of protective layer created by lubricant under

*i model in tribological tests* 

40 – kinematic viscosity of tested oil at 40 oC


μ- friction coefficient

This dependence points out that the mechanism of AW type of protective layer formation depends on base oil viscosity.

The EP values determined using above procedure were related to the second 4-ball test, where welding load Pw was the test result. This test was performed under the following operating conditions:


It was found that Pw values depend on durability of EP type of the boundary layer and was experimentally found the linear dependence between EP and Pw:

$$\mathbf{u} \pm \mathbf{p} = \mathbf{0}/48 - \mathbf{0}/00013 \,\mathrm{P\_W} \tag{15}$$

Using both these standard tests the value of AW and EP can be determined easily and than used to predict the results of another tribological tests (eg. FZG) or analyze the structure of protective layer created in various machines (participation of AW and EP structure in protective layer).

A New Attempt to Better Understand Arrehnius Equation and Its Activation Energy 55

References [11-12] allow to make an estimate that at room temperature the narrow maximum around the direction normal to the emitting surface contains approximately 1/3 of the total emission. Further work of Hrach [7,13] was of particular significance as he measured the energy characteristics of emitted electrons at different angles by means of a hemispherical collector and the retarding-grid technique. He found that at room temperature (300K) anisotropic energy spectra of emitted electrons were in the range of 0 and 7eV, but for emission angles closer to the normal to the surface the measured energy

**Figure 3.** Typical experimental angular and energy distribution of electrons emitted from Al-Al2O3-Au

Reference [3] includes more detailed information concerning figures 2 and 3. More recent study [14] was on the angular distribution of thermo-stimulated exoelectron emission (TSEE) from alpha-Al2O3. Anisotropic angular distributions were strongly directed normally to the surface; the energy of the emitted electrons was in the range of 0 to 4 eV. Jablonski and Zemek [8] found highly anisotropic distribution similar to one of electron-type for the

In Arrhenius equation T equals Ta, (T = Ta) and it relates to ambient temperature of reaction

 When catalyst is used ambient energy RTa is less than energy in space near catalysts particles surface = RTs. Ts is the calculated temperature near catalyst surface and it should be higher than ambient temperature of reaction mixture. RTs is real energy acted

sandwich cathode structures. Temperature: 300K, applied voltage: 10V [13]

**3.5. Disscussion on the basic Arrehnius equation** 

mixture. Conclusions from i model are as follows:

photon emission from X-ray-irradiated thin polycrystalline aluminum foils.

was between zero and 4 eV. Figure 3. depicts some of these findings.
