## Thermally Activated Self Organization

With describing the total entropy rates and surface entropy rate during friction (i.e. Eqs. 3.2 and 3.6), their application to the tribosystems will be discussed in this section. Frictional sliding and wear are irreversible processes, since they are inhomogeneous and often non-stationary. The transition from the steady-state (stationary) sliding regime to the regime with self-organized structures occurs through the destabilization of the steady-state regime. At the steady state, the rate of entropy production is at minimum. The stability condition for the thermodynamic system is given in the variational form by number of the generalized forces and flows. Equation 3.7 states that the energy dissipation per unit time at the steady state should be at its minimum or the variations of the flow and the force should be of the same sign. Otherwise, the steady-state regime becomes unstable and the transition to the self-organized regime with patterns can occur. Equation 3.7 is valid for a wide range of interactions, including mechanical, thermal, and chemical, however, the corresponding terms in the entropy production rate should be considered. When Eq. 3.7 is not satisfied, the system is driven away from the equilibrium, which creates the possibility for self-organization.

In the situation when only mechanical interactions are significant, and the change of temperature T has a negligible effect on friction, the entropy is proportional to the dissipated energy divided by temperature dS = dQ/T. Consider first the situation when the production of entropy depends on the sliding velocity V. Considering that the rate of entropy production is given by Eq. 3.2, the stability condition (Eq. 3.7) now yields

If the slope of the i(V) curve (the partial derivative i'V = gV) is negative, then the steady-state sliding becomes unstable. And understandably so, since decreasing friction leads to increasing sliding velocity and to further increasing friction, and thus to the positive feedback loop.

Suppose that one contacting material has microstructure characterized by a certain parameter W, such as, for example, the size of reinforcement particles in a composite material. Such values of W that 1V (W) > 0 correspond to steady-state sliding. However, 1V (W) = 0 corresponds to the destabilization of the steady-state solution. As a result, new equilibrium position will be found with a lower value of i. Suppose now that the coefficient of friction depends also on a microstructure parameter /, such as the thickness of the interface film (Fig. 3.2). The difference between W and / is that the parameter W is constant (the composition of the material does not change during the friction), whereas the parameter / can change during friction (the film can grow or decrease due to a friction-induced chemical reaction or wear). The stability condition is now given by

where d2S is the second variation of entropy production rate [55] and k is the

Fig. 3.2 a Self-organized protective film at the interface of a composite material. b The coefficient of friction as a function of film thickness for various values of the microstructure parameter W. Sub-critical values of W < Wcr result in the positive slope (no layer formed), whereas W > Wcr results in the instability and self-organization of the protective layer. The slope depends on the ratio of the bulk and layer values of i, which allows to find composite microstructure providing the self-organization of the layer [51]

Fig. 3.2 a Self-organized protective film at the interface of a composite material. b The coefficient of friction as a function of film thickness for various values of the microstructure parameter W. Sub-critical values of W < Wcr result in the positive slope (no layer formed), whereas W > Wcr results in the instability and self-organization of the protective layer. The slope depends on the ratio of the bulk and layer values of i, which allows to find composite microstructure providing the self-organization of the layer [51]

If the stability condition is violated for a certain value of /, then further growth of the film will result in decreasing friction and wear, which will facilitate the further growth of the film. The destabilization occurs at i'lp(W, /, = 0. Note that Eq. 3.9 becomes Eq. 3.8 if / = V. At this point, we are not discussing the question of which particular thermodynamic force is responsible for the growth of the film.

Since we are interested in the conditions of the formation of such a protective film, consider now the limit of the thin film (/ ! 0,. With increasing film

10 100 1000 10000 10 100 1000 10000 AlsOj Particle Size (nm), 15 vol"/« Al-Qj PartictBSiza{nm), 15 vol%

Fig. 3.3 A significant wear and friction reduction with decreasing particle size in Al-Al2O3 nanocomposite (based on [33] can be attributed to surface self-organization [51])

10 100 1000 10000 10 100 1000 10000 AlsOj Particle Size (nm), 15 vol"/« Al-Qj PartictBSiza{nm), 15 vol%

Fig. 3.3 A significant wear and friction reduction with decreasing particle size in Al-Al2O3 nanocomposite (based on [33] can be attributed to surface self-organization [51])

thickness the value of i changes from that of the bulk composite material to that of the film material. On the other hand, the value for the bulk composite material depends also on its microstructure W (Fig. 3.2). The critical value, Wcr, corresponds to 0) = 0. For the size of reinforcement particles finer than Wcr, the bulk

(no film, / = 0) values of the coefficient of friction are lower than the values of the film. That can lead to a sudden destabilization (formation of the film with thickness /0) and reduction of friction to the value of i(W, /0) as well as wear reduction. Here, we do not investigate the question of why the film would form and how its material is related to the material of the contacting bodies. However, it is known that such a reaction occurs in a number of situations when a soft phase is present in a hard matrix, including Al-Sn and Cu-Sn-based alloys [14].

An experimental example of such sudden decrease of friction and wear with a gradual decrease of the size of reinforcement particles, which could be attributed to the destabilization, is presented in Fig. 3.3 for Al2O3 reinforced Al matrix nanocomposite friction and wear tests (steel ball-on-disk in ambient air) based on Jun et al. [33]. The abrupt decrease of friction and wear occurs for reinforcement particles smaller than Wcr = 1 im in size and can be attributed to the changing sign of the derivative i/ (WCT, 0) = 0. The decrease is sudden and dramatic, so it can be explained by the loss of stability (cf. Eq. 3.7) rather than by a gradual change of properties; although additional study is required to prove it.

For the entropy production governed by Eq. 3.6, the stability condition of Eq. 3.7 yields

The coefficient of friction and the thermal conductivity depend upon material's microstructure, /, so that l = 1(W; /)

The stability condition given by Eq. 3.20 takes the form of

The stability condition can be violated if

It is known from non-equilibrium thermodynamics that when the secondary structure is formed, the rate of entropy production reduces [26]. Therefore, if Eq. 3.13 is satisfied, the fictional force and wear can reduce. By selecting appropriate values of W (e.g., the density of a micro pattern), the condition of Eq. 3.13 can be satisfied. Note, that the wear rate is related to the rate of surface entropy production dW = B T = YJ (3.14)

dt dt

It is suggested to use the theory presented in this section to optimize the microstructure of a composite material in order to ensure that the self-organized regime occurs. To that end, the dependencies Eq. 3.11 should be investigated experimentally and their derivatives obtained. Following that, the value of W should be selected, which provides the best chances for the transition to the self-organized regime [51].

A particular field where this approach has been applied is the electrical contact, involving the current collection (e.g., for a railroad locomotive [29]). It is noted that when Tribology, which is today considered the science and technology of friction, wear, and lubrication, emerged in the 1960s, it was meant to include the fourth component, namely, the electrical contact. The electrical current is important in many applications involving the mechanical contact, such as the microelectromechanical systems (MEMS). Besides that, electromechanical contact is a typical example of a system, where two coupled processes (friction and electrical current) take place simultaneously which creates a potential for feedback, destabilization, and pattern formation. Frictional self-organization during electromechanical contact was investigated by Gershman [30]. Other areas where this approach has been successfully applied include cutting tools and the theory of ''tribological compatibility'' of materials [14].

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