## Abstract

Despite the fact that self-organization during friction has received relatively little attention from tribologists so far, it has the potential for the creation of self-healing and self-lubricating materials, which are important for green or environment-friendly tribology. The principles of the thermodynamics of irreversible processes and of the nonlinear theory of dynamical systems are used to investigate the formation of spatial and temporal structures during friction. The transition to the self-organized state with low friction and wear occurs through destabilization of steady-state (stationary) sliding. The criterion for destabilization is formulated and several examples are discussed: the formation of a protective film, microtopography evolution and slip waves. The pattern formation may involve self-organized criticality and reaction–diffusion systems. A special self-healing mechanism may be embedded into the material by coupling the corresponding required forces. The analysis provides the structure–property relationship, which can be applied for the design optimization of composite self-lubricating and self-healing materials for various ecologically friendly applications and green tribology.

## 1. Introduction

The emergence of green tribology makes it important to reconsider some conventional approaches to friction and wear. Friction and wear are usually viewed as irreversible processes, which lead to energy dissipation (friction) and material deterioration (wear). On the other hand, it is known that under certain circumstances frictional sliding can result in the formation (self-organization) of spatial and temporal patterns. Friction-induced self-organization has a significant potential for the development of self-lubricating, self-healing and self-cleaning materials, which can reduce environmental contamination, and thus it is relevant to green tribology.

Historically, the first attempts to investigate friction-induced self-organization were made in Russia starting in the 1970s. Several groups can be mentioned. First, Bershadsky (1992, 1993) in Kiev investigated the formation of the so-called self-organized ‘secondary structures’. According to Bershadsky, friction and wear are two sides of the same phenomenon, and they represent the tendency of energy and matter to achieve the most disordered state. However, the synergy of various mechanisms can lead to self-organization of the secondary structures, which are ‘non-stoichiometric and metastable phases’, whereas ‘the friction force is also a reaction on the informational (entropic) excitations, analogous to the elastic properties of a polymer, which are related mostly to the change of entropy and have the magnitude of the order of the elasticity of a gas’ (Bershadsky 1993).

These ideas were influenced by the theory of self-organization developed by I. Prigogine, a Belgian physical chemist of Russian origin and the winner of the 1977 Nobel Prize in Chemistry. Prigogine (1968) used the ideas of non-equilibrium thermodynamics to describe the processes of self-organization (Nicolis & Prigogine 1977). Later, he wrote several popular books about the scientific and philosophical importance of self-organization (Prigogine 1980; Prigogine & Stengers 1984). At the same time, at the end of the 1970s, the concept of ‘synergetics’ was suggested by Haken (1983) as an interdisciplinary science investigating the formation and self-organization of patterns and structures in open systems, which are far from thermodynamic equilibrium. The very word ‘synergetics’ was apparently coined by the architect and philosopher R. B. Fuller (the same person after whom the C_{60} molecule was later called ‘fullerene’). The books by Prigogine and Haken were published in the USSR in Russian translations and became very popular in the 1980s.

The second group to mention is that of N. Bushe and I. Gershman in Moscow. Their results were summarized in the book edited by Fox-Rabinovich & Totten (2006). In 2002, Bushe won the most prestigious tribology award, the Tribology Gold Medal, for his studies of tribological compatibility and other related effects. The third group is that of Garkunov (2004) and co-workers, who claimed the discovery of the synergetic ‘non-deterioration effect’, also called ‘selective transfer’. Garkunov’s mostly experimental research also received international recognition when he was awarded the 2005 Tribology Gold Medal ‘for his achievements in tribology, especially in the fields of selective transfer’.

In the English language literature, the works by Klamecki (1980) were the first to use the concepts of non-equilibrium thermodynamics to describe friction and wear. His work was extended by Zmitrowicz (1987), Dai *et al*. (2000), Doelling *et al*. (2000) and others. An important entropic study of the thermodynamics of wear was conducted by Bryant *et al*. (2008), who introduced a degradation function and formulated the degradation–entropy generation theorem in their approach intended to study the friction and wear in combination. They note that friction and wear, which are often treated as unrelated processes, are in fact manifestations of the same dissipative physical processes occurring at sliding interfaces. The possibility of the reduction of friction between two elastic bodies due to a pattern of propagating slip waves was investigated by Adams (1998) and Nosonovsky & Adams (2001), who used the approach of the theory of elasticity.

A completely different approach to friction-induced self-organization is related to the theory of dynamical systems and involves the investigation of friction test results as time series. Since the 1980s, it has been suggested that a specific type of self-organization, called ‘self-organized criticality’ (SOC), plays a role in diverse ‘avalanche-like’ processes, such as the stick–slip phenomenon during dry friction. The research of Zypman and Ferrante (Zypman *et al.* 2003; Adler *et al.* 2004; Buldyrev *et al.* 2006; Fleurquin *et al*. 2010) and others deals with this topic.

A different approach based on the theory of dynamical systems was suggested by Kagan (2010), which uses the so-called Turing systems (the diffusion–reaction system) to describe the formation of spatial and time patterns induced by friction. Nosonovsky & Bushan (2009, 2010) and Nosonovsky *et al*. (2009) suggested treating self-lubrication and surface healing as manifestations of self-organization. They noted that the orderliness at the interface can increase (and, therefore, the entropy is decreased) at the expense of the entropy either in the bulk of the body or at the microscale. They also suggested that self-organized spatial patterns (such as interface slip waves) can be studied by the methods of the theory of self-organization.

The reduction of friction and wear due to self-organization at the sliding frictional interface can lead to self-lubrication, i.e. the ability to sustain low friction and wear without the external supply of a lubricant. Since lubricants constitute environmental hazards, while friction and wear often lead to heat and chemical contamination of the environment, self-lubrication has potential for green tribology. Self-lubrication is also common in living nature, and therefore it is of interest to scientists and engineers who are looking for a biomimetic approach, which also has potential for green tribology.

In this paper, I present a model of the formation of self-organized patterns during friction through destabilization of the steady state, and discuss the methods of investigating such patterns and their relation to self-lubrication and environment-friendly tribology.

## 2. Friction-induced self-organization

In this section, I will discuss phenomenological observations of friction-induced self-organization, suggest a model for the transition from stationary sliding to that with self-organized structures and review several examples.

### (a) Qualitative studies

Bershadsky (1992) was apparently the first to pay attention to friction-induced self-organization and to suggest a classification of various self-organization effects. In the search for self-organization during friction, he investigated quite a diverse range of processes and phenomena: autohydrodynamic effects, the evolution of microtopography, the formation of chemical and convective patterns and the oscillation of various parameters measured experimentally during friction. Some of these phenomena had well-investigated principles and mechanisms, while others were studied in a phenomenological manner, so it was not possible to approach them all in a uniform manner. He, therefore, suggested that the state and evolution of a self-organized tribosystem might be described using different methods, including the equations of motion, a statistical description, measurement of certain parameters, etc. Depending on the method of description, different features of self-organization (‘synergism’) were observed, but in most situations a self-regulated parameter existed and the governing principle or target function could be identified.

Nosonovsky & Bhushan (2009, 2010), Nosonovsky *et al.* (2009) and Nosonovsky (2010) suggested entropic criteria for friction-induced self-organization on the basis of the multi-scale structure of the material (when self-organization at the macroscale occurs at the expense of deterioration at the microscale) and coupling of the healing and degradation thermodynamic forces. Table 1 summarizes his interpretation of various tribological phenomena that can be interpreted as self-organization. In addition, self-organization is often a consequence of the coupling of friction and wear with other processes, which creates a feedback in the tribosystem.

In the consequent subsections, I will discuss quantitative criteria of friction-induced self-organization and concentrate on three examples: the formation of a protective film at the interface (a spatial ‘chemical’ pattern), the adjustment of sliding surfaces during the running-in transient period (steady-state microtopography) and friction-induced elastic vibrations (patterns).

### (b) Stability criteria for the transition from the steady-state to the self-organized regime

In non-equilibrium (or irreversible) thermodynamics, a process that does not depend on time is called stationary, whereas a process that does not depend on the spatial coordinates is called homogeneous. A process that is both homogeneous and stationary is called equilibrium, whereas a process that is either non-stationary or inhomogeneous is called non-equilibrium. When a non-equilibrium process, which can be characterized by a parameter *q* (a so-called generalized coordinate), occurs, a generalized thermodynamic force *X* that drives the process can be introduced in such a manner that the work of the force is equal to d*Q*=*X*d*q*. The flux (or flow rate) is associated with the generalized coordinate, so the entropy production rate per unit time is . For sliding friction of a body that is characterized by its position *x*, the flow rate is equal to the sliding velocity *J*=*V* , and the thermodynamic force is equal to the friction force *X*=*μW*/*T*.

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
2.1
where is the second variation of entropy production rate (Fox-Rabinovich *et al*. 2007; Nosonovsky & Bhushan 2009) and *k* is the number of generalized forces and flows. Equation (2.1) states that the energy dissipation per unit time in 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 solution becomes unstable and transition to the self-organized solution with patterns can occur. Equation (2.1) is valid for a wide range of interactions, including mechanical, thermal and chemical, but the corresponding terms in the entropy production rate should be considered.

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 d*S*=d*Q*/*T* (figure 1*a*). Consider first the situation when the production of entropy depends on the sliding velocity *V* . The rate of entropy production is given by
2.2
where *W* is the normal load and *μ*(*V*) is the coefficient of friction. The stability condition is given by
2.3
If the slope of the *μ*(*V*) curve (the partial derivative *μ*′_{V}≡∂*μ*/∂*V*) is negative, then the steady-state sliding becomes unstable (figure 1*b*), understandably so, because the decrease of friction with increasing force leads to the increase of sliding velocity and to a further increase of friction, and thus creates a positive feedback loop (figure 1*c*). There are many practical examples of decreasing coefficient of friction with increasing sliding velocity (so-called ‘negative viscosity’) due to time-dependent deformation (viscoelastic, elastoplastic, creep) at the interface, which leads to an increasing real area of contact between the two bodies and, therefore, increasing friction (Nosonovsky & Adams 2004).

Suppose that one contacting material has microstructure characterized by a certain parameter *ψ*, such as, for example, the size of reinforcement particles in a composite material. Values of *ψ* such that *μ*′_{V}(*ψ*)>0 correspond to steady-state sliding. However, *μ*′_{V}(*ψ*)=0 corresponds to destabilization of the steady-state solution. As a result, a new equilibrium position will be found with a lower value of *μ*.

Suppose now that the coefficient of friction depends also on a microstructure parameter *ϕ*, such as the thickness of the interface film (figure 2*a*). The difference between *ψ* and *ϕ* is that the parameter *ψ* is constant (the composition of the material does not change during friction), whereas the parameter *ϕ* 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
2.4
If the stability condition is violated for a certain value of *ϕ*, 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 *μ*′_{ϕ}(*ψ*,*ϕ*)=0. Note that equation (2.4) becomes equation (2.3) if *ϕ*=*V* . At this point, I will not discuss the question of which particular thermodynamic force is responsible for the growth of the film.

Since the conditions of the formation of such a protective film are of interest, consider now the limit of a thin film (*ϕ*→0). With increasing film thickness the value of *μ* changes from that for 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 *ψ* (figure 2*b*). The critical value, *ψ*_{cr}, corresponds to *μ*′_{ϕ}(*ψ*,0)=0. For the size of reinforcement particles finer than *ψ*_{cr}, the bulk (no film, *ϕ*=0) values of the coefficient of friction are lower than the values of the film. That can lead to a sudden destabilization (formation of a film with thickness *ϕ*_{0}) and reduction of friction to the value of *μ*(*ψ*,*ϕ*_{0}) as well as wear reduction. Here I do not investigate the question 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 (Gershman & Bushe 2006).

An experimental example of such a sudden decrease of friction and wear with a gradual decrease of the size of reinforcement particles, which could be attributed to destabilization, is presented in figure 3 for Al_{2}O_{3}-reinforced Al matrix nanocomposite friction and wear tests (steel ball on disc in ambient air; Jun *et al*. 2006). The abrupt decrease of friction and wear occurs for reinforcement particles smaller than *ψ*_{cr}=1 μm in size and can be attributed to the changing sign of the derivative *μ*′_{ϕ}(*ψ*_{cr},0)=0, although additional study is required to prove this.

The model suggested in this section can easily be generalized for the case of several parameters *ψ*_{k} and *ϕ*_{k}. In addition, if heat conduction is taken into account, the rate of entropy production per unit interface area is given by (Fox-Rabinovich *et al.* 2007)
2.5
where *λ* is the thermal conductivity. Note that *S*_{i} in equation (2.5) is entropy per unit surface area and is thus measured in joules per kelvin per square metre, unlike the total entropy in equation (2.2), which is measured in joules per kelvin (Nosonovsky & Bhushan 2009). Now the stability condition is given by
2.6
and destabilization occurs at and .

### (c) Running-in

When friction is initiated, there is a certain transient period, referred to as ‘running-in’, during which friction and wear rate decrease to their stationary values. Furthermore, when the load or sliding velocity changes, the friction force usually increases at first, and then decreases to the steady-state value (Blau 1989). This is an experimental observation, and it is not obvious at all *a priori* why the opposite tendency (increasing friction during the transient period) is almost never observed (figure 4).

During the running-in period, the surfaces roughness of the contacting solid surfaces changes, until it reaches a certain equilibrium value and thus the adjustment of the surfaces to each other occurs. This process can be viewed as self-organization that leads to minimized energy dissipation and thus minimum friction and wear. However, the particular mechanism of this process remains to be explained.

Assume that one of the surfaces in contact is harder than the other, so most wear occurs on the softer surface. The surface roughness of the softer surface at any moment of time is characterized by a certain distribution of microtopography parameters. For simplicity, I will assume that the surface roughness is sufficiently characterized by only one parameter, *R*, for example, the average roughness of the rough profile of the softer material, and this parameter is identified with *ϕ* from equation (2.4). In §2*b*, the mechanism that leads to the change of *ϕ* was not addressed and it was assumed that there is such a mechanism that leads to the minimum entropy rate for the system. Here a simple phenomenological model is suggested for the microtopography evolution and the coefficient of friction (Mortazavi *et al.* 2010):
2.7
and
2.8
The model assumes that there are two mechanisms involved in friction, wear and microtopography evolution, i.e. adhesion and deformation. For the deformation mechanism, the coefficient of friction and wear rate grow with increasing roughness, whereas for the adhesion mechanism, friction and wear grow with decreasing roughness. A linear proportionality for *R* and 1/*R* is assumed, with *C*_{def} and *C*_{adh} being the corresponding proportionality constants. Roughness increases due to the adhesional wear and decreases due to the deformational wear, so that there is a minimum, and the corresponding proportionality constant is *A*. Using *R*=*ϕ* and substituting equation (2.7) into equation (2.4) yields
2.9
The stationary point corresponding to minimum friction is given by
2.10
Owing to the feedback loop present in the system (figure 5*a*), there are two competing processes, and, as a result, roughness tends to decrease to its equilibrium value given by equation (2.10), which also corresponds to the minimum coefficient of friction (figure 5*b*).

The natural parameter that characterizes the degree of surface disorder is the Shannon (information) entropy of a rough profile,
2.11
where *p*_{j} is the probability of appearance of a height in the bin *j*, and *B* is the total number of bins. For a less disordered profile, the value of *S* is lower (Fleurquin *et al.* 2010; Nosonovsky 2010). For example, for a smooth surface profile with a constant height *y*(*x*)=*y*_{0}, the value of the probability of the bin *n* that includes *y*_{0} will be unity, *p*_{n}=1, whereas *p*_{j}=0 otherwise, so the total entropy is *S*=0, since both and the limit of . For a profile consisting of two height levels with equal probabilities *p*_{1}=*p*_{2}=0.5, the entropy is . The entropy rate of a stochastic process can be a more adequate parameter to investigate spatial patterns. The interplay between the surface roughness, dissipation and mass transfer is of interest (table 2).

### (d) Self-organized elastic structures

In the preceding sections, I have discussed the self-organization of interface films and microtopography evolution. Another type of self-organized microstructure is elastic slip waves. The mathematical formulation of quasi-static sliding of two elastic bodies (half-spaces) with a flat surface and a frictional interface is a classical contact mechanics problem. Interestingly, the stability of such sliding was not investigated until the 1990s, when Adams (1995) showed that the steady sliding of two elastic half-spaces is dynamically unstable, even at low sliding speeds. Steady-state sliding was shown to give rise to a dynamic instability in the form of self-excited oscillations with exponentially growing amplitudes. These oscillations were confined to a region near the sliding interface and could eventually lead to either partial loss of contact or to propagating regions of stick–slip motion (slip waves). The existence of these instabilities depends upon the elastic properties of the surfaces; however, it does not depend upon the friction coefficient, nor does it require a nonlinear contact model. The same effect was predicted theoretically by Nosonovsky & Adams (2004) for the contact of rough periodic elastic surfaces.

It is well known that two types of elastic waves can propagate in an elastic medium: shear and dilatational waves. In addition, surface elastic waves (Rayleigh waves) may exist, the amplitude of which decreases exponentially with distance from the surface. For two slightly dissimilar elastic materials in contact, generalized Rayleigh waves (GRWs) may exist at the interface zone. The instability mechanism described above is essentially one of GRW destabilization; that is, when friction is introduced, the amplitude of the GRW is no longer constant but grows exponentially with time.

The stability analysis involves the following scheme. First, a steady-state solution should be obtained. Second, a small arbitrary perturbation of the steady-state solution is considered. Third, the small arbitrary perturbation is presented as a superposition of modes, which correspond to certain eigenvalues (frequencies). Fourth, the equations of the elasticity (Navier equations) with the boundary conditions are formulated for the modes, and solved for the eigenvalues. Positive real parts of the eigenvalues show that the solution is unstable.

For the GRWs, the two-dimensional displacement field at the interface is given by
2.12
where *k* stands for the *x*- or *y*-component of the displacement field at the interface (*y*=0), *A*_{k} is the complex amplitude, *k*/*l* is the wavenumber, *l* is the wavelength of the lowest wavenumber and *Λ* is the complex frequency. It can be shown that for the frictionless case *Λ*=±i*λ* is pure imaginary and thus, for real *A*_{k}, the displacement is a propagating generalized Rayleigh wave, . It can be shown also that, if small friction with the coefficient *μ* is present, then *Λ*=±(i*λ*+*αμ*), where *α* is a real number, and thus one root of *Λ* always has a positive real component, leading to instability (Adams 1995; Ranjith & Rice 2001). As a result, the amplitude of the interface waves grows with time. In a real system, of course, the growth is limited by the limits of applicability of the linear elasticity and linear vibration theory. This type of friction-induced vibration may be, at least partially, responsible for noise (such as car brake squeal) and other effects during friction, which are often undesirable (Nosonovsky & Adams 2004).

Whereas the GRWs occur for slightly dissimilar (in the sense of their elastic properties) materials, for very dissimilar materials, waves would be radiated along the interfaces, providing a different mechanism of pumping the energy away from the interface (Nosonovsky & Adams 2001). These waves can form a rectangular train of slip pulses that propagates, so that the two bodies shift relative to each other in a ‘caterpillar’ or ‘carpet-like’ motion (figure 6). This microslip can lead to a significant reduction of the observed coefficient of friction, as the slip is initiated at a shear load much smaller than *μW* (Nosonovsky & Adams 2001; Bhushan & Nosonovsky 2003).

The motion is observed as the reduction of the coefficient of friction (in comparison with the physical coefficient of friction *μ*) to the apparent value of *μ*_{app}=*q*/*p*, where *q* is the applied shear force per unit area and *p* is the normal pressure. The slip pulses can be treated as ‘secondary structures’ self-organized at the interface, which result in the reduction of the observed coefficient of friction. Note that the analysis in this case remains linear and it just shows that the equations of elasticity with friction are consistent with the existence of such waves. The amplitude of the slip waves cannot be determined from this analysis, since they are dependent on the initial and boundary conditions. In order to investigate whether slip waves will actually occur, it is important to ask the question whether it is energetically profitable for them to exist. To that end, the energy balance should be calculated for the work of the friction force and the energy dissipated at the interface and radiated away from the interface. A stability criterion based on equation (2.4) can be used.

### (e) Self-organized criticality

The methods in the preceding subsections describe the onset of instability, which can lead to the self-organization of spatial and temporal structures. The linear stability analysis is independent of the amplitudes of small vibrations and in the case of instability it predicts an exponential growth of the amplitudes until they become so large that the linear analysis cannot apply any more. It is much more difficult to perform the nonlinear analysis to find the amplitudes and actual finite motion. Several approaches have been suggested.

One is that a specific type of self-organization, known as SOC, plays a role in frictional systems. SOC is a concept in the theory of dynamic systems that was introduced in the 1980s (Bak 1996). The best-studied example of SOC is the ‘sandpile model’, representing grains of sand randomly placed into a pile until the slope exceeds a threshold value, transferring sand into the adjacent sites and increasing their slope in turn. There are typical external signs of an SOC system, such as power-law behaviour (the magnitude distribution of the avalanches) and a ‘one-over-frequency’ noise distribution. The concept has been applied to such diverse fields as physics, cellular automata theory, biology, economics, sociology, linguistics and others (Bak 1996).

In the case of dry frictional sliding, it has been suggested that a transition between the stick and slip phases during dry friction may be associated with SOC, since the slip is triggered in a similar manner to the sandpile avalanches and earthquake slides. Zypman *et al*. (2003) showed that in a traditional pin-on-disc experiment, the probability distribution of slip zone sizes follows a power law. In a later work, the same group found nanoscale SOC-like behaviour during atomic force microscopy studies of at least some materials (Zypman *et al.* 2003; Buldyrev *et al.* 2006). Thus ‘stick’ and ‘slip’ are two phases, and the system tends to achieve the critical state between them: in the stick state elastic energy is accumulated until slip is initiated, whereas energy release during slip leads, again, to the stick state. Entropic methods of analysing frictional systems with SOC were used by Zypman and co-workers (Fleurquin *et al.* 2010), who calculated the Shannon entropy of the surface profile and showed that the entropy of the profile decreased, indicating self-organization.

A different approach was suggested by Kagan (2010), who studied friction as a reaction–diffusion system (a Turing system) with the exchange of mass and heat. A Turing system is defined as the following system of *n* partial differential equations:
where Δ is the Laplace operator, *u*=(*u*_{1},*u*_{2},…,*u*_{n}) is interpreted as a vector of reagent concentrations, and *D* is a diagonal diffusion matrix. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of travelling waves and wave-like phenomena as well as other self-organized patterns like stripes, hexagons, etc. During friction, the exchange of heat (due to the friction) and matter (due to the wear) occurs, and thus a tribosystem with coupled friction and wear can be viewed as a Turing system.

## 3. Self-lubrication

The term ‘self-lubrication’ implies the ability of details and components to operate without lubrication and refers to several methods and effects, in addition to the above-mentioned, that reduce friction or wear. Among these methods are the deposition of self-lubricating coatings that are either hard (to reduce wear) or with low surface energy (to reduce adhesion and friction). Besides coatings, self-lubrication refers to the development of metal-, polymer- or ceramic-based composite self-lubricating materials, often with a matrix that provides structural integrity and a reinforcement material that provides low friction and wear. Nanocomposites have become a focus of this research, as well as numerous attempts to include carbon nanotubes and C_{60} fullerene molecules (Rapoport *et al.* 2003). Simple models assume that these large molecules and nanosized particles serve as ‘rolling bearings’ that reduce friction; however, it is obvious now that the mechanism can be more complicated and can involve self-organization.

Dynamic self-organization is thought to be responsible for self-lubrication in atomic force microscopy experiments with atomic resolution. A protective layer can also be formed due to a chemical oxidation reaction or reaction with water vapour. For example, a self-lubricating layer of boric acid (H_{3}BO_{3}) is formed as a result of the reaction of water molecules with a B_{2}O_{3} coating (Donnet & Erdemir 2004). Another type of self-lubricating material involves lubricant embedded into the matrix, e.g. inside microcapsules that rupture during wear and release the lubricant. Surface microtexturing that provides holes and dimples that serve as reservoirs for the lubricant can be viewed as another method of providing self-lubrication. In addition, I should mention that self-lubrication is also observed in many biological systems (e.g. human joints) and that the term ‘self-lubrication’ is used also in geophysics where it refers to anomalously low friction between tectonic plates that is observed during some earthquakes.

## 4. Surface healing

Self-healing or surface-healing materials constitute a special class of novel smart materials, usually composites, which are capable of repairing minor damage caused by wear and deterioration (van der Zwaag 2009). Self-healing is a non-equilibrium process. When a system exchanges mass and energy with its surroundings, various irreversible processes inside the system may interact with each other. These interactions are called thermodynamic couplings and they provide a mechanism for a process without its primary driving force or they may even move the process in a direction opposite to the one imposed by its own driving force. For example, in thermodiffusion a substance diffuses because of a temperature gradient rather than a concentration gradient. When a substance flows from a low- to a high-concentration region, it must be coupled with a compensating process. The principles of thermodynamics allow the progress of a process without or against its primary driving force *X*_{n} only if it is coupled with another process. In the widely accepted linear approximation, the thermodynamic flows are related to the forces by
4.1
where *L*_{kn} are Onsager coefficients (De Groot & Mazur 1962). Many linear empirical laws of physics, such as Ohm’s law of electrical resistance, Fourier’s law of heat conduction or Fick’s law of diffusion, which relate thermodynamic flows *J*_{k} with thermodynamic forces *X*_{n}, are consequences of the equations of linear thermodynamics given in equation (4.1). The heat production per unit time is given by
4.2
and the rate of entropy production is
4.3
which is a linear function of thermodynamic flows.

In most self-healing schemes, the self-healing material is driven away from thermodynamic equilibrium either by the deterioration process itself or by an external intervention, such as heating. After that, the composite slowly restores thermodynamic equilibrium, and this process of equilibrium restoration drives the healing.

In order to characterize degradation, it is convenient to introduce a so-called ‘degradation parameter’ *ξ*, to represent, for example, the wear volume or the total volume of cracks. When a self-healing mechanism is embedded in the system, another generalized coordinate, the healing parameter *ζ*, can also be introduced, to represent, for example, the volume of released healing agent. The corresponding thermodynamic flows are linearly related to the thermodynamic forces (equation (4.1)) and entropy rates (equation (4.3)). The generalized degradation and healing forces are external forces that are applied to the system, and flows are related to the forces by the governing equations
4.4
where *L*, *M*, *N* and *H* are corresponding Onsager coefficients (figure 7). The degradation force *X*^{deg} is an externally applied thermodynamic force that causes the degradation. In most self-healing mechanisms, the system is placed out of equilibrium and the restoring force *X*^{heal} emerges. Since the restoring force is coupled with the degradation parameter *ξ* by the negative coefficients *N*=*M*, it also causes degradation decrease or healing.

The surface is the most vulnerable part of a material sample, and, not surprisingly, deterioration often occurs at the surface (wear, fretting, etc.) and is induced by friction. The empirical Coulomb (or Amontons–Coulomb) law of friction states that the dry friction force *F* is linearly proportional to the normal load force *W*,
4.5
where *μ* is the coefficient of friction, which is independent of load, sliding velocity and the nominal area of contact. Unlike many other linear empirical laws, the Coulomb law cannot be directly deduced from the linear equations of non-equilibrium thermodynamics, such as equation (4.1). Indeed, in the case of dry or lubricated friction, the sliding velocity is the thermodynamic flow, *V* =*J*, which, in accordance with equation (2.4), should be proportional to the friction force *F*=*X* (as is the case for viscous friction), so that the energy dissipation rate is given by the product of the thermodynamic flow and force,
4.6
However, the Coulomb friction force is independent of sliding velocity.

To overcome this difficulty, consider the normal degree of freedom *y*, in addition to the sliding coordinate *x* (figure 8). Introducing the normal degree of freedom is a standard procedure in the study of dynamic friction, where normal vibrations are often coupled with the in-plane vibrations. Further, define the generalized flows as , and forces as *X*_{1}=*F*, *X*_{2}=*W* (Nosonovsky 2010).

The thermodynamic equations of motion (4.1) immediately yield the law of viscous friction in the form of 4.7 Note that equation (4.7) is valid, in the general case, for the bulk of a three-dimensional deformable medium and not necessarily for the interface between two solids.

The interface between sliding bodies has highly anisotropic properties, because a small force in the direction of the interface causes large displacements, whereas a small force in the normal direction causes only small displacements. To compensate for this anisotropy, substitute coordinates using a small parameter *ε* as (*x*,*y*)→(*εx*,*y*). The force–displacement relationships are now given by
4.8
In the limit *ε*→0, equation (4.8) yields
4.9
*Any* velocity satisfies equation (4.9), provided *L*_{11}*F*+*L*_{12}*W*=0, which is exactly the case of Coulomb friction if *μ*=−*L*_{12}/*L*_{11} (Nosonovsky 2010).

Thus friction can be included into the linear framework of non-equilibrium thermodynamics as the force driving the degradation process (e.g. wear). The force that leads to healing should be related to the material microstructure in order to design an optimized material with the highest potential for self-healing and self-lubrication.

## 5. Conclusions

Despite the fact that self-organization during friction has received relatively little attention in the tribological community so far, it has the potential for the creation of self-healing and self-lubricating materials, which are important for green or environment-friendly tribology. The principles of the thermodynamics of irreversible processes and of the nonlinear theory of dynamical systems are used to investigate the formation of spatial and temporal structures during friction. These structures lead to friction and wear reduction (self-lubrication). The self-organization of these structures occurs through the destabilization of stationary sliding. The stability criteria involve minimum entropy production. A self-healing mechanism, which provides the coupling of degradation and healing, can be embedded into the material. Structure–property relationships were formulated and can be used for the optimized design of self-lubricating and self-healing materials and surfaces for various ecologically friendly applications and green tribology.

## Acknowledgements

The author acknowledges the support of the INSIC TAPE and UWM Research Growth Initiative research grant. In addition, he would like to thank Mr Chuannfeng Wang and Mr Vahid Mortazavi from the University of Wisconsin-Milwaukee for their help in producing figures 4 and 5.

## Footnotes

One contribution of 11 to a Theme Issue ‘Green tribology’.

- © 2010 The Royal Society