## Qualitative Studies

The earlier works on self-organization are characterized by qualitative, rather than quantitative analysis of self-organization mechanisms. For example, it is noted in many publications that typical tribosystems possess the qualities which, according to Prigogine, are required (but not sufficient) for the self-organization, for example, the system should be thermodynamically open, nonlinear, and it should operate far from the equilibrium.

Bershadsky [8] suggested a classification of various friction-induced self-organization effects. In search of self-organization during friction he investigates quite a diverse range of processes and phenomena—auto-hydrodynamic effects, the evolution of micro topography, 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, statistical description, measurement of a certain parameter, 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. Table 3.1 summarizes the features of "synergism" in various tribological phenomena and the corresponding governing principles and target function, based on Bershadsky [8].

An important entropic study of the thermodynamics of wear was conducted by Bryant et al. [12], who introduced a degradation function and formulated the Degradation-Entropy Generation theorem in their approach intended to study the friction and wear in complex. 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 [3] and Nosonovsky and Adams [49], who used the approach of the theory of elasticity.

Nosonovsky [51, 52], Nosonovsky and Bhushan [55], and Nosonovsky et al. [57] suggested entropic criteria for friction-induced self-organization on the basis of the multiscale structure of the material (when self-organization at the macro-scale occurs at the expense of the deterioration at the microscale) and coupling of the healing and degradation thermodynamic forces. Table 3.2 summarizes their interpretation of various tribological phenomena, which can be interpreted as self-organization. In addition, self-organization is often a consequence of coupling of friction and wear with other processes, which creates a feedback in the tribosystem.

In addition, self-organization is often a consequence of coupling of friction and wear with other processes, which create a feedback in the tribosystem. These "other processes'' may include radiation, electricity, ultrasound, electric field etc. Following Haken and Prigogine, Bershadsky considered self-organization as a general property of matter, which is complimentary to wear and degradation. These ideas, while interesting from the philosophical point of view, caused the criticism of the synergetics as not being a sufficiently "scientific" field in terms of quantitative analysis. It took several decades until the investigation of spatial and temporal pattern formation during friction found a foundation in the thermodynamics and the theory of dynamical systems.

### 3.2.2 Entropy During Friction and Dissipation

Before going through quantitative investigation of self-organization phenomena, we will discuss entropic methods of the description of friction and wear. Consider a rigid body sliding upon a flat solid surface with the sliding velocity V = dx/dt (Fig. 3.1). The normal load W is applied to the body and the friction force F = iW is generated [55]. The work of the friction force is equal to the dissipated energy, and, therefore, we will assume for now that all dissipated energy is converted into heat

Table 3.1 Self-organization effects in tribosystems [52]

Effect

Description of the state or evolution Features of synergism

Self-regulated parameter

Target function and/or governing principle

Auto-hydrodynamic effects (wedges, gaps, canyons)

Self-reproducing micro-topography, waviness

Equations of motion, competing processes for entropy and negentropy production

Equations of motion or kinetics

Bifurcation; self-excited vibrations and waves; feedback and target functions

Bifurcations; self-excited vibrations and waves

Gap thickness, temperature, and microtopography distributions Rough surface microtopography

Steady state microtopography of worn surfaces ("natural wear shape") Self-excited vibrations of wear, electric resistance, stresses, etc. Spatial or periodic chemical pattern

Periodic or concentric structures, such as Bénard cells Decrease in macrofluctuation of temperature, particle size and other parameters

Competing processes for entropy and negentropy (information) production

Feedback and target function Shape of the profile

Measurements of a parameter of the Instabilities and self-excited system (friction force, electrical resistance, wear rate, etc.)

Molecular, atomic, or dislocation structure

Molecular, atomic, or dislocation structure; Entropy is measured

Order-parameter dependent on generalized coordinate. Measurements of a parameter of the system oscillations of the measured parameter Large-scale ordered structures

Corresponding parameter

Secondary heterogeneity at the surface

Large-scale order structures; a sudden decrease in entropy production

Microfluctuations; phase transitions; instabilities and self-excited vibrations of the measured parameter

Minimum friction

Minimum energy dissipation; pressure or heat flow distribution Minimum energy dissipation

Minimum entropy production

Dissipative principles

Minimum entropy production

Sub-minimal friction

Table 3.2 Self-organization effects in tribosystems [51]

Effect

Mechanism/ driving force

Condition to initiate

Final configuration

Stationary microtopography distribution after running in In situ tribofilm formation

Slip waves

Self-lubrication

### Surface-healing

Feedback due to coupling of friction and wear Chemical reaction leads to the film growth Dynamic instability Embedded self-lubrication mechanism Embedded self-healing mechanism

Wear affects microtopography until it reaches the stationary value Wear decreases with increasing film thickness Unstable sliding

Thermodynamic criteria

Proper coupling of degradation and healing

Minimum friction and wear at the stationary microtopography

Minimum friction and wear at the stationary film thickness

Reduced friction

Reduced friction and wear

Reduced wear

The rate of entropy generation during friction is given by dS = lWV (3

It is noted that friction is a non-equilibrium process. 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 dQ = Xdq. The flux (or flow rate) J = q is associated with the generalized coordinate. For many linear processes the flow rate is linearly proportional to Y. For sliding friction, the flow rate J = V, and the thermodynamic force X = iW/T. Note that, for the Coulombian friction, J is not proportional to Y, which is the case for the viscous friction. Nosonovsky [52] discussed in detail the problem of bringing the linear friction in compliance with the linear thermodynamics.

The net entropy growth rate for frictional sliding of rigid bodies is given by Eq. 3.2. However, if instead of the net entropy, the entropy per surface area at the frictional interface is considered, the rate equation becomes more complicated. We consider the 1D flow of entropy near the infinite interface in the steady-state situation, and suppose the flow is equal to entropy generation. Heat dQ is generated at the interface in accordance with Eq. 3.1. For simplicity, we assume that all generated heat is dissipated in one of the two contacting bodies and ignore the division of heat between the two bodies. The heat is flown away from the interface in accordance with the heat conduction equation

oz where z is the vertical coordinate (distance from the interface), and X is the heat conductivity. Consider a thin layer near the interface with the thickness dz. The temperature drop across the layer is dT = (fiWV/X)dz. The ratio of the heat released at the interface, dQ, to that radiated at the bottom of the layer, dQ/, is equal to the ratio of the temperatures at the top and at the bottom of the layers dQ/ T - iWVdz/X

Therefore, the energy released at the subsurface layer of depth dz is given by [55]

Thus the entropy in the subsurface layer, dS/dt = dq/T, is given by dS (iWV)2 dt = XT2

Note that S in Eq. 3.6 is entropy per unit surface area and thus it is measured in JK- m- , unlike the total entropy Eq. 3.2, which is measured in JK-1 [ , ].

The difference between Eqs. 3.2 and 3.6 is that the latter takes into account the thermal conductivity and that in Eq. 3.2 gives the net entropy rate, while Eq. 3.6 gives the rate of entropy in the subsurface layer. Note that the form for the ther-modynamic flow is now J = iWV, and the thermodynamic force is X = iWVKXT2).

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