Pattern Formation and Turing Systems

As discussed in the previous sections, under certain circumstances friction can also result in self-organization or formation of patterns and structures at the frictional interface. These self-organized patterns or ''secondary structures'' [25] can include a broad range of phenomena, such as in-situ formed tribofilms [5], patterns of surface topography, and other interfacial patterns including propagating trains of stick and slip zones formed due to dynamic sliding instabilities [39, 49]. Moreover, experimental observations demonstrate various structures formed by friction and wear, such as the Schallamach [42] and slip waves [69], wave-like topography patterns [46], honeycomb-like and other structures [40].

Frictional sliding generates heat, and heat tends to lead to the thermal expansion of materials. This can result in so-called thermoelastic instabilities (TEI), which have been studied extensively in the past 30 years [7]. The effect of TEI on sliding is similar to that of AMI discussed in Sect. 3.3.2. Besides heat generation, friction can be coupled with other effects, such as wear, which can lead to similar instabilities. In addition, the coefficient of friction can depend upon the sliding velocity, load, and interface temperature, which can also lead to complex dynamic behavior [52].

The so-called RD systems and their important class called the ''Turing systems'' constitute a different type of self-organization mechanism [33, 47]. These systems can describe certain types of friction-induced pattern formation involving heat transfer and diffusion-like mass transfer due to wear. While AMI and TEI involve wave propagation (hyperbolic) partial differential equations (PDE), which describes dynamic behavior of elastic media, the RD system involves parabolic PDE, typical for diffusion and heat propagation problems. The RD systems describe evolution of concentrations of reagents in space and time due to local chemical reactions and the diffusion of the product of reactions [39]. The RD system of PDEs is given by

where Wj is the vector of concentrations, fi represent the reaction kinetics and djj and A are a diagonal matrix of diffusion coefficients and Laplace operator, respectively. Alan Turing [65] showed that a state that is stable in the local system can become unstable in the presence of diffusion, which is counterintuitive, since diffusion is commonly associated with a stabilizing effect. The solutions of RD equations demonstrate a wide range of behaviors including the traveling waves and self-organized patterns, such as stripes, hexagons, spirals, etc.

While parabolic RD equations cannot describe elastic deformation, they may be appropriate for other processes, such as viscoplastic deformation and interface film growth. For a system of two components, u and v, Eq. 3.21 has the following form:

where f and g are the reaction kinetics functions and dy is a diagonal matrix (d11 = du and d22 = dv). Suppose u represents the non-dimensional temperature at the sliding interface and v is the local slip velocity, also non-dimensional. The non-dimensional values of u, v, x, t, and other parameters are obtained from the dimensional values by division of the latter by the corresponding scale parameters. Then Eq. 3.22 is interpreted as the description of heat transfer along the interface, and Eq. 3.23 describes the flow of viscous material along the interface. In this paper, we discuss several types of kinetic functions which can lead to the formation of periodic patterns.

One of the standard forms for functions f(u, v) and g(u, v) based on transferring original RD equations and proper scaling was proposed to be [19, 20]:

In order to investigate the possibility of pattern formation in a Turing system, a stability analysis should be performed [47]. A RD system exhibits diffusion-driven instability or Turing instability if the homogeneous steady state is stable to small perturbations in the absence of diffusion; however, it is unstable to small spatial perturbations when diffusion is introduced [49]. Diffusion results in the spatially inhomogeneous instability and determines the spatial patterns that evolve.

From the linear stability analysis one can show when a solution with pattern formation (Turing pattern) can exist [19, 47, 49]. The results of the stability analysis shown depend on the parameters of Eqs. 3.22 and 3.23 (i.e., dv, du, a, b and y); three regions of stability could be identified. In region 1, the steady state is stable to any perturbation. In region 2, the steady state exhibits an oscillating instability. In region 3, the steady state is destabilized by inhomogeneous perturbations, which is ''Turing space''.

In this section we discuss several exemplary cases of parameter values and initial conditions which can lead to patterns. Cases 1 and 2 are the classical

f (u, v) = y(a — u + u2v) g(U; v) = C(b — u2 v)

examples of Turing systems, whereas case 3 is justified by the frictional mechanism of heat generation and mass transfer.

Examples of Turing patterns for two different cases with the following values of the parameters in Eqs. 3.22 and 3.23 are shown in Figs. 3.8 and 3.9. For case 1:

c = 10; 000; dv = 20; du = 1, a = 0.02 and b = 1.77 (3.26)

and considering a random distribution function as the initial conditions for both values of u and v. For case 2:

C = 10,000, dv = 20, du = 1, a = 0.07 and b = 1.61 (3.27)

For initial conditions, we considered the following harmonic functions u(x, y) = 0.919145 + 0.0016 cos(2p(x + y)) (3.28)

The results in Figs. 3.8 and 3.9 present five consecutive snapshots of systems presented by Eqs. 3.22 and 3.23 corresponding to different time steps (t = 0, 0.001, 0.005, 0.01, and 0.1). It is observed in both cases that the initially random pattern (t = 0) evolves finally, into a so-called hexagonal-like (or honeycomb) pattern (t = 0.1), which indicates that the pattern formation occurs.

Whereas the two first cases showed that the model is capable of capturing the self-organized patterns in Turing systems, in the third case Mortazavi and Noso-novsky [47] tried to use more specific functions of reaction kinetics, which were expected to characterize friction-induced reaction mechanisms. They assumed functions f(u, v) and g(u, v) to be in the following forms:

where i0 = a1u + b1v is the coefficient of friction dependent on the temperature u and local slip velocity v, and the non-dimensional coefficients a1 and b1 and w0 are constant. The function g(u, v) characterizes the rheological properties of the material and depends on its viscous and plastic properties.

Such interpretation of Eqs. 3.13-3.22 can be used if the growth of a tribo-film (a thin interfacial layer activated by friction) is considered. Whereas u still represents the interfacial temperature, v can be interpreted as the non-dimensional thickness of the tribofilm formed at the interface. The tribofilm can grow due to the material transfer to the interface via diffusion activated by friction, due to precipitation of a certain component (e.g., a softer one) in an alloy or composite material, due to a chemical reaction, temperature gradient, etc. For example, during the contact of bronze versus steel, a protective Cu tribofilm can form at the

Fig. 3.8 Time evolution of amounts of u and v i 0.01, and 0.1 (First case) [47]
different time steps, at t = 0, 0.001, 0.005,

interface, which significantly reduces the wear. Such in situ tribofilm has protective properties for the interface since it is formed dynamically and compensates the effect of wear. Furthermore, if wear is a decreasing function of the tribofilm

Fig. 3.9 Time evolution of u and v values in different time steps, t = 0, 0.001, 0.005, 0.01, 0.05 and 0.1 (Second case) [47]

thickness, it is energetically profitable for the film to grow, since a growing film reduces wear and further stimulates its growth forming a feedback loop, until a certain equilibrium thickness is attained. Therefore, such tribofilms can be used for

Fig. 3.10 Time evolution of u and v values in two different time steps (Third case, using Eqs. 3.30 and 3.31) [47]

machine tool protection and other applications, as discussed in the literature [5]. The growth of the film is governed by interfacial diffusion and by a local kinetic function g(u, v) dependent upon the temperature and local film thickness.

Use the following equation as a temperature distribution for initial conditions

u(x, y) = 0.002 cos (2p(x + y)) + 0.01 ^ cos(2pjx) (3-33)

j=i and random distribution function as a initial roughness for v(x, y) and, moreover, consider the following values for parameters of Eqs. 3.22 and 3.23

W0 = 102, ai = 10~4, b1 = 10~4, i0 = 5 x 10"1,

The obtained results for different time steps are shown in Figs. 3.10 and 3.11 considering Eqs. 3.31 and 3.32 as g(u, v), respectively. While the patterns found in Figs. 3.10 and 3.11 are not the same as the patterns found in the two previous cases, similar trends in the evolution of tribofilm thickness (v) from an initially random distribution of roughness to a more organized and more patterned distribution could be observed. However, the investigation of pattern formation based on solving complete three-dimensional heat and mass transfer equations in tribofilm is needed to show a more realistic picture of how and under what conditions such patterns could occur.

The modeling analysis in the preceding sections shows that properly selected functions f(u, v) and g(u, v) can lead to pattern formation. The question remains as to whether any experimental data can be interpreted as friction-induced patterns formed

Fig. 3.11 Time evolution of u and v values in different time steps (Third case, using Eqs. 3.30 and 3.32) [47]

by the RD mechanism? It would be appropriate to look for this type of self-organization in processes involving viscoplastic contact or diffusion-dominated effects, e.g., in-situ tribofilms. There are several effects which can be interpreted in this way. The so-called ''secondary structures'' can form at the fictional interface due to the self-organization [25], and some of these structures can have a spatial pattern. The so-called ''selective transfer'' discussed in Sect. 3.2.4 could be another example.

The selective layer (an in situ tribofilm due to the selective transfer) formed during friction between steel and a copper alloy (bronze) was investigated experimentally by Ilie and Tita [32]. In the presence of glycerin or a similar lubricant, the ions of copper were selectively transferred from bronze to the frictional interface forming the copper tribofilm. This copper was different in its structure from the copper that falls out through normal electrolytic procedures. Ilia and Tita [32] investigated the selective layer using the AFM and found that the layer formed a micro-island pattern with the size on the order of 1 im, rather than a uniform film of constant thickness. A schematic figure of such layer can be seen in Fig. 3.12.

Another important example of patterns formation reported in the literature is related to the self-adaptive mechanisms improving the frictional properties of hard coatings, e.g., during dry cutting, by tailoring their oxidation behavior [21, 22, 41]. Thus, boric acid formation on boron carbide is a potential mechanism for reaching ultra-low friction. Such mechanism uses the reaction of the boric oxide (B2O3) with ambient humidity (H2O) to form a thin boric acid (H3BO4) film. The low friction coefficient of boric acid is associated with its layered triclinic crystal structure [23, 63]. The layers consist of closely packed and strongly bonded boron, oxygen, and hydrogen atoms, but the layers are widely separated and attracted by van der Waals forces only. During sliding, these atomic layers can align themselves parallel to the direction of relative motion and slide easily over one another [22].

Fig. 3.12 Schematic presentation of the selective layer obtained by friction between a surface of steel and a surface of copper alloy (bronze) [47]

The tribological behavior of protective coatings formed by both ex-situ and in situ transfer films were studied by Singer et al. [63]. Coatings that exhibit long life in sliding contact often do so because the so-called ''third body'' forms and resides in the sliding interface. The concept of the ''third body'' as a separate entity, different from the two contacting bodies, is very similar to the concept of the tribofilm. Ex-situ surface analytical studies identified the composition and structure of third bodies and provided possible scenarios for their role in accommodating sliding and controlling friction. In situ Raman spectroscopy clearly identified the third bodies controlling frictional behavior during sliding contact between a transparent hemisphere and three solid lubricants: the amorphous Pb-Mo-S coating was lubricated by an MoS2 transfer film; the diamond-like carbon/nanocomposite (DLC/DLN) coating by a graphite-like transfer film; and the annealed boron carbide by H3BO3 and/or carbon couples. In situ optical investigations identified third body processes with certain patterns responsible for the frictional behavior [63].

TiB2 thin films are well known for their high hardness which makes them useful for wear-resistant applications. Mayrhofer et al. [43] showed that overstoichio-metric TiB2.4 layers have a complex self-organized columnar nanostructure precursor. Selected area electron diffraction (SAED) pattern from a TiB2.4 layer showed a texture near the film/substrate interface with increased preferred orientation near the film surface [43]. The film has a dense columnar structure with an average column diameter of * 20 nm and a smooth surface with an average RMS roughness essentially equal to that of the polished substrate surface, * 15 nm.

Aizawa et al. [5] investigated in situ TiN and TiC ceramic coating films utilized as a protective coating for dies and cutting tools. They found that chlorine ion implantation assists these lubricious oxide (TiO and TinO2n_i) films to be in situ formed during wearing. They also performed the microscopic analysis and observed worn surfaces and wear debris and found microscale patterns.

Lin and Chu [40] described Benard cell-like surface structures found from the observation of the Transmission electron microscopy (TEM) images of the scuffed worn surface as a result of lubricated steel versus steel contact. They attributed the cells to high temperatures (800°C) and very strong fluid convection or even evaporation occurring inside the scuffed surface. However, the possibility of diffusion-driven based pattern formation should not be ruled out.

The experimental evidences of pattern formation, which can, at least theoretically attributed to the RD mechanism, are summarized in Table 3.3. Further evidence is needed to rule out alternative explanations.

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