# Entropy in Tribology: in the Search for Applications

## Abstract

**:**

## 1. Introduction

_{1}to a body with temperature T

_{2}, the entropy grows by dS = (1/T

_{2}– 1/T

_{1})dQ. This provides a convenient mathematical formulation for the Second Law of thermodynamics, which states that heat does not spontaneously flow from a colder body to a hotter body or, in other words, that the entropy does not decrease, just dS ≥ 0. A more formal thermodynamic definition involves the energy of a system U(S,V,N) as a function of several parameters, including entropy, volume V, and the number of particles, N. Then temperature is defined as a partial derivative of the energy of the system with respect to entropy

_{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 among critically-minded Soviet intellectuals, since the “synergetic” studies claimed to suggest a general methodology to investigate physical, biological, information and social phenomena, which, in a sense, was (or at least, was perfected in this way by some scientists) an alternative to the official Soviet Marxist methodology. On the other hand, many more traditionally-minded theoretical physicists opposed the “synergetics,” since the synergetic studies often contained a lot of rhetoric and little practical quantitative and verifiable results [11], and even the term “pseudo-synergetics” was coined to refer to these speculative studied.

## 2. Entropy During Friction and Wear

#### 2.1. Friction and Dissipation

^{−1}m

^{−2}, unlike the total entropy in Equation 6, which is measured in JK

^{−1}[22,25].

^{2}).

#### 2.2. Wear and Entropy

**Table 1.**Entropy change during various dissipative processes (based on [16]).

Process | Entropy change |
---|---|

Adhesion | $dS=\frac{\gamma dA}{T}$, where γ is surface energy, A area |

Plastic deformation | $dS=\frac{{U}_{p}dV}{T}$, where U_{p} is the work per volume, V volume |

Fracture | $dS=\frac{\left(\frac{\partial U}{\partial a}-2\gamma \right)da}{T}$, where $\frac{\partial U}{\partial a}$ is the energy release rate, a is crack length |

Phase transition | $dS=\frac{dH}{T}$, where H is enthalpy |

Chemical reaction | $dS=\frac{{\displaystyle \sum _{react}{\mu}_{i}d{N}_{i}}-{\displaystyle \sum _{products}{\mu}_{i}d{N}_{i}}}{T}$, where N_{i} are numbers of molecules and μ_{i} are chemical potentials for reactants and products. |

Mixing | $\Delta S=-R{\displaystyle \sum _{i}^{n}\frac{{N}_{i}}{N}\mathrm{ln}\frac{{N}_{i}}{N}}$, where N_{i} are numbers of molecules and R is the universal gas constant |

Heat transfer | $dS=\left(\frac{1}{{T}_{1}}-\frac{1}{{T}_{2}}\right)dQ$, where T_{1} and T_{2} are temperatures of the two bodies |

_{i}= p

_{i}(ζ

^{i}

_{1}, ζ

^{i}

_{2},… ζ

^{i}

_{n},), where ζ

_{j}are generalized coordinates (or “phenomenological variables”) associated with the processes, and for a degradation measure w(p

_{1}, p

_{2},… p

_{n}), which is non-negative and monotonic function of the process energies pi , the rate of degradation $\dot{w}={\displaystyle \sum _{i}^{n}{B}_{i}}{\dot{S}}_{i}$ is a linear function of the components of entropy production ${\dot{S}}_{i}={\displaystyle \sum _{j}^{n}{X}_{i}^{j}{J}_{i}^{j}}$ of the dissipative processes (where X and J are generalized forces and flows of the processes). Degradation components ${\dot{w}}_{i}={{\displaystyle \sum _{j}^{n}{Y}_{i}^{j}{J}_{i}^{j}}}_{i}$ proceed at rates ${J}_{i}^{j}$ determined by entropy production, whereas the generalized “degradation forces” ${Y}_{i}^{j}={B}_{i}{Y}_{i}^{j}$are linear functions of Y

_{i}= B

_{i}X

_{i}, and degradation coefficients B

_{i}are partial derivatives of w by entropy.

^{*}(σ/β

^{*})B/T yields

^{*}is the effective elastic modulus, σ is the standard deviation of rough profile height, and β

^{*}is the correlation length of profile roughness (in a sense, σ is the height, and β

^{*}is the length of a typical asperity). Equation 13 is widely used as the Archard equation for adhesive wear [22].

_{ik}and ε

_{ik}are the stress and strain tensors, N

_{i}and μ

_{i}are the number of particles and the chemical potential of the fraction i. The mass transfer in a frictional system depends upon the heat flow, dissipation, and chemical potentials of components in the system.

^{2}and it shows an agreement with the experimental values on the deterioration as a function of the numner of cycles for various loadings [29].

## 3. Entropic Methods of Study of Self-organized Tribological Systems

#### 3.1. Self-organization in Tribology

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) | Equations of motion, competing processes for entropy and negentropy production | Bifurcation; self-excited vibrations and waves; feedback and target functions | Gap thickness, temperature, and microtopography distributions | Minimum friction |

Self-reproducing micro-topography, waviness | Equations of motion or kinetics | Bifurcations; self-excited vibrations and waves | Rough surface microtopography | Minimum energy dissipation; pressure or heat flow distribution |

Steady state microtopography of worn surfaces (“natural wear shape”) | Competing processes for entropy and negentropy (information) production | Feedback and target function | Shape of the profile | Minimum energy dissipation |

Self-excited vibrations of wear, electric resistance, stresses, etc. | Measurements of a parameter of the system (friction force, electrical resistance, wear rate, etc) | Instabilities and self-excited oscillations of the measured parameter | Corresponding parameter | Minimum entropy production |

Spatial or periodic chemical pattern | Molecular, atomic, or dislocation structure | Large-scale ordered structures | Secondary heterogeneity at the surface | Dissipative principles |

Periodic or concentric structures, such as Bénard cells | Molecular, atomic, or dislocation structure; Entropy is measured | Large-scale order structures; a sudden decrease in entropy production | - | Minimum entropy production |

Decrease in macrofluctuation of temperature, particle size and other parameters | Order-parameter dependent on generalized coordinate, Measurements of a parameter of the system | Microfluctuations; phase transitions; instabilities and self-excited vibrations of the measured parameter | - | Sub-minimal friction |

**Table 3.**Self-organization effects in tribosystems [30]

Effect | Mechanism/ driving force | Condition to initiate | Final configuration |
---|---|---|---|

Stationary microtopography distribution after running in | Feedback due to coupling of friction and wear | Wear affects microtopography until it reaches the stationary value | Minimum friction and wear at the stationary microtopography |

In situ tribofilm formation | Chemical reaction leads to the film growth | Wear decreases with increasing film thickness | Minimum friction and wear at the stationary film thickness |

Slip waves | Dynamic instability | Unstable sliding | Reduced friction |

Self-lubrication | Embedded self-lubrication mechanism | Thermodynamic criteria | Reduced friction and wear |

Surface-healing | Embedded self-healing mechanism | Proper coupling of degradation and healing | Reduced wear |

#### 3.2. Thermally Activated Self-organization

_{cr}, corresponds to ${\mu}_{\varphi}^{\prime}\left(\psi ,0\right)=0$. 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 the film with thickness ϕ

_{0}) and reduction of friction to the value of μ(ψ, ϕ

_{0}) as well as wear reduction. Here we 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 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 [33].

**Figure 3.**(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 ψ. Sub-critical values of ψ < ψ

_{cr}result in the positive slope (no layer formed), whereas ψ < ψ

_{cr}results in the instability and self-organization of the protective layer. The slope depends on the ratio of the bulk and layer values of μ, which allows finding composite microstructure that provides the self-organization of the layer.

_{2}O

_{3}reinforced Al matrix nanocomposite friction and wear tests (steel ball-on-disk in ambient air, [30,31]). 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 ${\mu}_{\varphi}^{\prime}\left({\psi}_{cr},0\right)=0$. The decrease is sudden and dramatic, so it can be explained by the loss of stability (cf. Equation 17) rather than by a gradual change of properties; although additional study is required to prove it.

**Figure 4.**A significant wear and friction reduction with decreasing particle size in Al-Al

_{2}O

_{3}nanocomposite (based on [31]) can be attributed to surface self-organization

#### 3.3. The Concept of “Selective Transfer”

#### 3.4. The Concept of “Tribofatigue”

_{Σ}, the change of energy in the volume W

_{p}, where degradation occurs, the effective energy and the tribofatigue entropy production are given by

_{1}is a coefficient characterizing the proportionality.

_{p}/dt as the generalized flux and γ

_{1}ω

_{Σ}/T as the corresponding generalized force. The authors claim that, on the basis of Prigogine’s approach, the state of thermodynamic equilibrium is characterized either by the minimum entropy generation (in the self-organized state) or by the maximum entropy. They also distinguish several states of objects depending on the value of the tribofatigue damage parameter: (a) ω

_{Σ}= 0, (undamaged), (b) 0 < ω

_{Σ}< 1 (damaged) (c) ω

_{Σ}= 1 (critical), (d) 1 < ω

_{Σ}<∞ (supercritical), (e) ω

_{Σ}= ∞ (decomposition). Further investigation of structure-property relationship on the basis of this model can be promising.

## 4. Frictional Dynamical Systems: Self-Organization and Entropy

#### 4.1. Frictional Adjustment During the Running-in

**Figure 5.**(a) Variation of the steady state and stick-slip friction with sliding distance [35]; (b) a typical decrease of friction during the running-in.

#### 4.1.1. Feedback Loop Model for the Running-in

_{def}and μ

_{adh}are the deformational and adhesional components of the coefficient of friction, k

_{def}and k

_{adh}are the deformational and adhesional components of the wear coefficient, and C

_{def}, K

_{def}, C

_{adh}, and K

_{adh}are corresponding proportionality constants.

_{adh}/K

_{def}= C

_{adh}/C

_{def}. The assumption of A = B is justified if the rate of change of roughness is proportional to the wear rate. The assumption K

_{adh}/K

_{def}= C

_{adh}/C

_{def}is justified if wear is proportional to friction.

**Figure 6.**A feedback loop (a) model and (b) its presentation in Simulink. Two simultaneous processes (adhesion and deformation) affect surface roughness in different manners. (c) Total friction is the sum of the deformational and adhesional components and the equilibrium value of roughness R corresponds to the minimum value of friction. Consequently, an equilibrium value of roughness exists, which corresponds to minimum friction [36].

**Figure 7.**The time-dependence of the coefficient of friction and roughness parameter during the running-in simulated with Simulink for A = B and A ≠ B. For A = B, while roughness reaches its equilibrium value, the coefficient of friction always decreases. Therefore, self-organization of the rough interface results in the decrease of friction and wear. For A ≠ B the coefficient of can decrease or increase depending on the initial value of roughness [36].

**Figure 8.**The change of the coefficient of friction and the surface roughness in Cu substrate with the number of cycles during a ball-on-disk test with a tungsten carbide (WC) ball [36].

#### 4.1.2. Shannon Entropy and Entropy Rate of a Random Process as Measures of Surface Roughness

_{1}= 1) has the lowest possible entropy S = 0, a periodic profile with two values of equal probability (B = 2, p

_{1}= p

_{2}= 0.5) S = ln2, whereas for a random profile the value of S will be higher. Therefore, the Shannon entropy essentially constitutes a new surface roughness parameter appropriate for the description of the adjustment of sliding surfaces. Note that Equation 31 ignores the spatial correlation between the measurement points. In order to take the spatial correlation into account, the entropy of a random process (entropy rate) should be considered instead.

Information | Energy | Mass | |
---|---|---|---|

Surface roughness | Friction (dissipation) | Wear (mass flow) | |

Entropic description | Shannon entropy and entropy rate for a stochastic process $S=-{\displaystyle \sum _{j=1}^{B}{p}_{j}\ell n[{p}_{j}]}$ | Thermodynamic entropy $dS=\frac{dQ}{T}$ | Entropy of mixing (configurational) $\Delta S=-R{\displaystyle \sum _{i}^{n}\frac{{N}_{i}}{N}\mathrm{ln}\frac{{N}_{i}}{N}}$ |

#### 4.2. The Problems of Combining Friction with Dynamics and Linear Elasticity

#### 4.2.1. Paradoxes

#### 4.2.2. Frictional Dynamic Instabilities

**Instabilities due to velocity-dependence of friction:**When the coefficient of friction is dependent on sliding velocity, the positive feedback can occur in the system in the case when friction decreases with increasing velocity (the so-called “negative viscosity”). With increasing velocity, the resistance to the motion decreases and the velocity further grows, leading to the instability. This can be formally deduced from Equation 18, which states that if $\partial \mu /\partial V\ge 0$ (the positive viscosity), the motion is stable, δYδJ > 0 meaning that the change of the driving force of the process (with is opposite to the friction force) opposes the change of velocity. However, if $\partial \mu /\partial V<0$ (the “negative viscosity”), the motion is unstable, and the secondary structures can form. The “negative viscosity” effect is often found in systems with dry friction, since it is not just the kinetic coefficient of friction is smaller than the static, but the kinetic coefficient also decreases with the sliding velocity. This is due to the increase of the real area of contact with the age of contact as a result of viscosity. The physical meaning of this conclusion is clear: if the friction force decreases with increasing velocity, the velocity will further grow leading to the instability [42]. The process will continue until it will leave the linear region and enter a limiting cycle, which is likely to be manifested in the stick-slip motion. Such stick-slip motion can be viewed as a self-organized secondary structure.

**Elastodynamic instabilities:**Let us assume now that the coefficient of friction is constant; however, the contacting body can be deformed elastically. 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 has not been investigated until the 1990s, when Adams [43] 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 motion. These self-excited oscillations are confined to a region near the sliding interface and can 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 and Adams [44] for the contact of rough periodic elastic surfaces. These instabilities exist for slightly dissimilar materials in terms of their elastic properties (the shear modules G, the Poisons ratios ν, and densities ρ), whereas for significantly dissimilar materials dilatational and shear elastic waves can be radiated away from the frictional interface under the wave angles prescribed by the elastic properties of the materials (Figure 9).

**Figure 9.**Elastic waves radiated from the frictional interface between two elastic half-spaces. The bodies are shown pre-stressed with the pressure p

^{*}and shear q

^{*}applied at infinity [45].

**Thermoelastic instabilities:**Another type of friction-induced instability is thermoelastic instability [46]. Heat is generated during friction, and it leads to the thermal expansion of the material, which increases the contact pressure. The increased pressure results in increased friction force and excess heat generation, i.e., the instability.

**The role of wear:**Another mechanism that may provide instability is the coupling between friction and wear. As friction increases, so does the wear, which may result in an increase of the real area of contact between the bodies and in further increase of friction. The sliding bodies adjust to each other, and the process is known as the frictional self-organization. On the other hand, wear produces smoothening of the surface distorted by the TEI mechanism, and thus the wear and thermal expansion are competing factors, with the wear leading to stabilization of sliding and the thermal expansion leading to destabilization (Figure 10).

**Figure 10.**Positive feedback leading to the friction-induced instabilities [22]

#### 4.2.3. Self-organized Elastic Structures

**Figure 11.**Friction reduction due to propagating stick-slip zones. The shear force F is smaller than the force needed to initiate friction μW; however, due to many propagating slip regions the two contacting bodies shift relative to each other in what is observed as friction at the reduced apparent coefficient of friction μ

_{ap}= F/W<μ.

System | Single half-space | Two half-spaces, no friction | Two welded half-spaces | Two half-spaces, finite friction | |
---|---|---|---|---|---|

slightly dissimilar | significantly dissimilar | ||||

Waves | Surface (Rayleigh) waves | Interface (generalized Rayleigh) waves (GRW) | Stoneley waves | Instabilities confined at the interface (GRW with growing amplitude) | Radiated waves |

Derivative waves | Non-linear stick-slip waves | Linear slip waves |

#### 4.3. Non-linear Models

_{c}= 374 °C and P

_{c}= 218 atm, and the distinction between liquid and gas water at these conditions disappears, so that no energy is needed to convert liquid water into vapor. At the critical point the energy barrier vanishes. The systems with SOC have a critical point as an attractor, so that they spontaneously reach the vicinity of the critical point and exhibit power law scaling behavior. Since SOC allows a system to reach criticality spontaneously and without tuning the controlling parameter, it was suggested that it plays a major role in the spontaneous creation of complexity and hierarchical structures in various natural and social systems [50]. SOC was suggested to be responsible for landslides and earthquakes, because it is known that the number of earthquakes and their amplitude are related by a power law. In other words, a number of earthquakes with the amplitude greater than a certain level in a given area during a given period is related to that level by a power law. During earthquakes, the stress between two plates is accumulated for a long time and released suddenly in a catastrophic event, which is similar to the sandpile avalanche [22,51].

## 5. Non-equilibrium Thermodynamic Approach to Friction, Wear and Self-healing

#### 5.1. Linear Equations of the Non-equilibrium Thermodynamics and Friction

_{i}and a thermodynamic flow ${J}_{i}={\dot{q}}_{i}$ are associated with every generalized coordinate q

_{i}. In the widely accepted linear approximation, the flows are related to the forces by

_{ki}are Onsager coefficients [52]. The heat production per unit time is given by

_{k}with thermodynamic forces Y

_{k}can be treated in the view of the assumptions of the linear thermodynamic equations of motion (Equation 34). However, unlike many other linear empirical laws, the Coulomb law cannot be directly deduced from the linear equations of the non-equilibrium thermodynamics, such as Equation 34. Indeed, in the case of dry or lubricated friction, the sliding velocity is the thermodynamic flow, V = J, which, in accordance with Equation 34 should be proportional to the friction force F = Y (as it is the case for the viscous friction), so that the energy dissipation rate is given by the product of the thermodynamic flow and force

_{1}= F, Y

_{2}= W [53].

_{11}F + L

_{12}W = 0, which is exactly the case of Coulomb friction if μ = −L

_{12}/ L

_{11}[53].

#### 5.2. Linear Equations of the Non-equilibrium Thermodynamics and Wear

#### 5.3. Self-healing

**Figure 13.**Schematics of self-healing using (a) precipitation (figure provided by Mr. J. M. Lucci, from the UWM) (b) reinforcement with shape-memory alloy (c) embedding of a healing agent (e.g., low melting point solder).

Mechanism | Precipitation | SMA reinforcement | Healing agent encapsulation |
---|---|---|---|

Type (according to [56]) | Damage prevention | Damage management | |

Type (according to [57]) | Solid-state | Solid-state (possibly also liquid assisted) | Liquid-assisted |

Matrix Material | Al-Cu, Fe-B-Ce, Fe-B-N, etc. | Sn-Bi, Mg-Zn | Al |

Reinforcement Materials | - | NiTi | Sn-Pb |

Microstructure parameter, | Solute fraction | Concentration of microwires | Concentration of microcapsules or low-melting point alloy |

Degradation measure, | Volume of voids | Volume of voids | Volume of voids |

Healing measure, | Amount of precipitated solute | SMA strain | Amount of released healing agent |

Characteristic length of degradation | Void size (microscale) | Void/crack size (macroscale) | Void/crack size (macroscale) |

Characteristic length of the healing mechanism | Atomic scale (atomic diffusion) | Microwires diameter (macro or microscale) | Microcapsule size (microscale) |

Phase transition involved | Solute precipitation | Martensite/austenite | Solidification of the solder |

Healing temperature | Ambient | Martensite/austenite transition | Melting of the low-melting point alloy |

Property improved | Creep resistance | Restored strength | Restored strength and fracture toughness |

**Figure 14.**Block diagram of the healing process. Deterioration is caused by an external force. The deteriorated system is brought out of equilibrium so that the restoring (“healing”) force is created, which is coupled with the degradation flow through the parameter M.

^{deg}in Equation 44 is an externally applied thermodynamic force that results in the degradation. The healing force Y

^{heal}is an external thermodynamic force that is applied to the system. In most self-healing mechanisms the system is placed out of equilibrium and the restoring force emerges, so we can identify this restoring force with Y

^{heal}. Since the restoring force is coupled with the degradation parameter ξ by the negative coefficients N = M, it also causes degradation decrease or healing.

_{net}= ΔS

_{macro}+ ΔS

_{micro}

**Figure 15.**Self-healing observed at the macroscale (healed cracks and increased orderliness) and microscale (ruptured microcapsules and decreased orderliness) [23].

_{micro}than a body with such defects. Larger-scale defects such as cracks and voids contribute to the macroscale component of the entropy, ΔS

_{macro}. A material or a surface with a regular microstructure (e.g., a microtextured surface) is more ordered and thus it has lower microscale entropy, S

_{micro}than a material with an irregular microstructure [54]. At the same time, it is noted that the approach is limited in the general case, but not in the situation when “seeds” exist at the microlevel that could, in the future, give rise to macro defects.

_{macro}, associated with the macroscale defects, such as cracks or voids. Healing can be triggered by affecting the mesoscale structure, e.g., by release of microcapsules. The fracture of the microcapsules decreases the orderliness of the microstructure and thus increases the entropy for ΔS

_{micro}. In the case |ΔS

_{macro}| < |ΔS

_{micro}|, the healing is done by decreasing the macroscale component of entropy at the expense of the mesoscale component [23]. In other words, for most practical applications, the macroscale integrity and orderliness of material are of interest, and thus the expression for net entropy given by Equation 3 can be truncated at the macroscale level, ΔS

_{net}= ΔS

_{macro}. The orderliness of the material, as observed at the macroscale, can grow (and, therefore, entropy can decrease) at the expense of excess entropy production at the lower scales. This is analogous to the crystal grain growth (e.g., in aluminum) due to thermal fluctuations: with growing grains material structure becomes more ordered, however, dissipation and excess entropy production occurs at the nanoscale every time when a grain border propagates [58].

## 6. Future Directions

Phenomena | Principle | Application |
---|---|---|

Wear (friction-induced) | Proportionality of the wear rate and entropy flow | Wear reduction for various applications [15,16] |

Running-in | Microtopography adjustment observed as Shannon entropy and roughness reduction | Friction and wear reduction in the stationary regime [20,30] |

Formation of in-situ tribofilms | Friction-induced diffusion of the film-material to the interface due to the destabilization of the stationary state. | Friction and wear reduction due to protective tribofilm [12,13,22,25,31,32,33,34] |

Slip waves | Elastic waves at the interface which can result in friction reduction. Can result in self-organized critical behavior and stick-slip. | Novel ways of ultrasonic motors, etc. [43,44,45]; new theories of dislocation-assisted sliding [47]; geomechanical applications [17,18,19,20] |

Friction-induced instabilities | Coupling of friction with wear, thermal expansion, etc. Usually leads to the “negative viscosity” and similar types of frictional instabilities | Eliminating friction-induced vibrations and noise [39,42,43,44,45,46] |

Self-healing by embedding microstructures | A mechanism, which provides the coupling of healing with another relevant thermodynamic force, is embedded into material. Healing can occur due to the deterioration of embedded microstructure (e.g., microcapsules). | Self-healing materials and surfaces [22,23,24,30, 53, 56,57] |

Damage prevention | A mechanism to heal voids as they appear is embedded into material (e.g., nucleation of a solute at void points in supersaturated solid solution). | Wear-resistant and self-healing materials and surfaces [56,57] |

Self-lubrication | Various mechanisms, including embedded microstructure, to reduce friction and wear. | Self-lubricating materials [30,53] |

## 7. Conclusion

## Acknowledgements

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Nosonovsky, M.
Entropy in Tribology: in the Search for Applications. *Entropy* **2010**, *12*, 1345-1390.
https://doi.org/10.3390/e12061345

**AMA Style**

Nosonovsky M.
Entropy in Tribology: in the Search for Applications. *Entropy*. 2010; 12(6):1345-1390.
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**Chicago/Turabian Style**

Nosonovsky, Michael.
2010. "Entropy in Tribology: in the Search for Applications" *Entropy* 12, no. 6: 1345-1390.
https://doi.org/10.3390/e12061345