# Self-Organization, Entropy Generation Rate, and Boundary Defects: A Control Volume Approach

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Formalism

**S**e/dt + d

**S**gen/dt = d

**S**cv/dt

**S**is the entropy, d

**S**cv/dt [2] is the accumulation/reduction of entropy per unit time inside the control volume (cv), d

**S**e/dt is the flow of entropy per unit time between the environment and the system (recognized at the boundary of the control volume), and d

**S**gen/dt is the internal entropy production rate inside the control volume. Energy and mass exchanges, because of potential gradients, lead to entropy production [1,2,3,4,5,6,7,8,9]. The d

**S**eg/dt is a consequence of gradients of chemical potential, temperature, pressure, or reaction/transformation activity and their conjugate fluxes or flow of mass, heat, total-volume, and chemical-species, respectively [1,2,3,4,5,6,7,8,9].

#### Steady State

**S**cv/dt = 0. A system that is in a steady state may not necessarily be in a state of dynamic equilibrium (equilibrium implies that d

**S**gen/dt = 0), as several of the physical processes involved are not reversible, i.e., they produce entropy.

**S**gen/dt. Exergy is destroyed when entropy is produced, but that does not mean that there is no effective work done in the process. An amazing feature of entropy-producing, self-organization systems, particularly at steady state, is that a balance is struck between mechanisms which offer rapid entropy transport and entropy production.

## 3. Maximum Entropy Production Rate (MEPR) Principle

## 4. Patterns and Texture Examples from Directional Solidification, Wear, and Friction

_{13}H

_{10}O

_{3}, produced by the interaction of salicylic acid and phenol) by the MEPR calculation [13,75]. The plot in Figure 2 shows the transition from cellular faceted morphology to non-facet morphology with increasing velocity, as shown by the dotted black diagonal line. The inset shows the various crystal orientations that the interface adopts to cope with the increasing entropy generation rate and entropy dissipation requirement. The interface velocity and interface undercooling (T

_{l}− T

_{tip}) scale linearly, unless a new (111) configuration can replace the previous one [42]. Any increase in velocity results in the side-branch (SD) formation, which is a method of increasing the entropy generation [2]. This is a method of enhancing the entropy generation, as well as for creating new defects that can aid in entropy removal. Subsequently, non-faceted topographical forms, such as cells, can be noted [23,24,25,26]. Cellular and dendritic features may form with a further increase in the solidification velocity. These are the higher, entropy-producing variations [2].

#### 4.1. Low-Velocity Transitions, Facets to Smooth Curvature

_{L}) during solidification.

**V**/G

_{SLI}= (√

**N**)·

_{c}**d**

**V**is the interface velocity, G

_{SLI}is the temperature gradient in the diffuse region, and

**d**is the atomic spacing normal to the growth interface.

**N**is the parameter that contains the partition coefficient and the liquidus slope,

_{c}**c**indicates critical [13,75].

**N**(m

^{2}K

^{−2}s

^{−2}) is defined as $\left[\left(\frac{2\Delta {\mathrm{h}}_{\mathrm{sl}}}{\Delta {\mathsf{\rho}}_{\mathrm{k}}{\mathrm{T}}_{\mathrm{m}}^{2}}\right)-\left(\frac{\mathrm{V}\Delta {\mathrm{T}}_{0}{\mathrm{R}}_{\mathrm{g}}\mathrm{ln}\left(\frac{1}{{k}_{eff}}\right)}{2{\mathrm{G}}_{\mathrm{SLI}}{\mathrm{D}}_{\mathrm{L}}\Delta {\mathsf{\rho}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{L}}}\right)\right]$. Here, Δρ

_{k}(kg·m

^{−3}), is the density change (shrinkage) between the liquid and solid.

_{SLI,}the entropy generation rate for low-concentration binary alloys can be expressed in the following manner [13,75]:

_{av}·ζ. As ζ is small (often lattice dimensions), the entropy generation rate per unit volume for a plane front interface is high. For alloys, T

_{i}and T

_{s}are the liquidus and solidus temperatures, ${\dot{\mathsf{\phi}}}_{\mathrm{max}}$decreases as T

_{li}approaches T

_{l}. ${\dot{\mathsf{\phi}}}_{\mathrm{max}}\mathrm{is}$also further reduced by the second term in Equation (3). A peak $\mathrm{in}{\dot{\mathsf{\phi}}}_{\mathrm{max}}$ as a function of the imposed processing variables V or G is thus noted. For this condition, ${(\frac{\partial {\dot{\mathsf{\phi}}}_{\mathrm{max}}}{\partial \mathrm{V}})}_{\mathsf{\zeta},{\mathrm{C}}_{\mathrm{O}}}=0$, which yields:

#### 4.2. Medium Velocity Transitions: Cells, Cellular-Dendrites, and Dendrites

_{t}) of a dendrite, or the mean array tip temperature for a particular columnar morphology, lies between T

_{s}and T

_{l}, where T

_{t}depends on the imposed process conditions, namely the solidification velocity and temperature gradient (Figure 3 and Figure 4a). The spacings between individual dendrites, or the spacings of the secondary dendrite (SD) features, generally decreases with an increase in the solidification velocity. One of the important consequences of all such array structures is the occurrence of boundary defects between the cells and primary dendrites in the rigorous solid. These defects can range from low-angle to high-angle grain boundaries or even show new phase formation boundaries (because of microsegregation) between the dendrites [24,25]. The entropy production is also influenced by curvature and the enthalpy of transformation.

_{s}) can be calculated from an energy balance, described by Equations (5) and (6):

_{max}= −AV (∆h

_{sl}− ∆h

_{m})

_{gb}/λ

_{1}+ 6·γ

_{gb}·λ

_{2}/λ

_{1}

^{2}+ ω

_{D}]

_{1}and λ

_{2}are the primary and secondary spacings, respectively. Here, ω

_{D}is the defect energy associated with defect entropy ω

_{D}/Tav, not including boundary area defects but including energy/entropy terms associated with chemical species segregation, called microsgregation, and including the two-phase mixing of eutectics and solute gradients (for high alloy concentrations) that form between dendrites. W is the work done, in relation to W

_{L}, the loss in work potential. The sign in Equations (5) and (6) follow the standard thermodynamic conventions, where the work done on the system is positive. The symbol γ

_{gb}is the energy of the solid-solid interface (grain-boundary) between the of cells or dendrites in the arrays. This energy can vary between ~1 mJ/m

^{2}to 1000 mJ/m

^{2}for metals, depending on the type of grain boundary or the microsegregated boundary [2,8,46]. A steady state [2] entropy balance gives:

**S**gen/dt = −AV∆h

_{sl}(1 −T

_{l}/T

_{s})/T

_{l}+ (dW/dt)/T

_{av}

**S**gen/dt = AV∆h

_{sl}(1 − T

_{s}/T

_{l})/T

_{s}− AV [γ

_{gb}/λ

_{1}+ 6·γ

_{gb}·λ

_{2}/λ

_{1}

^{2}+ ω

_{D}]/T

_{av}

_{av}is a temperature between Ts and Tl. Here, Ti is the tip temperature for a generic morphology, whereas T

_{c}

_{tip}indicates cell tip temperature. Note that (1 − T

_{s}/T

_{l}) is the maximum work efficiency (Carnot efficiency) that is feasible for work between T

_{l}(liquidus) and T

_{s}(solidus). Figure 5 is a transition prediction, based on Equation (8), for cell to cellular dendrite and to a dendrite morphology, with an increase in the solidification velocity. Note that the second differential d

^{2}${\dot{\phi}}_{}$/dV

^{2}has a positive sign, when λ

_{1}and λ

_{2}are decreasing functions of velocity (a fact known known from experiments [23]), thus indicating that ${\dot{\phi}}_{}$ is, indeed, the ${\dot{\mathsf{\phi}}}_{\mathrm{max}}$, when (d${\dot{\phi}}_{}$/dV = 0).

_{s}does not include curved interfaces, such as secondary dendrite envelopes, the entropy generation rates for cellular arrays (Equation (9)) and dendritic arrays (Equation (10)) may be approximated as:

**S**gen/dt = AV∆h

_{sl}(1 − T

_{s}/T

_{c}

_{tip})/T

_{s}− AV (γ

_{cell(}

_{gb)}/λ + ω

_{D})/T

_{av}

**S**gen/dt = AV∆h

_{sl}(∆T

_{0}/T

_{l})/T

_{s}− A

_{SD}V (γ

_{dendrite(}

_{gb)}/λ + 6·γ

_{gb}·λ

_{2}/λ

_{1}

^{2}+ ω

_{D})/T

_{av}

_{1}will decrease with an increasing velocity and increasing temperature gradient, except under certain conditions of very high γ

_{gb}or low T

_{av}. If the γ

_{gb}changes abruptly with morphology, the spacing must also abruptly change between the scaling elements of a pattern that dissipates entropy. Assuming that the dendritic boundaries are of a much higher energy, this would imply that a cell(C)/cellular-dendrite (CD) to dendrite(D) transition is associated with an increased primary spacing at the transition, when plotted against velocity of solidification. This result agrees broadly with experiments [23]. When the secondary dendrites dominate the structure, a further approximation is possible:

**S**gen/dt = AV∆h

_{sl}(∆T

_{0}/T

_{l})/T

_{s}− A

_{SD}V (γ

_{dendrite(}

_{gb)}/λ + 6·γ

_{gb}·λ

_{2}/λ

_{1}

^{2}+ ωD)/T

_{av}

_{SD}is the surface area between the secondary dendrite features. Equating (9) and (10), and making the approximation that γ

_{gb}is the same for both cells and dendrites, with the approximation that A

_{SD}~A, yields the elements of the connections (approximately) between the various self-organizing scales at the cell to dendrite transition-region and gives:

_{1}

^{2}/λ

_{2(C−D)}) = 6

**Γ**·(T

_{s}/T

_{l})/(∆T

_{0}− ∆T

_{ctip(C−D)})

_{ctip}is the cell tip temperature, minus T

_{s}.

**Γ**is the boundary capillarity constant γ

_{gb}/∆s

_{sl}. Note that T

_{l}(known from the phase diagram) becomes smaller with increasing alloying concentrations (when the solute partition ratio, k, is less than one). With increasing velocity (leading to an increased tip undercooling for dendrites), it is possible to calculate (λ

_{2(}

_{C−D)}) or ∆T

_{ctip(C}

_{−D)}) with Equation (12). The secondary arm spacing at the cell dendrite transition (λ

_{2(}

_{C−D)}) is predicted to become smaller with increasing alloy concentration and with velocity, again in accordance with experimental observations [23,24,25]. As the T

_{ctip}is experimentally found to be only slightly lowered with an increasing temperature gradient, the (λ

_{2(}

_{C−D)}) is predicted to increases slightly with G

_{SLI}, depending on λ

_{2.}Note that the (λ

_{2(}

_{C−D)}) is the secondary spacing, measured at T

_{s}(and for conditions of the cell/cellular-dendrite to dendrite transition). Although numerous λ

_{2}measurements are available for several alloys, there are very few experimental reports for (λ

_{2(}

_{C−D)}). Regardless, some tests of the predictions can be made from published information.

_{2(C−D)}) is noted to be about ~100 micrometers. λ

_{1}at the C-D transition condition is reported to be ~300 micrometers for both alloys. The ∆T

_{0}is 369 K for the Rene-108 multi-component alloy but only about ~30 K for IN-718 multi-component alloy [61]. Therefore, the tip temperature differences of the cell/cellular-dendrite and the dendrite at the C-D transition should be very small (as noted) for a meaningful prediction by Equation (12). This also appears to be the general experimental finding with all metallic alloys [24,25,43], thus, further indicating that the entropy maximization principle may be employed as a key selection criterion [2,3,13,75] for the prediction of specific patterns during solidification. Figure 5 and Figure 6 are plots of the entropy generation rate per unit volume for several commonly identifiable solidification morphologies.

#### 4.3. High-Velocity Regimes Including Featureless Solids and Metallic Glass

_{l}) and solidus (m

_{s}), to establish a plane front at very high interface velocities. This is not necessarily so for the MS model. The experimental results by Trivedi et.al [36] and Sekhar [45] do show extremely fine cells, and even a planar interface at very high growth rates for low solute concentration alloys; however, the partition coefficient cannot be ascertained from the two reports, thus making it difficult to compare the MS and MEPR models in this high-velocity growth regime.

**Se**/dt is large (for example with rapid heat removal at the CV boundary) but d

**S**gen/dt from a particular morphology is inadequate, then the phase itself could alter to reestablish a steady state (e.g., metallic glass can result, instead of a crystalline solid). For freezing into a crystalline form, the typical entropy generation rate mechanisms are captured by crystal structure and shapes with defects, i.e., by crystal structure, reorientation and expansion, interface diffuseness, cell/dendrite patterns, or by creating a new phase (crystalline or glassy) [2]. For the freezing of a liquid into a glass, there is no need to invoke segregated area defects in the solid [36,37]. Note, in Figure 4d, that the transition from very fine cells to a more planar interface involves a drop in the interface temperature. This is required in MEPR but not required in the MS model.

_{sl}~0, the entropy generation is from the steep and non-linear temperature gradient over a very small thickness. For the control volume, the boundaries are at the T

_{l}and T

_{g}(glass transition temperature). Below T

_{g}, the rate of volume change for glass is very low with further cooling. The supercooled liquid (not yet a glass) generally has the coefficient of thermal expansion of a liquid. The faster the cooling rate, the higher is the molar volume and the molar enthalpy of the glass (solid) that forms. The rates of entropy generation per unit volume for a featureless crystalline feature phase (Equation (13)) or for a glassy phase (Equation (14)) are respectively:

_{av}is the thermal conductivity (of the diffuse interface in Equation (13) and of the supercooled liquid becoming solid- glass in Equation (14)). T

_{g}is the glass transition temperature and ζ

_{g}is the zone thickness between T

_{l}and T

_{g}. The very first metallic glass was made with a imposed cooling rate of ~10

^{6}K/s [40] giving ${\dot{\phi}}_{}$~4 × 10

^{10}(J/m

^{3}·K·s), based on a gradient of 10

^{6}K/m. Figure 6, shows an entropy generation rate per unit volume plot that includes metallic glass formation. Note that, in such severe conditions, extremely small thicknesses could experience a high temperature gradient because of a high Biot number with a high heat transfer coefficient. (The Biot number is the ratio of the thermal resistances inside of a body and at the surface of a body). Biot numbers much larger than 1 indicate a temperature gradient in the splat cooled material.

^{−3}) would be able to generate a much higher entropy (during cooling) for the same thin dimensions. In contrast, materials with a low heat of fusion per unit volume, such as glass-forming silicate ceramics, the glass formation is the preferred pattern morphology (atomic configuration) during a cooling process.

#### 4.4. Range of Solidification Morphological Transitions

_{l}− T

_{s})/T

_{l}, i.e., when d

**S**gen/dt = 0. Minimum work is when d

**S**gen/dt is maximized [2]. The minimum extracted work cannot be zero, because defects and curved interfaces can form within the patterns.

## 5. Discussions: The Utility of Self-Organization

**S**cv/dt = 0) is satisfied, i.e., a steady state is reached. With new patterns, a changed manner of energy and entropy fluxes become operative, as noted in the previous section and Appendix A. Work can also manifest during the energy exchange, which leads to ordering. This work, in turn, can be used to build better and more efficient features for the production and transport of entropy. The coordination in self-organized systems, seemingly arises out of the local interactions between smaller-sized, component–parts of a system, which can quickly, otherwise, disorganize if not ordered into a new pattern. The process and the rate of self-organization can be “spontaneous” [14], i.e., it is not necessarily controlled by any auxiliary agent outside of the system. It is often triggered by random fluctuations that are amplified by positive feedback, which allow maximum entropy production and provide a method for entropy storage and transport. This becomes the basis for controlled defect formation events. The resulting organization is, thus, in a sense, wholly decentralized or distributed, yet entangled over all the components of the system.

^{3}·K·s), assuming ~O(10

^{3}) J/K of entropy is dissipated by the human brain (volume ~10

^{−4}m

^{3}) over a year, during its growth/development stages [13,58,59,75]. The structure of the cells in the human brain [53,58,59] do appear to have resemblances to the defect-enveloped, entropic pathways envelopes seen in microsegregated solidified grains, cells, or dendrites [8,23,24,25,26,61]. Regardless, more studies that are warranted to establish the similarities of shape evolution between inanimate solidification studies and human cell development particularly because biological cell-multiplication and grain-nucleation happen by vastly differing mechanisms for chemical transport.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature and Abbreviations

Letter Symbols | |

A | area of an interface in a solid-liquid region (m^{2}) |

A_{SD} | surface area of the secondary dendrite features |

d | interplanar lattice spacing (m) |

∆C0 | change in concentration for the liquid from the position of rigorous liquid to the bulk. ∆Co = (Cl* − Cs*) where Cl* and Cs* are the composition of the local rigorous-liquid and solid, respectively |

D | diffusion coefficient (m^{2}·s^{−1}) |

f | facet |

F | is the friction force (kg·ms^{−2}) |

G_{SLI} | gradient across a diffuse interface (K·m^{−1}) |

G_{l} | temperature gradient in the liquid or supercooling liquid (K·m^{−1}) |

Δhm | heat of fusion of a solid with defects (J·m^{−3}) |

Δhsl | heat of fusion (J·m^{−3}) |

H | hardness (J·m^{−3} or MPa) |

Js | solute flux in a liquid entering a solid-liquid interface (mole·s^{−1}) |

k | equilibrium partition coefficient obtained from the phase diagram (dimensionless) |

keff | effective partition coefficient at a solid-liquid interface (dimensionless)—a function of the velocity and diffuse interface structure |

ΔKE | gain or loss in kinetic energy (J) |

K_{L} | thermal conductivity for a rigorous liquid (J·m^{−1}·K^{−1}·s^{−1}) |

K_{S} | thermal conductivity for a rigorous solid (J·m^{−1}·K^{−1}·s^{−1}) |

K_{av} | average thermal conductivity in the control volume (J·m^{−1}·K^{−1}·s^{−1}) |

K_{wear} | wear constant, dimensionless ~10^{−4} for metal pairs. |

Lo | sliding distance (length of object in Appendix A, in the direction of travel) |

m_{L} | slope of the equilibrium liquidus at the SLI for a binary material (K·m^{3}·mole^{−1}) |

m_{s} | slope of the equilibrium solidus line at the SLI for a binary material (K·m^{3}·mole^{−1}) |

MEPR | maximum entropy production rate per unit volume (J·m^{−3}·K^{−1}·s^{−1}) |

MS | Mullins and Sekerka criterion [44] |

N_{p} | normal force (kg·m·s^{−2}) |

nf | non-facet |

Rg | molar gas constant (J·mol^{−1}·K^{−1}) |

R_{RMS} | RMS height (m) |

Sf | Entropy flux rate (J·K^{−1}·s^{−1}) to and from a solid-liquid interface with its surrounding |

dSgen/dt | entropy generation rate in a diffuse region (J·K^{−1}·s^{−1}) |

dSin/dt | rate of entropy entering a control volume (J·K^{−1}·s^{−1}) |

dSout/dt | rate of entropy leaving a control volume (J·K^{−1}·s^{−1}) |

dsgen/dt | entropy produced/generated rate density (J·m^{−3}·K^{−1}·s^{−1}) |

S_{LG} | entropy generation rate density by the solute gradient in a liquid (J·m^{−3}·Ks^{−1}) |

(Sgen)max | maximum entropy generation (J·K^{−1}) |

dScv/dt | total steady state entropy rate in a control volume (J·K^{−1}·s^{−1}) |

dscv/dt | total steady state entropy rate density in a control volume ((J·K^{−1}·s^{−1}·m^{−3}) |

t | time (s) |

SLI | solid-liquid interface |

SD | side-branch secondary structures, such as secondary dendrites. |

q_{1}/τ | the heat transfer rate to the main body in a friction pair. |

Tli | liquidus interface temperature at a rigorous liquid interface (K); generally assumed to be T_{l} (the equilibrium liquidus temperature) |

Tsi | solidus interface temperature at a rigorous solid interface (K); generally assumed to be T_{s} (the equilibrium solidus or eutectic temperature) |

T_{ctip} | tip temperature of a cell |

T_{tip} | tip temperature of cells, dendrites, or facets in an array |

T_{i} | friction interface temperature of a friction pair |

T_{0} | room temperature for the friction problem |

T_{i} | friction interface temperature of a friction pair |

Tm | melting temperature (K) of the pure metal or species |

Tav | average temperature between Tli and Tsi in diffuse interface or solid-liquid region (K), when G is negative or close to zero |

T_{0} | room temperature (ambient) |

ΔTSLI | temperature difference across a solid-liquid interface (K) |

ΔT_{ctip(C−D)}) = T_{ctip} − Ts) (K) | at the cell or cellular dendrite transition to a dendrite |

(dclG/dz) or (ΔCO/δc) | change in solute gradient in a liquid (mole·m^{−4}) |

ΔTO | solidification temperature range (K) = −mL·ΔCo(1/k − 1) |

ΔTi | temperature difference between rigorous liquid and rigorous solid (K) = T_{l}i − T_{si} |

ΔTctip | (T_{ctip} − Ts) (K) |

V | solidification interface velocity (ms^{−1}) |

V | sliding pair velocity (ms^{−1}) (see Appendix A) |

WL | lost work (J) |

W | work (J) |

Greek symbols | |

ΔΩS | volume shrinkage (m^{3}) |

β | the autocorrelation length (m) of the surface |

|Δρk| | density shrinkage (kg·m^{−3}) |

ρl | density of rigorous liquid (kg·m^{−3}) |

ρs | density of rigorous solid (kg·m^{−3}) |

γ_{gb} | solid-solid boundary energy (e.g., between primary cells or primary dendrites) |

λ_{1} | primary spacing at the Ts isotherm |

λ_{2} | secondary spacing at the Ts isotherm |

λ_{2(C−D)} | secondary spacing at the Ts isotherm for the conditions of cell, to dendrite transition |

Δμc | chemical potential difference (J·mole^{−1}) |

Γ | boundary capillarity constant γ_{gb}/Δs_{sl}. |

μ | coefficient of friction |

θ | fraction of energy transferred to heat. |

ζ | solid-liquid interface thickness (m), i.e., the diffuse interface thickness |

ζ_{g} | is the zone thickness between T_{l} and T_{g} |

ξ | wear volume |

τ | (length of travel, Lo)/V (s) |

ζ^{3}_{cv} | thermal control volume for steady state friction (Appendix A) |

ω_{D} | energy of defects (J·m^{−3}) other than area defects |

${\dot{\phi}}_{}$ | maximum entropy generation rate density for a moving interface (control volume) (J·m^{−3}·K^{−1}·s^{−1}) |

## Appendix A. Self-Organization of Surface Texture during Friction and Wear

_{RMS}) and the autocorrelation lenght (β), of a rough surface [56]. When there is stability of the coefficient of friction, all the defining features of the asperities, such as the RMS height (R

_{RMS}) and the autocorrelation length (β), decay exponentially at the same rate [56], thus preserving the steady state, regardless of the size of the asperities, i.e., wear does not necessarily alter the coefficient of friction (at least in certain regimes of wear).

_{p}to the reactionary friction force F that must be overcome for pushing the object with a certain velocity; the law is written as F = μ N

_{p}. Here, μ is the coefficient of friction [56]. The maximum heat generation rate, q/τ, is given by [63]:

_{p}

_{p}, the sliding distance S, and the hardness H, through a proportionality constant K

_{wear}, i.e., the wear coefficient [63,64,65]:

_{wear}·SN

_{p}/H

_{wear}(q

_{1}/τ)/ημH

_{wear}~10

^{−3}to 10

^{−4}(dimensionless) for many metal pairs [63] (note that this term incorporates the Elastic Modulus/Hardness ratio to calculate the plasticity coefficient, plastic deformation, and heat release). The wear volume ξ is not the control volume for the entropy balance problem. The control voulme (ζ

^{3}

_{cv}) is the volume which is bound by the isotherms T

_{0}(room temperature) and the T

_{i}(the contacting interface temperature). Note that the entire Hertzian zone should be considered inside the control volume to apply MEPR.

_{1}/τ:

_{1}/τ = ημVN

_{p}θ

_{i,}the interface temperature. η is the heat partitioning constant between the solid pairs. If the sliding object is small, the object temperature is also T

_{i}, with η~1.

^{3}

_{cv}) is established, bound by the T

_{0}(room temperature isotherm). From references [66,67], we note that this steady state volume (ζ

^{3}

_{cv}) decreases with increasing velocity for a fixed-power heat source. Applying the steady state entropy equation [1] the entropy generation rate per unit volume inside a control volume can be calculated. If the entropy exchange by heat transfer leaves the body at the isotherm T

_{0}(the ambiant temperature), and that the entropy associated with the wear debris leaves at T

_{i}, the entropy balance may be written as:

**S**gen/dt = (q

_{1}/τ)/T

_{0}+ ω/Ti = ηθ μV N

_{p}/T

_{0}+ ω/Ti = ∫

_{cv}K·(ΔT/T)

^{2}dζ

^{3}+ ω/Ti

_{i}is thus the entropy that exits the control volume with defects/wear debris, i.e., the part that does not leave with the heat to the semi-infinite substrate. The MEPR principle uses the unit volume approach for the comparison between competing morphologies. The entropy generation rate per unit volume for a particular surface morphology at the region of contact (see Figure 1c and Figure A1a) at steady state is thus:

**S**gen/dt = (q

_{1}/τ)/T

_{0}/ζ

^{3}

_{cv}+ ω/Ti/ζ

^{3}= ηθμV N

_{p}/T

_{0}/(ζ

^{3}

_{cv}) + ω/Ti/(ζ

^{3}

_{cv})

= (∫

_{cv}K·(ΔT/T)

^{2}) dζ

^{3})/(ζ

^{3}

_{cv}) + ω/Ti/(ζ

^{3}

_{cv})

_{i}, the contact temperature (interface temperature), is set by the coefficient of friction and material properties that control deformation of the surface features during the sliding process [56,69,70]; ζ

^{3}

_{cv}is the control volume. Therefore, for a fixed surface-asperity configuration (i.e., morphology) at steady state, there is a fixed entropy generation rate, which increases with velocity, for a fixed dimension of the control volume. (Equation (A6)). When the control volme dimensions change [66,67] because of the severity of the heat produced, then correspondingly, the entropy generation rate per unit volume also changes, which could lead to inversions, shown in Figure A1c. Consequently, the morphology is altered by wear processes to one which displays a lower coefficient of frction. Equation A6 is simple but powerful application of MEPR to wear surface texture.

^{3}m

^{3}) (i) at the friction interface, which includes the Hertzian zone, i.e., the zone of contact of the two surfaces (ii) by the thermal gradients. The control volume is bound by T

_{i}and T

_{o}. Figure A1b shows the similarity of the heat transfer problem to one where an area source moves over a semi-infinite substrate. Figure A1c is a plot of the entropy generated per unit volume, for the comparison of two different friction coefficient of pairing materials (following Equation (A6)). Note that two lines are shown, representing two different scenarios of surface texture, i.e., for the two different coefficients of friction. The different surface morphologies are defined by the RMS height (R

_{RMS}) and the autocorrelation length (β) [56] of the surface texture (asperities). Note, again, that the surface texture features of interest (Figure A1a) are located inside the control volume, as only then is the MEPR applicable [73].

**Figure A1.**(

**a**) Schematic of a body moving with a velocity V, over a surface (friction pair). Entropy is generated in the control volume ζ

^{3}(m

^{3}) at the friction interface, which includes the Hertzian zone, i.e., the zone of contact of the two surfaces (the roughness in this zone is shown in the magnified inset). The control volume is bound by T

_{i}and T

_{o}. The entropy leaves the control volume with the heat q at T

_{o}and with the wear debris T

_{i}, respectively. The approximate entropy generation rate, as a function of the velocity (of sliding) for two different morphologies at the interface, is characterized by the asperity roughness height and the autocorrelation length; (

**b**) the thermal problem is similar to an area source moving on a semi-infinite substrate [56,65,66], for which a schematic is shown, T

_{0}is the ambient temperature; (

**c**) a plot of Equation (A6) for d

**S**gen/dt, as a function of the sliding velocity for two different surface morphologies, 1 and 2, corresponding to two different coefficient of friction values (μ) for the solid-solid pair; (

**d**) a plot showing extremely low friction for a self-organized surface asperity cluster, compared to other forms of surface features, modified from the data reported in Reference [54].

## References

- Kondepudi, D.; Prigogine, I. Modern Thermodynamics: From Heat Engines to Dissipative Structures; John Wiley & Sons: Hoboken, NJ, USA, 2015; ISBN-10:0471973947. [Google Scholar]
- Sekhar, J.A. The description of morphologically stable regimes for steady state solidification based on the maximum entropy production rate postulate. J. Mater. Sci.
**2011**, 46, 6172–6190. [Google Scholar] [CrossRef] - Martyushev, L.M. Maximum entropy production principle: History and current status. Uspekhi Fiz. Nauk
**2021**, 64, 586–613. [Google Scholar] [CrossRef] - Tzafestas, S.G. Energy, Information, Feedback, Adaptation, and Self-Organization, Intelligent Systems, Control and Automation: Science and Engineering 90; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar] [CrossRef]
- Pave, A.; Schmidt-Lainé, C. Integrative Biology: Modelling and Simulation of the Complexity of Natural Systems. Biol. Int.
**2004**, 44, 13–24. [Google Scholar] - Fath, B. Encyclopedia of Ecology, 2nd ed.; Elsevier: Amsterdam, The Netherlands, 2019. [Google Scholar]
- Sekhar, J. Thermodynamics, Irreversibility and Beauty. Available online: https://encyclopedia.pub/9548 (accessed on 23 June 2021).
- Ballufi, R.W.; Allen, M.A.; Carter, W.C. Kinetics of Materials; John Wiley & Sons: Hoboken, NJ, USA, 2005. [Google Scholar]
- Wikipedia. Differential Entropy. Available online: https://en.wikipedia.org/wiki/Differential_entropy (accessed on 27 June 2021).
- Martyushev, L.M.; Zubarev, S.N. Entropy production of stars. Entropy
**2015**, 17, 3645–3655. [Google Scholar] [CrossRef] [Green Version] - Martyushev, L.M.; Birzina, A. Entropy production and stability during radial displacement of fluid in Hele-Shaw cell. J. Phys. Condens. Matter
**2008**, 20, 465102. [Google Scholar] [CrossRef] - Turing, A.M. The molecular basis of morphogenesis. Philos. Trans. R. Soc.
**1952**, 37, 237. [Google Scholar] - Bensah, Y.D.; Sekhar, J.A. Solidification Morphology and Bifurcation Predictions with the Maximum Entropy Production Rate Model. Entropy
**2020**, 22, 40. [Google Scholar] [CrossRef] [Green Version] - Heylighen, F. The Science of Self-Organization and Adaptivity, Center; Free University of Brussels: Brussels, Belgium, 2001; p. 9. [Google Scholar]
- Sánchez-Gutiérrez, D.; Tozluoglu, M.; Barry, J.D.; Pascual, A.; Mao, Y.; Escuder, L.M. Fundamental physical cellular constraints drive self-organization of tissues. EMBO J.
**2016**, 35, 77–88. [Google Scholar] [CrossRef] - Lucia, U. Maximum entropy generation and K-exponential model. Phys. A Stat. Mech. Appl.
**2010**, 389, 4558–4563. [Google Scholar] [CrossRef] - Hill, A. Entropy production as the selection rule between different growth morphologies. Nature
**1990**, 348, 426–428. [Google Scholar] [CrossRef] - Martyushev, L.M.; Seleznev, V.D.; Kuznetsova, I.E. Application of the Principle of Maximum Entropy production to the analysis of the morphological stability of a growing crystal. Zh. Éksp. Teor. Fiz.
**2000**, 118, 149. [Google Scholar] - Ziman, J.M. The general variational principle of transport theory. Can. J. Phys.
**1956**, 35, 1256. [Google Scholar] [CrossRef] - Kirkaldy, J.S. Entropy criteria applied to pattern selection in systems with free boundaries. Metall. Trans. A
**1985**, 16, 1781–1796. [Google Scholar] [CrossRef] - Ziegler, H. An Introduction to Thermomechanics; Elsevier: Amsterdam, The Netherlands, 1983. [Google Scholar]
- Ziegler, H.; Wehrli, C. On a principle of maximal rate of entropy production. J. Non-Equilib. Therm.
**1978**, 12, 229. [Google Scholar] [CrossRef] - Trivedi, R.; Somboonsuk, K. Constrained Dendritic Growth and Spacing. Mater. Sci. Eng.
**1984**, 65, 65–74. [Google Scholar] [CrossRef] - Flemings, M.C. Solidification Processing; McGraw Hill: New York, NY, USA, 1974. [Google Scholar]
- Kurz, W.; Fisher, D.J. Fundamentals of Solidification, 4th ed.; Trans Tech Publications: Aedermannsdorf, Switzerland, 1989. [Google Scholar]
- Mehrabian, R.; Kear, B.H.; Cohen, M. Rapid Solidification Processing. Principles and Technologies, II. In Proceedings of the 2nd International Conference on Rapid Solidification Processing, Reston, VA, USA, 23–26 March 1980; Claitor’s Publishing Division: Baton Rouge, LA, USA, 1980; pp. 153–164. [Google Scholar]
- Tsallis, C. Nonadditive entropy: The concept and its use, Theoretical Physics. Eur. Phys.
**2009**, 40, 257–266. [Google Scholar] [CrossRef] [Green Version] - Haitao, Y.; Jiulin, D. Entropy Production Rate of Nonequilibrium Systems from the Fokker-Planck Equation. Available online: https://arxiv.org/ftp/arxiv/papers/1406/1406.4453.pdf (accessed on 15 June 2021).
- Schnakenberg, J. Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys.
**1976**, 48, 571. [Google Scholar] [CrossRef] - Glansdorff, P.; Prigogine, I. Structure, Stabilité et Fluctuations; Wiley-Interscience: Paris, France, 1971. [Google Scholar]
- Sekhar, J.A.; Li, H.P.; Dey, G.K. Decay-dissipative Belousov–Zhabotinsky nanobands and nanoparticles in NiAl. Acta Mater.
**2010**, 58, 1056–1073. [Google Scholar] [CrossRef] - Shohoji, N. Roles of Unstable Chemical Species and Non-Equilibrium Raction Routes on Properties of Reaction Product—A review. J. Surf. Interfaces Mater.
**2014**, 2, 182–205. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.; Servio, P.; Ray, A.D. Rate of Entropy Production in Evolving Interfaces and Membranes under Astigmatic Kinematics: Shape Evolution in Geometric-Dissipation Landscapes. Entropy
**2020**, 22, 909. [Google Scholar] [CrossRef] - Bilal, S. Finite element simulations for Entropy Generation. In Review.
**2020**. [Google Scholar] [CrossRef] - Utter, B.; Bodenschatz, E. Double Dendrite Growth in Solidification, August. Phys. Rev.
**2005**, 72, 011601. [Google Scholar] [CrossRef] [Green Version] - Trivedi, R.; Sekhar, J.A.; Seetharaman, V. Solidification Microstructures near the Limit of Absolute Stability. Met. Trans. A
**1989**, 20, 769–777. [Google Scholar] [CrossRef] - Fabietti, L.M.; Sekhar, J.A. Planar to Equiaxed Transition in the Presence of an External Wetting Surface. Metall. Trans.
**1992**, 23, 3361–3368. [Google Scholar] [CrossRef] - Fabietti, L.M.; Sekhar, J.A. Quantitative microstructure maps for restrained directional growth. J. Mater. Sci.
**1994**, 19, 473–477. [Google Scholar] [CrossRef] - Jones, H. The status of rapid solidification of alloys in research and application. J. Mater. Sci.
**1984**, 19, 1043–1076. [Google Scholar] [CrossRef] - Klement, W.; Willens, R.H.; Duwez, R. Non-crystalline structure in solidified gold–silicon alloys. Nature
**1960**, 187, 869–870. [Google Scholar] [CrossRef] - Shangguan, D.; Hunt, J.D. In situ observation of faceted cellular array growth. Metall Mater Trans A.
**1991**, 22, 941–945. [Google Scholar] [CrossRef] - Dey, N.; Sekhar, J.A. Interface Configurations during the directional growth of Salol-1 Morphology. Acta Metall. Mater.
**1993**, 41, 409–424. [Google Scholar] [CrossRef] - Biloni, H.; Boettinger, W.J. Solidification. In Physical Metallurgy, 4th ed.; Cahn, R.W., Haasen, P., Eds.; Elsevier: Amsterdam, The Netherlands, 1996; pp. 669–842. [Google Scholar]
- Mullins, W.W.; Sekerka, R.F. Stability of a Planar Interface during Solidification of a Dilute Binary Alloy. J. Appl. Phys.
**1964**, 35, 444. [Google Scholar] [CrossRef] - Sekhar, J.A. Laser Materials. Interaction. Ph.D. Thesis, University of Illinois, Urbana-Champaign, IL, USA, 1982. [Google Scholar]
- Glicksman, M.; Ankit, K. Thermodynamic behavior of solid-liquid grain boundary grooves. Philos. Mag.
**2020**, 100, 1789–1817. [Google Scholar] [CrossRef] - Baker, J.C.; Cahn, J.W. Thermodynamics of Solidification. In Solidification; ASM: Materials Park, OH, USA, 1971; p. 21. [Google Scholar]
- Kolmogorov, A.N. Entropy per unit time as a metric invariant of automorphisms. Dokl. Akad. Nauk SSSR
**1959**, 124, 754–755. [Google Scholar] - Sotolongo-Costa, O.; Rodriguez, I. Entropy variation in a fractal phase space. Acad. Lett.
**2021**, 662, 1–4. [Google Scholar] [CrossRef] - Trelles, J.P. Pattern formation and self-organization in plasmas interacting with surfaces. J. Phys. D Appl. Phys.
**2016**, 49, 393002. [Google Scholar] [CrossRef] [Green Version] - Kahn, D. Brain basis of self: Self-organization and lessons from dreaming. Front. Psychol.
**2013**, 4, 408. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ivanitskii, G.R. Self-organizing dynamic stability of far-from-equilibrium biological systems. Physics-Uspekhi
**2017**, 60, 705. [Google Scholar] [CrossRef] - Hiesinger, P.R. The Self-Assembling Brain: How Neural Networks Grow Smarter; Princeton University Press: Princeton, NJ, USA, 2021. [Google Scholar]
- Nosonovsky, M. Entropy in Tribology: In the Search for Applications. Entropy
**2010**, 12, 1345–1390. [Google Scholar] [CrossRef] - Nosonvsky, M.; Bhushan, B. Biomimetic superhydrophobic surfaces: Multiscale approach. Nano Lett.
**2007**, 7, 2633–2637. [Google Scholar] [CrossRef] [PubMed] - Sekhar, J.A. Tunable coefficient of friction with surface texturing in materials engineering and biological systems. Curr. Opin. Chem. Eng.
**2018**, 19, 94–106. [Google Scholar] [CrossRef] - Sekhar, J.A.; Mantri, A.S.; Saha, S.; Balamuralikrishnan, R.; Rao, P.R. Photonic, Low-Friction and Antimicrobial Applications for an Ancient Icosahedral/Quasicrystalline Nano-composite Bronze Alloy. Trans. Indian Inst. Met.
**2019**, 72, 2105–2119. [Google Scholar] [CrossRef] - De La Fuente, I.M.; Martínez, L.; Pérez-Samartín, A.L.; Ormaetxea, L.; Amezaga, C.; Vera-López, A. Global Self-Organization of the Cellular Metabolic Structure. PLoS ONE
**2008**, 3, e3100. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Blumenfeld, L.A.; Haken, H. Problems of Biological Physics; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Bensah, Y.D.; Sekhar, J.A. Morphological assessment with the maximum entropy production rate (MEPR) postulate. Curr. Opin. Chem. Eng.
**2014**, 3, 91–98. [Google Scholar] [CrossRef] - Lin, C.S.; Sekhar, J.A. Solidification Morphology and semi-solid deformation in Superalloy Rene 108 (Part IV). J. Mater. Sci.
**1994**, 29, 5005–5013. [Google Scholar] [CrossRef] - Nigmatullin, R.; Prokopenko, M. Thermodynamic Efficiency of Interactions in Self-Organizing Systems. Entropy
**2021**, 23, 757. [Google Scholar] [CrossRef] - Amiri, M.; Khonsari, M.M. On the Thermodynamics of Friction and Wear―A Review. Entropy
**2010**, 12, 1021–1049. [Google Scholar] [CrossRef] [Green Version] - Archard, J.F. Contact and rubbing of flat surfaces. J. Appl. Phys.
**1953**, 24, 981–988. [Google Scholar] [CrossRef] - Sista, B.; Vemaganti, K. Estimation of statistical parameters of rough surfaces suitable for developing micro-asperity friction models. Wear
**2014**, 316, 6–18. [Google Scholar] [CrossRef] - Rao, K.V.R.; Sekhar, J.A. Surface solidification with a moving heat source, a study of solidification parameters. Acta Metall.
**1987**, 3, 81. [Google Scholar] [CrossRef] - Basu, B.; Sekhar, J.A.; Schaefer, R.J.; Mehrabian, R. An analysis of the steady state molten pool obtained by heating a substrate with an electron beam. Acta Metall. Mater.
**1991**, 39, 725–733. [Google Scholar] [CrossRef] - Wang, Y.; Wang, Q.J.; Lin, C.; Shi, F. Development of a set of Stribeck curves for conformal contacts of rough surfaces. Tribol. Trans.
**2006**, 49, 526–535. [Google Scholar] [CrossRef] - Pasumarty, S.M.; Johnson, S.A.; Watson, S.A.; Adams, M.J. Friction of the human finger pad: Influence of moisture, occlusion and velocity. Tribol. Lett.
**2011**, 44, 117–137. [Google Scholar] [CrossRef] - Chen, G.S. Handbook of Friction-Vibration; Woodhead Publishing: Sawston, UK, 2014. [Google Scholar]
- Gersham, L.; Gersham, E.I.; Mironov, A.E.; Fox-Rabinovich, G.S.; Veldhuid, S.C. Application of the self-organization phenomenon in the development of wear resistant materials—A Review. Entropy
**2016**, 18, 385. [Google Scholar] [CrossRef] - Pavlos, G.P.; Illiopolous, A.C.; Zastenker, G.N.; Zeleny, L.M.; Karakatsanis, L.P.; Riazanteseva, M.; Xenalis, M.N.; Pavlov, E.G. Sudying Complexity in Solar Wind Plasma During Shock Events. arXiv
**2015**, arXiv:1310.0525. [Google Scholar] - Reis, A.H. Use and Validity of principles of extremum on entropy production in the study of complex systems. Ann. Phys.
**2014**, 346, 22–27. [Google Scholar] [CrossRef] - Pavlos, G.P.; Iliopoulos, A.C.; Karakatsanis, L.P.; Xenakis, M.; Pavlos, E. Complexity of Economical Systems, Special Issue on Econophysics. J. Eng. Sci. Technol. Rev.
**2015**, 8, 41–55. [Google Scholar] [CrossRef] - Bensah, Y.D.; Sekhar, J.A. Interfacial instability of a planar interface and diffuseness at the solid-liquid interface for pure and binary materials. arXiv
**2016**, arXiv:1605.05005. [Google Scholar] - Pontzer, H.; Brown, M.H.; Raichlen, D.A.; Dunsworth, H.; Hare, B.; Walker, K.; Luke, A.; Dugas, L.R.; Durazo-Arvizu, R.; Schoeller, D.; et al. Metabolic acceleration and the evolution of human brain size and life history. Nature
**2016**, 533, 390–392. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**The typical length scales that are studied (

**a**) for patterns and (

**b**) for defects for metallurgical assessments of engineering properties, (

**c**) shows the various shapes and arrays studied in this article for solidification on the left and for surface texture (with hierarchical nano features) on the right [56]. Note that the ripples at the peaks and valleys are also called nano-hierarchical structures depending on the scale.

**Figure 2.**(

**a**) A plot of the interface topography, as a function of the entropy generation rate for the solidification of salol (C

_{13}H

_{10}O

_{3}, produced by the interaction of salicylic acid and phenol), compared to predictions made by the MEPR calculation [13,75]. (

**a**) The plot shows the transition from a cellular faceted morphology to non-facet morphology for salol, with increasing velocity (dotted black line). The horizontal dotted red line is the prediction [13,75] of the boundary between the facet morphology and non-facet morphology [41]. (

**b**) Shows the facet morphological reorientation, with increasing driving force (velocity driven) [38,42]. This driving force, that establishes the entropy generation rate, scales with V/G or V.G [13,75]. Salol is an orthorhombic crystal structure, the (111) facet planes may take on 103.8, 67.6, or 31 degrees for the facet-tip angle. (

**c**) Facet reorientation with increasing entropy generation. The higher solidification rates lead to the finer facet tips [42]. The faceted tip undercooling and the entropy generation rate (per unit volume) increase with the imposed velocity of solidification (growth), (

**d**) from [42] the replacement of coarse tips by a finer tip, when required, with an increase in the velocity. As the solidification velocity is further increased, a side-branch formation feature is noted. This is a method of enhancing the entropy generation, as well as creating new defect-structures to enable entropy dissipation.

**Figure 3.**A schematic showing the directional solidification array features inside a control volume. The tip is at T

_{l}; the root is at T

_{s}. Only a few dendrites are shown in the mushy, solid liquid zone between T

_{l}and T

_{s}. The control volume is defined as the region between the rigorous liquid and rigorous solid.

**Figure 4.**Transparent material solidification patterns from [23,24,35,36,37,38], pictured by moving the glass slide enclosure, containing the transparent material from a hot to the cold zone, thereby affecting crystallization. The solid-liquid zone (the control volume) is bound by the isotherms for T

_{s}(Solidus or Eutectic temperature and T

_{t}(tips temperature). The maximum work efficiency possible is (T

_{l}− T

_{s})/T

_{l}, i.e., when d

**S**gen/dt = 0, alternately, the minimum work is when d

**S**gen/dt is maximized. The minimum extracted work cannot be zero, because defects and curved interfaces form within the patterns. The materials studied are succinonitrile, SCN (C₂H₄(CN)₂ in (

**a**), salol, (C

_{13}H

_{10}O

_{3}) in (

**b**), and carbon tetrabromide, CBr4, in (

**c**). In (

**a**), the SCN is grown along wetting (low boundary entropy) and non-wetting interfaces (high boundary entropy) in the same experiment [37,38]. Note that the higher tip temperatures are associated with larger, confused structure and boundary regions of the secondary dendrites. In (

**b**), from [42], the higher tip temperature is associated with sidearm-forming, faceted dendrites again; the higher, entropy-producing pattern require more defect area. As the demand on entropy production increases, because of a more severe imposed driving force (velocity of solidification), the solute partition function changes towards k

_{eff}~1. Consequently, very fine cells (i.e., with multiple boundaries), or a plane front (presumably with a large diffuse zone), re-emerge [36]. These are shown in (

**c**,

**d**), from [36]. Note in (

**d**) that the transition from very fine cells to a more planar interface involves a drop in the interface temperature.

**Figure 5.**Entropy generation rate per unit volume for the three morphologies namely, Cell (C) (with hemispherical tip) to cellular-dendrite (CD) (no side branches but with a paraboloid tip) transition to dendrite (D) with side-branches and paraboloid tip. The tip temperature T

_{c}

_{tip}increases discontinuously for transition from a cell with hemispherical tip to cellular dendrite with the paraboloid tip. For the comparisons, λ

_{1}is assumed to be constant.

**Figure 6.**Maximum entropy generation rate per unit volume for planar diffuse interfaces (P) (thin colored lines), cells (C), and dendrites (D). M is a possible maximum entropy rate generation curve for a new featureless crystalline material or metallic glass. Note that the curves for C and D may show a peak, as per Equation (10), similar to the behavior of the plane front with diffuse interface (P). The P (Plane Front) plots are from reference [13,75]. The M indicates metallic glass formation made by splat or ribbon cooling methods.

**Figure 7.**The scale of the typical features and patterns for self-organization in a solidified material, as a function of the rate of entropy generation per unit volume during transformation from liquid to a solid.

**Figure 8.**(

**a**) A self-organized efficient, one atmosphere plasma (from www.mhi-inc.com accessed on 26 July 2021). The plume length is about 800 mm. When this plasma interacts with a tool-bit surface, self-organized asperities form as a result of surface chemical reactions, as shown in (

**b**,

**c**). (

**b**) Gaussian-like (affine) distributed random asperities on M35 tool steel surface are shown (scale bar is 3 microns). In (

**c**) Quasi R (Quasi R is a trademark of MHI Inc., Cincinnati, OH, USA) asperities, which deviate from Gaussian distribution of RMS heights are shown (scale bar is 3 microns). Limited measurements for the coefficient of friction indicate that the texture in (

**c**) has a very low coefficient of friction [56]. The asperities in (

**b**,

**c**) are comprised of a nanoscale phase distribution of extremely fine nanoscale iron oxides and iron nitrides that display high elastic modulus along with a high hardness. The Vickers of the Hertzian zone is Hv~1300.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sekhar, J.A.
Self-Organization, Entropy Generation Rate, and Boundary Defects: A Control Volume Approach. *Entropy* **2021**, *23*, 1092.
https://doi.org/10.3390/e23081092

**AMA Style**

Sekhar JA.
Self-Organization, Entropy Generation Rate, and Boundary Defects: A Control Volume Approach. *Entropy*. 2021; 23(8):1092.
https://doi.org/10.3390/e23081092

**Chicago/Turabian Style**

Sekhar, Jainagesh A.
2021. "Self-Organization, Entropy Generation Rate, and Boundary Defects: A Control Volume Approach" *Entropy* 23, no. 8: 1092.
https://doi.org/10.3390/e23081092