Self-Organization, Entropy Generation Rate, and Boundary Defects: A Control Volume Approach
Abstract
:1. Introduction
2. Formalism
Steady State
3. Maximum Entropy Production Rate (MEPR) Principle
4. Patterns and Texture Examples from Directional Solidification, Wear, and Friction
4.1. Low-Velocity Transitions, Facets to Smooth Curvature
4.2. Medium Velocity Transitions: Cells, Cellular-Dendrites, and Dendrites
4.3. High-Velocity Regimes Including Featureless Solids and Metallic Glass
4.4. Range of Solidification Morphological Transitions
5. Discussions: The Utility of Self-Organization
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature and Abbreviations
Letter Symbols | |
A | area of an interface in a solid-liquid region (m2) |
ASD | 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 (m2·s−1) |
f | facet |
F | is the friction force (kg·ms−2) |
GSLI | gradient across a diffuse interface (K·m−1) |
Gl | 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) |
KL | thermal conductivity for a rigorous liquid (J·m−1·K−1·s−1) |
KS | thermal conductivity for a rigorous solid (J·m−1·K−1·s−1) |
Kav | average thermal conductivity in the control volume (J·m−1·K−1·s−1) |
Kwear | wear constant, dimensionless ~10−4 for metal pairs. |
Lo | sliding distance (length of object in Appendix A, in the direction of travel) |
mL | slope of the equilibrium liquidus at the SLI for a binary material (K·m3·mole−1) |
ms | slope of the equilibrium solidus line at the SLI for a binary material (K·m3·mole−1) |
MEPR | maximum entropy production rate per unit volume (J·m−3·K−1·s−1) |
MS | Mullins and Sekerka criterion [44] |
Np | normal force (kg·m·s−2) |
nf | non-facet |
Rg | molar gas constant (J·mol−1·K−1) |
RRMS | 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) |
SLG | 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. |
q1/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 Tl (the equilibrium liquidus temperature) |
Tsi | solidus interface temperature at a rigorous solid interface (K); generally assumed to be Ts (the equilibrium solidus or eutectic temperature) |
Tctip | tip temperature of a cell |
Ttip | tip temperature of cells, dendrites, or facets in an array |
Ti | friction interface temperature of a friction pair |
T0 | room temperature for the friction problem |
Ti | 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 |
T0 | room temperature (ambient) |
ΔTSLI | temperature difference across a solid-liquid interface (K) |
ΔTctip(C−D)) = Tctip − 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) = Tli − Tsi |
ΔTctip | (Tctip − 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 (m3) |
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) |
γgb | solid-solid boundary energy (e.g., between primary cells or primary dendrites) |
λ1 | primary spacing at the Ts isotherm |
λ2 | secondary spacing at the Ts isotherm |
λ2(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 γgb/Δssl. |
μ | coefficient of friction |
θ | fraction of energy transferred to heat. |
ζ | solid-liquid interface thickness (m), i.e., the diffuse interface thickness |
ζg | is the zone thickness between Tl and Tg |
ξ | wear volume |
τ | (length of travel, Lo)/V (s) |
ζ3cv | thermal control volume for steady state friction (Appendix A) |
ωD | energy of defects (J·m−3) other than area defects |
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
= (∫cv K·(ΔT/T)2) dζ3)/(ζ3cv) + ω/Ti/(ζ3cv)
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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
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 StyleSekhar, 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
APA StyleSekhar, J. A. (2021). Self-Organization, Entropy Generation Rate, and Boundary Defects: A Control Volume Approach. Entropy, 23(8), 1092. https://doi.org/10.3390/e23081092