First-principles computation of Onsager-like degradation–healing coefficients and healing kinetics for friction-induced surface self-organization and self-healing

Sliding surfaces degrade through friction and wear, yet some driven interfaces spontaneously build protective order: plant leaves regrow their waxy coating, and engineering contacts form tribofilms or selective-transfer layers. Nosonovsky and Bhushan (2010) cast these phenomena in a linear irreversible-thermodynamic framework in which degradation and healing are coupled Onsager flows with phenomenological coefficients $L$ , $M$ , $H$ and a healing-rate function $f(\xi)$ , yielding the healing-dominance criterion $M^{2}/(HL)>1$ , and fitted the model to plant-wax regeneration data. The framework, however, offers no way to compute $L$ , $M$ , $H$ , or $f(\xi)$ from material microstructure and microscopic physics, is restricted to linear, spatially homogeneous behavior, and its application to engineering self-organization regimes (tribofilm formation, selective transfer) remains qualitative. This problem asks for a first-principles determination of these coefficients and healing functions from microstructure, the nonlinear and heterogeneous extensions required for quantitative prediction, and predictive, mechanism-specific constitutive equations for tribofilm and selective-transfer self-organization. No published method derives the framework's Onsager-like coefficients from microstructure, and the problem remains open.

Background

Friction and wear are dissipative irreversible processes: mechanical work expended at a sliding interface is converted to heat and entropy. Classical tribology describes them through empirical laws (Coulomb–Amontons friction, Archard wear, in which wear volume grows in proportion to normal load and sliding distance) rather than from first principles. Nonequilibrium (irreversible) thermodynamics offers a complementary route: near local equilibrium (each small material element relaxing fast enough that equilibrium thermodynamics applies within it), irreversible flows $J_i$ (of heat, matter, or, here, degradation and healing 'rates') are linear functions of their conjugate thermodynamic forces $Y_i$ , $J_i=\sum_k L_{ik}Y_k$ , with a symmetric coefficient matrix (Onsager relations, from Onsager's 1931 reciprocal relations), and the entropy production rate is $\dot S=\sum_i J_i Y_i \ge 0$ .

Building on entropy-based views of wear, Nosonovsky and Bhushan (2009, 2010) formulated a thermodynamic framework for friction-induced self-organization and surface self-healing. For a sliding interface they estimate the local entropy production rate in a thin subsurface layer as $\mathrm{d}S/\mathrm{d}t=(\mu W V)^{2}/(\lambda T^{2})$ , where $\mu$ is the friction coefficient, $W$ the normal load, $V$ the sliding velocity, $\lambda$ the thermal conductivity, and $T$ the local temperature. A Glansdorff–Prigogine-type stability criterion — the second variation of entropy production with respect to a microstructural parameter $\psi$ becoming negative — signals instability of the nominal state and the possible emergence of self-organized secondary structures (for example tribofilms, the reaction-formed protective boundary layers created by lubricant additives or material transfer at sliding contacts) with lowered friction and wear. Degradation (wear, cracking) and healing are then represented as two coupled scalar flows: $J^{\mathrm{deg}}=L\,Y^{\mathrm{deg}}+M\,Y^{\mathrm{heal}}$ and $J^{\mathrm{heal}}=M\,Y^{\mathrm{deg}}+H\,Y^{\mathrm{heal}}$ , with diagonal Onsager-like coefficients $L,H>0$ and off-diagonal coupling $M$ . With a constant degradational force $Y^{\mathrm{deg}}=\kappa$ and a linearized healing force $Y^{\mathrm{heal}}=-\beta\zeta$ ( $\zeta$ a healing-state variable and $\beta$ the healing-force coefficient, distinct from the thermal conductivity $\lambda$ ), the model has closed-form solutions, and the degradation variable $\xi$ has negative drift — healing dominates degradation — precisely when $M^{2}/(HL)>1$ . The healing kinetics in general depends on a function $f(\xi)$ of the degradation/healing state $\xi$ (for instance, healed wax area), whose functional form encodes mechanism and geometry. Fitting the analytic solution to Koch, Bhushan, Ensikat and Barthlott's (2009) measurements of epicuticular wax regeneration (the self-repair of the waxy crystal layer on plant leaf surfaces, observed as concentric or strip-like regrowth saturating in time) captured the observed trends, but the fitted parameter combinations are phenomenological and non-unique.

The framework has clear precedents and neighbors. Fox-Rabinovich, Gershman and co-workers analyze self-organization during friction via the same second-variation (dissipative-structure) criteria for adaptive hard coatings and tribofilms; the degradation–entropy generation (DEG) theorem of Bryant, Khonsari and colleagues relates wear measures to entropy generation through experimentally calibrated degradation coefficients; and continuum damage mechanics (the continuum theory of progressive material deterioration through internal state variables) has produced thermodynamically consistent damage–healing constitutive models (Voyiadjis, Shojaei and Li; Alsheghri and Abu Al-Rub) for self-healing materials. All of these, like the source framework itself, leave the phenomenological coefficients to experiment: none derives them from microstructure. The unresolved target is the missing microscopic foundation of the thermodynamic theory of friction-induced self-organization and self-healing.

Problem Statement

Establish the microscopic foundation of the thermodynamic theory of friction-induced surface self-organization and self-healing formulated by Nosonovsky and Bhushan. Specifically: (1) develop a general first-principles method — starting from material microstructure and microscopic physics — to compute the Onsager-like coefficients $L$ , $M$ , $H$ coupling degradation and healing flows at sliding interfaces and the functional form of the healing dependence $f(\xi)$ , so that entropy-production rates and healing kinetics are predicted rather than fitted; (2) extend the analysis beyond the linear, spatially homogeneous regime by incorporating the neglected effects required for quantitative prediction: phase-transition entropy changes (for example, solidification of healing agents), detailed thermal transients, nonlinear Onsager behavior away from the linear regime, and spatially heterogeneous transport and reaction kinetics, thereby giving a mechanistic account of the nonlinear saturation observed in healing data such as plant epicuticular wax regeneration; and (3) derive predictive, mechanism-specific constitutive equations for the self-organization regimes of engineering materials — tribofilm formation and selective transfer (the formation of thin, low-shear, copper-rich protective films at bronze–steel sliding contacts under glycerin-based lubrication) — identifying the mechanisms and parameter ranges that lead to self-organized low-wear states. The common objective uniting these deliverables is to replace phenomenologically fitted coefficients with a microscopically grounded, predictive constitutive theory linking entropy production, microstructure, and healing, aimed at predicting when a driven sliding interface will self-organize or self-heal rather than degrade. Parts (1)–(3) are complementary components of this single objective rather than independent alternatives, and the list of effects in part (2) is illustrative scope rather than an exhaustive checklist.

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