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Dislocation Entropy: Temperature and Density Dependence

A. G. Sukharev

TL;DR

The paper addresses the thermodynamics of dislocations under laser shock peening by formulating a two-scale, locally equilibrated description and developing two entropy calculations: a combinatorial configurational entropy and a vibrational entropy from a physical pendulum (Peyrard-Nabarro) model with an energy barrier parameter $\alpha = \Delta/(2T)$. It provides analytic expressions for per-dislocation entropy $s_0$ and $s_1$, discusses their decomposition into configurational and field-induced contributions, and demonstrates how grain boundaries and external stress modify the entropy via both analytical formulas and numerical simulations of a 1D dislocation chain. The results reveal entropy saturation at high temperature and reductions under applied pressure, offering insights into dislocation-mediated hardening and grain refinement in polycrystals. Overall, the work links microscopic dislocation configurations and their vibrational spectra to macroscopic thermodynamic quantities relevant for material strength under dynamic loading.

Abstract

Laser hardening of metals occurs under the influence of a shock wave, which changes the distribution and density of one-dimensional defects - dislocations. There is a relationship between the density of dislocations, the grain size and the resistance of a single crystal to shear loading. The mechanism of hardening processes continues to be intensively studied, and the dynamics of defects plays a central role here. In this paper, the dislocation entropy is analyzed from a combinatorial point of view and from the point of view of a physical oscillator with a given energy reserve. Both contributions play an important role in describing the free energy of a one-dimensional ensemble of dislocations, and are necessary to take into account the dynamic processes inside the grain of a polycrystalline structure. Keywords: Laser Shock Peening, statistical mechanics

Dislocation Entropy: Temperature and Density Dependence

TL;DR

The paper addresses the thermodynamics of dislocations under laser shock peening by formulating a two-scale, locally equilibrated description and developing two entropy calculations: a combinatorial configurational entropy and a vibrational entropy from a physical pendulum (Peyrard-Nabarro) model with an energy barrier parameter . It provides analytic expressions for per-dislocation entropy and , discusses their decomposition into configurational and field-induced contributions, and demonstrates how grain boundaries and external stress modify the entropy via both analytical formulas and numerical simulations of a 1D dislocation chain. The results reveal entropy saturation at high temperature and reductions under applied pressure, offering insights into dislocation-mediated hardening and grain refinement in polycrystals. Overall, the work links microscopic dislocation configurations and their vibrational spectra to macroscopic thermodynamic quantities relevant for material strength under dynamic loading.

Abstract

Laser hardening of metals occurs under the influence of a shock wave, which changes the distribution and density of one-dimensional defects - dislocations. There is a relationship between the density of dislocations, the grain size and the resistance of a single crystal to shear loading. The mechanism of hardening processes continues to be intensively studied, and the dynamics of defects plays a central role here. In this paper, the dislocation entropy is analyzed from a combinatorial point of view and from the point of view of a physical oscillator with a given energy reserve. Both contributions play an important role in describing the free energy of a one-dimensional ensemble of dislocations, and are necessary to take into account the dynamic processes inside the grain of a polycrystalline structure. Keywords: Laser Shock Peening, statistical mechanics
Paper Structure (6 sections, 29 equations, 7 figures)

This paper contains 6 sections, 29 equations, 7 figures.

Table of Contents

  1. Введение
  2. Энтропия как результат подсчёта числа допустимых состояний системы
  3. Энтропия дислокации в рамках модели физического маятника
  4. Численные результаты
  5. Обсуждение вопросов применимости
  6. Заключение

Figures (7)

  • Figure 1: Возможное положение дислокаций и их реальное размещение. Места, закрашенные цветом, показывают реализованные размещения, прочие - возможные.
  • Figure 2: Энтропия (ось слева) и энергия (ось справа) на одну дислокацию при разных температурах. Кривая $s_1(T)$ (синяя) соответствует энтропии по формуле(\ref{['eq:nn8']}), вторая ${s_{app}}(T)$ - аппроксимации (\ref{['eq:nn7']})-(оранжевая) . Температура и энергия(\ref{['eq:nn9']})-(красная линяя) - измеряются в электронвольтах, теплоёмкость(коричневая) - безразмерная. Число вакансий на дислокацию взято $n/k = 2000$.
  • Figure 3: Линии уровня для положений дислокаций относительно левого края зерна в (в микронах, $(L + x(n))/d$) заданы цветовой шкалой как функция номера дислокации и наведённого давления $p0 = pd/Db$ ($d = 1\mu$)
  • Figure 4: Давление ${P_{{\rm{bound}}}}$ на границах зерна как функция давления внутри ${p_0}$. Давление нормировано на туже величину, как на предыдущем рисунке. Коэффициент ослабления примерно в сто раз (совпадает с числом дислокаций).
  • Figure 5: $s(T)$ - энтропия в пересчёте на одну дислокацию в зависимости от нормированной температуры, и ${s_{sp}}(T)$ - энтропия математического маятника с частотой, равной частоте колебаний дислокации на первом уровне энергии. Случай ${M_s} = 1$. Формулы для определения энтропии-(\ref{['eq:n23']})
  • ...and 2 more figures