#
Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Applicability of the Heat Equation with Dynamic Heat Capacity

## 3. Heat Equation with Dynamic Heat Capacity

## 4. Heat Equation with Dynamic Heat Capacity: Plane Geometry

**Example**

**1.**

^{3}, $\rho {c}_{0}=$ 2 × 10

^{6}J/m

^{3}K, $\lambda =$ 0.3 W/mK, ${D}_{0}=$ 1.5 × 10

^{−7}m

^{2}/s, and ${\Phi}_{0}=\rho {h}_{0}$, where ${h}_{0}=$ 200 J/g is the heat release during crystallization. We focused on the nanometer and nanosecond scales since they are close to real processes during subcritical crystal nucleation [59,60,61]. First, we verified that Equation (21) gave the correct solution $T\left(t,x\right)$ for a rectangular pulsed heat source. Indeed, the solutions ${\stackrel{\u02d8}{T}}_{p}\left(t,x\right)$ and ${T}_{p}\left(t,x\right)$ represented by Equations (23) and (25) coincided with the results $\stackrel{\u02d8}{T}\left(t,x\right)$ and $T\left(t,x\right)$ calculated using the Fourier series (see Equation (21)) for $m$ up to 45 (see Figure 1b). In addition, Figure 1b shows that the effect of the temporal dispersion of the dynamic heat capacity on the temperature was significant.

## 5. Heat Equation with Dynamic Heat Capacity: Spherical Geometry

**Example**

**2.**

^{−7}m

^{2}/s (see Figure 4). Note, in the case of glass-forming substances, the relaxation times ${\tau}_{0}$ have a broad distribution. Next, we considered the influence of this distribution on the solution $T\left(t,r\right)$ for the spherically symmetric problem.

## 6. Dependence of the Solution $T\left(t,r\right)$ on the Distribution of Relaxation Times ${\tau}_{0}$

**Example**

**3.**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Latin Symbols | |

${c}_{dyn}\left(t\right)$ | dynamic heat capacity (J·kg^{−1}·K^{−1}) |

${c}_{in}$ | initial part of ${c}_{dyn}\left(t\right)$ (J·kg^{−1}·K^{−1}) |

${c}_{0}$ | equilibrium heat capacity (J·kg^{−1}·K^{−1}) |

${D}_{0}$ | thermal diffusivity ${D}_{0}=\lambda /\rho {c}_{0}$ (m^{2}·s^{−1}) |

$d$ | sample thickness (m) |

$F\left(t\right)$ | heat flux time dependence (dimensionless) |

$H\left({\tau}_{0}\right)$ | distribution function (s^{−1}) |

${h}_{0}$ | heat release (J·kg^{−1}) |

${l}_{ph}$ | phonon mean-free-path (m) |

$r,x$ | space variables (m) |

$R$ | radius of spherical sample (m) |

${r}_{0}$ | radius of spherical heat source (m) |

$t$ | time (s) |

$T\left(t,x\right)$ | solution to non-equilibrium heat equation (K) |

$\stackrel{\u02d8}{T}\left(t,x\right)$ | solution to conventional heat equation (K) |

$\delta T\left(t,x\right)$ | non-equilibrium component of the solution $T\left(t,x\right)$ (K) |

${x}_{0}$ | thickness of the flat heat source (m) |

Greek Symbols | |

$\beta $ | Kohlrausch coefficient (dimensionless) |

${\gamma}_{n}$ | nth relaxation parameter (s^{−1}) |

${\epsilon}_{0}$ | $\left({c}_{0}-{c}_{in}\right)/{c}_{0}$ (dimensionless) |

$\theta \left(t\right)$ | Heaviside unit step function (dimensionless) |

$\lambda $ | thermal conductivity (W·K^{−1}·m^{−1}) |

${\mu}_{n}$ | nth relaxation parameter (s^{−1}) |

$\rho $ | density (kg·m^{−3}) |

${\tau}_{K}$ | Kohlrausch relaxation time (s) |

${\tau}_{n}$ | time constant of nth component (s) |

${\tau}_{0}$ | Debye relaxation time (s) |

${\tau}_{p}$ | duration of the heating pulse (s) |

$\Phi \left(\overrightarrow{r}\right)F\left(t\right)$ | volumetric heat flux (W·m^{−3}) |

$\Phi \left(\overrightarrow{r}\right)$ | heat flux space dependence (W·m^{−3}) |

$\frac{{\Phi}_{n}}{\rho {c}_{0}}$ | nth Fourier components (K/s) |

${\chi}_{n},{\phi}_{n}$ | nth Fourier components (K) |

${\psi}_{n,m}\left(t\right)$ | n,mth Fourier component (K) |

## Appendix A

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**Figure 1.**(

**a**) Rectangular heating pulse ${F}_{p}\left(t\right)$ with duration ${\tau}_{p}=$ 10 ns. (

**b**) Time dependences ${\stackrel{\u02d8}{T}}_{p}\left(t,\frac{d}{2}\right)$ and $\stackrel{\u02d8}{T}\left(t,\frac{d}{2}\right)$ (shown using squares and circles), as well as ${T}_{p}\left(t,\frac{d}{2}\right)$ and $T\left(t,\frac{d}{2}\right)$ for ${\epsilon}_{0}=1/3$ and ${\tau}_{0}=$ 30 ns (shown using up and down triangles).

**Figure 2.**(

**a**) Heating pulses ${F}_{1}\left(t\right)$, ${F}_{A}\left(t\right)$, and ${\mathit{\mathcal{F}}}_{A}\left(t\right)$ (shown using squares, circles, and crosses, respectively). (

**b**) Time dependences of ${\stackrel{\u02d8}{T}}_{1}\left(t,\frac{d}{2}\right)$ and ${\stackrel{\u02d8}{T}}_{A}\left(t,\frac{d}{2}\right)$ (shown using squares and circles), as well as ${T}_{1}\left(t,\frac{d}{2}\right)$ and ${T}_{A}\left(t,\frac{d}{2}\right)$ for ${\epsilon}_{0}=1/3$ and ${\tau}_{0}=$ 30 ns (shown using up triangles and down triangles). The inset shows the ratios $\delta {T}_{1}$ and $\delta {T}_{A}$ (shown using squares and circles).

**Figure 3.**(

**a**) Time dependences of ${\stackrel{\u02d8}{T}}_{p}\left(t,0\right)$, ${\stackrel{\u02d8}{T}}_{2}\left(t,0\right)$, and ${\tilde{T}}_{A}\left(t,0\right)$ (shown using filled squares, circles, and up triangles, respectively), as well as ${T}_{p}\left(t,0\right)$, ${T}_{2}\left(t,0\right)$, and ${T}_{A}\left(t,0\right)$ for ${\epsilon}_{0}=1/3$ and ${\tau}_{0}=$ 5 ns (shown using open squares, circles, and up triangles, respectively) and (

**b**) the corresponding spatial temperature distributions at ${t}_{0}$ = 1.5 ns and $R$ = 300 nm. Similar dependences for ${\stackrel{\u02d8}{T}}_{A}$ and ${T}_{A}$ were calculated at $R$ = 1000 nm (shown using down triangles).

**Figure 4.**Time dependences of ${T}_{A}\left(t,0\right)$ at ${\tau}_{0}=$ 0.2 ns, 0.5 ns, 1 ns, 2 ns, 5 ns, 30 ns, and 300 ns (shown using squares, circles, up triangles, down triangles, diamonds, stars, and crosses, respectively), as well as ${\stackrel{\u02d8}{T}}_{A}\left(t,0\right)$ (shown using filled squares). The inset shows ${T}_{A}\left({t}_{0},0\right)$ as a function of ${\tau}_{0}$ at ${t}_{0}$ = 1.4 ns.

**Figure 5.**(

**a**) Time dependences of ${\stackrel{\u02d8}{T}}_{1}\left(t,0\right)$, ${T}_{1}\left(t,0,{\tau}_{K}\left({T}_{in}\right)\right)$, and ${T}_{1}\left(t,0,{\tau}_{K}\left({T}_{max}\right)\right)$ (shown using squares, circles, and triangles, respectively) (

**b**) and corresponding spatial temperature distributions at ${t}_{0}$ = 1.5 ns for PS at ${T}_{in}=$ 500 K and a sinusoidal heat source ${F}_{1}\left(t\right)$. Similar dependences were found for ${T}_{1}\left(t,r,{\tau}_{K}\left({T}_{in}\right)\right)$ (shown using stars), which were calculated using the uniform distribution ${H}_{u}\left({\tau}_{0}\right)$. PS: polystyrene.

**Figure 6.**(

**a**) Time dependences of ${\stackrel{\u02d8}{T}}_{1}\left(t,0\right)$, ${T}_{1}\left(t,0,{\tau}_{K}\left({T}_{in}\right)\right)$, and ${T}_{1}\left(t,0,{\tau}_{K}\left({T}_{max}\right)\right)$ (shown using squares, circles, and triangles, respectively) and (

**b**) the corresponding spatial temperature distributions at ${t}_{0}$ = 1.5 ns for PS at ${T}_{in}=$ 400 K and a sinusoidal heat source ${F}_{1}\left(t\right)$. Similar dependences were found for ${T}_{1}\left(t,r,{\tau}_{K}\left({T}_{in}\right)\right)$ (shown using stars), which were calculated using the uniform distribution ${H}_{u}\left({\tau}_{0}\right)$.

**Figure 7.**(

**a**) Time dependences of ${\stackrel{\u02d8}{T}}_{1}\left(t,0\right)$, ${T}_{1}\left(t,0,{\tau}_{K}\left({T}_{in}\right)\right)$, and ${T}_{1}\left(t,0,{\tau}_{K}\left({T}_{max}\right)\right)$ (shown using squares, circles, and triangles, respectively) and (

**b**) the corresponding spatial temperature distributions at ${t}_{0}$ = 1.5 ns for PMMA at ${T}_{in}=$ 700 K and a sinusoidal heat source ${F}_{1}\left(t\right)$. PMMA: poly(methyl methacrylate).

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**MDPI and ACS Style**

Minakov, A.A.; Schick, C.
Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions. *Symmetry* **2021**, *13*, 256.
https://doi.org/10.3390/sym13020256

**AMA Style**

Minakov AA, Schick C.
Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions. *Symmetry*. 2021; 13(2):256.
https://doi.org/10.3390/sym13020256

**Chicago/Turabian Style**

Minakov, Alexander A., and Christoph Schick.
2021. "Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions" *Symmetry* 13, no. 2: 256.
https://doi.org/10.3390/sym13020256