# Temperature Distribution through a Nanofilm by Means of a Ballistic-Diffusive Approach

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

## 2. Methods: Constitutional Equation for Ballistic Heat Transport

^{2}, the unity of ${\mathit{Q}}^{\left(2\right)}$ is W/(m·s), according to (13): $\left[{\mathit{Q}}^{\left(n+1\right)}\right]=\frac{\left[L\right]}{\left[t\right]}\left[{\mathit{Q}}^{\left(n\right)}\right]$, where [$L$] is a unity in length. As a direct consequence of the property that ${\mathit{Q}}^{\left(2\right)}$ is the flux of $\mathit{q}$, one has in (8) that $\frac{{\beta}_{1}}{{\gamma}_{1}}=-1$ or ${\beta}_{1}=-{\gamma}_{1}=-{\tau}_{1}{\nu}_{1}$. For further application, we neglect higher orders of the relaxation times, i.e., ${\tau}_{2}={\tau}_{3}=\dots ={\tau}_{n}=0$, which means that the phenomenological coefficients ${\beta}_{n}$ and ${\gamma}_{n}$, with $n\ge 2$, and ${\nu}_{n}$, and $n\ge 3$ (note that although ${\beta}_{2}$ and ${\gamma}_{2}$ are related to ${\tau}_{2}$, ${\nu}_{2}$ is rather related to ${\tau}_{1}$), can be omitted. Equation (15) becomes:

## 3. Temperature Distribution in Nanoscaled Materials

#### 3.1. Two-Temperature Model

#### 3.2. Boundary and Initial Conditions

#### 3.3. Comparison with Fourier and Cattaneo Laws

#### 3.4. Numerical Method

**,**a vector of dimension $\left(\mathrm{n}-2\right)$ containing the unknown temperature values at the spatial nodes at time ${t}^{*}+1,$ and $B$ being a vector of dimension $\left(\mathrm{n}-2\right),$ containing the known temperature values at time ${t}^{*}$. Due to the spatial gradients in the bulk equations, the values at nodes $2$ and $\mathrm{n}-1$ depend on the boundary values at nodes $1$ and $\mathrm{n}$, respectively. The vector $V$ is calculated as $V={A}^{-1}B$. The boundary condition for the ballistic temperature at the hot side is imposed and held at that value at each time step. The boundary condition for the ballistic temperature at the cold side as well as the boundary conditions of the diffusive temperature at both sides at time ${t}^{*}+1$ are then obtained through the bulk values at time ${t}^{*}+1$. This procedure is repeated until the relative errors of the bulk and boundary values of both the ballistic and diffusive temperatures between the previous and present loops become smaller than ${10}^{-4}$. The obtained values at time ${t}^{*}+1$ are stocked in the output matrix, and used as the known values for the next time step ${t}^{*}+2$. This procedure is outlined in a schematic form in Figure 1.

## 4. Results

#### 4.1. Temperature Distribution of the Two-Temperature Model

#### 4.2. Temperature Distribution of the Fourier and Cattaneo Laws with an Effective Thermal Conductivity

## 5. Discussion

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

$c$ | heat capacity per unit volume [J/m^{3}K] |

${f}_{c}$ | correction factor |

${\mathit{J}}^{s}$ | entropy flux [Wm/K] |

$k$ | wavenumber vector |

$Kn$ | Knudsen number |

$\mathcal{l}$ | mean free path [m] |

$L$ | characteristic length [m] |

$\mathit{q}$ | heat flux [W/m^{2}] |

${\mathit{Q}}^{\left(n\right)}$ | heat flux of order n [W/m^{2}(m/s)^{n−1}] |

$r$ | position vector |

$r$ | source term [W/m^{3}] |

$\U0001d4c8$ | entropy per unit volume [J/Km^{3}] |

$T$ | temperature [K] |

$t$ | time [s] |

${t}^{*}$ | dimensionless time |

$u$ | internal energy per unit volume [J/m^{3}] |

$v$ | phonon velocity [m/s] |

$z$ | spatial coordinate [m] |

${z}^{*}$ | dimensionless spatial coordinate |

Greek symbols | |

${\eta}^{s}$ | rate of entropy production [W/K] |

$\theta $ | dimensionless temperature |

$\lambda $ | thermal conductivity [W/Km] |

$\tau $ | relaxation time [s] |

Subscript | |

$0$ | initial state |

$b$ | ballistic |

$c$ | Cattaneo |

$d$ | diffusive |

$eff$ | effective value |

$f$ | Fourier |

Upperscript | |

$^$ | Fourier transformed variable |

## References

- Machrafi, H.; Lebon, G. General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport. Appl. Math. Lett.
**2016**, 52, 30–37. [Google Scholar] [CrossRef] [Green Version] - Hill, T.L. Thermodynamics of Small Systems; Dover: New York, NY, USA, 1994. [Google Scholar]
- Niemann, J.; Härter, S.; Kästle, C.; Franke, J. Challenges of the miniaturization in the electronics production on the example of 01005 components. In Tagungsband des 2. Kongresses Montage Handhabung Industrieroboter; Springer Vieweg: Berlin, Germany, 2017; pp. 113–123. [Google Scholar]
- Moore, A.L.; Shi, L. Emerging challenges and materials for thermal management of electronics. Materialstoday
**2014**, 17, 163–174. [Google Scholar] [CrossRef] - Cahill, D.G.; Ford, W.K.; Goodson, K.E.; Mahan, G.D.; Majumdar, A.; Maris, H.J.; Merlin, R.; Phillpot, S.R. Nanoscale thermal transport. J. Appl. Phys.
**2003**, 93, 793. [Google Scholar] [CrossRef] - Dinh, T.; Phan, H.P.; Kashaninejad, N.; Nguyen, T.K.; Dao, D.V.; Nguyen, N.T. An on-chip SiC MEMS device with integrated heating, sensing, and microfluidic cooling systems. Adv. Mater. Interfaces
**2018**, 5, 1800764. [Google Scholar] [CrossRef] - Zheng, W.; Huang, B.; Li, H.; Koh, Y.K. Achieving huge thermal conductance of metallic nitride on graphene through enhanced elastic and inelastic phonon transmission. ACS Appl. Mater. Interfaces
**2018**, 10, 35487–35494. [Google Scholar] [CrossRef] [PubMed] - Yang, L.; Chen, Z.G.; Dargusch, M.S.; Zou, J. High performance thermoelectric materials: Progress and their applications. Adv. Energy Mater.
**2018**, 8, 1701797. [Google Scholar] [CrossRef] - Yang, N.; Xu, X.; Zhang, G.; Li, B. Thermal transport in nanostructures. AIP Adv.
**2012**, 2, 041410. [Google Scholar] [CrossRef] [Green Version] - Kuleyev, I.I.; Kuleyev, I.G.; Bakharev, S.M. Phonon focusing and features of phonon transport in silicon nanofilms and nanowires at low temperatures. Phys. Status Solidi B
**2015**, 252, 323–332. [Google Scholar] [CrossRef] - Terris, D.; Joulain, K.; Lacroix, D.; Lemonnier, D. Numerical simulation of transient phonon heat transfer in silicon nanowires and nanofilms J. Phys. Conf. Ser.
**2007**, 92, 012077. [Google Scholar] [CrossRef] - Gao, Y.; Bao, W.; Meng, Q.; Jing, Y.; Song, X. The thermal transport properties of single-crystalline nanowires covered with amorphous shell: A molecular dynamics study. J. Non-Cryst. Solids
**2014**, 387, 132–138. [Google Scholar] [CrossRef] - Kaiser, J.; Feng, T.; Maassen, J.; Wang, X.; Ruan, X.; Lundstrom, M. Thermal transport at the nanoscale: A Fourier’s law vs. phonon Boltzmann equation study. J. Appl. Phys.
**2017**, 121, 044302. [Google Scholar] [CrossRef] - Guo, Y.; Wang, M. Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures. Phys. Rev. B
**2018**, 97, 035421. [Google Scholar] [CrossRef] - Tang, D.S.; Hua, Y.C.; Cao, B.Y. Thermal wave propagation through nanofilms in ballistic-diffusive regime by Monte Carlo simulations. Int. J. Therm. Sci.
**2016**, 109, 81–89. [Google Scholar] [CrossRef] - Li, H.L.; Hua, Y.C.; Cao, B.Y. A hybrid phonon Monte Carlo-diffusion method for ballistic-diffusive heat conduction in nano- and micro-structures. Int. J. Heat Mass Transf.
**2018**, 127, 1014–1022. [Google Scholar] [CrossRef] - Guyer, R.A.; Krumhansl, J.A. Solution of the linearized Boltzmann phonon equation. Phys. Rev.
**1966**, 148, 766–778. [Google Scholar] [CrossRef] - Guyer, R.A.; Krumhansl, J.A. Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic Crystals. Phys. Rev.
**1966**, 148, 778–788. [Google Scholar] [CrossRef] - Yu, Y.J.; Li, C.L.; Xue, Z.N.; Tian, X.G. The dilemma of hyperbolic heat conduction and its settlement by incorporating spatially nonlocal effect at nanoscale. Phys. Lett. A
**2016**, 380, 255–261. [Google Scholar] [CrossRef] - Koh, Y.K.; Cahill, D.G.; Sun, B. Nonlocal theory for heat transport at high frequencies. Phys. Rev. B
**2014**, 90, 205412. [Google Scholar] [CrossRef] - Tzou, D.Y. Macro to Microscale Heat Transfer: The Lagging Behaviour; Taylor and Francis: New York, NY, USA, 1997. [Google Scholar]
- Ordonez-Miranda, J.; Alvarado-Gil, J. On the stability of the exact solutions of the dual-phase lagging model of heat conduction. Nanoscale Res. Lett.
**2011**, 6, 327. [Google Scholar] [CrossRef] [Green Version] - Cattaneo, C. Sulla conduzione del calore. Atti del Seminario Matematico e Fisico dell’ Universita di Modena
**1948**, 3, 83–101. [Google Scholar] - Jou, D.; Casas-Vazquez, J.; Lebon, G. Extended Irreversible Thermodynamics, 4th ed.; Springer: Berlin, Germany, 2010. [Google Scholar]
- Lebon, G.; Machrafi, H.; Grmela, M.; Dubois, C. An extended thermodynamic model of transient heat conduction at sub-continuum scales. Proc. R. Soc. A
**2011**, 467, 3241–3256. [Google Scholar] [CrossRef] [Green Version] - Longshaw, S.M.; Borg, M.K.; Ramisetti, S.B.; Zhang, J.; Lockerby, D.A.; Emerson, D.R.; Reese, J.M. mdFoam+: Advanced molecular dynamics in OpenFOAM. Comput. Phys. Commun.
**2018**, 224, 1–21. [Google Scholar] [CrossRef] - Machrafi, H. An extended thermodynamic model for size-dependent thermoelectric properties at nanometric scales: Application to nanofilms, nanocomposites and thin nanocomposite films. Appl. Math. Model.
**2016**, 40, 2143–2160. [Google Scholar] [CrossRef] [Green Version] - Lebon, G.; Jou, D.; Casas-Vazquez, J. Understanding Non-Equilibrium Thermodynamics; Springer: Berlin, Germany, 2008. [Google Scholar]
- Modest, M.F. Radiative Heat Transfer; McGraw-Hill: New York, NY, USA, 1993. [Google Scholar]
- Alvarez, F.X.; Jou, D. Boundary conditions and Evolution of Ballistic Heat transport. ASME J. Heat Transf.
**2010**, 132, 0124404. [Google Scholar] [CrossRef] - Swartz, E.T.; Pohl, R.O. Thermal boundary resistance. Rev. Mod. Phys.
**1981**, 61, 605–668. [Google Scholar] [CrossRef] - Hess, S. On Nonlocal Constitutive Relations, Continued Fraction Expansion for the Wave Vector Dependent Diffusion Coefficient. Z. Naturforsch.
**1977**, 32a, 678–684. [Google Scholar] [CrossRef] - Machrafi, H. Heat transfer at nanometric scales described by extended irreversible thermodynamics. Commun. Appl. Ind. Math.
**2016**, 7, 177–195. [Google Scholar] [CrossRef] [Green Version] - Lebon, G.; Machrafi, H. Effective thermal conductivity of nanostructures: A review. Atti della Accademia Peloritana dei Pericolanti
**2019**, 96, A14. [Google Scholar] [CrossRef] - Joshi, A.A.; Majumdar, A. Transient ballistic and diffusive phonon heat transport in thin films. J. Appl. Phys.
**1993**, 74, 31–39. [Google Scholar] [CrossRef]

**Figure 1.**Numerical scheme for $\theta \left({z}^{*},{t}^{*}\right)$. This scheme is general, and applied to all of the temperatures in this study.

**Figure 2.**Non-dimensional temperature profiles $\theta \left({z}^{*},{t}^{*}\right)$ as a function of distance ${z}^{*}=z/L$ at different times ${t}^{*}=t/{\tau}_{b}$ (${t}^{*}=1,10$, and $100$, respectively) for $K{n}_{d}=K{n}_{b}=Kn=0.1$, $1$, and $10$. The respective contributions of the ballistic, diffusive, and total temperatures are shown and compared to the ones obtained from Cattaneo’s and Fourier’s equations.

**Figure 3.**Non-dimensional temperature profiles $\theta \left({z}^{*},{t}^{*}\right)$ as a function of distance ${z}^{*}=z/L$ at different times ${t}^{*}=t/{\tau}_{b}$ (${t}^{*}=1,10$, and $100$, respectively) for $K{n}_{d}=K{n}_{b}=Kn=0.1$, $1$, and $10$. The total temperatures from Figure 2 are compared to the ones obtained from Fourier’s and Cattaneo’s equations ((39) and (40)), using the effective thermal conductivity (38).

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Machrafi, H.
Temperature Distribution through a Nanofilm by Means of a Ballistic-Diffusive Approach. *Inventions* **2019**, *4*, 2.
https://doi.org/10.3390/inventions4010002

**AMA Style**

Machrafi H.
Temperature Distribution through a Nanofilm by Means of a Ballistic-Diffusive Approach. *Inventions*. 2019; 4(1):2.
https://doi.org/10.3390/inventions4010002

**Chicago/Turabian Style**

Machrafi, Hatim.
2019. "Temperature Distribution through a Nanofilm by Means of a Ballistic-Diffusive Approach" *Inventions* 4, no. 1: 2.
https://doi.org/10.3390/inventions4010002