# Nanoscale Quantum Thermal Conductance at Water Interface: Green’s Function Approach Based on One-Dimensional Phonon Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Extraction of Phonon Dispersion Curves of Liquid Water from Experimental Results

- Extracting wave numbers ${k}_{b}$ and ${k}_{e}$ at yellow and red broken lines of absorbance maxima in the experimental absorption spectrum of water in the liquid phase shown in Figure 3.
- Calculating angular frequencies ${\omega}_{q}$ and ${\omega}_{r}$ as $2\pi {k}_{b}c$ and $2\pi {k}_{e}c$, respectively, with c being the light velocity, so that the one-dimensional phonon model is consistent with the experimental spectrum.
- Calculating angular frequency ${\omega}_{p}$ by using Equation (2c) above.
- Calculating the wave number k of phonon by using the dispersion relation Equation (1) above.
- Obtaining dispersion curves in the upper part of Figure 4 below with the frequency on the abscissa and the normalized wave number $k{l}_{j}/\pi $ on the ordinate.

## 3. Formulation of the Phonon Transmission Function through H${}_{2}$O

## 4. Thermal Conductance in Water

#### 4.1. Validation of Calculated Thermal Conductance

#### 4.2. Classical Limit of Thermal Conductance

- variable heat sources in stimulated cells (order 10${}^{1}$);
- length scales (order 10${}^{1}$–10${}^{2}$);
- micro- and nanoscale thermal parameters (order 10${}^{1}$–10${}^{2}$).

- reservoir model representing heat and bath reservoirs;
- one-dimensional phonon model with water molecules sandwiched between the reservoirs;
- phonon transports at atomic scale in water.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GFM | Green’s function method |

Trad/s | unit of tera radians per second |

tr | trace of matrix |

Im | imaginary part of complex number |

## Appendix A. Phonon Transport Equations in a One-Dimensional Chain Model of Water

#### Appendix A.1. Derivation of Transport Equations of Phonon in Water

#### Appendix A.2. Derivation of Eigenvalue Equations in Left and Right Reservoirs

#### Appendix A.3. Non-Equilibrium Green’s Functions

#### Appendix A.4. Quantum Thermal Conductance in Water

- Transforming wave functions consisting of the tight binding (TB) basis into those consisting of the plane wave (PW) basis.
- Finding the PW basis solutions in semi-infinite (${N}_{L}$, ${N}_{R}$→∞) heat reservoir at both ends of water with using the periodic characteristics of the heat reservoir in thermal equilibrium.
- Inverse transformation from PW basis wave functions to TB basis ones.

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Sample Availability: Not available. |

**Figure 1.**Definitions of displacements, masses, and spring constants of the elements H${}_{2}$ (blue) and O (yellow) in the one-dimensional phonon model.

**Figure 2.**One-dimensional phonon model of reservoir-water-reservoir structure. Each unit cell j has length ${l}_{j}$, ($j=-{N}_{L}+1,-{N}_{L}+2,\cdots N+{N}_{R}$) along the x-direction and temperature ${\theta}_{j}$, ($j=0,1,2,\cdots N+1$), where ${\theta}_{-{N}_{L}+1}={\theta}_{-{N}_{L}+2}=\cdots ={\theta}_{0}$ and ${\theta}_{N+1}={\theta}_{N+2}=\cdots ={\theta}_{N+{N}_{R}}$ [23].

**Figure 3.**Absorption spectrum of water in the liquid phase [24]. Yellow and red broken lines represent the vibration modes (b) and (e), respectively.

**Figure 4.**Complex dispersion curves (

**upper**) and transmission probability spectrum of the phonon (

**lower**). The upper right inset plots enlarged dispersion curves in the frequency range of 642–645 Trad/s.

**Figure 5.**Modes of H${}_{2}$O molecular motions of (

**a**) anti-symmetric stretching; (

**b**) symmetric stretching; (

**c**) bending; (

**d**) libration; and (

**e**) intermolecular vibration.

**Figure 7.**Temperature dependences of thermal conductance calculated by Equation (7). The parameters 1, 2, ⋯ 10 with unit of degrees Kelvin in the graph mean temperature differences ${\theta}_{0}-{\theta}_{N+1}$ between the left and right reservoirs.

**Table 1.**Wavenumbers and angular frequencies for various molecular vibration modes of water (c: light velocity).

Wave Number | Angular Frequency | ||
---|---|---|---|

Mode | $\mathit{\alpha}$ | ${\mathit{k}}_{\mathit{\alpha}}$ (cm${}^{-1}$) | 2 $\mathit{\pi}$${\mathit{k}}_{\mathit{\alpha}}\mathit{c}$ (Trad/s) |

OH anti-symmetric | a | 3509 | 661.4 |

stretching | |||

OH symmetric | b | 3410 | 642.8 |

stretching | |||

OH bending | c | 1660 | 312.9 |

Binding rotation | d | 700 | 131.9 |

(libration) | |||

Intermolecular | e | 190 | 35.8 |

vibration |

**Table 2.**Each vibrational mode (left), angular frequency range (center) and transmission probability (right) ${}^{\u2020}$.

Mode | Angular Frequency | Transmission | |
---|---|---|---|

Range | Probability | ||

$\left(\phantom{\rule{0.166667em}{0ex}}\mathrm{I}\phantom{\rule{0.166667em}{0ex}}\right)$ | acoustic wave | $0<\omega <{\omega}_{r}$ | $\cong 1$ |

$\left(\mathrm{I}\phantom{\rule{-0.09995pt}{0ex}}\mathrm{I}\right)$ | attenuation | ${\omega}_{r}<\omega <{\omega}_{q}$ | $\cong 0$ |

$\left(\mathrm{I}\phantom{\rule{-1.49994pt}{0ex}}\mathrm{I}\phantom{\rule{-1.49994pt}{0ex}}\mathrm{I}\right)$ | optical wave | ${\omega}_{q}<\omega <{\omega}_{p}$ | $\cong 1$ |

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**MDPI and ACS Style**

Umegaki, T.; Tanaka, S.
Nanoscale Quantum Thermal Conductance at Water Interface: Green’s Function Approach Based on One-Dimensional Phonon Model. *Molecules* **2020**, *25*, 1185.
https://doi.org/10.3390/molecules25051185

**AMA Style**

Umegaki T, Tanaka S.
Nanoscale Quantum Thermal Conductance at Water Interface: Green’s Function Approach Based on One-Dimensional Phonon Model. *Molecules*. 2020; 25(5):1185.
https://doi.org/10.3390/molecules25051185

**Chicago/Turabian Style**

Umegaki, Toshihito, and Shigenori Tanaka.
2020. "Nanoscale Quantum Thermal Conductance at Water Interface: Green’s Function Approach Based on One-Dimensional Phonon Model" *Molecules* 25, no. 5: 1185.
https://doi.org/10.3390/molecules25051185