# Anisotropic Thermal Conductivity of Inkjet-Printed 2D Crystal Films: Role of the Microstructure and Interfaces

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}(glass). ${A}_{Kth}$ is found to be ∼30, independent of the chemical composition of the 2D crystal and the films thickness (<400 nm). Ab-initio modeling shows that even for such low ${K}_{\perp}$, energy transport is essentially ballistic across near-ideal interfaces. This is a remarkable result considering that previous reports, demonstrating comparable thermal conductivities, are obtained for either the disordered amorphous limit [11,31] or by maximizing atomic mass contrast in layered compounds [12,14]. At the same time, the measured ${K}_{\Vert}$ of these films are found to be very similar to one another, while this may at first be surprising considering the nearly 2 orders of magnitude difference in intrinsic thermal conductivities of the constituent crystals, the weak flake bonding filters high-energy phonon modes and limits the phonon spectrum contributing to the overall ${K}_{\Vert}$. This highlights fundamental differences from previous studies on single crystal 2D materials.

## 2. 2D Crystal Film Preparation and Characterization

_{2}and 6.5 nm for graphene and h-BN, which however cannot be converted directly into number of layers, as the value also includes the presence of residual stabilizer on the surfaces of the nanosheets. A more precise estimate of the actual thickness was obtained by transmission electron microscopy, which indicates 4–7 layers on average [32]. Taking a value of 6 layers, this corresponds to average thickness of 2 nm for h-BN and the two graphene dispersions, and 4 nm for MoS${}_{2}$, and this is the flake thickness used in the theoretical analysis.

## 3. Results

**Table 1.**Summary table of reported thermal conductivity for films made by solution processing. In the table, the following abbreviations are used: rGO for reduced graphene oxide, LPE for liquid-phase exfoliation, ECE for electro-chemical exfoliation, FLG for few-layer graphene, GNP for graphene nano-platelet, IJP for ink-jet printed, NSs for nano-sheets, VF for vacuum filtration.

Materials | Flake Thickness (nm) | Lateral Size ($\mathsf{\mu}$m) | Thickness ($\mathsf{\mu}$m) | ${\mathit{K}}_{\Vert}$ (Wm${}^{-1}$K${}^{-1}$) | ${\mathit{K}}_{\perp}$ (Wm${}^{-1}$K${}^{-1}$) | ${\mathit{A}}_{\mathit{Kth}}$ | Method | Reference |
---|---|---|---|---|---|---|---|---|

rGO | 1.1 | — | 4.3–12 | 1100 | — | — | 2000 ${}^{\xb0}$C annealed | [49] |

rGO | ∼1 | Avg. area 23 $\mathsf{\mu}$m^{2} | 7.5 | 1390 ± 65 | — | — | VF, HI acid reduced | [50] |

rGO | ∼1 | Avg. area 1 $\mathsf{\mu}$m^{2} | — | 900 ± 45 | — | — | VF, HI acid reduced | [50] |

rGO | — | 25 | — | 1434 | — | — | Electrospray deposition, 2850 ${}^{\xb0}$C annealed | [51] |

rGO | 1–7 | 108 | 10 | 1940 ± 113 | — | — | Scraping deposition, compressed and 3000 ${}^{\xb0}$C annealed | [36] |

rGO | <1 | >6 | 0.8 | 3200 | — | — | 2850 ${}^{\xb0}$C annealed, compressed | [52] |

rGO | — | — | 170 | 62 | 0.09 | 675 | 1000 ${}^{\xb0}$C annealed | [23] |

rGO + Carbon nanorings | — | — | — | 890 | 5.8 | 15 | VF, in situ growth of CNR (800 ${}^{\xb0}$C) | [53] |

LPE graphene | <10 layers | — | 30 | 110 | 0.25 | 440 | VF | [24] |

LPE graphene | — | 0.96–1.24 | 9–44 | 40–90 | — | — | VF, compressed | [21] |

ECE graphene | 4 | 3–4 | — | 1023 | — | — | VF, 2500 ${}^{\xb0}$C annealed | [43] |

ECE graphene | ≤8 layers | — | 33 | 674 | — | — | VF | [54] ^{1} |

ECE graphene | ∼2.2 | >10 | 5–10 | 3390 | 5.5 | 616 | VF | [46] |

Fluorinated graphene | 0.8–2.3 | 0.8 | 10–100 | 88–242 | 0.4–22 | 220–11 | Ball milling, VF | [26] |

Functionalized FLG | 7.35 | — | 1050 | 112–123 | 1.62–1.81 | 69–68 | VF | [27] |

GNP | ≤10 | 15 | — | 178 ± 12 | 1.28 ± 0.12 | 139 | Microwave exfoliation, VF, 340 ${}^{\xb0}$C annealed | [28] |

GNP | 4–5 layers | 0.648 | 30–70 | 1529 | — | — | Ball milling, 2850 ${}^{\xb0}$C annealed | [44] |

IJP graphene | ∼2 | ∼0.2 | 0.08–0.4 | ∼12 | 0.3 | ∼27–40 | LPE, IJP, 150 ${}^{\xb0}$C annealed | This work |

hBN laminate | 10 | 1 | 10–100 | 20 | — | — | LPE, VF | [22] |

BN NSs | 2.9 ± 0.3 | 1.8 ± 0.1 | 10–30 | 58 | 3.3 | 18 | Molten alkali-assisted exfoliation, VF, 450 ${}^{\xb0}$C annealed | [29] |

IJP hBN | ∼2 | ∼0.2 | 0.2–1 | ∼11 | 0.5 | ∼22 | LPE, IJP, 150 ${}^{\xb0}$C annealed | This work |

MXene (Ti${}_{3}$C${}_{2}$Tx) | — | — | 3000 | 55 | — | — | Chemical etching, VF | [55] |

IJP MoS${}_{2}$ | ∼4 | ∼0.05 | 0.06–0.25 | ∼9.5 | 0.3 | ∼32 | LPE, IJP, 150 ${}^{\xb0}$C annealed | This work |

^{1}https://doi.org/10.1109/ICEPT.2015.7236587, accessed on 25 October 2022.

## 4. Ab-Initio Modeling

#### 4.1. Ballistic Model and Ideal Interface

#### 4.2. Ballistic vs. Diffusive Transport

#### 4.3. Disorder Limit for the BTE

#### 4.4. Hard Limits of the Phonon Models

#### 4.5. Ideal vs. Dirty Interfaces

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Kim, W.; Wang, R.; Majumdar, A. Nanostructuring expands thermal limits. Nano Today
**2007**, 2, 40–47. [Google Scholar] [CrossRef] - Pop, E. Energy dissipation and transport in nanoscale devices. Nano Res.
**2010**, 3, 147–169. [Google Scholar] [CrossRef][Green Version] - Stan, M. Discovery and design of nuclear fuels. Mater. Today
**2009**, 12, 20–28. [Google Scholar] [CrossRef] - Zhao, L.D.; Lo, S.H.; Zhang, Y.; Sun, H.; Tan, G.; Uher, C.; Wolverton, C.; Dravid, V.P.; Kanatzidis, M.G. Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature
**2014**, 508, 373–377. [Google Scholar] [CrossRef] - Wu, S.; Yan, T.; Kuai, Z.; Pan, W. Thermal conductivity enhancement on phase change materials for thermal energy storage: A review. Energy Storage Mater.
**2020**, 25, 251–295. [Google Scholar] [CrossRef] - Baur, J.; Silverman, E. Challenges and Opportunities in Multifunctional Nanocomposite Structures for Aerospace Applications. MRS Bull.
**2007**, 32, 328–334. [Google Scholar] [CrossRef] - Shi, X.; Kong, H.; Li, C.P.; Uher, C.; Yang, J.; Salvador, J.R.; Wang, H.; Chen, L.; Zhang, W. Low thermal conductivity and high thermoelectric figure of merit in n-type BaxYbyCo4Sb12 double-filled skutterudites. Appl. Phys. Lett.
**2008**, 92, 182101. [Google Scholar] [CrossRef] - Tian, X.; Itkis, M.E.; Bekyarova, E.B.; Haddon, R.C. Anisotropic Thermal and Electrical Properties of Thin Thermal Interface Layers of Graphite Nanoplatelet-Based Composites. Sci. Rep.
**2013**, 3, 1710. [Google Scholar] [CrossRef][Green Version] - Bain, J.A.; Malen, J.A.; Jeong, M.; Ganapathy, T. Nanoscale thermal transport aspects of heat-assisted magnetic recording devices and materials. MRS Bull.
**2018**, 43, 112–118. [Google Scholar] [CrossRef] - Slack, G.A. Nonmetallic crystals with high thermal conductivity. J. Phys. Chem. Solids
**1973**, 34, 321–335. [Google Scholar] [CrossRef] - Cahill, D.G.; Pohl, R.O. Lattice Vibrations and Heat Transport in Crystals and Glasses. Annu. Rev. Phys. Chem.
**1988**, 39, 93–121. [Google Scholar] [CrossRef] - Chiritescu, C.; Cahill, D.G.; Nguyen, N.; Johnson, D.; Bodapati, A.; Keblinski, P.; Zschack, P. Ultralow Thermal Conductivity in Disordered, Layered WS2 Crystals. Science
**2007**, 315, 351. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lindroth, D.O.; Erhart, P. Thermal transport in van der Waals solids from first-principles calculations. Phys. Rev. B
**2016**, 94, 115205. [Google Scholar] [CrossRef][Green Version] - Hadland, E.; Jang, H.; Falmbigl, M.; Fischer, R.; Medlin, D.L.; Cahill, D.G.; Johnson, D.C. Synthesis, Characterization, and Ultralow Thermal Conductivity of a Lattice-Mismatched SnSe2(MoSe2)1.32 Heterostructure. Chem. Mater.
**2019**, 31, 5699–5705. [Google Scholar] [CrossRef] - Qian, X.; Gu, X.; Dresselhaus, M.S.; Yang, R. Anisotropic Tuning of Graphite Thermal Conductivity by Lithium Intercalation. J. Phys. Chem. Lett.
**2016**, 7, 4744–4750. [Google Scholar] [CrossRef] - Fugallo, G.; Cepellotti, A.; Paulatto, L.; Lazzeri, M.; Marzari, N.; Mauri, F. Thermal Conductivity of Graphene and Graphite: Collective Excitations and Mean Free Paths. Nano Lett.
**2014**, 14, 6109–6114. [Google Scholar] [CrossRef] [PubMed] - Erhart, P.; Hyldgaard, P.; Lindroth, D.O. Microscopic Origin of Thermal Conductivity Reduction in Disordered van der Waals Solids. Chem. Mater.
**2015**, 27, 5511–5518. [Google Scholar] [CrossRef] - Dechaumphai, E.; Lu, D.; Kan, J.J.; Moon, J.; Fullerton, E.E.; Liu, Z.; Chen, R. Ultralow Thermal Conductivity of Multilayers with Highly Dissimilar Debye Temperatures. Nano Lett.
**2014**, 14, 2448–2455. [Google Scholar] [CrossRef] - Shahzadeh, M.; Andriyevska, O.; Salikhov, R.; Fallarino, L.; Hellwig, O.; Pisana, S. Nondiffusive Transport and Anisotropic Thermal Conductivity in High-Density Pt/Co Superlattices. ACS Appl. Electron. Mater.
**2021**, 3, 1931–1936. [Google Scholar] [CrossRef] - Sood, A.; Xiong, F.; Chen, S.; Cheaito, R.; Lian, F.; Asheghi, M.; Cui, Y.; Donadio, D.; Goodson, K.E.; Pop, E. Quasi-Ballistic Thermal Transport Across MoS
_{2}Thin Films. Nano Lett.**2019**, 19, 2434–2442. [Google Scholar] [CrossRef] - Malekpour, H.; Chang, K.H.; Chen, J.C.; Lu, C.Y.; Nika, D.L.; Novoselov, K.S.; Balandin, A.A. Thermal Conductivity of Graphene Laminate. Nano Lett.
**2014**, 14, 5155–5161. [Google Scholar] [CrossRef] [PubMed] - Zheng, J.C.; Zhang, L.; Kretinin, A.V.; Morozov, S.V.; Wang, Y.B.; Wang, T.; Li, X.; Ren, F.; Zhang, J.; Lu, C.Y.; et al. High thermal conductivity of hexagonal boron nitride laminates. 2D Mater.
**2016**, 3, 011004. [Google Scholar] [CrossRef][Green Version] - Renteria, J.D.; Ramirez, S.; Malekpour, H.; Alonso, B.; Centeno, A.; Zurutuza, A.; Cocemasov, A.I.; Nika, D.L.; Balandin, A.A. Strongly Anisotropic Thermal Conductivity of Free-Standing Reduced Graphene Oxide Films Annealed at High Temperature. Adv. Funct. Mater.
**2015**, 25, 4664–4672. [Google Scholar] [CrossRef] - Zhang, Y.; Edwards, M.; Samani, M.K.; Logothetis, N.; Ye, L.; Fu, Y.; Jeppson, K.; Liu, J. Characterization and simulation of liquid phase exfoliated graphene-based films for heat spreading applications. Carbon
**2016**, 106, 195–201. [Google Scholar] [CrossRef] - McManus, D.; Vranic, S.; Withers, F.; Sanchez-Romaguera, V.; Macucci, M.; Yang, H.; Sorrentino, R.; Parvez, K.; Son, S.K.; Iannaccone, G.; et al. Water-based and biocompatible 2D crystal inks for all-inkjet-printed heterostructures. Nat. Nanotechnol.
**2017**, 12, 343. [Google Scholar] [CrossRef][Green Version] - Vu, M.C.; Thi Thieu, N.A.; Lim, J.H.; Choi, W.K.; Chan Won, J.; Islam, M.A.; Kim, S.R. Ultrathin thermally conductive yet electrically insulating exfoliated graphene fluoride film for high performance heat dissipation. Carbon
**2020**, 157, 741–749. [Google Scholar] [CrossRef] - Liang, Q.; Yao, X.; Wang, W.; Liu, Y.; Wong, C.P. A Three-Dimensional Vertically Aligned Functionalized Multilayer Graphene Architecture: An Approach for Graphene-Based Thermal Interfacial Materials. ACS Nano
**2011**, 5, 2392–2401. [Google Scholar] [CrossRef] - Xiang, J.; Drzal, L.T. Thermal conductivity of exfoliated graphite nanoplatelet paper. Carbon
**2011**, 49, 773–778. [Google Scholar] [CrossRef] - Fu, L.; Wang, T.; Yu, J.; Dai, W.; Sun, H.; Liu, Z.; Sun, R.; Jiang, N.; Yu, A.; Lin, C.T. An ultrathin high-performance heat spreader fabricated with hydroxylated boron nitride nanosheets. 2D Mater.
**2017**, 4, 025047. [Google Scholar] [CrossRef] - Worsley, R.; Pimpolari, L.; McManus, D.; Ge, N.; Ionescu, R.; Wittkopf, J.A.; Alieva, A.; Basso, G.; Macucci, M.; Iannaccone, G.; et al. All-2D Material Inkjet-Printed Capacitors: Toward Fully Printed Integrated Circuits. ACS Nano
**2019**, 13, 54–60. [Google Scholar] [CrossRef] - Balandin, A.A. Thermal properties of graphene and nanostructured carbon materials. Nat. Mater.
**2011**, 10, 569–581. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yang, H.; Withers, F.; Gebremedhn, E.; Lewis, E.; Britnell, L.; Felten, A.; Palermo, V.; Haigh, S.; Beljonne, D.; Casiraghi, C. Dielectric nanosheets made by liquid-phase exfoliation in water and their use in graphene-based electronics. 2D Mater.
**2014**, 1, 011012. [Google Scholar] [CrossRef] - Rahman, M.; Shahzadeh, M.; Braeuninger-Weimer, P.; Hofmann, S.; Hellwig, O.; Pisana, S. Measuring the thermal properties of anisotropic materials using beam-offset frequency domain thermoreflectance. J. Appl. Phys.
**2018**, 123, 245110. [Google Scholar] [CrossRef] - Rahman, M.; Shahzadeh, M.; Pisana, S. Simultaneous measurement of anisotropic thermal conductivity and thermal boundary conductance of 2-dimensional materials. J. Appl. Phys.
**2019**, 126, 205103. [Google Scholar] [CrossRef][Green Version] - Schmidt, A.J.; Collins, K.C.; Minnich, A.J.; Chen, G. Thermal conductance and phonon transmissivity of metal-graphite interfaces. J. Appl. Phys.
**2010**, 107, 104907. [Google Scholar] [CrossRef] - Jiang, P.; Qian, X.; Gu, X.; Yang, R. Probing Anisotropic Thermal Conductivity of Transition Metal Dichalcogenides MX2 (M = Mo, W and X = S, Se) using Time-Domain Thermoreflectance. Adv. Mater.
**2017**, 29, 1701068. [Google Scholar] [CrossRef] - Wei, Z.; Chen, Y.; Dames, C. Negative correlation between in-plane bonding strength and cross-plane thermal conductivity in a model layered material. Appl. Phys. Lett.
**2013**, 102, 011901. [Google Scholar] [CrossRef] - Klein, C.A.; Holland, M.G. Thermal Conductivity of Pyrolytic Graphite at Low Temperatures. I. Turbostratic Structures. Phys. Rev.
**1964**, 136, A575–A590. [Google Scholar] [CrossRef] - Slack, G.A. Anisotropic Thermal Conductivity of Pyrolytic Graphite. Phys. Rev.
**1962**, 127, 694–701. [Google Scholar] [CrossRef] - Schmidt, A.J.; Chen, X.; Chen, G. Pulse accumulation, radial heat conduction, and anisotropic thermal conductivity in pump-probe transient thermoreflectance. Rev. Sci. Instrum.
**2008**, 79, 114902. [Google Scholar] [CrossRef] - Lindsay, L.; Broido, D.A. Enhanced thermal conductivity and isotope effect in single-layer hexagonal boron nitride. Phys. Rev. B
**2011**, 84, 155421. [Google Scholar] [CrossRef][Green Version] - Liu, J.; Choi, G.M.; Cahill, D.G. Measurement of the anisotropic thermal conductivity of molybdenum disulfide by the time-resolved magneto-optic Kerr effect. J. Appl. Phys.
**2014**, 116, 233107. [Google Scholar] [CrossRef][Green Version] - Kwon, Y.J.; Kwon, Y.; Park, H.S.; Lee, J.U. Mass-Produced Electrochemically Exfoliated Graphene for Ultrahigh Thermally Conductive Paper Using a Multimetal Electrode System. Adv. Mater. Interfaces
**2021**, 6, 1900095. [Google Scholar] [CrossRef] - Teng, C.; Xie, D.; Wang, J.; Yang, Z.; Ren, G.; Zhu, Y. Ultrahigh Conductive Graphene Paper Based on Ball-Milling Exfoliated Graphene. Adv. Funct. Mater.
**2021**, 27, 1700240. [Google Scholar] [CrossRef] - Ong, W.L.; Rupich, S.M.; Talapin, D.V.; McGaughey, A.J.H.; Malen, J.A. Surface chemistry mediates thermal transport in three-dimensional nanocrystal arrays. Nat. Mater.
**2013**, 12, 410–415. [Google Scholar] [CrossRef] - Gee, C.M.; Tseng, C.C.; Wu, F.Y.; Lin, C.T.; Chang, H.P.; Li, L.J.; Chen, J.C.; Hu, L.H. Few layer graphene paper from electrochemical process for heat conduction. Mater. Res. Innov.
**2014**, 18, 208–213. [Google Scholar] [CrossRef] - Costescu, R.M.; Cahill, D.G.; Fabreguette, F.H.; Sechrist, Z.A.; George, S.M. Ultra-Low Thermal Conductivity in W/Al2O3 Nanolaminates. Science
**2004**, 303, 989–990. [Google Scholar] [CrossRef] - Muratore, C.; Varshney, V.; Gengler, J.J.; Hu, J.J.; Bultman, J.E.; Smith, T.M.; Shamberger, P.J.; Qiu, B.; Ruan, X.; Roy, A.K.; et al. Cross-plane thermal properties of transition metal dichalcogenides. Appl. Phys. Lett.
**2013**, 102, 081604. [Google Scholar] [CrossRef] - Shen, B.; Zhai, W; Zheng, W. Ultrathin flexible graphene film: An excellent thermal conducting material with efficient EMI shielding. Adv. Funct. Mater.
**2014**, 24, 4542–4548. [Google Scholar] [CrossRef] - Kumar, P.; Shahzad, F.; Yu, S.; Hong, S.M.; Kim, Y.-H.; Koo, C.M. Large-area reduced graphene oxide thin film with excellent thermal conductivity and electromagnetic interference shielding effectiveness. Carbon
**2015**, 94, 494–500. [Google Scholar] [CrossRef] - Xin, G.; Sun, H.; Hu, T.; Fard, H.R.; Sun, X.; Koratkar, N.; Borca-Tasciuc, T.; Lian, J. Large-area freestanding graphene paper for superior thermal management. Adv. Mater.
**2014**, 26, 4521–4526. [Google Scholar] [CrossRef] - Wang, N.; Samani, M.K.; Li, H.; Dong, L.; Zhang, Z.; Su, P.; Chen, S.; Chen, J.; Huang, S.; Yuan, G.; et al. Tailoring the thermal and mechanical properties of graphene film by structural engineering. Small
**2018**, 29, 1801346. [Google Scholar] [CrossRef] - Zhang, J.; Shi, G.; Jiang, C.; Ju, S.; Jiang, D. 3D bridged carbon nanoring/graphene hybrid paper as a high-performance lateral heat spreader. Small
**2015**, 11, 6197–6204. [Google Scholar] [CrossRef] [PubMed] - Zhao, B.; Zhang, K.; Yuen, M.M.F.; Fu, X.-Z.; Sun, R.; Wong, C.P. Electrochemically exfoliated graphene nanosheets for thermal and electrical conductions. In Proceedings of the 2015 16th International Conference on Electronic Packaging Technology (ICEPT), Changsha, China, 11–14 August 2015; 2015; pp. 253–255. [Google Scholar]
- Liu, R.; Li, W. High-thermal-stability and high-thermal-conductivity Ti3C2Tx MXene/poly(vinyl alcohol) (PVA) composites. ACS Omega
**2018**, 3, 2609–2617. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jiang, P.; Qian, X.; Yang, R.; Lindsay, L. Anisotropic thermal transport in bulk hexagonal boron nitride. Phys. Rev. Mater.
**2018**, 2, 064005. [Google Scholar] [CrossRef] - Fugallo, G.; Lazzeri, M.; Paulatto, L.; Mauri, F. Ab initio variational approach for evaluating lattice thermal conductivity. Phys. Rev. B
**2013**, 88, 045430. [Google Scholar] [CrossRef][Green Version] - Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G.L.; Cococcioni, M.; Dabo, I.; et al. QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter
**2009**, 21, 395502. [Google Scholar] [CrossRef] - Baroni, S.; de Gironcoli, S.; Dal Corso, A.; Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys.
**2001**, 73, 515–562. [Google Scholar] [CrossRef][Green Version] - Paulatto, L.; Mauri, F.; Lazzeri, M. Anharmonic properties from a generalized third-order ab initio approach: Theory and applications to graphite and graphene. Phys. Rev. B
**2013**, 87, 214303. [Google Scholar] [CrossRef][Green Version] - Swartz, E.T.; Pohl, R.O. Thermal boundary resistance. Rev. Mod. Phys.
**1989**, 61, 605–668. [Google Scholar] [CrossRef] - Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Chen, G. Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices. Phys. Rev. B
**1998**, 57, 14958–14973. [Google Scholar] [CrossRef] - Hua, C.; Minnich, A.J. Semi-analytical solution to the frequency-dependent Boltzmann transport equation for cross-plane heat conduction in thin films. J. Appl. Phys.
**2015**, 117, 175306. [Google Scholar] [CrossRef][Green Version] - Maassen, J.; Lundstrom, M. Steady-state heat transport: Ballistic-to-diffusive with Fourier’s law. J. Appl. Phys.
**2015**, 117, 035104. [Google Scholar] [CrossRef][Green Version] - Minnich, A.J. Determining Phonon Mean Free Paths from Observations of Quasiballistic Thermal Transport. Phys. Rev. Lett.
**2012**, 109, 205901. [Google Scholar] [CrossRef] - Zhang, H.; Chen, X.; Jho, Y.D.; Minnich, A.J. Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite. Nano Lett.
**2016**, 16, 1643–1649. [Google Scholar] [CrossRef][Green Version] - Ziman, J. Electrons and Phonons; Clarendon Press: Oxford, UK, 1960. [Google Scholar]
- Simoncelli, M.; Marzari, N.; Mauri, F. Unified theory of thermal transport in crystals and glasses. Nat. Phys.
**2019**, 15, 809–813. [Google Scholar] [CrossRef][Green Version] - Isaeva, L.; Barbalinardo, G.; Donadio, D.; Baroni, S. Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach. Nat. Commun.
**2019**, 10, 3853. [Google Scholar] [CrossRef][Green Version] - Allen, P.B.; Feldman, J.L. Thermal conductivity of disordered harmonic solids. Phys. Rev. B
**1993**, 48, 12581–12588. [Google Scholar] [CrossRef] - Chen, Z.; Dames, C. An anisotropic model for the minimum thermal conductivity. Appl. Phys. Lett.
**2015**, 107, 193104. [Google Scholar] [CrossRef] - Cahill, D.G.; Watson, S.K.; Pohl, R.O. Lower limit to the thermal conductivity of disordered crystals. Phys. Rev. B
**1992**, 46, 6131–6140. [Google Scholar] [CrossRef] [PubMed] - Hernandez, Y.; Nicolosi, V.; Lotya, M.; Blighe, F.M.; Sun, Z.; De, S.; McGovern, I.T.; Holland, B.; Byrne, M.; Gun’Ko, Y.K.; et al. High-yield production of graphene by liquid-phase exfoliation of graphite. Nat. Nanotechnol.
**2008**, 3, 563. [Google Scholar] [CrossRef] [PubMed][Green Version] - Coleman, J.N.; Lotya, M.; O’Neill, A.; Bergin, S.D.; King, P.J.; Khan, U.; Young, K.; Gaucher, A.; De, S.; Smith, R.J.; et al. Two-dimensional nanosheets produced by liquid exfoliation of layered materials. Science
**2011**, 331, 568. [Google Scholar] [CrossRef] [PubMed][Green Version] - Takahashi, Y.; Azumi, T.; Sekine, Y. Heat capacity of aluminum from 80 to 800 K. Themochimica Acta
**1989**, 139, 133. [Google Scholar] [CrossRef] - Butland, A.T.D.; Maddison, R.J. The specific heat of graphite: An evaluation of measurements. J. Nucl. Mater.
**1973**, 49, 45. [Google Scholar] [CrossRef] - Dworkin, A.S.; Sasmor, D.J.; van Artsdalen, E.R. The thermodynamics of boron nitride: Low-temperature heat capacity and entropy; heats of combustion and formation. J. Chem. Phys.
**1954**, 22, 837. [Google Scholar] [CrossRef] - Volovik, L.S.; Fesenko, V.V.; Bolgar, A.S.; Drozdova, S.V.; Klochkov, L.A.; Primachenko, V.F. Enthalpy and heat capacity of molybdenum disulfide. Sov. Powder Metall. Met. Ceram.
**1978**, 17, 697. [Google Scholar] [CrossRef] - Foss, C.J.; Aksamija, Z. Quantifying thermal boundary conductance of 2D-3D interfaces. 2D Mater.
**2019**, 6, 025019. [Google Scholar] [CrossRef][Green Version] - Gabourie, A.J.; Koroglu, C.; Pop, E. Substrate-dependence of monolayer MoS2 thermal conductivity and thermal boundary conductance. arXiv
**2022**, arXiv:2204:11381. [Google Scholar] [CrossRef] - Lide, D.R. CRC Handbook of Chemistry and Physics; Taylor & Francis: Abingdon, UK, 2007. [Google Scholar]
- Perdew, J.P.; Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B
**1981**, 23, 5048. [Google Scholar] [CrossRef][Green Version] - Chen, S.; Sood, A.; Pop, E.; Goodson, K.E.; Donadio, D. Strongly tunable anisotropic thermal transport in MoS2 by strain and lithium intercalations: First-principles calculations. 2D Mater.
**2019**, 6, 025033. [Google Scholar] [CrossRef] - Serrano, J.; Bosak, A.; Arenal, R.; Krisch, M.; Watanabe, K.; Taniguchi, T.; Kanda, H.; Rubio, A.; Wirtz, L. Vibrational properties of hexagonal boron nitride: Inelastic x-ray scattering and ab initio calculations. Phys. Rev. Lett.
**2007**, 98, 095503. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nemanich, R.J.; Solin, S.A.; Martin, R.M. Light scattering study of boron nitride microcrystals. Phys. Rev. B
**1981**, 23, 6348. [Google Scholar] [CrossRef] - Wakabayashi, N.; Smith, H.G.; Nicklow, R.M. Lattice dynamics of hexagonal MoS2 studied by neutron scattering. Phys. Rev. B
**1975**, 12, 659. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the sample preparation for thermal conductivity measurements by pump-probe frequency-domain thermoreflectance. The 2D-material based ink is first prepared by assisted-liquid phase exfoliation, and then inkjet printed on silicon substrate and coated with an Al metal layer. The films are characterized by a dense array of 2D crystal nanosheets (see inset, showing the film cross section, taken by scanning electron microscopy; scale bar = 1 $\mathsf{\mu}$m).

**Figure 2.**Thermal conductivity and thermal boundary conductance (TBC) of inkjet-printed 2D crystal films. Each value is the average of the measurements obtained for films of varying thickness, where each film thickness is measured on several different locations. As-deposited films before annealing are labeled as “pristine”, otherwise films were annealed in air at 150 ${}^{\xb0}$C. Smaller diameter flakes obtained through a longer sonication treatment of the ink solution (no annealing) are labeled as “small flakes”. The in-plane thermal conductivity (

**a**) shows remarkable similarity for crystals having intrinsically very different bulk thermal conductivities. This quantity is affected mostly by flake size and quality of interface among flakes (see text and Figure 4). The out-of-plane thermal conductivities (

**b**) are ultra-low, a repercussion of the small thickness of the flakes, but associated with high transmissivity interfaces (Figure 4). The TBC of the printed film with Al (

**c**) shows values that are typical of metal interfaces with 2D materials.

**Figure 3.**Thickness dependence of sheet resistance and in-plane thermal conductivity of inkjet-printed graphene films. The sheet resistance (

**a**) shows a marked change with film thickness, size of the flakes and annealing. The in-plane thermal conductivity (

**b**) shows negligible dependence on film thickness and annealing, whereas flake size has a more marked contribution. The thermal conductivity is expected to stay constant with film thickness if the microstructure is unaltered. Annealing increases the electrical conductivity more than the thermal conductivity. Flake size alters the boundary scattering length scale, as indicated in Figure 4.

**Figure 4.**Modelling of the thermal conductivity of assembled flakes with size L along two possible transport directions (out-of-plane, ${K}_{\perp}$, and in-plane, ${K}_{\Vert}$) for the three studied materials. Black solid dots are the measurements in this work of non-annealed samples (the abscissa is the average thickness of a single flake within the film), while open dots are from Refs. [20,38,39] (see main text). The lines correspond to various models labeled as B (ballistic), RS (resistance in series), BT (Boltzmann Transport), Generalized Boltzmann (GB), and are obtained assuming an “ideal” (K, blue lines) or “dirty” (${K}^{\ast}$, red lines) interface among the flakes. Vertical and horizontal grey lines are defined in the text.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rahman, M.; Parvez, K.; Fugallo, G.; Dun, C.; Read, O.; Alieva, A.; Urban, J.J.; Lazzeri, M.; Casiraghi, C.; Pisana, S.
Anisotropic Thermal Conductivity of Inkjet-Printed 2D Crystal Films: Role of the Microstructure and Interfaces. *Nanomaterials* **2022**, *12*, 3861.
https://doi.org/10.3390/nano12213861

**AMA Style**

Rahman M, Parvez K, Fugallo G, Dun C, Read O, Alieva A, Urban JJ, Lazzeri M, Casiraghi C, Pisana S.
Anisotropic Thermal Conductivity of Inkjet-Printed 2D Crystal Films: Role of the Microstructure and Interfaces. *Nanomaterials*. 2022; 12(21):3861.
https://doi.org/10.3390/nano12213861

**Chicago/Turabian Style**

Rahman, Mizanur, Khaled Parvez, Giorgia Fugallo, Chaochao Dun, Oliver Read, Adriana Alieva, Jeffrey J. Urban, Michele Lazzeri, Cinzia Casiraghi, and Simone Pisana.
2022. "Anisotropic Thermal Conductivity of Inkjet-Printed 2D Crystal Films: Role of the Microstructure and Interfaces" *Nanomaterials* 12, no. 21: 3861.
https://doi.org/10.3390/nano12213861