# Modeling of Catalytic Centers Formation Processes during Annealing of Multilayer Nanosized Metal Films for Carbon Nanotubes Growth

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermal Stresses in the Structure in the Course of Annealing

_{Dyn}) is determined by the rate of temperature increase at the heating stage:

_{Int}) depends on the difference in thermal expansion coefficients of materials [22]:

_{sub}, α

_{sl},α

_{cl}—thermal expansion coefficients of the substrate material, sublayer and catalytic layer, respectively.

_{Dyn}has a maximum value at the initial stage of heating, therefore, the appearance of this stress can cause cracking of the film in a still unheated structure.

## 3. Mass Transfer in the Course of Annealing

_{out}is a specific surface energy of the outer layer, K—surface curvature, Ω—atom volume of the outer layer material.

_{m}is the radius of the profile element curvature.

_{0}is thickness of the outer metal layer.

_{*}is a contact angle of the outer layer breaking, ϕ

_{k}—contact angle of the CC stationary profile).

## 4. The Formation of Catalytic Centers

_{k}is a critical radius) will not be faceted in accordance with the crystallographic parameters of the material. Therefore, we can assume that the isolated profile elements created during the film breaking will be formed as spherical segments. Since the geometric dimensions of the catalytic centers are nanoscale objects, its shape factor (SF) will have a significant effect on the dynamics of formation and the final profile of the CC. After the film ruptures, the dominant factor controlling the formation of the CC profile is the specific surface energy. Thermal stress only supports the transfer of matter through the film fragment, on which the developing CC is based.

_{in}is a specific surface energy of the inner layer.

## 5. Evolution of the Catalytic Center Profile

_{0}, we can demonstrate that the intensity of the diffusion flow (F) to the base of the CC will be determined as:

_{0}= ρ·N

_{A}xM

^{−1}is a concentration of atoms in the lattice of the outer layer material, ρ—density of the outer layer material; M—atomic weight of the outer layer material; N

_{A}—Avogadro number, k—Boltzmann constant.

_{D}) and substance loss rate (β

_{C}) upon sublimation of atoms from the outer surface of the CC, respectively; P

_{0}is an equilibrium pressure of saturated vapor of the outer layer material above a flat surface at annealing temperature [24]; $\chi =\frac{{\sigma}_{0}\times \mathsf{\Omega}}{\mathit{kT}}$ = 5.339 × 10

^{−8}– dimensionless energy factor, m—molar mass.

_{D}+ χxβ

_{C}·) and θ

_{1}= (β

_{D}− β

_{C}).

_{*}) to the CC steady-state profile formation moment (ϕ

_{k}), we can determine the time of its formation:

_{k}is an estimated time of formation of a CC of a radius R

_{k}(ϕ

_{k}).

_{k}, only the matter flow to the CC base is blocked, while the sublimation process at a constant annealing temperature will continue until the center disappears completely. The equation of the CC relaxation process at this stage of annealing will have the form of:

_{k}, we can find the time during which the CC completely sublimates:

## 6. Experiments and Methods

## 7. Results and Discussion

_{0}– predetermined heating temperature.

_{0}, the characteristic parameter can be estimated by the formula:

_{0}, ξ = 100).

^{−4}m and the initial heating rate of 3 deg/s, the dynamic thermoelastic stresses are 160 Pa (Figure 2a), which is significantly less than the tensile strength of silicon substrate. During the heating time (t* = 1200 s), dynamic thermoelastic stresses decrease almost to zero. In this case, the stress arising at the metal/substrate interface σ

_{Int}(t) continuously increases with increasing temperature according to (2), (25) and after reaching the stationary heating mode remains constant σ

_{Int}= 2.2 × 10

^{9}Pa (Figure 2b). The dependence of the change in SF on the contact angle in accordance with Equation (11) is shown in Figure 3.

_{0}= 20 × 10

^{−9}m and the formation of CC with a height of h* = 2.001 × 10

^{−8}m and a radius of the base R* = 3.052 × 10

^{−8}m. Moreover, the estimation of the CC shape factor with the profile of the spherical segment of radius R

_{*}was δ(ϕ

_{*},${R}_{*}$) = 1.25 × 10

^{8}m

^{−1}.

_{к}= 1.555 rad (89°). Based on this, using Formulas (8), (9) and (13), we calculated the radius of the stationary CC profile R(φ

_{k}) = 1.006 × 10

^{−7}m, diameter of its base (2.012 × 10

^{−7}m) and its height h

_{k}= 9.903 × 10

^{−8}m. The shape factor in accordance with (15) δ(ϕ

_{k}) = 9.089 × 10

^{7}m

^{−1}.

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Iijima, S. Helical microtubules of graphitic carbon. Nature
**1991**, 354, 56–58. [Google Scholar] [CrossRef] - Ebbesen, T.W.; Lezec, H.J.; Hiura, H.; Bennett, J.W.; Ghaemi, H.F.; Thio, T. Electrical conductivity of individual carbon nanotubes. Nature
**1996**, 382, 54–56. [Google Scholar] [CrossRef] - Treacy, M.M.J.; Ebbesen, T.W.; Gibson, J.M. Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature
**1996**, 381, 678–680. [Google Scholar] [CrossRef] - Il’ina, M.V.; Il’in, O.I.; Blinov, Y.F.; Smirnov, V.A.; Kolomiytsev, A.S.; Fedotov, A.A.; Konoplev, B.G.; Ageev, O.A. Memristive switching mechanism of vertically aligned carbon nanotubes. Carbon
**2017**, 123, 514–524. [Google Scholar] [CrossRef] - Lim, Y.D.; Kong, Q.; Wang, S.; Tan, C.W.; Tay, B.K.; Aditya, S. Enhanced field emission properties of carbon nanotube films using densification technique. Appl. Surf. Sci.
**2019**, 477, 211–219. [Google Scholar] [CrossRef] - Chen, Z.; Cao, G.; Zhang, Q.; Lan, P.; Zhu, B.; Yu, T.; Lin, Z. Large current carbon nanotube emitter growth using nickel as a buffer layer. Nanotechnology
**2007**, 18, 095604. [Google Scholar] [CrossRef] - Ageev, O.A.; Blinov, Y.F.; Il’in, O.I.; Konoplev, B.G.; Rubashkina, M.V.; Smirnov, V.A.; Fedotov, A.A. Study of the resistive switching of vertically aligned carbon nanotubes by scanning tunneling microscopy. Phys. Solid State
**2015**, 57, 825–831. [Google Scholar] [CrossRef] - Tsai, C.-L.; Xiong, F.; Pop, E.; Shim, M. Resistive Random Access Memory Enabled by Carbon Nanotube Crossbar Electrodes. ACS Nano
**2013**, 7, 5360–5366. [Google Scholar] [CrossRef] [PubMed] - Srivastava, A.; Liu, X.H.; Banadaki, Y.M. Overview of Carbon Nanotube Interconnects. In Carbon Nanotubes for Interconnects; Springer International Publishing: Cham, Switzerland, 2017; pp. 37–80. [Google Scholar]
- Dao, V.-D.; Vu, N.H.; Yun, S. Recent advances and challenges for solar-driven water evaporation system toward applications. Nano Energy
**2020**, 68, 104324. [Google Scholar] [CrossRef] - Dao, V.-D.; Choi, H.-S. Carbon-Based Sunlight Absorbers in Solar-Driven Steam Generation Devices. Glob. Chall.
**2018**, 2, 1700094. [Google Scholar] [CrossRef] [PubMed] - Dao, V.-D.; Vu, N.H.; Choi, H.-S. All day Limnobium laevigatum inspired nanogenerator self-driven via water evaporation. J. Power Sources
**2020**, 448, 227388. [Google Scholar] [CrossRef] - Dang, H.-L.T.; Tran, N.A.; Dao, V.-D.; Vu, N.H.; Quang, D.V.; Vu, H.H.T.; Nguyen, T.H.; Pham, T.-D.; Hoang, X.-C.; Nguyen, H.T.; et al. Carbon nanotubes-ruthenium as an outstanding catalyst for triiodide ions reduction. Synth. Metals
**2020**, 260, 116299. [Google Scholar] [CrossRef] - Il’ina, M.; Il’in, O.; Blinov, Y.; Konshin, A.; Konoplev, B.; Ageev, O. Piezoelectric Response of Multi-Walled Carbon Nanotubes. Materials
**2018**, 11, 638. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mitra, M. Characteristics of Carbon Nanotube Relay. Sci. J. Res. Rev.
**2019**, 1. [Google Scholar] [CrossRef] - Liu, J.; Jiang, D.; Fu, Y.; Wang, T. Carbon nanotubes for electronics manufacturing and packaging: From growth to integration. Adv. Manuf.
**2013**, 1, 13–27. [Google Scholar] [CrossRef] [Green Version] - Gupta, B.K.; Kedawat, G.; Gangwar, A.K.; Nagpal, K.; Kashyap, P.K.; Srivastava, S.; Singh, S.; Kumar, P.; Suryawanshi, S.R.; Seo, D.M.; et al. High-performance field emission device utilizing vertically aligned carbon nanotubes-based pillar architectures. AIP Adv.
**2018**, 8, 015117. [Google Scholar] [CrossRef] - Klinke, C. Analysis of Catalytic Growth of Carbon; EPFL: Lausanne, Switzerland, 2003. [Google Scholar] [CrossRef]
- Cherepanov, G.P. Physics of Sintering. In Methods of Fracture Mechanics: Solid Matter Physics; Springer: Dordrecht, The Netherlands, 1997; pp. 84–123. [Google Scholar] [CrossRef]
- Pines, B.Y.; Geguzin, Y.E. Self-diffusion and heterodiffusion in heterogeneous porous bodies. J. Tech. Phys.
**1953**, 23, 1559–1572. [Google Scholar] - Antman, S.S. Nonlinear Problems of Elasticity; Applied Mathematical Sciences; Springer New York: New York, NY, USA, 1995; Volume 107, ISBN 978-1-4757-4149-0. [Google Scholar]
- Mullins, W.W. Theory of linear facet growth during thermal etching. Philos. Mag.
**1961**, 6, 1313–1341. [Google Scholar] [CrossRef] - Bello, I. Vacuum and Ultravacuum: Physics and Technology; CRC Press: Boca Raton, FL, USA, 2017; ISBN 9781498782050. [Google Scholar]
- Frenkel, J. Kinetic Theory of Liquids; Clarendon Press: Oxford, UK, 1946. [Google Scholar] [CrossRef]
- Kikoin, I. Tables of Physical Constants; Atomizdat: Moscow, Russia, 1976. (In Russian) [Google Scholar]
- Grigoriev, I.; Meilikhov, E.; Radzig, A. Handbook of Physical Quantities; CRC Press: Boca Raton, FL, USA, 1996; ISBN 9780849328619. [Google Scholar]
- Il’in, O.I.; Rudyk, N.N.; Il’ina, M.V.; Osotova, O.I.; Fedotov, A.A. Influence of annealing temperature and activation time on the catalytic centers formation for carbon nanostructures growth. J. Phys. Conf. Ser.
**2019**, 1410, 012231. [Google Scholar] [CrossRef]

**Figure 1.**SEM image of film before heating (

**a**) and catalytic centers (CC) obtained at a heating temperature of 750 °C (

**b**).

**Figure 2.**The dependence of stress on time arising in the substrate (

**a**) and in the contact plane (

**b**).

**Figure 4.**Dependences of the diameter (

**a**) and height (

**b**) of the catalytic centers on the contact angle obtained at different temperatures.

Parameters of Catalytic Centers | Heating Temperature, °C | ||
---|---|---|---|

700 | 750 | 800 | |

Diameter, nm | 91 ± 17 | 110 ± 11 | 95 ± 10 |

Height, nm | 28 ± 12 | 40 ± 7 | 30 ± 7 |

Physical Quantity | Estimation Formula | Si | Cr | Ni | Units | Ref. |
---|---|---|---|---|---|---|

Thermal conductivity, λ | constant | 36.5 | – | – | $\frac{J}{s\times m\times \xb0K}$ | [25] |

Density, ρ | constant | 2.33 × 10^{3} | 7.1 × 10^{3} | 8.75 × 10^{3} | $\frac{kg}{{m}^{3}}$ | [25] |

Molar weight, M | constant | 28.06 × 10^{−3} | 51.99 × 10^{−3} | 58.71 × 10^{−3} | $\frac{kg}{mol}$ | [25] |

Heat capacity, c | constant | 19.79 | – | – | $\frac{J}{\xb0K}$ | [25] |

Thermal diffusivity, a_{TD} | ${a}_{TD}=\frac{\lambda}{\rho \times c}$ | 7.92 × 10^{4} | – | – | $\frac{{m}^{2}}{s}$ | |

Atom concentration in the lattice, N | $N=\frac{\rho}{M}\times {N}_{A}$ | 5 × 10^{28} | 8.22 × 10^{28} | 8.98 × 10^{28} | ${m}^{-3}$ | |

Atom volume, Ω | $\mathsf{\Omega}=\frac{1}{N}$ | 2 × 10^{−29} | 1.22 × 10^{−29} | 1.11 × 10^{−29} | ${m}^{3}$ | |

Atom radius, r | constant | 1.68 × 10^{−10} | 1.7 × 10^{−10} | 1.24 × 10^{−10} | $m$ | [25] |

Atom diameter, a | $a=2\times r$ | 3.36 × 10^{−10} | 3.4 × 10^{−10} | 2.48 × 10^{−10} | $m$ | |

Young’s modulus, E | constant | 110 × 10^{9} | 297 × 10^{9} | 200 × 10^{9} | $\frac{N}{{m}^{2}}$ | [26] |

Poisson’s ratio, υ | constant | 0.288 | 0.21 | 0.3 | – | [25] |

Coefficient of thermal expansion, α | constant | 4.65 | 10.98 | 18.2 | $\frac{{10}^{-6}}{\xb0K}$ | [26] |

Specific surface energy, γ | $\gamma =\frac{E\times a}{2}$ | 18.52 | 50.58 | 24.8 | $\frac{N}{m}$ | [25] |

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**MDPI and ACS Style**

Il’in, O.I.; Rudyk, N.N.; Fedotov, A.A.; Il’ina, M.V.; Cherednichenko, D.I.; Ageev, O.A.
Modeling of Catalytic Centers Formation Processes during Annealing of Multilayer Nanosized Metal Films for Carbon Nanotubes Growth. *Nanomaterials* **2020**, *10*, 554.
https://doi.org/10.3390/nano10030554

**AMA Style**

Il’in OI, Rudyk NN, Fedotov AA, Il’ina MV, Cherednichenko DI, Ageev OA.
Modeling of Catalytic Centers Formation Processes during Annealing of Multilayer Nanosized Metal Films for Carbon Nanotubes Growth. *Nanomaterials*. 2020; 10(3):554.
https://doi.org/10.3390/nano10030554

**Chicago/Turabian Style**

Il’in, Oleg I., Nikolay N. Rudyk, Alexandr A. Fedotov, Marina V. Il’ina, Dmitriy I. Cherednichenko, and Oleg A. Ageev.
2020. "Modeling of Catalytic Centers Formation Processes during Annealing of Multilayer Nanosized Metal Films for Carbon Nanotubes Growth" *Nanomaterials* 10, no. 3: 554.
https://doi.org/10.3390/nano10030554