# Nanotechnology Questions and Answers – Heat Transport in Carbon Nanotubes

This set of Nanotechnology Multiple Choice Questions & Answers (MCQs) focuses on “Heat Transport in Carbon Nanotubes”.

1. Identify a method not used for calculating ζph(w) for nanoscale objects.
a) Green’s function method
b) Scattering boundary method
c) Sword-in-sheath method
d) Phonon wave packet scattering method

Explanation: There are multiple methods for calculating the transmission function for phonons at energy ħw, which is denoted by ζph(w), for nanoscale objects with complicated atomic structures. The methods involved are Green’s function method, scattering boundary method and the phonon wave packet scattering method.

2. Which among the following factors causes a hindrance in heat transport of CNTs?
a) Structure of CNTs
b) Presence of extraneous contaminants
c) Phonon conduction mechanism
d) Crosslinking between individual CNTs

Explanation: Heat transport through CNTs is not always smooth. There are certain hindrances. First, CNTs have extraneous contaminants that interfere with thermal energy transport. Second, topological defects such as the Stone-Wales, vacancies etc., cause scattering that reduce the mean free path for phonons. Hence, phonon conduction mechanism is hampered.

3. For T > θ(1/D), how does the thermal conductivity of MWCNTs vary?
a) As T2.3
b) As T-1
c) As T2
d) As T-5

Explanation: For T < θ (1/D) (interwall Debye temperature), determined by the Van der Waals interactions between the MWCNTs, the thermal conductivity increases as λph ~ T2.5. While for T > θ(1/D), where the weak interwall interactions are negligible, thermal conductivity increases as λph ~ T2.

4. Thermal conductance of CNTs are quantized at low temperature
a) True
b) False

Explanation: Recent studies have revealed that the thermal conductance of CNTs Кph(t) remain quantized. This indicates that the thermal conductance of CNTs have definite values in terms of К0 = (π2kB2T/3h) per acoustic mode at low temperatures.

5. What is the expression for the thermal conductance of a SWCNT in Landauer Theory for phonon transport?
a) Кph(T) = kB ∫$$_0^{\infty}$$ [dw (ħw/kBT)3 {(eħw/kBT)/(eħw/kBT-1)2} ζph(w)
b) Кph(T) = (kB/4π2) ∫$$_0^{\infty}$$ [dw (ħw/kBT) {(eħw/kBT)/(eħw/kBT-1)2} ζph(w)
c) Кph(T) = (2π/kB) ∫$$_0^{\infty}$$ [dw (ħw/kBT)2 {(eħw/kBT)/(eħw/kBT-1)2} ζph(w)
d) Кph(T) = (kB/2π) ∫$$_0^{\infty}$$ [dw (ħw/kBT)2 {(eħw/kBT)/(eħw/kBT-1)2} ζph(w)

Explanation: The thermal conductance for ballistic phonon transport in SWCNTs at low temperatures using Landauer theory is given by
Кph(T) = (kB/2π) ∫$$_0^{\infty}$$ [dw (ħw/kBT)2 {(eħw/kBT)/(eħw/kBT-1)2} ζph(w)
Here, T is the average temperature, T = (Th+Tc)/2. Th and Tc are the temperatures of the hot and cold reservoirs to which the SWCNTs are connected. ζph(w) is the transmission function for phonons at energy ħw.

6. What happens to the transmission function for phonons, at energy ħw, when SWCNTs contain defects?
a) Deviates from stepwise structure
b) Remains unchanged with temperature variation
c) Phonon scattering is prevented
d) A stepwise structure is exhibited

Explanation: The structure of SWCNTs often contains defects, for example: single-atom vacancies, isotope impurities etc. These deviate the ζph(w) from stepwise structure due to the phonon scattering with defects. This mainly occurs at higher energies while it remains unchanged in the lower – energy regions.

7. Which expression indicates phonon transmission function ζph(w), in Landauer theory, in the quasiballistic region?
a) ζph(w) = L02(w)/[1+L0(w)]
b) ζph(w) = L0(w)/[1-L0(w)]
c) ζph(w) = L0(w)/[L0(w)+L]
d) ζph(w) = 1/[L0(w)+L]

Explanation: The phonon transmission function ζph(w) in Landauer theory in the quasiballistic region is approximately given by ζph(w) = L0(w)/[L0(w)+L]. Here, L0(w) is the phonon mean free path.

8. Which of the following theory is used to describe the ballistic phonon transport in SWCNTs?
a) Kubo’s linear response theory
b) Landauer theory
c) Ainsworth attachment theory
d) Max Planck’s Quantum theory

Explanation: The heat transport through SWCNTs occurs both in a ballistic and quasi-ballistic way. The ballistic heat transport involving phonons at low temperatures is discussed using the Landauer theory for phonon conduction. The conventional transport theories for bulk systems, such as Kubo’s linear response theory are not suitable for phonon transport in small systems far from equilibrium.

9. Thermal conductivity of SWCNTs remains unaffected by the presence of single-atom vacancy defects.
a) True
b) False

Explanation: CNTs possess single – atom vacancies that excessively reduce their thermal conductivity. This happens as the defects locally disrupt the hexagonal network of CNTs. In SWCNTs these defects are energetically metastable, a rebonding of two dangling – bond atoms take place and only one dangling – bond atom is left as a result of thermal treatments.

10. Identify the expression that is in accordance with the Wiedemann Franz Law.
a) Кel = (4π3kB4T)/h
b) Кel = (4πkB3T)/2h
c) Кel = 4πkBT
d) Кel = (4π2kB2T)/3h

Explanation: In accordance with the Wiedemann Franz law, the electron-derived thermal conductance remains quantized by the same value as the phonon conductance which is given as Кel = 4π2kB2T/3h.

Sanfoundry Global Education & Learning Series – Nanotechnology.

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