3. Results and discussion
The steady-state temperature profiles along the length of the monolayer
graphene nanoribbon are presented in Figure 3 with substantially smooth
edges. Graphene nanoribbons, as defined herein, refer to, for example,
single or multiple layers of graphene that have an aspect ratio of
greater than about 5, based on their length and their width. Graphene
nanoribbons may be prepared in either oxidized or reduced forms. When
not otherwise specified herein, the term graphene nanoribbons should be
interpreted to encompass both oxidized graphene nanoribbons and reduced
graphene nanoribbons. Longitudinally opening, as defined herein, refers
to, for example, opening of carbon nanotubes along their longitudinal
axis to form graphene nanoribbons. Such longitudinal opening may be
thought of as an unzipping reaction of the carbon nanotubes. Narrow
graphene nanoribbons, as defined herein, refers to, for example,
graphene nanoribbons having widths less than about 10 nanometers. Wide
graphene nanoribbons, as defined herein, refers to, for example,
graphene nanoribbons having widths greater than about 10 nanometers. In
some cases, wide graphene nanoribbons have widths greater than about 100
nanometers. The graphene nanoribbon has transverse edges that have a
so-called armchair configuration. The longitudinal edges of the graphene
nanoribbon have a perfect zigzag configuration. The nanoribbon length is
assumed to be 40 nanometers. The temperature has a nonlinear dependence
on the distance from the given reference point in the direction of heat
flow. More specifically, in the vicinity of the hot and cold regions,
there exists a nonlinear dependence of the temperature with respect to
the distance. This nonlinear dependence is caused by the finite-size
effect arising from the graphene nanoribbon, given the fact that the
characteristic length scale of the monolayer graphene nanoribbon is much
smaller than the mean free path of phonons in graphene. The mean free
path of phonons in the graphene nanoribbon can vary depending on the
exact structure and dimensions of the monolayer graphene nanoribbon. For
example, the mean free path of phonons in graphene can be up to 700
nanometers at room temperature. As a result, the resulting temperature
gradient is very steep in the vicinity of the hot and cold regions so
that Fourier’s law is no longer applicable to the monolayer graphene
nanoribbon. This indicates that phonon transport within the monolayer
graphene nanoribbon is not fully diffusive, and the thermal transport
within the monolayer graphene nanoribbon is dominated by the mechanism
of diffusive-ballistic heat conduction. In the regions between the hot
slab and the cold slabs, the temperature has a more or less linear
dependence on the distance, and thus the thermal conductivity can be
determined by the temperature gradient.