Microwave absorption by an array of carbon nanotubes: A phenomenological model Page: 2
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PHYSICAL REVIEW B 74, 075425 (2006)
aspect ratio is from 10 to 50, a rigid rod model should be
used; if the ratio is larger than 500, a wire or string model is
suitable. We considered the aspect ratio to be 1000 and as-
sumed a straight, classical, extensible, and flexible string
with instantaneous mechanical response to the microwave
field parallel to the string. The resulting tension is assumed
to be uniform along the string, with the ends free with atomic
degrees of freedom neglected. The CNT motions that open
an energy input window for microwaves are considered
while neglecting the motions that cause heating. (Many au-
thors have systematically studied electron-phonon interaction
of CNTs. See Ref. 18 and references contained therein.) Here
we assume that microwave heating of the CNTs begins with
resonant excitation of transverse wave motion followed by
electron-phonon and phonon-phonon interactions. Periodic
variation of the tension of the string leads to transverse para-
metric resonance and unstable motion. Using the third-order
longitudinal-transverse, coupled, nonlinear wave equations
given by Morse and Ingard,19 the resonant solutions are
shown to be pseudounstable. The displacements are solutions
of a nonlinear Mathieu equation derived from the third-order
wave equations constrained to two-dimensional wave mo-
tion. Initial conditions of the nonlinear Mathieu's equation
are considered given (referred to as a "seed" by Rowland20)
and initiated by thermal excitations. (Experiments 6-8 were
done at room temperature in the work cited in Ref. 20.)
II. THE MODEL
A CNT interacting with a spatially uniform electric field
at a microwave frequency WE pointing along the tube axis
induces polarization that responds instantly to the field. This
polarization causes an antenna effect in illuminated arrays of
CNTs21 that produces forces in opposite directions on the
ends of the tube, giving rise to a mechanical frequency 2WE.
If the equilibrium length L0 of the stretched CNT is set at
Lr+ (LM - Lr)/2, for the frequency of the driving force 2wOE,
the CNTs under a microwave field behaves almost the same
as that of an elastic string in Melde's acoustic exper-
iment.22-25 Herein, both ends are free to move transversely,
whereas in the acoustic experiment both ends are fixed.
Rowland's analysis of the Melde experiment to the micro-
wave-driven CNT problem20 is adopted in this work.
If the x axis is chosen to lie along the resting CNT, a point
on the string initially at (x, 0,0), is found at time t> 0 to be
at the pointr(t) = {x + u(x, t), v (x, t),w(x, t)}.
(1)
At the free ends, the boundary conditions are
u(O,t) = uo cos(2wEt), (2)
u(Lo,t) = - Uo cos(2wEt), (3)
and
vx(0,t) = vx(Lo, t) = wx(O, t) = wx(Lo, t) = 0, (4)
where uo= (LM-Lr)/2 is a function of the microwave power
and is dependent on the electronic energy structure,14 vx= v v/x, etc. Boundary conditions, Eqs. (2) and (3), cause the
tension 7 in the tube to change in time according to Hooke's
law,
[L0 - 2u0 cos(2wEt) - Lr]
= SY = To[1 - b cos(2oEt)],
Lr
(5)
with(Lo - Lr)
0= SY Lr
Lr(6)
and b= 2uo/(Lo-Lr)= 2, from the definitions of u0 and Lo; S
is the cross-sectional area of the CNT (or nanorope, nano-
strand), Y is Young's modulus, and To is the tension of the
nanotube in equilibrium. If E is the equilibrium strain, a
(17,17)-type SWNT with a length of 1 j/m and strain of 10-4
(Lo-Lr=1 A) with Y= 1 TPa, the fundamental transverse
mode frequency is approximately 2.1 GHz, which is near the
microwave frequency of 2.45 GHz used in the experiments.
A. The linear model
The equation for the motion of the nanotube for time-
dependent tension using Eq. (5) gives2 - CT [1 - b cos(2WEt)] - =0,
(7)
where CT= T0/P0 is the speed of transverse waves. Expand-
ing the solutions over odd harmonics with suitable replace-
ments gives the Mathieu equation for vn(t),(8)
d2u,
+ [a-,p cos(T)]vn = 0,
dT2where a= P/2= on2/(4wE2), wfn=CTkn, T= 2wEt; the integer n
is odd to keep the center of mass stationary. A resonance
occurs for on-, l/2, where f( is the frequency of the driving
force.26 In our case f=2OE and resonance occurs for an
WE= 2.45 GHz with n= 1, so that a=/3/2 1/4. The solu-
tions of Eq. (8) are stable or unstable depending on the val-
ues of a and /3, as shown in the (a- /) parameter space in
Fig. 1. The boundaries between stable and unstable regions
near the values a=1/4, /3=0 are given by a 1/4//2
-/2/820,27,28 It is seen that the point a= 1/4, /3=1/2 lies in
an unstable region and the general solution has the form
vn(t) - c le/ql(T) + c2e- "q2(T), (9)
where ju is a real number depending on a and /3, and q l and
q2 are periodic functions. The exponentially growing solu-
tion in Eq. (9) holds for continuous ranges of values of a and
p/ (Fig. 1). For parametric resonance, the result is either un-
limited growth for damping K less than a critical value K
<2/4, or overdamped transverse motion for damping larger
than the critical value. If the interaction between the CNT
and its surroundings is included through the addition in Eq.
(8) of a linear damping term of the form Kdv/dT with the
transformation vn=exp(-KT/2) (T) made to eliminate the
first derivative term, satisfies another Mathieu equation, of
the form075425-2
YE et al.
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Ye, Z.; Deering, William D.; Krokhin, Arkadii A. & Roberts, James A. Microwave absorption by an array of carbon nanotubes: A phenomenological model, article, August 29, 2006; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc103271/m1/2/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.