Experimental determination of the spring constant of an individual
multiwalled carbon nanotube cantilever using fluorescence
measurement
Soongeun Kwon, Hyojun Park, Hyung Cheoul Shim, Hyung Woo Lee, Yoonkeun Kwak,
and Soohyun Kim
EPAPS Material
Optimization of the bending stiffness of the MWNT cantilever.
The bending stiffness of a MWNT cantilever was optimized to make a difference of gap
distance between simulation and experimental results minimum. From the equation (1),
it is known that the deflection of the MWNT at a given voltage is dependent on the
bending stiffness of the MWNT. Figure S1 shows the simulation result of the gap
distance at various applied voltages, assuming different bending stiffness. Based on the
simulation results, we can investigate the optimized bending stiffness of an individual
MWNT cantilever to match the experimental result of the gap distance.
As it is difficult to come up with the analytical solution of the bending deflection for the
nonlinear differential equation (1), a genetic algorithm was employed for the
optimization. Here, the gene to be optimized was chosen as the bending stiffness of a
MWNT cantilever. The initial generation of the bending stiffness consists of 50 genes in
a range of [7, 11] × 10-20 [N·m2], and maximum number of generation was set to be 100.
The lower boundary value for genes was chosen for nanotube not to contact the counter
electrode at maximum voltage due to the low bending stiffness.
1
Figure S1 Plot of gap distance against applied voltage assuming different bending stiffness:
(a) EI = 7×10-20 [N·m2] and (b) EI = 11×10-20 [N·m2].
The objective function was computed as,
n
(Objective function)   (ri  rie ) 2
i 1
, where ri is the theoretical gap distance calculated by selected genes and the equation
(1), rie is the experimental gap distance and n is the number of experiments. Figure
S2 shows the optimization result of bending stiffness after 100 generations. A best
fitness value is the output of the objective function calculated by the best individual
gene while an average fitness is the average value of the objective functions calculated
by 50 genes comprising each generation. The average fitness value, an indicator of the
gene’s quality, is approaching the best fitness value, which indicates that highly fit
genes are selected and survive as the generation proceeds (see figure S2(a)). The
average error of objective function for optimized bending stiffness makes 14nm error in
deflection between simulation and experimental results, resulting 1.3% error in
evaluation of bending stiffness. Also, the values of L and Do were measured with the
perturbation of ±1% from scanning electro microscope image. The optimization results
show that the best fit of bending stiffness (EI) is (8.22±0.27) × 10-20 [N·m2],
determining the spring constant (k) of (1.26±0.02) pN/nm. (see figure S2(b))
In figure 2(d) of the manuscript, a solid line indicated as “simulation” shows the
simulation result of the gap distance by solving equation (1) for the optimized bending
stiffness.
2
Figure S2 Optimization result of bending stiffness using genetic algorithm: (a) Fitness
evaluation of each generation during optimization and (b) Best fitting result of gap
distance using the optimum bending stiffness.
3