[Pw_forum] Young's Modulus

[Eduardo Ariel Menendez Proupin]

Dear Mohsen

Let me comment on your procedure

>At first i relaxed nano-tube and obtain the position of carbon atoms in the
>ground state. Then i decrease the tube length. A in the relation is the
area of >nano-tube (A=pi*R*R which R is the radius of tube). F is the atomic
force in >the Z axis which can be calculated by using Q.E.
>I think it is better to calculate total energy in some configuration and
use >(F=dE/dz, where E is the total energy in different configuration).
>Is there any problem in this procedure?

First, are you calculating the Young modulus of an isolated nanotube, or of
a material composed of (3,3) nanotubes perfectly oriented and with a well
defined density ?

The Young modulus is a concept that is well suited for a homogeneous
material, e.g., a bulk of nanotubes. It will depend on the density and
orientations of the nanotubes, but will be indendent of the transverse area
(the force and the energy of deformation is prportional to the area). In a
bulk of nanotubes perfectly oriented, the relevant area A would be the
transverse area of the unit cell, and the size and shape of the  unit cell
depends on the density of nanotubes, or must be optimized by a mimimum
energy criterium. If you are studying the Young modulus of a single isolated
nanotube, then note that the Young modulus is well defined only if everybody
use the same definition of A. Hence, be sure of considering the same
definition as the other results that you compare. Second, be sure to use a
unit cell sufficiently wide,  so that the preriodic replicas of the nanotube
are far enough and  the results do not depend on the transversal dimensions
of the unit cell.


>If we relaxed the tube after compression there is no net force on carbon
>atoms so I did not relax the system after compression.
*>Have you used a finite nanotube and computed the forces upon the >atoms of
the edge?*.....> A finite nano-tube? I consider a unit cell (which can >make
an infinite tube ) and calculate force between atoms.
>Is there any problem?
Yes. The state with the atoms not relaxed is not a state of equilibrium, and
according to Boltzmann law, if its energy difference with the relaxed state
is larger than kT (0.026 eV at 300 K), this state may happen only once in
the lifetime of the universe. Then, the property you calculate may have
nothing to do with a real situation. Hence, relax the atoms.

If you relax, of course  there will be no net force upon the atoms, but the
tube have stress. Use tstress=.true. and use the appropriate  component of
the stress tensor as F/A. Remember to do it for various transverse sizes of
the simulation cell and be sure that the stress is independent of the cell
size, or extrapolate. Of course, also must be independent of the cutoffs and
smearing parameters (degauss). Alternatively, you may use the second
derivative of the total energy to obtain the Young modulus. It is healthy to
use both methods to check that the results are fine. The numbers will not be
exactly equal due to numerical reasons, and you can systematically improve
the agreement increasing the cutoffs and the number of kpoints.


When I asked about a finite nanotube, I was thinking in an alternative way
to do it. Use a large cubic cell, with a finite nanotube inside, made a
constrained relaxation fixing the z coordinate (I assume that z is the
nanotube axis) of the edge atoms, relax the other atoms, and sum the forces
acting upon the atoms at one edge (that must be the negative of the total
force upon the other edge). With that you can obtain the Young modulus of a
finite nanotube. If you want it for the infinite nanotube you must do it
with several lengths and extrapolate to infinite length. Of course this is
much more  expensive (really brute force) if your goal is the infinite
nanotube, but is is what you should do if your interest is in short
nanotubes.

Best regards
Eduardo

-- 


Eduardo Menendez
Departamento de Fisica
Facultad de Ciencias
Universidad de Chile
Phone: (56)(2)9787439
URL: http://fisica.ciencias.uchile.cl/~emenendez

Let's pray for the 33 trapped miners! Four months to rescue.
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