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Effect of Surface Energy on Mechanical Behaviour of Nano Structural Elements

M. Shaat's picture

Extremely small size of nano-structures such as beams, sheets and
plates, which are commonly used as components in Nanoelectromechanical Systems
(NEMS), presents a significant challenge to the researchers of nano-mechanics.
Several studies have been developed on the mechanical behavior of nano-sized
bars, tubes, sheets and plates. The results of these studies show that the
elastic modulii of such nano-structural elements depend on their size.
Unfortunately, classical elasticity lacks an intrinsic length scale and thus
cannot be used to model the size effect. Atomistic simulation, however it is
very powerful, needs intensive computations.

All physical theories possess a certain domain of applicability outside
which they fail to predict the physical phenomena with reasonable accuracy.
While the boundaries of these domains are not known precisely, often the
failure of a given mathematical model is indicated by its prediction that
deviate considerably from experimental results or obviously displayed
mathematical singularities. The domain of applicability of a theory is a
formation of some internal characteristic length and time scales of the media
for which it is constructed. When these scales are sufficiently small compared
to the corresponding external scales then the classical field theories give
successful results; otherwise they fail to apply. For each theory, the domain
of application defines the level of the considered constituents and the
appropriate processes of interactions between these constituents. The
components below this level would not be accounted for and consequently, the
interaction process between these components and the other ones would be
avoided also. As an obvious example, for a macro-scale body the surface
component of the body is very small relative to the volume of the solid. Thus,
we can neglect the surface, as component of the continuum and focus our
attention only on the bulk solid. On the contrary, for a tiny body the surface
is very comparable to the bulk volume. Therefore, it should be taken into
consideration and deserves to pay a considerable attention to its
characteristics and the processes of interactions with the bulk of the

The same issue can be observed when we study the mechanical deformation
of a macro –scale elastic continuum. In this case, it will be sufficient to
investigate the behavior on the level of particles as already happened in the
classical continuum mechanics theories. On the contrary, for nano-scale systems
we have to deal with the atomic discrete nature of the system. Thus, we have to
concern primarily with the level of microstructure elements and investigate
different interaction processes between those elements.

Unfortunately, classical continuum mechanics is explicitly designed to
be size-independent which may reflect the breakdown of classical continuum
mechanics at nano-scale sizes. One of the physical reasons for the breakdown of
classical continuum mechanics at nano-scale sizes is the surface effects.

Atoms at a free surface experience a different local environment than do
atoms in the bulk of a material. As a result, the energy associated with these
atoms will be different from that of the atoms in the bulk. The excess energy
associated with surface atoms is called surface free energy. In classical
continuum mechanics, such surface free energy is typically neglected because it
is associated with only a few layers of atoms near the surface and the ratio of
the volume occupied by the surface atoms and the total volume of material of
interest is extremely small. However, for ultra-thin films, the thickness of
the films reduces to micro/nano scales; hence the effects of surface energy
become significant and need more interest.

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