Biology Reference
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An alternative model was developed by Sneddon in 1965. It considers
indentation by rigid indenters with arbitrary axis-symmetry proiles. This
model is commonly used in AFM indentation experiments. However, neither
the Hertz nor the Sneddon model takes into account the effects of surface
energy on the contact deformation. These surface forces cannot be neglected,
therefore more sophisticated models, such as those propounded by Bradley,
Derjaguin-Müller-Toporov and Johnson-Kendall-Roberts, have to be
implemented. For an in-depth analysis of these models, the interested reader
is referred to the review by Cappella and Dietler. 1
It has to be emphasized that all these models consider idealized tips
and samples which approximate the experimental conditions. Numerical
approaches, such as inite-element or molecular dynamics modelling, permit
a more precise analysis of the experimental situation, but at the cost of
increased complexity and of sophisticated computational requirements.
16.3 FINITE ELEMENTS AND MOLECULAR DYNAMICS
Finite-element modelling is an analytical technique which was irst
developed for the ield of structural engineering. It is based on the
assumption that any structure can be subdivided into smaller regions
(elements) for which the differential equations describing the deformation
under a load can be solved numerically. By assembling the sets of equations
for each region, the behaviour of the entire problem domain can be
approximated. This method is particularly well adapted for samples that
have a complex geometrical shape and in situations where a substantial
deformation occurs during the indentation process. More and more
investigators are now using inite-element modelling to interpret their AFM
data. For a comprehensive overview of the applications of inite-element
modelling in conjunction with the AFM, the interested reader is referred to
the review by Ikai
et al. 2
Molecular dynamics modelling is another analytical tool, which involves
calculating the interactions between single atoms as a function of time.
Equations in which the Newtonian motion of each individual atom is
integrated describe the dynamics of the particles. The technique is widely
used to study the conformational changes of proteins. The drawback of
this method is its high computational requirements. A lavour of these
requirements may be obtained from the following example: a 50 ns
simulation of the million atoms that comprise a single tobacco mosaic virus
would require several dozen years using a single home computer.
 
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