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Models
SPIP™ provides two indentation models for extracting Young’s modulus from force
curves and from force volume images. The first model is the classic rigid sphere
on flat surface model, known as the Hertz1 model. The other model derived by Sneddon2
assumes a rigid cone indenting a soft flat surface. Both models do not include
adhesion and visco-elasticity. The Hertz model is valid for indentations significantly
smaller than the sphere radius. For the Sneddon model the indentation has to be
so large that the cone apex can be considered infinitely sharp.
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Figure 1.
Left:
The hard sphere on soft surface known as the Hertz model.
Right:
The rigid cone on soft surface model by Sneddon. |
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Single curve analysis
Figure 1 shows a force curve pair after transformation into force versus separation.
The Hertz model has been fitted to the approach curve. The fitted Young’s modulus
is about 18 MPa. |
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Figure 2.
Approach (blue) and retract (red) curves transf-ormed into force versus separation,
S, (opposite sign of indentation) with the Hertz model fitted to the approach
curve. |
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Mapping the elastic modulus
SPIP™ not only fits the indentation models to single force curves but also makes
a fit to each curve in a force volume image producing as the result an elasticity
map. An example is shown in Figure 2. The left image is a topography image. The
visible structure is a Primary Cilium from epithelial kidney cells. The image
to the right is the corresponding Young’s modulus (elastic modulus) map. A section
profile over the structure is shown below the images.
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Figure 3.
Left:
Topography image.
Right:
Young’s modulus image.
Bottom:
Section profile from the Young’s modulus image. |
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Careful choice and estimation of sensor properties
The indentation fits are sensitive to the input parameters. Besides the explicit
parameters in the model, input parameters also include the cantilever sensitivity
and the spring constant. So for the tip/cantilever there are three parameters
which have to be determined or estimated – all fairly difficult to assess. In
particular the detector sensitivity factor which is the conversion factor of detector
signal in volts to cantilever deflection in nanometers constitutes a potential
source to errors. This has to be calculated from force curves recorded on a rigid
surface. If such areas are present in the force volume image no extra experiments
are required. In the example reported here, we have merely guessed the tip radius
to 25 nm, the spring constant to 0.05 N/m and the sensitivity factor to 0.1 V/nm.
A careful look at Figure 1 reveals that the fit range used here is of the same
magnitude as the assumed tip radius – according to the model it should be smaller! Hence
restricting the fitting range to e.g. 0.01 nN would have been more correct. For
the Poisson’s ratio we have used a value of 0.5 (as for perfect rubber).
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Acknowledgements
The image, of which the structure is a Primary Cilium from epithelial kidney cells, has kindly been provided by Dr. Terry McMaster,
Dr. David Sheppard and Jo Evangelides, University of Bristol, UK. |
