The above method requires accurate
knowledge of indentation. Indentation is measured as a difference between
the Z coordinate and the deflection and the accurate determination of
the indentation value requires accurate determination of the position
where tip contacts the surface. However, finding this position might
be not obvious in the presence of the long range repulsive forces as
illustrated in the figure to the right. The error in measuring the contact
point will result in the error in the Young's modulus that is determined
from the abscissa intercept. Moreover, in the method described above,
the abscissa intercept depends on units of measurements and therefore
if the contact point is not determined accurately, the error in the
Young's modulus will be also units-dependent. |
|
The error due to uncertanty in the contact point
can be avoided if the force-indentation dependence is written in a different
form by taking the power of 2/3 of the both sides of the equatiion,
as shown on the right.
Using this equation, the Young's modulus can be determined from the
slope of F^(2/3) dependence on the tip-sample separation. This method
avoids the error associated with the uncertainty in the contact point
determination. |
Here 
is determined from the difference between deflection and Z coordinate
and C* is some arbitarary constant that depends on the position of
the contact point along the Z coordinate.
|
The use of this dependence to determine the modulus
is illustrated in the figure to the right. The Young's modulus is determined
from the slope of the straight line in these coordinates. Co nverting
the slope to the modulus requires measurement or estimate of the tip
radius of curvature and the Poisson's ratio. Since the Poisson's ratio
ranges from 0 to 0.5 (it equals to the latter value for incompressible
solids), the error introduced by guessing the Poisson's ratio might
introduce ~10-15% error in the Young's modulus. |
|
Other important
issues to consider |
Sensitivity calibration
Accuracy with wich the tip-sample separation is measured strongly depends
on the accuracy of deflection sensitivity. Indeed, since the small indentation
is determined as a difference between two much larger values of the
Z travel and cantilever deflection, the latter should be determined
accurately to minimize the error in the indentation. Equations to the
right show that the relative error in the indentation might significantly
exceed the relative error in the sensitivity, this amplification depends
on the ratio of the Z travel to the indentation. In these equations
Vdefl - is the measured deflection voltage and InvOLS
is the inverse lever sensitivity.
The best solution is to measure the sensitivity on the hard substrate
at the point as close as possible to the location of the indentation
measurement. However, this does not eliminate the random error in the
sensitivity that might cause broadening of the measured elastic modulus
of relatively hard sampes. This problem is considered in detail below.
|
|
Measurements of the tip radius of curvature
A) Use electron microscopy to image the tip after the measurements.
Due to low electron density of silicon nitride the resulting pictures
have low contrast. Contrast can be dramatically enhanced if the tip
is coated with a very small amould of metal. Image to the right shows
SEM image of the AFM tip after the imaging as well as parabola fitted
to match the shape of the tip.
B) After the elasticity measurements, tip can be scanned across the
test sample with sharp features. The resulting map can be used to determine
the tip radius of curvature. One concern here is to be certain that
the features on the surface are sharper than the tip, otherwise blind
ceconvolution method can be used to estimate the tip shape. |
|
Confirming the elastic response of the
sample
Since the above equations were derived for the elastic deformation,
this type of responce should be confirmed for the sample. This means
that tha approach curve should overlap the withdra curve. If hysteresis
is observed it usually can be explained by adhesion, or by plastic/viscoelastic
deformation of the sample, or by sample motion across the surface during
indentation measurement.
Figure to the right shows one approach-withdraw pair collected on top
of the amyloid fibril. Close overlap between the curves supports the
elastic nature of measured indentation. |
|
Effects due to finite sample thickness
Substrate with elastic modulus different than that of the sample can
influence tip indentation. This influence can be estimated and corrected
using the appropriate mechanical models. |
|
Adhesion
there should be no strong adhesion, otherwise the above theory does not
apply. |
|
Uncertainty
in the optical lever sensitivity and its effect on the measured elastic
modulus |
Mechanical properties of proteing crystals might be expected to be
uniform across the surface of a crystal along a particular crystallographic
axis. However, when measured by AFM as described above,
the elastic modulus of insulin crystal exhibited a large variation in
values as illustrated in the histogram to the right. It can be noted
that the distribution is not only wide but it is also asymmetric.
Important observation here is that if modulus were measured by collecting
very few force-indentation curves, the resulting values might differ
by a factor of ~3. Therefore it is important to collect representative
statistics on each sample. |
Elastic modulus measured on 001 face of insulin crystal. |
It can be shown that the uncertainty
in the sensitivity is the major contribution to the measured width and
shape of the modulus distribution. In particular this is important when
spring constant of the sample noticeably exceeds the spring constant
of the cantilever (see below).
Figure to the right shows typical distribution of the sensitivity values
measured on the hard substrate and that is typical in the nanoindentation
measurements. It can be noted that ±5% uncertainty in the sensitivity
might cause ±30% uncertainty in the modulus as in the graph above.
|
Typical distribution of relative sensitivity values measured on hard
substrate
|
We have developed a model
(Langmuir 2008) that relates the uncertainty in the sensitivity
to the apparent distribution of the elastic modulus. Equations are a
bit cumbersome and therefore are not shown here. Figure to the right
shows distributions predicted based on this model. Different lines correspond
to different ratio of the surface spring constant to the cantilever
spring constant. Surface spring constant is evaluated at the half of
the maximum applied force during indentation (values are shown next
to the lines).
This model shows that the true modulus corresponds to the most probable
modulus value, not to the mean modulus value. Using the mean modulus
value might overestimate the true modulus (particularly for hard samples).
This model suggests that for stiff samples the unbiased modulus can
be estimated using the following steps:
- Measure statistically significant number of force curves on the
hard substrate close to the sample
- Measure statistically significant number of force curves on the
sample
- From the measurements on the hard substrate obtain the distribution
of the optical lever sensitivity values and determine the most probable
sensitivity
- Use this most probable sensitivity to obtain distribution of the
elastic modulus for the sample using approach as desrcibed above.
- Determine the most probable modulus value.
Thus determined most probable modulus value gives the unbiased estimate
of the true modulus. It should be noted that this procedure does not
eliminate other possible sources of systematic errors in the modulus:
bias due to deviation of the tip shape from the assumed paraboloidal
profile and bias due to the finite sample thickness. |
Distributions of the relative elastic modulus (relative modulus is
the measured modulus devided over true value of the modulus). Lines
correspond to different values of the surface stiffnes. The assumed
distribution in the relative sensitivity is shown in the inset (relative
sensitivity is the measured sensitivity divided by the most probable
sensitivity value).
|
