Download AHMED-DISSERTATION

Texas Tech University, H. S. Tanvir Ahmed, December 2010
C=
z
(4.22c)
kc
Hence, equation (4.22b) is simplified to:
α=
f0 R *
1
E
kc
1 + C RE *
(4.23)
Solving for the reduced modulus (E*) yields (only positive sign is taken into
consideration):
 f R
C R + (C R ) + 4  0

 α kc 
2
2
E* =
 f R
2 0

 α kc 
2
(4.24)
Using this formula of equation (4.24), the reduced elastic modulus (E*) of
sample can be derived from the slope of frequency shift versus probe displacement
plot, provided that the values of free standing frequency of oscillation of the cantilever
f0, cantilever bending stiffness kc and tip radius R is exactly known. However, in
reality, these values can only be determined with limited accuracy. Hence, calibration
method for measurement of elastic modulus is more frequently put into use. In the
calibration method, α values are measured for a number of materials with known
elastic modulus, for example, materials with standard values are used such as fused
silica, gold, nickel, sapphire and tungsten. Those α values from the standards are
plotted as a function of the corresponding modulus values which can then be fitted
with a power law relationship, as shown in Figure 4.7.
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