W. Richard Bowen and Nidal Hilal 4

20 1. BAsIC PRINCIPLEs OF ATOMIC FORCE MICROsCOPy
pressed against a hard surface, which will not appreciably deform under
the pressure exerted upon it, will allow the calculation of the unknown
spring constant, providing that the same optical lever set-up and hence
its sensitivity remains unchanged between each set of measurements:
k � k
c ref
⎛ �hard � � ⎞
⎜
ref
⎜
⎝⎜
�ref � cosϕ
⎠⎟
⎛
�
⎞
⎜ hard
kc � k ⎜ ref ⎜ � 1
⎝⎜
�ref
⎠⎟
(1.9)
(1.10)
where k c and k ref are the spring constants of the unknown and known
reference levers, � hard and � ref the gradients of the contact regions of force
curves against a hard surface and the reference lever, respectively, ϕ the
angle between the two levers.
Typically levers are mounted onto the AFM with an in-built tilt angle of
approximately 10–12°, which is liable to give a value of k varying from the
true value by less than the experimental uncertainty. In addition, if care is
taken to maintain the same angle between the lever and the experimental
sample when calibrating the lever, then the apparent spring constant calculated
by this method will be identical to the effective spring constant. As such
the term cos ϕ can be ignored (equation (1.10)). This is a quick and simple
method to use and can be carried out where the instrumentation being used
does not allow the measurement of the resonance spectrum of the cantilever.
One area where caution must be taken with the reference lever method
is in the precise positioning of the two levers (Figure 1.7). As seen from
equation (1.2), k will vary inversely in relation to l 3 . This means that if the
cantilever to be calibrated overlaps with the reference lever, effectively
reducing the length of the reference, the measurements obtained will be
as though taken against a stiffer reference lever, leading to an underestimation
of k c. Another important factor to consider is that the stiffness of
the unknown cantilever and the reference must be similar in order to get
a truly accurate result. If one cantilever is much stiffer than the other, then
the slope obtained from a force curve of the cantilevers pressed together
k ref
k c
ϕ
fIgure .7 Static def-
lection of a cantilever of
unknown spring constant
against a reference cantilever.
The slope of the contact
region of the resultant
force curve is dependent
upon the stiffness of the two
levers combined as well as
the angle between them.