On a hard surface, the cantilever deflection on approach will be equal to the movement of the piezo in the
z direction (denoted as
z), with
z =
d. On a soft sample, however, the cantilever will both deflect and indent into the sample surface as
z is increased. Values obtained from the force curve are
z,
zo,
d, and
do, where
z is the piezo displacement,
d is the cantilever deflection, and
zo and
do are the values at initial contact of the tip with the sample. These values can be used to calculate the indentation depth, which is given by:
The elastic modulus of the biological sample can be calculated by fitting the force curve with an appropriate theoretical model. Most biological materials can be modeled as perfectly elastic at typical indentation rates and depths, and the Hertz model is most often used to determine modulus. The assumptions and concerns associated with the use of this equation are discussed in the following section. The Hertz model provides a relationship between the loading force and the indentation and has a specific form for each tip shape.
83,84 The two most common tip geometries used for AFM nanoindentation are conical (
equation 2) and spherical (
equation 3):
where
F is the loading force, ν is Poisson's ratio (assumed to be 0.5), δ is the indentation depth, and
E is the elastic modulus. The half- opening angle of a conical indenter is α, and for spherical indenters
R is the radius of the tip. With these equations and the knowledge that
F =
k (
d −
do), the equations can be solved for the elastic modulus
E. The value of
E can then be plotted for each point on the force curve as a function of indentation depth. The elastic modulus will be constant as a function of indentation depth when there is no contribution from the substrate or a stiffening of the sample. The advantages and disadvantages of the different tip geometries will be discussed in a later section.