232 6.5  Scanning Probe Microscopy and Force Spectroscopy

UME (which is due to the mechanical elastic potential of the cantilever bending), and UIP

(which is the interaction potential between the sample and the tip), resulting in the total force

F on the AFM tip:

(6.18)

F

U

U

U

U

F

F

F

total

ME

SD

IP

ME

SD

IP

= −∇

= −∇

= ∇

+ ∇

=

+

+

)







Sample deformation forces are not trivial to characterize due to the often heterogeneous and

nonlinear elastic response of soft matter. The Hertz model offers a reasonable approxima­

tion, however, for a small local sample indentation zS such that the restoring force FSD (see

Figure 6.6b) is given by

(6.19)

F

Y

R z

SD

sample

tip

s

≈

−

(

)

4

3 1

2

3

µ

where Rtip is the radius of curvature of the AFM tip being pushed toward a sample of Young’s

modulus Ysample, with the Poisson ratio of the soft-​matter material given by μ.

The restoring mechanical force resulting from deflections of the cantilever is easier to

characterize since it can be modeled as a simple Hookean spring, such that FME =​ −kzz where

kz is the cantilever’s vertical stiffness and z is the small vertical displacement of the cantilever

from its equilibrium zero force position. Thus, this mechanical elastic potential has a simple

FIGURE 6.6  Atomic force microscopy imaging. (a) Samples are raster scanned laterally rela­

tive to an AFM tip, such that the cantilever on the tip will deflect due to topographical features,

which can be measured by the deflection of a laser beam. (b) Schematic for the Hertz model of

biological material deformation. (c) Standard AFM cantilever size and shape. (d) Shape of energy

potential curve for a tip–​cantilever system as a function of the tip–​sample distance.