Revisiting the theory behind AFM indentation procedures. Exploring the physical significance of fundamental equations

Abstract

Fundamental contact mechanics models concerning the interaction of an axisymmetric indenter and an elastic half-space are usually employed in atomic force microscopy (AFM) indentation methods. In this paper, a simplified ‘equivalent’ physical system is used to correlate basic magnitudes such as the applied force on an elastic half space, the Young’s modulus, the contact radius and the indentation depth. More specifically, the equations correlating the above magnitudes are derived using fundamental physics instead of the typical rigorous mathematical process with a small error. In addition, the relation between a force-indentation curve and the indenter’s shape is also presented in detail in order to help students and non-specialists in contact mechanics to obtain a strong background to the AFM indentation theory.

Atomic Force Microscopy Nanoindentation Method on Collagen Fibrils

“…In other words, the sample is significantly bigger than the AFM tip and presents a linear elastic behavior. Thus, they can be used only if the radius of the collagen fibril is extremely big compared to the tip radius [ 56 , 57 ]. A rational approximation to apply the equations of Section 2.2 is the fibril radius to be at least 5 times bigger compared to the tip radius [ 56 , 57 ].…”

“…Thus, they can be used only if the radius of the collagen fibril is extremely big compared to the tip radius [ 56 , 57 ]. A rational approximation to apply the equations of Section 2.2 is the fibril radius to be at least 5 times bigger compared to the tip radius [ 56 , 57 ]. However, from a rigorous mathematical perspective, when indenting a collagen fibril using an AFM tip the interaction can be modelled as the interaction between a rigid sphere, and a cylinder-shaped sample under the restriction that the tip apex can be considered as spherical ( Figure 4 ).…”

Is It Possible to Directly Determine the Radius of a Spherical Indenter Using Force Indentation Data on Soft Samples?

“…As it has been previously reported, the applied force on a half space when using an axisymmetric indenter is directly proportional to the contact radius r c between the indenter and the sample for a specific indentation depth [ 16 ]. In other words, F ~ r c h .…”

“…As a result, while the indentation depth increases when using a spherical indenter, the contact radius tends to a limit value which will be equal to the indenter's radius [ 15 , 16 ]. Thus, the applied force F = f ( h ) should be at first proportional to h 3/2 (very small indentation depths) and will become linear (i.e., proportional to h ) for very big indentation depths [ 15 , 16 ]. Hence, the parameter Z is a “correction factor” in order to apply the Hertz equation for big indentation depths (i.e., h > R /10) [ 15 , 16 ].…”

“…Thus, the applied force F = f ( h ) should be at first proportional to h 3/2 (very small indentation depths) and will become linear (i.e., proportional to h ) for very big indentation depths [ 15 , 16 ]. Hence, the parameter Z is a “correction factor” in order to apply the Hertz equation for big indentation depths (i.e., h > R /10) [ 15 , 16 ]. In other words, Z is a mathematical quantity that accounts for the change of the slope of the F = f ( h ) curve as the indentation depth increases.…”