Surface chemical analysis - Scanning probe microscopy - Determination of cantilever normal spring constants
1 Scope
This International Standard describes five of the methods for the determination of normal spring constants for atomic force microscope cantilevers to an accuracy of 5 % to 10 %. Each method is in one of the three categories of dimensional, static experimental, and dynamic experimental methods. The method chosen depends on the purpose, convenience, and instrumentation available to the analyst.
This document is not applicable to an accuracy higher than 5 % to 10 %. For accuracies better than 5 % to 10 %, more sophisticated methods not described here are required.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 18115-2:2013 Surface chemical analysis - Vocabulary - Part 2: Terms used in scanning-probe microscopy
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 18115-2 and the following apply.
3.1
normal spring constant
spring constant
force constant
DEPRECATED: cantilever stiffness
kz
quotient of the applied normal force at the probe tip (3.2) by the deflection of the cantilever in that direction at the probe tip position
Note 1: See lateral spring constant, torsional spring constant.
Note 2: The normal spring constant is usually referred to as the spring constant. The full term is used when it is necessary to distinguish it from the lateral spring constant.
Note 3: The force is applied normal to the plane of the cantilever to compute or measure the normal force constant, kz. In application, the cantilever in an AFM may be tilted at an angle, θ, to the plane of the sample surface and the plane normal to the direction of approach of the tip to the sample. This angle is important in applying the normal spring constant in AFM studies.
3.2
probe tip
tip
probe apex
structure at the extremity of a probe, the apex of which senses the surface
Note: See cantilever apex (3.3).
3.3
cantilever apex
end of the cantilever furthest from the cantilever support structure
Note: See probe apex (3.2), tip apex (3.2).
4 Symbols and abbreviated terms
The following symbols and abbreviated terms apply to this document.
The abbreviated terms are:
AFM Atomic force microscopy
FEA Finite element analysis
PSD Power spectral density
SEM Scanning electron microscopy
SPM Scanning probe microscopy
Note: In the list of abbreviated terms below, note that the final “M”, given as “Microscopy”, may be taken equally as “Microscope” depending on the context.
The symbols for use in the formulae are:
A amplitude of cantilever at a certain frequency
A0 amplitude of a cantilever at its fundamental resonant frequency
Awhite mean amplitude of a cantilever associated with white noise
BΦ gradient determined from a straight line fit to values of Lx versus Φx1/3
Bk gradient determined from a straight line fit to values of Lx versus (kzLx)-1/3
C1 correction factor for the thermal vibration method described in 8.2
C2 correction factor for the thermal vibration method described in 8.2
d distance between the probe tip and the cantilever apex
D height of the probe tip
e width of the V-shaped cantilever at a distance L0 from the apex
E Young’s modulus of the material of a cantilever
EB Young’s modulus of the base material of a cantilever
EC Young’s modulus of the coating material on a cantilever
f frequency
f0 fundamental resonant frequency of a cantilever
F force of a nanoindenter
h displacement of a nanoindenter
i index of Pi, where i = 1 to 5
kB Boltzmann constant
kz normal spring constant
kzLx normal spring constant at the position Lx along a cantilever
kzR normal spring constant of a reference cantilever
kzW normal spring constant of a working cantilever
kz(tc=0) normal spring constant of a cantilever with a coating thickness of 0
L length of a rectangular cantilever or the effective length of a V-shaped cantilever
Lx distance between the base of a cantilever and the effective position of a V-shaped cantilever
L0 length of a V-shaped cantilever between the apex and the start of the arms
L1 length of a V-shaped cantilever between the base and the start of the arms
Pi label of one of the five positions on the reference cantilever axis
Q quality factor of a cantilever
r term defined by Formula (7)
t thickness of a cantilever
tB thickness of the bulk material of a cantilever
tC thickness of a coating on a cantilever
T absolute temperature of the cantilever measured in Kelvins
uA0 standard uncertainty in A0
uB standard uncertainty in B
uC1 standard uncertainty in C1
uC2 standard uncertainty in C2
ud standard uncertainty in the distance between the probe tip and the cantilever apex
uE standard uncertainty in the Young’s modulus of a cantilever
uF standard uncertainty due to the calibration of force in the nanoindenter
uf0 standard uncertainty in the resonant frequency
uh standard uncertainty due to the calibration of displacement in the nanoindenter
ukz standard uncertainty in the normal spring constant
ukzR standard uncertainty in the normal spring constant of the reference cantilever
uL standard uncertainty in the length of a cantilever
uQ standard uncertainty in the quality factor of a cantilever
ut standard uncertainty in the thickness of a cantilever
uT standard uncertainty in the absolute temperature
uw standard uncertainty in the width of a cantilever
ux1 standard uncertainty in x1
uα1 standard uncertainty in α1
uρ standard uncertainty in the density of a cantilever
w width of a cantilever
w1 width of one side of a trapezium
w2 width of one side of a trapezium
wt w cosθ
x1 offset to account for the small uncertainty in the true position of the base of the cantilever com-pared to an arbitrary reference point
x2 offset to account for the uncertainty in the true position of the probe tip compared to an arbi-trary reference point
Z1 term defined by Formula (4)
Z2 term defined by Formula (5)
α angle of the working cantilever with respect to the reference cantilever or surface
α1 numeric constant used in Formula (11)
δR average inverse gradient of the force-distance curve obtained with the working cantilever pressing on the reference cantilever or device
δW average inverse gradient of the force-distance curve obtained with the working cantilever pressing on a stiff surface
υ half angle between the arms of a V-shaped cantilever
θ2 term defined by Formula (6)
ν Poisson’s ratio of the cantilever material
ρ density of a cantilever
φx term defined by Formula (16)
