Spring Constant Calibration

Atomic force microscopy (AFM) is used in a great variety of force measurement applications, including investigating the unfolding pathways of native membrane proteins, probing the structure of single polysaccharide molecules, and monitoring the response of living cells to biochemical stimuli. Although the cantilever deflection can be measured with great accuracy and sub-Angstrom sensitivity, converting these measurements to units of force requires that the spring constant, k, be determined for each cantilever.

When cantilever deflection is sufficiently small, as is the case in most Contact Mode and TappingMode applications, the relation of the force on the tip to the position of the cantilever's free end is elastic, that is, linear, given by Hooke’s Law applied to a spring:

Spring constants can vary greatly from the values quoted by their manufacturers. In fact, these values are only provided as nominal indications of the cantilever properties and the manufacturers often specify the spring constant in a wide range that may span values up to four times smaller and four times larger than the nominal value. This is because the techniques used to fabricate the probes can result in substantially different cantilever dimensions, especially thickness, from wafer to wafer and smaller variations within a single wafer. While it is sometimes possible to achieve tighter tolerances, this generally is not practical for the economical production of probes for general imaging and force measurement applications.

Fortunately,many techniques have been proposed to characterize cantilever spring constants. These can generally be grouped into three categories:

• Dimensional models, where fully theoretical analysis or semi-empirical formulas are used to calculate the cantilever spring constants based on their dimensions and material properties
  • Eg, Estimation from Cantilever Geometry
• Static deflection measurements, where the spring constant is determined by loading the cantilever with a known static force
  • Eg, The Added Mass Method
• Dynamic deflection measurements, where the resonance behavior of the cantilever is related back to its spring constant
  • Eg, Rectangular Cantilever, Hydrodynamic Model
  • Eg, Measuring Thermal Noise

These techniques are reviewed in detail in Bruker Application Note 94: Practical Advice on the Determination of Cantilever Spring Constants.

The most widely used and commonly applicable of these techniques is the “thermal tune method”, an algorithm that makes use of thermal noise data. This technique is implemented in the NanoScope software, which allows the probe spring constant to be calibrated in either air or in fluid by simply calibrating the deflection sensitivity and then making a measurement in the Thermal Tune window. The thermal tune technique is best applied to probes with spring constants less that approximately 5 N/m (e.g. FESP) all the way down to the very softest silicon nitride probes (e.g. MLCT). For stiffer probes, please refer to the above application note for recommendations.

Other methods, notably the added mass method, can be more accurate if done with great care. However, the thermal tune method is very simple as implemented in NanoScope, and offers accuracy comparable to that normally obtained by other methods.

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