What is a spring constant?

As the name implies the spring constant *k***[N/m]** describes how much force [N] is needed to squeeze a spring system by a certain distance [m].

In case of the FluidFM probes, the spring constant relates the probe deflection in [m] with the corresponding force [N] that acts on the probe in order to achieve this deflection.

Measuring the spring constant

In the FluidFM system we rely on the commonly used Sader method to measure the spring constant of the FluidFM probe. It relies on the probe dimensions and its first resonance peak. Details can be found here.

It is crucial to note that the Sader Method gives the spring constant for a force acting perpendicular to the very end of the probe.

In reality the probe is however at an angle to the surface and interacts not at its very end, but at the pyramid, a few microns from its edge. This is where we come to:

The effective spring constant

The force needed to displace the cantilever by 100 nm will increase as the place of interactions get further away from the cantilever edge. For FluidFM nanopipettes the place where the force is acting is the pyramid. The relation is can be gained through simple beam mechanics and is as follows (Sader et al. 1995. Rev.Sci.Instrum, 66):

Where *kp* is the spring constant at the pyramid, *kmeas* the measured spring constant according to Sader, *l* the length of the cantilever (typically 200 um) and *lp* the length of the cantilever until the pyramid (typically 190 um), or the length until the opening in case of the micro pipettes (193 um).

As can be seen through beam mechanics, pressing an inclined cantilever against a surface, will increase the required force to bend it. This is the case for FluidFM systems, where the cantilever comes with an angle of 11° towards the surface. The relation is (Heim et al. 2004. Langmuir, 20(7)):

Where *ka* is the spring constant due to the angle *a *and the pyramid height D (7 um). In case of colloidal probes the colloid radius has to be added to D.

So the total correction equals:

Typical correction factors

For typical FluidFM probes the total correction factor for the spring constant is as follows:

FluidFM Probe type |
keff/kmeas |

FluidFM Nanopipette | 1.229 |

FluidFM Micropipette | 1.155 |

FluidFM Micropipette with 10 um colloid | 1.167 |

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