AFM Force Measurements

The main part of this work deals with the measurement of force by expand-ing actin gels. The type of AFM experiment that allows insight in the forces generated by the actin system differs from conventional AFM force measure-ments in some ways. The first conventional AFM force measuremeasure-ments, also known as ‘force spectroscopy measurements’, were published in 1989 [100]. A major part of AFM force measurements is dedicated to study interaction forces of interfaces. These will only be discussed briefly, because the measurements performed here deal with the micromechanical analysis of colloidal objects, in which interfacial forces are not sensed deliberately. AFM mechanical studies have been done on numerous colloidal objects e.g. cells, thin films or artificial

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capsules [99, 101, 102]. In this mode usually force-displacements curves, i.e. a plot of the applied force versus the probe-sample distance, are recorded. In the following we discuss the acquisition of such a plot and its meaning from a gen-eral viewpoint. In order to obtain a force displacement curve, the probe (or the sample) is displaced along the vertical axis, while the cantilever deflection  , as well as the z-piezo position is measured. To study mechanical material proper-ties, like the sample spring constant , it is required that the force imposed by the AFM probe actually deforms the sample. The force acting on the AFM can-tilever  , is described by Hooke’s law:

Eq. 3-22

where is the spring constant of the cantilever. Upon approaching the sample the probe will impose a force and a sufficiently compliant sample will deform by   (Figure 3-12). Where is the thickness of the uncompressed sample and the thickness upon compression. The only distance that is con-trolled or measured in a force-displacement experiment is the z-piezo position and the cantilever deflection . The sample deformation can be calculated with and with the z-piezo shift  . is the z-piezo movement after first contact with the sample, , where is the final approach posi-tion and is the z-position at which the probe touches the surface only very slightly. The sample deformation then reads

. Eq. 3-23

Figure 3-12 The probe-sample system: Deformation and lengths upon deformation of the sample. Z is the vertical coordinate that is controlled by the z-piezo. D is the thickness of the soft sample

An AFM force-displacement curve reflects of two contributions: the tip-sample interaction and the spring force of the cantilever. This is schematically depicted in Figure 3-13A. Here the curve represents the tip-sample inte-raction force. At this point we assume a simple mechanical spring potential for the sample that is compressed by the AFM probe, 1/2    , hence . Note that the potential functions are usually more complicated in reality. The straight lines represent the elastic force of the sample and cantilever as expressed in Eq. 3-22. Upon driving the probe further down after probe-sample contact, the sample generates the restoring force   . In me-chanical equilibrium the total force acting on the cantilever is zero. That means for each piezo displacement the cantilever is deflected until the elastic force of the cantilever equals the tip-sample interaction force, that is:  

0. In equilibrium of forces and with a deformation of the sample , plus a given cantilever deflection , we can therefore write the basic equation of AFM force measurements:

  ^Eq.

3-24

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Figure 3-13 A) The cantilever deflection is, at every position, the result of two contributions: the probe-sample interaction (black curve) and the elastic force − (blue line). In force balance, the two forces must compensate and give   . For an arbitrary intersection point P of and , there exists a deflection  and   , which represent the elastic force of the canti-lever and sample. Both  and   are given by difference of the force curve inter-sections with the axis   and  . B)  is the z-piezo motion to any equilibrium point (P in this case) after probe-sample contact at . The graphical reconstruction shows the respective experimental force-displacement curve. C: The intersection of the force curves in A) can be seen as the combina-tion of two springs of length and which is the thickness of the deformable sample. The total spring constant   can be calculated accordingly and is found the slope of the experimental force curve.

Force balance is given by the intersection points of with   , see Figure 3-13A. The relation between the tip-sample interac-F

tion and the resulting deflection-displacement curve measured with AFM is shown by the graphical reconstruction in Figure 3-13B. The representative force curve shows that the sample is first approached by the probe ( =0), then con-tact is made at the z-position . Now, the two quantities recorded by an AFM are and , while force balance is always given. Reading these two parameters allows calculation of the sample deformation by using Eq. 3-24.

As a prerequisite the spring constant of the cantilever has to be known, which is the case after calibration the cantilevers with for example the Sader [103] or thermal noise method [104]. For the simple elastic spring potential discussed here, knowledge of the sample deformation would directly lead to the apparent spring constant of the sample . The total spring constant is represented as the slope of the force-displacement curve in Figure 3-13B. The graphical recon-struction descriptively shows how changes in the spring constants change the resulting deflections/deformations of the probe and sample as well as the slope of the force displacement curve.

A next step in the AFM force measurement procedure is relating the applied force with the sample deformation. With the appropriate model one obtains the mechanical properties of the sample, e.g. the elastic modulus of the material.

This is a critical part in an AFM force measurement. Contradictory to the simple example shown in Figure 3-13, the force curves are in general not linear, al-ready because the probe-sample contact area is changing as the sample deforms.

For a perfectly elastic and planar sample that is deformed by a spherical probe the AFM experiments can be described by the Hertz model [99]. In any case, analysis of the force curves requires geometrical control of the probe – sample contact. In order to exclude uncertainties due to the probe geometry or the tilt angle between sample and cantilever, spherical colloidal probes (see paragraph 3.7.3) are very frequently used as AFM force probes. For the sake of simplicity the discussion of the force-displacement curves covered only the elastic forces of the probe and the sample. Typically there are a number of other sample-probe interactions, such as adhesion forces, that can be studied using with AFM force-displacement curves