Parameters#
Material parameters#
Definition and physical meaningful range of all material parameters.
Probe’s spring constant \(k_c\)#
The AFM probe’s cantilever spring constant \(k_c\) describes the force \(F\) needed to bend the cantilever by a deflection \(δ\). Since this is considered an elastic deflection, it is described by Hooke’s law (1). Commercially available AFM Probes have an \(k_c\) range of 0.01 - 100 N/m.
Probe’s tip radius \(R\)#
The AFM probe’s tip radius \(R\) describes the curvature at the very end of the tip. In cases of a sharp tip (\(R\) << 100nm) the tip has a height of up to 20µm and is rather a paraboloid or a pyramid. However, for the mechanical contact (6) only the tip’s shape at the very end is of consequence. For the calculation of attractive forces this radius is usually underestimated which is accepted in SyFoS for the time being. In the future SyFoS might use an effective radius \(R_{eff}\) >> \(R\) which represents the influence of the tip size more realistically in respect to the attractive forces. In the case of colloidal probes (100nm < \(R\) < 20µm) the whole tip is of spherical shape and \(R\) represents the radius for both cases, attractive and repulsive forces, correctly.
Modulus \(E\)#
The Young’s modulus \(E\) is a characteristic property of every material, which describes the elastic responds to applied Force \(F\). Since the mechanical contact of an AFM measurement involves two different materials, the Young’s modulus of the tip material \(E_{tip}\) and the Young’s modulus of the sample’s material \(E_{sample}\) are required parameters.
Poisson ratio \(ν\)#
The Poisson ratio \(ν\) describes the deformation perpendicular to the direction of applied force \(F\). It is the negative ratio of \(ν=-\frac{e_\perp}{e_\parallel}\), with \(e_\perp\) being the relative change of dimension perpendicular to the applied force \(F\) and \(e_\parallel\) being the relative change of dimension in direction of applied force \(F\). Poisson ratios of common materials are expected to be between 0 and 0.5, which are accepted values in SyFoS. There is a class of materials which exhibits negative Poisson ratios (basically getting thicker when stretched), but those are not considered in SyFoS.
Hamaker constant \(A\)#
Van der Waals described the force needed to bring single neutral molecules from infinte to finite separation, now well known as van der Waals force \(F_{vdW}\). Hamaker transferred this concept to bulk materials by using pairwise summation approximation. This way the interactions and attractive forces of bodies can be described for different geometries, such as sphere/plane as appropriate for AFM measurements (tip/sample), by means of the Hamaker constant, which ranges from 1 to 450 zJ. All material specific Hamaker constants in SyFoS are for ambient conditions.
Force spectroscopy experimental set-up#
For creating synthetic force spectroscopy data parameters which concern the experimental acquisition of data need to be defined.
Start Distance \(Z_0\)#
The start distance \(Z_0\) is the initial distance between the AFM probe and the sample surface. It also defines the beginning of the Z-scale and hence the lowest value in the abscisse of the force-distance curve. The start distance should exceed by far the distance at which attractive forces are relevant. Very common values are between -1µm and -100nm.
Step Size \(dZ\)#
The step size \(dZ\) is the inverse of the point rate (\(nm^{-1}\)) of the data.
Maximum piezo displacement \(Z_{max}\)#
The maximum piezo displacement \(Z_{max}\) marks the end of the Z-scale and hence the highest value on the abscisse of a force-distance curve. As the maximum piezo displacement \(Z_{max}\) a positive value should be chosen, so contact between tip and sample is established.
Size of Force Volume – number of curves#
It is very common to record force distance curves in a x-y grid (mapping, also called force volume) rather than having single force-distance curves at arbitrary points. When recorded in a grid, the force spectroscopy data has an additional spatial information which is important for inhomogeneous samples. It is also useful for homogeneous samples because one acquires an array of curves which can be averaged and statistically treated for errors. In order to achieve a reasonable signal to noise ratio between 30 and 100 curves are averaged.
Artefact parameters#
Experimentally acquired force curves show typically artefacts, which are not present in ideal synthetic curves. These artefacts can be added to the ideal curves in SyFoS and their magnitude can be controlled by the following parameters.
Virtual Deflection#
In a force spectroscopy experiment deflection is assumed to be \(δ≡0\) when the cantilver is in its equilibrium position. Due to either thermal drift, creep, offset of deflection by AFM software or interference of the laser with the sample surface an additional value for the deflection is superimposed, the virtual deflection. In SyFoS this is simulated by an offset \(Δδ\). In real experiments this can be also a function of Z \(Δδ(Z)\).
Topography offset#
In a realistic force spectroscopy experiment the distnce between AFM tip and sample surface can differ, depending on the x,y position on the sample. Is is due to the samples topography. Even smooth smaples are tilted in respect to the normal plane (X,Y-plane) or show a certain roughness in the nanometer range. Considering selfassempled or cut samples the topography’s roughness is in the range of micrometer. Therefore the point of contact \(Z≡0\) is not known when the tip starts to approach the sample, merely estimated. This leeds to an offset \(ΔZ\) of the point of contact due to sample topography. In SyFoS this offset is a constant \(ΔZ\), since it is a control value for the subsequently tested correction.
Noise#
In AFM measurements signal to noise ratio is an important factor, since thermal noise of the probe and the sample stage can not be avoided when working at ambient conditions and is in the range of the measured values. In SyFoS this noise can be simulated by adding Gaussian noise to the synthetic data. The propability density function of the noise equals the normal distribution with the standard deviation σ, which can be varied by the user.
Auxiliary Parameters#
From all parameters given by the user auxiliary parameters as tip-sample distance ζ, reduced modulus \(E_{tot}\), jump to contact JTC and combined Hamaker constant \(A_{tot}\) can be calculated for creating a synthetic force curve. The auxiliary parameters are also given as output in the gui.
Tip-Sample distance#
For all theories, describing the different regimes of a force distance curve the true tip sample distance needs to be known. During the regime of attractive forces, the cantilever deflects towards the sample surface by \(δ\), thereby decreasing the tip sample distance \(ζ\) additionally to the z-pizo displacement, as described by equation (2). During the contact or the repulsive regime, the tip sample distance \(ζ\) should be 0, but it is actually increased by the deformation \(D\) that is caused by the contact between tip and sample, see equation (5).
In SyFoS \(ζ\) and \(D\) are calculated continuously for each iterative step. Only between JTC and contact the tip sample distance is assumed to be zero. This is a simplification but since this part of the data is not relevant for any automated analysis the effect of this simplification is neglectable.
Reduced modulus \(E_{tot}\)#
The reduced Young’s modulus \(E_{tot}\) is the resulting Young’s modulus of two materials - tip and sample - in contact. It is calculated from the Young’s moduli of tip and sample and the Poisson ratio of tip and sample with the given equation (7).
Jump to contact#
The attractive forces \(F_{attr}\) are dependent on the tip-sample distance (2). At a certain tip-sample distance the attractive forces \(F_{attr}\) between sample and tip increase up to a point when their gradient exceeds the spring constant kc. Figure 1 (ii) jump to contact (JTC): a discontinuity where the system is not in equilibrium and the tip snaps onto the sample.
Combined Hamaker constant \(A_{tot}\)#
The Hamaker constant A describes the attractive forces acting between matter depending on its distance, in case the distance is much smaller than the size of the bodies and is specific for a material pairing. Only if two bodies are of the same material, the Hamaker constant is considered material specific. In case SyFoS needs to estimate the Hamaker constant for a mixed pairing of tip and sample material it calculates the combined Hamaker constant: in the case of an Si-tip versus a PMMA-sample the combined Hamaker constant would be calculated as \(A_{SI,PMMA} = \sqrt{A_{Si}} * \sqrt{A_{PMMA}}\).