# HertzFit

## Hertz Fit

## Data Import

This application imports ASCII files, which have two columns of data: column 1 is the z position and column 2 is the deflection value, separated by a space. Data format is float, possible with an exponent excluding units. One typical line may look like: 1.023e-9 234.7e-9.

So, the units of the data are "m".

If your data, have differents, e.g. they are scaled in nanometers, you can use the xscaling or yscaling for correcting this, e.g. by using a scaling factor of 1e-9.

If the orientation of data is different then the example data, because your instrument uses a different definition for the sign of height or deflection data, you may use a scaling factor of "-1" to correct this. The data should look like below.

The file should include approach and retract part of the data as a continuous sequence of data as they are recorded by the AFM. So the z data will look like a triangular wave.

## Using the Application

The application will analyze the force curve in a region of interest determined by delta1 and delta2. It will calculate the slope and try to fit a Hertz model to the data within this range. During the procedure it will display the part of the force curve of interest (approach or retract), the force versus indentation calculated from the data on a linear and a log-log scale, the application will also add theoretical curves based on results of the fit.

Every time you change a relevant parameter of the fit, it will redo the procedure, so the DoFit button is mostly obsolete.

If you type in a Young modulus value it will calculate a simulated force curve and force versus indentation curve based on this value

## Understanding the application

The details of the fit procedure have been documented in the following book chapter (Radmacher, M., Studying the mechanics of cellular processes by atomic force microscopy, in Cell Mechanics, Y.-l. Wang and D.E. Discher, Editors. 2007, Academic Press. p. 347-372.).

The procedure is summarized briefly here:

Subtracting Deflection Offset

Since the laser spot is usually not perfectly centred on the laser diode, there will be a deflection value other than zero when the cantilever is off the surface. The application averages the first 10% of the data points, assuming they are off the surface and subtracts this value. The Raw Force Data panel shows approach and retract curves before offset subtraction.

The Force Data to Analyse panel shows either approach or retract curve after offset subtraction.

Analysis Range

The parameters delta1 and delta2 define the range of deflection values, which are included in the analysis. Delta1 may become as small as zero. The analysis range is depicted by two green horizontal lines in the Force Data to Analyze panel and in the Force vs Indentation panel, where the deltas have been converted to the corresponding force values.

The analysis range is used for calculating the slope as well as for the Hertz fit.

Hertz Fit

The Hertz fit is implemented using a so called Levenberg Marquardt algorithm. In short, χ2 (the difference between the fit function and the data points) is minimized by varying the fit parameters. The Levenberg Marquard algorithm does this minimization in an optimized way since in addition to the fit function the derivatives of the fit function with respect to the fit parameters are used to obtain a fast and stable optimization. However the algorithm can only work reasonably, when start values sufficiently close to the expected results are provided.

## Fit Functions

The only fit parameters in each relation are the elastic modulus E and the contact point δ0. To include the contact point as a fit parameter is essential since it can not otherwise be determined accurately. E.g. the simplest way to define the contact point as the z height, where the deflection gets larger than zero does not work due to noise. The next idea to define the contact point as the point where the deflection gets larger than some threshold value, is actually used by the application to determine a first guess (see turquoise vertical line in Force Data to Analyse panel). However this guess will systematically be wrong.

The model for a pyramidal indenter is an adaption the well known conical model and has been suggested by Felix Rico (Rico, F., Roca-Cusachs, P., Gavara, N., Farré, R., Rotger, M., Navajas, D. 2005. Probing mechanical properties of living cells by atomic force microscopy with blunted pyramidal cantilever tips. *Phys Review E 72*:021914-021911-021910)

## Advanced Fit Functions

We have also included BEC (Bottom Effect Correction) fit functions, which correct for finite thickness of samples. As a general rule of thumb, when the indentation becomes larger than 10% of the sample thickness, the AFM tip will "feel" the underlying substrate. Thus the Young moduli calculated from a simple Hertz fit will be systematically to high. This artefact can change values by an order of magnitude or even two. Richard Chadwick has therefore developed a correction factor, which can be added to the Hertz fit to incorporate this effect. The only parameter is the sample thickness, which has to be determined independently, e.g. by comparing the contact point of the force curve of interest on a cell, and another force curve nearby, but on the substrate. The thickness is a parameter and cannot be determined by the fit routine itself.

To allow the user to play around with this parameter, entering a new value will recalculate the simulated force curve, but not invoke a fit. If you wish to restart the fit, you will have to press the DoFit button.

For details, you may refer to the original publication: Gavara, N. and R. S. Chadwick. 2012. Determination of the elastic moduli of thin samples and adherent cells using conical atomic force microscope tips. *Nature Nanotechnol.* 30. Sept:1-4.

## The Fit Process Step by Step

1. Subtract deflection offset by averaging first 10% of data to be analysed

2. Use threshold value to determine a first guess of the contact point

3. On basis of this first guess for a contact point the force and indentation values at the two data points, where the deflection corresponds to delta1 or delta2. The fit function is inverted for these two value pairs to obtain a new guess for the contact point and the elastic modulus.

4. These two values are the starting point for the Levenberg Marquard algorithm, which will result in new values for E and δ_{0}.

5. On the basis of this new contact point δ_{0} the indentation is calculated again and the Levenberg Marquard algorithm is started a second time.

6. The results of this second round are the final results

7. In addition to the numerical values, a theoretical force vs. indentation curve and a theoretical force curve are calculated and displayed at the corresponding panels.

## System Requirements

We have tested both applets in the following configuration:

JavaRE 1.7.0.51

Mac OS X 10.7, 10.8, 10.9

Windows XP, Windows 7

Ubuntu Linux 10.04, 12.04

Since the applet loads data from your system, you need to grant them the necessary privileges and change the certificate.

This requires to include our website on the Java exception list and then accept the certificate.

## Java Exception List

In order to do so, you have to set the appropiate exemption in the second tab of the Java Control Panel. If you feel uncomfortable with this procedure, you may download the source code and run it locally. We have used Eclipes for development.

## Accept the Certificate

## 10th November 2017

*2017, Nov. 10*

*New Version 1.0 released*

HertzFit is no longer available as an Applet, because most browsers

do no more accept running Applets. You can download the application

and run it locally.

Bruker force curves exported as two textfiles for approach and retract

are now loadable and also single force curves from JPK instruments.

## 25th Feburary 2014

*New Version 0.3 released*

Bugs with the graphical diplay fixed.

Applet has a valid certificate now.

## 17th June 2013

*New Version 0.xx releaesed*

*Bug fixed: *

Selecting retract curve works now.

*F**eature added: *

xscaling and yscalin added. This allows now to use the applet with data, which have different units, or where x or y data have the wrong orientation. (see instructions for details)

## Credits

The first version of this applet has been created by Philipp Haupt from the Department of Informatics at the University of Bremen.

The current version has been developed by Holger Doschke and Manfred Radmacher at the Institute for Biophysics.

The Levenberg Marquard algorithm has been adopted from *Numerical Recipes.*

The XY plots are created with the open source library JFreeChart.

## Hertz Fit

You can download the application here.

You can download the source code for both applications here.

You can download a sample data set here.

The applet has been developed with Eclipse.

Plots are created with the open source library jFree Chart.

The software publicated here is free, may be modified and used for non-commercial purposes according to the GNU (General Public Universe).