Pacific Nanotechnology Inc.

Introduction

This application note describes a method for measuring lateral forces as a function of applied load with a Pacific Nanotechnology Nano-R™ SPM using a LabView interface to the Nano-R™ Software supplied with the AFM. Conventional friction measurements with realistic, large-area contacts are generally characterized by a friction coefficient μ = FL/FN, the ratio of the lateral force FL to the normal force FN and, according to Amontons’ law, this parameter is generally relatively independent of the normal force FN. Since the lateral force depends on the contact area AC and the shear strength of the contacting interface S, as FL = ACS, this implies that the contact area is proportional to the normal force. When measuring the friction force using an atomic force microscope, the contact pressures are invariably below the plastic yield strength of the interface, which therefore deforms elastically. In this case, the contact area does not vary linearly with the normal force. In addition, measurement of a force-distance curve in AFM indicates that there is adhesion between the tip and substrate that contributes to the resulting normal force experienced by the tip-surface contact, which must be taken into account.

The elastic contact area in AFM is given either by JKR [1,2] or DMT [3] theory, where JKR theory applies to compliant materials with large, short-range attractive forces, while DMT theory applies to stiff materials with small, long-range attractive forces. The nondimensional parameter μ defines which theory is applicable where:

Equation 1:

Where: R is the tip radius
γ is the work per unit area red elastic modulus
z0 represents the equilibrium squired to separate the tip from the surface
E* is the reducepacing for the interaction potential of the surfaces [4].

If μ<0.1, DMT theory should apply, while if μ>5, JKR theory should be valid. For DMT theory, the contact area A^DMT is given by:

Equation 2:

Where: Lcorr = L + 2πγR and is the normal load corrected for adhesion between the tip and substrate [3]
2πγR is the pull-off force measured from the force distance curve.

In the case of JKR theory [1,2], the contact radius a is related to the load L as:

Equation 3:

Where: the pull-off force is now 3πγR

Measurement of an interfacial shear strength in AFM requires a knowledge of the tip radius and the normal and lateral forces. However, comparative measurements of frictional forces can be made, using the same AFM tip, without calibrating these values.

The method is similar to MFM. Images can be collected in DC (contact mode) by recording the deflection of the cantilever or by AC mode (close contact mode) where the cantilever is oscillated above the surface and either the phase or amplitude of the cantilever is recorded. In this work we chose to do DC imaging. However, there is no reason that AC imaging cannot be performed and by following the MFM application note this can easily be done.

Background

The position of the cantilever is monitored by the quadrant detector, where normal forces cause a vertical deflection of the cantilever and a corresponding vertical motion of the “red dot” in the Red Dot Alignment screen shown in Figure 1, while a lateral force results in a lateral motion of the “red dot” in the alignment screen. The corresponding vertical and normal voltages from the quadrant detector are therefore proportional to the normal and lateral forces respectively. In order to completely decouple the normal and lateral forces from each other, ideally the quadrants of the detector should be precisely aligned with the normal motion of the cantilever. In practice, the quadrant detector is aligned at some, generally small, angle Θ with respect to the vertical motion of the cantilever so that pure vertical motion of the cantilever can give rise to a lateral voltage change, while a purely lateral motion can give rise to a vertical voltage change. Thus, in order to precisely measure normal and lateral forces using AFM the angle Θ must be determined. The method for performing this measurement is outlined in reference 5.

A friction force versus normal force curve is measured by initially bringing the tip into contact with the surface with some initial normal force (given by some value of the quadrant detector voltage). The frictional force is then measured by moving the cantilever laterally in one direction at some sliding velocity, and by measuring the lateral deflection voltage in the quadrant detector. Since the sample is unlikely to be completely flat, a tilt of the sample with respect to the horizontal trajectory of the tip can add contribute to the measured forces. The lateral force is therefore also measured for sliding in the opposite direction resulting in a “friction loop” as shown in the figure. The friction force for a particular normal force is taken as the average of the forward and reverse lateral forces. The normal force is then decreased by some amount and the friction loop collected once again. This procedure is repeated until the friction force decreases to zero. The resulting plot yields values of lateral quadrant detector deflection voltage versus normal quadrant detector voltage. Since the resulting force at the surface is a sum of the applied force and adhesion between the tip and substrate, for part of the measurement, for resulting forces less that the pull-off force, the cantilever is in tension. In this case, the system is susceptible to vibrations that can cause the tip to some out of contact resulting in a sudden drop to zero lateral force and the tip moves freely in space. This effect is minimized by carrying out the experiment as rapidly as possible. If such effects are found, changing the location of the AFM or mounting the system on an anti-vibration table may be required.

Precise measurements of contact areas also require a knowledge of the tip radius, the elastic modulus E* of the interface and the normal and lateral force constants k_N and k_L respectively. The value of E* is obtained from:

Equation 4:

Where: E₁ and E₂ are the bulk moduli of the tip and substrate materials and v₁ and v2 their Poisson’s ratios

Tip radii can be measured directly using transmission electron microscopy (TEM). Note that the resolution available in scanning electron microscopy (SEM) is generally not sufficiently high to allow precise radii to be measured. Alternatively it is possible to reconstruct tip shapes by scanning a surface of known topology using SPIP software (http://www.nanoscience.com/products/SPIP.html). This latter strategy suffers from the problem that the measurement itself may result in a change in tip shape or in contamination of the tip. The normal force constant is determined from the from the resonance behavior of the tip [5], which can be measured using the Nano-R™ Software. The lateral force constant can me measured by collecting a lateral force versus normal force curve on a surface of known topology.

Materials Needed

AFM tip. This may be either Unmounted or pre-mounted. Note that mounted tips may be too large to fit between the poles of a TEM. Lateral force calibration grid, for example, that available from MikroMasch, number TGG01.

Requirements

• Pacific Nanotechnology AFM
• Pacific Nanotechnology Signal Access Control
• LabView Software
• LabView Interface

AFM Setup

• In the Signal access Control connect the Mon10 Z(T-B) to AUX1-
• Open SPM Cockpit
• Set Channels:

Channel 1: Z(HGT)
Channel 2: Z(ERR)
Channel 3: Z(L-R)
Channel 4: AUX(IN)

• Align laser
• Go into contact
• Record a force distant curve monitoring the AUX(IN) signal and record pull-off force.
• Set the resolution to the desired value
• Set the scan size. Example: 10 μm.

Starting LabView Interface

• Open Friction_vs_load.vi
• On the front panel window select the file path where all the resulting files will be stored. Example: C:\friction data
• Input the desired file names of the forward and reverse raw data. Example: uwm_forwad.txt and uwm_reverse.txt
• Input the initial load, the step size (absolute value) and the final load in mV.
• Input the corresponding scan resolution
• Run the VI
• After the run, five files will be placed in the corresponding file path.
• (uwm_forward.txt, uwm_reverse.txt, for.txt, rev.txt and aux.txt)

Procedure for Friction versus Load (FvL) Data Analysis

Based on a method by D. Flater and R. Cannara

I. Calibrating F versus L data- Calibration Experiments

A. Lateral calibration (Based on a method suggested by Ogletree et al., Rev. Sci. Instrum. 67, 9 (1996))

1. After performing FvL experiments, the tip should be scanned across a lateral calibration grating. This sample has long parallel ridges of well-defined crystallographic orientation, such as a SrTiO3 sample or Mikro Masch’s TGG01. The TGG01 works best for all types of tips, even tips with a large radius.

2. Scan an image with load variation so that at least 2 sloped facets (one up, one down) can be seen during the scan. Scanning at loads between - FPO and 2FPO, where FPO is the pull-off force, is sufficient.

B. Normal calibration (Based on a method suggested by Sader et al. Rev. Sci. Instrum. 70, 10 (1999))

1. Determine the resonance frequency of the cantilever.

a) Measure amplitude of oscillation of the cantilever as a function of frequency of forced oscillation. Find the first significant resonance peak. This can be performed by the Pacific Nanotechnology software. This should correspond to the first resonance of the cantilever. Record the frequency at which the amplitude is a maximum within this resonance peak.

b) Double check this frequency with that which is given by the manufacturer.

(1) Your measured frequency should be in the same range as the manufacturer’s value.

(2) If not, you may have found a harmonic of the resonance frequency. Change the frequency range so that this resonance frequency may be found.

2. Determine the Q factor of the cantilever.

a) Some software programs can calculate Q automatically.

b) If it is not calculated automatically, record the resonance peak to be imported into a graphing software package (e.g. KaleidaGraph).

(1) Determine the full width at half the maximum.

(2) Q = w/δw, where w is the resonance frequency and δw is the full width at half maximum of the resonance peak.

3. If possible, record ambient temperature and relative humidity, during the above steps.

4. Measure in-plane dimensions of the cantilever.

a) Use an optical microscope with a calibrated measuring device to determine the length and width of the cantilever.

b) Be careful to measure the length from the appropriate base of the cantilever; this position is not always obvious, and it may be necessary to check with the manufacturer (or obtain a side-view image in an SEM or TEM).

II. Analyzing F versus L Data

A. Normal Force Constant Calibration

1. To determine the normal spring constant of the cantilever.

a) Go to http://www.ampc.ms.unimelb.edu.au/afm/calibration.html

b) Input values for the resonance frequency, Q factor, and the in-plane dimensions of the cantilever.

c) The program will automatically calculate the normal spring constant of your cantilever, in N/m. If you know the torsional resonance properties, the section at the bottom will calculate the torsional spring constant.

2. Determine the normal sensitivity of the cantilever.

a) From force distance curves measured on a stiff surface, such as silicon or diamond, calculate the normal sensitivity of the cantilever (i.e., the slope of deflection voltage versus z displacement). Some AFMs calculate this slope for you from a force distance plot.

b) The normal sensitivity, SN, is basically the conversion factor between the number of vertical nanometers traveled by the piezo and the change in volts measured by the movement of the laser spot on the photodiode. Units: [nm/V]

C. Lateral Force Constant Calibration

1. Take the friction_v_load.m file, including friction calibration images (rev.txt, for.txt, revaux.txt), and place them in the same file folder.

2. Open up MatLab and make the current directory the same as the one that holds the above files.

3. Run friction_v_load.m to calibrate the lateral sensitivity of the cantilever. (The prompts for the program should be self-explanatory, but here are the step-by-step actions. Italics here indicate what you see in the MatLab command window, or what you should type.)

a) Type the following at the MatLab prompt, where for.txt, rev.txt and revaux.txt are the files that you wish to analyze: friction_v_load(‘auxrev.txt’,’for.txt’,’rev.txt’)

b) Input offset value to be subtracted from norm (none = 0):

(1) A window will appear showing a force distance curve obtained from the variation of the load. Forces should be given here in volts and there will be an offset due to the initial alignment of the photodiode. The out-of-contact region of the force curve should be at 0V but may be offset due to this alignment issue. Therefore the data need to be shifted accordingly.

(2) Enter the value in volts that this line is away from 0. You can use the zoom-in tool to help with this.

(3) For example, if the out-of-contact line is at –1.2V, type in –1.2 at the prompt.

c) Do you want to refine the offset? (yes = 1)

(1) Type 0 or hit ENTER if offset value is correct.

(2) If you type the number incorrectly or wish to modify the offset, type 1 at this prompt. Then repeat process as in b2, according to the offset that you now see in the plot.

d) Perform lateral calibration using this image? (yes = 1) Type 1

e) Normal and lateral forces will remain in volts for lateral calibration routine. Pull_off_force_V =-2.2084 Your image should contain 2 sloped features/facets parallel to the edge of the image.
image_size =512 (pixels) and scan_size_nm =100 (nm) Specify facet 1. See plot for (lateral offset) image (rotate the image for a top view.) Input the range of pixels to include (smallest value first.) Select the first edge (integer between 1 and image pixel size):

(1) A window will appear showing in 3D the trace minus retrace friction image. Rotate the image using the rotate tool to see the image as 2D with colors giving the 3rd dimension.

(2) The colors should show you the friction image captured on the faceted surface. You should have two facets visible. Make sure you know which region the tip is going up on during the trace (this is the UP
slope) and which the tip is going down on during the trace (DOWN slope).

(3) Also the edges of n. Note the pixel numbers that enclose this region.

(5) Type in the corresponding pixel numbers to eliminate data before the first edge facet. For example, if there are about 20 pixels of unwanted data on the edge, then type 20 at first prompt.

f) Select the second edge (integer between 1 and image pixel size):

(1) Again determine what pixel number you would like to cut the facet off at.

(2) For example, if the data are good from pixel 20 to pixel 190, type 190.

g) Do you want to revise this region? (yes = 1)

(1) This prompt allows you to go back and fix things is you made an error in choosing the region.

(2) With this prompt will appear the region you have chosen, with the graph axes modified so that the beginning edge starts at pixel 1.

(3) If you hit ENTER or type anything but 1, the program will continue.

(4) If you type 1, care needs to be taken to redo the region. You will either need to remember what you think it should be in relation to the original pixel numbers, or you can type 1 and 512 to see the original image with its original numbers, and then redo the region once more.

h) Calculated lateral offset and halfwidth from each of the two sloped features. The program will output a text file with the name results.txt. This will have the space-delimited data in 5 columns and a row of data for every line of the image (512 for this example), where the columns are:
Deflection[V] Halfwidth(1)[V] LatOffset(1)[V] Halfwidth(2)[V]LatOffset(2)[V].
These columns are cantilever deflection, friction loop halfwidth on facet 1, friction loop lateral offset on facet 1, friction loop halfwidth on facet 2, and friction loop lateral offset on facet 2, respectively. This file will be created in the same folder as the original file.

4. Repeat process (step 3) for additional friction calibration files.

5. Using the output file from friction_v_load.m, plot deflection versus the other variables. Determine the slopes of each of these curves. Use Ogletree et. al. RSI (1996) to determine the lateral sensitivity factor of the cantilever/AFM system for the experiment. This will be S (non dim.).

D. Analyzing F versus L Data

1. Once the normal stiffness and normal and lateral sensitivities have been calculated, you are ready to analyze data to create fully calibrated F versus L plots. You may also go ahead an analyze F versus L data without the calibration numbers.

2. Run friction_v_load.m on actual F versus L experiment files for your sample.

a) Follow steps in C1,2,3a-c

b) Perform lateral calibration using this image? (yes = 1) Type 0 or hit ENTER.

c) Enter the normal force calibration in nN/V (none = 1 or return):

(1) Enter the value of the spring constant of the cantilever in N/m and multiply it by the normal sensitivity of the cantilever in nm/V. Enter this value at this prompt. You may multiply the numbers together at the prompt if you like.

(2) For example, if the normal spring constant of the lever is 0.05 N/m and the normal sensitivity is 100nm/V, then you can either type in 5 or 0.05*100.

(3) If the normal calibration is unknown, type 1 or hit ENTER at the prompt.

d) pull_off_force_V = -2.2084

(1) The pull-off force is measured automatically and given in either nN or V, depending on if the normal forces were calibrated or not.

e) Eliminate edges from friction trace and retrace? (yes = 1)

(1) Hit ENTER or type 0 if you do not wish to perform this operation. Note that this will need to be performed on most images, unless the scan size is so large that these edge regions can be ignored. Skip to part f).

(2) Type 1 if you wish the analysis of the image to take place on the sliding region of the images. This means that at the edges, where the tip initially sticks before sliding, will be eliminated from the analysis.

(a) Calculating friction force (or energy dissipation) from region to be specified as follows:
image_size = 512 (in pixels)
scan_size_nm = 100 (nm)
See plot for trace-retrace image. Note: rotate the image for a top view. Input the range of pixels to include (smallest value first.) Select the first edge (>=1):

(i) A window will appear showing in 3D the trace minus retrace friction image. Rotate the image using the rotate tool to see the image as 2D with colors giving the 3rd dimension.

(ii) At the beginning and end of the image, the friction trace minus retrace values will be slightly smaller than the rest of the image. We want to eliminate these regions because this is where the tip has not yet begun sliding. Note the pixel numbers where this region begins.

(iii) Type in the corresponding pixel number to eliminate the first edge. For example if there are about 20 pixels of unwanted data along the edge, then type 20 at the first prompt.

(b) Select the second edge (>=1):

(i) Note the pixel numbers where the sliding region of the image begins.

(ii) Type in the pixel number corresponding to eliminate the second edge of the image. For example if there are about 20 pixels of unwanted data along the edge, then type 490 at this prompt.

(i) The modified image will appear with this prompt. Rotate the image again using the rotate tool to see if the edge regions have been eliminated.

(ii) If the edges have successfully been eliminated, type 0 or hit ENTER.

(iii) If the edges have not been completely eliminated, type 1. You will either need to remember what you think it should be in relation to the original pixel numbers, or you can type 1 and 512 to see the original image with its original numbers, and then redo the region once more.

f) Choose friction or energy dissipation (energy = 1): Type 0 or hit ENTER.

g) Enter the lateral calibration factor in nN/V (none = 1 or return):

(1) Enter the value of the lateral calibration factor of the cantilever in nN/V. Use the value of S determined in C6 and multiply it by the normal calibration factor of the cantilever in nN/V. Enter this value at this prompt. You may multiply the numbers together at the prompt if you like.

(2) For example, if the normal spring constant of the lever is 0.05 N/m, the normal sensitivity is 100nm/V, and S=10, then you can either type in 50 or 0.05*100*10.

(3) If the lateral calibration is unknown, type 1 or hit ENTER at the prompt.

h) The program will output a text file with the name results.txt. This will have the space-delimited data in 2 columns: load data for every line (512 for this example) and friction data (trace-retrace) for every line. This file will be created in the same folder as the original file.

3. Repeat process (step 2) for additional friction calibration files.

III. Plotting Friction Data

A. Now the data is ready to be plotted in a software analysis program such as KaleidaGraph or Origin.

B. When the data are plotted you may notice some stray data points clustered around 0 nN applied load. These stray points are due to a false frictional signal measured while the tip is out of contact with the surface. From this plot, eliminate all the data points within the out-of-contact region of the curve. Then replot the FvL curve and you will have eliminated these stray points.

Acknowledgements

The software in this application note has been adapted from a program originally written by Professor Robert Carpick of the University of Wisconsin-Madison. We are grateful for him allowing us to use this method as part of this application note. The original files are available at http://mandm.engr.wisc.edu/faculty_pages/carpick/toolbox.htm.

References

1. K.L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1985)
2. K.L. Johnson, K. Kendall and A.D. Roberts, Proc. Roy. Soc, London A, 324, 301 (1975)
3. B.V. Derjaguin, V.M. Muller and Y.P. Toporov, J. Colloid Interface Sci., 53, 314 (1975)
4. V.M. Muller and V.S. Yushchenko, Colloid, J. USSR., 42, 803 (2003)
5. J.E. Sader, J.W.M. Chon and P.