AFM probe for measuring ∼10−5 ultra-low friction coefficient: Design and application

Superlubricity provides a novel approach to addressing friction and wear issues in mechanical systems. However, little is known regarding improving the atomic force microscope (AFM) friction coefficient measurement resolution. Accordingly, this study established the theoretical formula for the AFM friction coefficient measurement and deduced the measurement resolution. Then, the formula was applied to the AFM probe with a rectangular cross-section cantilever. The measurement resolution is associated with the dimensional properties of the AFM probe, the mechanical properties of the cantilever material, the properties of the position-sensitive detector (PSD), and probably the anti-vibration performance of the AFM. It is feasible to make the cantilever as short as possible and the tip as high as possible to improve the measurement resolution. An AFM probe for measuring an ultra-low friction coefficient was designed and fabricated. The cantilever’s length, width, and thickness are 50, 35, and 0.6 µm, respectively. The tip height is 23 µm. The measurement resolution can reach 7.1×10−6 under the maximum normal force. Moreover, the AFM probe was applied to measure the superlubricity between graphene layers. The friction coefficient is 0.00139 under 853.08 nN. This work provides a promising method for measuring a ∼10−5 friction coefficient of superlubricity.


Introduction
Friction and wear always exist in mechanical systems.They would lower the service performance and life of mechanical systems and cause tremendous energy loss.According to the statistics in 2017, the consumption caused by friction and wear accounted for about 23% of the global energy consumption [1].Superlubricity provides a novel approach to addressing this issue [2][3][4], which Hirano and Shinjo [5] first reported in 1990.Superlubricity is a phenomenon with a sliding friction coefficient of < 0.01 [2].Because of the ultralow friction coefficient, friction and wear can be remarkably reduced, which is significant to energy conservation [3].
The friction coefficient is critical to assessing the performance of superlubricity.Table S1 in the Electronic Supplementary Material (ESM) summarizes the current friction coefficient measurement methods and the corresponding equipment.There are two primary methods to calculate the friction coefficient.One is based on the formula: μ = F L /F N , where μ is the friction coefficient, F L is the frictional force, and F N is the normal force.The premise of this method is that the adhesion force can be neglected compared with the normal force [6].The other is based on the experimental data fitting of the frictional and normal forces.The slope can be considered as the friction coefficient.In addition, two main types of equipment are used to measure the friction coefficient.One is the macroscale tribometer, such as Universal Mechanical Tester (UMT; Bruker).The other is the microscale atomic force microscope (AFM), such as MFP-3D (Asylum Research).In terms of the measuring principle, the tribometer uses a force transducer to convert mechanical signals to electrical signals [7].In contrast, AFM employs the optical lever detection method to monitor the frictional and normal forces [8].Generally, AFM can achieve a higher friction coefficient measurement resolution.Besides, AFM can achieve a single-asperity contact state in a more controllable environment, effectively avoiding outside interference [9].Therefore, AFM is becoming the primary method to investigate superlubricity.
Table S1 in the ESM also lists the friction pairs used in superlubricity and the corresponding friction coefficients achieved.The friction coefficient is reaching the magnitude of 10 −4 .With the continuous development of superlubricity, the friction coefficient will undoubtedly accomplish a much smaller value.Therefore, it is demanding to improve the friction coefficient measurement resolution.To our knowledge, little work was performed.Only a few attempts have investigated the measuring principle of the frictional force or the normal force with AFM, such as the AFM probe, the position-sensitive detector (PSD), and the optical level detection system.For instance, Daeinabi et al. [10] analyzed the lateral, longitudinal, and normal spring constants of AFM probes with rectangular, V-shaped, and dagger cantilevers.Woltring [11] reported an approximately linear relationship between the light spot position and the output current along the pertinent axis, providing substantial support for applying the silicon photodiode in the position measurement.Liu et al. [12] theoretically studied the relationship between the displacement of the laser spot on the PSD and the bending and torsional deformation of the AFM probe.
This study established the theoretical formula for the AFM friction coefficient measurement, and deduced the measurement resolution.Then, the formula was applied to the AFM probe with a rectangular cross-section cantilever.The measures to improve the measurement resolution were put forward accordingly.Moreover, an AFM probe for measuring a ~10 −5 friction coefficient was designed, fabricated, and applied to measure the superlubricity between graphene layers.

Experimental
The experiments were conducted on an AFM (MFP-3D, Asylum Research) under 40%-60% relative humidity and ~23 °C in the air.Various AFM probes were used to verify the theoretical formula for the AFM friction coefficient measurement.Table S2 in the ESM lists the dimensions of the test AFM probes.The cantilever material is silicon.The Young's modulus E is 169 GPa, and the shear modulus G is 50 GPa [13].The lateral sensitivity factor α (the unit is N/V) was calibrated with the wedge method using a commercial trapezoidal grating (TGF11, Mikromasch) [14].The angle of the sloped surface is 54.74°, the step height is 1.75 μm, and the pitch is 10 μm.The lateral stiffness K T (the unit is N/rad) was calculated by Eq. (15).The inverse of the lateral optical lever sensitivity InvOLS L (the unit is rad/V) was calculated with the formula: InvOLS L = α/K T .The normal stiffness K N (the unit is N/m) was obtained with the thermal noise method [14].The inverse of the normal optical lever sensitivity InvOLS N (the unit is m/V) was acquired from the force-distance curve on the monocrystalline silicon (111) sample.In addition, the total output voltage U sum (the unit is V) was collected with the AFM.
For the application in measuring the superlubricity, a freshly cleaved highly oriented pyrolytic graphite (HOPG) substrate (SPI-1 Grade, SPI Supplies) and a silica microsphere tip of 23 μm in diameter (Thermo Fisher Scientific) were used as the friction pairs.Firstly, the silica microsphere tip slid on the HOPG substrate to transfer multiple graphene nanoflakes to the silica surface.The F N was 175 nN, the sliding speed was 2 μm/s, and the sliding area was 10 μm × 10 μm.Then, superlubricity could be obtained between graphene layers [15].Because the diameter of the silica microsphere is as large as 23 μm, α was calibrated with a custom trapezoidal grating.The angle of the sloped surface is 54.74°, the step height is 7 μm, and the pitch is 28 μm.

Basic formula for AFM friction coefficient measurement
As shown in Fig. 1, an AFM probe primarily consists of a cantilever and a tip located at the front part of the ), H is the total length of the photosensitive surface of the PSD (the unit is m), h tip is the tip height (the unit is m), φ is the torsional angle of the AFM probe (the unit is rad), θ is the bending angle of the AFM probe (the unit is rad), l is the length of the cantilever of the AFM probe (the unit is m), w is the width of the cantilever of the AFM probe (the unit is m), and t is the the thickness (the unit is m).
cantilever.A laser beam focuses on the front part of the backside of the cantilever, and then is reflected to the PSD to form a laser spot.The vertical and horizontal positions of the laser spot on the PSD, reflecting the deformation in the normal and torsional directions, can be obtained through the output voltages of the PSD.Consequently, F N and F L can be obtained by multiplying the normal and lateral output voltages of the PSD with the corresponding coefficients, respecticely.
Based on the AFM force measurement principle, F L is sensed by measuring the torsion of the AFM probe, while F N is sensed by measuring the vertical bending of the AFM probe.On the premise that the adhesion force can be neglected compared with the normal force, the μ measured with AFM can be expressed by Eq. ( 1) [6,16]: If the formulas of K T , InvOLS L , K N , and InvOLS N are obtained, Eq. ( 1) can be further simplified.To our knowledge, the formulas of K T and K N have been thoroughly researched [17,18].However, the formulas of InvOLS L and InvOLS N remain to be elucidated.

Derivation and validation of InvOLS L
Section 3.2 derived and verified the formula of InvOLS L .According to the definition, InvOLS L equals  divided by U L .Therefore, InvOLS L can be expressed by Eq. ( 2) [19]: As shown in Fig. 2(a), the horizontal position of the laser beam on the PSD changes with the torsion of the cantilever of the AFM probe.a 1 b 1 c 1 and a 2 b 2 c 2 are the laser beams reflected by the backside of the cantilever before and after torsion, respectively.It should be noted that a 1 and a 2 coincide in this case.n 1 and n 2 are the normal lines of the cantilever before and after torsion, respectively.According to the geometrical relationship (the inset of Fig. 2(a)), the angle between n 1 and n 2 equals  .β is the angle between the incident laser beam and n 1 .Generally, β is relatively small, and the angle between a 1 b 1 and   [12,20,21].Therefore, h L caused by the torsional deformation of the AFM probe can be expressed by Eq. ( 3): where d is the distance of the laser beam reflected from the backside of the cantilever to the PSD (the unit is m).
Based on the principle of the PSD, the lateral output current i L (the unit is A) caused by h L can be depicted by Eq. ( 4) [11,22]: where i sum is the total output current of the PSD generated by the laser beam (the unit is A).
Based on the equivalent electric circuit of the PSD, the relationship between the output voltage and the output current can be given by Eqs. ( 5) and ( 6) [23]: where U sum is the total output voltage generated by the laser beam (the unit is V), and R sum and R L are the amplification coefficients of the total output current and the lateral output current of the I/V conversion circuit, respectively (both of the units are Ω).Substituting Eqs. ( 3)-( 6) into Eq.( 2), InvOLS L can be simplified as Eq. ( 7): InvOLS L is associated with U sum that represents the reflection performance of the backside of the cantilever.Besides, InvOLS L is associated with R sum , R L , and H of the PSD and d of the laser beam.
To verify the relationship between InvOLS L and U sum , InvOLS L of different types of AFM probes with different U sum were recorded.Figure 2(b) shows the results.InvOLS L is linearly dependent on 1/U sum , confirming that Eq. ( 7) of InvOLS L is valid.

Derivation and validation of InvOLS N
Section 3.3 derived and verified the formula of InvOLS N .
According to the definition, InvOLS N equals the normal bending displacement  (the unit is m) divided by U N [24].Therefore, InvOLS N can be expressed by Eq. ( 8): The relationship between  and θ can be given by Eq. ( 9) [19]: As shown in Fig. 3(a), the vertical position of the laser beam on the PSD changes with the normal bending of the cantilever of the AFM probe.a 1 b 1 c 1 and a 3 b 3 c 3 are the laser beams reflected from the backside of the cantilever before and after bending, respectively.According to the geometrical relationship (the inset of Fig. 3(a)), the angle between a 1 b 1 and a 3 b 3 equals 2θ [12,25].Therefore, h N caused by the normal bending deformation of the AFM probe can be expressed by Eq. ( 10): Based on the principle of the PSD, the normal output current of the PSD i N (the unit is A) can be expressed by Eq. ( 11) [11,22]: Based on the equivalent electric circuit of the PSD, the relationship between the output voltage and the output current can be expressed by Eq. ( 12) [23]: where R N is the amplification coefficient of the normal output current of the I/V conversion circuit (the unit is Ω).Substituting Eqs. ( 9)-( 12) and Eq. ( 6) into Eq.( 8), InvOLS N can be described by Eq. ( 13): InvOLS N is associated with the properties of the AFM probe, including l and U sum .Besides, like InvOLS L , InvOLS N is associated with R sum , R N , and H of the PSD and d of the laser beam.
To confirm the relationship between InvOLS N and l or U sum , InvOLS N of different types of AFM probes with different l and U sum were collected.Figure 3(b) shows the relationship between InvOLS N × U sum and l.As can be seen, InvOLS N × U sum is linearly related with l.Moreover, Figure 3(c) shows the relationship between InvOLS N /l and 1/U sum .InvOLS N /l is also linearly related with 1/U sum .In conclusion, InvOLS N is linearly dependent on l and 1/U sum , confirming that Eq. ( 13) of InvOLS N is valid.

Theoretical formula for AFM friction coefficient measurement
The formulas of InvOLS N and InvOLS L were derived and confirmed above.Section 3.4 further derives the theoretical formula for the AFM friction coefficient measurement.
For an AFM probe, K N can be expressed by Eq. ( 14) [14]: where x I is the moment of inertia against the centroidal axis (the unit is m 4 ).
K T can be expressed by Eq. ( 15) [19,26]: where I t is the polar moment of inertia (the unit is m 4 ), and e is the distance from the center of the cross-section to the bottom of the cross-section (the unit is m).Substituting Eq. ( 7) and Eqs. ( 13)-( 15) into Eq.( 1), μ can be expressed by Eq. ( 16): When the normal output voltage reaches the maximum value U N,max , the AFM friction coefficient measurement resolution Δμ can be derived by Eq. ( 17): where ΔU L is the lateral output voltage measurement resolution (the unit is V).

AFM probe with rectangular cross-section cantilever for measuring ~10 −5 friction coefficient
The AFM probes with rectangular cross-section cantilevers have been widely used in tribological experiments because they can realize relatively large C x I can be expressed by Eq. ( 19) [26]: Substituting Eqs. ( 18) and ( 19) into Eq.( 17), Δμ under U N,max can be simplified as Eq. ( 20): ) According to Eq. ( 20), the following measures can be taken to reduce Δμ: 1) For the cantilever dimensions, l should be as small as possible, while h tip should be as large as possible.
2) For the cantilever material, G should be as small as possible, while E should be as large as possible.
3) For the AFM hardware, the PSD should be highprecision and wide-range, and the AFM noise should be as small as possible.
Generally, the cantilever of the AFM probe is made of silicon or silicon nitride.Meanwhile, the PSD is fixed unless the AFM is custom-made.Therefore, the latter two measures are highly challenging.It can be feasible to follow the first measure to reduce Δμ by optimizing the AFM probe.
On the basis, Fig. 4 demonstrates the design process of an AFM probe with a rectangular cross-section cantilever for measuring an ultra-low friction coefficient.At first, preliminarily design the dimensions of the AFM probe based on the conventional constraints of l, w, t, and h tip .Then, find a commercial AFM probe with close dimensions.According to our survey, a tipless AFM probe (Hydra6R-200NG-TL, AppNano; l of 200 μm, w of 35 μm, and t of 0.6 μm) can basically meet the constraints except for l.Afterward, determine the final dimensions of the AFM probe as follows: l of 50 μm, w of 35 μm, t of 0.6 μm, and h tip of 23 μm.
Figure 5 displays the fabrication process of the designed AFM probe.Firstly, the cantilever was cut into the designed l of 50 μm by the focused ion beam (FIB).Secondly, silica microspheres of 23 μm in diameter and methyl methacrylate structural adhesive (1918-2, Ergo) were spread on a clean silicon wafer.Thirdly, the tipless AFM probe was installed on the AFM, and then engaged to the adhesive and the Fig. 4 Design process of AFM probe with rectangular cross-section cantilever for measuring ultra-low friction coefficient.In particular, for the laser spot size, please refer to Ref. [28]; for the maximum normal force F N,max that can achieve graphene superlubricity, please refer to Ref. [29]; for the mechanical properties of silicon nitride, please refer to Ref. [30].| https://mc03.manuscriptcentral.com/frictionsilica microsphere successively.After 24 h of curing, the silica microsphere was stably fixed on the front part of the cantilever.The designed AFM probe was fabricated successfully.
Afterward, the fabricated AFM probe was calibrated.Figure 6 shows the calibration results.K N is 4.19 nN/nm, InvOLS N is 50.90 nm/V, and α is 1,507.8nN/V.
As aforementioned, ΔU L can be associated with the PSD and probably the anti-vibration performance of AFM.The ΔU L of the AFM (MFP-3D, Asylum Research) used in this study was measured.An AFM probe with a silica microsphere tip (2.5 μm in diameter, 16 N/m normal stiffness, Novascan Technologies) slid on the HOPG substrate.In practice, the average lateral output voltage is used to eliminate the deviation, which equals half of the lateral output voltage difference between the trace and retrace.Figure 7 shows the average lateral output voltages under different normal forces.As can be seen, when the F N is less than 0.54 μN, the average lateral output voltage slightly fluctuates and does not change significantly with the F N , indicating that the signal can be considered as noise.In contrast, when the F N is higher than 0.54 μN, the average lateral output voltage almost linearly increases with the F N , which corresponds with Eq. ( 1).Therefore, it is inferred that the F N of 0.54 μN can be the turning point, and the corresponding average lateral output voltage can be considered as ΔU L , which is 0.003 mV.In addition, Liu et al. [29] and Li et al. [31] reported that the ΔU L of the AFM (MFP-3D, Asylum Research) is approximately 0.001 mV.In this study, 0.01 mV is used as ΔU L to make Δμ more reliable.
In addition, Fig. S1 in the ESM compares the lateral output voltages measured with our fabricated AFM probe and a commercial AFM probe (TL-FM, Nanosensors).Under the F N of 1,000 nN, the U L obtained with our fabricated AFM probe is nearly  100 times higher than that obtained with the commercial AFM probe (TL-FM, Nanosensors), which can be probably ascribed to the differences in h tip , l, and U N corresponding to the applied normal force and the mechanical properties E and G of the cantilever material.In addition, as shown in Fig. S1(c) in the ESM, the comparison results are nearly in line with Eq. ( 20), indicating that the theoretical formula of Δμ is valid.

Application in measuring ultra-low friction coefficient of superlubricity
Section 4 used the fabricated AFM probe to measure the ultra-low friction coefficient between graphene layers.At the beginning of the experiment, the friction coefficient between the silica microsphere tip and the HOPG substrate could not reach superlubricity.After a running-in stage, graphene nanoflakes were transferred to the silica surface.As a result, the friction pairs became graphene layers, significantly reducing the friction coefficient and entering superlubricity.Meanwhile, the adhesion force was reduced to 13.3 nN, which can be neglected compared with the normal force [15,29].Therefore, Eq. ( 1) can be used to calculate μ, and Eq. ( 17) can be used to calculate Δμ. Figure 8 shows the results.Under F N of 426.54 nN, μ can reach 0.00164 with Δμ of 3.5×10 −5 ; while under F N of 853.08 nN, μ can reach 0.00139 with Δμ of 1.8×10 −5 .The ultra-low friction coefficients are consistent with Ref. [29], confirming the reliability of the experimental results.In sum, the theoretical formula for the AFM friction coefficient measurement was established, and Δμ was deduced, filling in the gap.According to Eq. ( 17), it is highly possible that Δμ can be further reduced by decreasing C t / x I I via optimizing the cross-section shape of the cantilever, and the friction coefficient of 10 −6 and even less than 10 −6 can be measured.

Conclusions
This study deduced the theoretical formula for the AFM friction coefficient measurement.Based on the formula, an AFM probe for measuring a ~10 −5 ultra-low friction coefficient was designed, fabricated, and applied to measure the superlubricity between graphene layers.The conclusions can be drawn as follows: 1) Based on the principle of AFM force measurement, the theoretical formulas of μ and Δμ were deduced.The formula of Δμ was applied to the AFM probe with a rectangular cross-section cantilever.Based on the formula, to reduce Δμ, l should be as small as possible, while h tip should be as large as possible; G should be as small as possible, while E should be as large as possible; PSD should be high-precision and wide-range, and AFM should have excellent antivibration performance.

Fig. 1
Fig.1Schematic diagram of AFM friction coefficient measurement.Note: U L is the lateral output voltage of the PSD caused by F L (the unit is V), U N is the normal output voltage of the PSD caused by F N (the unit is V), h N is the vertical displacement of the laser spot on the PSD (the unit is m), h L is the horizontal displacement of the laser spot on the PSD (the unit is m), H is the total length of the photosensitive surface of the PSD (the unit is m), h tip is the tip height (the unit is m), φ is the torsional angle of the AFM probe (the unit is rad), θ is the bending angle of the AFM probe (the unit is rad), l is the length of the cantilever of the AFM probe (the unit is m), w is the width of the cantilever of the AFM probe (the unit is m), and t is the the thickness (the unit is m).

Fig. 2
Fig. 2 Derivation and validation of InvOLS L .(a) Horizontal position of laser beam on the PSD changing with the torsion of the cantilever of the AFM probe.(b) Relationship between InvOLS L and 1/U sum .

Friction 12 ( 1 )Fig. 3
Fig. 3 Derivation and validation of InvOLS N .(a) Vertical position of laser beam on the PSD changing with the normal bending of the cantilever of the AFM probe.(b) Relationship between InvOLS N × U sum and l.(c) Relationship between InvOLS N /l and 1/U sum .

Fig. 6
Fig. 6 Calibration results of fabricated AFM probe.Note: W is the half-width of the lateral output voltage in the friction loop.Δ is the offset of the lateral output voltage in the friction loop.W' and Δ' are the slopes of the linear fits of W and Δ with respect to the normal force, respectively.is the angle of the sloped surface of the trapezoidal grating.

Fig. 8
Fig. 8 Application in measuring ultra-low friction coefficient of superlubricity.
[26]tionnormal forces and install various types of tips[27].Generally, w of the AFM probe is much larger compared to t (w/t is larger than 10), and thus I t can be expressed by Eq. (18)[26]: www.Springer.com/journal/40544|