The atomic force microscope (AFM) can measure nanoscale morphology and mechanical properties and has a wide range of applications. The traditional method for measuring the mechanical properties of a sample does so for the longitudinal and transverse properties separately, ignoring the coupling between them. In this paper, a data processing and multidimensional mechanical information extraction algorithm for the composite mode of peak force tapping and torsional resonance is proposed. On the basis of a tip–sample interaction model for the AFM, longitudinal peak force data are used to decouple amplitude and phase data of transverse torsional resonance, accurately identify the tip–sample longitudinal contact force in each peak force cycle, and synchronously obtain the corresponding characteristic images of the transverse amplitude and phase. Experimental results show that the measured longitudinal mechanical characteristics are consistent with the transverse amplitude and phase characteristics, which verifies the effectiveness of the method. Thus, a new method is provided for the measurement of multidimensional mechanical characteristics using the AFM.
HIGHLIGHTS
Based on a composite method of peak force tapping and torsional resonance, a multidimensional mechanical signal measurement system for an atomic force microscope (AFM) is constructed that overcomes the inability of the peak force tapping mode to measure the transverse mechanical properties of a sample and the inability of the traditional torsional resonance mode to realize closedloop feedback control of stable morphology. It can thus obtain both longitudinal and transverse mechanical information of a sample simultaneously.
Based on a model of the interaction between the AFM tip and a sample, longitudinal peak force data is used to synchronize and decouple transverse torsional resonance amplitude and phase data, enabling accurate identification of the longitudinal contact force between tip and sample in each peak force cycle, as well as synchronous acquisition of the corresponding characteristic images of transverse amplitude and phase.
The improved contact point registration algorithm accurately identifies the signal boundary for each peak force cycle, providing a better way of dealing with background noise and obtaining more accurate image data.
I. INTRODUCTION
The atomic force microscope (AFM) is an important tool in the field of nanoscience and, since its invention jointly by Binnig of IBM and Quate and Gerber of Stanford University in 1986,^{1} it has become widely used owing to its low requirements on the working environment and few restrictions on the samples to be measured.^{2–5} The earliest AFMs were mainly used to obtain information about the morphology of samples, providing atomiclevel highresolution images. With developments in equipment and techniques, AFMs have become able to perform a much wider range of tasks. In 1994, the force volume mode AFM (FVAFM) was born.^{6} By acquiring a force curve at each scanning point, the FVAFM can not only examine the morphology of a sample, but also measure its mechanical properties, such as adhesion, deformation, and elastic modulus, in the imaging area. However, owing to its low potential for automation and its slow imaging speed, the FVAFM is not an ideal method for measuring mechanical characteristics. The advent of the peak force tapping mode AFM (PFTAFM) and of the technique of peak force quantitative nanomechanical mapping (PFQNM) in 2009 provided a powerful alternative approach for the measurement of mechanical properties.^{7–9} In recent years, these have become important means of simultaneously obtaining information on sample morphology and sample surface mechanical properties.
Owing to its fine control of the vertical maximum interaction force and its highresolution imaging capability, the PFTAFM is currently the most favored imaging mode. It is widely used in the measurement of longitudinal mechanical properties of samples such as cells and has led to many advances.^{10,11} In 2011, the invention of the peak force tunneling AFM (PFTUNA) by the Bruker company in Germany opened the way to other physical field composite modes in combination with the peak force tapping mode. The PFTUNA combines the peak force tapping technique and electrical characteristic measurements to provide simultaneous plotting of graphs of the morphology, modulus, adhesion, and conductivity of advanced materials as fine samples that cannot be imaged by traditional conductive AFM, thereby allowing direct comparison of morphology and electrical characteristics on the nanoscale.^{12} The application of the AFM peak force tapping mode characterization technique to the study of morphology and related mechanical, electrical, chemical, and biological properties of samples on the nano and microscales constitutes an important contemporary example of interdisciplinary integration.
Many AFM techniques have been developed to investigate the transverse mechanical properties of samples with the aim of determining the inplane anisotropy of materials and measuring nanoscale dynamic friction.^{13–15} Among these, the torsional resonance mode AFM (TRAFM) shows much promise.^{16} It uses torsional resonance amplitude (or phase) to control feedback and maintains the relative position of the needle tip and the sample surface throughout transverse interaction, providing supplementary information on the peak force tapping mode for imaging and other studies of sample surfaces.^{17}
However, individual measurement of the longitudinal and transverse mechanical properties of a sample ignores the coupling between them. To overcome this problem, this paper presents a data processing and multidimensional mechanical information extraction algorithm based on the composite mode of peak force tapping and torsional resonance. First, the peak force tapping mode is used to control the interaction between needle tip and sample, and the sample morphology is measured and longitudinal mechanical information obtained. While the peak force tapping mode is in operation, the torsional resonance mode is used to obtain transverse mechanical information about the sample, and thus the sample is dynamically examined in both the longitudinal and transverse dimensions.^{18} Furthermore, the coupling between longitudinal and transverse mechanical properties is analyzed. Through decoupling of the transverse mechanical information, both quantitative measurement of longitudinal mechanical properties and qualitative analysis of transverse mechanical properties are realized. Based on this approach, an optimized algorithm for synchronizing AFM images is developed, and a method is proposed to accurately identify the contact area between tip and sample for each peak force cycle, thereby enabling more accurate information about morphology and mechanical properties to be obtained.
II. MULTIDIMENSIONAL MECHANICAL INFORMATION EXTRACTION ALGORITHM BASED ON COMPOUND PEAK FORCE TAPPING
The peak force tapping mode uses a sinusoidal signal to drive a piezoelectric ceramic and thereby produce periodic stretching movements. The typical cycle frequency is about 2 kHz, and therefore the vibrational frequency is far lower than the probe resonance frequency, and so this is also called the “nonresonance mode.” The peak force tapping mode produces a force curve in each cycle of sinusoidal motion of the piezoelectric ceramic. Compared with the traditional method of using triangular waves to drive a piezoelectric ceramic to generate a force curve, this process has a higher frequency and smaller amplitude. The composite peak force tapping mode is a new measurement method proposed by the authors’ team, which aims to overcome the problems caused by the inability of the peak force tapping mode to measure the transverse mechanical characteristics of a sample and the inability of the traditional torsional resonance mode to realize closedloop feedback control of stable morphology. As shown in Fig. 1, in the process of peak force tapping, the probe is activated by a torsional resonance driving signal generated by a torsional piezoelectric controller to generate a torsional resonance state. The torsional resonance signal is affected by the interaction between the needle tip and the sample, which will change the amplitude and phase. A position sensor detects the longitudinal and transverse movements of the laser spot, which are transformed into a signal representing the longitudinal interaction force between needle tip and sample and a signal representing the transverse torsional resonance and containing amplitude and phase information. To realize this process, the realtime closedloop feedback control loop of the probe consists of three main components: a lockin amplifier (LIA), proportion–integral–derivative (PID) controller, and a phaselocked loop (PLL). As shown in Fig. 1, during the peak force tapping mode, the transverse torsional resonance signal from the cantilever beam is sent to the lock detection module and the PLL probe resonance tracking module. The frequency of the PLL output signal controls the frequency of the output signal of the torsional piezoelectric controller so that the frequency is the same as that detected by the PLL. The locking detection module parses the amplitude of the torsional resonance signal according to the reference signal and sends the output signal to the PID controller adjustment module. The amplitude set value is input into the PID controller and compared with the amplitude signal in the output signal of the lock detection module to generate a control signal and output it to the torsional piezoelectric controller, which then controls the amplitude of the output signal of the torsional piezoelectric controller so that it remains the same as the amplitude set value. Thus, the compound peak force tapping mode combines the good longitudinal force control of the peak force tapping mode with the transverse force measurement capability of the torsional resonance mode and is thereby able to simultaneously obtain longitudinal and transverse mechanical information.
The multidimensional mechanical signals obtained using the compound peak force tapping mode include peak force data and torsional resonance data. First, the longitudinal mechanical information is extracted from the peak force data. Then, with the aid of the longitudinal/transverse tip–sample interaction model of the AFM, the torsional resonance data are decoupled, and the transverse mechanical information is extracted using the peak force data. Finally, the morphology and multidimensional mechanical information of the sample are obtained by analyzing the force curve of each pixel. The overall flow of the algorithm is shown in Fig. 2.
A. AFM longitudinal/transverse tipsample interaction model
To obtain mechanical information and images from the interaction between the probe and the sample, a theoretical model must be used to describe these physical processes and obtain quantitative mechanical information and images according to the measured data.^{19}
1. Longitudinal interaction model
The Sneddon model is usually used for the deeper and larger samples (such as for measurement of softer cells and hydrogels). As shown in Fig. 3(a), the Sneddon probe model is based on a cone structure pressed into a solid surface and considers the indentation depth, the cone halfangle, the force on the tip, Young’s modulus, and Poisson’s ratio. The mathematical expression for the force F on the needle tip is
where E is Young’s modulus, v is Poisson’s ratio (which for water and incompressible material is v = 0.5), α is the halfangle of the cone, and δ is the indentation depth of the needle tip in the sample.
The Sneddon model does not consider the influence of adhesion. If this influence is taken into account, the Derjaguin–Muller–Toporov (DMT) model [Fig. 3(b)] is usually used when the sample is hard and the indentation depth is small. This model gives an analytical relationship between the force on the object and the distance between tip and sample under weak adhesion and hard contact for small tip radius. In the DMT model, it is assumed that a gravitational force is acting before direct contact between the objects, but there is no deformation, and so the contact radius of the separate objects is zero. The mathematical expression for the force F on the needle tip in this model is
where R is the radius of the probe tip and γ is the surface energy density of the sample [the rest of the notation is the same as in Eq. (1)].
2. Transverse mechanical model
We consider forced torsional vibration of the cantilever beam under a linear viscoelastic interaction between needle tip and sample, as shown in Fig. 4(a), where two piezoelectric elements are connected to the cantilever support and vibrate out of phase to drive the cantilever into torsional motion.^{15} The differential equation of motion of the cantilever beam is
where G is the shear modulus, J is the torsion constant, θ(x, t) is the torsional angle of the cantilever, x is the position on the long axis of the cantilever, t is time, ρ is the mass density, I_{p} is the polar moment of inertia of the cantilever section, and c is the viscous damping coefficient of the cantilever. Taking θ(x, t) = Θ(x)e^{iΩt} in Eq. (3), we obtain
where $\eta \theta 2=\rho Ip\Omega 2/(GJ)$ and η_{c} = cΩ/(GJ). The two limiting conditions of the cantilever are
where k_{lat} is the transverse contact stiffness between the tip and the sample surface and η_{lat} is the viscosity. Considering the boundary conditions in Eq. (5), Eq. (4) can be solved, and the torque at the tip when x = L can then be calculated:
Here H_{h}(Ω) is the frequency response function of the cantilever at the tip and is given by
where R_{a} and I_{a} are the real and imaginary parts of $\u2212\eta \theta 2+i\eta c$, respectively. The torsional amplitude and phase difference of the cantilever beam at x = L are
where $r1=Ia2+Ra2$, r_{2} = R_{a}k_{lat} + I_{a}η_{lat}Ω, r_{3} = I_{a}k_{lat} − R_{a}η_{lat}Ω, $r4=e2RaL+1$, and $r5=e2RaL\u22121$.
In the torsional resonance mode, the torsional amplitude and phase difference are the amplitude and phase difference between the torsional signal from the cantilever and the driving signal. It can be seen from Eq. (8) that the amplitude and phase difference are determined by the viscous damping encountered by the cantilever when it vibrates away from the sample surface and by the transverse contact between the needle tip and the sample surface.
3. Coupled mechanical model
The transverse contact stiffness describes the force required to compress the sample per unit displacement in the transverse direction when the needle tip contacts the sample surface. In the DMT model, the transverse contact stiffness between the tip and the sample surface is defined as
Here the contact radius a_{c} is given by
where R is the radius of the probe tip, F_{nor} is the force in the vertical direction between probe and tip, and E* is the effective elastic modulus. The expression for E* is
where E_{t} and E_{s} are the elastic moduli of the tip and sample, respectively, and v_{t} and v_{s} are their Poisson’s ratios. The effective shear modulus G* in Eq. (9) is given by
where G_{t} and G_{s} are the shear moduli of the tip and sample, respectively. The shear modulus of the tip is given by
Therefore, from Eqs. (8) and (9), if the influence of the interaction force between the needle tip and the sample in the vertical direction is eliminated, which means that the transverse force is decoupled from the longitudinal force, then transverse mechanical properties of the sample, namely, the viscosity η_{lat} and shear modulus G_{s}, can be analyzed from the transverse amplitude and phase difference, thereby providing a new basis for the measurement and characterization of the transverse mechanical properties of a sample.
B. Longitudinal mechanical information extraction algorithm
Noise will inevitably be introduced during the processes of data acquisition and transmission. To acquire quantitative data from AFM images clearly and accurately, it is necessary to design a program to preprocess the original AFM image data. First, the contact area between the needle tip and the sample in each peak force cycle is determined from the peak force data by using a synchronization algorithm. The traditional method of drawing a window by percentage is to add a window at the halfcycle of the sinewave driving signal corresponding to the received signal and select 20% of the whole peak force cycle as the contact area between needle tip and sample.^{20} However, because the peak force periods of different samples can be very different, this traditional percentage window drawing method will inevitably be prone to errors, resulting in failure. In this paper, optimization of the traditional synchronization algorithm leads to the proposal of a new synchronization algorithm that is able to accurately identify the contact area between the tip and the sample for each peak force cycle, thereby obtaining more accurate image data for determination of morphology and mechanical properties. The algorithm flow chart is shown in Fig. 5.
Second, a background subtraction algorithm is used to fit and subtract the aperiodic parasitic deflection signal from the periodic noncontact area data of each peak force, to eliminate the influence of the interaction force between the sample and the cantilever beam. The specific procedure is as follows:
Determine the contact area between the needle tip and the sample on the force curve by using the synchronization algorithm.
Perform polynomial curve fitting based on least squares on the noncontact area to obtain the background noise signal.
Calculate the difference between the actual deflection signal data and the corresponding data from the fitting curve to obtain the deflection signal due to the interaction between needle tip and sample.
Subtract the background noise of all peak force cycles to obtain the corrected data of the complete image.
Hardware background signal subtraction can effectively eliminate interference by the periodic parasitic deflection signal due to environmental influence, thereby allowing more accurate control. Software background signal subtraction can fit aperiodic parasitic deflection signals, such as that due to the interaction force between a sample with large fluctuations and the cantilever beam. An original signal collected in the experiment and signals after background subtraction are shown in Fig. 6. Before scanning, the scanning head should be kept away from the sample by a certain distance such that there is no interaction between needle tip and sample, and the scanning head should be vibrated up and down to obtain the hardware background signal [Fig. 6(b)] and remove it [Fig. 6(c)]. The software background signal subtraction algorithm described in detail above can then be used to obtain the final signal shown in Fig. 6(d).
Finally, the normal peak force is extracted by the traditional peak force tapping data analysis algorithm,^{21} and the sample morphology is characterized. A series of mechanical properties such as deformation, elastic modulus, adhesion force, and energy dissipation are obtained from the force curves at each pixel point, and finally the longitudinal nanomechanical properties of the sample are characterized.
C. Decoupling and acquisition of transverse mechanical information
Tip–sample interaction is a multidimensional coupling of longitudinal force and transverse force. For torsional resonance data, the effect of the transverse force cannot be considered alone, but also needs to be decoupled with the help of correlated peak force data. In this paper, the peak force data are used to synchronize the torsional resonance data, as shown in Fig. 7, and the transverse amplitude and phase under the same longitudinal force in the force curve of each pixel are extracted to eliminate the influence of the longitudinal force. The specific procedure for decoupling the transverse mechanical information and obtaining the transverse amplitude and phase from characteristic image data under different longitudinal mechanical loads is as follows:

Calculate the start and end times of each row of data in the longitudinal mechanical characteristic image.

Calculate the starting position of the peak force period contained in the data of each row of the longitudinal mechanical characteristic image.

Select the peak force data and each peak force cycle data point according to the longitudinal force and record the position.

Obtain the original data positions of each row of the transverse amplitude and phase characteristic image by using the peak force data and the position of each peak force period data point.

Acquire the original data of the transverse amplitude and phase characteristic image.

Acquire the forward/backward scanning image data of the transverse amplitude and phase characteristics.

Perform thirdorder Bspline interpolation on the original row data of the transverse amplitude and phase characteristic image to eliminate the image distortion caused by the nonlinear sine wave.

Perform data conversion to obtain the final image data array of the transverse amplitude and phase characteristics.

Generate a twodimensional amplitude diagram and phase diagram of the transverse amplitude and phase characteristics of the sample.

Generate a threedimensional volume data plot of the transverse amplitude and phase characteristics of the sample related to the longitudinal force.
III. RESULTS AND DISCUSSION
To verify the effectiveness of the proposed method, extensive experiments were conducted on a Dimension Icon (Bruker) AFM [Fig. 8(a)]. The TR probe clamp used in the experiment is shown in Fig. 8(b). The probe selected for the experiments was an RFESP75 (Bruker), with a calibrated resonance frequency of 75 kHz and an elastic coefficient of 3 N/m. The main parameters of the probe are shown in Table I. The experiments were conducted in an atmospheric environment, at a temperature of about 23 °C. A composite sample (from Bruker) composed of two different materials, namely, polystyrene (PS) and lowdensity polyethylene (LDPE), was used for imaging.
Length  225 μm (215–235 μm) 
Width  35 μm (33–37 μm) 
Thickness  2.8 μm (2.05–3.55 μm) 
Elastic coefficient  3 N/m (1.5–6 N/m) 
Resonance frequency  75 kHz (50–100 kHz) 
Length  225 μm (215–235 μm) 
Width  35 μm (33–37 μm) 
Thickness  2.8 μm (2.05–3.55 μm) 
Elastic coefficient  3 N/m (1.5–6 N/m) 
Resonance frequency  75 kHz (50–100 kHz) 
Because the original data obtained by an AFM do not directly provide the morphology and physical parameters of each pixel, but rather indirect information such as the scanning table position and cantilever deflection of the point, further calculations are needed to obtain the desired image. Therefore, to obtain an AFM image, it is necessary to store data several times the size of an ordinary image and carry out many operations. An appropriate data acquisition, storage, and operation scheme is particularly important for an AFM system. To achieve realtime lossless acquisition of highthroughput multichannel signals, we used an M2p.5923x4 highspeed eightchannel data acquisition card with a sampling frequency of 500 kHz [Fig. 8(c)].
A. Measurement results for longitudinal mechanical properties
A PSLDPE composite sample with a calibrated height of 540.4 nm, a DMT modulus of 1.4–3.4 TPa, an adhesion force of 19.3–29.7 nN and an energy dissipation of 1.3–2.4 keV was tested. A sine wave with a frequency of 500 Hz was used as the excitation signal for the peak force tapping module scanner, and the scanning range was 2.5 × 0.6 μm^{2}. The line scanning frequency was set to 0.9 Hz. The morphology of the PSLDPE measured in the experiment is shown in Fig. 9(a), the DMT modulus in Fig. 9(b), the adhesion in Fig. 9(c), and the energy dissipation in Fig. 9 (d). It can be seen that the measurement system developed in this paper can clearly characterize the morphology of PSLDPE and accurately obtain the height of its surface morphology. The nanomechanical measurement results are basically consistent with the calibrated values for PSLDPE, which verifies the effectiveness and accuracy of the system. It therefore provides a powerful means to study the relationship between the mechanical properties and microstructure of PSLDPE and other composite samples.
B. Experimental results for extraction of transverse mechanical characteristics
Figures 10(a) and 10(b) present twodimensional plots of the transverse amplitude and phase, respectively, extracted under different longitudinal force conditions. The transverse amplitude and phase depend significantly on the longitudinal force. When the probe is pressed into the sample, the torsional resonance amplitude decreases with increasing longitudinal interaction force between tip and sample, and the amplitude image contrast increases gradually. The torsional resonance phase increases gradually, as does the phase image contrast. These experimental results are consistent with the theoretical analysis.
To provide a more intuitive representation of the influence of the longitudinal force on transverse mechanical properties, Fig. 11 presents threedimensional volume plots of the transverse amplitude and phase. Figures 11(aI) and 11(bI) are volume plots of the transverse amplitude and phase, respectively, extracted under different longitudinal force conditions. Figures 11(aII) and 11(bII) are crosssectional slices of these volume plots in the XZ plane, and Figs. 11(aIII) and 11(bIII) are crosssectional slices in the YZ plane. The slice positions are shown by the dashed lines in Figs. 11(aI) and 11(bI).
Although it is impossible to quantitatively determine the mechanical parameters of a sample from the plots of torsional resonance amplitude and phase, the image contrast arises from the changes in these quantities caused by the variations of mechanical properties in the scanning area during the imaging process, and so it can provide a qualitative reflection of differences in mechanical properties between different areas of the sample. Thus, the contrast can be used as a good means to qualitatively characterize the distribution of mechanical properties of samples on a nanoscale, and the imaging of torsional resonance amplitude and phase is directly affected by the magnitude of the longitudinal force load. With threedimensional imaging, it is possible to generate threedimensional volume maps of the mechanical properties of samples, as well as performing tomography and image rotation. The contrast in these volume maps provides an intuitive representation of changes in the mechanical properties of samples.
IV. CONCLUSIONS
To measure and characterize the multidimensional mechanical properties of samples using an AFM, a new data processing and multidimensional mechanical information extraction algorithm for the peak force tapping and torsional resonance composite mode has been proposed. This method takes full account of the coupling of the longitudinal and transverse interaction forces between needle tip and sample. Together with a multidimensional mechanical model of the AFM, longitudinal peak force data are used to decouple transverse torsional resonance amplitude and phase data, thereby realizing quantitative measurements of the longitudinal mechanical properties of the sample and a qualitative analysis of transverse mechanical properties. Based on this approach, a new synchronization algorithm for AFM characterization has been proposed. The synchronization algorithm accurately identifies the contact area for each peak force cycle, thereby enabling elimination of the aperiodic parasitic deflection signal generated by the interaction between the sample and the cantilever beam and improving the accuracy of the scanned image.
Experimental results show that the longitudinal mechanical properties measured by the method proposed in this paper are consistent with those predicted theoretically, and the changes in transverse mechanical properties under different longitudinal forces further verify the method. Our future research will focus on quantitative measurement and characterization of the transverse mechanical properties of samples.
ACKNOWLEDGMENTS
This project is supported by the General Program of the National Natural Science Foundation of China (62073227) and the National Natural Science Foundation of China (61927805 and 61903359).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The authors confirm that the data supporting the findings of this study are available within the article.
REFERENCES
Shoujin Wang was born in Rizhao, Shandong Province. He has a Master’s degree and is an Associate Professor in the School of Information and Control Engineering, Shenyang Jianzhu University, Shenyang. His research interests are applications of computer technology and data mining.
Shuai Yuan was born in Hohhot, Inner Mongolia. He has a Doctor’s degree and is a Professor in the School of Information and Control Engineering, Shenyang Jianzhu University, Shenyang and the State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang. His research interests are nanooperations and deep learning.
Jialin Shi was born in Linghai, Liaoning Province. He has a Doctor’s degree and is an Associate Researcher in the State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang and the Institute of Robotics and Intelligent Manufacturing Innovation, Chinese Academy of Sciences. His research interests are micro/nanorobotics and atomic force microscopy, focusing on the development of advanced scientific instruments.