Extraction algorithm for longitudinal and transverse mechanical information of AFM

Chunxue Hao, Shoujin Wang,Shuai Yuan, Boyu Wu, Peng Yu, and Jialin Shi,a)

ABSTRACT 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 verifie the effectiveness of the method.Thus,a new method is provided for the measurement of multidimensional mechanical characteristics using the AFM.©2022 Author(s).All article content,except where otherwise noted,is licensed under a Creative Commons Attribution(CC BY)license(http://creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/10.0010252

KEYWORDS Atomic force microscope,Peak force tapping,Torsional resonance,Mechanical characteristic measurement,Background subtraction algorithm,Coupled mechanical model

I.INTRODUCTION

The atomic force microscope(AFM)is an important tool in the fiel of nanoscience and,since its invention jointly by Binnig of IBM and Quate and Gerber of Stanford University in 1986,1it has become widely used owing to its low requirements on the working environment and few restrictions on the samples to be measured.2-5The earliest AFMs were mainly used to obtain information about the morphology of samples,providing atomic-level high-resolution 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(FV-AFM)was born.6By acquiring a force curve at each scanning point,the FV-AFMcan 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 FV-AFM is not an ideal method for measuring mechanical characteristics.The advent of the peak force tapping mode AFM(PFT-AFM)and of the technique of peak force quantitative nanomechanical mapping(PF-QNM)in 2009 provided a powerful alternative approach for the measurement of mechanical properties.7-9In recent years,these have become important means of simultaneously obtaining information on sample morphology and sample surface mechanical properties.

Owing to its fin control of the vertical maximum interaction force and its high-resolution imaging capability,the PFT-AFM 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.10,11In 2011,the invention of the peak force tunneling AFM(PF-TUNA)by the Bruker company in Germany opened the way to other physical fiel composite modes in combination with the peak force tapping mode.The PF-TUNA 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 fin samples that cannot be imaged by traditional conductive AFM,thereby allowing direct comparison of morphology and electrical characteristics on the nanoscale.12The application of the AFMpeak 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 in-plane anisotropy of materials and measuring nanoscale dynamic friction.13-15Among these,the torsional resonance mode AFM(TR-AFM)shows much promise.16It 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.18Furthermore,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 closed-loop 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 real-time closed-loop feedback control loop of the probe consists of three main components:a lock-in amplifie(LIA),proportion-integral-derivative(PID)controller,and a phase-locked 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.

FIG.1.Schematic illustration of the driving and measurement capabilities ofthe compound peak force tapping mode.

FIG.2.Schematic illustration of how the mechanicalproperties of the sample are obtained based on the force curve.

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 flo of the algorithm is shown in Fig.2.

A.AFM longitudinal/transverse tip-sample 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 half-angle,the force on the tip,Young’s modulus,and Poisson’s ratio.The mathematical expression for the forceFon the needle tip is

FIG.3.(a)Sneddon model.(b)DMT model.

whereEis Young’s modulus,vis Poisson’s ratio(which for water and incompressible material isv=0.5),αis the half-angle of the cone,andδis the indentation depth of the needle tip in the sample.

FIG.4.(a)Schematic of the AFM cantilever torsionally excited by the holder.(b)Tip–cantilever assembly under viscoelastic tip–sample interaction.

The Sneddon model does not consider the influenc of adhesion.If this influenc 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 forceFon the needle tip in this model is

whereRis the radius of the probe tip andγis 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.15The differential equation of motion of the cantilever beam is

whereGis the shear modulus,Jis the torsion constant,θ(x,t)is the torsional angle of the cantilever,xis the position on the long axis of the cantilever,tis time,ρis the mass density,Ipis the polar moment of inertia of the cantilever section,andcis the viscous damping coefficien of the cantilever.Takingθ(x,t)=Θ(x)eiΩtin Eq.(3),we obtain

whereklatis the transverse contact stiffness between the tip and the sample surface andηlatis the viscosity.Considering the boundary conditions in Eq.(5),Eq.(4)can be solved,and the torque at the tip whenx=Lcan then be calculated:

HereHh(Ω)is the frequency response function of the cantilever at the tip and is given by

whereRaandIaare the real and imaginary parts of,respectively.The torsional amplitude and phase difference of the cantilever beam atx=Lare

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 define as

Here the contact radiusacis given by

whereRis the radius of the probe tip,Fnoris the force in the vertical direction between probe and tip,andE＊is the effective elastic modulus.The expression forE＊is

whereEtandEsare the elastic moduli of the tip and sample,respectively,andvtandvsare their Poisson’s ratios.The effective shear modulusG＊in Eq.(9)is given by

whereGtandGsare 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 influenc 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 viscosityηlatand shear modulusGs,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 half-cycle of the sine-wave 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.20However,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 flo chart is shown in Fig.5.

Second,a background subtraction algorithm is used to fi and subtract the aperiodic parasitic deflectio signal from the periodic noncontact area data of each peak force,to eliminate the influenc of the interaction force between the sample and the cantilever beam.The specifi procedure is as follows:

Step 1:Determine the contact area between the needle tip and the sample on the force curve by using the synchronization algorithm.

Step 2:Perform polynomial curve fittin based on least squares on the noncontact area to obtain the background noise signal.

Step 3:Calculate the difference between the actual deflectio signal data and the corresponding data from the fittin curve to obtain the deflectio signal due to the interaction between needle tip and sample.

Step 4: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 deflectio signal due to environmental influence thereby allowing more accurate control.Software background signal subtraction can fi aperiodic parasitic deflectio signals,such as that due to the interaction force between a sample with large fluctuation 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 fina signal shown in Fig.6(d).

FIG.5.Flow chart of improved synchronization algorithm.

Finally,the normal peak force is extracted by the traditional peak force tapping data analysis algorithm,21and 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 finall 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 influenc of the longitudinal force.The specifi 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:

Step 1:Calculate the start and end times of each row of data in the longitudinal mechanical characteristic image.

Step 2:Calculate the starting position of the peak force period contained in the data of each row of the longitudinal mechanical characteristic image.

Step 3:Select the peak force data and each peak force cycle data point according to the longitudinal force and record the position.

FIG.6.Background subtraction process for a peak force periodic signal:(a)originalsignal;(b)hardware background noise;(c)signalafter hardware background subtraction;(d)signalafter software background subtraction.

Step 4: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.

Step 5:Acquire the original data of the transverse amplitude and phase characteristic image.

Step 6:Acquire the forward/backward scanning image data of the transverse amplitude and phase characteristics.

Step 7:Perform third-order B-spline 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.

Step 8:Perform data conversion to obtain the fina image data array of the transverse amplitude and phase characteristics.

Step 9:Generate a two-dimensional amplitude diagram and phase diagram of the transverse amplitude and phase characteristics of the sample.

Step 10:Generate a three-dimensional volume data plot of the transverse amplitude and phase characteristics of the sample related to the longitudinal force.

FIG.7.Synchronization of torsionalresonance data using peak force data.

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 RFESP-75(Bruker),with a calibrated resonance frequency of 75 kHz and an elastic coefficien 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 low-density polyethylene(LDPE),was used for imaging.

FIG.8.(a)Dimension Icon(Bruker)AFM.(b)TR probe clamp.(c)M2p.5923-x4 high-speed eight-channel data acquisition card.

TABLE I.Main parameters of RFESP-75 probe.

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 deflectio 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 real-time lossless acquisition of high-throughput multichannel signals,we used an M2p.5923-x4 high-speed eightchannel data acquisition card with a sampling frequency of 500 kHz[Fig.8(c)].

A.Measurement results for longitudinal mechanical properties

FIG.9.Measurementresults for longitudinalmechanicalproperties:(a)height;(b)DMT modulus;(c)adhesion;(d)energy dissipation.

A PS-LDPE 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 2.5×0.6μm2.The line scanning frequency was set to 0.9 Hz.The morphology of the PS-LDPE 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 PS-LDPE and accurately obtain the height of its surface morphology.The nanomechanical measurement results are basically consistent with the calibrated values for PS-LDPE,which verifie the effectiveness and accuracy of the system.It therefore provides a powerful means to study the relationship between the mechanical properties and microstructure of PS-LDPE and other composite samples.

B.Experimental results for extraction of transverse mechanical characteristics

Figures 10(a)and 10(b)present two-dimensional plots of the transverse amplitude and phase,respectively,extracted under different longitudinal force conditions.The transverse amplitude and phase depend significantl 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.

FIG.10.Two-dimensionalplots of(a)the transverse amplitude and(b)the transverse phase for different longitudinalforces.

FIG.11.(aI)Three-dimensionalvolume plot of transverse amplitude for differentlongitudinalforces.(aII)and(aIII)Cross-sectionalslices of this volume plot in the XZ and YZ planes,respectively.(bI)Three-dimensionalvolume plot of transverse phase for differentlongitudinalforces.(bII)and(bIII)Cross-sectionalslices ofthis volume plotin the XZ and YZ planes,respectively.The dashed lines in(aI)and(bI)indicate the slice positions.

To provide a more intuitive representation of the influenc of the longitudinal force on transverse mechanical properties,Fig.11 presents three-dimensional 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 cross-sectional slices of these volume plots in theXZplane,and Figs.11(aIII)and 11(bIII)are cross-sectional slices in theYZplane.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 reflectio 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 three-dimensional 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 identifie the contact area for each peak force cycle,thereby enabling elimination of the aperiodic parasitic deflectio 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 conflict to disclose.

DATA AVAILABILITY

The authors confir that the data supporting the finding of this study are available within the article.