Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 3147863, 10 pages
https://doi.org/10.1155/2019/3147863

Research Article

Resonance-Size Parameter Relationship and Dynamics of an AFM Subjected to Multimode Excitation and Based on the Modified Couple Stress Theory

Shueei-Muh Lin and Ching-Yao Chang

Mechanical Engineering Department, Kun Shan University, Tainan 710-03, Taiwan

Correspondence should be addressed to Shueei-Muh Lin; smlin45@gmail.com

Received 24 March 2019; Accepted 20 May 2019; Published 16 June 2019

Academic Editor: Weizhong Dai

Copyright © 2019 Shueei-Muh Lin and Ching-Yao Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The mathematical model of AFM probe subjected to multimode excitation based on the modified couple stress theory is presented. The semianalytical solution of the system is proposed. The transient behavior and response spectrum of AFM probe subjected to multimode excitation are investigated. It is very helpful to predict the nanotopography and surface properties based on the response of multimodes excitation. The effects of the root excitation, size parameter, and interacting distance on the response spectrum and frequency shift are investigated. The resonant frequency relation of the two systems with different size parameters is discovered and expressed in a formula. The natural frequencies predicted via the formula and those determined by the semianalytical method are significantly consistent.

1. Introduction

Atomic force microscopy (AFM) is a powerful device for scanning the atomic-scale topography and property of a sample’s surface [1].

In convention, the literatures investigated the steady behavior of AFM probe [2–9]. Definitely, the investigation of the amplitude and phase behavior in the transient regime will increase significantly the speed of imaging and accuracy. Santos and Gaderlab [10] investigated the tip-sample force in transient behavior. Sahoo et al. [11] increased the speed of imaging and control of AFM by studying the transient motion of cantilever signal. Chang et al. [12] presented a new algorithm for high speed AFM imaging of biopolymers by investigating the transition from transient part to steady state regime. Payam [9] studied the transient behavior of tapping mode-AFM in the mass-spring-damper model.

Some literatures [13–16] investigated the multifrequency excitation of FM-AFM systems in fast force spectroscopy and material characterization at the atomic scale as well. The topography and material properties of sample’s surface can be measured by using AFM subjected to the simultaneous multimodes excitations. The amplitude signal of the first mode is used to image the surface topography. The phase shift signal of the second mode is usually used to map changes in material properties of the sample’s surface [17–19].

In convention, the structure’s size effect is not considered. Several literatures [20–23] found that the size-dependent effect on the dynamic behavior of the structures in the order of microns or submicrons is significant. Some nonclassical theories with size dependency are the couple stress theory [16, 24, 25], the strain gradient theory [26], the nonlocal theory [27], and the surface elasticity model [28]. Mindlin and Tiersten [24] and Toupin [25] presented the classical couple stress theory with the gradients of rotation and displacement. Yang et al. [29] modified the classical stress theory (MCST) by considering that the density of strain energy is a function of the strain and curvature tensors.

Several literatures [30–34] investigated the size effect on the behavior of beam. Ansari et al. [35] investigated the size-dependent resonant frequency and flexural sensitivity of AFM based on the modified strain gradient theory by using the force gradient method. However, it is well known that the interpretation of frequency shift by using the force gradient method is unsatisfactory [4].

So far, no literature is devoted to investigate the transient behavior of an AFM probe subjected to multimodes excitations based on the modified couple stress theory. In this study, the mathematical model of AFM probe subjected to two-mode excitation and the van der Waals force based on the modified couple stress theory is constructed. The semianalytical method is presented. The effects of several parameters on the transient behavior are investigated.

2. Dynamic System Subjected to Multi-Mode Excitation

2.1. Governing Equation and Boundary Conditions

Consider the transient response of atomic force microscopy excited at the root measuring the interatomic van der Waals force, as shown in Figure 1. The multimode harmonic excitation is presented, as shown in Figure 1. The governing equation is

The boundary conditions are as follows.

Figure 1: Geometry and coordinate system of AFM.

At x = 0:

At x = L:

where and denote the cross-sectional area and the area moment of inertia, respectively. is the mass density per unit volume, the tip mass, the flexural displacement, Young's modulus, the coordinate along the beam, the length of the beam, and the material length scale parameter. The van der Waals force [11] is

in which is the Hamaker constant, the tip-surface distance, and the tip radius.

In terms of the following dimensionless quantities:

where is the characteristic length. A small value of is introduced to avoid the numerical transaction error. The dimensionless governing differential equation of the system is

The associated boundary conditions are as follows.

At = 0:

At = 1:

The initial conditions are

2.2. Solution Method

Because the system composed of (8)-(13) is nonlinear and the boundary condition is time-dependent, it is difficult to directly solve. The semianalytical method is presented here.

2.2.1. Linearization of System

Due to its complexity, the nonlinear general system is approximated by infinite linearized subsystems. At first, the time variable is divided into infinite sections and the dynamic performance of the system is derived step by step. The methodology is described here. The tip displacement is presented by the Taylor series as

where is an unknown transaction difference to be determined. Based on this relation and the following methods one can determine the displacement . Moreover, the error of the tip displacement must approach zero:

Substituting (14) into the nonlinear nonhomogeneous boundary condition (12), the linearized time-dependent boundary condition is obtained:

So far, the nonlinear system is linearized in each time domain. The approximated system can be exactly solved by the solution method presented in the following sections. Therefore, the overall displacement can be derived step by step.

So far, the linearized subsystem in the time domain can be expressed as follows:

The associated time-dependent boundary conditions are as follows.

At = 0:

At = 1:

2.2.2. Solution Method of Linearized Subsystem

(A) Transform of Variable. The linearized subsystem includes two nonhomogeneous time-dependent boundary conditions. Solving this subsystem, the following transformation of variable is considered:

Substituting relation (22) into (17)-(21), two following subsystems are obtained. The first subsystem is expressed in terms of the transformed variable as follows: the transformed governing equation is

The associated boundary conditions are as follows.

At ξ = 0:

At = 1:

The corresponding initial conditions are

The second subsystem is expressed in terms of the shifting function . The second transformed governing equation is

At = 0:

At = 1:

The third subsystem is expressed in terms of the shifting function . The third transformed governing equation is

At = 0:

At = 1:

The solutions of the two shifting functions are easily discovered as follows:

(B) Solution of the First Subsystem

(B-1) Orthogonality of Eigenfunctions. The frequency equation of the first subsystem composed of (23)-(27) is

where . The n mode shapes are

which is the same as that given by Rao [36]. The orthogonality of the eigenfunctions is presented as follows:

(B-2) Modes Superposition Method. Based on the orthogonality conditions (44), the mode superposition method is used to derive the solution of the first subsystem composed of (23)-(27). The transformed variable is expressed in terms of the eigenfunctions

Substituting (45) into (23) and multiplying it by and integrating it from 0 to 1, one obtains

where

in which

The solution of (40) is

Substituting (48a) and (48b) back into (22) results in the displacement of the cantilever. It should be noted that the tip displacement must satisfy condition (15).

2.3. Numerical Results of Transient Response and Discussion

The influences of the excitation frequency , the excitation amplitude , and tip-sample distance on the tip transient response are investigated and shown in Figures 2(a) and 2(b). Figure 2(a) demonstrates that the interacting distance is nm. The initial displacement and velocity are zero and are listed in (13). One has the natural frequency of the cantilever without the interacting force, f₁ = 53525.9 Hz. When the frequency of root excitation is far from the natural frequency such as 48000 and 60000 Hz the small vibration of the cantilever is harmonically excited. When the frequency of excitation such as 53000 Hz approaches the natural frequency f₁, the vibration response is significantly increased. It is obvious that the frequency shift occurs due to the interacting force. Figure 2(b) demonstrates that the interacting distance is D₀=2 nm. When the frequency of excitation is 52000 Hz, the vibration response is significantly increased. It is obvious from Figures 2(a) and 2(b) that the smaller the interacting distance D₀ is, the greater the frequency shift is.

(a)

(b)

Figure 2: The influences of the excitation frequency , the excitation amplitude , and tip-sample distance on the tip transient response [, B = 45 μm, H = 2.50 μm, E = 70.3×10⁹ Pa, ρ = 2.5×10³ kg/m³, , L = 200 μm, k = 1.5447 N/m, and f₁ = 53525.9 Hz,].

Figure 3 demonstrates the influence of the distance between the tip and a sample surface on the tip response spectrum. It is shown that the smaller the interacting distance is, the larger the resonance frequency shift is.

Figure 3: The influence of the distance between the tip and a sample surface on the tip response spectrum [, B = 45 μm, H = 3.50 μm, E = 70.3×10⁹ Pa, ρ = 2.5×10³ kg/m³, , L = 200 μm, k = 4.2386 N/m, f₁ = 74941.4 Hz, , , and d0=3, 5, 10 nm].

Figures 4(a) and 4(b) demonstrate the influence of the size parameter on the tip response spectrum for D₀= 3, 5 nm. It is observed that the larger the size parameter is, the greater the frequency shift is. Moreover, the smaller the interacting distance D₀ is, the greater the frequency shift is.

(a)

(b)

Figure 4: The influence of the size parameter δ on the tip response spectrum [, B = 45 μm, H = 2.50 μm, E = 70.3×10⁹ Pa, ρ = 2.5×10³ kg/m³, , L = 200 μm, k = 1.5447 N/m, f₁ = 53525.9 Hz, =0.01 nm, and =0.0 nm].

Further, the influence of two excitation modes on the tip response spectrum is investigated; Figure 5 demonstrates that if the second frequency of excitation = 76500 Hz is far from the natural frequency f₁ = 53525.9 Hz, the response spectrum is as shown Figures 3 and 4 with one-mode excitation. Moreover, the influence of the interacting distance D₀ on the response spectrum is slight. However, if the second frequency of excitation = 56500 Hz is close to the natural frequency f₁ = 53525.9 Hz, several resonant responses occur. Moreover, the influence of the interacting distance D₀ on the response spectrum is significant. It is concluded that the response spectrum of two-mode excitations is significantly different to that of one-mode excitation.

Figure 5: The influence of the two-mode excitation on the tip response [, , B = 45 μm, H = 2.50 μm, E = 70.3×10⁹ Pa, ρ = 2.5×10³ kg/m³, , L = 200 μm, k = 1.5447 N/m, f₁ = 53525.9 Hz, and = =0.01 nm].

3. Prediction of Natural Frequencies

3.1. The System with the Size Effect

The dimensionless governing equation and boundary conditions are

The clamped boundary conditions are as follows.

At = 0:

At = 1:

where and , and the tip-surface distance is .

(A) Solution Method. Assume

Substituting (54) into (49), one obtains

where . Substituting (54) into the boundary conditions (50)-(52), one obtains the following.

At = 0:

At = 1:

Substituting (54) into the boundary conditions (53) and multiplying it by and integrating it from 0 to the period T, , (53) becomes

where . Further, the semianalytical solution can be determined by using the method by Lin [2].

3.2. The Classical System without the Size Effect

The dimensionless governing equation and boundary conditions are

The clamped boundary conditions are as follows.

At = 0:

At = 1:

where and , and the tip-surface distance is .

(A) Solution Method. Assume

Substituting (65) into (60), one obtains

Substituting (65) into the boundary conditions (61)-(63), one obtains the following.

At = 0:

At = 1:

Substituting (65) into the boundary conditions (64) and multiplying it by and integrating it from 0 to the period T, , (64) becomes

Further, the semianalytical solution can be determined by using the method by Lin [2].

3.3. Relation between Natural Frequencies and Size Parameter

It is discovered that if the following relations are satisfied

the system composed of (55)-(59) is similar to that composed of (66)-(70) and .

In other words, if the parameters of the two systems are the same, the tip radius relation and , the relation of natural frequencies (71) must be satisfied.

Tables 1 and 2 demonstrate the comparison of the first two natural frequencies determined by using the analytical method [2, 4] and formula (71). It is found that these results are significantly consistent. When the interacting distance is large, the results are the same. When the interacting distance D₀=5nm, the errors of the first two frequencies are 0.08% and 0.002%, respectively.

Table 1: Effects of the interacting distance D₀, the axial deformation, and the size dependency parameter on the first frequency [SiO2: E = 70.3×10⁹Pa, ρ = 2.5×10³ kg/m³, B =45 μm, H = 3.5 μm, L=200 μm, R = , AH = , and ; the first natural frequency of beam with =0, f₁ = 74355.803 Hz, and ].

(nm)
f₁(Hz)
Error
(%)

Semianalytical method
Based on (71)

5
73686.61
77347.30
77283.17
0.0829

6
74051.34
77694.80
77665.68
0.0375

7
74187.16
77824.20
77808.15
0.0206

8
74251.28
77885.32
77875.40
0.0127

9
74286.01
77918.47
77911.82
0.0085

10
74306.66
77938.20
77933.48
0.0061

11
74319.82
77950.69
77947.28
0.0044

12
74328.64
77959.10
77956.54
0.0033

13
74334.81
77964.93
77963.01
0.0025

14
74339.14
77969.18
77967.55
0.0021

15
74342.38
77972.18
77970.95
0.0016

20
74350.29
77979.84
77979.24
0.0008

40
74354.29
77983.50
77983.43
0.0001

80
74355.73
77984.95
77984.95
0.0000

Table 2: Effects of the interacting distance D₀, the axial deformation, and the size dependency parameter on the second frequency f₂ [SiO2: E = 70.3×10⁹Pa, ρ = 2.5×10³ kg/m³, B =45 μm, H = 3.5 μm, L=200 μm, R = , AH = , and ; the first natural frequency of beam with δ=0, f₁ = 74355.803 Hz, and ].

(nm)
f₂(Hz)
Error
(%)

Semianalytical method
Based on (71)

5
465927.01
488678.25
488668.37
0.0020

6
465983.22
488731.79
488727.32
0.0009

7
466004.20
488751.78
488749.33
0.0005

8
466014.11
488761.27
488759.72
0.0003

9
466019.52
488766.43
488765.40
0.0002

10
466022.69
488769.43
488768.72
0.0001

11
466024.77
488771.43
488770.90
0.0001

12
466026.19
488772.76
488772.39
0.0001

13
466027.02
488773.59
488773.26
0.0001

14
466027.77
488774.26
488774.05
0.0000

15
466028.27
488774.76
488774.57
0.0000

20
466029.52
488775.93
488775.88
0.0000

40
466030.18
488776.44
488776.58
0.0000

80
466030.34
488766.74
488766.74
0.0000

4. Conclusion

The transient response of AFM probe subjected to two-mode excitations based on the modified couple stress theory is investigated. The semianalytical solution is presented. The similarity of the classical Bernoulli-Euler beam system and the system based on the modified couple stress theory is discovered. The relation between the natural frequencies of the two systems is . Several trends about the response spectrum due to two-mode excitations are obtained as follows.

(1)The response spectrum of two-mode excitations is significantly different to that of one-mode excitation. If the two frequencies of excitation are close to the natural frequency, several resonant responses occur.(2)The larger the size parameter is, the greater the frequency shift is.(3)The smaller the interacting distance is, the greater the frequency shift is.

Data Availability

All the data used to support the findings of this article are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The support of the Ministry of Science and Technology of Taiwan, R. O. C., is gratefully acknowledged (Grant number: MOST 106-2221-E-168-005).

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