Design of Harmonic AFM Probe Subjected to van der Waals Force in the Modified Couple Stress Theory

The conventional design of harmonic AFM probe geometry is made in neglect of the effects of the size-dependency factor and the tip-sample interacting force. Obviously, the effect of these two factors on the natural frequencies of a probe is significant. In this study, the effects of the two factors on the integer-multiples relation among frequencies are investigated. In this study, the effects of the two factors on the integer-multiples relation among frequencies are investigated. It is discovered that, in general, the integer-multiples relations of the probe’s frequencies in the classical model does not be kept as the same as that in the system with the effect of the size-dependency factor under the same material and geometry properties of probe. In addition, when the probe is used to measure the sample, the deviation of the relations will happen. The smaller the tip-sample distance is, the larger the deviation of integer-multiples frequencies is. The analytical method is presented here such that during scanning a sample at some tip-sample distance, the material and geometry properties of the probe can be tuned to the integer-multiples relation of resonant frequencies. Moreover, five similarity conditions among the systems with and without the effects of size-dependency and the tip-sample interacting force are discovered. According to these conditions, the integer-multiples relation is kept in different systems.


Introduction
For high-speed AFM imaging, atomic force microscopy is often employed to measure the surface topography and material property of specimens, simultaneously. e dynamic response of the fundamental mode is conventionally used to measure the topography of the sample. e response of higher flexural modes of a cantilever is taken to measure some material properties [1][2][3][4][5][6]. It is found that if the probe's higher-order resonance frequencies match the integer multiples of its fundamental frequency, the probe's responses at such harmonic frequencies will be enhanced. e harmonic amplitude is sensitive to local mechanical properties of the specimen [5][6][7]. In general, the design methods of a harmonic probe include (1) the attached mass method [8], (2) the removed mass method [6,9,10], (3) the cutting-boundary method [5,11,12], and (4) the varying cross section method [13]. e literatures are introduced as follows.
Li et al. [8] proposed attaching micro/nanoparticles with specific mass and position on the cantilever. Unfortunately, precise positioning of the attached mass could be technically not so easy to realize. e literatures [6,9,10] investigated the selective removal of the lever materials is more convenient than the previous one, which can be accomplished by changing the cantilever shape and cutting hole structures in the probe. e literatures [5,11,12] studied the cutting the boundary of probe cross section to tailor the frequency characteristics. In the above literatures, the finite element method is used to determine the frequencies of the probe due to no general rule about the geometry. Moreover, the method is just for some special case. For example, Cai et al. [11] changed the geometry of the commercial probe to tune the integer-multiples relation. In the literatures [5,6,[8][9][10][11][12], the classical theory of elasticity is used to investigate the harmonic AFM probe. Moreover, the effect of structure's size dependency is not considered. Lin and Chang [13] designed the harmonic AFM probe with varying cross section in polynomial series based on the modified couple stress theory, neglecting the effect of the tip-sample interacting force. However, it is hard to manufacture the varying cross section of the probe precisely due to the complicated geometry. In addition, the integer-multiples relations without the effect of the tip-sample interacting force will not be kept during scanning the sample.
Several literatures [14][15][16][17] discovered that when the structures are of the order of microns or submicrons, the size-dependent effect on the behavior of structure is significant. Some literatures investigated mechanical properties of nanostructures in nonclassical theories. ese theories are (1) the couple stress theory [18,19], (2) the strain gradient theory [20], (3) the nonlocal theory [21], and (4) the surface elasticity model [22]. Lin et al. [18,19] investigated the dynamic behaviors of a conventional AFM uniform probe subjected to the van der Waals force and the multimode excitation based on the modified couple stress theory. Further, Lin et al. [20] made assessment of the classical, modified couple stress, and strain gradient theories about size dependency effect on the resonant frequency of a conventional AFM uniform probe subjected to van der Waals force. e literatures [18][19][20] investigated the dynamic behavior of the conventional probe with the size dependency effect only.
So far, no literature is devoted to design the harmonic probe with the integer-multiples relation of frequencies considering the effects of the size dependency and the tipsample interacting force during an AFM probe measuring a sample's topography and properties. In this study, the design of linearly varying cross section of the probe with the integer-multiples relation of frequencies and these effects is proposed. e cross section of the probe is easily and precisely manufactured. Moreover, the similarity conditions among the systems with and without the effects of sizedependency and the tip-sample interacting force are investigated.

Governing Equation and Boundary Conditions in Ham-
ilton's Principle. In this study, the conventional AFM slender probe has a large aspect ratio and its deflection during measuring the topography of the sample is very small; the Bernoulli-Euler beam theory is usually used to simulate the system. e dynamic response of a Bernoulli-Euler beam with tip mass subjected to the van der Waals interacting force F v is investigated, as shown in Figure 1.
Application of Hamilton's principle [13,23] yields the coupled governing differential equations and the associated general elastic boundary conditions. e governing equation of an AFM cantilever probe as shown in Figure 1 is e clamped boundary conditions are expressed as follows.
At x � 0: At x � L: where F v is the van der Waals tip-sample interacting force [5] which is expressed as in which the tip-surface distance D � D 0 − W(L, t), A H is the Hamaker constant, D 0 is the distance between the tip of the undeformed beam and the surface of sample, and R is the tip radius, as shown in Figure 1.

Dimensionless Governing Equation and Boundary
Conditions. In terms of the dimensionless parameters in the nomenclature, the dimensionless governing equation and boundary conditions are e clamped boundary conditions are expressed as follows.

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where f v � − (c v /6d 2 ) and the tip-surface distance is Because boundary condition (16) is nonlinear, it is hard to derive the solution of the nonlinear system. In this study, the analytical method is proposed.

Separation of Variable.
e dynamic solution of harmonic motion is Substituting equation (11) into governing equation (5) and boundary conditions (6)-(8), the transformed governing equation is obtained.
and the associated boundary conditions are expressed as follows.

Frequency Equation.
e solution of the characteristic governing equation (12) can be written as where V i (ξ), i � 1, 2, 3, 4 are the four linearly independent fundamental solutions of equation (14), which are assumed to satisfy the following normalized condition: e coefficient C i , i � 1, 2, 3, 4, is dependent on characteristic boundary conditions (13)- (16). Substituting equation (17) into boundary conditions (13)-(16), the amplitude is where e associated frequency equation is It is well known that the resonant frequency depends significantly on the amplitude of probe vibration w given (1).
One can determine the resonant frequency via frequency equation (21).

Semianalytical Fundamental Solutions.
e length of the beam is divided into several continuous subdomains as Mathematical Problems in Engineering shown in Figure 2. e displacement in the (j + 1) th subdomain can be expressed in terms of the global fundamental solutions V i and local ones V j+1,i : e local fundamental solutions in the (j + 1) th subdomain satisfy the following normalized condition: e four continuity conditions at the interface between two neighboring subdomains are Substituting equation (22) into continuity condition (24), the coefficients of the (j + 1) th subdomain can be expressed in the fundamental solutions of the j-th subdomain: Finally, the global semianalytical fundamental solutions are derived in terms of the local ones: (26)

Probe with Constant ickness in Classical eory.
It is well known that if the size-dependency parameter δ of equations (27)-(31) is zero, the characteristic system in the classical theory is obtained: and the associated boundary conditions are expressed as follows.

Similarity Formula.
It is found that if the dimensionless bending rigidity b(ξ) is the same as the dimensionless mass m(ξ), the characteristic system in the modified couple stress theory is composed of equations (33)-(37). Based on the characteristic systems in the two different theories, it is discovered that if the following relations are satisfied, the two characteristic systems are same.
Equation (38) is the geometry similarity of the probe. Equation (39) is the tip similarity. Equations (40) and (41) are the similarity of measurement conditions. Equation (42) is the similarity of frequency.
In the absence of the effect of the tip-sample interacting force, only two similar relations (38) and (42) is required [15]. When the dimensionless bending rigidity b(ξ) is the same as the dimensionless mass m(ξ), one can predict the resonant frequencies in the modified couple stress theory by a similar equation (43) and the frequencies in the classical theory.
Considering the effect of the tip-sample interacting force, similar relations (38)-(41) must be satisfied. One can predict the resonant frequencies in the modified stress theory based on the frequencies in the classical theory via similarity relation (42). So far, no literature is devoted to the similarity mechanism. In reality, the effect of nano size depends on the dimension and geometry. e size-dependency parameter δ changes with different conditions and is not accurately obtained so far. In general, the integermultiples relation among frequencies depends on the sizedependency parameter δ. However, if conditions (38)-(42) are satisfied, the integer-multiples relation is preserved in spite of the size-dependency parameter δ.

Numerical Results
Consider the stepwise tapered rectangular cross-sectional probe as shown in Figure 3. e constant thickness is e dimensionless mass and bending rigidity are

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Because the coefficients (42) are in polynomial series, the local fundamental solutions can be exactly determined by the Frobenius method [13]. Further, the global semianalytical fundamental solutions can be determined via equation (26).
Without the effect of the interacting force between the tip and sample and considering the probe with the positions of interface {ξ 1 � 0.3, ξ 2 � 0.6}, the taper ratios λ 1 , λ 2 , λ 3 of AFM cantilevers in the classical theory are tuned such that the second and the third resonance frequencies are integer multiples of the fundamental one for higher harmonic enhancement.
One discovers three types of probe satisfying the integermultiples relation among frequencies which are listed in Table 1 Figure 3 demonstrates the three profiles of the probe width with different integer multiplication of frequency.
In a similar way, one also discovers three types of probe with the positions of interface {ξ 1 � 0.4, ξ 2 � 0.7}, satisfying the integer-multiples relation among frequencies which are listed in Table 2 { }. When the probe designed without the effect of interacting force between the tip and sample is used to measure the sample topography and material properties, the effect of Table 1: ree types of probe satisfying the integer-multiples relation among frequencies without the effects of the tip-sample force and the size dependency (μ � 4.0381 × 10 − 3 , ξ 1 � 0.4, and ξ 2 � 0.7).  interacting force on the deviation of integer multiples of frequencies is investigated. Figure 4(a) demonstrates that when the tip-sample distance D 0 is very large, the deviations of integer multiples of frequencies are very small. e smaller the tip-sample distance D 0 is, the larger the deviations of integer multiples of frequencies are. e effect of the tip-sample distance D 0 on the deviation of (ω 3 /ω 1 ) is significantly more than that of (ω 2 /ω 1 ).
Moreover, the larger the size-dependency parameter δ is, the smaller the deviations of integer multiples of frequencies are.
It is observed from Figures 4(a) with tip amplitude w(L) � 1 nm and Figure 3(b) with tip amplitude w(L) � 3 nm that the larger the tip amplitude w(L) is, the larger the deviations of integer multiples of frequencies are.
Comparing Figure 4 with n 21 � 7, n 31 � 21 , Figure 5 with n 21 � 8, n 31 � 26 , and Figure  6 with n 21 � 10, n 31 � 34 , it is discovered that the larger the frequency ratios are, the larger the deviations of integer multiples of frequencies are during measuring some samples.
Considering the effect of interacting force at D 0 � 5 nm and in the absence of the effect of size dependency, δ � 0, the taper ratios λ 1 , λ 2 , λ 3 of AFM cantilevers are 1.096436, 1.68730, 0.24847 { } such that the second and the third resonance frequencies are integer multiples of the fundamental one, n 21 � 7, n 31 � 21 . Figure 7 shows that if the tip-sample distance D 0 is smaller than 5 nm, the deviations of integer multiples of frequencies are negative. If the tip-sample distance D 0 is larger than 5 nm, the deviations of integer multiples of frequencies are positive. If the taper ratios λ 1 , λ 2 , λ 3 of AFM cantilevers are 1.022422, 1.68730, 0.24847 { }, the deviation at D 0 � 8 nm is zero.
Considering the effects of size-dependency, δ � 0.5, and the interacting force at D 0 � 5 nm, Figure 8 shows that if the taper ratios λ 1 , λ 2 , λ 3 of AFM cantilevers are 0.80000, 3.99794, − 0.90650 { }, the second and the third resonance frequencies are integer multiples of the fundamental Table 2: ree types of probe satisfying the integer-multiples relation among frequencies without the effects of the tip-sample force and the size dependency (μ � 4.0381 × 10 − 3 , ξ 1 � 0.3, and ξ 2 � 0.6).
Mathematical Problems in Engineering one, n 21 � 10, n 31 � 34 . If the tip-sample distance D 0 is smaller than 5 nm, the deviation of integer multiples of frequencies is negative. If the tip-sample distance D 0 is larger than 5 nm, the deviation of integer multiples of frequencies is positive. If the taper ratios λ 1 , λ 2 , λ 3 of AFM cantilevers are 0.80000, 3.99794, − 0.85049 { }, the deviation at D 0 � 8 nm is zero.
Considering the tip-sample distance D 0 � 5 nm, the taper ratios λ 1 , λ 2 , λ 3 of AFM cantilevers in the classical theory are tuned such that the second and the third resonance frequencies are integer multiples of the fundamental one, n 21 � 7, n 31 � 21 . With the effect of size dependency, Table 3 reveals that the proposed probe has the same integermultiples relation of frequency at the tip-sample distance D 0 � 5 nm. e numerical results determined by the proposed method and similarity formula (44) are consistent. e fact is also demonstrated in Table 4 with the tip-sample distance D 0 � 8 nm.

Conclusion
With the effect of the interacting tip-sample force, the integermultiples relations among frequencies of nonuniform rectangular AFM probes in the classical theory and the modified couple stress one are discovered. When the thickness of rectangular cross section of the beam is constant, the integermultiples relations in different theories are the same in spite of the size-dependency parameter δ. e stepwise width of probe can be tuned to match the requirement of integer multiples of frequencies. Based on the conventional harmonic probe designed without the effect of the tip-sample force, the effect of interacting force on the deviation of integer multiples of frequencies is investigated as follows: (1) e smaller the tip-sample distance D 0 is, the larger the deviations of integer multiples of frequencies are. (2) e effect of the tip-sample distance D 0 on the deviation of (ω 3 /ω 1 ) is significantly more than that of (ω 2 /ω 1 ). (3) e larger the size-dependency parameter δ is, the smaller the deviations of integer multiples of frequencies are. (4) e larger the tip amplitude w(L) is, the larger the deviations of integer multiples of frequencies are.