Binary coded cantilevers for enhancing multi-harmonic atomic force microscopy

Sensors and Actuators A 300 (2019) 111668

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Binary coded cantilevers for enhancing multi-harmonic atomic force microscopy Yaoping Hou a,b , Chengfu Ma a,b , Wenting Wang a,b , Yuhang Chen a,b,∗ a

Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, 230027, China Key Laboratory of Precision Scientific Instrumentation of Anhui Higher Education Institutes, University of Science and Technology of China, Hefei, 230027, China b

a r t i c l e

i n f o

Article history: Received 15 July 2019 Received in revised form 30 September 2019 Accepted 9 October 2019 Available online 13 October 2019 Keywords: Atomic force microscopy Harmonic imaging Cantilever dynamics Genetic algorithm Structural optimization

a b s t r a c t Harmonic atomic force microscopy (AFM) has been widely applied to characterize local mechanical properties of the specimens under imaging. However, the harmonic amplitude is commonly quite weak owing to the fast decay at an off-resonance state for a conventional cantilever and the internal resonance between eigenfrequencies and higher harmonics can amplify the harmonic response. Therefore, specific dynamic characteristics of AFM cantilevers are frequently required. Here, we proposed a coding scheme of a cantilever where the structure is described by a two-dimensional binary matrix to tailor its dynamic characteristics. Genetic algorithms combined with finite element analysis were used to optimize the code configuration. Cantilever properties including eigenfrequencies, stiffness and quality factor can be conveniently tuned to satisfy multiple requirements. AFM cantilevers were then re-designed with the second and the third resonance frequencies being moved to integer multiples of the fundamental one for higher harmonic enhancement, yet with minor stiffness alteration. Harmonic AFM imaging on a polymer mixture demonstrated that the amplitude difference between two material phases was increased up to at least 2.7 times for the 6th harmonic and 1.6 folds for the 18th harmonic using the optimized cantilevers. The enhancement of harmonic contrast can benefit the discrimination of different materials. The proposed binary coding method has advantages such as flexible tailoring of dynamic characteristics, multiple outputs of optimal structures, and automatic optimization. Consequently, it could be a promising way to design cantilevers for enhanced performances in various dynamic AFM imaging applications and also cantilever-based sensing. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In atomic force microscopy (AFM) and numerous AFM-based sensing techniques, the mechanical characteristics of the cantilever including its stiffness k, resonance frequency f and quality factor Q are of critical importance [1]. In dynamic AFM characterization, for example, a larger quantity ωQ/k where ω = 2f is the angular frequency is always required to achieve better force resolution [2,3]. For harmonic AFM imaging, the ratio of a higher eigenmode frequency to the fundamental one near an integer is necessary to enhance the corresponding harmonic signals [4]. In mass sensing, the frequency shift induced by the mass attachment depends on the cantilever’s stiffness and resonance frequency [5]. In all these

∗ Corresponding author at: Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, 230027, China. E-mail address: [email protected] (Y. Chen). https://doi.org/10.1016/j.sna.2019.111668 0924-4247/© 2019 Elsevier B.V. All rights reserved.

applications, it is desirable to have the cantilever properties k, f and Q satisfying certain specified requirements to improve the sensing sensitivity, signal strength, and so on [6]. Therefore, versatile methods to tailor the dynamic characteristics are highly demanded. For conventional dynamic AFM operation, the cantilever is usually excited at or near its first resonance frequency and the amplitude at the fundamental frequency is used to track the topography [7]. To further extend the AFM capability in characterizing surface properties, various multifrequency methods including multi-harmonic AFM, bimodal AFM and some other alternatives have been proposed [8]. In intermittent-contact AFM mode, the tip touches the sample surface during each oscillation period and the nonlinear tip-sample contact forces can induce higher harmonic responses of the cantilever [9]. The amplitude and phase at a higher harmonic frequency are also lock-in analyzed in the so-called harmonic AFM. It has been demonstrated that the harmonic amplitude is sensitive to local mechanical properties of the specimen [10,11]. Therefore, higher harmonic AFM has been

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widely applied to discriminate different materials [12–14] and scan soft specimens at an ultra-high spatial resolution [15]. In liquid environment, the harmonic response is much obvious owing to the low-Q characteristic and the local sample modulus can be quantitively recovered from the dynamic responses of the first two higher-harmonics, which benefits the quantitative mechanical characterization of biological specimens [13]. In addition to monitor the flexural oscillation, torsional dynamics are analyzed in torsional harmonic AFM to reconstruct the time-varying interactions [16]. This method has the advantages of enhancing a larger number of higher torsional harmonics and subsequent accurate determination of time-varying forces. Consequently, the sample’s mechanical properties can be measured with a high accuracy [17]. A special cantilever whose tip is offset from the central axis is required to magnify the cantilever torsion. In the bimodal AFM, two flexural eigenmodes of the cantilever are excited simultaneously. The first mode is usually used to image the topography and the second mode is used to determine the sample’s local properties [8]. In comparison, harmonic AFM is compatible with the conventional intermittent-contact operation, which enables easy implementation. The harmonic amplitude, however, is considerably weak because of the fast decay at an off-resonance state. It has been proved that the internal resonance between eigenfrequencies and higher harmonics can be an effective approach to improve the harmonic signals [4]. For a common AFM cantilever, however, the integer multiples of its fundamental frequency do not naturally align with any higher-order eigenmode resonance frequencies. For instance, the second and third eigenfrequencies of an ideal rectangular cantilever are 6.27ω0 and 17.55ω0 . They are both not enough close to the sixth harmonic (6ω0 ) and the eighteenth harmonic (18ω0 ), respectively [8]. Here, ω0 denotes the fundamental angular resonance frequency. As a result, one or a few of the cantilever’s higher eigenfrequencies should be tuned to integer times of the fundamental one. Altering the cantilever mass distribution is a convenient way to tailor the frequency characteristics [4,18]. For instance, attaching micro/nano-particles with specific mass and position on the cantilever was proposed to be an effective method [19]. Unfortunately, precise positioning of the attached mass could be technically not so easy to realize. In contrast, selective removal of the lever materials is more convenient, which can be accomplished by changing the cantilever shape [20,21], altering its cross-section [22,23], cutting hole structures [24–26]. Another harmonic cantilever design is based on integrating an inner paddle structure so that it vibrates like two linearly coupled oscillators [27,28]. The cantilever is optimized to support a 1: n internal resonance between two eigenmodes and thus the nth harmonic response is enhanced. Among the above cantilever’s structural modifications for the enhanced harmonic imaging, complex geometries with either step cross-section or varying width may increase the fabrication difficulty. By introducing an opening slot with appropriate dimension and position, a specific higher-order flexural eigenmode can be made resonant at an integer multiple of the fundamental frequency [25,26]. In principle, a few suitable cantilever structures exist that can meet multiple constraints. However, the optimization process remains somewhat complicated and time-consuming because such an approach requires manual sweeping of a number of variables together with subsequent computations to find the most satisfactory positional and dimensional parameters of the cutting structure [25]. The number of optimal parameters for a single cutting structure may be relatively limited. Employing a multiple cutting approach provides other adjustable parameters, for example, the adjacent distance of two cutting elements and so it could release such limitations. However, the cantilever optimization process will be much more complex.

Fig. 1. Schematic illustration of the binary coded cantilever. (a) Code “0 denotes the element to be removed while code “1 means the one to be kept. The elements within the range L near the free end are all coded with “1 to leave a space for positioning the tip and the laser spot. (b) Generated cantilever structure after the connectivity evaluation.

Here, we proposed a binary-coded cantilever scheme for tailoring its dynamic and mechanical properties. The AFM cantilever was first coded by a two-dimensional binary matrix and the coding was iteratively optimized with genetic algorithm (GA) until the required properties were acquired. During each iteration, the cantilever structure was constructed according to the resulted code matrix and its properties were calculated by using finite element analysis (FEA). As a prototype, focused ion beam (FIB) etching was utilized to fabricate the final optimized structures. The conventional rectangular cantilevers were then modified for enhancing multi-harmonic AFM imaging. Spectroscopy characterization and harmonic imaging on a polystyrene (PS) and low-density polyethylene (LDPE) blend were performed to verify the enhancement of higher-order harmonic signals. Though FIB modification was applied in this work, batch fabrication of the resulted harmonic cantilevers should be easily realizable by employing well-developed microfabrication techniques such as photolithography. Moreover, it should be emphasized that the binary coding approach can be conveniently extended to satisfy other cantilever’s specific requirements to improve the detection sensitivity and the signal strength in numerous AFM-based applications including sensing [5,29–32] and imaging [25–28].

2. Cantilever design and optimization 2.1. Binary coding As schematically illustrated in Fig. 1, the cantilever is binarycoded with a two-dimensional m × n matrix, where code “0 denotes the corresponding element to be removed while code “1 means the one to be kept. The coding area is L × W with the elements always set to “1 in a length range L at the free end to leave a space for positioning the tip and the laser spot. Each coding element has then a length of L/m, a width of W/n and a thickness of H. With a smaller element size, the total number of codes increases and the probability of finding appropriate coding should increase subsequently. In practice, the element size can be selected according to the fabrication resolution of the hole structures. Since a binary code matrix uniquely represents a certain solid structure, the structural design of the cantilever is equivalent to optimizing the code arrangement to meet the predefined objectives. It should be mentioned that for a generated binary matrix, connectivity evaluation is required to exclude invalid coding. For blocks with one or more “1 elements that are separated by “0 elements, see for example those orange blocks in Fig. 1(a), their elements are then converted to code “0 . In addition, the structure is designed to be symmetric about the longitudinal axis to avoid coupling of the torsional dynamics.

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Using such a procedure, the cantilever structure is determined as illustrated in Fig. 1(b). 2.2. Finite element analysis of cantilever properties To evaluate the dynamic characteristics of the cantilever, FEA calculations are employed. Because the air damping has little influence on the resonance frequencies of an AFM cantilever oscillating in ambient air, it is therefore ignored. The free vibrations for an undamped system can be described as [33], 2

∂ ∂x2



2

∂ z EI ∂x2



2

+ As

∂ z =0 ∂t 2

(1)

Where E is Young’s modulus of the cantilever material. I denotes the moment of inertia.  is the mass density and As is the section area. Solution of Eq. (1) is in the form of [33], z (x, t) =

n 

ıi (t)∅i (x)

(2)

i=1

In the above equation, functions ıi (t) are the displacements along the cantilever coordinate, and functions ∅i (x) are the mode shapes satisfying all the geometric and natural boundary conditions. Considering a clamped cantilever, the displacement and slope are zero at the fixed end while the moment and shear are zero at the free end. The corresponding boundary conditions are then, At x = 0, z = 0,

∂z =0 ∂x

(3)

d3 z ∂ z = 0, =0 2 dx3 ∂x

small as possible to facilitate the FIB drilling in the followed experiments. Considering all these constraints, the objective function is described as,

 3      ω Ac   k  i  = 1/ ˛i  − ri  + ˇ  + A ω k i=2

2

At x = L,

Fig. 2. GA-optimization flowchart of the binary code configuration of the cantilever.

(4)

(6)

1

2.3. Cantilever coding optimization

Here, ωi = 2fi is the angular resonance frequency of the ith eigenmode. Symbol ri denotes the target integer frequency ratio. k/k and Ac /A are respectively the relative stiffness alteration after the structural modification and the normalized cutting area, with k and A the stiffness and the top area of the unmodified cantilever. Parameters ˛i (i = 2, 3), ˇ and are the weights of the frequency tuning accuracy, the stiffness alteration and the cutting area, respectively. The weights are normalized with {˛2 , ˛3 , ˇ, } ∈ [0, 1] and ˛2 + ˛3 + ˇ + = 1. A larger weight indicates that the corresponding quantity has a higher priority in optimization. The above objective function is oriented for the applications in higher harmonic AFM imaging and it is employed here for the demonstration purpose. Certainly, other constraints of cantilever stiffness, resonance frequency and quality factor can be also included if necessary. Now, we need to maximize the objective function with proper arrangements of the binary codes. For a m × n matrix, the total amount of different coding is 2mn , which is usually a large number. The computation will be rather costly to find the best configuration via going through all the possible code arrangements. The GA method mimicking the natural evolution is then an alternative approach [35], which is recognized as an effective tool in stochastic global optimization. The GA method does not require a continuum objective function and avoids complex mathematical processes in searching the best-fitted solutions. In designing the coded harmonic cantilevers, the GA operations are performed iteratively on a series of cantilever codes, called as a population, by applying the general principle of the fittest survival. The design flowchart is presented in Fig. 2 and the main procedures are briefly described as follows.

The main objective in optimizing the cantilever is that its two higher eigenmode resonance frequencies are integer multiples of the fundamental one. In addition, the induced stiffness alteration maintains the minimum and the material removal volume is kept as

1) An initial population containing a number of random individuals is generated with each individual a binary-coded matrix representing the current approximation of the cantilever structure. The GA parameters are initialized. The amounts of the total gen-

Substituting Eqs. (2)–(4) into Eq. (1), the characteristic equation for the modal analysis is,

 

 

[M] ı¨ + [K] ı

=0

(5)

Where [M] and [K] are the global mass matrix and the global T stiffness matrix, respectively. {ı} = [ı1 , ı2 , · · ·, ıi ] where ıi = ı0i cos(ωi t + ϕi ) is the displacement vector of the cantilever from the equilibrium position with ı0i the amplitude and ϕi the phase of the ith eigenmode. Using these expressions in Eq. (5), it is clear that the eigenmode frequencies can be computed by solving the equation [K] − ω2 [M] = 0. To obtain the global stiffness matrix [K] and the global mass matrix [M], the cantilever is discretized as a set of four node rectangular Mindlin plate elements in FEA [34]. The element mass matrix [M]e and the element stiffness matrix [K]e of a Mindlin plate should be solved firstly and the details can be referred to literature [34]. Then, these element matrices are integrated to construct the global mass matrix [M] and the global stiffness matrix [K]. Subsequently, each eigenmode frequency is numerically obtained. For a better approximation of the realistic AFM cantilever, FEA simulations could be more accurate than analytical simplifications. With the calculated eigenmode frequencies and stiffness, the properties of the cantilevers having different coding structures are evaluated to ensure whether the pre-assigned design objectives are reached or not.

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Fig. 3. Higher harmonic amplitude images of a PS/LDPE blend with the corresponding amplitude histograms. The marked n in each diagram denotes the harmonic order.

2)

3)

4)

5)

6)

7)

erations and the individuals in each generation are set to 50 and 20, respectively. Connectivity evaluation is performed on each individual to adjust unreasonable local codes as schematically illustrated in Fig. 1. The individual coding binary matrix is exported to construct the cantilever’s structural model accordingly and FEA-computation is carried out to obtain the eigenfrequencies and the spring constant of the generated cantilever. The fitness of every individual is evaluated quantitatively by using Eq. (6). Conventional roulette wheel selection is applied on the individuals, which means the code matrices that have better fitness will have higher probabilities to pass on their genes to the next generation. The elite mechanism is adopted to directly select the best individuals in the previous generation. New individuals are generated by applying crossover and mutation operations. In GA processes, crossing the individuals with higher fitness values is likely to create new individuals better than either parent and mutation can maintain diversity within the population and inhibit premature convergence. A crossover rate of 0.8 and a mutation rate of 0.1 are applied, which are typically used in GA optimizations [36]. With the newly obtained individuals of the next generation, the above steps 2)-5) are repeated until a predefined GA generation or a specified convergence criterion is reached. The individuals with the best fitness, which represent the most favorable cantilever structures, are adopted for FIB fabrication and further experimental characterizations.

3. Results and discussion The harmonic cantilever design in this work is based on a kind of rectangular silicon cantilevers (ContAl-G, Budget Sensors). First, we need to determine the optimization objectives mainly the target integer ratios r2 and r3 since other parameters such as k and A are determined for a specified cantilever. For this purpose, we performed harmonic imaging by using the original unmodified cantilever. The experiments were carried out on a commercial AFM (MFP-3D Origin, Asylum Research). The sample was a PS/LDPE blend spin coated on a silicon substrate (Bruker, Santa Barbara CA), which was consisted of PS with phase separated LDPE islands. The spring constant of the cantilever was calibrated to be 0.39 N/m and the free resonance frequency of the first eigenmode was 14.46 kHz. In harmonic AFM imaging, the free amplitude A0 was about 178.7 nm and the feedback set-point for tapping was approximately 0.72 × A0 . The 2nd to 7th harmonic amplitude images together with the histograms are presented in Fig. 3. From the results, the harmonic amplitude difference between the two polymer phases can be hardly discriminated from the 2nd to 4th harmonic images while the amplitude differences are respectively 65, 525 and 350 pm from the 5th to 7th harmonics. It is obvious that the 6th harmonic image has the maximum contrast and the largest amplitude difference. For the used rectangular cantilevers, its second and third eigenfrequencies are about 6.3ω0 and 17.8ω0 . They are near the 6th and 18th harmonics, respectively. Therefore, the 6th and 18th harmonic amplitudes are much stronger than other orders. In the following cantilever’s structural optimization, it is

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Fig. 4. Typical trace of the individual fitness during the GA iteration. The inset shows an optimized harmonic cantilevers satisfying the multiple requirements. Table 1 Frequencies and stiffnesses of the cantilevers before and after structural optimization. Cantilever #1

f1 (kHz) f2 (kHz) f3 (kHz) f2 /f1 f3 /f1 k (N/m) a b

Cantilever #2

ORGa

FEA

MODb

ORG

FEA

MOD

14.46 90.92 255.03 6.29 17.64 0.39

12.78 77.03 229.93 6.03 17.99 0.23

12.85 77.63 232.33 6.04 18.08 0.21

13.84 87.81 246.37 6.34 17.80 0.26

14.05 83.88 238.97 5.97 17.01 0.22

14.10 83.93 240.76 5.95 17.08 0.24

ORG denotes the experimentally measured values of the original cantilever. MOD denotes the measured values of the cantilever after modification.

more effective to set the target eigenfrequency ratios to r2 = 6 and r3 = 18. Hereafter, we call such a harmonic cantilever as cantilever #1. It should be mentioned that other frequency ratios can be also applied. For demonstration, we set the ratios to r2 = 6 and r3 = 17 for additional optimization and the resulted cantilever is called as cantilever #2. A typical trace in optimizing the harmonic cantilever is shown in Fig. 4. The original rectangular cantilever (ContAl-G, Budget Sensors) has an effective dimension for coding L×W×H of 450 ␮m × 50 ␮m × 2 ␮m. These parameters are determined in accordance with the manufacturer’s specifications. The cantilever is then 45 × 5 coded with each coding element having a dimension of 10 ␮m × 10 ␮m × 2 ␮m. The weight constants ˛2 , ˛3 , ˇ and in the objective function are set to 0.35, 0.35, 0.20 and 0.10, respectively. The main material parameters used in the FEA calculations are density of 2329 kg/m3 , elastic modulus of 170 GPa and Poisson’s ratio of 0.28 for silicon. As can be seen from Fig. 4, the best fitness of each generation increases along with the GA iteration indicating a much better structure to satisfy the assigned properties. In addition, multiple outputs can be obtained from the last several generations. Here, only one resulted optimal cantilever #1 with the best fitness is illustrated in the inset. The FEA-computed eigenfrequencies and stiffness after structural modifications are summarized in Table 1. For other optimization objectives (e.g. in optimizing cantilever #2), the typical evolution fitness and the optimal results are presented in the Supplementary Material (see Fig. S1). Besides the rectangular drilling element, other element shapes are also applicable and the binary coding method is able to deal with arbitrary cantilever geometries in principle. For example, the optimization results of harmonic AFM cantilevers with circular and diamond drilling elements are demonstrated in the Supplementary Material (see note 2 and Table S1). In all cases, it can be found that the obtained opti-

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mal cantilever structures well satisfy the design objectives. The frequency ratios are tuned very close to the assigned integers and the stiffness changes are kept rather small. The proposed GA-based binary coding method has several remarkable advantages. First, the design process is automatic and it does not require the sweep of any dimensional and positional parameters to find the suitable ones. Second, cantilever characteristics become closer to meet the multiple objectives along with the GA iteration, which means that a much better structure than the original one can be always obtained. Third, the design is flexible and it can handle cantilevers as well as coding elements with different shapes. Last, a set of structures with the same level of fitness can be simultaneously acquired. After the structural optimization, FIB milling (FEI Helios, Nanolab 650) was applied on several fresh ContAl-G cantilevers to fabricate the optimal binary-coded elements. The coding structures have already been presented in the inset of Fig. 4 and the Supplementary Material (see Fig. S1). In processing, the ion beam energy and the current were respectively 30 keV and 0.1 nA. The scanning electron microscopy (SEM) images of the resulted harmonic cantilevers are shown in Figs. 5(a) and (c). The first three resonance frequencies of the cantilevers were measured before and after the FIB processing, and their spring constants were determined by using the thermal calibration method. The obtained free oscillation spectra of the cantilevers before and after the structural modification are shown in Figs. 5(b) and (d), with the resonance frequencies and stiffnesses being listed in Table 1. It can be seen that the second and third eigenmodes have been shifted to align precisely with the 6th and 18th higher harmonics (or 17th higher harmonics), respectively. Fabrication errors emerge in both the element dimension and shape. However, the accuracy of frequency tuning remains satisfactory. FEA simulations have been further performed to analyze the possible influence of dimensional and positional errors of the cutting elements on the frequency tuning accuracy. A series of cantilevers based on the coding of cantilever #1 were generated. The size and position of the cutting elements were set to have errors in a range from −5 ␮m to 5 ␮m as compared with the assigned values. The corresponding eigenfrequency ratios were analyzed. The typical results are shown in Fig. 6. Assuming that the deviation of the frequency ratio from the target integer within a range from −0.1 to 0.1 is applicable, the allowed dimensional error is in a range from −0.7 ␮m to 1.5 ␮m as illustrated in the shaded region (Fig. 6(a)). Similarly, the permitted positional error is from −1.2 ␮m to 1.2 ␮m (see Fig. 6(b)). The FIB accuracy is far smaller than the maximum allowable fabrication error. Furthermore, such a tolerance in the fabrication error should be also reachable by conventional photolithography. In general, the target frequency ratios do not deviate much from the pre-assigned integers when a slight fabrication error no larger than approximately 1 ␮m presents. Additionally, the deviation can be eliminated by adopting more accurate FEA models in design based on practical SEM measurements. To ascertain the enhanced functional performances of the optimized cantilever, higher harmonic AFM imaging of the cantilever #1 at the 6th and 18th harmonics were performed. The typical results are demonstrated in Fig. 7 and the harmonic imaging results of cantilever #2 are shown in the Supplementary Material (see Fig. S2). The free amplitude and the feedback set-point in the experiments were the same as those presented in Fig. 3. The amplitude images of the 6th and 18th harmonics are shown in Figs. 7(a) and (b) for the before modification case and in Figs. 7(e) and (f) for the after modification one. Moreover, quantitative analyses were made by taking the histograms of the amplitude images, see Figs. 7(c) and (g) for the 6th harmonic and Figs. 7(d) and (h) for the 18th harmonic. Tapping on the softer LDPE (E LDPE ≈ 100 MPa) usually leads to much smaller harmonic amplitudes than on the stiffer PS (EPS ≈ 2 GPa). From the amplitude histograms in Figs. 7(c) and (g), we can evalu-

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Fig. 5. Experimental frequency spectra of the binary-coded harmonic cantilevers. (a) and (c) are the SEM images of the fabricated harmonic cantilevers #1 and #2, respectively. The scale bar is 100 ␮m. (b) and (d) are frequency spectra of the two AFM cantilevers before and after the FIB processing. The arrows guide the first three flexural resonance frequencies.

Fig. 6. Influence of (a) dimensional and (b) positional errors of the cutting elements on frequency tuning accuracy. Symbols x and y in the legend denote the positional errors along the longitudinal direction and the beam width direction, respectively.

ate that the 6th harmonic amplitudes on the LDPE and PS domains are respectively 1.33 ± 0.07 nm and 1.88 ± 0.04 nm when the original cantilever is employed. The harmonic amplitudes increase to respectively 2.00 ± 0.06 nm and 3.50 ± 0.03 nm when the modified cantilever is adopted. It is quite evident that the amplitude difference between the two materials has a 2.7-fold increment. The harmonic amplitude also increases obviously. The 18th harmonic amplitude differences before and after the cantilever modification are 0.21 nm and 0.34 nm, respectively. Thus, the amplitude difference has almost a 1.6-fold enhancement. Using the optimized cantilevers, the harmonic signals and the sensitivities to the local mechanic properties have significant enhancement as expected. Other processed cantilevers demonstrate similar results. Theoretical calculations based on modelling the cantilever dynamics were additionally performed [25]. Simulations verify that the target harmonic amplitudes have obvious improvement by using the optimized harmonic cantilever, in reasonable agreement with the experimental observations (see Fig. S3 and note 3 in the Supple-

mentary Material). When the target harmonics are aligned with the vibration eigenmodes, the sensitivity of the corresponding harmonic amplitude to the sample’s local elasticity is enhanced besides amplitude increment as determined from the amplitude-modulus relation. These theoretical modelling results can well interpret the experimental enhancement of harmonic amplitude difference between on PS and on LDPE. It is worth mentioning that all the experimental settings in harmonic AFM imaging before and after optimization, including set-point and free amplitude, are kept the same for comparison. The integration of the binary-coded elements does not have significant influence on the optical lever sensitivity. The settings are the optimal parameters for the harmonic imaging using the original cantilever, but they are not the optimal parameters for the processed harmonic cantilever. By further adjusting the parameter settings in harmonic AFM imaging, even much better amplitude difference can be achieved. In addition, all the harmonic amplitudes are distinguishable against the noise floor, which is estimated to be approximately 10 pm.

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Fig. 7. Harmonic imaging of a PS/LDPE blend using the cantilever #1 before and after the optimization. (a)(e) The 6th harmonic amplitude images acquired with the cantilever before and after the structure modification, respectively. (c)(g) Amplitude histograms of (a) and (e). (b)(f) The 18th harmonic amplitude images acquired with the original cantilever and the modified one. (d)(h) Amplitude histograms of (b) and (f).

4. Conclusion In summary, a binary coding scheme of cantilevers was proposed to tune their dynamic characteristics flexibly. Combining GA-guided code optimization and FEA-aided computations of dynamic characteristics, many cantilever structures with wellarranged micro-holes can be designed and optimized. As a proof-of-concept, optimization of multi-harmonic AFM cantilevers was demonstrated to have the capability of satisfying multiple objectives. The optimized cantilevers can meet the demands that the ratios of higher eigenfrequencies to the fundamental resonance to be integers together with minimum stiffness alteration and material removing. The resulted structures were acquired via FIB fabrications in this work and mass production of the optimized cantilevers could be also possible by using conventional microfabrication techniques. Spectroscopy characterization and harmonic AFM imaging on a PS/LDPE copolymer verified the effectiveness of the fabricated harmonic cantilevers. The experimental results show

that the amplitude differences between PS and LDPE domains are enhanced by at least 2.7-fold for the 6th harmonic and 1.6-fold for the 18th harmonic. The sensitivities of the target harmonics to the sample’s local modulus are therefore significantly improved. The main advantages of the proposed approach include flexible tuning of various dynamic characteristics by appropriate code arrangements, fully automatic optimization and multiple resulting optimal structures. Owing to all these characteristics, such a binary-coding method is supposed to have broad applications in numerous cantilever-based sensors and also dynamic AFMs.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 51675504). We acknowledge the USTC Center for Micro- and Nanoscale Research and Fabrication for technical support in FIB fabrication of the binary-coded harmonic cantilevers.

Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.sna.2019. 111668.

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Biographies Yaoping Hou received her B.E. degree in Measurement and Control Technology and Instrument from Hefei University of Technology, in 2016. Thereafter, she joined the Department of Precision Machinery and Precision Instrumentation at University of Science and Technology of China. Her research interests include harmonic atomic force microscopy and cantilever-based sensing. Chengfu Ma received his Ph.D. degree in 2017 from the Department of Precision Machinery and Precision Instrumentation at University of Science and Technology of China. He is currently doing postdoctoral research at the same university. His research interests are focused on nanomechanical characterization and subsurface imaging by atomic force microscopy. Wenting Wang received his B.E. degree in Measurement and Control Technology and Instrument from Anhui University, in 2017. After that, he joined the Department of Precision Machinery and Precision Instrumentation at University of Science and Technology of China as a Ph.D. student. His research interests include nanoscale subsurface imaging methods and contact resonance atomic force microscopy. Yuhang Chen received his Ph.D. degree in 2005 from University of Science and Technology of China, Hefei, China. He is currently an associate professor with the Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China. His research interests include advanced atomic force microscopy, micro-and-nano sensors, and nano-optics.