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Vibration analysis of dynamic mode scanning thermal microscope nanomachining probe
Results in Physics 13 (2019) 102164
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Vibration analysis of dynamic mode scanning thermal microscope nanomachining probe Mohammad Sohrabia, Kaveh E. Torkanpourib, a b
T
⁎
Department of Mechatronic Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran Department of Mechanical Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Scanning thermal microscope Base excitation Nanomachining depth Shifted resonance frequency Probe response amplitude Surface finish
Here dynamic mode Scanning Thermal Microscope (SThM) nanomachining technique is presented. The hypothesis states that “Excitation on the base can control the depth of nanomachining process by SThM probe”. The mathematical model is determined based on the Euler-Bernoulli beam. Partial differential equation of motion is solved by separating variables for linear equations and assumed modes method for nonlinear equations. Thermal vibrations is associated with a temperature dependent axial force. Constant, linear and quadratic temperature distribution over the probe are investigated while the probe is harmonically excited on the base at its approximated fundamental resonance frequency. The shifted resonance frequencies and amplitude of response in the excited frequency and shifted resonance frequencies are determined. The results indicate that temperature difference shifts the resonance frequencies, and its influence on vibration amplitudes are significant. It is explained that the effect of temperature distribution function is significant on resonance frequencies shift but negligible on amplitudes. It is showed, the amplitude is decreased in all frequencies with increasing the temperature difference. Finally it is demonstrated that with adding a known excitation on the base the depth of nanomachining process can be controlled. It is declared increasing the temperature improves the quality of final nanomachined surface.
Introduction Scanning Thermal Microscope (SThM) is a type of scanning probe microscope with different applications in nanoscale such as thermal imaging [1–5], conductivity imaging [6–8], measuring: thermal diffusivity [9,10], thermal resistant [11], thermal transport [12] and also nanomachining [13]. Initial SThM was conducted by Williams and Wickramasinghe [14] where a new high resolution profile-meter was introduced. The presented technique was based on non-contact technique where the interaction forces could ignored. By this the proximity between the sample and the probe was studied/measured. In order to overcome the limitations of the initial SThM, use of an Atomic Force Microscope (AFM) cantilever consisting of a pair of non-homogeneous metal wires, was proposed. Thus the quantitative thermal conductivity imaging at the material surface was evaluated [15]. Nanomachining is an important aspect in nanotechnology. Different methods were used for this achievement including electrochemical [16], ultrasonic [17], mechanical [18,19] techniques and etc. [20]. Some researchers have used the SThM based technique for nanomachining of polymer materials. For this based on thermo-mechanical
⁎
lithography the researchers used a warm probe and scratched a linear patterns of 40 nm in wide and about 2 μm in length on a thin film of poly methyl methacrylate in non-contact regime and in static mode [13]. Static mode here states on that there is no dynamic excitation on SThM and thus there are some piezoelectric layers that just control the coordinate of probe without any wanted vibration in cantilever. In this technique the point lithography (something like nano-drilling) is more accurate than line scratching. But due to unwanted vibration generated from the thermal variation between probe and sample, depth of nanomachining could not be controlled/predict in process [13,21–23]. This unwanted vibration was named thermal vibration that also affects the quality of nanomachining final surface [21–23]. To study this phenomena, some research on SThM probe were conducted, in which the effect of thermal vibration was modeled. The effect of thermal variation was considered as the axial force that depends on the distribution of temperature on the probe. As the SThM for nanomachining used in static mode, thus a free vibration of SThM with initial conditions was studied. It was showed, thermal variation will shift the resonance frequency of SThM [21–23] and thus will affect the final surface of nanomachined sample. Obviously, the response of free vibration is
Corresponding author. E-mail address: [email protected] (K.E. Torkanpouri).
https://doi.org/10.1016/j.rinp.2019.102164 Received 26 October 2018; Received in revised form 9 February 2019; Accepted 27 February 2019 Available online 28 February 2019 2211-3797/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
separation method the mode shapes of linear model is solved analytically. Accordingly, based on the assume mode method and assuming enough number of modes, the nonlinear PDEs of SThM are solved. In order to obtain a high quality nanomachining, it is essential to preestimate the shifted resonance frequency of the probe and its amplitude for machining depth control. Thus, the shifted resonance frequencies and amplitude of response in base excitation and shifted resonance frequencies are investigated. Results are validated and discussed. Finally the defined hypothesis is confirmed theoretically.
depended on the amount of initial conditions (ICs). These ICs are the source of unwanted vibration. The ICs were assumed constant in [21–23], however these ICs are randomly depended to sample specifications. The unwanted vibration in this technique is the sole vibration of the probe. Regarding to the physical limitations for nano-scale, applying controlled initial conditions on the tip/probe is not a simple target for today’s technology and so the amplitude of probe and thus the depth of nanomachining is un-controllable for operator in static mode. From other side presciently controlled nanomachining is an important vision for nanotechnology. Here a techniqueally possible idea is proposed to control the depth and final surface of nanomachined sample using base excitation. The base excitation is a well-known technique in AFMs [24,25] that are applied to base of probe by piezoelectric layers harmonically [26,27]. The base excitation is highly controllable both in frequency and amplitude. Currently, more complicate excitations either on base or sample have been investigated [26,28–31]. Adding a set point and also excitation to the base of SThM are caused to reduce the scale of unwanted/free vibration in comparison with the tip’s total displacement. The configuration of static mode nanomachining technique [13] and dynamic mode nanomachining technique (that is proposed in current paper) and schematic of final surfaces after nanomachining are figured in 1. As it is seen in Fig. 1a the final surface of nanomachined area because of unwanted thermal vibration is not controlled. Temperature variation causes changes in resonance frequencies [22,23] and subsequently on the amplitude. Increasing the frequency decrease the distance between hills and fluctuating of tip amplitude will generate disturbances in depth of nanomachined area. The hypothesis of current research states that “Excitation on the base can predict/control the depth of nanomachining process by SThM probe”. It is showed schematically in Fig. 1b that adding an excitation on the base will cause controlled frequency (or equal distance between hills) and amplitude (or more uniform nanomachining depth). Consequently, in this study, assuming a harmonic excitation in the base the mathematical model of dynamic mode nanomachining SThM probe is determined with extended Hamilton’s principle. The temperature distribution along the probe is assumed as constant, linear, and quadratic functions. To solve the mathematical model, using the variable
Mathematical model A SThM probe underlying the harmonic excitation at the base in a non-contact mode and far away from the sample surface is studied, in fact it is assumed there is always a secured distance between tip and sample so neglecting the vertical intermolecular interaction force do not affects dynamics of probe [22,23]. The probe assumed to have a rectangular, uniform section (wide b and height h) with length L. The axial force is dependent on the temperature distribution; i.e. the temperature changes causes this force along the probe [23]. The Schematic of problem is figured in 2. For mathematical modeling, the Euler-Bernoulli beam theory assumptions and extended Hamilton's principle [32] are used. It should be noted it was showed in the absence of large interaction forces the EulerBernoulli beam model have enough accuracy [25,33] and it is not needed to use some advanced beam theories like Timoshenko beam. t2
δ ∫ (K − U + W nc ). dt = 0.
(1)
t1 nc
here, K is kinetic energy, U is potential energy, W is the work done by non-conservative forces, t1 and t2 are initial and final times, respectively. The kinetic energy of the probe is due to the vibration of the system as well as the rigid body movement of the probe:
K=
∫0
L
ρA ⎛ ∂y (x , t ) dhb ⎞2 dx . + 2 ⎝ ∂t dt ⎠
(2)
where, ρ is the probe specific mass, A is area of probe, hb is base
Fig. 1. The schematic of nanomachining SThM (a) Static mode (with unwanted vibration) [13] (b) Dynamic mode (with controlled vibration). 2
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Fig. 2. Schematic configuration of dynamic mode nanomachining SThM probe and defined relative displacement of probe.
will be used as assumed modes shapes for non-linear PDE), the separation of variables is used. To solve these equations analytically, the excitation of the base should be ignored and for linearizing θ(x) = 1 is assumed that means a constant temperature distribution over probe.
excitation and y (x , t ) is the relative displacement of the SThM probe relative to the base as showed in Fig. 2. The tip mass is ignored in mathematical model as it is not impact the hypothesis of current manuscript. The potential energy resulting from the bending of the probe is obtained in the form of Eq. (3).
U=
∫0
L
ρA
2
EI ⎛ ∂2y (x , t ) ⎞ . dx . 2 ⎝ ∂x 2 ⎠ ⎜
=
∫0
L
(3)
The constant value for P(x) is defined as P. Defining the separated variables Y(x) as shape modes and q(t) as generalized coordinates
P (x ) ⎛ ∂y (x , t ) ⎞2 . dx . 2 ⎝ ∂x ⎠
y (x , t ) = Y (x ) q (t ) .
P (x ) = −αEAΔTθ (x ).
(4)
ρA (5)
(7)
∂y (0, t ) = 0. ∂x
(8)
q¨ (t )
P Y″(x ) + EI Y (x )
βn4 ρA
⎜
(15)
= ωn .
(16)
−P ±
P 2 + 4βn4 EI 2EI
.
(17)
Y(x) is calculated as
Y (x ) = Y01 e s1 x + Y02 e s2 x + Y03 e s3 x + Y04 e s4 x .
⎟
(10)
(18)
where Y01 … Y04 are contacts that should be find from boundary conditions of problem. Using the Euler equation, the imaginary terms are transformed into harmonic terms.
It should remarked, as y(x,t) is the relative displacement of probe to the base, hb is absent in boundary conditions. In Eq. (6) ρA (d 2hb/ dt 2) is equivalent to fictitious force per unit length originated in the frame of reference attached to the base. Eq. (10) states that the force P(x) generated from ΔT will cause a vertical force at the tip. Mathematically the first term of Eq. (10) is the component of P(L) in vertical direction that will show itself when there is a slope at the probe. The excitation of the base is assumed to be a simple harmonic motion in the form of Eq. (11).
hb (t ) = y° cosωt .
= βn4 .
From Eq. (15), the roots of the characteristic equation are obtained in the below form
(9)
3 ⎛ ∂y (L, t ) ⎞ P (L) + EI ⎛ ∂ y (L, t ) ⎞ = 0. 3 ∂ x ⎝ ⎠ ⎠ ⎝ ∂x
Y (4) (x ) Y (x )
where βn is a constant. From solution of the time domain, the resonance frequency of the SThM probe is obtained for linear PDE and homogenous equation (free vibration), with the resonance frequency being shown with ωn where n denotes the mode number.
s1,2,3,4 = ±
∂2y (L, t ) = 0. ∂x 2
(14)
ρA q (t ) = −βn4 ,
(6)
y (0, t ) = 0.
q¨ (t ) Y (4) (x ) Y ″ (x ) + EI +P = 0. q (t ) Y (x ) Y (x )
By separating the spatial and temporal domains in Eqs. (14) and (15) are obtained.
above α is the coefficient of thermal expansion, ΔT is the temperature difference between sample and the tip and θ(x) is defined as the temperature distribution function. Because of conductive heat transfer through probe length, temperature distribution can be a function of × while the ΔT is still constant. θ(x) shows how the temperature is distributed along the length of the probe. By substituting Eqs. (2), (3) and (4) in Eq. (1) and applying the variation calculation principle, the PDE of motion and its boundary conditions are determined as below after some mathematical efforts:
∂2y (x , t ) ∂y (x , t ) ⎞ ∂ 4y (x , t ) ∂ ⎛ d 2h + ρA 2b + P (x ) = 0. + EI 2 ∂t ∂x ⎝ dt ∂x ⎠ ∂x 4
(13)
and substituting Eq. (13) in Eq. (12) and doing some simple mathematical operations, Eq. (14) is determined:
where the P(x) is determined [23] as
ρA
(12)
⎟
where E is elastic modulus and I is the second moment of the probe cross section. The work resulting from the axial force due to the temperature distribution is calculated in the form below.
W nc
∂2y (x , t ) ∂2y (x , t ) ∂ 4y (x . t ) +P + EI = 0. ∂x 2 ∂x 4 ∂t 2
Y (x ) = Y01 e s1 x + Y02 e s2 x + Y03sin(s3 x ) + Y04 cos(s4 x ).
(19)
By introducing Eqs. (7)–(10) into Eq. (19), Eqs. (20)–(23) must be satisfied:
(11)
in which y0 is the base excitation amplitude and ω is the base excitation frequency, which is assumed to be approximately equal to the resonance frequency of the first mode in the absence of thermal loading (ω = 0.8ω1 @ ΔT = 0).
Y01 + Y02 + Y04 = 0.
(20)
s1 Y01 + s2 Y02 + s3 Y03 = 0.
(21)
s12 Y01 e s1 L + s22 Y02 e s2 L − s32 Y03 sin(s3 L) − s42 Y04 cos(s4 L) = 0.
(22)
EI [s13 Y01 e s1 L + s23 Y02 e s2 L − s33 Y03 cos(s3 L) + s43 Y04 sin(s4 L)] + P [s1 Y01 e s1 L + s2 Y02 e s2 L + +s3 Y03 cos(s3 L) − s4 Y04 sin(s4 L)] = 0. (23)
Separating variables
Non-trivial solution of Eqs. (20)–(23) leads to
|Λ| = 0.
In order to determine the frequency equation and mode shapes (that 3
(24)
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Λ is matrix of multipliers of Eqs. (20)–(23). Eq. (24) should be solved numerically to find ωn. The constants Y01 … Y04 are determined for each ωn. Thus, the shifted resonance frequencies and mode shapes of SThM probe for θ(x) = 1 in free vibration (static mode) are achieved.
Table 1 Specification of SThM nanomachining probe adopted from [23] except noted.
Assumed modes method To solve the governing equations for nonlinear distribution of temperature and also for the case of base excitation (dynamic mode), the Assumed Modes Method is used. The Assumed Modes Method is closely related to the Rayleigh–Ritz method. In the assumed modes method, the solution of the vibration problem of the continuous system is assumed in the form of a series composed of a linear combination of admissible functions which are functions of the spatial coordinates, multiplied by time-dependent generalized coordinates [32,34]. In the previous part, the mode shape equations were obtained for a special case witch satisfy the boundary conditions and thus is an admissible function. These mode shapes are assumed as mode functions of nonlinear problem.
∑
Yi (x ) qi (t ).
m is an integer and it is clear increasing m causes better estimation. Yi(x) are the mode functions and qi(t) are the generalized coordinates of the problem. Introducing Eqs. (5) and (11) into Eq. (6) leads to:
ρA
∂2y (x , t ) ∂ 4y (x , t ) ∂y (x , t ) ⎞ ∂ + EI − αEAΔT ⎛θ (x ) = A0 cos ωt . 2 ∂t ∂x 4 ∂x ⎝ ∂x ⎠ (26)
where A0 is defined as
A0 = y0 ρAω2 .
(27)
By applying the Eqs. (25) and (26), and after some mathematical computations Eq. (28) is determined. The terms t and x in functions are discarded for compactness of equations. m
m
m
m
⎞ ⎛ ρA ∑ Yi q¨i + EI ∑ Yi(4) qi − αEAΔT ⎜θ′ ∑ Yi′ qi + θ ∑ Yi″ qi⎟ = A° cos ωt . i=1 i=1 i=1 ⎠ ⎝ i=1 (28) where dot and prime denotes partial derivative with respect to t and x respectively. By multiplying Yp to Eq. (28), integrating over length of probe and applying orthogonality condition of shape modes, Eq. (29) is obtained. m
L
L
∫ ρAYp2 q¨p . dx + ∫ EIYp ∑ Yi(4) qi . dx 0
0
i=1 m
m
L ⎞ ⎛ − αEAΔT ∫ Yp ⎜θ′ ∑ Yi′ qi + θ ∑ Yi″ qi⎟. dx = ∫ Yp A° . dx cos ωt p 0 0 i=1 ⎠ ⎝ i=1 (29) = 1, 2, ...,m L
Size
Unit
L E b h ρ A I α *ω *y0
Probe length Elastic Module Probe width Probe thickness Probe density Probe area Second moment of area Thermal expansion coefficient Base excitation frequency Base excitation amplitude
120 170 25 5 2300 125 260.417 2.6e−6 ω ≃ 0.8ω1 20
µm Gpa µm µm Kg.m−3 µm2 µm4 1/°C Hz nm
difference is assumed from 0 to 800°c . To investigate the effect of nonlinear distribution of temperature, Eq. (29) is expanded. The sensitivity of number of modes are studied. The difference between results of 4 modes and 3 modes assumptions is investigated. The order of change by adding 4th mode is in 1e−9 nm order that may be discarded for applied science, thus 3 modes assumption has enough accuracy. Therefore three first mode shapes determined from Eqs. (16)–(23) are assumed as mode functions in Eq. (29). The non-linear coupled ODEs of motion are solved using 4th order Runge-Kutta method by coding in Matlab. The achieved time domain response is transformed to frequency response by Fast Fourier Transform (FFT). The frequency accuracy is assumed 5 Hz for FFT (small enough to prevent aliasing effect), then the shifted resonance frequencies and amplitudes of response in shifted resonance frequencies and base excitation frequency are extracted for constant, linear and quadratic distributions. The details of temperature distribution function for three mentioned model are tabulated in Table 2. The obtained normalized shifted resonance frequency for constant distribution function are compared with results presented in [23] in Figs. 3 and 4. The normalization is done by dividing shifted resonance frequencies to resonance frequency of the first mode in the absence of thermal loading (Here 482.2 KHz). It is noted that the results in [23] were presented for free unwanted thermal vibration that has been solved by Galerkin method. It is clear that resonance frequency is same in free and forced vibration, thus the results in [23] are used for verification of presented. Figs. 3 and 4 indicate an increase in the resonance frequencies of the probe, depended on ΔT. The results of Timoshenko beam [23] is a little higher especially in larger ΔT, but the difference is negligible. It is demonstrated that in the absence of large interaction forces (in noncontact regime) the Euler-Bernoulli beam model has enough accuracy [25,33] and it is not essential to use some advanced beam theories like Timoshenko beam for SThM. The mode shapes depended on the amount of ΔT are figured in 5 and 6 for first and third mode respectively for θ(x) = 1. It is seen in Fig. 5, as the temperature difference rises, due to increasing of axial force, the mode shapes are changed and the amplitude of probe is generally reduced at end. Fig. 6 shows that the position of nodes are also shifted. The position of nodes are important for laser’s initial set point in AFM based SThM. The shifted resonance frequencies for constant, linear and quadratic
(25)
i=1
Description
*Applied by the authors.
m
y (x , t ) =
Sym.
Eq. (29) can be expanded to a system of m non-linear second order coupled ordinary differential equations where m is the number of estimated mode functions in Eq. (25). Eqs. (29) should be solved simultaneously for qi. With known values of Yi from Eqs. (19)–(23) the vibration response of dynamic mode nanomachining SThM probe is achieved in any point of probe. As the nanomachining is performed at the end of probe, thus amplitude of response at x = L is studied in next section.
Table 2 Details of constant, linear and quadratic temperature distribution function.
Discussion and results The governing equations are solved numerically for a case study. The specifications of SThM nanomachining probe are tabulated in Table 1 that are adopted from [23] except noted. The temperature 4
Models
θ(x)
Constant Linear Quadratic
1 0.2(x / L) + 0.8
0.15(x / L)2 + 0.05(x / L) + 0.8
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Fig. 6. Third mode shapes for different temperature difference for θ(x) = 1.
Fig. 3. Comparison between normalized shifted resonance frequencies of current study vs. [23] for θ(x) = 1.
Fig. 7. The first shifted resonance frequency for constant, linear and quadratic distributions.
Fig. 4. Comparison between normalized shifted resonance frequencies current study vs. [23] θ(x) = 1.
effect of θ(x) is also significant in frequency shifts. The constant distributions has the highest frequency increase, and quadratic has the lowest frequency shift. Therefore knowing the accurate temperature distribution of the probe is important for frequency estimation. It opens new visions for study in the behavior of SThM probe if anybody can introduce more realistic estimation for temperature distribution function for presented mathematical model as the used θ(x) in current research are some rough estimates. Fig. 8 shows the normalized amplitude of vibration at end/tip (x = L) in base excitation frequency and Figs. 9 and 10 show the normalized amplitude of response in shifted resonance frequencies of probe for constant, linear, and quadratic distributions. Ae and Aish are defined as normalized amplitude of response in base excitation frequency and ith shifted resonance frequency, respectively. The normalization is done by dividing the amplitude of response in ΔT = 0 in base frequency (Here 432.95 nm). It is stated that, in all cases, with increasing ΔT the normalized amplitudes decrease. These changes are significant both in base excitation and first shifted resonance frequency. As the base is excited in the first resonance frequency range (ω ≃ 0.8ω1), order of magnitude is high between first and second mode (and higher modes). In comparison between temperature distribution function, maximum amplitude is seen in the quadratic distribution. It is because the quadratic model causes lower axial force in probe comparing with constant model. Thus the amplitude is less controlled with temperature difference. In general, the effect of θ(x) can be neglected in amplitude
Fig. 5. First mode shapes for different temperature difference for θ(x) = 1.
distributions are compared in Fig. 7. It is seen in all temperature distribution, the shifted resonance frequencies are increased by increasing of ΔT as seen previously for linear case. It means, by increasing the ΔT the distance between hills decrease and it can improve the quality of final surface (if the depth of nanomachining can be controlled). The 5
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Fig. 8. Normalized amplitude of response in base excitation frequency for constant, linear and quadratic distributions.
Fig. 11. Simulation of nanomachined cross section in ΔT = (a) 200 °C (b) 500 °C (c) 700 °C.
200 nm thickness sample in three different temperatures (200, 500 and 700 °C). The travelling speed of 400 μm/s is assumed for calculation of machining length and set point is assumed as 40 nm. Set point is the distance that is initially set for SThM probe from the sample free surface i.e. it is the equilibrium point of base. It is clear decreasing amount of set point will cause deeper scratch similar to static mode. Fig. 11 demonstrate that the depth of nanomachined area is significantly dependent on ΔT that is controlled by base excitation. If the free surface of sample assumed as datum, Root Mean Square (RMS) value for final surface are calculated 126.7, 55.5 and 38.1 nm in negative direction for ΔT = 200, 500 and 700 °C respectively for the simulated length. The RMS is used due to nonlinear behavior of tip that cause non-smooth nanomachined surface finish. It is clear increasing the travelling speed can improve the final surface. Fig. 12 shows lower peak envelope of tip response. This figure can be used as a rough estimation of final surface of nanomachined area. It is seen the final surface is improved with increasing ΔT. The reason is increasing of shifted resonance frequency of SThM probe with increasing ΔT. It is clear increasing the frequencies of vibrational response causes better surface finish in constant travelling speed. A point should not be ignored, that in current research effect of tip dimension (tip radius) was ignored i.e. it was assumed the nanomachining was
Fig. 9. Normalized amplitude of vibration in first shifted resonance frequency for constant, linear and quadratic distributions.
Fig. 10. Normalized amplitude of vibration in second shifted resonance frequency for constant, linear and quadratic distributions.
(however the expected accuracy of nanomachining should be concerned). Fig. 11 shows the simulation of nanomachined cross section of
Fig. 12. Lower peak envelope of tip response in ΔT = (a) 200 °C (b) 500 °C (c) 700 °C. 6
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Appendix A. Supplementary data
applied with a zero tip radius (point tip). This parameter can also affect the final surface practically. The results state that with adding a known excitation on the base the depth of nanomachining process can be controlled with ΔT that demonstrate the hypothesis of research. It is obvious that controlling the magnitude of ΔT is not a simple effort as it depends on the temperature of sample and generally there isn’t constant temperature distribution on the samples. Thus a control system should be designed (that is not the issue of current paper) to track the magnitude of desired depth by controlling input amplitude of base (y0). It seems the error of control loop may be used for thermal imaging of sample while nanomachining process.
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Conclusion In this research paper, a techniqueally possible idea was proposed to control the depth of nanomachining process with SThM probe. It was named dynamic mode SThM nanomachining technique. Afterwards a mathematical model for dynamic mode SThM nanomachining probe based on Euler-Bernoulli beam was determined. The temperature distribution function was roughly assumed accordance to constant, linear and quadratic functions. The mode shapes were figured for various temperature. The shifted resonance frequencies were verified with literature for some special cases. To find the amplitude of response the assume mode method was used. The PDE of motion was reduced to a system of coupled non-linear ODEs. The equation of motion were solve numerically by coding in Matlab. FFT was applied to find the frequency domain response of non-linear system. The shifted resonance frequencies were figured for temperature difference between 0 and 800 °C in constant, linear and quadratic models. Amplitude of response in base frequency and shifted resonance frequencies were determined at tip location. It was showed in all temperature distribution, the shifted resonance frequencies are increased by increasing of temperature difference. It was explained that the effect of temperature distribution function is significant on resonance frequencies shift but negligible on amplitude. It was showed, the amplitude is decreased in all temperature distribution with increasing the temperature difference. Finally it was stated that with adding a known excitation on the base the depth of nanomachining process can be controlled with temperature that demonstrate the hypothesis of research. Also it was declared increasing the temperature improves the quality of final nanomachined surface for dynamic mode SThM nanomachining technique.
7
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Results in Physics journal homepage: www.elsevier.com/locate/rinp
Vibration analysis of dynamic mode scanning thermal microscope nanomachining probe Mohammad Sohrabia, Kaveh E. Torkanpourib, a b
T
⁎
Department of Mechatronic Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran Department of Mechanical Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Scanning thermal microscope Base excitation Nanomachining depth Shifted resonance frequency Probe response amplitude Surface finish
Here dynamic mode Scanning Thermal Microscope (SThM) nanomachining technique is presented. The hypothesis states that “Excitation on the base can control the depth of nanomachining process by SThM probe”. The mathematical model is determined based on the Euler-Bernoulli beam. Partial differential equation of motion is solved by separating variables for linear equations and assumed modes method for nonlinear equations. Thermal vibrations is associated with a temperature dependent axial force. Constant, linear and quadratic temperature distribution over the probe are investigated while the probe is harmonically excited on the base at its approximated fundamental resonance frequency. The shifted resonance frequencies and amplitude of response in the excited frequency and shifted resonance frequencies are determined. The results indicate that temperature difference shifts the resonance frequencies, and its influence on vibration amplitudes are significant. It is explained that the effect of temperature distribution function is significant on resonance frequencies shift but negligible on amplitudes. It is showed, the amplitude is decreased in all frequencies with increasing the temperature difference. Finally it is demonstrated that with adding a known excitation on the base the depth of nanomachining process can be controlled. It is declared increasing the temperature improves the quality of final nanomachined surface.
Introduction Scanning Thermal Microscope (SThM) is a type of scanning probe microscope with different applications in nanoscale such as thermal imaging [1–5], conductivity imaging [6–8], measuring: thermal diffusivity [9,10], thermal resistant [11], thermal transport [12] and also nanomachining [13]. Initial SThM was conducted by Williams and Wickramasinghe [14] where a new high resolution profile-meter was introduced. The presented technique was based on non-contact technique where the interaction forces could ignored. By this the proximity between the sample and the probe was studied/measured. In order to overcome the limitations of the initial SThM, use of an Atomic Force Microscope (AFM) cantilever consisting of a pair of non-homogeneous metal wires, was proposed. Thus the quantitative thermal conductivity imaging at the material surface was evaluated [15]. Nanomachining is an important aspect in nanotechnology. Different methods were used for this achievement including electrochemical [16], ultrasonic [17], mechanical [18,19] techniques and etc. [20]. Some researchers have used the SThM based technique for nanomachining of polymer materials. For this based on thermo-mechanical
⁎
lithography the researchers used a warm probe and scratched a linear patterns of 40 nm in wide and about 2 μm in length on a thin film of poly methyl methacrylate in non-contact regime and in static mode [13]. Static mode here states on that there is no dynamic excitation on SThM and thus there are some piezoelectric layers that just control the coordinate of probe without any wanted vibration in cantilever. In this technique the point lithography (something like nano-drilling) is more accurate than line scratching. But due to unwanted vibration generated from the thermal variation between probe and sample, depth of nanomachining could not be controlled/predict in process [13,21–23]. This unwanted vibration was named thermal vibration that also affects the quality of nanomachining final surface [21–23]. To study this phenomena, some research on SThM probe were conducted, in which the effect of thermal vibration was modeled. The effect of thermal variation was considered as the axial force that depends on the distribution of temperature on the probe. As the SThM for nanomachining used in static mode, thus a free vibration of SThM with initial conditions was studied. It was showed, thermal variation will shift the resonance frequency of SThM [21–23] and thus will affect the final surface of nanomachined sample. Obviously, the response of free vibration is
Corresponding author. E-mail address: [email protected] (K.E. Torkanpouri).
https://doi.org/10.1016/j.rinp.2019.102164 Received 26 October 2018; Received in revised form 9 February 2019; Accepted 27 February 2019 Available online 28 February 2019 2211-3797/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
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M. Sohrabi and K.E. Torkanpouri
separation method the mode shapes of linear model is solved analytically. Accordingly, based on the assume mode method and assuming enough number of modes, the nonlinear PDEs of SThM are solved. In order to obtain a high quality nanomachining, it is essential to preestimate the shifted resonance frequency of the probe and its amplitude for machining depth control. Thus, the shifted resonance frequencies and amplitude of response in base excitation and shifted resonance frequencies are investigated. Results are validated and discussed. Finally the defined hypothesis is confirmed theoretically.
depended on the amount of initial conditions (ICs). These ICs are the source of unwanted vibration. The ICs were assumed constant in [21–23], however these ICs are randomly depended to sample specifications. The unwanted vibration in this technique is the sole vibration of the probe. Regarding to the physical limitations for nano-scale, applying controlled initial conditions on the tip/probe is not a simple target for today’s technology and so the amplitude of probe and thus the depth of nanomachining is un-controllable for operator in static mode. From other side presciently controlled nanomachining is an important vision for nanotechnology. Here a techniqueally possible idea is proposed to control the depth and final surface of nanomachined sample using base excitation. The base excitation is a well-known technique in AFMs [24,25] that are applied to base of probe by piezoelectric layers harmonically [26,27]. The base excitation is highly controllable both in frequency and amplitude. Currently, more complicate excitations either on base or sample have been investigated [26,28–31]. Adding a set point and also excitation to the base of SThM are caused to reduce the scale of unwanted/free vibration in comparison with the tip’s total displacement. The configuration of static mode nanomachining technique [13] and dynamic mode nanomachining technique (that is proposed in current paper) and schematic of final surfaces after nanomachining are figured in 1. As it is seen in Fig. 1a the final surface of nanomachined area because of unwanted thermal vibration is not controlled. Temperature variation causes changes in resonance frequencies [22,23] and subsequently on the amplitude. Increasing the frequency decrease the distance between hills and fluctuating of tip amplitude will generate disturbances in depth of nanomachined area. The hypothesis of current research states that “Excitation on the base can predict/control the depth of nanomachining process by SThM probe”. It is showed schematically in Fig. 1b that adding an excitation on the base will cause controlled frequency (or equal distance between hills) and amplitude (or more uniform nanomachining depth). Consequently, in this study, assuming a harmonic excitation in the base the mathematical model of dynamic mode nanomachining SThM probe is determined with extended Hamilton’s principle. The temperature distribution along the probe is assumed as constant, linear, and quadratic functions. To solve the mathematical model, using the variable
Mathematical model A SThM probe underlying the harmonic excitation at the base in a non-contact mode and far away from the sample surface is studied, in fact it is assumed there is always a secured distance between tip and sample so neglecting the vertical intermolecular interaction force do not affects dynamics of probe [22,23]. The probe assumed to have a rectangular, uniform section (wide b and height h) with length L. The axial force is dependent on the temperature distribution; i.e. the temperature changes causes this force along the probe [23]. The Schematic of problem is figured in 2. For mathematical modeling, the Euler-Bernoulli beam theory assumptions and extended Hamilton's principle [32] are used. It should be noted it was showed in the absence of large interaction forces the EulerBernoulli beam model have enough accuracy [25,33] and it is not needed to use some advanced beam theories like Timoshenko beam. t2
δ ∫ (K − U + W nc ). dt = 0.
(1)
t1 nc
here, K is kinetic energy, U is potential energy, W is the work done by non-conservative forces, t1 and t2 are initial and final times, respectively. The kinetic energy of the probe is due to the vibration of the system as well as the rigid body movement of the probe:
K=
∫0
L
ρA ⎛ ∂y (x , t ) dhb ⎞2 dx . + 2 ⎝ ∂t dt ⎠
(2)
where, ρ is the probe specific mass, A is area of probe, hb is base
Fig. 1. The schematic of nanomachining SThM (a) Static mode (with unwanted vibration) [13] (b) Dynamic mode (with controlled vibration). 2
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Fig. 2. Schematic configuration of dynamic mode nanomachining SThM probe and defined relative displacement of probe.
will be used as assumed modes shapes for non-linear PDE), the separation of variables is used. To solve these equations analytically, the excitation of the base should be ignored and for linearizing θ(x) = 1 is assumed that means a constant temperature distribution over probe.
excitation and y (x , t ) is the relative displacement of the SThM probe relative to the base as showed in Fig. 2. The tip mass is ignored in mathematical model as it is not impact the hypothesis of current manuscript. The potential energy resulting from the bending of the probe is obtained in the form of Eq. (3).
U=
∫0
L
ρA
2
EI ⎛ ∂2y (x , t ) ⎞ . dx . 2 ⎝ ∂x 2 ⎠ ⎜
=
∫0
L
(3)
The constant value for P(x) is defined as P. Defining the separated variables Y(x) as shape modes and q(t) as generalized coordinates
P (x ) ⎛ ∂y (x , t ) ⎞2 . dx . 2 ⎝ ∂x ⎠
y (x , t ) = Y (x ) q (t ) .
P (x ) = −αEAΔTθ (x ).
(4)
ρA (5)
(7)
∂y (0, t ) = 0. ∂x
(8)
q¨ (t )
P Y″(x ) + EI Y (x )
βn4 ρA
⎜
(15)
= ωn .
(16)
−P ±
P 2 + 4βn4 EI 2EI
.
(17)
Y(x) is calculated as
Y (x ) = Y01 e s1 x + Y02 e s2 x + Y03 e s3 x + Y04 e s4 x .
⎟
(10)
(18)
where Y01 … Y04 are contacts that should be find from boundary conditions of problem. Using the Euler equation, the imaginary terms are transformed into harmonic terms.
It should remarked, as y(x,t) is the relative displacement of probe to the base, hb is absent in boundary conditions. In Eq. (6) ρA (d 2hb/ dt 2) is equivalent to fictitious force per unit length originated in the frame of reference attached to the base. Eq. (10) states that the force P(x) generated from ΔT will cause a vertical force at the tip. Mathematically the first term of Eq. (10) is the component of P(L) in vertical direction that will show itself when there is a slope at the probe. The excitation of the base is assumed to be a simple harmonic motion in the form of Eq. (11).
hb (t ) = y° cosωt .
= βn4 .
From Eq. (15), the roots of the characteristic equation are obtained in the below form
(9)
3 ⎛ ∂y (L, t ) ⎞ P (L) + EI ⎛ ∂ y (L, t ) ⎞ = 0. 3 ∂ x ⎝ ⎠ ⎠ ⎝ ∂x
Y (4) (x ) Y (x )
where βn is a constant. From solution of the time domain, the resonance frequency of the SThM probe is obtained for linear PDE and homogenous equation (free vibration), with the resonance frequency being shown with ωn where n denotes the mode number.
s1,2,3,4 = ±
∂2y (L, t ) = 0. ∂x 2
(14)
ρA q (t ) = −βn4 ,
(6)
y (0, t ) = 0.
q¨ (t ) Y (4) (x ) Y ″ (x ) + EI +P = 0. q (t ) Y (x ) Y (x )
By separating the spatial and temporal domains in Eqs. (14) and (15) are obtained.
above α is the coefficient of thermal expansion, ΔT is the temperature difference between sample and the tip and θ(x) is defined as the temperature distribution function. Because of conductive heat transfer through probe length, temperature distribution can be a function of × while the ΔT is still constant. θ(x) shows how the temperature is distributed along the length of the probe. By substituting Eqs. (2), (3) and (4) in Eq. (1) and applying the variation calculation principle, the PDE of motion and its boundary conditions are determined as below after some mathematical efforts:
∂2y (x , t ) ∂y (x , t ) ⎞ ∂ 4y (x , t ) ∂ ⎛ d 2h + ρA 2b + P (x ) = 0. + EI 2 ∂t ∂x ⎝ dt ∂x ⎠ ∂x 4
(13)
and substituting Eq. (13) in Eq. (12) and doing some simple mathematical operations, Eq. (14) is determined:
where the P(x) is determined [23] as
ρA
(12)
⎟
where E is elastic modulus and I is the second moment of the probe cross section. The work resulting from the axial force due to the temperature distribution is calculated in the form below.
W nc
∂2y (x , t ) ∂2y (x , t ) ∂ 4y (x . t ) +P + EI = 0. ∂x 2 ∂x 4 ∂t 2
Y (x ) = Y01 e s1 x + Y02 e s2 x + Y03sin(s3 x ) + Y04 cos(s4 x ).
(19)
By introducing Eqs. (7)–(10) into Eq. (19), Eqs. (20)–(23) must be satisfied:
(11)
in which y0 is the base excitation amplitude and ω is the base excitation frequency, which is assumed to be approximately equal to the resonance frequency of the first mode in the absence of thermal loading (ω = 0.8ω1 @ ΔT = 0).
Y01 + Y02 + Y04 = 0.
(20)
s1 Y01 + s2 Y02 + s3 Y03 = 0.
(21)
s12 Y01 e s1 L + s22 Y02 e s2 L − s32 Y03 sin(s3 L) − s42 Y04 cos(s4 L) = 0.
(22)
EI [s13 Y01 e s1 L + s23 Y02 e s2 L − s33 Y03 cos(s3 L) + s43 Y04 sin(s4 L)] + P [s1 Y01 e s1 L + s2 Y02 e s2 L + +s3 Y03 cos(s3 L) − s4 Y04 sin(s4 L)] = 0. (23)
Separating variables
Non-trivial solution of Eqs. (20)–(23) leads to
|Λ| = 0.
In order to determine the frequency equation and mode shapes (that 3
(24)
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Λ is matrix of multipliers of Eqs. (20)–(23). Eq. (24) should be solved numerically to find ωn. The constants Y01 … Y04 are determined for each ωn. Thus, the shifted resonance frequencies and mode shapes of SThM probe for θ(x) = 1 in free vibration (static mode) are achieved.
Table 1 Specification of SThM nanomachining probe adopted from [23] except noted.
Assumed modes method To solve the governing equations for nonlinear distribution of temperature and also for the case of base excitation (dynamic mode), the Assumed Modes Method is used. The Assumed Modes Method is closely related to the Rayleigh–Ritz method. In the assumed modes method, the solution of the vibration problem of the continuous system is assumed in the form of a series composed of a linear combination of admissible functions which are functions of the spatial coordinates, multiplied by time-dependent generalized coordinates [32,34]. In the previous part, the mode shape equations were obtained for a special case witch satisfy the boundary conditions and thus is an admissible function. These mode shapes are assumed as mode functions of nonlinear problem.
∑
Yi (x ) qi (t ).
m is an integer and it is clear increasing m causes better estimation. Yi(x) are the mode functions and qi(t) are the generalized coordinates of the problem. Introducing Eqs. (5) and (11) into Eq. (6) leads to:
ρA
∂2y (x , t ) ∂ 4y (x , t ) ∂y (x , t ) ⎞ ∂ + EI − αEAΔT ⎛θ (x ) = A0 cos ωt . 2 ∂t ∂x 4 ∂x ⎝ ∂x ⎠ (26)
where A0 is defined as
A0 = y0 ρAω2 .
(27)
By applying the Eqs. (25) and (26), and after some mathematical computations Eq. (28) is determined. The terms t and x in functions are discarded for compactness of equations. m
m
m
m
⎞ ⎛ ρA ∑ Yi q¨i + EI ∑ Yi(4) qi − αEAΔT ⎜θ′ ∑ Yi′ qi + θ ∑ Yi″ qi⎟ = A° cos ωt . i=1 i=1 i=1 ⎠ ⎝ i=1 (28) where dot and prime denotes partial derivative with respect to t and x respectively. By multiplying Yp to Eq. (28), integrating over length of probe and applying orthogonality condition of shape modes, Eq. (29) is obtained. m
L
L
∫ ρAYp2 q¨p . dx + ∫ EIYp ∑ Yi(4) qi . dx 0
0
i=1 m
m
L ⎞ ⎛ − αEAΔT ∫ Yp ⎜θ′ ∑ Yi′ qi + θ ∑ Yi″ qi⎟. dx = ∫ Yp A° . dx cos ωt p 0 0 i=1 ⎠ ⎝ i=1 (29) = 1, 2, ...,m L
Size
Unit
L E b h ρ A I α *ω *y0
Probe length Elastic Module Probe width Probe thickness Probe density Probe area Second moment of area Thermal expansion coefficient Base excitation frequency Base excitation amplitude
120 170 25 5 2300 125 260.417 2.6e−6 ω ≃ 0.8ω1 20
µm Gpa µm µm Kg.m−3 µm2 µm4 1/°C Hz nm
difference is assumed from 0 to 800°c . To investigate the effect of nonlinear distribution of temperature, Eq. (29) is expanded. The sensitivity of number of modes are studied. The difference between results of 4 modes and 3 modes assumptions is investigated. The order of change by adding 4th mode is in 1e−9 nm order that may be discarded for applied science, thus 3 modes assumption has enough accuracy. Therefore three first mode shapes determined from Eqs. (16)–(23) are assumed as mode functions in Eq. (29). The non-linear coupled ODEs of motion are solved using 4th order Runge-Kutta method by coding in Matlab. The achieved time domain response is transformed to frequency response by Fast Fourier Transform (FFT). The frequency accuracy is assumed 5 Hz for FFT (small enough to prevent aliasing effect), then the shifted resonance frequencies and amplitudes of response in shifted resonance frequencies and base excitation frequency are extracted for constant, linear and quadratic distributions. The details of temperature distribution function for three mentioned model are tabulated in Table 2. The obtained normalized shifted resonance frequency for constant distribution function are compared with results presented in [23] in Figs. 3 and 4. The normalization is done by dividing shifted resonance frequencies to resonance frequency of the first mode in the absence of thermal loading (Here 482.2 KHz). It is noted that the results in [23] were presented for free unwanted thermal vibration that has been solved by Galerkin method. It is clear that resonance frequency is same in free and forced vibration, thus the results in [23] are used for verification of presented. Figs. 3 and 4 indicate an increase in the resonance frequencies of the probe, depended on ΔT. The results of Timoshenko beam [23] is a little higher especially in larger ΔT, but the difference is negligible. It is demonstrated that in the absence of large interaction forces (in noncontact regime) the Euler-Bernoulli beam model has enough accuracy [25,33] and it is not essential to use some advanced beam theories like Timoshenko beam for SThM. The mode shapes depended on the amount of ΔT are figured in 5 and 6 for first and third mode respectively for θ(x) = 1. It is seen in Fig. 5, as the temperature difference rises, due to increasing of axial force, the mode shapes are changed and the amplitude of probe is generally reduced at end. Fig. 6 shows that the position of nodes are also shifted. The position of nodes are important for laser’s initial set point in AFM based SThM. The shifted resonance frequencies for constant, linear and quadratic
(25)
i=1
Description
*Applied by the authors.
m
y (x , t ) =
Sym.
Eq. (29) can be expanded to a system of m non-linear second order coupled ordinary differential equations where m is the number of estimated mode functions in Eq. (25). Eqs. (29) should be solved simultaneously for qi. With known values of Yi from Eqs. (19)–(23) the vibration response of dynamic mode nanomachining SThM probe is achieved in any point of probe. As the nanomachining is performed at the end of probe, thus amplitude of response at x = L is studied in next section.
Table 2 Details of constant, linear and quadratic temperature distribution function.
Discussion and results The governing equations are solved numerically for a case study. The specifications of SThM nanomachining probe are tabulated in Table 1 that are adopted from [23] except noted. The temperature 4
Models
θ(x)
Constant Linear Quadratic
1 0.2(x / L) + 0.8
0.15(x / L)2 + 0.05(x / L) + 0.8
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Fig. 6. Third mode shapes for different temperature difference for θ(x) = 1.
Fig. 3. Comparison between normalized shifted resonance frequencies of current study vs. [23] for θ(x) = 1.
Fig. 7. The first shifted resonance frequency for constant, linear and quadratic distributions.
Fig. 4. Comparison between normalized shifted resonance frequencies current study vs. [23] θ(x) = 1.
effect of θ(x) is also significant in frequency shifts. The constant distributions has the highest frequency increase, and quadratic has the lowest frequency shift. Therefore knowing the accurate temperature distribution of the probe is important for frequency estimation. It opens new visions for study in the behavior of SThM probe if anybody can introduce more realistic estimation for temperature distribution function for presented mathematical model as the used θ(x) in current research are some rough estimates. Fig. 8 shows the normalized amplitude of vibration at end/tip (x = L) in base excitation frequency and Figs. 9 and 10 show the normalized amplitude of response in shifted resonance frequencies of probe for constant, linear, and quadratic distributions. Ae and Aish are defined as normalized amplitude of response in base excitation frequency and ith shifted resonance frequency, respectively. The normalization is done by dividing the amplitude of response in ΔT = 0 in base frequency (Here 432.95 nm). It is stated that, in all cases, with increasing ΔT the normalized amplitudes decrease. These changes are significant both in base excitation and first shifted resonance frequency. As the base is excited in the first resonance frequency range (ω ≃ 0.8ω1), order of magnitude is high between first and second mode (and higher modes). In comparison between temperature distribution function, maximum amplitude is seen in the quadratic distribution. It is because the quadratic model causes lower axial force in probe comparing with constant model. Thus the amplitude is less controlled with temperature difference. In general, the effect of θ(x) can be neglected in amplitude
Fig. 5. First mode shapes for different temperature difference for θ(x) = 1.
distributions are compared in Fig. 7. It is seen in all temperature distribution, the shifted resonance frequencies are increased by increasing of ΔT as seen previously for linear case. It means, by increasing the ΔT the distance between hills decrease and it can improve the quality of final surface (if the depth of nanomachining can be controlled). The 5
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Fig. 8. Normalized amplitude of response in base excitation frequency for constant, linear and quadratic distributions.
Fig. 11. Simulation of nanomachined cross section in ΔT = (a) 200 °C (b) 500 °C (c) 700 °C.
200 nm thickness sample in three different temperatures (200, 500 and 700 °C). The travelling speed of 400 μm/s is assumed for calculation of machining length and set point is assumed as 40 nm. Set point is the distance that is initially set for SThM probe from the sample free surface i.e. it is the equilibrium point of base. It is clear decreasing amount of set point will cause deeper scratch similar to static mode. Fig. 11 demonstrate that the depth of nanomachined area is significantly dependent on ΔT that is controlled by base excitation. If the free surface of sample assumed as datum, Root Mean Square (RMS) value for final surface are calculated 126.7, 55.5 and 38.1 nm in negative direction for ΔT = 200, 500 and 700 °C respectively for the simulated length. The RMS is used due to nonlinear behavior of tip that cause non-smooth nanomachined surface finish. It is clear increasing the travelling speed can improve the final surface. Fig. 12 shows lower peak envelope of tip response. This figure can be used as a rough estimation of final surface of nanomachined area. It is seen the final surface is improved with increasing ΔT. The reason is increasing of shifted resonance frequency of SThM probe with increasing ΔT. It is clear increasing the frequencies of vibrational response causes better surface finish in constant travelling speed. A point should not be ignored, that in current research effect of tip dimension (tip radius) was ignored i.e. it was assumed the nanomachining was
Fig. 9. Normalized amplitude of vibration in first shifted resonance frequency for constant, linear and quadratic distributions.
Fig. 10. Normalized amplitude of vibration in second shifted resonance frequency for constant, linear and quadratic distributions.
(however the expected accuracy of nanomachining should be concerned). Fig. 11 shows the simulation of nanomachined cross section of
Fig. 12. Lower peak envelope of tip response in ΔT = (a) 200 °C (b) 500 °C (c) 700 °C. 6
Results in Physics 13 (2019) 102164
M. Sohrabi and K.E. Torkanpouri
Appendix A. Supplementary data
applied with a zero tip radius (point tip). This parameter can also affect the final surface practically. The results state that with adding a known excitation on the base the depth of nanomachining process can be controlled with ΔT that demonstrate the hypothesis of research. It is obvious that controlling the magnitude of ΔT is not a simple effort as it depends on the temperature of sample and generally there isn’t constant temperature distribution on the samples. Thus a control system should be designed (that is not the issue of current paper) to track the magnitude of desired depth by controlling input amplitude of base (y0). It seems the error of control loop may be used for thermal imaging of sample while nanomachining process.
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Conclusion In this research paper, a techniqueally possible idea was proposed to control the depth of nanomachining process with SThM probe. It was named dynamic mode SThM nanomachining technique. Afterwards a mathematical model for dynamic mode SThM nanomachining probe based on Euler-Bernoulli beam was determined. The temperature distribution function was roughly assumed accordance to constant, linear and quadratic functions. The mode shapes were figured for various temperature. The shifted resonance frequencies were verified with literature for some special cases. To find the amplitude of response the assume mode method was used. The PDE of motion was reduced to a system of coupled non-linear ODEs. The equation of motion were solve numerically by coding in Matlab. FFT was applied to find the frequency domain response of non-linear system. The shifted resonance frequencies were figured for temperature difference between 0 and 800 °C in constant, linear and quadratic models. Amplitude of response in base frequency and shifted resonance frequencies were determined at tip location. It was showed in all temperature distribution, the shifted resonance frequencies are increased by increasing of temperature difference. It was explained that the effect of temperature distribution function is significant on resonance frequencies shift but negligible on amplitude. It was showed, the amplitude is decreased in all temperature distribution with increasing the temperature difference. Finally it was stated that with adding a known excitation on the base the depth of nanomachining process can be controlled with temperature that demonstrate the hypothesis of research. Also it was declared increasing the temperature improves the quality of final nanomachined surface for dynamic mode SThM nanomachining technique.
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