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Role of mechanical and thermal nonlinearities in imaging by Atomic Force Microscope
Author’s Accepted Manuscript Role of mechanical and thermal nonlinearities in imaging by Atomic Force Microscope Sadegh Sadeghzadeh, M.H. Korayem
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PII: DOI: Reference:
S0020-7403(17)30098-X http://dx.doi.org/10.1016/j.ijmecsci.2017.01.021 MS3559
To appear in: International Journal of Mechanical Sciences Received date: 11 May 2016 Revised date: 16 December 2016 Accepted date: 13 January 2017 Cite this article as: Sadegh Sadeghzadeh and M.H. Korayem, Role of mechanical and thermal nonlinearities in imaging by Atomic Force Microscope, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2017.01.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Role of mechanical and thermal nonlinearities in imaging by Atomic Force Microscope 1
2
Sadegh Sadeghzadeh1*, M. H. Korayem2 Assistant Professor, School of New Technologies, Iran University of Science and Technology, Tehran, Iran
Professor, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
[email protected]
Abstract In this paper, the effects of nonlinearities such as hysteresis, creep and thermal drift of piezoelectric elements on imaging by Atomic Force Microscope have been studied via a semi-analytical approach. The Generalized Differential Quadrature Method (GDQM) has been used to find the dynamic response of a system. Bouc-Wen and PI hysteresis models have been incorporated with the presented linear model to obtain the effects of nonlinearities; and thermal drift has been augmented in an offline scheme. All the sub-models have been validated by comparing their results with the findings of previously reported experiments. Finally, these nonlinear effects have been applied on AFM-based imaging operations, and the obtained results have been evaluated. In comparison with the linear case in which the hysteresis effects are not taken into consideration, in the nonlinear imaging model, depending on the scan direction, the asperities on a rough substrate are sensed at different locations. In the creep case, the high level of error obtained during scan at the ascending points of substrate profile is due to the delay in the time interval in which the input increases. According to the results, the imaging of a standard sample substrate while assuming a 2º temperature change during a 10 min scan of the substrate has yielded a maximum thermal drift error of about 3.3 nm. At the end, based on the presented comprehensive nonlinear imaging model, the coupled effects of creep-hysteresisthermal drift on the final image have been reported and discussed.
KeyWords; Nonlinearities, Atomic Force Microscope, Imaging, Hysteresis, creep, Thermal drift
1- Introduction The researchers in various technological fields are now able to more effectively control and manipulate the structures of different materials at increasingly lower scales. Today, the new
investigation methods make it possible to perform engineering operations on single atoms, molecules and atomic bonds by means of nanotools ]1[. The science and technology of using such techniques is known as nanorobotics. In general, nanorobotic systems have been classified into two groups of nanoscale and microscale nanorobots ]2[. Microscopes with the ability of performing various operations at nano dimensions and lower have been called ‘large-scale nanorobots’. Macroscale tools are needed to access, identify and manipulate various materials at the nanoscale. Remote-controlled systems are the usual tools which are used for this purpose. Among these tools, the SPMs, and specially the Atomic Force Microscope (AFM), are considered as powerful tools for manipulation and assembly tasks at the nanoscale. With the invention of AFM ]3[, many types of nanorobotic systems, with newly found applications, have been built. Identifying the frictional and surface properties of metals, polymers and biopolymers at the micro/nanoscale, displacing and selfassembling of single molecules, cells, viruses, and acquiring instantaneous images of nanoparticles during manipulation by using simulations based on the relevant model and the measured forces are some of the applications of these nanorobots. Fig. 1 shows the schematic of a type of AFM. The capabilities of nanorobots could be generally classified into two areas of detection and manipulation. Imaging by the AFM at the nano and lower scales is a clear example of the detection process; which can be defined for all the operating modes of nanorobots, and which has its own particular application in each specific operating mode. The influential parameters on manipulation at the nanoscale have been summarized and presented in Table 1.
Fig. 1: The schematic of an AFM nanorobot made by the Bruker Company (left), Enlarged schematic of small parts (right) Table 1: Various areas of manipulation process and its general effective parameters Type of manipulation
Pushing/Pulling
Lifting/Placing
Environment
Air (simple, multibody, ambient temperature, etc.)
Vacuum (vacuum level, sensitivity, etc.)
Cutting
Bending/Buckling
Liquid (plain liquid, Biological and cellular blood, non(soft, hard, genetic, etc.) Newtonian liquid,
Microscope
Object
etc.) Cantilever shape Operating mode Environmental (pincer-shaped, V(contact, non-contact, conditions (humid, shaped, rectangular, tapping) dry, PH range, etc.) etc.)
Tip shape (triangular, rectangular, T-shaped, etc.)
Physical field (metallic, biological, magnetic, etc.) Physical field Special conditions Mechanical behavior (metallic, biological, (humid, dry, PH range, (rigid, flexible) magnetic, etc.) etc.) Dynamics (constant Dimensions (2D, Straight or curved velocity, constant 3D) path acceleration
Shape (sphere, rod, Dimensions (2D, 3D, Mechanical behavior tube, wire, cell, etc.) small, large, etc.) (rigid, flexible)
Substrate
Roughness (low, high)
Process dynamics
Dominant forces
With respect to its nature and functionality, a nanorobot can be considered as equivalent to an ordinary large-scale robot; nevertheless, there is a major difference in the dynamics of contact between components at the two scales.
2- History of developments in nanorobotics field Although, nanorobots can be essentially considered equivalent to ordinary robots, due to the existing limitations in the measuring of forces and in instantaneous sensing, and also because of the presence of unknown physical phenomena, contrary to ordinary robots, numerous challenges are involved in nanorobots, which inherently makes the nature of the existing problems totally different. This has prompted the scientists to develop more powerful and realistic models that can shed more light on the nature of the examined problems, before costly and expensive experiments are undertaken. For example, computational procedures for investigating AFM-based manipulations, especially for the pushing of nanoparticles and nanorods ]4[ have been presented. Considering the existence of surface and intermolecular forces at this scale, there are numerous variables and parameters that affect the motion of these nanoparticles during manipulation. Due to the large number of unknown variables involved in the calculations, it is difficult to perform a quantitative comparison with empirical data; although these comparisons could be very informative ]4[. In order to achieve conceptual and exact models, Sitti and Hashimoto ]5[ presented, for the first
time, a 2D dynamic model for the assembly of micro/nanospheres. Then, based on this model, Taffazoli et al. developed a model that described the dynamic motion modes of nanospheres ]6[. In that work, they had analyzed the conditions of sliding and rolling on a substrate. Some modification were also applied on the mechanism of AFM. For example, the resonant frequencies and flexural sensitivities of an atomic force microscope (AFM) with assembled cantilever probe (ACP) were studied [7]. Korayem et al. obtained the maximum amount of load that could be carried by a microscope, and presented an algorithm for calculating this load ]8[. In [9], the 3D modeling of motion dynamics by using different cantilevers (rectangular, V-shaped and dagger-shaped) was
focused and was simulated the kinematics and dynamics of a manipulation operation. A comprehensive algorithm by which the critical time and force can be determined was developed in [10]. By using the proposed algorithm, the maximum angle of spinning was specified. An algorithm was used in this paper [11] to show in 3D, the manner by which the dynamic forces vary in the manipulation process. In parallel with these works, and as the significance of such research endeavors became evident, other research groups with defined plans concentrated on this subject. A relatively comprehensive account of the research works conducted by these groups and their achievements have been presented in Table 2. These works reflect the scientific and experimental progresses made in the field of nanorobotics. Many of these investigators have specifically focused on experimental works and the development of these works for industrial applications. Such inclinations and outlooks by the researchers have made it quite clear that nanorobotics will play an effective role in the forthcoming progresses and achievements of nanotechnology in the world. The concurrent theoretical and experimental developments of large-scale nanorobots, so far as their applications did not involve biological objects, were good and effective. However, with the development of these tools and the various mediums in which these instruments operated, continuum mechanics-based theories left many questions unanswered. As an approach for answering these questions, Korayem et al. suggested using the molecular dynamics method ]13 ,12[. Nevertheless, there are still challenges regarding the connection between the large-scale and small-scale worlds; which should be addressed. The authors of this paper have recently made a great effort in using multi-scale methods for the analysis of large-scale nanorobots. By employing a multi-scale model, they have even studied the effects of hysteresis on the imaging and manipulation of nano-objects and have revealed that the nonlinear effects should be seriously considered if precise manipulations are intended. However, no work has simultaneously modeled the nonlinear effects of hysteresis, creep and thermal drift; so that the mutual effects of these parameters could be evaluated. In this regard, the present work will attempt to provide an appropriate groundwork for the simultaneous investigation of these aspects. We hope that this study can provide a comprehensive and exact model for nanorobots; so that, anywhere, and by any specific mechanism, an arbitrary nanorobotic system can be analyzed by considering the nonlinear effects. Of course, there are still traces of many errors, which are suggested to be examined; but the groundwork presented in this paper can ensure, with a high degree of reliability, the addition of other error effects in the future.
Table 2: Groups active in the nanorobotics field, and their important innovations and achievements
Name
Research area
Ning Xi A. Requicha S. Devasia M. Sitti T. Fukuda S. Fatikow M. H. Korayem A. Meghdari C. Zhou J. Tamayo
D. Spinosa R. Carpick Y. Sun
Important innovations and achievements
Pushing of nanoparticles, DNA Obtaining instant images from the nanorobotic manipulation, automatic assembly operations on microobjects Control of nanorobotic Experimental studies of nonlinearity effects on manipulations the results of nanorobotic operations Design and development of Implementing all the hysteresis modeling piezoelectric actuators methods in nanorobotic operations Remote control of nanorobotic Presenting micro/nano remote-operating systems manipulations Manipulation of biological Controlled manipulation and deformation of systems living samples Making various AFM-based Combining two nanorobots of AFM and SEM nanotools Modeling nanorobotic tools and Presenting nanorobot models for liquid and operations, Modeling the vacuum environments, Modeling large-, smallnanorobots in various mediums, and multi-scale dynamics etc. Modeling nanorobotic tools and Small-scale dynamic modeling and large-scale manipulations chaos dynamics Building nanostructures by using Making nanometric transistors nanowires and graphene Exciting and measuring the produced responses at a very Making and analyzing mechanical nanosensors small scale Analyzing, designing and controlling of piezoelectric Designing nano-electro-mechanical switches nanowires Investigating system behaviors at Analyzing the friction behavior at a small scale nano and lower scales Controlling the behavior of biological Controlled displacement nanostructures
B. J. Inkson
Special nanomanipulations
Nanomanipulation on different prostheses
Period of activity
Sample of works
2000 till now
]41[
1995-2011 ]41 ,41[ 1993 till now
]41[
1998 till now
]41[
2000 till now
]41[
2002 till now
]02[
2000 till now
]12[
2000 till now
]00[
2003 till now
]02[
1998 till now
]01[
1995 till now
]01[
1996 till now
]01[
1998 till now
]01[
1995 till now
]01[
Designing and making of different controllers Large-scale nonlinear effects of based on the modeling methods of hysteresis and 2003 till now nanorobots creep Modeling and control at nano and Producing nano indentation and controlled 1990 till now A. Ferreira lower dimensions scratches on nanometric surfaces K. Leang
]01[ ]22[
3- Nonlinear dynamics of large-scale nanorobots As precise tools, piezoelectric materials play a vital role in micro/nanorobotics, and they are usually used as actuators and distributed sensors. Mostly, it is these piezo-actuators which are referred to as the large-scale parts of nanorobots. In this paper, the equations of motion for actuators and distributed sensors are obtained by means of the generalized Hamilton’s principle. The presented formulation has a general form and can be applied to all piezo-actuators and piezo-sensors. 3-1- Deriving the nonlinear motion equations
Piezoelectric materials are usually laminated over a base system; and therefore, the main components of nanorobots (microcantilever and piezotube) are made of a multilayer structure. Considering the piezoelectric effect and disregarding the thermal stresses, the structural equations for the kth layer in will be as follows ]24[:
the material coordinate system { }
[ ]{ }
[ ] { }{ }
[ ]{ }
[ ]{ }
{ }
{ }
)2 (
where [ ] is the reduced stiffness matrix, [ ] is the piezoelectric modulus, and [ ] is the dielectric constant of the kth layer in the material coordinate system. Also, { } is known as the residual dipoles factor (used to express the hysteresis), and { } denotes the creep factor. In addition, {E}, {S}, {σ} and { } represent the stress, strain, electrical field and electrical diplacement vectors, respectively. By using the first-degree plate and shell theories, the mechanical displacement field is written as Eq. 2 ]24[. )1 (
In the above equations, x3 is the distance to the neutral plane, u, v and w are the displacement components along directions 1, 2 and 3, u0, v0 and w0 are the displacements of the points of neutral plane along the mentioned directions, and
and
are the rotations about axes 1 and 2,
respectively. The potential energy of a layer of electromechanical plate can be expressed as ∫
∫
)3 (
where T is the surface reactions, u is the mechanical displacement, Q is the electrical charge, electrical potential,
is a volume element and ∫(
considering the kinetic energy (
̇
̇ )
is the
is a surface area element of the jth layer. By ), the Hamilton’s principle for a multilayer shell
is written as
(∑ ∑ ∫
in which
)
)4 (
is the work of external forces and δ, SN and LN denote the variation operator, number
of patches and number of layers, respectively. By applying the presented relations, the motion equations of a shell with two curvatures of radii R1 and R2 will be expressed as follows ]20[:
̈ ̈ ̈ ̈
(
)
̈
(5) ̈ ̈
̈ ̈
) {
(
}
{
∑∫
}
(6)
This is a general form of the equation. By assuming a plate, and by assuming
and
and
, the motion equation of
, the motion equation of a cylinder with radius
will be
obtained ]32[. The forces and moments are computed by integrating the stresses in the structural equations. By substituting these forces and moments into the motion equations, the equations are obtained with respect to displacements. In general, the forces and moments are derived as follows: { { {
} }
}
{ { {
} }
}
{ { {
} }
} { { {
{ } }
{ {
}
} }
}
{ {
} }
{
)7 (
}
)8 (
Considering the structural equations and the fact that the electrical field is applied along the thickness ( ), the electrical forces and moments for the ith element are defined as ̅ {
{
where
}
}
{
{
}
}
̅
{ ̅ } ̅ ̅ ∑∫ { ̅ } ̅
∑∫
∑{ ̅ } ̅ ̅ ∑{ ̅ } ̅
is the surface electrical potential of the kth layer,
the kth layer to the middle plane, and
(9)
is the distance from the mid-section of
is the distance between the middle plane of layers and the
middle planes of the multilayer shell. By using the load equation, and considering the fact that , we can write ]33[:
)21(
|
|
| ̅
̅
|
̅
̅
)22(
̅
̅
is the surface load density of the kth layer in the ith element. Also,
where , where
is the electrical potential on the surface of the considered layer.
is the linear ratio
coefficient, which indicates the direct effect of applied voltage on dipole orientation in piezo materials. When a completely linear system is assumed, this coefficient will be equal to one (1). Since nonlinear parameters are equivalent to electrical displacements and this type of load has been induced by voltage, we can write )21(
where
and
denote the equivalent voltages that produce hysteresis and creep, respectively;
and they are modeled by means of specific hysteresis and creep models. By computing the electrical forces and moments from Eq. 9 and substituting them into Eq. 5, the motion equations are obtained with respect to displacements. By substituting the forces into the motion equations, their operator form versus displacement can be expressed as Eq. 13. ̈ In the above equation,
)23(
and
represent the mechanical th
and electrical terms, respectively, and
is the inertia term of the i equation. The electrical effect
has two parts: reverse effect
and direct effect
. By considering
the damping effect, Eq. 14 can be presented, where c is the damping coefficient and ̂ represents the external load; i.e. ̂
. ̇
̈
̂
)24(
3-2- Solving the motion equations by the Differential Quadrature method
In view of the types of problems involved and the complexity of the equations in special cases (e.g. in multi-patch systems), finding an exact solution for the considered problem will not be possible; and so, it will be necessary to use numerical methods, especially those which produce the least error and involve a smaller number of degrees of freedom. Differential Quadrature (DQ) is an approach that adequately addresses such concerns. In this method, the partial derivative of the nth order of
continuous function f(x, z) with respect to x at a known point
⌋
∑
In the above equation,
(
is expressed by Eq. 15 ]24[.
)
)21(
are the weight function coefficients, N is the number of grid points, and f
represents the function values at the considered points. 3-2-1- The GDQE method
Fig. 2(a) shows a plate with a piezoelectric patch in the middle, and Fig. 2(b) illustrates the correct element configuration of this problem. The presence of a geometrical or force discontinuity makes it difficult to use the ordinary form of the GDQ method. In these cases, the physics and geometry of the problem should be divided into several distinct sections; in other words, domain decomposition should be implemented. In this case, the GDQ problems will be equal in number to the elements. This method is called the Generalized Differential Quadrature Element (GDQE) method. As shown in figure 2, when a patch s placed at the middle of the plate, the plate should be divided into nine parts and separate degrees of freedom should be taken for each of them.
Fig. 2: (a) a plate with a piezoelectric patch, (b) the correct configuration of elements in the GDQE method In the most general case, and with the consideration of proportionate damping, the dynamic motion equations are expressed as Eq. 16. ̇
{ By substituting
̇
̈
)21(
from the second equation into the first equation and converting the obtained
equation to state space, we can write
{ ̇ } { in which
( and
[
]{ ̇ } ̇ are chosen as follows:
)}
)27(
(
)
)28(
Eq. 17 is solved by using numerical methods such as Runge-Kutta; and then the boundary displacements can be determined from relation
.
3-3- The model of nonlinear effects
Because of using piezo-actuators in nanorobts, the nonlinear effects including creep, hysteresis and thermal drift should be considered and addressed in these systems. Following the early studies of creep in AFM, Richter et al. presented the first model for it ]34[. All the researchers have modeled the creep effect by combining several linear mass-spring models and then have incorporated the effect of hysteresis in this model. One of the first and most detailed analyses performed so far is the work of Prandtle and, later, Preisach ]35[. Some researchers have used the Prisach model for control and compensation purposes ]36[. Subsequently, many works have employed new methods for promptly adapting the hysteresis model ]33[. Not all these methods can be appropriately used; and the models do not properly match the empirical results and are even computationally sensitive to reasonable levels of accuracy. Conversely, analytical methods such as the Bouc-Wen method do not have many of these drawbacks. Here, in addition to the Bouc-Wen approach, we have attempted to use the Prandtle-Ishlinski (PI) method in cases in which simpler applications are considered. In this way, the model of nonlinear elements (creep and hysteresis) can be expressed and presented more effectively. 3-3-1- The hysteresis model
Here, the Prandtle-Ishlinski and Bouc-Wen models have been employed for nanorobotic applications. In the Prandtle-Ishlinski (PI) model, operator output
and input
[ ]
is used to define the relationship between
]38[.
∫ In the above equation,
[ ] and q are weight functions with conditions
)21(
.
To describe the system’s hysteresis behavior, an analytical equation is presented in the Bouc-Wen model. Eq. 20 expresses the Bouc-Wen equation for hysteresis displacement ]33[. )11( | ̇ || | | | ̇ ̇ In the above equation, A controls the amplitude of hysteresis, and β, γ and n change the shape and
̇
amplitude of hysteresis (n ≥ 1). 3-3-2- Creep
Each time delay operator can be defined as Eq. 21 ]15[.
̇
)12(
In the above equation, the poles ( ) are negative and
; where
and
denote the degree
of creep and the input, respectively. 3-3-3- The final equations
In view of the presented nonlinear model and the differential quadrature solution method, and with the consideration of the nonlinear effects of hysteresis and creep, and by developing the equations presented in ]44[, the final equations will be obtained as follows:
{
{ ̇}
[ ̇
̇
| ̇ ||
|
̇ |
]{ ̇} |
[
]{ }
)11(
Since, by using the calibration of input voltage, the creep model yields the effect of creep on resulting displacement, in Eq. 22,
represents the input excitation voltage of the system, and we
have ̇ )13(
3-4- Validating the model of nonlinear effects
The free vibration, static, and transient response models of the present method have been validated in ]41[. In view of those validations for piezo-tubes and for different plates and beams, only the
nonlinear model is validated here. 3-4-1- Validating the model of hysteresis in microcantilever
By using the generalized PI method, the results of a double-layer (piezoelectric and copper electrode layers) microcantilever have been compared with the findings of the experimental work in ]42[. The cantilever dimensions are 15 x 2 x 0.3 mm, and the applied voltage is in the form of 40 sin(2πft) or 20 sin(2πft). The width of hysteresis, as a measure of intensity, has been indicated by λ in Fig. 3; and for the rate-dependent case, the hysteresis loops resulting from different frequencies have been illustrated in Fig. 4.
1 Hz Linear Nonlinear
10
0
Experiment (Rakotondrabe, 2008)
20
Displacement ( m)
Displacement ( m)
20
-10
Present Approach
10
0
-10
-20
-20 0
0.5
1 Time (Sec)
1.5
-40
2
-30
-20
-10
(a)
0 10 Voltage(V)
20
30
40
(b)
Fig. 3: (a) comparing the linear and nonlinear dynamic responses, (b) relevant hysteresis loops with a frequency of 1 Hz 200 Hz
10 Hz
15 Experiment (Rakotondrabe, 2008)
Experiment (Rakotondrabe, 2008)
10
10 Displacement ( m)
Displacement ( m)
Present Approach 5
0
-5
Present Approach
5 0 -5 -10
-10 -20
-15
-10
-5
0
Voltage(V)
5
10
15
-15 -20
20
-15
-10
-5
0 5 Voltage(V)
(a)
10
15
20
(b)
300 Hz
600 Hz
30
100 Present Approach Experiment (Rakotondrabe, 2008)
20
Displacement( m)
Displacement ( m)
50 10 0 -10 -20
-50 Present Approach Experiment (Rakotondrabe, 2008)
-30 -40
0
-30
-20
-10
0 10 Voltage(V)
20
30
40
-100 -40
-30
-20
-10
0 10 Voltage(V)
20
30
40
(c) (d) Fig. 4: Hysteresis loops obtained by applying a voltage with (a) amplitude of 20 V and frequency of 10 Hz, (b) amplitude of 20 V and frequency of 200 Hz, (c) amplitude of 40 V and frequency of 300 Hz, (d) amplitude of 40 V and frequency of 600 Hz 3-4-2- Validating the model of hysteresis in the piezo-tube
Fig. 5 shows the hysteresis loops obtained in two cases with sinusoidal and triangular inputs. In this case, the voltage has been applied to a pair of electrodes facing each other. The exterior electrodes all have an angle of 90º; and the voltage is applied in such a way that the piezo-tube has a lateral motion. In both cases, the applied voltage has an amplitude of 1 V and frequency of 30 Hz. Fig. 6 illustrates
the hysteresis loops obtained by exciting the piezo-tube of a non-contact AFM (Burleigh Metris 2000) along the x direction by an excitation voltage with an amplitude of 5 V. 1
0.5
0.5
Displacement (nm)
1
0 -0.5 -1 -1.5 -2
NonLinear 0.02
-0.5 -1 -1.5
Linear 0
0
-2
0.04 0.06 Time(s)
0.08
0.1
Linear NonLinear 0
0.02
0.04 0.06 Time(s)
(a)
0.08
0.1
(b)
1.5 1
Displacement (nm)
1 0.5 0 -0.5
0.5 0 -0.5 -1
-1 -1.5 -1
-0.5
0 Voltage(V)
0.5
-1
1
-0.5
(c)
0 Voltage(V)
0.5
1
(d)
Fig. 5: (a, b) comparing the linear and nonlinear dynamic responses for sinusoidal and triangular inputs, (c, d) relevant hysteresis loops resulting from the sinusoidal and triangular inputs (frequency of 30 Hz) 50 40
Experimental (K. Leang, 2006) Present approach
Displacement( m)
30 20 10 0 -10 -20 -30 -40 -50 -5
-4
-3
-2
-1
0 Input(V)
1
2
3
4
5
Fig. 6: Comparing the hysteresis loops (displacement vs. voltage along the x direction) for the AFM piezo-tube ]43[ 3-4-3- Validating the creep model
The results of the creep model have been compared in Fig. 7(a) with the empirical results of ]11[ for the mentioned cantilever (excited by applying a voltage of 10 V). The direct application of the creep model does not model the initial vibrations of the system; therefore, the proportional damping model has been used for this purpose. Fig. 7(b) shows the enlarged transient response section, with an excitation voltage of 40 V and mass and stiffness coefficients of
and
,
respectively. 45
6
Present Approach
40
Experiment (Rakotondrabe, 2010)
35
Displacement ( m)
Displacement ( m)
8
4 4.5 4.45
2
4.4
30 25 20 15
Experiment (Rakotondrabe, 2010)
10
Present Approach
4.35 0
0
0
5
0.02
10
0.04
15 Time(s) (a)
0.06
20
0.08
0.1
25
5
30
0
0
0.01
0.02 0.03 Time (Sec)
0.04
0.05
(b)
Fig. 7: Comparing the results of the creep model with the empirical results of the cantilever; (a) with an excitation voltage of 10 V, (b) with an excitation voltage of 40 V (enlarged section) 3-5- Thermal drift
If the ambient temperature is not controlled, the variation of temperature in the environment can cause a thermal strain, which is known as thermal drift. The researchers believe that thermal drift is a major cause of error in nanorobots. Here, the general analysis of thermal drift is carried out by dividing the problem domain into macro and micro coordinates. For analyzing the problem in the macro-dimension coordinates, an analytical model has been presented for the thermal circuit, and then the amount of generated heat and the thermal resistance values have been computed for this circuit. Four general types of heat transfer are involved in the mentioned circuit: generated heat, conduction heat transfer, heat transfer via an intermediary medium, and convection heat transfer. In the next step, a numerical solution has been used to analyze the present problem; and by evaluating the solution stability, this method has been compared with the solution of ANSYS software program. It should be mentioned that, for a better estimation of the answer, an approximate solution obtained by simplifying the governing differential equation has also been used. To analyze the system in micro/nano dimensions, the effect of heat on atomic friction has been explored and then, similar to the macro sections, the transfer of heat in micro/nano parts has been evaluated. By writing the resistances and the generated heat values at these scales, approximations of thermal drift have been provided at these dimensions and finally, the total amount of thermal drift has been obtained by finding the resultant of macro and micro/nano drifts. Fig. 8 depicts the mentioned thermal circuits for
the AFM nanorobot. The model presented in ]11[ has interested the authors; and after validating it and identifying its system, a dynamic model has been proposed for it, and eventually compensated.
Rtip
Rtip Rsph Rcontact
Tip
Rcontact1
Tip
Rsph RSub
Rparticle
Particle
Rcontact Substrate RSub
Rcontact2
Visco adhesive layer
RAdh
Substrate Viscoelastic adhesive layer
RSub XY-Piezotube
RAdh
XYPiezotube
Fig. 8: Thermal circuits for large-scale and small-scale sections in nanomanipulation and nanoimaging mechanisms The thermal expansion coefficient of PZTs normally varies in the range of 10-50 µm/ºC. This means that one degree of temperature change will lead to a displacement of about 40-200 nm along the PZT (assuming a 40 mm length for the actuator). For example, the thermal drift of the piezotube of a Vico nanorobot with a PZT-5H type piezoelectric has been modeled and compensated in [45]. In the present work, thermal drift is applied, offline, to the whole system. Following a standard nanomanipulation process, a total drift of about 580 nm has occurred in the x direction. Of course, the amount of total drift obtained in ANSYS is 600 nm; and experimental works have reported a total drift of about 560 nm ]46[. Nevertheless, a low amount of error has also been reported, which is due to the possible differences in ambient conditions and the effects of electronic equipment operating in the environment ]46[. Figure 9 shows an algorithm for implementation of presented three nonlinear effects in the nano imaging simulations. A completely same algorithm could be presented for nano manipulation.
Make the Force vector on MC (Forces with asterisks)
State Space Representation
Solution
A, B, C and D construction Make the state dependent force vector on the MC end
Obtain Sensed Voltage
Discretization GDQ or GDQE Approaches
Input Voltage Existent Electrical Forces
+ +
Thermal Drift
+ +
Obtained Profile
Existent Mechanical Forces
V
Convert to Displacement
Calculate the forces on the MC end
+
Hysteresis and creep model
Select Displacement of MC end
The Substrate profile
Fig. 9: Algorithm of nano imaging by using hysteresis, creep and thermal drift models
4- Applying the nonlinear effects on the imaging operation 4-1- Effects of hysteresis 4-1-1- Validation through comparison with empirical results
In addition to validating the nonlinear dynamics of hysteresis in cantilever and piezotube, the effects of hysteresis on imaging can also be investigated through comparison with the empirical results in ]16[. Fig. 10 shows a comparison of the simulation results related to the forward and backward travel
paths of gold nanoparticles of 15 nm diameter on a mica substrate. Here, there are three nanoparticles with a specific arrangement next to each other. By using this image, proper positions have been considered for the nanoparticles, so that the images of their forward travel paths could be approximately obtained in this figure. Then, in the backward paths, the hysteresis effects will be prominently displayed, and the validity of the presented model can be evaluated. Regardless of the differences that inherently exist in the diagrams, a correct approximation of directional changes in the presence of hysteresis is revealed. The presence of hysteresis causes the nanoparticle to experience a totally new route in the backward travel path. This can be observed in Fig. 11 by simultaneously plotting the forward and backward paths.
20
20
Simulation result Experimental result [Requicha, 2010]
Simulation result Experimental result [Requicha, 2010] 15
Profile (nm)
Profile (nm)
15
10
10
5
5 Backward Scan 0
Forward Scan 0
0
10
20
30 40 X direction (nm)
50
60
0
10
20
30 40 X direction (nm)
50
60
70
Fig. 10: Comparing the forward and backward travel paths of gold particles of 15 nm diameter on a mica substrate with the experimental results 18 Experiment (Requicha, 2010)
Simulation (Present Approach)
16 14
Profile (nm)
12 10 8 6 4 2 0
0
10
20
30 40 X Direction (nm)
50
60
70
Fig. 11: Comparing the forward and backward travel paths and the hysteresis loops obtained in imaging gold nanoparticles 4-1-2- Applying the hysteresis on the imaging of a standard sample
Fig. 12 illustrates the results of imaging a sample surface containing three different types of asperities (sinusoidal, square, and triangular. The unit used for this figure and subsequent figures is the nm.
Linear 1.2 Profile
Obtained image
1
Amplitude (nm)
0.8 0.6 0.4 0.2 0 -0.2
0
2
4 6 Horizontal Position (nm)
8
10
Fig. 12: 2D and 3D images of a sample substrate with sinusoidal, square, and triangular profile sections obtained without considering the nonlinear effects (the units are in nm). As is observed, the present model has been able to adequately image the examined profile. Of course, in the presence of nonlinear effects such as hysteresis, creep and thermal drift, the result would be completely different; and the nonlinear model should be able to adequately reveal the difference between the results obtained by considering the nonlinear and linear effects. In comparison with the linear case in which the hysteresis effects are not taken into consideration, Fig. 13 shows the results obtained by imaging the profile presented in Fig. 12 and considering a weak hysteresis effect. The imaging has been performed by means of a non-contact AFM (Burleigh Metris 2000). The difference between the obtained results and those of the linear model has been illustrated in Fig. 13(b). When hysteresis is considered in the system, depending on the scan direction, the asperities on a rough substrate are sensed at different locations. Hysteresis and Linear comparison 1.2 Linear Model
Nonlinear Model (Hysteresis)
1
Amplitude (nm)
0.8 0.6 0.4 0.2 0 -0.2
0
2
4 6 Horizontal Position (nm)
8
10
Fig. 13: 2D and 3D images of a sample substrate with sinusoidal, square, and triangular profile sections obtained by considering the nonlinear hysteresis effect (the units are in nm). Fig. 14 illustrates the hysteresis loops produced in the scanning of three sinusoidal, square and triangular asperities by plotting the obtained image versus the considered profile. As is observed, due to the uniform changes of the images obtained for the sinusoidal and triangular sections, the loops
corresponding to these sections display a standard hysteresis loop; while for the square section, because of using a constant input in the maximum and minimum sections of the square wave, two zones with constant minimum and maximum values in the horizontal direction can be observed in this diagram. Hysteresis Loops 1.2 1
Hysteresis loop for Sin part Hysteresis loop for Square part Hysteresis loop for triangular part
Image
0.8 0.6 0.4 0.2 0 -0.2 -0.2
0
0.2
0.4 0.6 Profile
0.8
1
1.2
Fig. 14: Hysteresis loops produced in the scanning of three modeled sinusoidal, square and triangular asperities 4-2- Considering the creep effect in the imaging operation
By using the results of the validation section, for the nonlinear creep model, a system with the specifications given in Fig. 7 has been considered. By imaging a standard sample substrate with a combination of sinusoidal-square-triangular profile sections (presented in the preceding section), the 2D and 3D images of that sample obtained by considering the nonlinear creep effect have been depicted in Fig. 15. Creep and Linear comparison 1.2 Linear Model Nonlinear Model (Creep) 1
Amplitude (nm)
0.8
0.6
0.4
0.2
0
-0.2
0
1
2
3
4 5 6 7 Horizontal Position (nm)
8
9
10
Fig. 15: 2D and 3D images of a sample substrate with sinusoidal, square, and triangular profile sections obtained by considering the nonlinear creep effect (the units are in nm). According to Fig. 15, the creep effect behaves very similarly to a delay in a dynamic process. To observe the difference between the images acquired by considering the hysteresis and creep models, the errors resulting from the difference between the images obtained by considering the nonlinear and linear effects have been illustrated in Fig. 16. In the creep case, the high level of error at the points the sample ascends during scan is due to the delay in the time interval in which the input increases. Also, in the hysteresis case, the high level of error at the highest and lowest times is due to the discrepancy between the locations of asperities in the nonlinear and linear models.
Fig. 16: The differences between the images obtained from the linear model and the nonlinear models considering the creep effect (right) and hysteresis effect (left) 4-3- Applying the effect of thermal drift on the imaging operation
Using the system in ]46[, the imaging of the standard sample mentioned in the preceding sections while assuming a 2º temperature change during a 10 min scan of the sample substrate has yielded a maximum thermal drift error of about 3.3 nm. The image obtained by considering the thermal drift model and also the drift error relative to the image obtained without the consideration of drift have been shown in Fig. 17.
Fig. 17: The resulting 3D image obtained by considering the thermal drift due to a 2º change of temperature during a sample scan time of 10 min (left), and the difference between the resulting image and the image obtained from the linear model (right)
As is observed, with the passage of time and the scanning of different lines during the imaging process, the thermal drift error increases proportionately, and reaches its maximum value at the end of the operation. A considerable distortion is observed in the output image; which shows that, with regards to the high level of variation that exists in the considered image form, thermal drift can greatly affect the anticipated outcome and therefore, should be considered seriously. 4-4- Considering the simultaneous nonlinear effects of hysteresis, creep and thermal drift on the imaging operation
The coupling effects of creep-hysteresis-thermal drift cannot be obtained by summing the image results from the preceding three sections. Instead, by using the present model, the simultaneous effects of these three nonlinear effects on the final image can be well observed. Fig. 18 shows the obtained image and the error resulting from the simultaneous application of the nonlinear hysteresis, creep and thermal drift effects. Because of a large error due to thermal drift at the end of the operation, the drift error also dominates in the concurrent application of the three nonlinear effects; and the highest error is observed at the end of the process.
Fig. 18: The resulting 3D image obtained by simultaneously considering the nonlinear effects of hysteresis, creep and thermal drift (left image), and the difference between the resulting image and the image obtained from the linear model (right image) By employing the present model, the effects of different parameters on the imaging process can be studied. Also, this model provides an appropriate framework for adding the other nonlinear effects.
5- Conclusion In this paper, we have attempted to demonstrate the simultaneous consideration of the nonlinear effects of hysteresis, creep and thermal drift on the results of imaging by large-scale nanorobots. After presenting the linear equations and the new forms of nonlinear equations including the creep and hysteresis terms, a model was considered for thermal drift, and then the effect of each nonlinearity was validated in a separate work. By applying the presented model and imaging a gold nanoparticle sample with 15 nm diameter moving on two forward and backward travel paths on a mica substrate, it was demonstrated that the presented model can be adequately validated by the existing experimental works. In the backward paths of these images, the hysteresis effects were
clearly visible; and despite the inherent differences in the relevant diagrams, a correct approximation of changes in the presence of hysteresis could be obtained. After a complete verification of the presented model through comparison with experimental works, a sample surface containing three types of asperities (sinusoidal, square and triangular) was imaged as a standard sample, first by assuming the effect of each nonlinearity separately, and then by assuming the simultaneous presence of all three nonlinear effects. In comparison with the linear case, in which the hysteresis effects are not taken into consideration, it is observed that when hysteresis is considered in the system, depending on the scan direction, the asperities on a rough substrate are sensed at different locations. In sketching the hysteresis loops related to the different sections of the considered profile, because of the uniform variations of the images obtained for the sinusoidal and triangular sections, the loops corresponding to these sections show up as standard hysteresis loops; while for the square section, due to the constant input in the maximum and minimum sections of the square wave, two zones with constant minimum and maximum values are observed on the horizontal axis. It was also realized that the creep effect behaves very similarly to a delay in a dynamic process. Finally, by assuming a 2º temperature change during a 10 min scan of sample, the maximum thermal drift error for a standard system (which was used in an experimental work and modeled by the present method) was estimated to be about 3.3 nm. With the passage of time and the scanning of different lines in the imaging process, the drift error increased proportionately and reached its maximum value at the end of the operation. A substantial distortion was observed in the output image; which shows that, considering the high degree of variations that exist in the considered image form, thermal drift can greatly affect the expected outcome, and should be considered seriously. In considering the simultaneous nonlinear effects of hysteresis, creep and thermal drift on an imaging process, it is realized that, because of the large error resulting from thermal drift at the end of the imaging operation, this drift error also prevails in the concurrent application of the three mentioned nonlinear effects, and that the highest error is observed at the end of the process. Finally, considering the complete validation of the presented model, this model can provide an appropriate framework for adding other nonlinear effects in future works.
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Highlights
The simultaneous effects of nonlinearities on imaging by AFM have been studied via a semianalytical approach. All the sub-models have been validated by comparing their results with the findings of previously reported experiments. When hysteresis effects are studied, the asperities on a rough substrate are sensed at different locations. In the creep case, the high level of error obtained during scan at the ascending points of substrate profile. The imaging of a standard sample substrate while assuming a 2º temperature change during a 10 min scan of the substrate has yielded a maximum thermal drift error of about 3.3 nm.
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PII: DOI: Reference:
S0020-7403(17)30098-X http://dx.doi.org/10.1016/j.ijmecsci.2017.01.021 MS3559
To appear in: International Journal of Mechanical Sciences Received date: 11 May 2016 Revised date: 16 December 2016 Accepted date: 13 January 2017 Cite this article as: Sadegh Sadeghzadeh and M.H. Korayem, Role of mechanical and thermal nonlinearities in imaging by Atomic Force Microscope, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2017.01.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Role of mechanical and thermal nonlinearities in imaging by Atomic Force Microscope 1
2
Sadegh Sadeghzadeh1*, M. H. Korayem2 Assistant Professor, School of New Technologies, Iran University of Science and Technology, Tehran, Iran
Professor, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
[email protected]
Abstract In this paper, the effects of nonlinearities such as hysteresis, creep and thermal drift of piezoelectric elements on imaging by Atomic Force Microscope have been studied via a semi-analytical approach. The Generalized Differential Quadrature Method (GDQM) has been used to find the dynamic response of a system. Bouc-Wen and PI hysteresis models have been incorporated with the presented linear model to obtain the effects of nonlinearities; and thermal drift has been augmented in an offline scheme. All the sub-models have been validated by comparing their results with the findings of previously reported experiments. Finally, these nonlinear effects have been applied on AFM-based imaging operations, and the obtained results have been evaluated. In comparison with the linear case in which the hysteresis effects are not taken into consideration, in the nonlinear imaging model, depending on the scan direction, the asperities on a rough substrate are sensed at different locations. In the creep case, the high level of error obtained during scan at the ascending points of substrate profile is due to the delay in the time interval in which the input increases. According to the results, the imaging of a standard sample substrate while assuming a 2º temperature change during a 10 min scan of the substrate has yielded a maximum thermal drift error of about 3.3 nm. At the end, based on the presented comprehensive nonlinear imaging model, the coupled effects of creep-hysteresisthermal drift on the final image have been reported and discussed.
KeyWords; Nonlinearities, Atomic Force Microscope, Imaging, Hysteresis, creep, Thermal drift
1- Introduction The researchers in various technological fields are now able to more effectively control and manipulate the structures of different materials at increasingly lower scales. Today, the new
investigation methods make it possible to perform engineering operations on single atoms, molecules and atomic bonds by means of nanotools ]1[. The science and technology of using such techniques is known as nanorobotics. In general, nanorobotic systems have been classified into two groups of nanoscale and microscale nanorobots ]2[. Microscopes with the ability of performing various operations at nano dimensions and lower have been called ‘large-scale nanorobots’. Macroscale tools are needed to access, identify and manipulate various materials at the nanoscale. Remote-controlled systems are the usual tools which are used for this purpose. Among these tools, the SPMs, and specially the Atomic Force Microscope (AFM), are considered as powerful tools for manipulation and assembly tasks at the nanoscale. With the invention of AFM ]3[, many types of nanorobotic systems, with newly found applications, have been built. Identifying the frictional and surface properties of metals, polymers and biopolymers at the micro/nanoscale, displacing and selfassembling of single molecules, cells, viruses, and acquiring instantaneous images of nanoparticles during manipulation by using simulations based on the relevant model and the measured forces are some of the applications of these nanorobots. Fig. 1 shows the schematic of a type of AFM. The capabilities of nanorobots could be generally classified into two areas of detection and manipulation. Imaging by the AFM at the nano and lower scales is a clear example of the detection process; which can be defined for all the operating modes of nanorobots, and which has its own particular application in each specific operating mode. The influential parameters on manipulation at the nanoscale have been summarized and presented in Table 1.
Fig. 1: The schematic of an AFM nanorobot made by the Bruker Company (left), Enlarged schematic of small parts (right) Table 1: Various areas of manipulation process and its general effective parameters Type of manipulation
Pushing/Pulling
Lifting/Placing
Environment
Air (simple, multibody, ambient temperature, etc.)
Vacuum (vacuum level, sensitivity, etc.)
Cutting
Bending/Buckling
Liquid (plain liquid, Biological and cellular blood, non(soft, hard, genetic, etc.) Newtonian liquid,
Microscope
Object
etc.) Cantilever shape Operating mode Environmental (pincer-shaped, V(contact, non-contact, conditions (humid, shaped, rectangular, tapping) dry, PH range, etc.) etc.)
Tip shape (triangular, rectangular, T-shaped, etc.)
Physical field (metallic, biological, magnetic, etc.) Physical field Special conditions Mechanical behavior (metallic, biological, (humid, dry, PH range, (rigid, flexible) magnetic, etc.) etc.) Dynamics (constant Dimensions (2D, Straight or curved velocity, constant 3D) path acceleration
Shape (sphere, rod, Dimensions (2D, 3D, Mechanical behavior tube, wire, cell, etc.) small, large, etc.) (rigid, flexible)
Substrate
Roughness (low, high)
Process dynamics
Dominant forces
With respect to its nature and functionality, a nanorobot can be considered as equivalent to an ordinary large-scale robot; nevertheless, there is a major difference in the dynamics of contact between components at the two scales.
2- History of developments in nanorobotics field Although, nanorobots can be essentially considered equivalent to ordinary robots, due to the existing limitations in the measuring of forces and in instantaneous sensing, and also because of the presence of unknown physical phenomena, contrary to ordinary robots, numerous challenges are involved in nanorobots, which inherently makes the nature of the existing problems totally different. This has prompted the scientists to develop more powerful and realistic models that can shed more light on the nature of the examined problems, before costly and expensive experiments are undertaken. For example, computational procedures for investigating AFM-based manipulations, especially for the pushing of nanoparticles and nanorods ]4[ have been presented. Considering the existence of surface and intermolecular forces at this scale, there are numerous variables and parameters that affect the motion of these nanoparticles during manipulation. Due to the large number of unknown variables involved in the calculations, it is difficult to perform a quantitative comparison with empirical data; although these comparisons could be very informative ]4[. In order to achieve conceptual and exact models, Sitti and Hashimoto ]5[ presented, for the first
time, a 2D dynamic model for the assembly of micro/nanospheres. Then, based on this model, Taffazoli et al. developed a model that described the dynamic motion modes of nanospheres ]6[. In that work, they had analyzed the conditions of sliding and rolling on a substrate. Some modification were also applied on the mechanism of AFM. For example, the resonant frequencies and flexural sensitivities of an atomic force microscope (AFM) with assembled cantilever probe (ACP) were studied [7]. Korayem et al. obtained the maximum amount of load that could be carried by a microscope, and presented an algorithm for calculating this load ]8[. In [9], the 3D modeling of motion dynamics by using different cantilevers (rectangular, V-shaped and dagger-shaped) was
focused and was simulated the kinematics and dynamics of a manipulation operation. A comprehensive algorithm by which the critical time and force can be determined was developed in [10]. By using the proposed algorithm, the maximum angle of spinning was specified. An algorithm was used in this paper [11] to show in 3D, the manner by which the dynamic forces vary in the manipulation process. In parallel with these works, and as the significance of such research endeavors became evident, other research groups with defined plans concentrated on this subject. A relatively comprehensive account of the research works conducted by these groups and their achievements have been presented in Table 2. These works reflect the scientific and experimental progresses made in the field of nanorobotics. Many of these investigators have specifically focused on experimental works and the development of these works for industrial applications. Such inclinations and outlooks by the researchers have made it quite clear that nanorobotics will play an effective role in the forthcoming progresses and achievements of nanotechnology in the world. The concurrent theoretical and experimental developments of large-scale nanorobots, so far as their applications did not involve biological objects, were good and effective. However, with the development of these tools and the various mediums in which these instruments operated, continuum mechanics-based theories left many questions unanswered. As an approach for answering these questions, Korayem et al. suggested using the molecular dynamics method ]13 ,12[. Nevertheless, there are still challenges regarding the connection between the large-scale and small-scale worlds; which should be addressed. The authors of this paper have recently made a great effort in using multi-scale methods for the analysis of large-scale nanorobots. By employing a multi-scale model, they have even studied the effects of hysteresis on the imaging and manipulation of nano-objects and have revealed that the nonlinear effects should be seriously considered if precise manipulations are intended. However, no work has simultaneously modeled the nonlinear effects of hysteresis, creep and thermal drift; so that the mutual effects of these parameters could be evaluated. In this regard, the present work will attempt to provide an appropriate groundwork for the simultaneous investigation of these aspects. We hope that this study can provide a comprehensive and exact model for nanorobots; so that, anywhere, and by any specific mechanism, an arbitrary nanorobotic system can be analyzed by considering the nonlinear effects. Of course, there are still traces of many errors, which are suggested to be examined; but the groundwork presented in this paper can ensure, with a high degree of reliability, the addition of other error effects in the future.
Table 2: Groups active in the nanorobotics field, and their important innovations and achievements
Name
Research area
Ning Xi A. Requicha S. Devasia M. Sitti T. Fukuda S. Fatikow M. H. Korayem A. Meghdari C. Zhou J. Tamayo
D. Spinosa R. Carpick Y. Sun
Important innovations and achievements
Pushing of nanoparticles, DNA Obtaining instant images from the nanorobotic manipulation, automatic assembly operations on microobjects Control of nanorobotic Experimental studies of nonlinearity effects on manipulations the results of nanorobotic operations Design and development of Implementing all the hysteresis modeling piezoelectric actuators methods in nanorobotic operations Remote control of nanorobotic Presenting micro/nano remote-operating systems manipulations Manipulation of biological Controlled manipulation and deformation of systems living samples Making various AFM-based Combining two nanorobots of AFM and SEM nanotools Modeling nanorobotic tools and Presenting nanorobot models for liquid and operations, Modeling the vacuum environments, Modeling large-, smallnanorobots in various mediums, and multi-scale dynamics etc. Modeling nanorobotic tools and Small-scale dynamic modeling and large-scale manipulations chaos dynamics Building nanostructures by using Making nanometric transistors nanowires and graphene Exciting and measuring the produced responses at a very Making and analyzing mechanical nanosensors small scale Analyzing, designing and controlling of piezoelectric Designing nano-electro-mechanical switches nanowires Investigating system behaviors at Analyzing the friction behavior at a small scale nano and lower scales Controlling the behavior of biological Controlled displacement nanostructures
B. J. Inkson
Special nanomanipulations
Nanomanipulation on different prostheses
Period of activity
Sample of works
2000 till now
]41[
1995-2011 ]41 ,41[ 1993 till now
]41[
1998 till now
]41[
2000 till now
]41[
2002 till now
]02[
2000 till now
]12[
2000 till now
]00[
2003 till now
]02[
1998 till now
]01[
1995 till now
]01[
1996 till now
]01[
1998 till now
]01[
1995 till now
]01[
Designing and making of different controllers Large-scale nonlinear effects of based on the modeling methods of hysteresis and 2003 till now nanorobots creep Modeling and control at nano and Producing nano indentation and controlled 1990 till now A. Ferreira lower dimensions scratches on nanometric surfaces K. Leang
]01[ ]22[
3- Nonlinear dynamics of large-scale nanorobots As precise tools, piezoelectric materials play a vital role in micro/nanorobotics, and they are usually used as actuators and distributed sensors. Mostly, it is these piezo-actuators which are referred to as the large-scale parts of nanorobots. In this paper, the equations of motion for actuators and distributed sensors are obtained by means of the generalized Hamilton’s principle. The presented formulation has a general form and can be applied to all piezo-actuators and piezo-sensors. 3-1- Deriving the nonlinear motion equations
Piezoelectric materials are usually laminated over a base system; and therefore, the main components of nanorobots (microcantilever and piezotube) are made of a multilayer structure. Considering the piezoelectric effect and disregarding the thermal stresses, the structural equations for the kth layer in will be as follows ]24[:
the material coordinate system { }
[ ]{ }
[ ] { }{ }
[ ]{ }
[ ]{ }
{ }
{ }
)2 (
where [ ] is the reduced stiffness matrix, [ ] is the piezoelectric modulus, and [ ] is the dielectric constant of the kth layer in the material coordinate system. Also, { } is known as the residual dipoles factor (used to express the hysteresis), and { } denotes the creep factor. In addition, {E}, {S}, {σ} and { } represent the stress, strain, electrical field and electrical diplacement vectors, respectively. By using the first-degree plate and shell theories, the mechanical displacement field is written as Eq. 2 ]24[. )1 (
In the above equations, x3 is the distance to the neutral plane, u, v and w are the displacement components along directions 1, 2 and 3, u0, v0 and w0 are the displacements of the points of neutral plane along the mentioned directions, and
and
are the rotations about axes 1 and 2,
respectively. The potential energy of a layer of electromechanical plate can be expressed as ∫
∫
)3 (
where T is the surface reactions, u is the mechanical displacement, Q is the electrical charge, electrical potential,
is a volume element and ∫(
considering the kinetic energy (
̇
̇ )
is the
is a surface area element of the jth layer. By ), the Hamilton’s principle for a multilayer shell
is written as
(∑ ∑ ∫
in which
)
)4 (
is the work of external forces and δ, SN and LN denote the variation operator, number
of patches and number of layers, respectively. By applying the presented relations, the motion equations of a shell with two curvatures of radii R1 and R2 will be expressed as follows ]20[:
̈ ̈ ̈ ̈
(
)
̈
(5) ̈ ̈
̈ ̈
) {
(
}
{
∑∫
}
(6)
This is a general form of the equation. By assuming a plate, and by assuming
and
and
, the motion equation of
, the motion equation of a cylinder with radius
will be
obtained ]32[. The forces and moments are computed by integrating the stresses in the structural equations. By substituting these forces and moments into the motion equations, the equations are obtained with respect to displacements. In general, the forces and moments are derived as follows: { { {
} }
}
{ { {
} }
}
{ { {
} }
} { { {
{ } }
{ {
}
} }
}
{ {
} }
{
)7 (
}
)8 (
Considering the structural equations and the fact that the electrical field is applied along the thickness ( ), the electrical forces and moments for the ith element are defined as ̅ {
{
where
}
}
{
{
}
}
̅
{ ̅ } ̅ ̅ ∑∫ { ̅ } ̅
∑∫
∑{ ̅ } ̅ ̅ ∑{ ̅ } ̅
is the surface electrical potential of the kth layer,
the kth layer to the middle plane, and
(9)
is the distance from the mid-section of
is the distance between the middle plane of layers and the
middle planes of the multilayer shell. By using the load equation, and considering the fact that , we can write ]33[:
)21(
|
|
| ̅
̅
|
̅
̅
)22(
̅
̅
is the surface load density of the kth layer in the ith element. Also,
where , where
is the electrical potential on the surface of the considered layer.
is the linear ratio
coefficient, which indicates the direct effect of applied voltage on dipole orientation in piezo materials. When a completely linear system is assumed, this coefficient will be equal to one (1). Since nonlinear parameters are equivalent to electrical displacements and this type of load has been induced by voltage, we can write )21(
where
and
denote the equivalent voltages that produce hysteresis and creep, respectively;
and they are modeled by means of specific hysteresis and creep models. By computing the electrical forces and moments from Eq. 9 and substituting them into Eq. 5, the motion equations are obtained with respect to displacements. By substituting the forces into the motion equations, their operator form versus displacement can be expressed as Eq. 13. ̈ In the above equation,
)23(
and
represent the mechanical th
and electrical terms, respectively, and
is the inertia term of the i equation. The electrical effect
has two parts: reverse effect
and direct effect
. By considering
the damping effect, Eq. 14 can be presented, where c is the damping coefficient and ̂ represents the external load; i.e. ̂
. ̇
̈
̂
)24(
3-2- Solving the motion equations by the Differential Quadrature method
In view of the types of problems involved and the complexity of the equations in special cases (e.g. in multi-patch systems), finding an exact solution for the considered problem will not be possible; and so, it will be necessary to use numerical methods, especially those which produce the least error and involve a smaller number of degrees of freedom. Differential Quadrature (DQ) is an approach that adequately addresses such concerns. In this method, the partial derivative of the nth order of
continuous function f(x, z) with respect to x at a known point
⌋
∑
In the above equation,
(
is expressed by Eq. 15 ]24[.
)
)21(
are the weight function coefficients, N is the number of grid points, and f
represents the function values at the considered points. 3-2-1- The GDQE method
Fig. 2(a) shows a plate with a piezoelectric patch in the middle, and Fig. 2(b) illustrates the correct element configuration of this problem. The presence of a geometrical or force discontinuity makes it difficult to use the ordinary form of the GDQ method. In these cases, the physics and geometry of the problem should be divided into several distinct sections; in other words, domain decomposition should be implemented. In this case, the GDQ problems will be equal in number to the elements. This method is called the Generalized Differential Quadrature Element (GDQE) method. As shown in figure 2, when a patch s placed at the middle of the plate, the plate should be divided into nine parts and separate degrees of freedom should be taken for each of them.
Fig. 2: (a) a plate with a piezoelectric patch, (b) the correct configuration of elements in the GDQE method In the most general case, and with the consideration of proportionate damping, the dynamic motion equations are expressed as Eq. 16. ̇
{ By substituting
̇
̈
)21(
from the second equation into the first equation and converting the obtained
equation to state space, we can write
{ ̇ } { in which
( and
[
]{ ̇ } ̇ are chosen as follows:
)}
)27(
(
)
)28(
Eq. 17 is solved by using numerical methods such as Runge-Kutta; and then the boundary displacements can be determined from relation
.
3-3- The model of nonlinear effects
Because of using piezo-actuators in nanorobts, the nonlinear effects including creep, hysteresis and thermal drift should be considered and addressed in these systems. Following the early studies of creep in AFM, Richter et al. presented the first model for it ]34[. All the researchers have modeled the creep effect by combining several linear mass-spring models and then have incorporated the effect of hysteresis in this model. One of the first and most detailed analyses performed so far is the work of Prandtle and, later, Preisach ]35[. Some researchers have used the Prisach model for control and compensation purposes ]36[. Subsequently, many works have employed new methods for promptly adapting the hysteresis model ]33[. Not all these methods can be appropriately used; and the models do not properly match the empirical results and are even computationally sensitive to reasonable levels of accuracy. Conversely, analytical methods such as the Bouc-Wen method do not have many of these drawbacks. Here, in addition to the Bouc-Wen approach, we have attempted to use the Prandtle-Ishlinski (PI) method in cases in which simpler applications are considered. In this way, the model of nonlinear elements (creep and hysteresis) can be expressed and presented more effectively. 3-3-1- The hysteresis model
Here, the Prandtle-Ishlinski and Bouc-Wen models have been employed for nanorobotic applications. In the Prandtle-Ishlinski (PI) model, operator output
and input
[ ]
is used to define the relationship between
]38[.
∫ In the above equation,
[ ] and q are weight functions with conditions
)21(
.
To describe the system’s hysteresis behavior, an analytical equation is presented in the Bouc-Wen model. Eq. 20 expresses the Bouc-Wen equation for hysteresis displacement ]33[. )11( | ̇ || | | | ̇ ̇ In the above equation, A controls the amplitude of hysteresis, and β, γ and n change the shape and
̇
amplitude of hysteresis (n ≥ 1). 3-3-2- Creep
Each time delay operator can be defined as Eq. 21 ]15[.
̇
)12(
In the above equation, the poles ( ) are negative and
; where
and
denote the degree
of creep and the input, respectively. 3-3-3- The final equations
In view of the presented nonlinear model and the differential quadrature solution method, and with the consideration of the nonlinear effects of hysteresis and creep, and by developing the equations presented in ]44[, the final equations will be obtained as follows:
{
{ ̇}
[ ̇
̇
| ̇ ||
|
̇ |
]{ ̇} |
[
]{ }
)11(
Since, by using the calibration of input voltage, the creep model yields the effect of creep on resulting displacement, in Eq. 22,
represents the input excitation voltage of the system, and we
have ̇ )13(
3-4- Validating the model of nonlinear effects
The free vibration, static, and transient response models of the present method have been validated in ]41[. In view of those validations for piezo-tubes and for different plates and beams, only the
nonlinear model is validated here. 3-4-1- Validating the model of hysteresis in microcantilever
By using the generalized PI method, the results of a double-layer (piezoelectric and copper electrode layers) microcantilever have been compared with the findings of the experimental work in ]42[. The cantilever dimensions are 15 x 2 x 0.3 mm, and the applied voltage is in the form of 40 sin(2πft) or 20 sin(2πft). The width of hysteresis, as a measure of intensity, has been indicated by λ in Fig. 3; and for the rate-dependent case, the hysteresis loops resulting from different frequencies have been illustrated in Fig. 4.
1 Hz Linear Nonlinear
10
0
Experiment (Rakotondrabe, 2008)
20
Displacement ( m)
Displacement ( m)
20
-10
Present Approach
10
0
-10
-20
-20 0
0.5
1 Time (Sec)
1.5
-40
2
-30
-20
-10
(a)
0 10 Voltage(V)
20
30
40
(b)
Fig. 3: (a) comparing the linear and nonlinear dynamic responses, (b) relevant hysteresis loops with a frequency of 1 Hz 200 Hz
10 Hz
15 Experiment (Rakotondrabe, 2008)
Experiment (Rakotondrabe, 2008)
10
10 Displacement ( m)
Displacement ( m)
Present Approach 5
0
-5
Present Approach
5 0 -5 -10
-10 -20
-15
-10
-5
0
Voltage(V)
5
10
15
-15 -20
20
-15
-10
-5
0 5 Voltage(V)
(a)
10
15
20
(b)
300 Hz
600 Hz
30
100 Present Approach Experiment (Rakotondrabe, 2008)
20
Displacement( m)
Displacement ( m)
50 10 0 -10 -20
-50 Present Approach Experiment (Rakotondrabe, 2008)
-30 -40
0
-30
-20
-10
0 10 Voltage(V)
20
30
40
-100 -40
-30
-20
-10
0 10 Voltage(V)
20
30
40
(c) (d) Fig. 4: Hysteresis loops obtained by applying a voltage with (a) amplitude of 20 V and frequency of 10 Hz, (b) amplitude of 20 V and frequency of 200 Hz, (c) amplitude of 40 V and frequency of 300 Hz, (d) amplitude of 40 V and frequency of 600 Hz 3-4-2- Validating the model of hysteresis in the piezo-tube
Fig. 5 shows the hysteresis loops obtained in two cases with sinusoidal and triangular inputs. In this case, the voltage has been applied to a pair of electrodes facing each other. The exterior electrodes all have an angle of 90º; and the voltage is applied in such a way that the piezo-tube has a lateral motion. In both cases, the applied voltage has an amplitude of 1 V and frequency of 30 Hz. Fig. 6 illustrates
the hysteresis loops obtained by exciting the piezo-tube of a non-contact AFM (Burleigh Metris 2000) along the x direction by an excitation voltage with an amplitude of 5 V. 1
0.5
0.5
Displacement (nm)
1
0 -0.5 -1 -1.5 -2
NonLinear 0.02
-0.5 -1 -1.5
Linear 0
0
-2
0.04 0.06 Time(s)
0.08
0.1
Linear NonLinear 0
0.02
0.04 0.06 Time(s)
(a)
0.08
0.1
(b)
1.5 1
Displacement (nm)
1 0.5 0 -0.5
0.5 0 -0.5 -1
-1 -1.5 -1
-0.5
0 Voltage(V)
0.5
-1
1
-0.5
(c)
0 Voltage(V)
0.5
1
(d)
Fig. 5: (a, b) comparing the linear and nonlinear dynamic responses for sinusoidal and triangular inputs, (c, d) relevant hysteresis loops resulting from the sinusoidal and triangular inputs (frequency of 30 Hz) 50 40
Experimental (K. Leang, 2006) Present approach
Displacement( m)
30 20 10 0 -10 -20 -30 -40 -50 -5
-4
-3
-2
-1
0 Input(V)
1
2
3
4
5
Fig. 6: Comparing the hysteresis loops (displacement vs. voltage along the x direction) for the AFM piezo-tube ]43[ 3-4-3- Validating the creep model
The results of the creep model have been compared in Fig. 7(a) with the empirical results of ]11[ for the mentioned cantilever (excited by applying a voltage of 10 V). The direct application of the creep model does not model the initial vibrations of the system; therefore, the proportional damping model has been used for this purpose. Fig. 7(b) shows the enlarged transient response section, with an excitation voltage of 40 V and mass and stiffness coefficients of
and
,
respectively. 45
6
Present Approach
40
Experiment (Rakotondrabe, 2010)
35
Displacement ( m)
Displacement ( m)
8
4 4.5 4.45
2
4.4
30 25 20 15
Experiment (Rakotondrabe, 2010)
10
Present Approach
4.35 0
0
0
5
0.02
10
0.04
15 Time(s) (a)
0.06
20
0.08
0.1
25
5
30
0
0
0.01
0.02 0.03 Time (Sec)
0.04
0.05
(b)
Fig. 7: Comparing the results of the creep model with the empirical results of the cantilever; (a) with an excitation voltage of 10 V, (b) with an excitation voltage of 40 V (enlarged section) 3-5- Thermal drift
If the ambient temperature is not controlled, the variation of temperature in the environment can cause a thermal strain, which is known as thermal drift. The researchers believe that thermal drift is a major cause of error in nanorobots. Here, the general analysis of thermal drift is carried out by dividing the problem domain into macro and micro coordinates. For analyzing the problem in the macro-dimension coordinates, an analytical model has been presented for the thermal circuit, and then the amount of generated heat and the thermal resistance values have been computed for this circuit. Four general types of heat transfer are involved in the mentioned circuit: generated heat, conduction heat transfer, heat transfer via an intermediary medium, and convection heat transfer. In the next step, a numerical solution has been used to analyze the present problem; and by evaluating the solution stability, this method has been compared with the solution of ANSYS software program. It should be mentioned that, for a better estimation of the answer, an approximate solution obtained by simplifying the governing differential equation has also been used. To analyze the system in micro/nano dimensions, the effect of heat on atomic friction has been explored and then, similar to the macro sections, the transfer of heat in micro/nano parts has been evaluated. By writing the resistances and the generated heat values at these scales, approximations of thermal drift have been provided at these dimensions and finally, the total amount of thermal drift has been obtained by finding the resultant of macro and micro/nano drifts. Fig. 8 depicts the mentioned thermal circuits for
the AFM nanorobot. The model presented in ]11[ has interested the authors; and after validating it and identifying its system, a dynamic model has been proposed for it, and eventually compensated.
Rtip
Rtip Rsph Rcontact
Tip
Rcontact1
Tip
Rsph RSub
Rparticle
Particle
Rcontact Substrate RSub
Rcontact2
Visco adhesive layer
RAdh
Substrate Viscoelastic adhesive layer
RSub XY-Piezotube
RAdh
XYPiezotube
Fig. 8: Thermal circuits for large-scale and small-scale sections in nanomanipulation and nanoimaging mechanisms The thermal expansion coefficient of PZTs normally varies in the range of 10-50 µm/ºC. This means that one degree of temperature change will lead to a displacement of about 40-200 nm along the PZT (assuming a 40 mm length for the actuator). For example, the thermal drift of the piezotube of a Vico nanorobot with a PZT-5H type piezoelectric has been modeled and compensated in [45]. In the present work, thermal drift is applied, offline, to the whole system. Following a standard nanomanipulation process, a total drift of about 580 nm has occurred in the x direction. Of course, the amount of total drift obtained in ANSYS is 600 nm; and experimental works have reported a total drift of about 560 nm ]46[. Nevertheless, a low amount of error has also been reported, which is due to the possible differences in ambient conditions and the effects of electronic equipment operating in the environment ]46[. Figure 9 shows an algorithm for implementation of presented three nonlinear effects in the nano imaging simulations. A completely same algorithm could be presented for nano manipulation.
Make the Force vector on MC (Forces with asterisks)
State Space Representation
Solution
A, B, C and D construction Make the state dependent force vector on the MC end
Obtain Sensed Voltage
Discretization GDQ or GDQE Approaches
Input Voltage Existent Electrical Forces
+ +
Thermal Drift
+ +
Obtained Profile
Existent Mechanical Forces
V
Convert to Displacement
Calculate the forces on the MC end
+
Hysteresis and creep model
Select Displacement of MC end
The Substrate profile
Fig. 9: Algorithm of nano imaging by using hysteresis, creep and thermal drift models
4- Applying the nonlinear effects on the imaging operation 4-1- Effects of hysteresis 4-1-1- Validation through comparison with empirical results
In addition to validating the nonlinear dynamics of hysteresis in cantilever and piezotube, the effects of hysteresis on imaging can also be investigated through comparison with the empirical results in ]16[. Fig. 10 shows a comparison of the simulation results related to the forward and backward travel
paths of gold nanoparticles of 15 nm diameter on a mica substrate. Here, there are three nanoparticles with a specific arrangement next to each other. By using this image, proper positions have been considered for the nanoparticles, so that the images of their forward travel paths could be approximately obtained in this figure. Then, in the backward paths, the hysteresis effects will be prominently displayed, and the validity of the presented model can be evaluated. Regardless of the differences that inherently exist in the diagrams, a correct approximation of directional changes in the presence of hysteresis is revealed. The presence of hysteresis causes the nanoparticle to experience a totally new route in the backward travel path. This can be observed in Fig. 11 by simultaneously plotting the forward and backward paths.
20
20
Simulation result Experimental result [Requicha, 2010]
Simulation result Experimental result [Requicha, 2010] 15
Profile (nm)
Profile (nm)
15
10
10
5
5 Backward Scan 0
Forward Scan 0
0
10
20
30 40 X direction (nm)
50
60
0
10
20
30 40 X direction (nm)
50
60
70
Fig. 10: Comparing the forward and backward travel paths of gold particles of 15 nm diameter on a mica substrate with the experimental results 18 Experiment (Requicha, 2010)
Simulation (Present Approach)
16 14
Profile (nm)
12 10 8 6 4 2 0
0
10
20
30 40 X Direction (nm)
50
60
70
Fig. 11: Comparing the forward and backward travel paths and the hysteresis loops obtained in imaging gold nanoparticles 4-1-2- Applying the hysteresis on the imaging of a standard sample
Fig. 12 illustrates the results of imaging a sample surface containing three different types of asperities (sinusoidal, square, and triangular. The unit used for this figure and subsequent figures is the nm.
Linear 1.2 Profile
Obtained image
1
Amplitude (nm)
0.8 0.6 0.4 0.2 0 -0.2
0
2
4 6 Horizontal Position (nm)
8
10
Fig. 12: 2D and 3D images of a sample substrate with sinusoidal, square, and triangular profile sections obtained without considering the nonlinear effects (the units are in nm). As is observed, the present model has been able to adequately image the examined profile. Of course, in the presence of nonlinear effects such as hysteresis, creep and thermal drift, the result would be completely different; and the nonlinear model should be able to adequately reveal the difference between the results obtained by considering the nonlinear and linear effects. In comparison with the linear case in which the hysteresis effects are not taken into consideration, Fig. 13 shows the results obtained by imaging the profile presented in Fig. 12 and considering a weak hysteresis effect. The imaging has been performed by means of a non-contact AFM (Burleigh Metris 2000). The difference between the obtained results and those of the linear model has been illustrated in Fig. 13(b). When hysteresis is considered in the system, depending on the scan direction, the asperities on a rough substrate are sensed at different locations. Hysteresis and Linear comparison 1.2 Linear Model
Nonlinear Model (Hysteresis)
1
Amplitude (nm)
0.8 0.6 0.4 0.2 0 -0.2
0
2
4 6 Horizontal Position (nm)
8
10
Fig. 13: 2D and 3D images of a sample substrate with sinusoidal, square, and triangular profile sections obtained by considering the nonlinear hysteresis effect (the units are in nm). Fig. 14 illustrates the hysteresis loops produced in the scanning of three sinusoidal, square and triangular asperities by plotting the obtained image versus the considered profile. As is observed, due to the uniform changes of the images obtained for the sinusoidal and triangular sections, the loops
corresponding to these sections display a standard hysteresis loop; while for the square section, because of using a constant input in the maximum and minimum sections of the square wave, two zones with constant minimum and maximum values in the horizontal direction can be observed in this diagram. Hysteresis Loops 1.2 1
Hysteresis loop for Sin part Hysteresis loop for Square part Hysteresis loop for triangular part
Image
0.8 0.6 0.4 0.2 0 -0.2 -0.2
0
0.2
0.4 0.6 Profile
0.8
1
1.2
Fig. 14: Hysteresis loops produced in the scanning of three modeled sinusoidal, square and triangular asperities 4-2- Considering the creep effect in the imaging operation
By using the results of the validation section, for the nonlinear creep model, a system with the specifications given in Fig. 7 has been considered. By imaging a standard sample substrate with a combination of sinusoidal-square-triangular profile sections (presented in the preceding section), the 2D and 3D images of that sample obtained by considering the nonlinear creep effect have been depicted in Fig. 15. Creep and Linear comparison 1.2 Linear Model Nonlinear Model (Creep) 1
Amplitude (nm)
0.8
0.6
0.4
0.2
0
-0.2
0
1
2
3
4 5 6 7 Horizontal Position (nm)
8
9
10
Fig. 15: 2D and 3D images of a sample substrate with sinusoidal, square, and triangular profile sections obtained by considering the nonlinear creep effect (the units are in nm). According to Fig. 15, the creep effect behaves very similarly to a delay in a dynamic process. To observe the difference between the images acquired by considering the hysteresis and creep models, the errors resulting from the difference between the images obtained by considering the nonlinear and linear effects have been illustrated in Fig. 16. In the creep case, the high level of error at the points the sample ascends during scan is due to the delay in the time interval in which the input increases. Also, in the hysteresis case, the high level of error at the highest and lowest times is due to the discrepancy between the locations of asperities in the nonlinear and linear models.
Fig. 16: The differences between the images obtained from the linear model and the nonlinear models considering the creep effect (right) and hysteresis effect (left) 4-3- Applying the effect of thermal drift on the imaging operation
Using the system in ]46[, the imaging of the standard sample mentioned in the preceding sections while assuming a 2º temperature change during a 10 min scan of the sample substrate has yielded a maximum thermal drift error of about 3.3 nm. The image obtained by considering the thermal drift model and also the drift error relative to the image obtained without the consideration of drift have been shown in Fig. 17.
Fig. 17: The resulting 3D image obtained by considering the thermal drift due to a 2º change of temperature during a sample scan time of 10 min (left), and the difference between the resulting image and the image obtained from the linear model (right)
As is observed, with the passage of time and the scanning of different lines during the imaging process, the thermal drift error increases proportionately, and reaches its maximum value at the end of the operation. A considerable distortion is observed in the output image; which shows that, with regards to the high level of variation that exists in the considered image form, thermal drift can greatly affect the anticipated outcome and therefore, should be considered seriously. 4-4- Considering the simultaneous nonlinear effects of hysteresis, creep and thermal drift on the imaging operation
The coupling effects of creep-hysteresis-thermal drift cannot be obtained by summing the image results from the preceding three sections. Instead, by using the present model, the simultaneous effects of these three nonlinear effects on the final image can be well observed. Fig. 18 shows the obtained image and the error resulting from the simultaneous application of the nonlinear hysteresis, creep and thermal drift effects. Because of a large error due to thermal drift at the end of the operation, the drift error also dominates in the concurrent application of the three nonlinear effects; and the highest error is observed at the end of the process.
Fig. 18: The resulting 3D image obtained by simultaneously considering the nonlinear effects of hysteresis, creep and thermal drift (left image), and the difference between the resulting image and the image obtained from the linear model (right image) By employing the present model, the effects of different parameters on the imaging process can be studied. Also, this model provides an appropriate framework for adding the other nonlinear effects.
5- Conclusion In this paper, we have attempted to demonstrate the simultaneous consideration of the nonlinear effects of hysteresis, creep and thermal drift on the results of imaging by large-scale nanorobots. After presenting the linear equations and the new forms of nonlinear equations including the creep and hysteresis terms, a model was considered for thermal drift, and then the effect of each nonlinearity was validated in a separate work. By applying the presented model and imaging a gold nanoparticle sample with 15 nm diameter moving on two forward and backward travel paths on a mica substrate, it was demonstrated that the presented model can be adequately validated by the existing experimental works. In the backward paths of these images, the hysteresis effects were
clearly visible; and despite the inherent differences in the relevant diagrams, a correct approximation of changes in the presence of hysteresis could be obtained. After a complete verification of the presented model through comparison with experimental works, a sample surface containing three types of asperities (sinusoidal, square and triangular) was imaged as a standard sample, first by assuming the effect of each nonlinearity separately, and then by assuming the simultaneous presence of all three nonlinear effects. In comparison with the linear case, in which the hysteresis effects are not taken into consideration, it is observed that when hysteresis is considered in the system, depending on the scan direction, the asperities on a rough substrate are sensed at different locations. In sketching the hysteresis loops related to the different sections of the considered profile, because of the uniform variations of the images obtained for the sinusoidal and triangular sections, the loops corresponding to these sections show up as standard hysteresis loops; while for the square section, due to the constant input in the maximum and minimum sections of the square wave, two zones with constant minimum and maximum values are observed on the horizontal axis. It was also realized that the creep effect behaves very similarly to a delay in a dynamic process. Finally, by assuming a 2º temperature change during a 10 min scan of sample, the maximum thermal drift error for a standard system (which was used in an experimental work and modeled by the present method) was estimated to be about 3.3 nm. With the passage of time and the scanning of different lines in the imaging process, the drift error increased proportionately and reached its maximum value at the end of the operation. A substantial distortion was observed in the output image; which shows that, considering the high degree of variations that exist in the considered image form, thermal drift can greatly affect the expected outcome, and should be considered seriously. In considering the simultaneous nonlinear effects of hysteresis, creep and thermal drift on an imaging process, it is realized that, because of the large error resulting from thermal drift at the end of the imaging operation, this drift error also prevails in the concurrent application of the three mentioned nonlinear effects, and that the highest error is observed at the end of the process. Finally, considering the complete validation of the presented model, this model can provide an appropriate framework for adding other nonlinear effects in future works.
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Highlights
The simultaneous effects of nonlinearities on imaging by AFM have been studied via a semianalytical approach. All the sub-models have been validated by comparing their results with the findings of previously reported experiments. When hysteresis effects are studied, the asperities on a rough substrate are sensed at different locations. In the creep case, the high level of error obtained during scan at the ascending points of substrate profile. The imaging of a standard sample substrate while assuming a 2º temperature change during a 10 min scan of the substrate has yielded a maximum thermal drift error of about 3.3 nm.