Purdue University
Purdue e-Pubs
Birck and NCN Publications
Birck Nanotechnology Center
June 2008
Invited Article: VEDA: A web-based virtual
environment for dynamic atomic force microscopy
John Melcher
Purdue University, jmelcher@purdue.edu
Shuiqing Hu
Purdue University - Main Campus, shuiqing@purdue.edu
Arvind Raman
Birck Nanotechnology Center, Purdue University, raman@purdue.edu
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Melcher, John; Hu, Shuiqing; and Raman, Arvind, "Invited Article: VEDA: A web-based virtual environment for dynamic atomic force
microscopy" (2008). Birck and NCN Publications. Paper 149.
http://docs.lib.purdue.edu/nanopub/149
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REVIEW OF SCIENTIFIC INSTRUMENTS 79, 061301 共2008兲
Invited Article: VEDA: A web-based virtual environment for dynamic atomic
force microscopy
John Melcher, Shuiqing Hu, and Arvind Ramana兲
School of Mechanical Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette,
Indiana 47907, USA
共Received 22 January 2008; accepted 11 May 2008; published online 24 June 2008兲
We describe here the theory and applications of virtual environment dynamic atomic force
microscopy 共VEDA兲, a suite of state-of-the-art simulation tools deployed on nanoHUB
共www.nanohub.org兲 for the accurate simulation of tip motion in dynamic atomic force microscopy
共dAFM兲 over organic and inorganic samples. VEDA takes advantage of nanoHUB’s
cyberinfrastructure to run high-fidelity dAFM tip dynamics computations on local clusters and the
teragrid. Consequently, these tools are freely accessible and the dAFM simulations are run using
standard web-based browsers without requiring additional software. A wide range of issues in
dAFM ranging from optimal probe choice, probe stability, and tip-sample interaction forces, power
dissipation, to material property extraction and scanning dynamics over hetereogeneous samples can
be addressed. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2938864兴
I. INTRODUCTION
Dynamic atomic force microscopy 共dAFM兲 has emerged
as a powerful tool for imaging, measuring, and manipulating
matter at the atomic and molecular scale. It consists of a
oscillating nanoscale tip at the end of a resonant microcantilever that interacts with the sample via short and long range
forces. At the same time, as a dAFM probe interrogates a
sample, the resulting probe-tip dynamics may contribute to
image artifacts and unreliable property measurements. Thus,
the topographic images generated in dAFM scans are actually a cumulative result of nonlinear interaction forces between the probe tip and the sample1–7 cantilever probe
dynamics,8–12 tip-sample geometry convolution,13–15 and the
dAFM feedback control system.16–18 Ultimately, the limits of
dAFM nanometrology depend on the ability to correctly deconvolve these effects or eliminate them in some manner.
This is one critical need than can be addressed via accurate
mathematical simulations of the scanning process. Additionally, the measurement of material properties via dAFM requires model-based inversion of measured dynamics 共amplitude, phase, higher harmonics, power dissipation兲 to
nanoscale material properties 共surface potential, sample viscosity, sample elasticity, adhesion, charge density, and magnetic moment兲. High-fidelity mathematical simulations are
also critical for quantitative material property measurements
using dAFM. Finally, due to a large parameter space and
nonintuitive underlying nonlinear dynamics, the task of the
dAFM experimentalist to choose an optimal cantilever, amplitude and set point for an application often includes much
trial and error. Where intuition fails, simulations can provide
valuable insight into the fundamental physics of this
instrument.
Simulations of the oscillating probe-tip approaching and
a兲
Electronic mail: raman@purdue.edu.
0034-6748/2008/79共6兲/061301/11/$23.00
interacting with the sample have provided key insights into
aspects of the probe dynamics that are also present while
imaging.2,19–21 In addition, verification of analytical formulas
in dAFM are often achieved through simulation of approaching the sample.22,23 Many authors have also chosen to study
physical phenomena in dAFM with “virtual” microscopes
where the feedback control effort is modeled in simulations
of the cantilever interacting with the sample.24–30 By this
approach, one can study the effect of the controller dynamics
and various noise sources on topographic images, imaging
forces, power dissipation, etc. However, these efforts have
been largely concentrated on the noncontact 共frequency
modulated兲 community and have also not taken advantage of
the unique capabilities of cyberinfrastructure.
In this article, we introduce virtual environment for
dAFM 共VEDA兲, a suite of high-powered, web-based simulation tools for dAFM under development on nanoHUB 共Ref.
31兲. VEDA currently includes the dynamic approach curves
共DACs兲 tool, which simulates a dAFM probe excited near
resonance and introduced to a sample 关Fig. 1共a兲兴, and the
amplitude modulated scanning 共AMS兲 tool, which simulates
an amplitude modulated scan over heterogeneous samples
关Fig. 1共b兲兴. Both tools feature 共a兲 accurate models of microcantilever dynamics, 共b兲 realistic tip-sample interaction force
models, and 共c兲 accurate, high-speed, FORTRAN-based, numerical integration schemes that are well suited for nonsmooth, nonlinear differential equations. In addition, the AMS
tool includes controller dynamics based on the lock-in amplifier amplitude error signal. Both tools allow users to investigate the contributions of different operating conditions
and cantilever and sample properties and produce detailed
information about the probe dynamics and nature of the tipsample interaction. VEDA simulation tools are available on
nanoHUB, which is a web-based interface for researchers,
educators and students interested in science and technology
at the nanoscale. NanoHUB is the web portal for the
79, 061301-1
© 2008 American Institute of Physics
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061301-2
Rev. Sci. Instrum. 79, 061301 共2008兲
Melcher, Hu, and Raman
FIG. 1. 共Color兲 共a兲 A schematic of DACs simulation of an excited cantilever
approaching the sample. 共b兲 A schematic of the AMS simulation of a topographical scan over a sample.
National Science Foundation funded Network for Computational Nanotechnology 共NCN兲. NanoHUB allows its users an
open access to research seminars, educational modules, and
simulation tools such as VEDA.
In what follows, we describe the models for tip-sample
interaction, cantilever dynamics and controller dynamics,
and the numerical integration schemes which are implemented in VEDA and discuss the methodology behind the
each tool. Finally, we present several example simulations
for validation with prior work as well as demonstration of
the insight gained through simulations in dAFM.
II. MODELING
VEDA simulations are based on accurate models of tipsample interaction, cantilever probe dynamics and controller
dynamics. These models ensure realistic predictions of the
cantilever response as the dAFM probe vibrates in close vicinity of the sample.
A. Tip-sample interaction forces
The total tip-sample interaction force Fts共d , ḋ兲 = Ftsc共d兲
+ Ftsnc共d , ḋ兲 acting between the tip of the dAFM probe and
the sample consists of a conservative component Ftsc共d兲 and
a nonconservative component Ftsnc共d , ḋ兲. At present, in
VEDA, the conservative interaction model is continuous but
nonsmooth, consisting of a van der Waals force regime32,33
and a contact regime based on the Derjaguin–Müller–
Toporov 共DMT兲 model:33,34
Ftsc共d兲 =
冦
−
HR
,
6d2
d ⬎ a0 ,
HR 4
− 2 + E*冑R共a0 − d兲3/2 , d 艋 a0 ,
6a0 3
冧
共1兲
where d is the gap between the tip and the sample, a0 is the
intermolecular distance, H is the Hamaker constant, R is the
2
兲 / Etip + 共1
radius
of
the
tip,
and
E* = 关共1 − ␯tip
2
−1
− ␯sample兲 / Esample兴 . E and ␯ represent elastic moduli and
Poisson’s ratios. The nonconservative interactions can be included as an option in a simulation based on the Kelvin–
Voigt viscoelastic contact damping model19
Ftsnc共d,ḋ兲 = − ␩ḋ冑R共a0 − d兲,
共2兲
where ␩, is the sample viscosity. Equation 共1兲 is valid for
stiff samples with low adhesion and probes with sharp tip
radii.33 Viscoelastic contact damping 关Eq. 共1兲兴 is a common
FIG. 2. 共Color兲 Continuous, uniform, rectangular microcantilever with base
excitation y = Abase cos ␻t, linear mass density ␳c, elastic modulus Ec, and
area moment Ic and the corresponding point-mass model with equivalent
mass mi, stiffness ki, and base excitation y i.
feature on polymer surfaces19 and biological materials.35 In
many instances, these models may be use extended outside
their valid regimes to provide qualitative understanding.
B. Cantilever dynamics
The dynamics of the cantilever probe are modeled by a
single degree of freedom 共SDOF兲, point-mass model, which
is derived from the continuous beam equation. We begin
with the derivation of the acoustically 共dither peizo兲 excited
cantilever model for rectangular cantilevers in ambient conditions but later, we discuss how this model can be extended
arbitrary geometry cantilevers, ultra high-vacuum 共UVH兲 environments and other forms of excitation. From the classical
Bernoulli–Euler beam theory, partial differential equation
governing small deflections of the slender, rectangular
dAFM cantilever in a ground-fixed 共inertial兲 frame in the
absence of tip-sample interactions can be written as
␳cẅ共x,t兲 + ␥ f ẇ共x,t兲 + EcIc
⳵4w共x,t兲
= 0,
⳵x4
共3兲
where x is the axial coordinate along the cantilever’s longitudinal axis, Lc is the length, dots represent temporal derivatives, ␥ f is the hydrodynamic damping coefficient, and ␳c,
Ec, and Ic are the linear density, elastic modulus, and area
moment of inertia, respectively. The absolute deflection
w共x , t兲 = Z共t兲 + y共t兲 + u共x , t兲 is composed of the separation
from the sample Z共t兲, the base excitation y共t兲 = Abase cos ␻t,
and the deflection in the noninertial frame attached to the
base u共x , t兲 共Fig. 2兲.
Typically, in dAFM, a photodiode measures the bending
angle u⬘共L , t兲 and infers the deflection u共L , t兲 relative to the
base, therefore, we proceed to model the dynamics of the
cantilever in the noninertial, moving frame attached to the
cantilever’s base. Translating Eq. 共3兲 into a noninertial frame
attached to the moving base of the cantilever leads to36
␳cü共x,t兲 + ␥ f u̇共x,t兲 + EcIc
⳵4u共x,t兲
= − ␳cÿ共t兲 − ␥ f ẏ共t兲. 共4兲
⳵x4
Our goal is to discretize Eq. 共4兲 into a SDOF model
describing the dynamics of the dAFM probe tip in the noninertial frame. In ambient environments, a single eigenmode
of the cantilever beam ⌽i共x兲 is typically sufficient to describe
dynamics of the continuous probe.11 The appropriate discreti-
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061301-3
Rev. Sci. Instrum. 79, 061301 共2008兲
Virtual environment for dynamic AFM
zation of Eq. 共4兲 in terms of the generalized coordinate
q共t兲 = u共L , t兲, and including the tip-sample interaction force
Fts yields the equivalent point-mass model12
miq̈ + 共mi␻i/Qi兲q̇ + kiq = Fts共d,ḋ兲 + Fi共t兲,
共5兲
where mi = ␳cLc / 4, ki = EcIc␣4i / 4L3c , and Qi and ␻i are the
equivalent mass and stiffness, quality factor and natural frequency of the ith eigenmode, respectively 共Fig. 2兲.37 ␣i is ith
solution to the dispersion relation cos ␣ cosh ␣ + 1 = 0. Finally, for Qi ⲏ 10, the excitation force Fi共t兲 may be expressed
as38
Fi共t兲 = 共␻/␻i兲2kiy i共t兲,
共6兲
where y i共t兲 = Biy共t兲 is the equivalent base excitation and Bi
= 兰L0 c⌽i共x兲dx / 兰L0 c⌽2i 共x兲dx is a modal parameter.39
For the purpose of the simulation, we choose to nondimensionalize the key spatial parameters by the initial amplitude far from the sample A0 and temporal parameters by the
natural frequency ␻i. Allowing primes to indicate derivatives
with respect to the scaled time ␶ = ␻it, Eq. 共5兲 becomes
q̄⬙ + 共1/Qi兲q̄⬘ + q̄ = F̄ts共d̄,d̄⬘兲 + Ābase⍀̄2Bi cos ⍀̄␶ ,
共7兲
where q̄ = q / A0, d̄ = d / A0, Ābase = Abase / A0, and ⍀̄ = ␻ / ␻i. Finally, the nondimensional tip-sample interaction force F̄ts can
be written as
冦
− CvdW/d̄2 ,
d̄ ⬎ ā0 ,
F̄ts共d̄,d̄⬘兲 = − CvdW/ā20 + CDMT共ā0 − d̄兲3/2
冑
− Dtsd̄⬘ ā0 − d̄,
d̄ 艋 ā0 ,
冧
共8兲
where ā0 = a0 / A0, CvdW = HR / 6kiA30, CDMT = 4冑RA0 / 3kiE*,
and Dts = ␩␻i冑RA0 / ki. At present, Eq. 共7兲 is the model for
cantilever dynamics implemented in both VEDA tools.
While Eq. 共7兲 was formulated for acoustically 共dither
piezo兲 excited rectangular cantilevers oscillating in ambient
conditions, coincidentally, it may be extended to a few additional scenarios. The key observation to note is that the quality factor in ambient conditions is typically quite large 共on
the order of 100兲. This had two important ramifications. For
near resonant excitations, the base motion of cantilever y
does not contribute substantially to instantaneous tip-sample
gap d or its rate ḋ. Second, for large quality factors, the
bandwidth of the resonance peak is also very small. This
means the factor of 共␻ / ␻i兲2 in the excitation force 关Eq. 共6兲兴
can be considered constant for near resonant excitations.
This allows the model for cantilever dynamics 关Eq. 共7兲兴 to be
extended to UVH environments and for magnetic
excitations.40 Finally, experiments involving nonrectangular
dAFM probes may also be simulated reasonably using Eq.
共7兲 in VEDA if the correct probe stiffness is used.12 While
the modal parameter Bi may be unknown for a particular
nonrectangular cantilever, the precise value is not important
as long as the quality factor is sufficiently large.
FIG. 3. Geometric approximation for imaging artifacts due to finite probetip sizes. For the actual sample height h at a position x, a convolved sample
height h* is observed at position x due to a contact with the sample at
position x*.
C. Lock-in amplifier and controller models
Scanning simulations also require reasonable models for
the lock-in amplifier and controller dynamics as the dAFM
probe scans the sample. The amplitude and phase of the deflection signal are determined by a lock-in amplifier. For a
period of time referred to as the lock-in time constant, the
amplitude and phase at the excitation frequency are determined by a Fourier transform of the deflection signal over a
period of time ⌬t. To avoid leakage in the frequency spectrum, ⌬t is adjusted according to be the nearest integer of
␦t␻ / 2␲, where ␦t is the user specified lock-in time constant.
The AMS tool additionally models shot noise according to a
user specified signal-to-noise ratio in the deflection wave
form.
While scanning, the controller objective is to maintain a
constant tip amplitude as the cantilever probes the sample in
the presence of tip-sample interaction forces. We choose a
commonly used proportional-integral 共PI兲 controller to maintain the desired amplitude. More specifically, the feedback
parameter is the amplitude error e, which is defined as
共9兲
e = A/A0 − Asp ,
where A and Asp are the current amplitude and the set-point
amplitude ratio, respectively. Based on the amplitude error,
the Z distance from the sample is adjusted according to
i=n
⌬Z共tn兲 = − K Pe共tn兲 − KI⌬t 兺 e共ti兲,
共10兲
i=1
where K P and KI are the proportional and integral gain constants, respectively, and ti = i⌬t.
D. Tip-sample convolution
Scanning simulations also include the option to simulate
image artifacts due to the finite size of the tip probe. However, we stress that such approximations consider only
geometry14 and do not account for differences in the tipsample interaction potential itself or any resulting nonconservative interactions. Figure 3 illustrates the actual tip-sample
surface h and geometrically convolved surface h*. For continuous functions, h* may be determined algebraically, however, since the scanning tool includes the option of a sharp
discontinuous step, the convolved surface is found by itera-
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061301-4
Rev. Sci. Instrum. 79, 061301 共2008兲
Melcher, Hu, and Raman
and is around two orders faster than MATLAB solvers such as
46
ODE45 and ODE15S.
III. OVERVIEW
In this section we provide a brief overview of the DAC
and AMS simulation tools. We describe the graphical user
interface 共GUI兲 in terms of the input parameters and output
data available. We also discuss the general methodology followed in the simulations.
A. Graphical user interface
FIG. 4. 共Color兲 A simplified comparison between the DDASKR routine and
conventional integration routines. Both fixed and adaptive time step routines
integrate from A to C without detecting the boundary between contact and
noncontact regimes. The DDASKR routine solves for the intersection with
the sample surface and creates an intermediate B⬘ and appropriate subsequent intermediates until reaching point B.
tion. Not surprisingly, we find that the tip geometry has a
substantial effect on the measured topography for sample
features on the same order or smaller than the tip radius.
E. Numerical integration
Next, we discuss the numerical integration routine
implemented in VEDA and why it is particularly well suited
for dAFM simulations. The DMT tip-sample interaction
model described above and other modes, such as Johnson–
Kendall–Roberts 共JKR兲41 and the capillary force model,6 involve nonsmooth and even discontinuous tip-sample interactions. Numerical integration of nonsmooth and discontinuous
differential equations requires careful consideration to ensure
accurate results. Accordingly, the DDASKR routine with a
root finding algorithm based on DASPK differential algebraic equation 共DAE兲 software package is used.42–45
Figure 4 illustrates the key advantage of the DDASKR
routine over conventional rountines. For simplicity of argument, first consider explicit integration routines, such as the
explicit Euler method, where y ⬘ = F共t , y兲 are solved explicitly
by y n+1 = y n + h · F共tn , y n兲 for an initial condition y共t0兲 = y 0 and
time step h. As shown in Fig. 4, explicit fixed-step routines
will integrate from the initial position at A across the sample
surface to C based on the noncontact forces at A. Similarly,
explicit adaptive time steps will also pass through the sample
surface to C before realizing the presence of stiff contact
forces. In reality, more advanced techniques, such as fourth
order Runge–Kutta routines, implicit backward differentiation formulas, and numerical differentiation formulas are often used for stiff differential equations,46 however, these
techniques generally do not solve for the intersection of the
tip and the sample boundary.
The DDASKR routine is particularly well suited for such
equations because of a root finding algorithm that solves for
the precise intersection of the tip with the sample surface and
creates the appropriate intermediates thereafter. This is a key
advantage in simulation of dAFM probe dynamics where the
tip-sample interaction forces change drastically when crossing the boundary between noncontact and contact force regimes. Additionally, the DDASKR routine is FORTRAN based
The GUI allows the user to specify input parameters for
a simulation as well as view the results. Rappture,47 an opensource toolkit, is used to construct each GUI. To provide
flexibility, the GUI contains several input parameters which
are sorted into the panels. Input panels for the DAC tool are
shown in Figs. 5共a兲–5共c兲 and for the AMS tool in Figs.
5共d兲–5共f兲. Each panel contains a group of related parameters
required for the simulation. Default values and limits are
imposed on each parameter, however, some knowledge of
what is practical may be required.
Once a simulation has concluded, the results are plotted
in the GUI. Some manipulation of these plots is possible
within the GUI, however, users are encouraged to download
the data for postprocessing. Subsequent simulations may be
performed 共and plotted together兲 simply by returning to the
input panels of GUI and changing one or more parameters.
B. Dynamic approach curves tool
The DAC tool simulates a dAFM cantilever excited and
approaching a sample 关Fig. 1共a兲兴. The simulation is performed over some range of the approach distance Z from Z0
to Z f . The Z datum is consistent with the d datum defined for
the tip-sample interaction models, i.e., the sample surface is
located at Z = a0. Choosing Z0 ⬎ Z f causes the cantilever to
approach the sample, while choosing Z0 ⬍ Z f results a retraction curve. The default Z range is Z0 = A0 + 5 nm and Z f = 0.
The initial conditions of the dAFM probe are chosen to
be close to the steady state solution to minimize transients.
For Z − a0 艌 A0, the initial conditions 关q̄共0兲 = cos ␾ , q̄⬘共0兲
= ⍀̄ sin ␾兴, where, tan ␾ = ⍀̄ / 关Qi共1 − ⍀̄2兲兴 are approximated
by the unconstrained, linear vibrations. For Z − a0 艋 A0, the
magnitudes of q̄共0兲 and q̄⬘共0兲 are reduced by a factor of
共Z0 − a0兲 / A0. Prior to approaching the sample, at Z = Z0, 10Qi
transient excitation cycles are allowed to pass. Users are discouraged from choosing Z0 in the bistable region where two
amplitude solutions are possible.
C. Amplitude modulated scanning tool
The AMS tool simulates a dAFM cantilever scanning
over a sample controlled by amplitude modulation feedback
关Fig. 1共b兲兴. For the simulation the probe is placed at Z = A0
+ a0 from the sample with initial conditions 共q̄0 = 0 , q̄0⬘ = ⍀̄兲.
20Qi transient excitation cycles are computed prior to the
beginning of the scanning simulation during that time, the PI
controller is adjusting the Z position to achieve the set-point
amplitude. Users are encouraged to first perform a DAC
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061301-5
Virtual environment for dynamic AFM
Rev. Sci. Instrum. 79, 061301 共2008兲
FIG. 5. 共Color兲 Screenshots from the GUIs of the DAC and AMS tools showing the input panels. The input panels of DAC are 共a兲 operating conditions and
cantilever properties, 共b兲 tip-sample interaction properties, and 共c兲 simulation parameters. The input panels of AMS include 共e兲 operating conditions, cantilever,
simulation, 共f兲 tip and substrate properties, and 共g兲 feature properties. Additional information about an individual input parameter can be found in the GUI
itself or in the comprehensive online manual.
simulation with the same relevant parameters to determine a
set-point amplitude which is in the monostable repulsive regime of oscillation.
IV. EXAMPLE SIMULATIONS
In this section, we present several example simulations
which were performed on VEDA according to parameter values listed in the corresponding table. For brevity, we have
only presented select results in each example. The full set of
output data available for the user to download in the DACs
tool includes amplitude and phase of oscillation, mean 共time
averaged兲, peak attractive and repulsive interaction forces,
power dissipated by nonconservative interactions, and indentation all versus the Z separation of the base of the cantilever.
In addition, there is the option of including the option of
displaying and recording the tip deflection waveform and
accompanying tip-sample interaction history for a specified Z
location and duration. The AMS tool provides the user with
plots of the measured topography, measurement error, amplitude error, mean, peak attractive and peak repulsive interaction forces, power dissipated by nonconservative forces, and
indentation versus the displacement in the scan direction. For
each plot, the data are available for download in user specified formats.
A. Attractive and repulsive regimes of oscillation
The first example examines jumps in amplitude due to
the coexistence of attractive and repulsive regimes of oscillation and provides a comparison between simulations in2
and a DAC simulation using the input parameter values in
Table I. The results are consistent for the large amplitude
simulations, however, for A0 = 10 nm VEDA does not predict
the jump to repulsive oscillations 共Fig. 6兲.
As previously mentioned, conventional integration routines will tend to overpredict indentations and corresponding
interaction forces. Under certain circumstances, this overpreTABLE I. Simulation parameter values for the attractive and repulsive regimes of oscillation example 共Sec. IV A兲.
Operating conditions and cantilever properties
Initial amplitude 共nm兲
10, 30, 60
Cantilever stiffness 共N/m兲
40
Cantilever oscillation mode
1
Quality factor
400
Resonance frequency 共kHz兲
350
Excitation frequency 共kHz兲
350
Z approach velocity 共nm/s兲
200
Tip-sample interaction properties
Hamaker constant 共J兲
Tip radius 共nm兲
Sample viscosity 共Pa·s兲
Elastic modulus 共tip兲 共GPa兲
Elastic modulus 共sample兲 共GPa兲
Poisson’s ratio 共tip兲
Poisson’s ratio 共sample兲
Adhesion force 共nN兲
Simulation parameters
Number of points plotted
Deflection points per cycle
Specify Z Range
7.1⫻ 10−20
20
0
130
1.2
0.3
0.3
8.8
500
500
No
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061301-6
Melcher, Hu, and Raman
Rev. Sci. Instrum. 79, 061301 共2008兲
TABLE II. Simulation parameters for the viscoelastic dissipation identification 共Sec. IV B兲
Operating conditions and cantilever properties
Initial amplitude 共nm兲
Cantilever stiffness 共N/m兲
Cantilever oscillation mode
Quality factor
Resonance frequency 共kHz兲
Excitation frequency 共kHz兲兲
Z approach velocity 共nm/s兲
FIG. 6. 共Color兲 Section IV A: comparison between asimulation performed
with DACs tool in VEDA and bsimulations performed by García and San
Paulo 共Ref. 2兲. Data from 共Ref. 2兲 was loosely interpreted graphically.
diction of tip-sample interaction forces may lead to false amplitude jumps. However, the jump predicted in Ref. 2 for A0
is most likely due to the choice of initial conditions in, which
must be imposed at each data point. A subsequent DAC
simulation of retracting from the sample revealed a large
bistable region for A0 = 10 nm. When performing simulations
at discrete Z positions, initial conditions must be specified
for each Z position, which becomes problematic in the
bistable regime. Therefore, it is necessary to have Z changing
continuously and initial position Z0 to be in a monostable
regime.
B. Viscoelastic dissipation identification
The second example demonstrates the power dissipation
curve for viscoelastic interactions shown in Ref. 19. Although simulations in Ref. 19 considered only Hertzian contact with van der Waals forces, DMT contact simulations are
similar for the cases where the effect of attractive regime is
small 共i.e., small adhesion forces or large energy oscillations兲. Choosing a sharp tip radius 共5 nm兲 results in a small
adhesion force since Fad = 4␲R␥, where ␥ is the surface energy. Parameter values required to reproduce this example
are given in Table II.
Figure 7 contains the key results from the simulation.
The curves in Fig. 7 were constructed by downloading the
amplitude and power dissipation data from DAC and postprocessing, such as computing A / A0 and Pts⬘ ⬅ ⳵ Pts / ⳵共A / A0兲.
The results are plotted in MATLAB. The trend of Pts⬘ versus
A / A0 is clearly indicative of viscoelastic energy dissipation.
Tip-sample interaction properties
Hamaker constant 共J兲
Tip radius 共nm兲
Sample viscosity 共Pa·s兲
Elastic modulus 共tip兲 共GPa兲
Elastic modulus 共sample兲 共GPa兲
Poisson’s ratio 共tip兲
Poisson’s ratio 共sample兲
Adhesion force 共nN兲
15
40
1
800
285
285
200
7.1⫻ 10−20
5
800
130
1.2
0.3
0.3
2.2
Simulation parameters
Number of points plotted
Deflection points per cycle
Specify Z Range
500
500
No
high quality factors20 simply by taking the quality factor to
be the effective quality factor modified in the feedback
scheme.
A recent publication by Holscher and Schwarz20 demonstrates the improved sensitivity to tip-sample interaction
achieved by increasing the effective quality factor of the
dAFM cantilever. Simulations performed with VEDA using
parameters from Ref. 20 共listed in Table III兲 show good
agreement 共Fig. 8兲. For the effective quality factor of 1500,
the probe remains in the attractive regime without tapping on
the sample for the entire range of approach.
D. Material properties in heterogeneous samples
In this example, we investigate the phase contrast observed while scanning heterogeneous samples using the
C. Q-control simulations
Active feedback control has been used to modify the
quality factor of dAFM cantilevers either enhance the probe
sensitivity48 or improve imaging speeds.49 Such feedback
schemes, commonly referred to as Q control, can be reasonably simulated in VEDA for near resonant excitations and
FIG. 7. Section IV B: power dissipation, Pts, and its derivative with respect
to A / A0, Pts⬘ ⬅ ⳵ Pts / ⳵共A / A0兲. Curves predicted consistent with the signature
viscoelastic curves demonstrated in Ref. 19 共see Table II兲.
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061301-7
Rev. Sci. Instrum. 79, 061301 共2008兲
Virtual environment for dynamic AFM
TABLE III. Simulation parameter values for the Q-control example 共see
Sec. IV C兲
TABLE IV. Input parameters for the material properties in heterogeneous
samples example 共Sec IV D兲.
Operating conditions and cantilever properties
Initial amplitude 共nm兲
Cantilever stiffness 共N/m兲
Cantilever oscillation mode
Quality factor
Resonance frequency 共kHz兲
Excitation frequnecy 共kHz兲
Z approach velocity 共nm/s兲
Operating conditions, cantilever, simulation
Initial amplitude 共nm兲
Cantilever stiffness 共N/m兲
Cantilever oscillation mode
Quality factor
Resonance frequency 共kHz兲
Excitation frequency 共kHz兲兲
Sampling frequency 共MHz兲
Scanning velocity 共nm/s兲
Set-point ratio
Signal/Noise ratio 共dB兲
Proportional gain
Integral gain 共kHz兲
Lock-in time constant 共␮s兲
Number of points plotted
Tip-sample interaction properties
Hamaker constant 共J兲
Tip radius 共nm兲
Sample viscosity 共Pa s兲
Elastic modulus 共tip兲 共GPa兲
Elastic modulus 共sample兲 共GPa兲
Poisson’s ratio 共tip兲
Poisson’s ratio 共sample兲
Adhesion force 共nN兲
Simulation parameters
Number of points plotted
Deflection points per cycle
Specify Z Range
Initial Z separation
Final Z separation
10
40
1
1500
300
300
20
2 ⫻ 10−19
10
0
130
1
0.3
0.3
3.7
1000
500
Yes
12共0兲a
0共12兲
a
Shown in parentheses are the parameter values for a second simulation in
which the cantilever retracts from the sample.
AMS tool. From previous work,50 we know that the the
power dissipated by nonconservative tip-sample interactions
is strongly related to the phase of the probe oscillation. In
this example, we show that phase contrast can result from
either variation in sample viscosity or elasticity. Table IV
lists the input parameter values used in the simulation. The
sample consists of a hypothetically flat surface with 30 nm
length, with contrasting material properties 共either elasticity
or viscosity兲 specified for the center 10 nm. This is achieved
by selecting the “step” feature in the GUI, entering a height
Tip and substrate properties
Hamaker constant 共J兲
Tip radius 共nm兲
Elastic modulus 共tip兲 共GPa兲
Elastic modulus 共sample兲 共GPa兲
Poisson’s ratio 共tip兲
Poisson’s ratio 共sample兲
Sample viscosity 共Pa·s兲
Adhesion force 共nN兲
Length of substrate 共nm兲
Feature properties
Select geometric feature
Feature height 共nm兲
Feature Length 共nm兲
Specify material properites
Elastic modulus 共sample兲 共GPa兲
Poisson’s ratio 共sample兲
Sample viscosity 共Pa s兲
Adhesion force 共nN兲
30
40
1
400
350
350
10
2000
0.9
20
0.1
1 ⫻ 10−4
30
500
5 ⫻ 10−20
10
130
2
0.3
0.3
500
5
30
Step
0
10
Yes
2共1兲a
0.3
1000共500兲
5
a
Shown in parentheses are input parameter values for a second simulation
where only the sample elasticity is varied.
of 0 nm and choosing to specify separate properties for the
feature.
From the simulation results 共Fig. 9兲, we find that the
scan involving contrast in sample viscosity results in negligible measurement error 共within the 20 dB noise兲 but yields
a non-negligible phase contrast. For a second scanning simulation involving contrast in sample elasticity, we see a small
measurement error due an increased indentation over the soft
material. The larger indentations result in an increase in
power dissipation 共corresponding to the phase兲 even though
viscosity of the sample has not changed. Interestingly, the
larger of the two phase contrasts corresponds to the simulation with varying elasticity, even though phase contrast is
usually associated with viscosity contrast alone.
E. Strong and weak controller parameters
FIG. 8. 共Color兲 Section IV C: approach and retraction curves simulation
with DAC for a Q-controlled cantilever 共Ref. 20兲. From the mean force, we
see the that the probe remains in the attractive regime 关defined by the mean
force 共Ref. 2兲兴 for the entire approach/retaction. Input parameter values for
the simulation are listed in Table III.
A second scanning simulation is presented here to show
the effect controller settings on scanned images. One simulation is performed with strong control parameters 共large proportional and integral gains兲 while a second simulation is
performed with weak controller settings 共small proportional
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061301-8
Melcher, Hu, and Raman
Rev. Sci. Instrum. 79, 061301 共2008兲
FIG. 9. 共Color兲 Section IV D: scanning simulation over
a hypothetically flat but heterogeneous surface containing with a patch of contrasting material from X
= 10 nm to X = 20 nm. A larger phase contrast occurs in
the case on contrasting elasticity than for the case of
contrasting viscosity. Results in this figure may be recreated with the input parameter values in Table IV.
and integral gains兲. The parameters used for the simulation
are listed in Table V. A sinusoidal feature type with a 50 nm
length and 30 nm height with homogeneous properties is
chosen for the simulation. Finally, our goal in this simulation
is to isolate the effects of the controller parameters. To do so,
we reduced the noise in deflection signal by choosing a
50 dB signal-to-noise ratio. In addition, we choose not to
include the effects of tip-sample geometry convolution 共Sec.
II D兲.
The measured topography observed with the strong controller settings well approximates the true sample surface
共Fig. 10兲. The subsequent simulation performed with weak
controller settings allows a small error in the measured to-
pography. A larger deviation is seen in the phase of oscillation as the weakly controlled dAFM scans over the feature.
The gradual drift in phase is a result of the small integral
gain while the change in phase seen while scanning over the
sample is a result of the small proportional gain. Also, the
phase exhibits a striking resemblance with the amplitude error signal, which confirms that the phase contrast did not
result from variation in sample properties. Interestingly, for
the weak controller settings, the amplitude error strongly resembles the first derivative of the sample height with respect
to the scanning distance X, while for the strong controller
settings, the amplitude error follows the general trend of the
second derivative of the sample height. Finally, we note that
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061301-9
Virtual environment for dynamic AFM
Rev. Sci. Instrum. 79, 061301 共2008兲
FIG. 10. 共Color兲 Section IV E: scanning simulations
with strong controller settings 共large proportional and
integral gains兲 and weak controller settings 共small proportional and integral gains兲. Weak controller settings
allow transient oscillations which can damage the
sample as well as produce phase contrast on homogeneous samples 共see Table V兲.
peak interaction forces may be influenced significantly by
the amplitude transients, which means the controller settings
play some role in the peak interaction forces. This implies
that beyond the amplitude setpoint, scanning feedback parameters also need to be fine tuned to minimize imaging
forces while scanning a sample.
V. SUPPORTING MATERIALS
Additional supporting materials for VEDA simulation
tools are available on the nanoHUB website. Selecting a
simulation tool on nanoHUB first leads to the tool’s information site where information about the tool, its contributors
and other related resources can be found. Information pages
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061301-10
Rev. Sci. Instrum. 79, 061301 共2008兲
Melcher, Hu, and Raman
TABLE V. Input parameters for the strong and weak controller parameters
example 共Sec. IV E兲.
Operating conditions, cantilever, simulation
Initial amplitude 共nm兲
30
Cantilever stiffness 共N/m兲
40
Cantilever oscillation mode
1
Quality factor
400
Resonance frequency 共kHz兲
350
Excitation frequency 共kHz兲
350
Sampling frequency 共MHz兲
10
Scanning velocity 共nm/s兲
2000
Set point ratio
0.9
Signal/Noise ratio 共dB兲
50
Proportional gain
0.15共0.05兲a
Integral gain 共kHz兲
1 ⫻ 10−4共1 ⫻ 10−6兲
30
Lock-in time constant 共␮s兲
Number of points plotted
500
Tip and substrate properties
Hamaker constant 共J兲
Tip radius 共nm兲
Elastic modulus 共tip兲 共GPa兲
Elastic modulus 共sample兲 共GPa兲
Poisson’s ratio 共tip兲
Poisson’s ratio 共sample兲
Sample viscosity 共Pa s兲
Adhesion force 共nN兲
Length of substrate 共nm兲
5 ⫻ 10−20
10
130
1.2
0.3
0.3
500
5
100
Feature properties
Select geometric feature
Feature height 共nm兲
Feature Length 共nm兲
Specify material properties
Sinusoid
30
50
No
a
Shown in parentheses are the parameter values for a second simulation with
weak controller settings, i.e., small proportional and integral gains.
for VEDA currently include links to a comprehensive user’s
manual, learning modules for each simulation tool51 共downloadable in breeze format兲, and other resources for dAFM.52
VI. UPCOMING FEATURES AND CONCLUSIONS
In this article, we have introduced two VEDA simulation
tools for dAFM that are part of a suite being developed on
nanoHUB. We have demonstrated a good agreement with
prior work as well as provided some new insight into dAFM
operation and interpretation.
There are many possibilities for future developments in
VEDA. An expansion to to the DAC with capabilities in
liquid environments and multiple frequency excitations53 is
currently underway and is slated for release in late summer
2008. Eventually, VEDA is likely to include additional tipsample interaction force models and eventually ab initio molecular dynamics.27 Finally, we may include filter models in
the lock-in amplifier and additional realistic noise
sources.24,25,30
Accurate simulations of dAFM can have a profound effect on our understanding of what is really a deceptively
complex instrument. While the versatility of dAFM has attracted researchers from a variety fields, accurate simulation
software is not accessible to most experimentalists. Through
the aid of a web-based infrastructure, NCN’s nanoHUB provides dAFM experimentalists with ready access via desktop,
laptop/handheld web-enabled device to research-grade simulation software, lectures, and learning modulus for dAFM.
High-fidelity simulation tools for dAFM offer both qualitative and quantitative insight into the nonintuitive, underlying
physics of dAFM. Such an understanding is a critical for
effective use of the instrument, accurate interpretation of
data and the overall development of dAFM.
ACKNOWLEDGMENTS
Financial support for this research provided by the
Network of Computational Nanotechnology and the National
Science Foundation 共Grant No. 0700289兲 is gratefully acknowledged. We also thank Professor R. Reifenberger
共Physics, Purdue兲 and other colleagues and VEDA users who
have provided valuable input for the design of VEDA.
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37
Equation 共5兲 is intended for the range of Z where 兩共⳵Fts / ⳵d兲兩d=Z
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38
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