Statistical Work in Nanomaterial
Research
C. F. Jeff Wu
Industrial and Systems Engineering
Georgia Institute of Technology
• A Statistical Approach to Quantifying the Elastic
Deformation of Nanomaterials.
(X. Deng, C. F. J. Wu, V. R. Joseph, W. Mai*, Z. L. Wang*)
* Material Science and Engineering, Geogia Tech.
• Efficient Experimentation for Nanostructure Synthesis
using Sequential Minimum Energy Designs (SMED).
(V. R. Joseph, T. Dasgupta, C. F. J. Wu)
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A Statistical Approach to Quantifying the
Elastic Deformation of Nanomaterials
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Existing method of experimentation and modeling
A general modeling and selection procedure
Demonstration on nanobelt data
Model with general error structure
Discussions and conclusions
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Introduction
• One-dimensional (1D) nanomaterials: fundamental
building blocks for constructing nanodevices and
nanosystems.
• Important to quantify mechanical property such as elastic
modulus of 1D nanomaterials.
• A common strategy is to deform a 1D nanostructure
using an AFM tip.
Schematic diagram of AFM
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Method of Experimentation and Modeling
• Mai and Wang (2006, Appl. Phys. Lett.) proposed a new
approach for measuring the elastic modulus of ZnO
nanobelt (NB).
– Based on a continuous scan of an NB along its direction using
an AFM tip in contact mode.
– Fitting the elastic bending shape of the NB as a function of the
bending force.
AFM images of a suspended ZnO nanobelt
• A series of bending images of the NB are recorded by
sequentially changing the magnitude of the contact
force.
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Example: Nanobelt 1 (NB1)
Figure 1
• (a) AFM image profiles of NB1 under different load forces from low
106 nN to high 289 nN.
• (b) Normalized profiles: subtracting the profile acquired at 106 nN
(nano Newton) from the profiles in (a).
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Initial Bias of Nanobelt 1
• The NB is not perfectly straight: initial bending
during sample manipulation,  shift and
deformation.
• The profile curves in Figure 1(a) are not smooth:
caused by a small surface roughness (around 1
nm) of the NB.
• Some ripples appear in the middle of the NB.
• Eliminate the initial bias: Mai and Wang suggest
subtracting the first profile from those measured
at higher applied forces.
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Free-Free Beam Model
• Mai and Wang (2006) suggested a free-free beam model
(FFBM) to quantify the elastic deflection.
• The deflection v of NB at x is determined by
where E is the elastic modulus, L is the width of trench,
and I is the moment of inertia.
• The elastic modulus is estimated by fitting the normalized
AFM image profiles using the FFBM. (MW method)
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Problem with MW Method
• Subtracting the first profile to normalize the data can result in poor
estimation if the first profile behaves poorly.
• Systematic biases can occur during the measurement,
normalizing the data doesn’t help.
• (a) AFM image profiles of nanobelt 2 (NB2) under different load forces.
(b) Normalized image profiles by subtracting the first profile acquired at
78 nN from the profiles in (a).
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Problem with MW Method
• Subtracting the first profile to normalize the data can result in poor
estimation if the first profile behaves poorly.
• Systematic biases can occur during the measurement,
normalizing the data doesn’t help.
157 nN
170 nN
235 nN
183 nN
248 nN
261 nN
209 nN
131 nN
222 nN
144 nN
• Inconsistent (order reversal) pattern: profiles at applied force 235,
248 and 261 nN lie above on those obtained at lower force F =
209 and 222 nN. This pattern persists in the normalized profiles.
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Why the Proposed Method
• The FFBM itself cannot explain the inconsistency.
– Requires a more general model to include other factors
besides the initial bias.
• Propose a general model to incorporate the initial
bias and other systematic biases.
• Use model selection to choose an appropriate
model.
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Performance Comparison on Nanobelt 2
• The fitting in MW method is obtained by adding the initial profile
back into the fitted normalized data using FFBM.
• Residuals for proposed method in narrow band.
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Performance Comparison on Nanobelt 1
• The fitting in MW method is obtained by adding the initial profile
back into the fitted normalized data using FFBM.
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