Statistical Work in Nanomaterial Research

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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

• 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.
• A series of bending images of the NB are recorded
  by sequentially changing the magnitude of the
  contact force.

AFM images of a suspended ZnO nanobelt
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Example Nanobelt 1 (NB1)

• (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).

Figure 1
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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 doesnt 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 doesnt help.
• 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.

157 nN 170 nN 183 nN
235 nN 248 nN 261 nN
131 nN 144 nN
209 nN 222 nN
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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.