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Statistical Work in Nanomaterial Research

• 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)

C. F. Jeff WuIndustrial and Systems Engineering

Georgia Institute of Technology

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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 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.

• 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.

235 nN

248 nN

261 nN 209 nN

222 nN

157 nN

170 nN

183 nN

131 nN

144 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.