0
Research Papers

# Size Distribution Estimation of Three-Dimensional Particle Clusters in Metal-Matrix Nanocomposites Considering Sampling Bias

[+] Author and Article Information
Jianguo Wu

Department of Industrial, Manufacturing
and Systems Engineering,
University of Texas at El Paso,
500 W University Avenue,
Engineering Building, A-244,
El Paso, TX 79968
e-mail: jwu2@utep.edu

Yuan Yuan

IBM Research, Singapore,
10 Marina Boulevard,
Marina Bay Financial Centre Tower 2,
Singapore 18983, Singapore
e-mail: polarisyy@gmail.com

Xiaochun Li

Department of Mechanical
and Aerospace Engineering,
University of California,
Los Angeles, 48-121G Eng IV,
Los Angeles, CA 90095
e-mail: xcli@seas.ucla.edu

1Corresponding author.

Manuscript received February 20, 2017; final manuscript received April 21, 2017; published online May 25, 2017. Assoc. Editor: Donggang Yao.

J. Manuf. Sci. Eng 139(8), 081017 (May 25, 2017) (11 pages) Paper No: MANU-17-1109; doi: 10.1115/1.4036642 History: Received February 20, 2017; Revised April 21, 2017

## Abstract

Nanoparticle clustering phenomenon is a critical quality issue in metal-matrix nanocomposites (MMNCs) manufacturing. Accurate estimation of the 3D cluster size distribution based on the 2D cross section images is essential for quality assessment, quality control, and process optimization. The existing studies often draw conclusions with observable samples, which are inherently biased because large clusters are more likely to be intersected by scanning electron microscope (SEM) images compared with small ones. This paper takes into account this sampling bias and proposes two statistical approaches, namely, the maximum likelihood estimation (MLE) and the method of moments (MM), to estimate the distribution parameters accurately. Numerical studies and real case study demonstrate the effectiveness and accuracy of the proposed approaches.

<>

## References

Cheah, L. W. , 2010, “ Cars on a Diet: The Material and Energy Impacts of Passenger Vehicle Weight Reduction in the US,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Yang, Y. , Lan, J. , and Li, X. , 2004, “ Study on Bulk Aluminum Matrix Nano-Composite Fabricated by Ultrasonic Dispersion of Nano-Sized SiC Particles in Molten Aluminum Alloy,” Mater. Sci. Eng. A, 380(1–2), pp. 378–383.
Li, X. , Yang, Y. , and Weiss, D. , 2008, “ Theoretical and Experimental Study on Ultrasonic Dispersion of Nanoparticles for Strengthening Cast Aluminum Alloy A 356,” Metall. Sci. Technol., 26(2), pp. 12–20.
Yang, Y. , and Li, X. , 2007, “ Ultrasonic Cavitation Based Nanomanufacturing of Bulk Aluminum Matrix Nanocomposites,” ASME J. Manuf. Sci. Eng., 129(3), pp. 497–501.
Cao, G. , Konishi, H. , and Li, X. , 2008, “ Mechanical Properties and Microstructure of Mg∕SiC Nanocomposites Fabricated by Ultrasonic Cavitation Based Nanomanufacturing,” ASME J. Manuf. Sci. Eng., 130(3), p. 031105.
Casati, R. , and Vedani, M. , 2014, “ Metal Matrix Composites Reinforced by Nano-Particles—A Review,” Metals, 4(1), pp. 65–83.
Chen, X. , Baburaj, E. , Froes, F. , and Vassel, A. , 1997, “ Ti-6 Al-4 V/SiC Composites by Mechanical Alloying and Hot Isostatic Pressing,” Fifth International Conference on Advanced Particulate Materials and Processes, West Palm Beach, FL, Apr. 7–9, pp. 185–192.
Groza, J. R. , 1999, “ Sintering of Nanocrystalline Powders,” Int. J. Powder Metall., 35(7), pp. 59–66.
Mussert, K. , Vellinga, W. , Bakker, A. , and Van Der Zwaag, S. , 2002, “ A Nano-Indentation Study on the Mechanical Behaviour of the Matrix Material in an AA6061-Al2O3 MMC,” J. Mater. Sci., 37(4), pp. 789–794.
Singh, S. , and Singh, R. , 2016, “ Study on Tribological Properties of Al–Al2O3 Composites Prepared Through FDMAIC Route Using Reinforced Sacrificial Patterns,” ASME J. Manuf. Sci. Eng., 138(2), p. 021009.
Choi, H. , Cho, W.-H. , Konishi, H. , Kou, S. , and Li, X. , 2013, “ Nanoparticle-Induced Superior Hot Tearing Resistance of A206 Alloy,” Metall. Mater. Trans. A, 44(4), pp. 1897–1907.
Sun, Y. , 2012, “ Microstructure Modification by Nanoparticles in Aluminum and Magnesium Matrix Nanocomposites,” M.S. thesis, University of Wisconsin-Madison, Madison, WI.
Wu, J. , Zhou, S. , and Li, X. , 2013, “ Acoustic Emission Monitoring for Ultrasonic Cavitation Based Dispersion Process,” ASME J. Manuf. Sci. Eng., 135(3), p. 031015.
Wu, J. , Zhou, S. , and Li, X. , 2015, “ Ultrasonic Attenuation Based Inspection Method for Scale-Up Production of A206–Al2O3 Metal Matrix Nanocomposites,” ASME J. Manuf. Sci. Eng., 137(1), p. 011013.
Wu, J. , Chen, Y. , Zhou, S. , and Li, X. , 2016, “ Online Steady-State Detection for Process Control Using Multiple Change-Point Models and Particle Filters,” IEEE Trans. Autom. Sci. Eng., 13(2), pp. 688–700.
Hou, Y. , Wu, J. , and Chen, Y. , 2016, “ Online Steady State Detection Based on Rao‐Blackwellized Sequential Monte Carlo,” Qual. Reliab. Eng. Int., 32(8), pp. 2667–2683.
Liu, Y. , Wu, J. , Zhou, S. , and Li, X. , 2016, “ Microstructure Modeling and Ultrasonic Wave Propagation Simulation of A206–Al2O3 Metal Matrix Nanocomposites for Quality Inspection,” ASME J. Manuf. Sci. Eng., 138(3), p. 031008.
Wu, J. , Liu, Y. , and Zhou, S. , 2016, “ Bayesian Hierarchical Linear Modeling of Profile Data With Applications to Quality Control of Nanomanufacturing,” IEEE Trans. Autom. Sci. Eng., 13(3), pp. 1355–1366.
Wu, J. , Chen, Y. , and Zhou, S. , 2016, “ Online Detection of Steady-State Operation Using a Multiple-Change-Point Model and Exact Bayesian Inference,” IIE Trans., 48(7), pp. 599–613.
Isaza, C. , Sierra, G. , and Meza, J. M. , 2016, “ A Novel Technique for Production of Metal Matrix Composites Reinforced With Carbon Nanotubes,” ASME J. Manuf. Sci. Eng., 138(2), p. 024501.
Diggle, P. J. , 2013, Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, CRC Press, Boca Raton, FL.
Zeng, L. , Zhou, Q. , De Cicco, M. P. , Li, X. , and Zhou, S. , 2012, “ Quantifying Boundary Effect of Nanoparticles in Metal Matrix Nanocomposite Fabrication Processes,” IIE Trans., 44(7), pp. 551–567.
Ganguly, P. , and Poole, W. J. , 2002, “ Characterization of Reinforcement Distribution Inhomogeneity in Metal Matrix Composites,” Mater. Sci. Eng. A, 332(1–2), pp. 301–310.
Gong, H. , 2013, “ Generation and Detection of Defects in Metallic Parts Fabricated by Selective Laser Melting and Electron Beam Melting and Their Effects on Mechanical Properties,” Ph.D. dissertation, University of Louisville, Louisville, KY.
Zhou, Q. , Zhou, J. , De Cicco, M. , Zhou, S. , and Li, X. , 2014, “ Detecting 3D Spatial Clustering of Particles in Nanocomposites Based on Cross-Sectional Images,” Technometrics, 56(2), pp. 212–224.
Liu, H. , Zhou, S. , and Li, X. , 2013, “ Inferring the Size Distribution of 3D Particle Clusters in Metal Matrix Nanocomposites,” ASME J. Manuf. Sci. Eng., 135(1), p. 011013.
Russ, J. C. , 2011, The Image Processing Handbook, CRC Press, Boca Raton, FL.
Khodashenas, B. , and Ghorbani, H. R. , 2015, “ Synthesis of Silver Nanoparticles With Different Shapes,” Arabian J. Chem., epub.
Can, W. , and Log, O. , 2001, “ Log-Normal Distributions Across the Sciences: Keys and Clues,” BioScience, 51(5), pp. 341–352.
Kiss, L. B. , Söderlund, J. , Niklasson, G. A. , and Granqvist, C. G. , 1999, “ New Approach to the Origin of Lognormal Size Distributions of Nanoparticles,” Nanotechnology, 10(1), p. 25.
Dempster, A. P. , Laird, N. M. , and Rubin, D. B. , 1977, “ Maximum Likelihood From Incomplete Data Via the EM Algorithm,” J. R. Stat. Soc. Ser. B, 39(1), pp. 1–38.
Wei, G. C. , and Tanner, M. A. , 1990, “ A Monte Carlo Implementation of the EM Algorithm and the Poor Man's Data Augmentation Algorithms,” J. Am. Stat. Assoc., 85(411), pp. 699–704.
Wu, C. J. , 1983, “ On the Convergence Properties of the EM Algorithm,” Ann. Stat., 11(1), pp. 95–103.
Mátyás, L. , 1999, Generalized Method of Moments Estimation, Vol. 5, Cambridge University Press, Cambridge, UK.
Massey, F. J., Jr. , 1951, “ The Kolmogorov-Smirnov Test for Goodness of Fit,” J. Am. Stat. Assoc., 46(253), pp. 68–78.

## Figures

Fig. 3

Geometric relationship among R, Zc, and Rc

Fig. 2

Illustration of randomly sampled SEM images showing particles and clusters: (a) no clustering and (b) clustered

Fig. 1

SEM images of Al2O3 reinforced MMNCs [14]: (a) nanoparticles are uniformly distributed and (b) nanoparticles form two big clusters

Fig. 4

Comparison of the proposed MLE method with Liu et al. [26] for uniform distribution (left column—(a)–(d)), truncated normal distribution (middle column—(e)–(h)), and log-normal distribution (right column—(i)–(l)). The first row ((a), (e), and (i)) shows the true and estimated pdfs; the second row ((b), (f), and (j)) shows the normalized histograms of the generated samples from the true distribution; the third row ((c), (g), and (k)) shows the normalized histograms of the samples cut by the microscopic images; the bottom row shows the normalized histograms of the circular cross sections.

Fig. 7

(a) Probability density functions of the simulated 2D sizes from three MLE-estimated models in comparison with the normalized histogram of real observations and (b)–(d) the empirical distribution function of the 30 observations in comparison with these of simulated ones from three candidate models

Fig. 5

(a) Histogram of the MLE estimation of ru of the uniform distribution U(3,5), where the sample size n=100, and the vertical dashed line denotes the true value and (b) the RSE of the MLE estimate ru* and the estimate r̂u=max{rc1,…,rcn}

Fig. 6

Nanoparticle clusters shown in SEM images of MMNCs fabricated using ultrasonic cavitation-assisted casting process

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections