Quantifying Nanoparticle Mixing State to Account for Both Location and Size Effects

ABSTRACT

Ripley’s K function is commonly used to characterize the homogeneity of spatial point distribution. Not surprisingly, it becomes a favored tool in quantifying the nanoparticles mixing state in composite materials, a parameter that material scientists believe is of close relevance to certain properties of the nanoparticle-embedding material. Ripley’s K function assumes that the spatial points are dimensionless. In reality, the nanoparticles, once mixed in a host material, form clusters or agglomerates of various sizes and shapes. Our analysis shows that using the original K function falls short of ranking or distinguishing the homogeneity of nanoparticle mixing. We therefore propose to revise the K function to account for both particle location and size effects. We apply the revised function to electron microscopy images of material samples and conduct analysis and comparison of nanoparticle mixing. The analysis shows that the revised function is a better index to quantify the mixing states.

KEYWORDS:

• Dispersion and distribution
• Nano images
• Permutation test
• Ripley’s K function
• Spatial point pattern

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Acknowledgments

The authors acknowledge the generous support from their sponsors. Qian and Ding are partially supported by NSF under grant no. CMMI-1000088. The authors are grateful to Professor Xiaoying Qi and Mr. Pengfei Chu of the National Center for Nanoscience and Technology, Beijing, China, for providing the image data used in the article.