1. Characterization of Heterogeneity in Nano-structures of Co-Copolymers using Two point Statistical Functions Gail Jefferson Mechanical Engineering FAMU-FSU College of Engineering & H. Garmestani (FAMU), B. L. Adams (CMU-BYU), Rina Tannenbaum (Georgia Tech) Presented to the Collaborative in Research and Education National Science Foundation Site Visit

2. Statistical Mechanics Modeling of Heterogeneous Materials To characterize heterogeneity in micro and nanostructures To characterize heterogeneity in micro and nanostructures Application in Application in CompositesComposites Layered structuresLayered structures Magnetic domainsMagnetic domains Polycrystalline materialsPolycrystalline materials Use of probability functions Use of probability functions Volume fraction as a one point probability functionVolume fraction as a one point probability function Two and three correlation functions up n-point correlations to include more complexitiesTwo and three correlation functions up n-point correlations to include more complexities

5. Probability Functions Different forms for the probability function of a composite material has been suggested by many authors Different forms for the probability function of a composite material has been suggested by many authors Corson i=1, 2; j=1, 2; represents the probability occurrence of one point in phase i and the other point which is located a distance r away in phase j  ij and  ij depend on the volume fractions V 1 and V 2 of the two phases

6. Probability Functions c ij, and n ij are empirical constants determined by a least squares fit for the measured data and  ij and  ij determine the limiting value of at r=0 and r->∞

7. 2 Probability Function For Increasing Number Of Phases For anisotropic materials an orientation dependant c and n can be introduced Here, k is aspect ratio,  is the angle between the direction being considered and axial direction, and are constants and will be determined by measurement.

8. Two point function by Two point function by Torquota For a two-phase random and homogeneous system of impenetrable spheres -where  is the number density of spheres, V 1 and V 2 are the volume fractions, r is the distance between two points

9. Two point functions for a cobalt-copolymer nano-structure magnetic nanocrystals have profound applications in information storage, color imaging, bioprocessing, magnetic refrigeration, and ferrofluids. In Summary: Both the crystalline size (compared to the domain size) and the inter particle distance should not be too small! Using two point functions both the size distribution and the inter-particle distance can be modeled and characterized

10. Two point functions for a cobalt-copolymer nano-structure Using Solution Chemistry Nanoscale colloidal Co particles with an average diameter of 3.3 nm have been prepared by a microemulsion technique at Georgia Tech

11. Goal: To digitize the images of the nano- structures To digitize the images of the nano- structures To extract two point probability functions, P 11 (r), P 12 (r),P 22 (r) To extract two point probability functions, P 11 (r), P 12 (r),P 22 (r) Produce a model which incorporates these in order to find the effective magnetic properties as a function of the microstructure Produce a model which incorporates these in order to find the effective magnetic properties as a function of the microstructure

12. Results: Probability functions for the Co- nanostructure for 1000 measurements Probability functions for the Co- nanostructure for 1000 measurements

13. Results: Investigation of the results show that the probability functions follow an exponential (Coron’s) behavior Investigation of the results show that the probability functions follow an exponential (Coron’s) behavior With X and Y described by With X and Y described by