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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 Statistical Mechanics Modeling of Heterogeneous Materials To characterize heterogeneity in micro and nanostructures Application in • Composites • Layered structures • Magnetic domains • Polycrystalline materials Use of probability functions • Volume fraction as a one point probability function • Two and three correlation functions up n-point correlations to include more complexities TWO POINTS PROBABILITY FUNCTION h l ß Randomly drop a line of lengt r into the materia many e times and observe into which phase ach end falls b r a ß There are four outcomes: ß P11 , P22 ,P12, P21 8 TWO POINTS PROBABILITY FUNCTION ß The normalization of probabilities requires that the following equations. P 1 1 P 1 2 P 2 1 P2 2 1 P 1 1 P 1 2 V 1 1 P 2 1 P22 V 2 1 9 Probability Functions Different forms for the probability function of a composite material has been suggested by many authors Corson nij Pij (r) ij ij exp( c ijr ) 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 V1 and V2 of the two phases Probability Functions nij Pij (r) ij ij exp( c ijr ) cij, and nij 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->∞ Tab le 1 Limi ting cond itions on two-point probab ili ty functions Bound ary cond iti ons Resultant coefficients Pij r=0 r ij = ij = P11 V1 V12 V12 V1 V2 P12 0 V 1V2 V1 V2 -V1V2 P21 0 V1V2 V1 V2 -V1V2 P22 V2 V22 V22 V1 V2 2 Probability Function For Increasing Number Of Phases nij Pij (r) ij ij exp( c ijr ) For anisotropic materials an orientation dependant c and n can be introduced 1 1 c ij ,k c 0 1 sin ij k k 1 n ij ,k n 0 1 1 sin ij k 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. Two point function by Torquota For a two-phase random and homogeneous system of impenetrable spheres P11r 1 Vr Mr 2 P22 V2 V1 P11 P12 P21 V1 P11 -where is the number density of spheres, V1 and V2 are the volume fractions, r is the distance between two points 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 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 Goal: To digitize the images of the nano- structures To extract two point probability functions, P11(r) , P12 (r),P22(r) Produce a model which incorporates these in order to find the effective magnetic properties as a function of the microstructure Results: Probability functions for the Co- nanostructure for 1000 measurements For horizontal vectors P11 at different angles 0.20 1.00 p11 0 Probability Probability 0.15 p11 5 0.50 p11 0 0.10 p11 10 p12 p11 15 0.00 p21 0.05 p11 30 p22 p11 45 0 10 20 0.00 p11 60 Vector Length 0 10 20 p11 90 Vector Length Results: Investigation of the results show that the probability functions follow an exponential (Coron’s) behavior With X and Y described by X ln r 1i j V1V2 Y ln ln Blockco polymer 4: determining cij & nij Pij Vi Vj usin g 1000 r and om [1,2] vecto rs 2.00 1.50 ln|ln |(Pij-Vi^2)/(V1*V2)|| p11' 1.00 p12' y = 0.1997x - 0.2293 p21' y = 0.4159x - 0.5098 p22' 0.50 p11' p12' y = 0.1417x - 0.0076 y = 0.0456x - 0.0493 p21' 0.00 p22' 0 0.5 1 1.5 2 2.5 3 3.5 -0.50 -1.00 ln |r |

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posted: | 2/24/2013 |

language: | English |

pages: | 13 |

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