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 Statistical Mechanics Modeling of Heterogeneous Materialsof Heterogeneous Materials

To characterize heterogeneity in To characterize heterogeneity in micro and nanostructuresmicro and nanostructures

Application in Application in

• CompositesComposites

• Layered structuresLayered structures

• Magnetic domainsMagnetic domains

• Polycrystalline materialsPolycrystalline materials Use of probability functionsUse of probability functions

• Volume fraction as a one point Volume fraction as a one point probability functionprobability function

• Two and three correlation Two and three correlation functions up n-point correlations functions up n-point correlations to include more complexitiesto include more complexities

8

TWO POINTS PROBABILITYTWO POINTS PROBABILITYFUNCTIONFUNCTION

ß Randomly drop a line of length r into the material manytimes and observe into which phase e ach end falls

ß There are four outcomes:

ß P11 , P22 ,P12, P21

r

a

b

9

TWO POINTS PROBABILITYTWO POINTS PROBABILITYFUNCTIONFUNCTION

ß The normalization of probabilities requiresthat the following equations.

P 1 1 P1 2 P 2 1 P2 2 1

P 1 1 P 1 2 V 1 1

P 2 1 P 2 2 V 2 1

Probability FunctionsProbability Functions

Different forms for the probability function of a Different forms for the probability function of a composite material has been suggested by composite material has been suggested by many authorsmany authors

Corson

Pij(r) ij ij exp( c ijrn ij )

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 FunctionsProbability Functions

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->∞

Pij(r) ij ij exp( c ijrn ij )

Table 1 Limi ting conditions on two-point probability functions

Boundary conditi ons Resultant coefficientsPij r=0 r ij= ij=

P11 V1 V12 V1

2 V1V2

P12 0 V1V2 V1V2 -V1V2

P21 0 V1V2 V1V2 -V1V2

P22 V2 V22 V2

2 V1V2

2 Probability Function For Increasing Number Of Phases

For anisotropic materials an orientation dependant c and n can be introduced

Pij(r) ij ij exp( c ijrn ij )

c ij ,k c ij0 1

k 1

1

k

sin

n ij ,k n ij0 1 1

1

k

sin

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 Two point function by Torquota

For a two-phase random and homogeneous system of impenetrable spheres

P11 r 1 V r 2M r

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 Two point functions for a cobalt-copolymer nano-structure 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 Two point functions for a cobalt-copolymer nano-structure 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:Goal:

To digitize the images of the nano-To digitize the images of the nano-structuresstructures

To extract two point probability To extract two point probability functions, Pfunctions, P1111(r)(r) , P, P12 12 (r),P(r),P2222(r)(r)

Produce a model which Produce a model which incorporates these in order to find incorporates these in order to find the effective magnetic properties the effective magnetic properties as a function of the as a function of the microstructuremicrostructure

Results:Results:

Probability functions for the Co-Probability functions for the Co-nanostructure for 1000 measurementsnanostructure for 1000 measurements

For horizontal vectors

0.00

0.50

1.00

0 10 20

Vector Length

Pro

bab

ility

p11 0p12p21p22

P11 at different angles

0.00

0.05

0.10

0.15

0.20

0 10 20

Vector Length

Pro

bab

ilit

y p11 0p11 5p11 10p11 15p11 30p11 45p11 60p11 90

Results:Results:

Investigation of the results show that the probability Investigation of the results show that the probability functions follow an exponential (Coron’s) behaviorfunctions follow an exponential (Coron’s) behavior

With X and Y described by With X and Y described by

Blockcopolymer 4: determining cij & nij using 1000 random [1,2] vectors

y = 0.1417x - 0.0076 y = 0.0456x - 0.0493

y = 0.1997x - 0.2293

y = 0.4159x - 0.5098

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0 0.5 1 1.5 2 2.5 3 3.5

ln |r|

ln|ln

|(P

ij-V

i^2)

/(V

1*V

2)|| p11'

p12'p21'p22'p11'p12'p21'p22'

X ln r

Y ln ln 1 i jV1V2

Pij ViVj