Characterization of Heterogeneity in Nano-structures of
Co-Copolymers using Two point Statistical Functions
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

ß Randomly drop a line of lengt h r into the materia l many times and observe into which phase e ach end falls b r a
ß There are four outcomes:
ß P
11 ,
P
22 ,
P
12,
P
21
8

ß The normalization of probabilities requires that the following equations.
P
1 1

P
1 2

P
2 1

P
2 2


P
P
2 1
1 1

P
1 2

P
2 2

V

V
2
1

1

1

1
9



Different forms for the probability function of a composite material has been suggested by many authors
Corson
P ij
(r)   ij
  ij exp(  c ij r n i j ) 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

P ij
(r)   ij
  ij exp(  c ij r n ij )
 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->∞
Table 1 Limi ting cond iti ons on two-point probab ili ty func tions
P ij
Bound ary cond iti ons Resultant coe fficients
r=0 r   ij
=  ij
=
P
11
V
1
V
1
2 V
1
2 V
1
V
2
P
12
0 V
1
V
2
V
1
V
2
-V
1
V
2
P
21
0 V
1
V
2
V
1
V
2
-V
1
V
2
P
22
V
2
V
2
2 V
2
2 V
1
V
2

P ij
(r)   ij
  ij exp(  c ij r n ij )
 For anisotropic materials an orientation dependant c and n can be introduced c ij
   c 0 ij
 1
 k



1 
1 k

 sin 

 n ij
   n 0 ij


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.

 For a two-phase random and homogeneous system of impenetrable spheres
P
11
   1      2 M r
P
22
P
12
 V
2
 P
21
 V
1
 V
1
 P
11
 P
11
-where  is the number density of spheres, V
1 and V
2 are the volume fractions, r is the distance between two points

 magnetic nanocrystals have profound applications in information storage, color imaging, bioprocessing, refrigeration, and ferrofluids.
magnetic

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


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

 To digitize the images of the nanostructures
 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

 Probability functions for the Conanostructure for 1000 measurements
For horizontal vectors
1.00
0.50
0.00
0 10
Vector Length
20 p11 0 p12 p21 p22
0.20
0.15
0.10
0.05
0.00
0
P11 at different angles
10
Vector Length
20 p11 0 p11 5 p11 10 p11 15 p11 30 p11 45 p11 60 p11 90



Investigation of the results show that the probability functions follow an exponential (Coron’s) behavior
With X and Y described by
X  ln r
Blo ckcopo lym er 4: de ter m ini ng cij & nij usi ng 1000 rando m [1,2] vector s
Y 
 ln ln

 
P ij i  j
 V i
V
1
V
2
V j


2. 00
1. 50
1. 00
0. 50
0. 00 y = 0.1417x - 0.0076
0 0. 5
-0. 50
-1. 00
1 1. 5 2 y = 0.1997x - 0.2293
y = 0.4159x - 0.5098
2. 5 y = 0.0456x - 0.0493
3 3. 5 p11' p12' p21' p22' p11' p12' p21' p22' ln |r |