# gail's talk by haijuangao1

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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

P11r  1 Vr  Mr    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
 1i  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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