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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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posted:2/24/2013
language:English
pages:13