Materials Science Lecture #3

							                     Materials Science
                       Lecture #3

                     Prof. David W. Steuerman
                            September 16, 2009




Topics:   More Defects

          Types of Solids

          Packing of Spheres


                      Department of Chemistry
                       University of Victoria
                   Crystals and Defects


Last Time: 0-D Defects, Diffusion,…

                          Au
                          Cu




             Antisite                     Schottky Defect

                  Real crystals have defects.
                     Screw Dislocation


Last Time: 1-D Defects




                Hard to see - think spiral ham
                                                 3
                   Dislocation Density


Definition: ρd = ld/V units of cm-2
          total dislocation length per unit volume

          Typical #’s:   108 to 1010 cm-2 for metals
                         10 to 105 cm-2 for semiconductors

                   Why the disparity?




                                                             4
                 2-Dimensional Defects


Grain Boundary: separation of regions with the same
                crystal structure but different orientations




                             Grain Boundaries in Iron

                                                               5
                Comment on Magnets


Magnetic Domains: separation of regions with the same
              crystal structure and orientation but
              different magnetic orientations




                                 Crystal with no
                                structural defects
                                                        6
                Comment on Magnets


Magnetic Domains: separation of regions with the same
              crystal structure and orientation, but
              different magnetic orientations




                                   Disorder in
                                  electron spin
                                                        7
                3-Dimensional Defects

Precipitates or Agglomerates: More of a concern in more
complex materials with multiple constituents.




                             3-D Aggregate of Atoms


                                                          8
                 3-Dimensional Defects

Example: High Resolution Transmission Electron Microscopy
         of Nb-doped SrTiO3




          http://www.iwe.rwth‐aachen.de/emrl/r_p_1.html     9
                   Classification of Defects


    Zero - Dimensional                    One - Dimensional
- Intrinsic/native point defects      - Dislocation
- Impurity atoms                      - Disinclination




     Two - Dimensional                   Three - Dimensional
- Stacking Faults,                    - Precipitates
- Grain boundaries, twin boundaries   - Inclusions
- Domain walls (magnetic)             - Agglomerates of point defects
                                                                10
                 What holds solids together?

4 Fundamental Forces:
                         Range (m)      Relative Magnitude

        Strong              10-15              1

   Electromagnetic           ∞                 10-2

        Weak                10-18              10-6

    Gravitational            ∞                 10-36


Electromagnetic interactions govern molecules and materials.
   Classification of Solids



 Ionic            Van der Waals




Covalent         Metallic
                            Ionic Bonding


                        ‐      +        2 Isolated Ions


               −e   2
                                       −e   2
        F=                         V=
             4πε o r    2
                                      4πε o r

    − e2                 (1.60 × 10−19 C ) 2
V=         =
   4πε o r   4π (8.85 × 10−12 F × m −1 )(0.236 × 10−9 m)
             = - 9.75 x 10-19 J
                                        Potential Energy for a
             = - 6.10 eV                single NaCl molecule
         Reminder: Work, Forces, and Potential


Concept:      V ( r ) = − ∫ F ( r )dr         From Introductory Physics

Conservative Force - Work done by a force on a particle is only
                     dependent upon the initial and final position.
                             rf
                  W = − ∫ F ( r )dr = −V f + Vi
                            ri

                                       rf
                  ΔE = −W = − ∫ F ( r )dr
                                      ri

              r                   Rearrange
  V ( r ) = − ∫ F ( r )dr                       F ( r ) = − dVdrr )
                                                               (
              ∞

            At ∞ potential is 0.
               Ionic Bonding Continued

Consider 1D Array        ‐   +   ‐   +    ‐    +    ‐   +
of Ions:


Consider 3D Array                              How many next
of Ions:                                      nearest neighbors?

                    e2
       E = −6 ×
                  4πε o ao           6 nearest neighbors of Na

                      e2             12 next nearest
       E = +12 ×
                 4πε o ao 2          neighbors of Cl
                    e2               8 nearest neighbors of Na
       E = −8 ×
                4πε o ao 3
                           Ionic Bonding



Consider 3D Array
of Ions:



           e2    ⎡    12   8         ⎤
    E=         × ⎢6 −    +    .......⎥
       4πε o ao ⎣      2    3        ⎦
                                    Convergent Infinite Series
                  e2                        e2
    E = −M d                   = −1.748
                4πε o ao                  4πε o ao
     Madelung Constant
                 Van der Waals Interactions


Van der Waals Forces: interaction of dipoles
 3 Categories:
                                      Can you name them and
    dipole-dipole
                                      give an example of each?
    dipole-induced dipole
    induced dipole-induced dipole

Recall from Electrostatics:

 Coulombic Potentials scale as 1/R
 Dipole Potentials scale as 1/R2
                   Lennard-Jones Potential


Model applies to interacting pairs of neutral atoms or molecules.



                                                   r
  Short Range:
  Repulsion – Pauli Exclusion Principle


  Long Range:
  Attraction – Van der Waals forces – multipole contributions
         Lennard-Jones Potential


                       Attractive Term



V   ao               V ( r ) = −( Bao )6 + ( Aao )12
                                   r          r




                                    Repulsion Term
                Metallic Bonding


            +        +        +        +
                                                -
                 -       -

                 +        +        +        +
                              -

            +    -   +        +        +


- Metal atoms readily give up electrons
- These “free” electrons are treated as an electron gas
- Electrons delocalized over entire crystal