Transcript Slide 1

Formal Crystallography
• Crystalline
– Periodic arrangement of atoms
– Pattern is repeated by translation
c
• Three translation vectors define:
– Coordinate system
– Crystal system
– Unit cell shape
• Lattice points
b
a
g
b
a
– Points of identical environment
– Related by translational symmetry
– Lattice = array of lattice points
•
•
•
•
•
space filling
defined by 3 vectors
parallelipiped
arbitrary coord system
lattice pts at corners +
Crystal system
Lattices
triclinic
simple
base-centered
monoclinic
Convention:
b = 90°
instead of a
a = 90°
simple
base-centered body-centered face-centered
orthorhombic
a = b = g = 90°
6 or 7
crystal
systems
hexagonal
g = 120°
c
14 lattices
a
a
rhombohedral
(trigonal)
Can be redefined as a nonprimitive hexagonal cell
a = b = ga
simple
body-centered
simple
body-centered face-centered
tetragonal
a = b = g = 90°
a=b
cubic
(isometric)
a = b = g = 90°
a=b=c
Crystallographic notation
cubic system
• Position:
a3
– x, y, z
– fractional coordinates
(x2, y2, z2)
x, y, z
• Direction:
– x2-x1, y2-y1, z2-z1
– no magnitude
– specific: [t u v]
– family: t u v
(x1, y1, z1)
a1
a2
Specific vs. family
square lattice
specific: [1, 2]
family: <1, 2> includes
a2
[1 2],
[2 1],
[1 2],
[2 1]
 
a1


Correction to lecture:
family <2, 1> is distinct

[2 1],
[1 2],
[2 1],
[1 2]
 

Do not include those permutations that imply a change in handedness
Generically: <t,u> = [t, u], [u, -t], [-t, -u], [-u, t] also for planes
Crystallographic notation
cubic system
a3
• Plane
– find intercepts
– compute 1/intercept
– clear fractions
• Example
– (0 2 1)
a1
axis
intercept
inverse
clear fractions
a1

0
0
a2
½
2
2
a3
1
1
1
a2
½
specific: (h k l)
family: {h k l}
Crystallographic notation
• Plane
cubic system
– find intercepts
– compute 1/intercept
– clear fractions
a3
½
• Example
– (0 2 1)
a1
axis
intercept
inverse
clear fractions
a1

0
0
a2
1
1
2
a3
2
½
1
a2
Crystallographic notation
a3
• Plane
– find intercepts
– compute 1/intercept
– clear fractions
cubic system
½
• Example
– (0 1 2)
a1
axis
intercept
inverse
clear fractions
a1

0
0
a2
2
½
1
a3
1
1
2
a2
More examples
a3
a3
front
edge
1/2
1/2
a2
a1
1/2
a2
a1
_
(2 0 2)
(1 1 -2)
(1 1 2)
Specific vs. Family
cubic
ac3
tetragonal
a3
1/2
1/2
a2
a1
1/2
{2 0 2}

  
(2 0 2) (0 2 2) (2 X
2 0) (2 0 2) (2 0 2) (2 0 2)
  




(0 2 2) (0 2 2) (0 2 2) (2 X
2 0) (2 X
2 0) (2 X
2 0)
a2
a1
1/2
{2 0 2}
Types of Bonds  Types of Materials
• Metallic
Isotropic, filled outer shells
– Electropositive: give up electrons
+
+
e-
e+
+
+
+
+
+
+
+
-
+
-
+
-
+
-
+
e-
• Ionic
– Electronegative/Electropositive
• Colavent
– Electronegative: want electrons
– Shared electrons along
bond direction
Close-packed
structures
What’s Missing?
units
many
methane
H
Long chain molecules with repeated units
Molecules formed by covalent bonds
Secondary bonds link molecules into solids
H H
H
C
C
C
H
H H
H
C
C
H
H H
H
H
H
H
C
C
H
H
H
C
C
C
C
H
H
http://en.wikipedia.org/wiki/File:Polyethylene-repeat-2D.png
Polymer Synthesis
• Traditional synthesis
H H
C=C
H H
– Initiation, using a catalyst that creates a free
radical
unpaired electron
R  + C=C  R – C – C 
– Propagation
R…… C – C  + C=C  R……C – C – C – C 
– Termination
R…… C – C  +  C – C……R  R –(C-C)n– R
Polydispersity
• Traditional synthesis  large variation in chain length
# of polymer chains
Average chain molecular weight
Mn
width is a measure
of polydispersity
M n  xi M i
number average
Mw
molecular weight
# of polymer chains of Mi
total number of chains
weight average
M w  wi Mi
weight of polymer chains of Mi = weight
fraction
total weight of all chains
molecular weight
• Degree of polymerization
– Average # of mer units/chain
Mn
nn 
m
nw 
Mw
m
by number
mer molecular weight
by weight
New modes of synthesis
• “Living polymerization”
–
–
–
–
Initiation occurs instantaneously
Chemically eliminate possibility of random termination
Polymer chains grow until monomer is consumed
Each grows for a fixed (identical) period
Polymers
• Homopolymer
– Only one type of ‘mer’
• Copolymer
– Two or more types of ‘mers’
• Block copolymer
– Long regions of each type of ‘mer’
• Bifunctional mer
– Can make two bonds, e.g. ethylene  linear polymer
• Trifunctional mer
– Can make three bonds  branched polymer
Polymers
• Linear
H
H
C=C
C
C
C
C
• Branched
• Cross-linked
C
C
H
C
C
C
H
Polymers
H
H out
C=C
H in
H
C
C
C
H
C
C
109.5°
C
H
C
C
C
R
Placement of side groups is fixed once polymer is formed
Example side group: styrene
R=
Cl
Isotactic
C=C
H
C
R R
R
C
C
C
R
C
C
H
C
R
R
H
C
R
C
C
C
C
C
R
C
C
C
Syndiotactic
C
C
R
R
C
Atactic
C
R
C
C
R
C
C
C
C
C
• Thermal Properties
– Thermoplastics
• Melt (on heating) and resolidify (on cooling)
• Linear polymers
– Thermosets
• Soften, decompose irreversibly on heating
• Crosslinked
• Crystallinity
• Linear: more crystalline
than branched or crosslinked
• Crystalline has higher
density than amorphous