1. Liquid Crystals
Liquid crystals have lost most of their translational order, but have retained some orientational order.
Smectic:
A, (C)
All translational order is lost.
Orientational order in 1 direction.
Translational order in 2 dimensions.
Translational order in 1 dimension.
Orientational order in 1, (2) directions.

2. Nematic Liquid Crystals
Example: Pairs of amphiphilic molecules (hydrophilic + hydrophobic)
8CB dimer
Nematic liquid crystals behave like a quadrupole , not like a dipole -

3. Various Mathematical Descriptions of Angular Symmetry
Angular momentum: quantum numbers l , m
l = l = l = 2
s – level p – level d – level
Angular wave functions:
spherical harmonics Ylm(,) Shown here for m = 0
Orbitals s pz dz2
- - + - + +
Multipoles Monopole Dipole Quadrupole
Tensors Scalar Vector Tensor
Polynomials
x0 = 1 (x,y,z)
x·x x·y x·z
y·x y·y y·z
z·x z·y z·z

4. Liquid Crystal Display (LCD): Twisted Nematic (TN)
Bright Dark
Bright Dark
A nematic liquid crystal is forced into a twisted structure by oriented polymers on the top and bottom plates. The thickness is adjusted such that the twist rotates the light polarization by 900, i. e., light is transmitted between crossed polarizers.
A voltage applied between top and bottom aligns the nematic molecules. In this “homeotropic” configuration they don’t rotate the polarization. The light is blocked.

5. Liquid Crystal Display (LCD): In-Plane Switching (IPS)
Dark Bright
More recently, the IPS geometry was introduced to LCDs , where brightness and color do not change as much with the viewing angle as for TN . The molecules are aligned by an in-plane electric field, applied between the side electrodes e1, e2 . Polarizer and analyzer are parallel, and an applied field makes the display bright.

6. Quasicrystals
These are crystals in higher dimensions, projected onto 3 dimensions. Translation and rotation invariance are both lost in 3D. However, local rotation symmetry remains. That allows 5- and 10-fold symmetry, which is impossible in real crystals. The diffraction spots remain sharp, but are infinitely dense. That makes it impossible to define a Brillouin zone.

7. Phase Diagrams of Alloys
Fully miscible
Partially miscible
eutectic
iquid
olid
Ge Si Ag Cu
Partially miscible alloys have a very peculiar phase diagram, where the melting point for the alloy is lower than the melting points of both components. The mixture with the lowest melting point is called a eutectic. The  phase is Ag-rich, the  phase Cu-
rich. The + phase consists of separate regions with sub-micrometer dimensions.

8. Catalytic Nanowire Growth Using Gold NanoparticlesGe Au
eutectic
AuGe
solid+solid
liquid+solid
liquid
Growing a Ge nanowire with a Au particle on top
ZnO
Path in the Au-Ge phase diagram during growth of Ge nanowires. One moves from left to right, as more Ge is dissolved in the Au. After crossing the 2nd boundary of the liquid, Ge precipitates out as solid while Au remains liquid.

9. Defects 0-dimensional: Vacancies, Interstitials
1-dimensional: Dislocations
2-dimensional: Stacking Faults, Grain Boundaries, Twins A
A dislocation (end view) and its Burgers vector b (pointing from A to B). This vector is obtained by comparing a loop in an undis-turbed region with a loop taken around a dislocation.