Jones, chapter 2: Forces, energies and time scales
The ordering in soft materials is intermediate between that of crystalline solid and that of liquid. The lack of crystalline order leads to 'soft' mechanical response of materials. Soft materials are usually partially ordered or amorphous.
Molecules are ’locked’ in a crystal lattice as a result of intermolecular bonds. The atoms vibrate around their equilibrium positions. This thermal motion increases with the temperature.
The energy and nature of the bonds are related to the properties of the materials. For instance, the energy of the bond and the stiffness of the material are related.
One-dimensional periodic structure is described with one base vector a. In crystallography the base vectors are called lattice vectors. The position vectors of the lattice points can be written as
R = pa,
where p is an integer.
Two-dimensional periodic structure is described with two lattice vectors a and b. Position vectors of the lattice points are given as
R = pa + qb,
where p and q are integers.
Two-dimensional lattices
|
crystal system
|
symbol
|
cell parameters
|
|
oblique
|
m
|
γ ≠ 90
, a ≠ b
|
|
rectangular
|
o
|
γ = 90
|
|
square
|
t
|
γ = 90
a = b
|
|
hexagonal
|
h
|
γ = 120
a = b
|
Example. Hydrophobin protein http://pubs.acs.org/doi/abs/10.1021/la2001943
Hydrophobin at air-water interface http://pubs.acs.org/doi/abs/10.1021/la803252g
For presenting three dimensional lattices three lattice vectors a, b, and c are needed. The position vectors of the lattice points may be written as
R = pa + qb + rc,
where p, q, and r are integers.
The unit cell geometries for seven crystal systems
http://en.wikipedia.org/wiki/Bravais_lattice
| cubic | |
| tetragonal | a = b ≠ c, α = β = γ = 90º |
| rhombohedral | a = b = c, α = β = γ ≠ 90º |
| orthorombic | a ≠ b ≠ c, α = β = γ = 90º |
| hexagonal | a = b ≠ c, α = β = 90, γ = 120º |
| monoclinic | a ≠ b ≠ c, α = β = 90, γ ≠ 90º |
| triclinic | a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90º |
In addition to the knowledge of the lattice, the positions of atoms (or other structural units) in a unit cell are needed for characterization of the structure of a crystalline material.
The position of an atom j in the unit cell l in a crystal may be written as rj + Rl, where rj gives the positions of atoms in the unit cell a and Rl = ∑ zl al the position of the unit cell in the crystal.
Example. Base centered cubic lattice. The unit cell is cubic and for the lattice constants hold: a = b = c, α = β = γ = 90 degrees. Atoms in the corners and in the middle of the unit cell, thus atomic positions in the unit cell are (0, 0, 0) and ( ½, ½, ½ ).
Example. Face centered cubic lattice. The unit cell is cubic and for the lattice constants hold: a = b = c, α = β = γ = 90 degrees.
Atoms are at positions r1 = (00,0), r2 = (a + b)/2,
r3 = (b + c)/2, r4 = (c + a)/2.
Example. Input file for the powder diffraction program PowCell which computes powder diffraction patterns on the basis of the crystal structure. The material is CsCl. The first line contains the six lattice parameters. The next line contains the number of atoms. The third and fourth line contain the name, Z, and coordinates of the atom. The last line contains the space group number.
cell 4.123 4.123 4.123 90 90 90W. Kraus, G. Nolze. Federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87, D-12205 Berlin
natom 2
Cs 55 0.0 0.0 0.0
Cl 17 0.5 0.5 0.5
rgnr 221
Example. Polymeric material may show several crystalline phases. block-co-polymers, lipids ...
http://www.almaden.ibm.com/st/chemistry/ps/self_assembly/block_copolymer/
Alexandridis et al. Record (Four Cubic, Two Hexagonal, and One Lamellar Lyotropic Liquid Crystalline and Two Micellar Solutions) in a Ternary Isothermal System of an Amphiphilic Block Copolymer and Selective Solvents (Water and Oil) Langmuir 1998; 14(10); 2627-2638
Example. Polymers and surfactants
In a liquid molecules are held together by intermolecular forces ín a state of high density. Molecules are not locked rigidly into well-defined positions of a lattice. Molecules change their positions relative to their neighbors on characteristic relaxation time. The relaxation time determines how easily the liquid will flow when a stress is applied.
The structure of a liquid is described by the atomic (molecular) radial distribution function.
Liquid-like order and solid-like elastic properties are typical for glasses. Glasses may show both short range order and intermediate range order, but no long range order. The structure of a glass is described by atomic (molecular) radial distribution function.
Example. Polystyrene forms an amorphous structure due to its flexibility.
http://en.wikipedia.org/wiki/Polystyrene
The radial distribution function
RDF = 4πr2 ρ(r),
where ρ is the atomic density, gives the average number of atom centers lying between distances r and r+dr from the centre of an arbitrary origin atom. For one component disordered system RDF is a suitable way to describe its structure.
For multi-component systems instead of only one radial distribution function several partial radial distribution functions are needed to describe the structure of the system.
The function ρab(r) presents the average atomic density of b-type of atoms at the distance r from an a-type of atom.
Important structural parameters describing the structure are the atomic distances and coordination numbers. Coordination numbers are the numbers of neighbors at the given distance from an average central atom.
The number of partial structure factors in n-component system is n(n+1)/2.
Example. Molten platinum.
Waseda & Ohtani, Z.Phys.B 21 229 (1975)
Example: Coordination number in a chain of equidistant atoms.
Soft materials are typically only partially ordered. On the other hand, such a material may show order at several length scales. Nanoscale ordering is 1-1000 nm. Mesoscopic ordering is between structural units with sizes between atomic and macroscopic scales.
Parameters describing partially ordered materials are, for instance: the crystallinity, which is the volume fraction of crystalline material, the crystallite size, and order parameters characterizing the preferred orientation of crystallites in the material with respect to a given direction in a sample.
Example. A liquid crystal (LC) may flow like a liquid, but have the molecules in the liquid arranged and oriented in a crystal-like way.
Example. Many natural polymers like cellulose or starchAlso synthetic polymers are rarely fully crystalline or amorphous. High density polyethylene may have a crystallinity of about 80 %.