# Introduction to Mathematical Physics/N body problem and matter description/Other matter arrangements

## Quasi crystals

Quasi crystals ([#References|references]) discovered by Israeli physicist Shechtman in 1982 correspond to a non periodical filling of a volume by atoms or molecules. Pentagon is a forbidden form in crystallography: a volume can not be filled with elementary bricks of fifth order symmetry repartited periodically. This problem has a two dimensional twin problem: a surface can not be covered only by pentagons. English mathematician Penrose \index{Penrose paving}discovered non periodical paving of the plane by lozenges of two types (see figure ---figpenrose--- and figpenrose2).

figpenrose

File:Penrose
Penrose paving: from two types of lozenges, non periodical paving of the plane can be constructed.

figpenrose2

File:Penrose2
Penrose paving: same figure as previous one, but with only two gray levels to distinguish between the two types of lozenges.

Obtained structures are amazingly complex and it is impossible to encounter some periodicity in the paving.

secristliquides

## Liquid crystals

Liquid crystals ([#References|references]), also called mesomorphic phases\index{mesomorphic phase}, are states intermediary between the cristaline perfect order and the liquid disorder. Molecules that compound liquid crystals\index{liquid crystal} have cigar--like shape. They arrange themselves in space in order to form a "fluid" state more or less ordered. Among the family of liquid crystals, several classes can be distinguished.

In the nematic \index{nematic} phase (see figure ---fignematique---), molecular axes stays parallel. There exists thus a privileged direction. Each molecule can move with respect to its neighbours, but in a fish ban way.

File:Nematique
Nematic material is similar to a fish ban: molecules have a privileged orientation and located at random points.}
fignematique

When molecules of a liquid crystal are not superposable to their image in a mirror, a torsion of nematic structure appears: phase is then called cholesteric \index{cholesteric} (see figure ---figcholesteric---).

figcholesteric

File:Cholesteric
In a cholesteric phase, molecules are arranged in layers. In each layer, molecule are oriented along a privileged direction. This direction varies from one layer to another, so that a helicoidal structure is formed.

Smectic \index{smectic} phase is more ordered : molecules are arranged in layers (see figure figsmectiqueA). The fluid character comes from the ability of layers to slide over their neighbours.

figsmectiqueA

File:SmectiqueA
Smectic A

To describe deformations of nematics, a field of vector is used. To describe deformations of smectics, each state is characterised by set of functions ${\displaystyle u_{i}(x,y)}$ that describes the surface of the i${\displaystyle ^{th}}$ layer. Energical properties of nematic materials are presented at section secenernema.

## Colloids

Colloids \index{colloid} are materials finely divided and dispersed. Examples are emulsions \index{emulsion} and aerosols. Consider a molecule that has two parts: a polar head which is soluble in water and an hydrophobe tail. Such molecules are called amphiphile\index{amphiphile molecule}. Once in water, they gather into small structures called micelles\index{micelle} (see figure fighuileeau). Molecules turn their head to water and their tail to the inside of the structure.

Remark: Cells of live world are very close to micelles, and it may be that life\index{life (origin of)}\index{origin of life} appeared by the mean of micelles.

fighuileeau

File:Huileeau
Amphiphile molecules (as oil molecules) that contains a hydrophile head and an hydrophobe tail organize themselves into small structures called micelles.

Micelles can be encountered in mayonnaise ([#References|references]). Tensioactive substances allow to disolve micelles (application to soaps). Foams are similar arrangements ([#References|references]). For a mathematical introduction to soap films and minimal surfaces, check ([#References|references]).

secglassyspin

## Glass

Glass state ([#References|references]) is characterized by a random distribution of molecules (see figure figglass). Glass\index{glass} is solid, that implies that movements of different constituents are small.

figglass

File:Glass
Glass is characterized by a great rigidity as crystals. However position of atoms are random.}

Phase transition between liquid state and glassy state is done progressively. A material close to glass can be made by compressing small balls together. Under pressure forces, those small balls are deformed and stick to each other. Following question arises: are remaining interstices sufficiently numerous to allow a liquid to pour trough interstices ? This pouring phenomena is a particular case of percolation phenomenon\index{percolation} ([#References|references]). Figure ---figpercol--- illustrates this phenomenom in an experiment presented in ([#References|references]).

figpercol

File:Percol
Example of percolation : black balls are conducting and white balls are insulating. Current running trough the circuit is measured as a function of proportion ${\displaystyle p}$ of white balls. There exists a critical proportion ${\displaystyle p_{c}}$ for which no more current can pass. Study of this critical point allows to exhibit some universal properties encountered in similar systems.

Another example is given by the vandalized grid ([#References|references]) where connections of a conducting wire is destroyed with probability ${\displaystyle p}$.

Spin glasses are disordered magnetic materials. A good example of spin glass is given by the alloy coper manganese, noted CuMn where manganese atoms carrying magnetic moments are dispersed at random in a coper matrix. two spins tend to orient them selves in same or in opposite direction, depending on distance between them (see figure ---figspinglass---).

figspinglass

File:Spinglass
In a spin glass, interactions between neighbour spins are at random ferromagnetic type or anti ferromagnetic type, due to the random distance between spin sites.

The resulting system is called "frustrated": there does not exist a configuration for which all interaction energies are minimal. The simplest example of frustrated system is given by a system constituted by three spins labelled 1,2 and 3 where interactions obey the following rule: energy decreases if 1 and 2 are pointing in the same direction, 2 and 3 are pointing in the same direction and 1 and 3 are pointing in opposite direction. Some properties of spin glasses are presented at section secverredespi.

## Sand piles, orange piles

Physics of granular systems is of high interest and is now the subject of many researches. Those systems, as a sand pile, exhibit properties of both liquids and solids. The sand in a hour-glass doesn't pour like liquids: the speed of pouring doesn't depend on the high of sand above the hole. The formation of the sand pile down of the hole is done by internal convection and avalanche \index{avalanche} at the surface of the pile (see for instance ([#References|references])).