Download Let us consider a cube, of length c. The area of its surface is . Let`s

Some easy nanophysics (preliminary document, please do not
distribute)
Hervé This
Septembre 2013
Let us consider a cube, of length c. The area of its surface is .
Let's now divide this cube in 8 smaller cubes, so that the side is divided by 2.
We get
cubes of lenght c/2, having each an area of
=
and a volume
= .
If we divide the length by n :
- the number of small cubes is equal to
- the total area is
=
In short, the area can increase very much.
In particular, after a big number of divisions (but please don't forget that we could arrive to the same
state bottom up instead of up bottom),when the size becomes nanometric (1-100 nm), quantum effects
appear. For example, when going from
to
(m, always SI units), the total area is multiplied by...
(How much ? This is an exercise, for you).
The various nanoemulsions
The small size of nanoparticules does not only give them high specific area and improved stability
(colloidal nanosystems don't sediment of cream), but they are also able to escape the immune system,
and target some specific celles, so that they are very promising vectors of drugs.
Liposomes are vesicles made of one or many concentric double layers of phospholipids and cholestrerol
molecules including an aqueous medium. The size is between 30 nm and many micrometres.
Niosomes are supramolecular assemblies similar to liposomes, but the molecules which make up the
double layer are not phospholipids, but rather synthetic surfactants (non ionic lipids).
This is also the case for polymersomes, for which block copopymers (with an hydrophilic and an
hydrophobic part) form the structure in which the aqueous solution is trapped.
Micelles are self-organized structures made of amphiphiles molecules, with a core-shell structure,
dispersed in an aqueous medium. Micelles form when the concentration in surfactants is higher than the
critical micellar concentration. In this case, amphiphiles molecules sef organize in order to put their
hydrophobic part together, and expose solely their hydrophilic moieties. Micelles are supramolecular
systems in equilibrium with the other amphiphilic molecules in solution.
Some useful basis for calculating the properties of nanoobjects
In quantum physics, objects are not considered as particles. Instead wave functions are used: remember
the de Broglie equation :
between the wavelength of the wave corresponding to an object and the momentum p = mv, for the
"particle" that is the same object.
Remember also that this can applies to electrons as well as to fullerenes, which have been used for
making "microscopes" !
In short, let's consider a particular wave function
The probability P(x,t) that a particle is in an intervalle of length dx around x at time t is equal to :
This implies that the wave function is "normalized" (a probability!):
One can understand easily that I used here a 1 dimensional expression, but it would be very similar with
3 dimensions.
When you consider a non relativistic case, the evolution of the system can be described with the time
dependant Schrödinger equation :
[please do observe that this equation has some similarities with the heat equation, or with the second
Fick equation]
Here, V(x,t) is a potential energy function. When it does not depend on t, time and position can be
separated, so that:
When you put this expression in the Schrödinger equation, you get:
And this can be written:
(using an hamiltonian operator H).
The wave function is a proper function of the Hamiltonian, and the energy is a proper value
corresponding to this proper function. Often, the equation has many solutions, with energies
Energies for electons in atoms and solids
Whe the Schrödinger equation is used for the description of isolated atoms, the potential function V(x)
describes the Coulomb interactions (you remember that between two charged particles, you have the
force
) between the electrons and the nucleus, as well at interactions between electrons.
This gives a "potential well", applied to electons :
The solution of the Schrödinger equation are a set of discontinuous energies, as you could see in atomic
physics:
When many atoms are linked, the potential is different :
The Coulomb potential applied only at the edges of the group. At the center, there is a series of minima
and maxima. When the number atoms is big, on the contrary, the edges can be neglected (except when
you consider the surface, of course) and you can consider that the material is obtained by a repetitinon of
a motif: this is the idea of solid state physics. The behavior can then be described (see the wonderful
book by Kittel), and energy levels are broadened, overlap, so taht "bands" are formed. One can have
empty and full bands, depending on the number of electrons in atoms, and the behavior of the solid can
be described using the energy distances between bands.
When the higher energy bands are not full, then electrons can move between all atoms, being weakly
bound: this is the case for metals.
Electrons in nanostructures
For nanostructures, this energy function has to be considered : :
Here you have to imagine between 10 to 100 atomes (or molecules, for organic nanoparticles, for
example) forming a "particle". On can show that the most external electrons in the conduction bands are
weakly bound, and move as is they were in a constant potential, as in a metal; on the other hand, the
presence of edges induces particular effects.
For a first approach, let's consider the simple squere energy well.
The potential energy for a square well
Let's consider the following well:
Its equation is simply:
if
, and
if
If the energy E of the particle is <
are stationary waves:
, then energies are quantizied, discrete. In the well region, solutions
if
Out of the well, the solution have to be 0 when
becomes large; they are of the shape:
if
if
function at the limit of the well (4 unknown variables, 4 equations).
In the general case, the solutions are difficult to write, but one can approximatd the solution for
statioanry wave function in an infinitely deep well. In this case, the condition is
for
.
Then the solution of minimum energy is
It is associated to a fundamental energy:
And for energy in general :
With 3 dimensions, spherical symetry, one would have :
where l is the angular quantum number, and
the n-th zero of the l-th spherical function of Bessel
The confinement of electrons in a nanoparticles well is called quantum confinement. As there are
quantification, as for atoms, the nanostructures are sometimes called "artificial atoms".
Zêta potentiel, a frequently used measurement
The zeta potential is used to characterize the surface charge of colloids. It is not the real charge at the
surface of the dispersed structures in a colloid, but the one it has in solution.
Solvatation of a charges particle induces the formation of a double layer of ions. Le layer near the
particle is made of ions whose charge is opposite to the charge of the surface. Such ions are firmly
bound to the particle, and this is why "Stern layer" is also called bound layer. This layer attracts ions of
opposite charge: this is the diffuse layer, or Gouy layer. The zeta potential is frequently measured in
order to give a quantitative determination of stability of particles in suspension: the more the zeta
potential, the more the particles repel each other, nd the less the aggregation.
(to be continued)
Questions of stability of nanoemulsions
(see sedimentation and creaming, part II and III)
References
biodistribution par imagerie de fluorescence in vivo, Mathieu Goutayer, Thèse de de l'Université Paris
VI, 11, dec, 2008
Charles Kittel, Solid State Physics.
D. McQuarrie, Physical chemistry.
And a wealth of scientific papers, on almost any aspect
Bernadene A. Magnuson, Tomas S. Jonaitis, and Jeffrey W. Card, A Brief Review of the Occurrence,
Use, and Safety of Food-Related Nanomaterials, Journal of Food Science r Vol. 76, Nr. 6, 2011
In particular, there is now a whole journal on nano!