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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!