The way we can describe a behaviour in low scales is based on (i) a particle related kinematic and on (ii) the evolution of a probability distribution on the configuration space of the particle [3]. Based on a probability distribution, one can make the macroscopic approach by a strains tensor analysis giving the microscopic contribution. Each particle kinematic is described by the hydrodynamic term and interactions' effort term. The terms that result from the brownians effects are taken into account in the convection-diffusion equation diffusive term which characterises the probability distribution evolution (called Fokker-Planck equation, see [6]for example).
Some nanometric models offer the possibility to take into account the polymer chain flexibility phenomena, like for example the multi-dumbell type model. These models are characterized by N dumbells formed chains (N+1 nodes and N extensible connectors). We can write the evolution equation [4] as a Fokker-Planck equation characterized by a high dimensin probability space. For example, when associating two dumbells (N=2) we define a 6 dimension problem. In fact, for each 3 dimension space first dumbell position, there are an infinity of second dumbell position in such a space. For a 3 dumbells' chain we define a 9 dimension problem, and so on.
Recently, we have suggested [1,2,5] a new numerical technique that enables as to solve models that have never been solved yet, and defined in a high space dimension.
Under the nanometric scale, quantum effects govern the materials state and consequently affect the finest materials' description. In this scale, we face an infinite number of elementary particles, which must be considered as waves when considering the quantum mechanics. The Schrödinger equation describes this type of behaviour. The highly multidimensional character of the Schrödinger equation justifies why there are no solutions neighter analitic nor numeric al to evaluate the electronic density [7], and also the true inter-atmoic potential. By determining the inter-atomic potentials we can understand the structure and the materials' mechanics in the nanomteric scale.
Untill now, such a solution has never been possible, and scientists have just used other inter-atomic potentials, empirical or quantum theory inspired, in a molecular dynamic simulation context. To efficiently solve the Schrödinger equation, one may firstly solve some numerical problems, we have begun to establish its basis by presenting multidimensional problems' specific strategies.
The Schrödinger equation solution can provide a better understanding of macromolecular constituted materials, where are encountered strong links in the molecule and weak links between molecules. A reflexion on the interaction : structure - mechanical properties in the molecule scale - properties in the molecule grouping scale (where confinement effects are present) must be carried out.
[1] Ammar A., Mokdad B., Chinesta F., Keunings R., «A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids », Journal of Non-Newtonian Fluid Mechanics, vol. 139, 2006, p. 153-176.
[2] Ammar A., Mokdad B., Chinesta F., Keunings R., «A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space-time separated representation», Journal of Non-Newtonian Fluid Mechanics, A paraître, 2007.
[3] Ammar A., Prulière E., Chinesta P., Laso M., «Reduced Numerical Modeling of Flow Involving Liquid-Crystalline Polymers», soumis au Journal of Non-Newtonian Fluid Mechanics.
[4] Bird R.B., Curtiss, C.F., Armstrong, R.C., Hassager, O., Dynamic of polymeric liquid, Volume 2: Kinetic Theory, John Wiley and Sons, 1987.
[5] Mokdad B., Pruliere E., Ammar A., Chinesta F., «On the simulation of kinetic theory models of complex fluids using the Fokker-Planck approach», Applied Rheology, A paraître, 2007.
[6] Ottinger H.C. Stochastic processes in polymeric fluids, Springer, 1996.
[7] Weiner J.H., Statistical mechanics of elasticity, Ed. Dover, 2002.
Laboratoire de Rhéologie - UMR 5520 / UJF / INPG
Amine AMMAR
Chinesta Francisco, Laboratoire : LMSP, ENSAM Paris.
