NUMERICAL MODELING OF DIFFUSION IN COMPLEX MEDIA WITH SURFACE INTERACTION EFFECTS
Diffusion in natural, technological and biological systems is very common and most important process. Within these systems, which contain complex media, diffusion may depend not only on internal geometry, but also on the chemical interactions between solid phase and transported particles. Modeling remains a challenge due to this complexity. Here we first present a new hierarchical multiscale microstructural model for diffusion within complex media that incorporates both the internal geometry of complex media and the interaction between diffusing particles and surfaces of microstructures. Hierarchical modeling approach, which was introduced in , is employed to construct a continuum diffusion model based on a novel numerical homogenization procedure. Using this procedure, we evaluate constitutive material parameters of the continuum model, which include: equivalent bulk diffusion coefficients and the equivalent distances from the solid surface. Here, we examined diffusion of glucose through water using the following two geometrical/material configurations: silica nanofibers, and a complex internal structure consisting of randomly placed nanospheres and nanofibers. This new approach, consisting of microstructural model, numerical homogenization and continuum model, offer a new platform for modeling diffusion within complex media, capable of connecting micro and macro scales.
 A. Ziemys, A. Grattoni, D. Fine, F. Hussain, M. Ferrari, Confinement effects on monosaccharide transport in nanochannels, The Journal of Physical Chemistry B 114−34 (2010) 1117−11126.
 N. Aggarwal, J. Sood, K. Tankeshwar, Anisotropic diffusion of a fluid confined to different geome-tries at the nanoscale. Nanotechnology, Vol. 18−33 (2007) 5.
 D. C. Rapaport, The Art of Molecular Dynamics Simulation, Cambridge University Press, (2004).
 A. Jr MacKerell, et al., CHARMM: The Energy Function and Its Parameterization with an Over-view of the Program, J. Phys. Chem. B, Vol. 102−18 (1998) 3586-3616
 A. Ziemys, M. Ferrari, C. N. Cavasotto, Molecular Modeling of Glucose Diffusivity in Silica Nanochannels. J. Nanosci. Nanotechnol., Vol. 9 (2009) 6349–6359.
 J. C. Phillips, R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, L. Kale´, K. Schulten, Scalable molecular dynamics with NAMD. J. Comput. Chem., Vol. 26 (2005) 1781–1802.
 W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, M. L. Klein, Comparison of simple potential functions for simulating liquid water. J. Chem. Phys., Vol. 79 (1983) 926–935.
 E. R. Cruz-Chu, A. Aksimentiev, K. Schulten, Water-silica force field for simulating nanodevices. J. Phys. Chem. B, Vol. 110 (2006) 21497–21508.
 J. K. Gladden, M. Dole, Diffusion in supersaturated solution-II: glucose solutions. J. Am. Chem. Soc., Vol. 75 (1953) 3900−3904.
 K. J. Bathe, Finite Element Procedures. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1996.
 T. Hughes, The finite element method: linear static and dynamic finite element analysis. 2000, New York: Dover Publications.
 N. Kojic, A. Kojic, and M. Kojic, Numerical determination of the solvent diffusion coefficient in a concentrated polymer solution. Communications in Numerical Methods in Engineering, Vol. 22−9 (2006) 1003−1013.
 M. Kojic, N. Filipovic, B. Stojanovic, and N.Kojic, Computer Modeling in Bioengineering, Theo-retical Background, Examples and Software, J Wiley and Sons, Chichester, 2008.
 M. Kojić, M. Milošević, N. Kojić, M. Ferrari, A. Ziemys, On diffusion in nanospace, JSSCM, Vol. 5−1 (2011) 84−109.
 M. Kojić, M. Milošević, N. Kojić, M. Ferrari, A. Ziemus, Diffusion in composite materials with surface interaction effects: microstructural and continuum models, Nature Materials (to be submitted), 2012.
 C. Boutin, C. Geindreau, Periodic homogenization and consistent estimates of transport parame-ters through sphere and polyhedron packings in the whole porosity range, Phys. Rev. E, Vol. 82 (2010) 036313.
 Z. Hashin, Assessment of the self-consistent scheme approximation, J. Compos. Mater., Vol. 2 (1968) 284.
 M. O. Nicolas, J. T. Oden, K. Vemagantia and J. F. Remacle, Simplified methods and a posteriori estimation for the homogenization of representative volume elements, Comput. Methods Appl. Mech. Engrg., Vol. 176 (1999) 265−278
 S. J. Chapman, R. J. Shipley, R. Jawad, Multiscale modelling of ﬂuid ﬂow in tumours, Bull. Math. Biology, Vol. 70 (2008) 2334–2357
 W. Yu, T. Tnag, Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials, Int. J. Solids Struct., Vol. 44 (22−23), (2007) 7510–7525.