Misura del grado di disordine e dell’indice di frattalità mediante tecniche NMR di diffusione anomala Silvia Capuani*, Marco Palombo°, Alessandra Caporale*, Andrea Gabrielli* *CNR-IPCF UOS Roma Physics Dpt. “Sapienza” Univ, Rome, Italy °MIRcen, I 2 BM, Commissariat à l'Énergie Atomique, CNRS, Fontaney-aux-Roses, Paris, France 101° Congresso della SIF. Roma, settembre 2015

In disordered and complex systems and highly heterogeneous media, diffusion is often non-Gaussian or anomalous And many other examples... Non-ordinary diffusion: anomalous diffusion

(l) = pdf of jumps lenght l w(t) = pdf of waiting time t, between two consecutive jumps If, <  If =  Gaussian diffusion, ordinary diffusion equation:  t Anomalous sub-diffusion, time fractional derivative equation:  t , with 0<  <1 Anomalous super-diffusion, space fractional derivative equation:  t 1/ , with 0<  <1 If = , <  Metzler R., Klafter J., Phys. Rep. 339 (2000) CTRW approach

Why and How to measure diffusion by NMR By using radio-frequency and magnetic field gradient pulses. Pulse Gradient Spin Echo (PGSE) sequence. The PGSE signal is proportional to the characteristic function of the diffusion propagator in time   :  varying PGSE

Biophysical interpretation of  and  parameters Length scale(s) of magnetic heterogeneity Disorder Degree Anomalous diffusion in 3D porous media: experiments Palombo M., Gabrielli A., De Santis S., Cametti C., Ruocco G., Capuani S., J. Chem. Phys. 135 (2011)

Steinhardt P.J., Nelson D.R., Ronchetti M., Phys. Rev. B 28 (1983) Analysis of sub-diffusion processes and disorder degree in crowded media Anomalous diffusion in 3D porous media Q = Q 6 system /Q 6 fcc Bond-orientational order parameters Sphere packing Φ = 0.74 Q = 1 Φ = Q = 0.65 Φ = Q = 0.70 Spatial distribution of obstacles

Φ = 0.74Φ = 0.68Φ = 0.64 Occupied volume Free volume Q =1 <1 fcc random config. Anomalous diffusion in 3D porous media: simulations Palombo M., Gabrielli A., Servedio V.D.P., Ruocco G., Capuani S., Scientific Report, 3 (2013)

Anomalous diffusion in 3D porous media: NMR experiments comparison of Monte Carlo simulation results to dNMR experimental data From geometrical structure characterization to anomalous diffusion properties: Palombo M., Gabrielli A., Servedio V.D.P., Ruocco G., Capuani S., Scientific Report, 3 (2013)

Highly porous polymeric matrices with randomly oriented interconnected pores obtained from a solution of polyvinyl alcohol and etyltrimethylammonium bromide (PVA scaffolds) void size distribution:  m interconnection size distribution: 4-50  m the three scaffolds differ in the roughness of the walls of their voids and interconnections Application to multiscale porous materials Barbetta A. et al., Soft Matter,6 (2010) PVA1PVA2PVA3 PVA3 PVA2 PVA1

Gaussian approach Imaging: PGSTE sequence TR = (  ) ms TE = 15 ms;  = 2 ms g = 74 mT/m  = (20  520) ms PVA 3 PVA 1 PVA 2 PVA 3 PVA 1 PVA 2 3 S A = 45 m 2 /g S A = 68 m 2 /g S A = 106 m 2 /g Palombo M., Barbetta A., Cametti C., Dentini M., Capuani S., in preparation

Imaging: PGSTE sequence TR = (  ) ms TE = 15 ms;  = 2 ms g = 74 mT/m  = (20  520) ms PVA 3 PVA 1 PVA 2 PVA 3 PVA 1 PVA 2 3 S A = 45 m 2 /g S A = 68 m 2 /g S A = 106 m 2 /g Anomalous diffusion approach Palombo M., Barbetta A., Cametti C., Dentini M., Capuani S., in preparation

Application to multiscale porous materials Palombo M., Barbetta A., Cametti C., Dentini M., Capuani S., in preparation M.J. Saxton Biophys, J. 66 (1994) X=S/  fractal dimension

Conclusions  ( which quantifies subdiffusion processes) can be measured by non invasive and non destructives diffusion NMR techniques. , is affected by both the density (Φ) and the spatial distribution (Q) of obstacles:  =  (Q, Φ)  value quantifies global structural complexity. It enables a classification of different kinds of disorder and it allows to monitor structural transitions. Our results indicate that , which quantifies the crowding effect on diffusion process, can be used to measure the fractal dimension of the random path (d w ):  = 2/d w  MAPS dramatically increase NMR sensitivity to micro and submicrostructures The present work suggests that  may be used to quantify unresolved effects due to heterogeneities and disorder in soft materials and living tissues.

Thank you for your attention attention NMR Laboratory, CNR-ISC Physics Dpt. Sapienza University of Rome, Italy