1. – Bad Honnef – 2006 –  – Anomalous Transport – Bad Honnef – 2006 –  – Anomalous Transport – Bad Honnef – 2006 – – Bad Honnef – 2006 –  – Anomalous Transport – Bad Honnef – 2006 –  – Anomalous Transport – Bad Honnef – 2006 – Rustem Valiullin Department of Interface Physics University of Leipzig, Germany Bad Honnef, 2006 New perspectives on anomalous dynamics during sorption hysteresis

2. Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions

3. Adsorption hysteresis phenomenon Micropores < 2 nm Reversible adsorption m m P P Mesopores 2-50 nm Irreversible adsorption vapor porous material Adsorption hysteresis in mesoporous materials P

4. The simplest view on adsorption hysteresis Kelvin equation Cohan LH. Sorption hysteresis and the vapor pressure of concave surfaces. J. Am. Chem. Soc. 1938;60:433-435.

5. Two metastable phases Equilibrium liquid-vapour transition equality of the potentials Upper limit of the metastable vapour zero barrier between the local and global potential minima P Liquid filled Empty

6. H1 and H2 type isotherms H1 - Hysteresis due to metastable pore fluid,narrow pore-size distribution no percolation effects! H2 - Hysteresis due to both metastable statesbroad pore-size distribution of the pore fluid and percolation effects. Pore blocking Cavitation

7. Multiplicity of metastable states Given that the occurrence of hysteresis represents a departure from equilibrium, what is the nature of the relaxation processes in the hysteresis region and why are hysteresis loops so easily reproducible in the laboratory? Kierlik E. et al Capillary condensation in disordered porous materials: Hysteresis versus equilibrium behavior. Phys. Rev. Lett. 2001;87:055701-4. Disordered lattice-gas model: Multiplicity of local mean-field solutions.The solid lines represent the equilibrium curves obtained by connecting the states of lowest grand potential.

8. Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions

9. Experimental method Nuclear magnetic resonance Spin angular momentum Magnetic moment radio waves in radio waves out microscopic macroscopic

10. spin-echo signal intensity S 90° g  g  diffusion time – t d (  ) z 0 Pulsed Field Gradient NMR

11. spin-echo signal intensity S 90° g  g  diffusion time – t d (  ) z 0 Pulsed Field Gradient NMR

12. Direct probe of diffusion propagator Stimulated echo NMR pulse sequence spin-echo signal intensity S 90° g  g  diffusion time – t d (  ) q =  g - wave number Gaussian propagator t d = 10 -3  1 s

13. NMR summary spin-echo signal intensity S 90° g  g  diffusion time – t d (  ) 90° FID intensity – amount adsorbed and uptake kinetics PFG NMR method – self-diffusivity in the same sample at the same conditions

14. - spinodal decomposition of alkali- borosilicate glasses - random structure - average pore diameter between 4 and 6 nanometers Porous Materials Vycor porous glass (Corning Inc.) Pellenq, R. J. M.; Rodts, S.; Pasquier, V.; Delville, A.; Levitz, P. Adsorpt.-J. Int. Adsorpt. Soc. 2000, 6, 241. Pore size distribution provided by the manifacturer. 3 mm 12 mm

15. Experimental setup V res >> V pore initial pressure – P  10 -5 atm temperature – T = 24° C LiquidP s (atm) M (g/mol)  (kg/m 3 ) Acetone0.293580.79 n-Hexane0.193860.66 Cyclohexane0.124840.78 turbo-molecular pumpmagnet

16. Experimental protocol FID signal intensity after pressure step Self-diffusion study after equilibration P

17. Normalized isotherm P FID 0 z 1  Concentration; Pore filling

18. Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions

19. Cyclohexane in Vycor porous glass  - adsorption  - desorption

20. Effective diffusivity: Fast exchange limit  - adsorption  - desorption D eff = p a D a + p g D g adsorbed phase gaseous phase Detailed balance principle d ~ 6 nm

21.  - adsorption  - desorption D eff = p a D a + p g D g This is not enough! Capillary condensed phase differently distributed on adsorption and desorption Effective diffusivity: Concentration dependence

22. Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions

23. Micro via Macro m P 1  P 2  eq Diffusion-controlled uptake Cylindrical samples with radius a  time 00

24. Example 1: Nitrogen in Vycor Rajniak, P.; Soos, M.; Yang, R. T. AICHE J. 1999, 45, 735. Slowing down of the uptake in the hysteresis region Due to decreasing diffusivity? No Experimental desorption diffusivity data

25. Adsorption kinetics in Vycor 3 mm 12 mm Diffusion-controlled model

26. Example 2: Nitrogen in porous silicon Wallacher, D.; Kunzner, N.; Kovalev, D.; Knorr, N.; Knorr, K. Phys. Rev. Lett. 2004, 92, 195704. Adorption kinetics follows stretched-exponential law with   0.5. Authors regard it as an indication of disorder.

27. Kinetics in the hysteresis region  = 0.66  = 0.37 Diffusion-controlled uptake Kohlrausch relaxation

28. Two mechanisms of the uptake Early times Diffusion-controlled uptake - Equilibrating concentrations in the intrapore gaseous phase and in reservoir - Building up next layers – polylayer adsorption - Formation of some bridges – capillary condensation quasi-equilibrium regime

29. Two mechanisms of the uptake Later times - System is in a metastable or quasi-equilibrium regime - Local free energy minimum corresponding to a certain density arrangement - Thermally activated density fluctuations resulting in density redistribution - Activated barrier crossing between local free energy minima - Slow relaxation towards the global free energy minimum quasi-equilibrium regime

30. Evidence of the activated character Density fluctuations around at equilibrium as observed in Glauber dynamics. Woo HJ, Monson PA. Phase behavior and dynamics of fluids in mesoporous glasses. Phys Rev E 2003;67:041207. Different realizations of density evolution in a slit-like pore after quench from low-pressure to high- pressure state. Restagno F, Bocquet L, Biben T. Metastability and nucleation in capillary condensation. Phys Rev Lett 2000;84:2433-2436.

31. Activated dynamic scaling Free energy barriers ~   (  > 0) Huse, D. A. Phys. Rev. B 1987, 36, 5383 Typical relaxation time Expected scaling function Experimental and computer simulations p = 3 Ogielski AT, Huse DA. Critical-Behavior of the 3-Dimensional Dilute Ising Antiferromagnet in a Field. Phys Rev Lett 1986;56:1298-1301. Dierker SB, Wiltzius P. Random-Field Transition of a Binary-Liquid in a Porous-Medium. Phys Rev Lett 1987;58:1865-1868.

32. Adsorption kinetics in Vycor Diffusive part Activated part Overall density equilibration function A diff ~ 0.8 ; t 0 ~ 600 s ;  ~ 4500 s

33. Conclusions  Equilibrium and non-equilibrium molecular dynamics in mesoporous materials in different regions of the adsorption isotherm are indepenedently probed using nuclear magnetic resonance methods.  Comparative analysis of the obtained experimental results yields a two-step mechanism of the molecular uptake in the adsorption hysteresis region.  These two mechanisms are identified as diffusion-controlled uptake at short times and uptake controlled by very slow activated density redistribution at longer times. The latter prevents the system from reaching equilibrium on laboratory time scale.

34. Acknowledgements Prof. J. Kärger – University of Leipzig Prof. P. Monson – University of Massachusets Prof. H.-J. Woo – University of Nevada, Reno PhD Students: P. Kortunov, S. Naumov

35. Self-diffusion Adsorption kinetics Two mechanisms of adsorption 90° NMR method Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions

36. a·nom·a·lous (ə-nŏm'ə-ləs) adj. 1. Deviating from the normal or common order, form, or rule. 2. Equivocal, as in classification or nature. [From Late Latin anōmalos, from Greek, uneven : probably from an-, not; see a– + homalos, even (from homos, same).] Anomalous transport