Bad Honnef 2006 Anomalous Transport Bad Honnef 2006
– Bad Honnef – 2006 – – Anomalous Transport – Bad Honnef – 2006 – New perspectives on anomalous dynamics during sorption hysteresis Rustem Valiullin Department of Interface Physics University of Leipzig, Germany Bad Honnef, 2006
Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions
Adsorption hysteresis phenomenon Adsorption hysteresis in mesoporous materials m Micropores < 2 nm vapor Reversible adsorption P porous material m P Mesopores 2 -50 nm Irreversible adsorption P
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.
Two metastable phases Equilibrium liquid-vapour transition equality of the potentials P Upper limit of the metastable vapour zero barrier between the local and global potential minima Liquid filled Empty
H 1 and H 2 type isotherms Pore blocking Cavitation H 1 - Hysteresis due to metastable pore fluid, narrow pore-size distribution no percolation effects! H 2 - Hysteresis due to both metastable states of the pore fluid and percolation effects. broad pore-size distribution
Multiplicity of metastable states 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. 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.
Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions
Experimental method Nuclear magnetic resonance Magnetic moment Spin angular momentum radio waves in microscopic radio waves out macroscopic
Pulsed Field Gradient NMR 90° 90° g z 0 spin-echo signal intensity S diffusion time – td ( ) g
Pulsed Field Gradient NMR 90° 90° g z 0 spin-echo signal intensity S diffusion time – td ( ) g
Direct probe of diffusion propagator Stimulated echo NMR pulse sequence 90° g td = 10 -3 1 s diffusion time – td ( ) q = g - wave number Gaussian propagator 90° spin-echo signal intensity S g
NMR summary FID intensity – amount adsorbed and uptake kinetics 90° in the sample at the same conditions PFG NMR method – self-diffusivity 90° 90° g spin-echo signal intensity S diffusion time – td ( ) g
Porous Materials Vycor porous glass (Corning Inc. ) - spinodal decomposition of alkaliborosilicate glasses - random structure - average pore diameter between 4 and 6 nanometers 12 mm 3 mm Pore size distribution provided by the manifacturer. Pellenq, R. J. M. ; Rodts, S. ; Pasquier, V. ; Delville, A. ; Levitz, P. Adsorpt. -J. Int. Adsorpt. Soc. 2000, 6, 241.
Experimental setup Vres >> Vpore initial pressure – P 10 -5 atm temperature – T = 24° C Liquid Ps (atm) M (g/mol) (kg/m 3) Acetone 0. 293 58 0. 79 n-Hexane 0. 193 86 0. 66 Cyclohexane 0. 124 84 0. 78 turbo-molecular pump magnet
Experimental protocol FID signal intensity after pressure step P Self-diffusion study after equilibration
Normalized isotherm FID P 0 z 1 Concentration; Pore filling
Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions
Cyclohexane in Vycor porous glass - adsorption - desorption
Effective diffusivity: Fast exchange limit Detailed balance principle - adsorption - desorption d ~ 6 nm Deff = pa Da + pg Dg adsorbed phase gaseous phase
Effective diffusivity: Concentration dependence - adsorption - desorption Deff = pa Da + pg Dg This is not enough! Capillary condensed phase differently distributed on adsorption and desorption
Outline Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions
Micro via Macro Diffusion-controlled uptake P 1 P 2 m Cylindrical samples with radius a eq 0 time
Example 1: Nitrogen in Vycor Experimental desorption diffusivity data Slowing down of the uptake in the hysteresis region Due to decreasing diffusivity? No Rajniak, P. ; Soos, M. ; Yang, R. T. AICHE J. 1999, 45, 735.
Adsorption kinetics in Vycor 12 mm 3 mm Diffusion-controlled model
Example 2: Nitrogen in porous silicon Adorption kinetics follows stretched-exponential law with 0. 5. Authors regard it as an indication of disorder. Wallacher, D. ; Kunzner, N. ; Kovalev, D. ; Knorr, N. ; Knorr, K. Phys. Rev. Lett. 2004, 92, 195704.
Kinetics in the hysteresis region Kohlrausch relaxation Diffusion-controlled uptake = 0. 66 = 0. 37
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
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
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 highpressure state. Restagno F, Bocquet L, Biben T. Metastability and nucleation in capillary condensation. Phys Rev Lett 2000; 84: 2433 -2436.
Activated dynamic scaling Free energy barriers ~ ( > 0) 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. Huse, D. A. Phys. Rev. B 1987, 36, 5383
Adsorption kinetics in Vycor Diffusive part Activated part Overall density equilibration function Adiff ~ 0. 8 ; t 0 ~ 600 s ; ~ 4500 s
Conclusions Ø Equilibrium and non-equilibrium molecular dynamics in mesoporous materials in different regions of the adsorption isotherm are indepenedently probed using nuclear magnetic resonance methods. Ø Comparative analysis of the obtained experimental results yields a two-step mechanism of the molecular uptake in the adsorption hysteresis region. Ø 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.
Acknowledgements Prof. J. Kärger – University of Leipzig Prof. P. Monson – University of Massachusets Prof. H. -J. Woo – University of Nevada, Reno Ph. D Students: P. Kortunov, S. Naumov
NMR method 90° Self-diffusion Adsorption kinetics 90° Adsorption hysteresis Experimental part Equilibrium dynamics Non-equilibrium dynamics Conclusions Two mechanisms of adsorption
Anomalous transport 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). ]
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