– bad honnef – 2006 – – anomalous transport – bad honnef – 2006 – – anomalous...

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Rustem ValiullinDepartment of Interface Physics

University of Leipzig, Germany

Bad Honnef, 2006

New perspectives on New perspectives on anomalous dynamics anomalous dynamics

during sorption hysteresisduring sorption hysteresis

Outline

Adsorption hysteresis

Experimental part

Equilibrium dynamics

Non-equilibrium dynamics

Conclusions

Adsorption hysteresis phenomenon

Micropores < 2 nm

Reversible adsorption

m

m

P

P Mesopores 2-50 nm

Irreversible adsorption

vapor

porousmaterial

Adsorption hysteresis in mesoporous materials

P

The simplest view on adsorption hysteresis

21

11ln

rrRTP

P llv

s

Kelvin equation

rRTP

P llv

s

2ln

rRTP

P llv

s

ln

rrr 21

2

1

r

rr

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 transitionequality of the potentials

Upper limit of the metastable vapourzero barrier between the local and global potential minima

P

Liquidfilled

Empty

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 states broad pore-size distribution of the pore fluid and percolation effects.

Pore blocking Cavitation

Multiplicity of metastable states

Given that the occurrence of hysteresis represents a departure fromequilibrium, 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.

Outline


Experimental part



Conclusions

Experimental methodExperimental method

Nuclear magnetic resonance

Spin angular momentum

Magnetic moment

0B

00 B

0B

radio waves in

radio waves out

0microscopic

macroscopic

spin-echo signal intensity S

90° 90° 90°

g

g

diffusion time – td ()

z

0

zz

B

Pulsed Field Gradient NMR

)( 00 gzB


90° 90° 90°

g

g


z

0

zz

B

e

d

Pulsed Field Gradient NMR

)( ed

Direct probe of diffusion propagator

Stimulated echo NMR pulse sequence


90° 90° 90°

g

g


q = g - wave number

000 exp);,(, rdrdrriqrtrPtqS dd

sdd DtqtqS 2exp),(

Gaussian propagator

td = 10-3 1 s

NMR summary


90° 90° 90°

g

g


90°

FID intensity – amount adsorbed and uptake kinetics

PFG NMR method – self-diffusivity

in the same sampleat the same conditions

- 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

Experimental setup

Vres >> Vpore

initial pressure – P 10-5 atm

temperature – T = 24° C

Liquid Ps (atm)

M (g/mol)

(kg/m3)

Acetone 0.293 58 0.79

n-Hexane 0.193 86 0.66

Cyclohexane 0.124 84 0.78turbo-molecular pump magnet

Experimental protocol

FID signal intensity after pressure step

Self-diffusion study after equilibration

P

Normalized isotherm

P

FID

0 z 1

)(IntensitySignalFID

)(IntensitySignalFID

SP

P

Concentration; Pore filling

sP

Pz

Outline


Experimental part



Conclusions

0,0 0,2 0,4 0,6 0,8 1,0

0,1

0,2

0,3

0,4

0,5

0,6

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

D

eff (

10

-9 m

2 /s)

relative pressure, z

con

cen

tratio

n,

Cyclohexane in Vycor porous glass

- adsorption - desorption

Effective diffusivity: Fast exchange limit


Deff = pa Da + pg Dg

)(11

z

RT

MPp s

ag

adsorbed phase

gaseous phase

jjiiij PP 11 Detailed balance principle

d ~

6 nm

nm500 deff tDr

Knudseng DD 0,0 0,2 0,4 0,6 0,8 1,0

0,2

0,3

0,4

0,5

0,6

0,7

Def

f (1

0-9

m2 /s

)


0,0 0,2 0,4 0,6 0,8 1,0

0,1

0,2

0,3

0,4

0,5

0,6

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

De

ff (1

0-9 m

2 /s)


con

cen

tratio

n,


Deff = pa Da + pg Dg

)(11

z

RT

MPp s

ag

This is not enough!

Capillary condensed phase differently distributed on adsorption and desorption

Effective diffusivity: Concentration dependence

Outline


Experimental part



Conclusions

Micro via Macro

mm

P1 P2

eq

Diffusion-controlled uptake

2

2 ),(),(

r

trD

t

trs

const)0,( 0 tr

const),( eqsurfacer

tr

)(),()( 3 trdtrtmV

Cylindrical samples with radius a

1

22200 exp

141)(

nsn

neq tD

at

0)(0 aJ

time0

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

0.0 0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

Def

f (1

0-9

m2 /s

)


con

cen

tratio

n,

Adsorption kinetics in Vycor

1

22200 exp

141)(

nsn

neq tD

at

3 mm

12 mm

Diffusion-controlled model

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

t

t exp)(

with 0.5.

Authors regard it as an indication

of disorder.

1

222

exp14

1)(n

snn

tDa

t

Kinetics in the hysteresis region

= 0.66 = 0.37

1

22200 exp

141)(

nsn

neq tD

at


Kohlrausch relaxation

t

t eq exp1)()( 00

Two mechanisms of the uptake

Early times


- 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 high-pressure 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)

kT

rTbtr

),(exp)( 0

Huse, D. A. Phys. Rev. B 1987, 36, 5383

b

ttkTfStqS

)/ln()0,0(),0( 0

pxxf exp)(

Typical relaxation time

Expected scaling function

Experimental and computer simulations

ptttS )/ln(exp),0( 0

p = 3Ogielski 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.

Adsorption kinetics in Vycor

Diffusive part

Activated part

1

222

exp14

1)(n

snn

diff tDa

t

3

0

0

)/ln(

)/ln(exp)(

t

tttact

Overall density equilibration function

)()()( tAtAt actactdiffdiff

Adiff ~ 0.8 ; t0 ~ 600 s ; ~ 4500 s

s6008

2

s

average D

Rt

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.

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

Self-diffusion

Adsorption kinetics

Two mechanisms of adsorption

90° 90° 90°

NMR method

3

0

0

)/ln(

)/ln(exp)(

t

tttact

1

22200 exp

141)(

nsn

neqdif tD

at


Experimental part



Conclusions

0B

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

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