– Bad Honnef – 2006 –  – Anomalous Transport – 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
r1  r
r2  
r1  r2  r
P
2 lvl


ln    
rRT
 Ps 
Kelvin equation
P
 
ln     lv l
RT
 Ps 
1 1
  
 r1 r2 
P
 
ln     lv l
rRT
 Ps 
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
H1 and H2 type isotherms
Pore blocking
Cavitation
H1 - Hysteresis due to metastable pore fluid,
no percolation effects!
narrow pore-size distribution
H2 - 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

B0
Magnetic
moment

Spin angular
momentum
0  B0

B0
radio waves in
microscopic
0
macroscopic
radio waves out
Pulsed Field Gradient NMR
90°
90°
90°

spin-echo signal
intensity S

diffusion time – td ()
g
g
z


B

z
z
0
0   ( B0  gz )
Pulsed Field Gradient NMR
90°
90°
90°

spin-echo signal
intensity S

diffusion time – td ()
g
g
z
e
d
0

B
z
z
( d   e )
Direct probe of diffusion propagator
Stimulated echo NMR pulse sequence
90°
90°

g
td =
10-3
1 s
90°

diffusion time – td ()
q = g - wave number


   
S q, td    P( r , td ; r0 ) exp iq r  r0 dr dr0
Gaussian propagator
S ( q, td )  exp  q 2td Ds 
spin-echo signal
intensity S
g
NMR summary
FID intensity – amount adsorbed and uptake kinetics
90°
in the same sample
at the same conditions
PFG NMR method – self-diffusivity
90°
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/m3)
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
0 z 1
P

FID Signal Intensity ( P )
FID Signal Intensity ( PS )
Concentration; Pore filling
z
P
Ps
Outline
Adsorption hysteresis
Experimental part
Equilibrium dynamics
Non-equilibrium dynamics
Conclusions
Cyclohexane in Vycor porous glass
 - adsorption  - desorption
0,6
-9
2
0,4
0,3
0,2
0,1
0,0
0,2
0,4
0,6
relative pressure, z
0,8
concentration,
Deff (10 m /s)
0,5
2,6
2,4
2,2
2,0
1,8
1,6
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0,0
1,0
Effective diffusivity: Fast exchange limit
 - adsorption  - desorption
Detailed balance principle
 ij1Pi   ji1Pj
0,6
pg 
1   1 Ps M
z ( )
  a RT
Dg  DKnudsen
0,5
-9
2
Deff (10 m /s)
Deff = pa Da + pg Dg
0,7
0,4
0,3
0,2
0,0
d ~ 6 nm
r  Deff td  500 nm
0,2
0,4
0,6
0,8
relative pressure, z
adsorbed phase
gaseous phase
1,0
Effective diffusivity: Concentration dependence
 - adsorption  - desorption
Deff = pa Da + pg Dg
0,6
This is not enough!
0,4
2
-9
1   1 Ps M
z ( )
  a RT
0,3
0,2
0,1
0,0
0,2
0,4
0,6
0,8
relative pressure, z
Capillary condensed phase differently distributed on adsorption and desorption
concentration,
pg 
Deff (10 m /s)
0,5
2,6
2,4
2,2
2,0
1,8
1,6
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0,0
1,0
Outline
Adsorption hysteresis
Experimental part
Equilibrium dynamics
Non-equilibrium dynamics
Conclusions
Micro via Macro
Diffusion-controlled uptake


 ( r , t )
2 (r , t )
 Ds

t
r 2

 ( r , t  0)  0  const
P1  P2

 ( r , t ) rsurface   eq  const
m

m ( t )    ( r , t ) d 3r   ( t )
V
Cylindrical samples with radius a

eq


4  1
 (t )  0  eq  0 1  2  2 exp  n2 Dst
 a n 1 n

J 0 (a )  0
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
0.6
-9
2
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
concentration,
Deff (10 m /s)
0.5
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
relative pressure, z
Rajniak, P.; Soos, M.; Yang, R. T. AICHE J. 1999, 45, 735.
Adsorption kinetics in Vycor
12 mm
3 mm
Diffusion-controlled model


4  1
 (t )  0  eq  0 1  2  2 exp  n2 Dst
 a n 1 n

Example 2: Nitrogen in porous silicon
Adorption kinetics follows
stretched-exponential law
  t   
 (t )  exp    
    
with   0.5.
Authors regard it as an indication
of disorder.
4  1
 (t )  1  2  2 exp  n2 Dst
a n1 n
Wallacher, D.; Kunzner, N.; Kovalev, D.; Knorr, N.; Knorr, K. Phys. Rev. Lett. 2004, 92, 195704.
Kinetics in the hysteresis region



t

 
  

 (t )  0  (eq  0 ) 1  exp    



   


Kohlrausch relaxation
Diffusion-controlled uptake
 = 0.66


4  1
 (t )  0  eq  0 1  2  2 exp  n2 Dst
 a n 1 n

 = 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

 b(T , r )  

 ( r )  t0 exp 

kT


Expected scaling function
 kT ln( t / t0 ) 
S ( q  0, t )  S (0,0) f 


b



Experimental and computer simulations
f ( x)  exp  x p 
p=3

S (0, t )  exp  ln( t / t0 )
p

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
4  1
diff (t )  1  2  2 exp  n2 Dst
a n1 n
Activated part
  ln( t / t ) 3 
0
act (t )  exp  
 
ln(

/
t
)
 
0  

Overall density equilibration function
 (t )  Adiff diff (t )  Aact act (t )
Adiff ~ 0.8 ; t0 ~ 600 s ;  ~ 4500 s
R2
taverage 
 600 s
8Ds
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

B0
NMR method
90°
90°
Self-diffusion
Adsorption kinetics
90°
Adsorption hysteresis
Experimental part
Equilibrium dynamics
Non-equilibrium dynamics
Conclusions
Two mechanisms of adsorption


4  1
dif (t )  0  eq  0 1  2  2 exp  n2 Dst
 a n 1 n

  ln( t / t ) 3 
0
act (t )  exp  
 
  ln(  / t0 )  
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).]