1
Time dependent aspects of fluid adsorption in mesopores
Harald Morgner Wilhelm Ostwald Institute for Physical and Theoretical Chemistry
Leipzig University, Linnéstrasse 2, D-04103 Leipzig, [email protected]
Introduction to the phenomenon of adsorption hysteresis
Shortcomings of standard thermodynamics for confined systems
The alternative concept of COS (=Curves of States)
Time dependent treatment: • pressure jump experiments revealing dynamic behavior• properties & role of fluctuations
CompPhys2012Leipzig November 29-30,2012
2
exp. adsoption isotherm, Ar / SBA-15, 77.35K
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
6.E-09
0 0.2 0.4 0.6 0.8 1P/P0
adso
rpti
on
[m
ol/c
m^
2]
R.Rockmann PhD Thesis
2007U Leipzig
Adsorption Hysteresis in Porous Material
SBA-15:
silica pores with two open ends
narrow pore size distribution
Introduction Theoretical Simulation Results adsorption from vapor phase
3
Adsorption Hysteresis in Porous Material adsorption isotherm, model calculation
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
6.E-09
7.E-09
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P/P0
ad
so
rpti
on
[m
ol/c
m^
2]
adsorption
desorption
low pressure
H.Morgner (2010)
J.Phys.Chem.C 114
8877-83
pore size distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2 4 6 8
diameter [nm]
pro
bab
ility
Introduction Theoretical Simulation Results adsorption from vapor phase
4
curves of statesstraight pore with two open ends
desorptionswitch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
grand canonical boundary conditions
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
canonical boundary conditions
H. Morgner
J. Chem. Chem. Eng. 5 (2011) 456 - 472
Introduction Theoretical Simulation Results Curves of States: introduction
5H. Morgner
J. Chem. Chem. Eng. 5 (2011) 456 - 472
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
COS() two IF
COS() one IF
Introduction Theoretical Simulation Results Curves of States: filling pattern
6H. Morgner
J. Chem. Chem. Eng. 5 (2011) 456 - 472
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
straight pore with two open ends
desorption switch point
adsorption switch point
0
1E-18
2E-18
3E-18
-6400 -6350 -6300 -6250 -6200 -6150 -6100
chemical potential [J/mol]
gra
nd
po
ten
tia
l [J
]COS(beta)
COS(alfa)
grand potential
Introduction Theoretical Simulation Results Curves of States: relation to TD
7
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
Neimark&Vishnyakov J.Phys.Chem. B110 (2006) 9403-12
•both branches contain a set of microstates
•these sets are subsets of one common state (equilibrium). Thus, both branches form only one state
true GCE isotherm
„….the true GCE isotherm cannot be sampled in practical simulations….“
Introduction Theoretical Simulation Results attempts to save TD
8
Diffusion treated by
Onsager ansatz
gradT
LJ
driving force: gradient of chemical potentialcurves of states
straight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
pressure in gas reservoir
Introduction Theoretical Simulation Results pressure jump experiments
9
pressure jump, adsorption
start point
end point
1.5E-09
2.5E-09
3.5E-09
4.5E-09
-6450 -6400 -6350 -6300 -6250 -6200 -6150 -6100
chem. pot. [J/mol]
loa
d [
mo
l/c
m^
2]
max.chem.pot.
mue_aver
min. chem. pot.
COS(beta)
COS(alfa)
Introduction Theoretical Simulation Results pressure jump experiments
10-6400
-6350
-6300
-6250
-6200
-6150
0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02 3.5E-02 4.0E-02 4.5E-02 5.0E-02
t [s]
chem
.po
t. [
J/m
ol]
min. mue [J/mol]
max. mue [J/mol]
mue_aver [J/mol]
mue_start J/mol
mue_end J/mol
min:mue[COS(beta)]
1.5E-09
2.5E-09
3.5E-09
4.5E-09
0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02 3.5E-02 4.0E-02 4.5E-02 5.0E-02
t [s]
load
[m
ol/
cm^
2]
load [mol/cm^2]
load_start
load_end
Introduction Theoretical Simulation Results pressure jump experiments
11
pressure jump, adsorption
start point
end point
1.5E-09
2.5E-09
3.5E-09
4.5E-09
-6450 -6400 -6350 -6300 -6250 -6200 -6150 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
^2
]
max.chem.pot.
mue_aver
min. chem. pot.
COS(beta)
COS(alfa)
Introduction Theoretical Simulation Results pressure jump experiments
12
pressure jump, adsorption
start point
end point
1.5E-09
2.5E-09
3.5E-09
4.5E-09
-6450 -6400 -6350 -6300 -6250 -6200 -6150 -6100 -6050
chem. pot. [J/mol]
loa
d [
mo
l/c
m^
2]
max.chem.pot.
mue_aver
min. chem. pot.
COS(beta)
COS(alfa)
pressure jump, adsorption
start point
end point
1.5E-09
2.5E-09
3.5E-09
4.5E-09
-6450 -6400 -6350 -6300 -6250 -6200 -6150 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
^2
]
max.chem.pot.
mue_aver
min. chem. pot.
COS(beta)
COS(alfa)
pressure jump, adsorption
start point
end point
1.5E-09
2.5E-09
3.5E-09
4.5E-09
-6450 -6400 -6350 -6300 -6250 -6200 -6150 -6100 -6050
chem. pot. [J/mol]
loa
d [
mo
l/c
m^
2]
max.chem.pot.
mue_aver
min. chem. pot.
COS(beta)
COS(alfa)
Introduction Theoretical Simulation Results pressure jump experiments
13
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
eq
PONR
fluc
PONR time needed to
lifetime of
fluctuation: fluc
fluctuation effective if fluc > PONR
Introduction Theoretical Simulation Results fluctuations in gas reservoir
shift pore to PONR from pressure jump
14
occurrence = fluc / Pstat
r 200 nm
= 5 nm
Introduction Theoretical Simulation Results fluctuations in gas reservoir
pore wall
pore wall
15
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
eq fluc
average time elapsed until effective fluctuation occurs
occurrence
Stat. TD provides statistical probability Pstat
Pstat = fluc / occurrence
occurrence = fluc / Pstat
Introduction Theoretical Simulation Results fluctuations in gas reservoir
16
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
eq occurrence 101377 [s]
occurrence 101370 [y]
TD of confined systems:
COS (curves of states) are
appropriate concept
bistability rather than metastable
states
Introduction Theoretical Simulation Results lifetime of short lived state
occurrence 101360 [14·109y]
H. Morgner J. Chem. Chem. Eng. 5 (2011) 456 - 472occurrence = fluc / Pstat
17
GCE and CE isotherms of LJ fluid confined to a 15.8 spherical pore at subcritical temperature, kT/ =1.02. (a) Practical GCMC isotherm is monotonic and reversible, CE isotherm generated with the IGGC method, and true GCE isotherm calculated from the CE isotherm (eq 9). The position of VLE is determined from theMaxwell rule (eqs 6).
Neimark& Vishnyakov J.Phys.Chem. B110 (2006) 9403-12
Arguments from other authors system with virtual interface to reservoir
18
Arguments from other authors system with virtual interface to reservoir
(c) Fluctuation of the number of molecules during theGCMC run at point A.
Neimark& Vishnyakov J.Phys.Chem. B110 (2006) 9403-12
spherical pore with virtual interface gas reservoir:
fluid density is homogeneous with respect to 2 dimensions (both angles)
fluid density can vary only with respect to 1 dimension (radius)
19
Neimark&Vishnyakov J.Phys.Chem. B110 (2006) 9403-12
Comparison of systems system with virtual interface to reservoir vs. system with real interface to reservoir
spherical pore with virtual interface to gas reservoir:
boundary conditionsallowhomogeneous density with respect to 2 dimensions (both angles)
isotherm is one coherent curve (EOS)
cylindrical pore with real interface to gas reservoir:
boundary conditionsallow homogeneous density with respect to only 1 dimension (polar angle)
isotherm consists of two separated curves (COS)
concept of present work
Thank you
critical remarks welcome !
20
Simulation: adsorption in cylindrical pore of infinite length EOS
pore of infinite length vs. pore of finite length
0.00E+00
5.00E-10
1.00E-09
1.50E-09
2.00E-09
2.50E-09
3.00E-09
3.50E-09
4.00E-09
4.50E-09
5.00E-09
-7200 -7000 -6800 -6600 -6400 -6200 -6000
chem.pot [J/mol]
loa
d [
mo
l/c
m^
2] infinite pore length
GCE
COS(alfa)
COS(beta)
21
Neimark& Vishnyakov J.Phys.Chem. B110 (2006) 9403-12
Arguments from other authors system with virtual interface to reservoir
spherical pore with virtual interface to gas reservoir:
fluid density is homogeneous with respect to 2 dimensions (both angles)
fluid density can vary only with respect to 1 dimension (radius)
22
Introduction Theoretical Simulation Results resume
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
equilibrium from infinite pore (GCE)
(CE)
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6400 -6350 -6300 -6250 -6200 -6150
chem. pot. [J/mol]
loa
d [
mo
l/c
m2 ] COS(beta)
COS(alfa)
true GCE isotherm
equilibrium from kinetics
23
Introduction Theoretical Simulation Results resume
curves of statesstraight pore with two open ends
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6309.9 -6309.7 -6309.5 -6309.3 -6309.1 -6308.9 -6308.7 -6308.5 -6308.3 -6308.1 -6307.9
chem. pot. [J/mol]
loa
d [
mo
l/c
m2 ]
-6289.5 -6289.3 -6289.1 -6288.9 -6288.7 -6288.5 -6288.3 -6288.1 -6287.9 -6287.7 -6287.5
from kinetics
from GCE equilibrium
24
Introduction Theoretical Simulation Results temperature variation
pore with two open ends
width of hysteresis (ads - des)
critical temperature
crit. temp. of hysteresis
0
20
40
60
80
100
120
140
160
180
70 80 90 100 110 120 130 140 150 160
T [K]
wid
th o
f h
ys
tere
sis
[J
/mo
l]simulation
cubic fit
25
Introduction Theoretical Simulation Results temperature variation
adsorption hysteresispore radius constant, two open ends
0.0E+00
5.0E-10
1.0E-09
1.5E-09
2.0E-09
2.5E-09
3.0E-09
3.5E-09
4.0E-09
4.5E-09
5.0E-09
0 0.2 0.4 0.6 0.8 1
P/P0
loa
d [
mo
l/c
m^
2]
77.35K COS(beta)
77.35K COS(alfa)
130K
pore with two open ends
width of hysteresis (ads - des)
critical temperature
crit. temp. of hysteresis
0
20
40
60
80
100
120
140
160
180
70 80 90 100 110 120 130 140 150 160
T [K]
wid
th o
f h
ys
tere
sis
[J
/mo
l]
simulation
cubic fit
26
Introduction Theoretical Simulation Results resume
adsorption hysteresispore radius constant, two open ends
2.8E-09
2.9E-09
3.0E-09
3.1E-09
3.2E-09
3.3E-09
3.4E-09
3.5E-09
-6380 -6375 -6370 -6365 -6360 -6355 -6350
chem. pot. [J/mol]
loa
d [
mo
l/cm
^2
] 130K
eq
occurrence
10338 [14·109y]
TD of confined systems: COS
(curves of states) are appropriate
concept
bistability rather than metastable
states
H. Morgner J. Chem. Chem. Eng. 5 (2011) 456 - 472
occurrence[77.35K] 101000 . occurrence [130K]
27
Introduction Theoretical Simulation Results time dependence in QM
t,xti
t,xxVt,xxm2 2
22
xExxVxxm2 2
22
intrinsic=/2
exp intrinsic
exp intrinsicexp intrinsic xExxVx
xm
2
22
2
28
Adsorption Hysteresis in Porous MaterialQuotations from literature:
1 D.Wallacher et al., Phys. Rev. Lett. 92, (2004) 195704-1
.. in the experimental system the metastable states just do not have time enough to relax...1
Metastable states ...... appear to be the most important aspect.1
...a failure of the system to equilibrate.2
This explains why hysteresis, although representing a departure from equilibrium, is so reproducible in experiment.2
2 R.Valiullin et al., Nature Letters, 443 (2006) 965-8
29
Adsorption Hysteresis in Porous MaterialQuotations from literature:
1 J. Puibasset et al. J.Chem.Phys. 131 (2009) 124123-1/10
However, even in experiments in which accessible observation times are much longer than in simulations, a hysteresis is usually observed, whose properties are quite reproducible.1
30
curves of statesstraight pore with two open ends
1.9E-09
2.0E-09
2.1E-09
2.2E-09
2.3E-09
2.4E-09
2.5E-09
0.477 0.478 0.479 0.48 0.481 0.482 0.483 0.484 0.485
reduced pressure P/P0
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
curve of statesstraight pore with two open ends
2.77E-09
2.78E-09
2.79E-09
2.80E-09
2.81E-09
2.82E-09
2.83E-09
2.84E-09
2.85E-09
0.613 0.6132 0.6134 0.6136 0.6138 0.614 0.6142 0.6144
reduced pressure P/P0
loa
d [
mo
l/cm
2 ]
COS(alfa)
H.Morgner (2010)
curves of statesstraight pore with two open ends
0.0E+00
5.0E-10
1.0E-09
1.5E-09
2.0E-09
2.5E-09
3.0E-09
3.5E-09
4.0E-09
4.5E-09
5.0E-09
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
reduced pressure P/P0
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
-2.0E-18
-1.5E-18
-1.0E-18
-5.0E-19
0.0E+00
5.0E-19
1.0E-18
1.5E-18
2.0E-18
2.5E-18
3.0E-18
0.4 0.5 0.6 0.7
reduced pressure P/P0
gra
nd
po
ten
tial
[J]
a)
b)
c)
d)
Fig.2 a) The isotherm: load as function of reduced pressure in gas reservoir, b) the two curves of states COS() and COS(): pressure (or chemical potential) evaluated as function of load, the insert shows related values of the grand potential c) enlarged display of the unstable parts of the COS, COS() in the left panel, COS() in the right panel, d) filling patterns, left panel COS(), right panel COS()
31H. Morgner
J. Chem. Chem. Eng. 5 (2011) 456 - 472COS and concept of applying canonical boundary conditions
COS allow to retrieve isotherm
curves of statesstraight pore with two open ends
desorption switch point
adsorption switch point
0.E+00
1.E-09
2.E-09
3.E-09
4.E-09
5.E-09
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
curve of statesstraight pore with two open ends
2.78E-09
2.79E-09
2.80E-09
2.81E-09
2.82E-09
2.83E-09
2.84E-09
2.85E-09
-6177 -6176 -6175
chem. pot. [J/mol]
load
[m
ol/c
m2] COS(alfa)
curves of statesstraight pore with two open ends
1.9E-09
2.0E-09
2.1E-09
2.2E-09
2.3E-09
2.4E-09
2.5E-09
-6340 -6335 -6330 -6325 -6320
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
Introduction Theoretical Method Results Curves of States (shape)
32
H.Morgner (2010)
curve of statespore tapering towards bottom
1.0E-09
1.5E-09
2.0E-09
2.5E-09
3.0E-09
3.5E-09
4.0E-09
4.5E-09
-6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
load
[m
ol/c
m2]
COS
curve of statesstraight pore with one open end
1.0E-09
1.5E-09
2.0E-09
2.5E-09
3.0E-09
3.5E-09
4.0E-09
4.5E-09
-6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS
curves of statespore tapering towards open end
1.0E-09
1.5E-09
2.0E-09
2.5E-09
3.0E-09
3.5E-09
4.0E-09
4.5E-09
-6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
load
[m
ol/c
m2]
COS(alfa)
COS(beta)
COS(gamma)
curves of statesstraight pore with two open ends
1.0E-09
1.5E-09
2.0E-09
2.5E-09
3.0E-09
3.5E-09
4.0E-09
4.5E-09
5.0E-09
-6600 -6550 -6500 -6450 -6400 -6350 -6300 -6250 -6200 -6150 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(alfa)
COS(beta)
curves of statespore with bottle neck
1.0E-09
1.5E-09
2.0E-09
2.5E-09
3.0E-09
3.5E-09
4.0E-09
4.5E-09
-6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
loa
d [
mo
l/cm
2 ]
COS(gamma)
COS(alfa)
COS(beta)
Fig.6 Effect of pore shape on curves of states COS. Five different pore shapes are shown. The dotted line indicates the onset of the gas reservoir, i.e. homogeneous distribution of component A in the vapor phase. Below every pore shape, the corresponding COS are displayed. For clarity, the conventional isotherms are omitted, but can be easily reconstructed with the aid of the COS.
33
Thermodynamics: EOS equilibrium states, coexistence
vdW fit to water at 493K
-600000
-500000
-400000
-300000
-200000
-100000
10 100 1000 10000
molar volume [cm^3]
Ch
em.P
ot.
-300
-200
-100
0
100
200
300
400
500
P [
ba
r]
chem. Pot.
tie line (chem.pot.)
pressure
tie line (pressure)
Reihe7
34
Thermodynamics: EOS
vdW EOS. Fit to water at 493K
-350000
-250000
-150000
10 100 1000 10000
molar volume [cm^3]
Ch
em.P
ot.
chem. Pot.
tie line
Reihe5
vdW EOS. Fit to water at 493K
-350000
-250000
-150000
10 100 1000 10000
molar volume [cm^3]
Ch
em.P
ot.
chem. Pot.
tie line
Reihe5
vdW EOS. Fit to water at 493K
-350000
-250000
-150000
10 100 1000 10000
molar volume [cm^3]
Ch
em.P
ot.
chem. Pot.
tie line
Reihe5
lowering grand potential
lowering free energy
equilibrium states, coexistencedecay of metastable states
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