1 Time dependent aspects of fluid adsorption in mesopores Harald Morgner Wilhelm Ostwald Institute...

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


desorption 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




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




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



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




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



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)


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


11


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)


12


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)


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)


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)


13




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


pore wall

pore wall

15




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


16




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


(c) Fluctuation of the number of molecules during theGCMC run at point A.


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



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




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



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


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


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


loa

d [

mo

l/cm

2 ]

COS(alfa)

H.Morgner (2010)


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


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


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




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)


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)


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


-350000

-250000

-150000

10 100 1000 10000

molar volume [cm^3]

Ch

em.P

ot.

chem. Pot.

tie line

Reihe5


-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

1 Time dependent aspects of fluid adsorption in mesopores Harald Morgner Wilhelm Ostwald Institute...

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