Transcript Properties

Quantachrome
INSTRUMENTS
Micropore Size
Calculations
© 2004-2006 Quantachrome
QuantachromeInstruments
Instruments
Multilayer adsorption
Volume adsorbed
Types II, IV
Types II+I, IV+I
After the knee,
micropores cease to
contribute to the
adsorption process.
Low slope region in middle of
isotherm indicates first few
multilayers, on external surface
including meso and macropores…
before the onset of capillary
condensation
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Estimation of Micropores...
the t-plot method
This method uses a mathematical representation of
multi-layer adsorption. The thickness, t, of an
adsorbate layer increases with increasing pressure.
The t-curve so produced is very similar in
appearance to a type II isotherm.
t-plot
isotherm
© 2004-2006 Quantachrome Instruments
The t-plot
Statistical thickness
Resembles a type II isotherm
A statistical multilayer
A statistical monolayer
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
The t-plot method
Not Only Multilayer Correction to
the Kelvin Equation,
But Also Estimation of Micropores...
For every value of P/Po, the volume adsorbed is
plotted against the corresponding value of “t”.
If the model describes the experimental data a
straight line is produced on the t-plot...
© 2004-2006 Quantachrome Instruments
Statistical Thickness, t
• Halsey equation
• Generalized Halsey
• deBoer equation
• Carbon Black STSA
© 2004-2006 Quantachrome Instruments
t-plot Method
(mesoporous only)
Slope = V/t = A
1
2
3
4
()
t t(Å)
© 2004-2006 Quantachrome Instruments
5
6
7
t-plot Method
(in the presence of micropores)
Intercept = micropore volume
1
2
3
4
tt ()
(Å)
© 2004-2006 Quantachrome Instruments
5
6
7
Volume adsorbed
Micropore Size Determination
by Gas Sorption
Limiting value (plateau) due to filled
pores and essentially zero external area.
Type I or
pseudo-“Langmuir”
Steep initial region due to very strong
adsorption, for example in micropores.
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Comparisons
•
Gas Sorption Calculation Methods
P/Po range
1x10-7 to 0.02
0.01 to 0.1
0.05 to 0.3
> 0.1
> 0.35
0.1 to 0.5
© 2004-2006 Quantachrome Instruments
Mechanism
micropore filling
sub-monolayer formation
monolayer complete
multilayer formation
capillary condensation
capillary filling
in M41S-type materials
Calculation model
DFT, GCMC, HK, SF, DA, DR
DR
BET, Langmuir
t-plot (de-Boer,FHH),
BJH, DH
DFT, BJH
DR & DA
Dubinin-Radushkevic and Dubinin-Astakov
• DR
Simple log(V) vs log2(Po/P) relationship which linearizes the
isotherm based on micropore filling principles. “Best fit” is
extrapolated to log2(Po/P) (i.e. where P/Po = 1) to find
micropore volume.
•
DA
Closely related to DR calculation based on pore filling
mechanism. Equation fits calculated data to experimental
isotherm by varying two parameters, E and n. E is average
adsorption energy that is directly related to average pore
diameter, and n is an exponent that controls the width of the
resulting pore size distribution. The calculated pore size
distribution always has a skewed, monomodal appearance
(Weibull distribution).
© 2004-2006 Quantachrome Instruments
Estimation of Micropores
Dubinin-Radushkevich (DR) Theory
  T 2

P


2
W  W0 exp  B  log  0 
 P 
   
W = volume of the liquid adsorbate
W0 = total volume of the micropores
B = adsorbent constant
 = adsorbate constant
A linear relationship should be found between log(W) and log2(Po/P)...
© 2004-2006 Quantachrome Instruments
Estimation of Micropores
Log (W)
Dubinin-Radushkevich (DR) Plot
Extrapolation
yields Wo
0
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Log2(Po/P)
HK & SF
Horvath-Kawazoe & Saito-Foley
•
•
HK
Direct mathematical relationship between relative pressure
(P/Po) and pore size. Relationship calculated from modified
Young-Laplace equation, and takes into account parameters
such as magnetic susceptibility. Based on slit-shape pore
geometry (e.g. activated carbons). Calculation restricted to
micropore region ( 2nm width).
SF
Similar mathematics to HK method, but based on cylindrical
pore geometry (e.g. zeolites). Calculation restricted to
micropore region ( 2 nm diameter).
© 2004-2006 Quantachrome Instruments
DFT
Density Functional Theory
a)
b)
c)
d)
Provides a microscopic treatment .
Complex mathematical modelling of fluid
interactions plus geometrical considerations
(pore geometry).
Fluid interactions are “calibrated”.
“Kernel” consists of up to 100 theoretical,
individual pore isotherms.
© 2004-2006 Quantachrome Instruments
Gas- and Liquid Density Profiles
in a Slit Pore by GCMC
(Walton and Quirke,1989)
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Pore Filling Pressures for Nitrogen
in Cylindrical Silica Pores at 77 K
(Neimark et al, 1998)
© 2004-2006 Quantachrome Instruments
Pore Size Analysis of MCM 41
(Templated Silica)
by N2 Sorption at 77 K
600
Exp. Nitrogen sorption at 77 K in MCM 41
DFT- Isotherm
Volume [cc/g]
500
400
300
200
100
0
0.2
0.4
0.6
P/P0
© 2004-2006 Quantachrome Instruments
0.8
1
Pore Size Analysis of MCM 41:
Calculations Compared
0.3
Dv(d) [cc/Å/g]
0.25
BJH-Pore size distribution
DFT-Pore size distribution
0.2
0.15
0.1
0.05
0
15
23
31
39
Pore Diameter [Å]
© 2004-2006 Quantachrome Instruments
47
55
Quantachrome
I N S T R U M E N T S
Recent Advances in
Pore Size Characterization by
Physical Adsorption
Author: Dr. Matthias Thommes
Director of Applied Science, Quantachrome Instruments
Boynton Beach, Florida, USA
Presented by Dr. Martin A. Thomas
Director of Business Development and Applied Technology
Quantachrome Instruments
Boynton Beach, Florida, USA
© 2004-2006 Quantachrome Instruments
Adsorption Potentials : Planar Surface,
Meso- and Micropores
Planar Surface
© 2004-2006 Quantachrome Instruments
Mesopores (2-50 nm) Micropore (<2 nm)
IUPAC’s Classification of
Sorption Isotherms
© 2004-2006 Quantachrome Instruments
Adsorption in
Micro- and Mesopores
Micropores (pore size < 2 nm):
 Micropore filling (continuous process)
at very low relative pressures P/P0 < 0.15
Type I isotherm (IUPAC Classification)
Mesopores (pore size 2 - 50 nm):
 Multilayer adsorption, pore condensation
and hysteresis (pore condensation
reflects as 1st order phase transition,
i.e., discontinuous process)
in relative pressure (P/P0) range from 0.15 – 1
Type IV, and V isotherm (IUPAC Classification)
© 2004-2006 Quantachrome Instruments
Quantachrome
I N S T R U M E N T S
Recent Advances in
Micropore (< 2 nm) Analysis
© 2004-2006 Quantachrome Instruments
Commonly Used Adsorptives for Surface Area
and Pore size Analysis
Nitrogen:
at 77.35 K (liquid nitrogen temperature, T/Tc = 0.61)
 pore size analysis of micro-,meso and macropores
 surface area analysis
Argon:
at 77.35 K (T – Tr = - 6.5 K; Tr : bulk triple point
temperature; T/Tc = 0.50)
at 87.27 K (liquid argon temperature, T/Tc = 0.57 )
 pore size analysis of micro- , meso- and macropores
 surface area analysis
CO2 :
at 195 K (T/Tc = 0.63)
at 273 K (T/Tc = 0.89)  pore size analysis of micropores
of widths < 1.5 nm (particularly for microporous carbons)
Krypton : at 77.35 K (T – Tr = - 38.5 K)  measurement of very low
surface areas
at 87.27 K (T – Tr = - 28.5 K)  pore size analysis of thin
micro/mesoporous films (M. Thommes et al, 2005)
© 2004-2006 Quantachrome Instruments
Argon Adsorbate
© 2004-2006 Quantachrome Instruments
Adsorption of Nitrogen (77.35 K) and
Argon (87.27 K) on some Zeolites
350
N2/77K
Ar/87 K
Volume [cm3]
280
Faujasite: Ar and N2 Adsorption
210
.
N2/77.35 K
140
Ar/87.27 K
70
0
10-6
ZEOLITE | 10.5.2001
© 2004-2006 Quantachrome Instruments
5 10-5
5 10-4
5 10-3
P/P0
5 10-2
5 10-1
5 100
Adsorption of Nitrogen (77.35 K) and Argon
(87.27 K) on some Zeolites
360
13X
NaX
MCM-58
H-Mordenite
Linde 5A
Volume [cc/g]
300
240
Argon/87.27 K
180
MCM-58
120
1 3X
60 H-Mord.
NaX
5A
0
© 2004-2006 Quantachrome Instruments
5 10-6
5 10-5
5 10-4
5 10-3
P/P0
5 10-2
5 10-1
5 100
Argon Adsorption at 87.27 K
• Due to weaker attractive fluid-wall interactions
(and the lack of a quadrupole moment), argon fills
micropores of dimensions 0.4 nm – 0.8 nm at
much higher relative pressures, (.i.e., at least 1.5
decades higher in relative pressures) as compared
to nitrogen.
 High resolution adsorption isotherm of high
accuracy can be measured over the complete
micro-mesopore range, in less time.
© 2004-2006 Quantachrome Instruments
Carbon Dioxide Adsorbate
© 2004-2006 Quantachrome Instruments
CO2 Micropore Analysis of Porous
Carbons at 273.15 K
 At elevated temperatures and higher absolute pressure (P0 =
26200 Torr) CO2 can access micropores, which are not
accessible for nitrogen at 77 K.
 Fast analysis: due to higher diffusion rate equilibrium is
achieved faster as compared to nitrogen adsorption at 77 K
 dramatic decrease in analysis time i.e., 3-5 h for CO2
versus 30-50 h N2.
 No need for high vacuum system with turbomolecular
pump; 10-3 torr vacuum is sufficient.
 No need for a low-pressure transducer; 1000 Torr
transducer is sufficient.
© 2004-2006 Quantachrome Instruments
400
300
0.07
N2 (77 K)
Ar (77 K)
CO2 (273 K)
N2/77.35 K
200
100
CO 2
N2
0.06
Pore Volume, cc/g
Amount Adsorbed, cc(STP)/g
N2 , Ar (at 77.35 K) vs. CO2 (273.15 K) Adsorption
on Activated Carbon Fiber (ACF-10) and
NLDFT-PSD Histograms
N2
CO2,
Ar
CO2/273.15 K
0
1E-06 1E-05 0.0001 0.001
0.05
Analysis Time:
0.04
CO2 = 3 h
N2 = 40 h
0.03
0.02
0.01
0.01
0.1
1
0
4
6
8
Relative Pressure
12
Pore Size Å
Quantachrome’s Powder Technote 35
© 2004-2006 Quantachrome Instruments
10
14
16
18
20
Water Adsorbate
© 2004-2006 Quantachrome Instruments
Microporous Carbons:
the Standard way
700
600
Volume [cc/g] STP
Nitrogen, 77.35 K
A5
A10
A15
500
400
300
200
100
0
0
2.10-1
4.10-1
6.10-1
8.10-1
100
P/P0
Nitrogen (77.35 K and Water Sorption (298.4K) in Activated Carbon Fibers (ACF),
(M. Thommes, et al., FOA 8, 2004)
© 2004-2006 Quantachrome Instruments
Featureless Isotherms
Nitrogen, 77.35 K
600
A5
A10
A15
Volume [cc/g] STP
480
360
240
120
0
5 10-6
5 10-5
5 10-4
5 10-3
P/P0
5 10-2
5 10-1
5 100
Nitrogen (77.35 K and Water Sorption (298.4K) in Activated Carbon Fibers (ACF),
(M. Thommes, et al., FOA 8, 2004
© 2004-2006 Quantachrome Instruments
State of the Art Cryogenic
Differentiation
NLDFT Pore Volume [cc/g]
0.8
NLDFT
A5
A10
A15
0.64
0.48
A 15
A 10
0.32
A5
0.16
0
6
8
10
20
Pore Diameter [Å]
40
60
80
100
Nitrogen (77.35 K and Water Sorption (298.4K) in Activated Carbon Fibers (ACF),
(M. Thommes, et al., FOA 8, 2004
© 2004-2006 Quantachrome Instruments
The Special Behavior of Water
800
Water, 25 C
A15
A5 25C
A10 25C
A15 25C
700
600
A10
500
400
A5
300
200
100
0
0
0.2
0.4
0.6
0.8
1
Nitrogen (77.35 K and Water Sorption (298.4K) in Activated Carbon Fibers (ACF),
(M. Thommes, et al., FOA 8, 2004
© 2004-2006 Quantachrome Instruments
Hydrogen Adsorbate
© 2004-2006 Quantachrome Instruments
Hydrogen adsorption at 77K and 273 K
for Ultramicropore Characterization
Including H2 isotherms in the PSD analysis allows extending
the lower limit of the analysis to pore sizes of about 3 Å. This
pore size range may be useful for hydrogen storage
applications.
J. Jagiello, M. Thommes, Carbon 42 (2004) 1227
© 2004-2006 Quantachrome Instruments
400
300
200
100
Amount Adsorbed, cc(STP)/g
Amount Adsorbed, cc(STP)/g
H2, CO2 and N2 Adsorption and NLDFT Analysis
in ACF Activated Carbon Fibers
N2 (77 K)
Ar (77 K)
CO2 (273 K)
N2
Ar
0
1E-06 1E-05 0.0001 0.001
CO2
0.01
Relative Pressure
© 2004-2006 Quantachrome Instruments
0.1
1
250
H2,77 K
200
150
100
ACF10(K)
ACF10(OG)
ACF15(OG)
50
0
0
0.2
0.4
0.6
0.8
1
Pressure, Atm
J. Jagiello, M. Thommes, Carbon 42 (2004) 1227
400
N2 (77 K)
Ar (77 K)
CO2 (273 K)
300
0.25
200
N2
100
Ar
0
1E-06 1E-05 0.0001 0.001
H2
CO2
0.01
0.1
1
Relative Pressure
250
H2,77 K
200
ACF10(OG)-CO2
ACF10(OG)-H2
0.15
N2
0.1
ACF10(K)
ACF10(OG)
ACF15(OG)
50
0
2
0
0
0.2
0.4
0.6
Pressure, Atm
© 2004-2006 Quantachrome Instruments
0.8
NLDFT-PSD
CO2
0.05
150
100
ACF10(OG)-N2
0.2
PSD, cc/(g Å)
Amount Adsorbed, cc(STP)/g
Amount Adsorbed, cc(STP)/g
H2, CO2 and N2 Adsorption and NLDFT Analysis
in ACF Activated Carbon Fibers
1
4
6
8
10
12
14
16
Pore Width, Å
J. Jagiello, M. Thommes, Carbon 42 (2004) 1227
Pore Shape & Size Influence
© 2004-2006 Quantachrome Instruments
Pore Size Analysis by Gas Adsorption
Macroscopic, thermodynamic methods
Micropores (< 2 mn): e.g., Dubinin-Radushkevitch
or more advanced methods such as Horvath-Kawazoe
(HK) and Saito-Foley (SF) , t-method, alpha-s method
Meso/Macropores (2-100 nm): e.g., Kelvin equation based methods such
as BJH (Barrett,Joyner, Halenda)
Modern, microscopic methods, based on statistical mechanics
describe configuration of adsorbed molecules on a molecular
level : e.g., Density Functional Theory (DFT), Molecular Simulation
these methods are applicable for pore size analysis of both the
micro- and mesopore size range
 An accurate pore size analysis over the complete pore size range can
be performed by a single method.
© 2004-2006 Quantachrome Instruments
Pore Filling Pressures for Nitrogen in
Cylindrical Micropores at 77 K
C. Lastoskie and K.E.Gubbins, J. Phys. Chem 77, 9786 (1997)
© 2004-2006 Quantachrome Instruments
Pore Size Analysis of Zeolites with Novel NLDFT Kernels
based on argon adsorption at 87.27 K
(M.Thommes et al., presented at the International Zeolite Conference, Cape Town,
2004)
300
0.7
H-Mordenite
13X
NLDFT_Zeolite Fit_(spherical pore model)
NLDFT-Zeolite Fit (cylindrical pore model)
0.56
180
dV[cc/Å/g]
Volume [cc/g]
240
120
60
0
MCM-41
(NLDFT_Silica_cylindrical pore model)
Zeolite X_type (NLDFT_Zeolite spherical pore model)
Mordenite-type (NLDFT_Zeolite_cylindrical pore model)
0.42
0.28
0.14
10-6
5 10-5
5 10-4
5 10-3
P/P0
5 10-2
5 10-1
0
5 100
X-Zeolite structure
4
12
20
28
Pore Diameter Å
36
44
5 10-1
5 100
300
Zeolite X- type
DFT-Fitting : cylindrical pore model
DFT-Fitting : spherical pore model
Volume [cc/g]
240
180
120
60
Mordenite structure
© 2004-2006 Quantachrome Instruments
0
10-5
5 10-4
5 10-3
5 10-2
P/P0
Quantachrome
INSTRUMENTS
Mesopore Size
Calculations
© 2004
2004-2006
–2006Quantachrome
QuantachromeInstruments
Instruments
Pore Size Determination
Requires a recognition and
understanding of different basic
isotherm types.
© 2004-2006 Quantachrome Instruments
Volume adsorbed
Types of Isotherms
Limiting value (plateau) due to filled
pores and essentially zero external area.
Type I or
pseudo-“Langmuir”
Steep initial region due to very strong
adsorption, for example in micropores.
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Types of Isotherms
Volume adsorbed
Absence of hysteresis indicates adsorption
on and desorption from a non-porous
surface..
Type II
Low slope region in middle of
isotherm indicates first few
multilayers
Rounded knee
indicates approximate
location of monolayer
formation.
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Types of Isotherms
Volume adsorbed
Example: krypton on polymethylmethacrylate
Type III
Lack of knee represents extremely
weak adsorbate-adsorbent interaction
BET is not applicable
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Types of Isotherms
Volume adsorbed
Type IV
Closure at P/Po~0.4 indicates
presence of small mesopores
(hysteresis would stay open
longer but for the tensilestrength-failure of the nitrogen
meniscus.
Rounded knee
indicates approximate
location of monolayer
formation.
Hysteresis indicates capillary
condensation in meso and
macropores.
Low slope region in middle of
isotherm indicates first few
multilayers
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Types of Isotherms
Volume adsorbed
Example: water on carbon black
Type V
Lack of knee represents extremely
weak adsorbate-adsorbent interaction
BET is not applicable
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Comparisons
•
Gas Sorption Calculation Methods
P/Po range
1x10-7 to 0.02
0.01 to 0.1
0.05 to 0.3
> 0.1
> 0.35
0.1 to 0.5
© 2004-2006 Quantachrome Instruments
Mechanism
micropore filling
sub-monolayer formation
monolayer complete
multilayer formation
capillary condensation
capillary filling
in M41S-type materials
Calculation model
DFT, GCMC, HK, SF, DA, DR
DR
BET, Langmuir
t-plot (de-Boer,FHH),
BJH, DH
DFT, BJH
Meso/Macropore Size
Determination by Gas Sorption
Volume adsorbed
Type IV
Closure at P/Po~0.4 indicates
presence of small mesopores
(hysteresis would stay open
longer but for the tensilestrength-failure of the nitrogen
meniscus.
Rounded knee
indicates approximate
location of monolayer
formation.
Hysteresis indicates capillary
condensation in meso and
macropores.
Low slope region in middle of
isotherm indicates first few
multilayers
Relative Pressure (P/Po)
© 2004-2006 Quantachrome Instruments
Pore Size Distribution
Hysteresis is indicative of the presence of
mesopores and the pore size distribution can be
calculated from the sorption isotherm.
Whilst it is possible to do so from the adsorption
branch, it is more normal to do so from the
desorption branch...
Micropore (Greek micro = small): 0 nm - 2 nm diameter
Mesopore (Greek meso = middle): 2nm - 50 nm diameter
Macropore (Greek macro = large): >50 nm diameter
© 2004-2006 Quantachrome Instruments
Adsorption / Desorption
(macroscopic description)
Adsorption =
multilayer formation,
then…
© 2004-2006 Quantachrome Instruments
Desorption =
meniscus “control”
BJH & DH
Barrett, Joyner, Halenda and Dollimore-Heal
•
•
BJH
Modified Kelvin equation. Kelvin equation predicts pressure at
which adsorptive will spontaneously condense (and evaporate)
in a cylindrical pore of a given size. Condensation occurs in
pores that already have some multilayers on the walls.
Therefore, the pore size is calculated from the Kelvin equation
and the selected statistical thickness (t-curve) equation.
DH
Extremely similar calculation to BJH, which gives very similar
results. Essentially differs only in minor mathematical details.
© 2004-2006 Quantachrome Instruments
Kelvin* Equation
P 2V
ln

cos 
P0 rRT
4.15
rk ( A) 
log( P0 / P )
* Lord Kelvin a.k.a. W.T. Thomson
© 2004-2006 Quantachrome Instruments
BJH Pore Size
rp  rk  t
rp = actual radius of the pore
rk = Kelvin radius of the pore
t = thickness of the adsorbed film
Pore volume requires assumption of liquid density!
© 2004-2006 Quantachrome Instruments
Statistical Thickness, t
• Halsey equation
• Generalized Halsey
• deBoer equation
• Carbon Black STSA
© 2004-2006 Quantachrome Instruments
Pore Size Distribution
dV/dlogD
Artifact
40
© 2004-2006 Quantachrome Instruments
Pore Diameter (angstrom)
Pore Filling Pressures for Nitrogen
in Cylindrical Pores at 77 K
(Gubbins et al, 1997)
© 2004-2006 Quantachrome Instruments
Pore Filling Pressures for Nitrogen
in Cylindrical Silica Pores at 77 K
(Neimark et al, 1998)
© 2004-2006 Quantachrome Instruments
DFT & Phase Transitions
equilibrium transition
spinodal evaporation
0.05
0.04
0.03
Adsorption,
mmol/m2
spinodal condensation
0.02
Experimental (des)
0.01
Experimental (ads)
NLDFT in 4.8nm pore
0
0
0.2
0.4
0.6
Relative pressure, P/P0
0.8
1
NLDFT adsorption isotherm of argon at 87K in a cylindrical pore of diameter 4.8 nm in comparison with the appropriate
experimental sorption isotherm on MCM-41. It can be clearly seen that the experimental desorption branch is associated with
the equilibrium gas-liquid phase transition, whereas the condensation step corresponds to the spinodal spontaneous transition.
(a)Neimark A.V., Ravikovitch P.I. and Vishnyakov A. (2000) Phys. Rev. E 62, R1493; (b)Neimark A.V. and Ravikovitch P.I. (2001) Microporous and Mesoporous Materials
44-56,
697.
© 2004-2006
Quantachrome Instruments
Where Does Cavitation
Occur?
Adsorptive
Temperature
~p/po
Nitrogen
77K
0.42
Argon
87K
0.38
Argon
77K
0.23
© 2004-2006 Quantachrome Instruments
Quantachrome
I N S T R U M E N T S
Recent Advances in
Mesopore (2 – 50 nm) Analysis
© 2004-2006 Quantachrome Instruments
Mesopore Analysis
Significant progress in the pore size analysis of porous
materials was recently achieved, mainly because of the
following reasons:
•
(i) The discovery of novel ordered mesoporous molecular
sieves which were used as model adsorbents to test theories of gas
adsorption
•
•
(ii) The development of microscopic methods, such as the
Non-Local-Density Functional Theory (NLDFT) or computer simulation methods
(e.g. Monte-Carlo – and Molecular-Dynamic simulations), which allow to
describe the configuration of adsorbed molecules in pores on a molecular level;
(iii) Carefully performed adsorption experiments
© 2004-2006 Quantachrome Instruments
TEM of MCM-41 Silica
© 2004-2006 Quantachrome Instruments
Sorption, Pore Condensation and Hysteresis Behavior of a
Fluid in a Single Cylindrical Mesopore
From: M Thommes, “ Physical adsorption characterization of ordered and amorphous mesoporous materials”,
Nanoporous Materials- Science and Engineering” (edited by Max Lu, X.S Zhao), Imperial College Press, Chapter 11,
317-364 (2004)
© 2004-2006 Quantachrome Instruments
Pore Size Analysis of Mesoporous Solids:
The Modified Kelvin Equation
ln(P/P0) = -2cos /RT(rp – tc)
rp: pore radius
tc : adsorbed multilayer film prior to condensation
 : surface tension
: densities of the coexistent liquid (l ) and gas (g) ( =
l - g )
 : contact angle of the liquid meniscus against the pore wall
© 2004-2006 Quantachrome Instruments
SEM- of Mesoporous TiO2
© 2004-2006 Quantachrome Instruments
Nitrogen Sorption at 77 K into
Mesoporous TiO2
150
Sachtopore 60
Sachtopore 100
Sachtopore 300
Sachtopore 1000
Sachtopore 2000
Volume STP [cc/g]
120
6 nm
10 nm
90
30 nm
60
30
100 nm
0
0
0.2
0.4
0.6
P/P0
H. Kueppers, B. Hirthe, M.Thommes, G.I.T, 3 (2001) 110
© 2004-2006 Quantachrome Instruments
0.8
1
Pore Size Analysis of
Mesoporous Materials
(I) Methods based on (modified) Kelvin Equation
•
e.g., - Barett-Joyner-Halenda (BJH)
- Dollimore-Heal (DH)
- Broeckhoff de Boer (BdB)
- Kruk-Jaroniec-Sayari (KJS))
- Bhatia et al (mod. BdB)
- D.D.Do & Ustinov (mod. BdB)
© 2004-2006 Quantachrome Instruments
(1951)
(1964)
(1967/68)
(1997)
(1998/2004)
(2004/2005)
Results of Sorption Studies on Ordered Mesoporous Materials in Combination With
Advanced Theoretical and Molecular Simulation Approaches :
Problem: Conventional, macroscopic, thermodynamic methods (e.g, methods
based on the Kelvin equation such as BJH, BdB) assume bulk-fluid like
behavior for pore fluid and neglect details of the fluid-wall interactions
 Errors of 25 % and more in pore size analysis!!
Solutions:
• (1) Correction,and/or proper
calibration of classical methods
(e.g, KJS method):
Disadvantage: only valid over limited pore size range
•
(2) Application of microscopic methods based on statistical mechanics
(e.g., NLDFT, GCMC) which describe the configuration of the adsorbed
phase on a molecular level 
Accurate pore size analysis over complete
micro/mesopore size range
© 2004-2006 Quantachrome Instruments
Phase Diagrams of Pure Fluids
Confined to Porous Glasses
SF6/CPG
M. Thommes and G.H. Findenegg,
Langmuir 10 (1994), 4270
© 2004-2006 Quantachrome Instruments
CO2/Vycor
H. Fretwell et al, J. Phys. Condens. Matter
7 (1995) L717
Effect of Confinement on Sorption
and Phase Behavior
•
•
Pore size and temperature are complimentary variables
with regard to the occurrence of hysteresis
The shape of sorption isotherms is affected by both, the
texture of the material but also by the difference in
thermodynamic states of pore and bulk fluid phases
 In contrast to classical, macroscopic approaches modern
microcopic theories based on statistical mechanics (e.g
Density-Functional Theory and Molecular Simulation)
take these phenomena into account
© 2004-2006 Quantachrome Instruments
Pore Size Analysis by Microscopic Methods based
on Statistical Mechanics
(a) Density Functional Theory :
e.g.- Evans and Tarazona (1985/86)
- Seaton (1989),
- Lastoskie and Gubbins (1993)
- Sombathley and Olivier (1994)
- Neimark and Ravikovitch (1995 ……)
b) Monte Carlo (MC) and Moleculardyn. (MD),
e.g. - Gubbins et. al. (1986…. )
- Walton and Quirke (1989…)
- Gelb (1999- ….)
- Neimark and Ravikovitch (1995….)
© 2004-2006 Quantachrome Instruments
Theoretical Predictions Of The Pore Size
Dependence Of The Relative Pressure Of The
Equilibrium Condensation/Evaporation Transition
N2/77 K in cylindrical
silica pores
. Neimark AV, Ravikovitch P.I., Grün M., Schüth F., Unger K.K, (1998)
J. Coll. Interface Sci. 207,159
© 2004-2006 Quantachrome Instruments
Nitrogen sorption (77 K) in MCM-41 and
Pore Size Analysis by BJH and NLDFT
0.3
560
N2 (77 K): ads
N2 (77 K): des
490
0.25
BJH
420
Dv(d) [cc/Å/g]
Volume [10-6 m3/g]
DFT-Fitting
350
0.15
0.1
210
0.05
0
0.2
0.4
0.6
RELATIVE PRESSURE p/p0
0.8
1
NLDFT
0.2
280
140
BJH-Pore size distribution
DFT-Pore size distribution
0
15
23
31
Pore Diameter [Å]
NLDFT method: N2/77K cylindrical-silica pore model
© 2004-2006 Quantachrome Instruments
39
47
55
Nitrogen Adsorption and Pore Size
Analysis in CMK 3 Mesoporous Carbon
1000
0.12
N2 (77.35 K)
NLDFT- FIT
0.1
800
D(v)[cm3/Å/g]
Volume [cm3 g-1] STP
900
700
600
500
NLDFT-N2(77.4 K)
BJH-N2 (77.4 K)
BJH (3.5 nm)
0.08
NLDFT (5.1 nm)
0.06
0.04
400
0.02
300
200
0
0.2
0.4
0.6
0.8
1
0
20
40
Pressure P/P0
60
Pore Diameter [Å]
NLDFT Methods: N2/77K cylindrical carbon pore model
M.Thommes, H. Huwe, M. Froeba et al, to be published (2005)
© 2004-2006 Quantachrome Instruments
80
100
Other Factors
The influence of
 Pore Geometry
 Connectivity
 Disorder (geometrical and surface heterogeneity )
on Adsorption, Pore Condensation, Hysteresis, and
thus the shape of the sorption isotherm remains under
investigation
© 2004-2006 Quantachrome Instruments
Nitrogen Sorption at 77 K into
various Mesoporous Silica Materials
700
Vycor
SBA-15
Controlled-Pore Glass (CPG)
SE3030
VOLUME (STP) [cc/g]
600
500
400
300
200
100
0
0
© 2004-2006 Quantachrome Instruments
0.2
0.4
0.6
RELATIVE PRESSURE P/P0
0.8
1
IUPAC Classification of Hysteresis
Cylindr.Pores
Cylindr.&Spherical
Pores
Disordered.
lamellar pore
structures, slit &
wedge, shape
pores
Micro/Mesoporous
adsorbents
© 2004-2006 Quantachrome Instruments
Origin of Capillary
Condensation Hysteresis
Single Pore Model : Hysteresis occurs in a single pore and
reflects a intrinsic property of phase transition in a pore.
Hysteresis is due due to metastable pore fluid  H1
Hysteresis
Network Model: Pore blocking, percolation effects, on
desorption branch  H2 Hysteresis
Disordered Porous Materials Model:
Combination of kinetic and thermodynamic effects;
phenomena are spanning the complete disordered pore
system  H1 and H2 Hysteresis
© 2004-2006 Quantachrome Instruments
Ar (87 K) and N2 (77 K) sorption
in MCM 48 and NLDFT-Pore size analysis by
using the NLDFT-equilibrium method (kernel)
1000
700
600
500
400
ads
300
des
0.25
Ar: ads
Ar: des
0.20
Ads(Ar/87 K)
0.15
0.10
0.05
N2 (77 K)
Ar (87 K)
200
Des(Ar/87 K)
N2: ads
N2: des
3
N2/77K
-5
-6
3
VOLUME [10 m /g]
800
Ar(87K): Hysteresis
Ar/ 87 K
N2(77K): Reversible
Dv(d) [10 m /nm/g]
900
0.30
0.00
100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
RELATIVE PRESSURE p/p0
(M. Thommes et al, Applied Surface Science, 196 (2002) 239-249)
© 2004-2006 Quantachrome Instruments
3.0
3.5
4.0
4.5
5.0
PORE DIAMETER [nm]
5.5
6.0
Network Model: Pore blocking and
percolation effects in interconnected pore
systems
Type H2
Hysteresis
Problem for Pore Size Analysis:
Adsorption Branch: metastable pore fluid  delayed pore
condensation
Desorption Branch: pore blocking,percolation  delayed
evaporation
How to tackle:
 Application of approaches based on percolation theory
 Application of novel NLDFT approaches
© 2004-2006 Quantachrome Instruments
Mesopore-Analysis by NLDFT
NLDFT-method for Pore Size Distribution
Calculation from Adsorption and Desorption
Desorption Branch:: Equilibrium liquid-gas phase
transition (evaporation)
 NLDFT-Kernel of Equilibrium Isotherms
Adsorption Branch: NLDFT-spinodal- gas-liquid phase
transition (condensation)
 NLDFT- Kernel of (Metastable) Adsorption
Isotherms
By P. Ravikovitch, A.V. Neimark, Colloids and Surfaces A: Physicochem Eng. Aspects 187-188 (2001) 11
© 2004-2006 Quantachrome Instruments
Nitrogen adsorption/desorption at 77.35 K in SBA-15 and
pore size distributions from adsorption- (NLDFT spinodal
condensation kernel ) and desorption (NLDFT equilibrium
transition kernel)
700
0.22
0.2
600
500
0.16
Dv(d) [cc/Å/g]
Volume STP [cc/g]
0.18
Ads (NLDFT-spinodal condensation)
Des (NLDFT- equilibrium transition)
400
300
200
0.14
0.12
0.1
0.08
0.06
100
0
0.04
0.02
0
0.2
0.4
0.6
Relative Pressure P/P0
0.8
1
0
25
45
65
85
Pore Diameter [Å]
105
125
M. Thommes, in Nanoporous Materials- Science and Engineering” (edited by Max Lu), Imperial College Press, Chapter 11 p. 317 364 (2004)
© 2004-2006 Quantachrome Instruments
Nitrogen sorption at 77 K in porous CPG and Vycor Glasses and
pore size distributions from adsorption- (NLDFT spinodal
condensation kernel) and desorption (NLDFT equilibrium transition
kernel)
0.026
420
Ads (NLDFT-spinodal condensation)
Des (NLDFT- equilibrium transition)
CPG
280
Dv(d) [cc/Å/g]
Volume STP [cc/g]
350
210
H1 Hysteresis
140
0.013
70
0
0
0.2
0.4
0.6
Relative Pressure P/P0
0.8
0
40
1
90
140
Pore Diameter [Å]
190
240
0.04
150
Ads (NLDFT- spinodal condensation)
Des (NLDFT- equilibrium transition)
Vycor
90
0.032
Dv(d) [cc/Å/g]
Volume STP [cm3/g]
120
H2 Hysteresis
60
0.016
0.008
30
0
0.024
0
0.2
0.4
0.6
Relative Pressure p/p0
0.8
1
0
25
VYCOR(PSD) | 12.11.2002
50
75
100
Pore Diameter [Å]
125
M. Thommes, in Nanoporous Materials- Science and Engineering” (edited by Max Lu), Imperial College
Press, Chapter 11 p. 317 - 364 (2004)
© 2004-2006 Quantachrome Instruments
150
Conclusion: H1 Hysteresis
• Mechanism of hysteresis in single meso- pores
(e.g. MCM-41, SBA-15) and in materials
consisting of ordered pore networks (e.g., MCM48 , CPG) seems to be similar. In both cases H1
hysteresis is observed.
 In case of H1 hysteresis methods based on the
independent pore model are in principle applicable
for pore size analysis
© 2004-2006 Quantachrome Instruments
H2/H3 Hysteresis
 In case of
type H2 hysteresis, pore blocking, percolation,
and cavitation effects play an important role. The position
of the desorption branch does not reflect the equilibrium
liquid-gas transition. Hence, a method for pore size
analysis based on the equilibrium phase transition can here
not be applied
 NLDFT-spinodal condensation method can be
applied to the
adsorption branch (in case of cylindrical-like pores and silica
materials – pore size range up to 80 nm!)
 Application of a calibrated correlation between the position of
capillary condensation step and pore size.
© 2004-2006 Quantachrome Instruments
H2/H3/H4 Hysteresis: Lower Limit of
Hysteresis Loop –Tensile Strength
Effect ??



Hysteresis loop for N2 (77.35 K) always closes at relative pressures
> 0.42 and for argon at 87.27 K at relative pressures > 0.38.
The lower closure point of hysteresis is believed (in the classical
picture) to be determined by the tensile strength of the capillary
condensed liquid, i.e., there exists a mechanical stability limit below
which a macroscopic meniscus cannot exist anymore and which leads
to a spontaneous evaporation of the pore liquid.
This forced closure of the hysteresis leads to an artifical step in the
desorption isotherm  Pore size distribution artifact at ca. 4 nm
 Adsorption Branch should be selected for Pore-Size Analysis
© 2004-2006 Quantachrome Instruments
H3 Hysteresis: Lower limit of Hysteresis
Loop –Tensile Strength Effect?
240
1
Adsorption
Desorption
210
Adsorption
Desorption
BJH-PSD
180
N2/77K sorption on
disordered alumina
catalyst
150
120
Dv(log d) [cc/g]
Volume STP [cc/g]
0.8
0.6
Artifact
0.4
90
0.2
60
30
0
0.2
0.4
0.6
Relative Pressure P/P0
0.8
1
0
10
50
100
Pore Diameter [Å]
M. Thommes, In Nanoporous Materials Science and Engineering, (Max Lu and X Zhao, eds.), World
Scientific, in press (2004)
© 2004-2006 Quantachrome Instruments
500
1000
H4 Hysteresis: Nitrogen adsorption at
77.4 K in activated carbon
-Tensile Strength effect?
500
Nitrogen (77 K)
Volume STP [cc/g]
400
300
200
100
0
0
0.2
0.6
0.4
P/P0
© 2004-2006 Quantachrome Instruments
0.8
1
Pore Condensation/Evaporation in Ink-bottle Pores:
Pore Blocking and Cavitation Phenomena.
M. Thommes, B. Smarsly, P.I. Ravokovitch, A.V. Neimark et al.. Langmuir, 22, 765 (2006)
© 2004-2006 Quantachrome Instruments
N2 and Ar adsorption on micro-mesoporous silica (SE3030)
and pore size analysis by the NLDFT- method
Pore size distribution from
metastable adsorption branch
0,025
9.4 nm
1 nm
0.8
Cumulative Pore Volume [cc/g]
0,020
Dv (cc/(Angstr. g))
Nitrogen (77.35 K)
Argon (87.27 K)
0.7
0,015
0,010
0.6
Cumulative
pore volume
0.5
0.4
0.3
0.2
0.1
0
0,005
10
50
100
500
1000
Pore Diameter [Å]
M. Thommes, B. Smarsly, P.I. Ravokovitch, A.V. Neimark et al..
Langmuir, 22, 765 (2006)
0,000
10
100
Pore size (Angström)
Mesopore Size
N2/77K sorption (NLDFT)
Ar/87K sorption (NLDFT)
SANS (CLD)
TEM
: 9.4 nm
: 9.1 nm
: 9.5 nm
: ca. 9.5 nm
Micropore Size
N2/77K sorption (NLDFT) : ca. 1.1 nm
SANS
: ca. 1 – 1.2 nm
© 2004-2006 Quantachrome Instruments
Micropore Volume
N2/77K: 0.12 ml/g
SANS: 0.1 ml/g
Excellent agreement
between NLDFT
and SANS/SAXS
Nitrogen sorption of “KLE silica“ at 77K and NLDFT analysis
N2 sorption isotherm
Pore size distribution
360
0.015
RUN 1 (Ads)
RUN1 (Des)
RUN 2( Des)
RUN 2 (Ads)
KLE-Silica NLDFT-PSD(spherical pore model)
0.012
240
dv(cc/Å/g)
Volume [cc/g]
300
180
0.009
13.9 nm
1.3 nm
0.006
120
0.003
60
0
0
0
0.2
0.4
0.6
Relative Pressure P/P0
0.8
0
60
120
180
240
300
Pore Diameter
[Å]
Pore diameter
(Angström)
1
NLDFT analysis (spherical mesopores,
cylindrical micropores)
Mesopore Size:
N2-sorption: 13.9 nm
TEM: Ca. 13 nm
SAXS: 13.8 nm
Excellent agreement
between SAXS and
new NLDFT approach!
M. Thommes, B. Smarsly, M. Groenewolt, P. Ravikovitch, and A. Neimark, Langmuir, 22,756 (2006)
© 2004-2006 Quantachrome Instruments
360
Pore Blocking/Percolation and
Cavitation
Pore Blocking/Percolation:
“Pore size” distribution determined from desorption
branch should be independent of the choice of the
adsorptive or temperature
Cavitation :
Artificial “Pore” size distribution determined from
desorption branch of hysteresis loop should depend on
the choice of the adsorptive and temperature
© 2004-2006 Quantachrome Instruments
Poreblocking/Percolation As Dominant Evaporation
Mechanism: Nitrogen And Argon Sorption In Vycor
0.008
Nitrogen /77 K- NLDFT(Adsorption-Branch-Kernel)
Argon/87 K - NLDFT (Adsorption-Branch-Kernel)
200
Nitrogen 77 K
Argon 87 K
D(v) [cc/Å/g]
Volume [cc/g] STP
160
0.006
120
Analysis of N2 and
Ar adsorp. branches
 “Pore Size”
0.004
0.002
80
0
20
40
50
80
0.035
0.2
0.4
0.6
0.8
1
P/P0
M. Thommes, B. Smarsly, P.I.
Ravokovitch, A.V. Neimark et al..
Langmuir, 22, 765 (2006
200
230
260
290
320
Analysis of N2 and
Ar desorp. branches
0.02
0.015

0.01
“Neck Size”
0.005
0
© 2004-2006 Quantachrome Instruments
170
Nitrogen/77 K NLDFT (Desorption-Branch Kernel)
Argon/87 K NLDFT- (Desorption -Branch-Kernel)
0.025
D(v) [cc/Å/g]
0
140
Pore Diameter [Å]
0.03
0
110
0
40
80
120
160
200
Pore Diameter [Å]
240
280
320
Cavitation as dominant mechanism for pore
evaporation: N2 and Ar sorption in SE3030
micro/mesoporous silica
NLDFT-PSD (from Adsorption Branch)
600
0.025
Nitrogen 77 K
Argon 87 K
Argon (87 K), Ads
Nitrogen (77K),Ads
9.4 nm
0.02
400
[cm3/Å/g]
Volume [cc/g] [STP]
500
0.44
300
0.015
0.01
0.47
200
NLDFT pore
size from
adsorption
0.005
0
30
100
42
54
66
78
90
102
114
126
138
150
Pore Diameter [Å]
SE3030(PSD) | 17.2.2003
NLDFT PSD from Desorption
0
0
0.2
0.4
0.6
0.8
0.48
1
Argon 87 K
Nitrogen 77 K
P/P0
M. Thommes, B. Smarsly, P.I. Ravokovitch, A.V.
Neimark et al.. Langmuir, 22, 765 (2006)
D(v)[cc/Å/g]
0.4
No pore size info
from desorption!
0.32
0.24
0.16
0.08
© 2004-2006 Quantachrome Instruments
0
30
40
50
60
70
80
90
Pore Diameter [Å]
100
110
120
130
Summary : Sorption Hysteresis in
Micro/Mesoporous Pore networks
• Sorption experiments using different adsorptives (e.g. Argon, Nitrogen) allow to
identify pore blocking/percolation and cavitation mechanisms :
Pore Blocking controlled desorption (e.g. porous Vycor glass): Neck Size from
analysis of desorption branch; Pore (Cavity) size from adsorption branch
Cavitation controlled desorption: No pore size info from analysis of desorption
branch ; pore (cavity) size from adsorption branch
• Cavitation controlled desorption is observed in SE3030, KLE and KLE/IL,SLN-326 silica,
which consist of of micro/mesoporous networks of ink-bottle like pores.
- The rel. pressure where cavitation occurs does not depend on the actual neck size as long as
Wneck < W neck,(critical), and Wneck(critical) is found to be larger than 5 nm!
- If Wneck > W neck,(critical) then pore blocking can be observed
• We confirm the validity of novel N2/silica and Ar/silica NLDFT methods, applicable
to the adsorption branch of a hysteretic isotherm. The pore size data obtained with
this method for SE3030, KLE-silica are in excellent agreement with independently
obtained results from novel SANS/SAXS techniques.
© 2004-2006 Quantachrome Instruments
Conclusions &Recommendations:
Hysteresis and Pore Size Analysis
• H1 Hysteresis: Independent pore model applies. Pore size can in
principle be determined from both desorption branch and adsorption
branch if proper methods are available
• H2 Hysteresis: caused by pore blocking/percolation or cavitation
phenomena in mesoporous and micro/mesoporous pore networks
Pore blocking:  Pore (cavity) size from adsorption branch; Neck size
from desorption branch,
Cavitation:  Pore (Cavity Size) from adsorption branch; No pore size
information from desorption branch
• H3/H4 Hysteresis: observed in very disordered micro/mesoporous pore
networks and caused by a combination of various phenomena (incl.
cavitation, pore blocking)  Pore (Cavity Size) from adsorption branch
•
© 2004-2006 Quantachrome Instruments
Conclusions
The use of different probe molecules allows not only to
check for consistency in the pore size and surface area
analysis, but allows also to obtain a much more accurate
micro- and mesopore analysis.
The shape of sorption isotherms is affected by, surface

chemistry and the texture of the adsorbent but also by the
difference in thermodynamik states of pore and bulk fluid
phases. This has to be taken into account in order to obtain
a correct and comprehensive pore size analysis.
Microscopic methods (e.g, NLDFT, Molecular simulation)
allow to obtain an more accurate and comprehenisve pore
size analysis compared to macroscopic, thermodynamic
methods (e.g., BJH, HK, SF, DR).
© 2004-2006 Quantachrome Instruments
Some Selected References
(1) M Thommes, “ Physical adsorption characterization of ordered and amorphous
mesoporous materials”, Nanoporous Materials- Science and Engineering”
(edited by Max Lu, X.S Zhao), Imperial College Press, Chapter 11, 317-364
(2004)
(2) S. Lowell, J.E. Shields, M.A. Thomas and M. Thommes, Characterization of
porous solids and powders: surface area, pore size and density, Kluwer
Academic Publisher, 2004
(3) M. Thommes, B. Smarsly, P.I. Ravokovitch, A.V. Neimark et al.. Langmuir, 22,
765 (2006)
(4) B. Smarsly, M. Thommes, P.I Ravikovitch, A.V. Neimark, Adsorption 11 (2004),
653, (2005)
(5) J. Jagiello, M. Thommes, Carbon 42 (2004) 1227
(6) M. Thommes, R. Koehn and M. Froeba, Applied Surface Science 196 (2002) 239
(7) M. Thommes, R. Koehn and M. Froeba, J. Phys. Chem. B 104, 7932 (2000)
© 2004-2006 Quantachrome Instruments