CO2 maîtrisé | Carburants diversifiés | Véhicules économes | Raffinage propre | Réserves prolongées
Physisorption and pore size
analysis
Gerhard Pirngruber
© IFP
Direction Catalyse et Séparation
[email protected]
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
CV Gerhard Pirngruber

1995 – 1999


Universiteit Twente (NL)
2000 – 2005




R&D Scientist, project leader
IFP Energies nouvelles, Catalysis and Separation Division
Research activity

© IFP
zeolites
mesoporous silica
2005 – now

Oberassistent
ETH Zurich, Inst of Chemical and Bioengineering
Research activity


PhD in heterogeneous catalysis

2005 – 2012 : CO2 capture, separation of hydrocarbons
2012 – now: hydrotreating and hydrocracking catalysts
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Porous solids in catalysis

Supported catalysts


e.g. noble metals, transition metal sulfides
Porous support






provides surface for dispersing the catalyst
provides mechanical stability
has an influence on diffusion of reactants and products
has an influence on adsorption of reactants and products
determines reactor volume
Porous catalysts

Optimising porosity is often the
key to improving catalytic
performance.
© IFP

Zeolites
Oxydes
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Types of porosity

Porous cristalline structures



Zeolites,
Metal Organic Frameworks
Ordered or disordered stacking of small particles
generates an interparticles porosity

© IFP

oxides (silica, alumina, etc.)
activated carbon
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Characterisation of porosity

Physisorption (N2, Ar, Kr)




Hg intrusion




surface area
pore volume
pore size distribution
pore volume
pore size distribution
surface area
Imaging methods

© IFP

Transmission Electron Microscopy
Secondary Electron Microscopy
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Terminology





Physisorption: no formation of chemical bonds, no
transfer of electron density
Chemisorption: formation of a chemical bond
(exchange of electrons)
© IFP

Adsorption is a surface phenomenon.
Adsorbent = solid that provides a surface for
adsorption.
Adsorbate = molecule adsorbed on the surface.
Adsorptive = molecule susceptible of being adsorbed.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Terminology

IUPAC distinguishes 3 catagories of pore sizes


< 2 nm
2 – 50 nm
> 50 nm
© IFP

Micropores
Mesopores
Macropores
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Forces involved in physisorption

van der Waals forces

amplified by multiple interactions with atoms or pore wall (i)
5000
U LJ
4000
Energy
3000
   ij 12   ij 6 
 4        
 r   r  


2000
1000
adsorbate
j
i
0
-1000
0
2
4
6
8
pore wall
Distance

maximal when size of adsobate (j) close to pore size
© IFP
pore wall
adsorbate
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Forces involved in physisorption

Electrostatic forces



© IFP

Any non-symmetric charge distribution in the adsorbent
generates an electric field
adsorbates with an electric moment (dipole, quadrupole)
interact with the electric field
undesirable in analysis of porosity/pores size because
surface chemistry/chemical composition enter into the game
N2 has a small quadrupole moment → use of Ar, Kr is
preferable for precise micropore size analysis
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Adsorption in micropores

Gradual filling of pore volume starting
from surfcae
Adsorbed amount
Saturation zone
Transition zone
additional adsorbateadsorbate interactions
Initial slope depends on adsorbentadsorbate interactions
© IFP
Pressure
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Adsorption in mesopores
© IFP
monolayer
adsorption
multilayer
adsorption
pore filled by
onset of
capillary
capillary
condensation condensation
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Adsorption in macropores

© IFP

Opposite pore wall is too far away to influence
adsorption and provoke capillary condensation
First monolayer, then multilayer adsorption
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
How to measure an adsorption isotherm?

Volumetric method
Réservoir étalon : Vres
Hélium
Réservoir étalon : Vres
Etat initial : PriHe, TriHe
Etat final : PrfHe, TrfHe
Etat initial : PriN2, TriN2
Etat final : PrfN2, TrfN2
Azote
Bilans Matières : He et N2
nHe
nHe
Température
ambiante
Cellule mesure : Vcel
nN 2
Température
N2 liquide
77 K
nN 2
Etat initial : PciHe, TciHe
Etat final : PcfHe, TcfHe
i
f
 PrHe
V
PrHe

  i  f  res
 TrHe TrHe  R
V
 Pc f  Pci cel
RTm


 PrNi 2 PrNf 2  Vres
  i  f 
 TrN 2 TrN 2  R
V
 Pc f  Pci cel  nNads2
RTm


Etat initial : PciN2, TciN2
Etat final : PcfN2, TcfN2
© IFP
Masse d'adsorbant : Msol
Réservoir d'azote
liquide à 77 K
Quantité adsorbée : Qads=nN2ads/Msol
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Material balance


Reservoir : 1
Sample cell : 2
without adsorption (He), sample cell initially under
vacuum
V2 V1 pinit  p final
 
T2 T1
p final
pinitV1 p finalV1 p finalV2


RT1
RT1
RT2

with adsorption (N2)
ntot 
pV1 pV2

 nads
RT1 RT2
© IFP
quantity initially
present in the
reservoir
determined with He
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Experimental difficulties

Precise measurement of pressure required



difficult at low pressure
difficult to measure a micropore distribution
Isotherm !!

T2 in principle 77 K (temperature of liquid N 2 )


pay attention to the level of liquid N 2
© IFP

part of the cell is not plunged into liquid N2, but is at ambient
temperature
temperature gradient must be the same as during the initial
calibration of the volume with He
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Example zeolite NaX
220
powder
Adsorbed volume ( cm3/g)
200
180
after shaping with a
binder
160
N2 at 77 K NaX (ADS 57701, Powder)
N2 at 77 K NaX (ADS 48100) SPX 3003
140
N2 at 77 K NaX (ADS44956) SPX 3001
120
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
P/P0
© IFP


1
Micropores are entirely filled at very low pressure.
Isotherm is totally flat once micropores are filled.

saturation zone
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Examples - ZnO
Quantité adsorbée (cm3 STP/g)
180
capillary condensation
in the mesopores
160
140
120
100
multilayer adsorption on the
surface of the particles
80
60
ZnO
40
20
0
0
0.2
0.4
0.6
0.8
1
p/p0

© IFP

no micropores
mesopores generated by stacking of inidividual
particles
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Comparison of two ZnO samples
Quantité adsorbée (cm3 STP/g)
180
160
140
120
100
65811
70299
80
60
40
20
0
0
0.2
0.4
0.6
0.8
1
p/p0
© IFP


Surface : 65811 > 70299
Pore size : 70299 > 65811
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Example – Al2O3
Quantité adsorbée (cm3 STP/g)
300
capillary condensation
in mesopores
250
200
150
100
multilayer adsorption
on surface of particles
50
0
0
0.2
0.4
0.6
0.8
p/p0
© IFP


no micropores
high surface area, broad pore size distribution
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
1
Comparison Al2O3
Quantité adsorbée (cm3 STP/g)
500
450
400
350
300
79999
61399
47148
250
200
150
100
50
0
0
0.2
0.4
0.6
0.8
1
p/p0

© IFP

Surface : 79999 > 61399 > 47148
Pore size : 47148 > 61399 > 79999
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Quantitative treatment of N2 isotherms

Pore volume: Vmicro, Vmeso, Vtotal



Surface area



t-plot
Dubinin-Radushkevitch
BET (Brunauer-Emerett-Teller)
Langmuir
Pore size distribution

Micropores


Mesopores

© IFP
Horwath-Kawazoe, Saito-Foley

BJH (Brunauer-Joyner-Halenda)
DFT (Density Functional Theory)
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Determination of the surface area

BET (Brunauer Emmet Teller): multilayer adsorption


first layer: adsorption on the surface
all the other layers are considered like a condensed liquid
Kcondensation
Ksurface
© IFP
nm = number of surface adsorption sites (monolayer)
nads = total number of adsorbed molecules
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
BET equation
P
BET equation
Vads P0  P 
C





C 1  P 
 
V M C  P0 
K surface
Recommended
range of p/p0 :
0,05 - 0,35 max.
K condensation
VM = monolayer volume
Vads = adsorbed volume
Plot p/Vads(p0-p) vs. p/p0
Calculation of surface area:

VMC

NAv : Avogadro number
a : area of N2 molecule (16,2 Ǻ2)
1/[V.(P/P0-1)]

1

PNT. = (C-1)/(VM.C)
O.O. = 1/(VM.C)
© IFP
0,1
S BET  a.
p0VM
.N A
R  273K
0,2
0,3
S BET (m2 / g )  4.355 Vm (cm3 STP / g )
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
0,4
P/P0
Exemple ZnO
Isothermes
BET plot
S BET 
0.05
160
slope
4.355
CBET 
1
int ercept
slope  int ercept
0.045
140
y = 0.1196x + 0.0005
0.04
120
100
65811
70299
80
60
40
prel / Vads(1-prel)
Quantité adsorbée (cm3 STP/g)
180
0.035
0.03
0.02
0.015
20
0.01
0
0.005
0
0.2
0.4
0.6
0.8
1
65811
70299
0.025
y = 0.0594x + 0.0003
0
p/p0
0
0.1
0.2
0.3
prel = p/p0
CBET
65811
74
136
70299
36
237
© IFP
SBET (m2/g)
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
0.4
0.5
Example NaX

The multilayer adsorption
model does not apply well to
microporous solids when the
pores are filled.
Choose a lower pressure
range for microporous solids.


p/p0 = 0.05 – 0.10
C constant may be negative.
Does not make physical sense
lower the pressure range
further
0.0006
y = 0.0064x - 3E-05
0.0005
0.0004
0.0003
0.0002
0.0001
0
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
prel = p/p0
S BET 
4.355
 685m 2 / g
0.0064  0.00003
© IFP

0.0007
prel / Vads(1-prel)

Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
0.1200
Langmuir Surface Area

Langmuir theory



Do not use Langmuir surface areas !!
SLangmuir is always > SBET because multilayer
adsorption is treated like monolayer adsorption.
© IFP

is a model of monolayer adsorption
this hypothesis is never fulfilled
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Determination of the pore volume
Vads = 178
cm3 STP/g
→ Vmicro
Volume adsorbé (cm3 STP/g)
300
250
200
Vads = 240
cm3 STP/g
→ Vtotal
150
100
Vmeso = Vtotal - Vmicro
50
0
0
0.2
0.4
0.6
0.8
1
p/p0
In some cases possible without using any model.
© IFP

Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Conversion gas volume – pore volume

Volume of adsorbed gas correponds to a certain
number of moles

STP = Standard Temperature Pressure


T = 273 K, p = 1 bar = 100 kPa
The adsorbed phase is considered like a liquid phase

Density of liquid N2 at 77 K : 0.807 g/ml
Vp 
© IFP
Vads p
nads 
RT
mN 2
 N 2,liq

nads  M N 2
 N 2,liq
Vp (ml / g )  1.5468 103 Vads (cm3 STP / g )
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
t-plot

© IFP

Transform adsorbed amount in an average thickness
(t) of the adsorbed layer
Vads
For multilayer adsorption on a flat surface
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
t
S
Universal curve t vs. p/p0


The isotherms (Vads/SBET) of
many low surface area oxides
(SiO2, Al2O3, ZrO2, TiO2, MgO)
form a universal curve
t = f(p/p0).
Numerical description of that
curve


© IFP
Vads
S BET
valid for t = 3.5 – 10 Å
and p/p0 = 0.1 – 0.8
Harkins
Jura
Halsey
t
t (
13.99
)1/ 2
p
 log
 0.034
p0


 5 

t  3,54
 ln  P0  
  P  
1/ 3
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Isotherm expressed as Vads = f(t)
180
180
160
160
limit of validity
140
120
100
65811
80
60
Vads cm3 STP/g
Vads (cm3 STP/g)
140
S
120
100
Vads
t
80
60
40
40
20
20
0
0
0
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
t (nm)
p/p0
t = thickness that the
adsorbed phase
would have on a nonporous material
© IFP
13.99
t (
)1/ 2
p
 log
 0.034
p0
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
2
Interpretation of t-plots

Adsorption on a non-porous solid
Vads
S
p/p0

Adsorption on a microporous solid


micropores rapidly filled
then adsorption on external surface
t
Vads
© IFP
S
p/p0
Vads
t
Vads
t
Vmicro
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
t
Example zeolite NaX
t-plot
300
300
250
250
Vads (cm3 STP/g)
Volume adsorbé (cm3 STP/g)
Isotherm
200
150
100
50
0
0.2
0.4
0.6
p/p0
150
100
0.8
1
0
0.00
Vmicro
0.50
1.00
t (nm)
Vmicro = 175 cm3 STP/g = 0.271 ml/g
© IFP

200
50
0
limit of validity
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
1.50
2.00
Surface BET vs. Surface t-plot
St  plot (m2 g 1 )  1.5468  PNT (cm3 g 1nm1 )
70
Vads cm3 STP/g
60
50
y = 45.11x + 0.965
40
30
20
y = 21.83x + 0.685
10
0
0
0.2
0.4
0.6
0.8
1
1.2
© IFP
t (nm)
SBET (m2/g)
St-plot
65811
74
70
70299
36
34
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Micro- and mesoporous solids
Solide non poreux : Adsorption sur Surface Externe
------------------
µP
-
Pressions relatives P/P0 croissantes
Courbe t = t(P/P0)
------------------
mP
Vol. Ads.
Solide Non Poreux
Adsorption sur Surface Externe
Vol. Ads.
Solide poreux : Adsorption sur Surfaces Interne & Externe
Solide µ et m-Poreux
Adsorption sur
Surface Externe
VµP+VmP
© IFP
VµP
Epaisseur t
Adsorption sur
Surface Interne
(mésopores) => SB.E.T.
Epaisseur t
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
t-plot may be ambiguous
Zeolite beta with supermicropores


no clear linear region in the t-plot
micropore volume depends on the interval chosen for
extrapolation
200
H-MCBeta
H-MCBeta steamed and leached
3
200
3
N2 ads cm /g STP
220
Vads cm /g STP

180
160
© IFP
140
0.0
0.2
0.4
0.6
p/p0
0.8
1.0
180
160
140
120
100
0
2
4
6
8
10
Thickness - Halsey [A]
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
12
Méthode de Dubinin-Radushkevitch

Théorie de remplissage de micropores

Vads
2
 exp   A / E 
Vmicro



© IFP
A   RT ln
p
pvap, sat
A = potentiel d'adsorption
E = energie caracteristique de l'adsorbant
pvap,sat = pression vapeur saturante



pvap,sat = p0 (1 atm) pour N2 à 77K
Equation linéarisée
 p
log Vads  log Vmicro  D log 
 p0 
2
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Example NaX
-0.51
0
0.5
1
1.5
2
2.5
3
3.5
4
-0.52
log10(Vol. liquide)
-0.53
Pression élevée
Potentiel faible
Adsorption dans mesopores
-0.54
Pression faible
Potentiel élevée
Début remplissage
micropores
-0.55
-0.56
-0.57
-0.58
© IFP
log10(P/Po)^2
Vmicro
= 10^(-0.548) = 0.283 ml/g
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
4.5
Average pore size

From geometrical rules

Cylindrical Pore:



Spherical pore :
Slit pore :
S 2

V r
d
4V
S
4 3

V

r

3

S  4r 2

S 3

V r
d
6V
S
V  lhr

S  lh
S 1

V r
2V
d
S
© IFP

2
V = pore volume V  r h

S = surface
S  2rh
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Micropore size distribution

Based on the relation between adsorption strength
and the ratio between adsorbate size and pore size


Mathematical models



condition: adsorption controled by van der Waals interactions
only
Horvath-Kawazoe
Saito-Foley
Input parameters


pore geometry: slit-shaped, cylinder, sphere
parameters of the Lennard-Jones potential well: ε and σij

© IFP

Saito, Foley, AICHE Journal 1991, 37, 429.
not always well known for atypical solids
Input data

high precision isotherm at very low pressure !
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Mesopore size distribution

via the theory of capillary condensation
A
B
C->D
D
Vads
E
D->E
E->F
F
C
A
B
© IFP
P/P0
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
F->A
Kelvin equation

Capillary forces in the pore lower the vapor pressure of the
condensed liquid

Pressure at the concave side of an interface is higher than at the
convex side.
overpressure counteracts the surface tension, which tries to collapse the
interface area
pressure
 Application to interface between gas and
difference Δp
adsorbed liquid film in a pore



© IFP

Pressure in liquid is lower than gas pressure.
Means that chemical potential in liquid
is lower, in other words
that the equilibrium vapor pressure
is lower.
Capillary condensation occurs at a
lower pressure than condensation.
surface
tension
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Kelvin equation

Mathematic formalism

Work against interfacial tension = change in free enthalpy
dA    dncapliq  capliq  dnliq  liq
dncapliq  dnliq 
dA   
dVcapliq
Vm
dVcapliq
Vm
  capliq 
dVcapliq
Vm
dA Vm    dVcapliq (  capliq  liq )
© IFP
dA
( capliq  liq )  
Vm  
dVcap, g
RT ln
pcap
psat

 liq
capliq  0  RT ln
liq  0  RT ln
dA
Vm
dVcap, g
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
pcap
p0
psat
p0
Generalized Kelvin equation
ln

psat
V  dA
 m 
RT dV
dV/dA = change in volume per change in
interface area
Vm = molar volume
dV/dA depends on the pore geometry




pcap
Sphere: dV/dA = r/2
Cylinder dV/dA = r
Slit
dV/dA = d
distance between slits
Relation to curvature of the pore
psat
2 Vm

r RT
© IFP
ln
pcap
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Capillary condensation
p
2rp
2(rp – t)

© IFP


p
vl
pc  p0  exp(
)
RT  (rp  t )
Multilayer adsorption on the surface: layer thickness t 
At a certain effective pore radius r p-t, capillary condensation
occurs
Pore is filled. There is a step in the adsorption isotherm.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Example isotherm with hysteresis
Type H1
1200
capillary
evaporation
Volads (cm3/g)
1000
800
600
400
capillary condensation
200
0
0.0
0.2
0.4
0.6
0.8
1.0
© IFP
p/p0
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Application of the Kelvin equation
1200
Volads (cm3/g)
1000
Harkins-Jura t-plot equation
800
600
log
p
13.99
 0.034  2
p0
t
400
log
p  4.14

p0
rm
Kelvin equation
200
0
0.0
0.2
0.4
0.6
0.8
Cylinder !!
1.0
p/p0
t(pc)
rm
rc
rp= rc+t
Ads
0.671
8.2
23.9
11.9
20.2
Des
0.565
7.0
16.7
16.7
23.7
© IFP
pc/p0
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Origin of the hysteresis
Adsorption:
cylindrical meniscus
rm = 2reff



© IFP

Desorption:
hemisperical meniscus
rm = reff
The lower rm, the lower is the pressure of capillary
condensation/evaporation
rm,ads = 2 rm,des
Capillary evaporation at a lower pressure than capillary
condensation
Hysteresis loop in isotherm
L.H. Cohan, JACS 60 (1938)
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Origin of the hysteresis


© IFP

Truth seems to be complicated than in Cohan’s
theory.
Cohan’s theory, based on the shape of mensiscus,
theory suggests that different vapor liquid equilibria
exist in a spherical or cylindrical meniscus, which
leads to hysteresis => is an equilibrium picture.
Molecular simulations (DFT) suggest that adsorption
branch is not in thermodynamic equilibrium, but is a
metastable state.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Hysteresis and metastability



Adsorption branch comprises metastable states.
 grand free energy of filled pore is lower than
that of the empty pore in the hysteresis
Desorption branch is in thermodynamic
equilibrium.
Reason for metastability in adsorption branch
 barrier of nucleation by formation of a liquid
bridge across the pore
 condensation occurs when limit of
metastability is reached
limit of metastability
gas phase
© IFP
liquid phase
Monson, MMM, 2012.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Peterson, JCS Farad 2, 1986.
Pore size distribution – BJH model
p0 – all pores filled
p1 – capillary evaporation in
largest pore
reduction of layer thickness
© IFP
p2 – capillary evaporation in
2nd largest pore
reduction of layer thickness in
both pores
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Mathematic description - BJH
rp
rk
rk + t
(rk  t ) 2
V 
V p
2
rp
In each desorption step pn-1  pn, capillary
evaporation occurs from a pore of size r pn
 The volume desorbed in that step (Vn) can be related
to the pore volume by the geometrical relation given
above.
 Complication: Reduction in layer thickness in the
pores, which were already emptied, also contributes to
Vn. of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Characterization

© IFP
rp
The BJH-equation
V pn 
rpn
2
(rkn  tn )
2
 Vn 
E.P. Barret, L.G. Joyner, P.P. Halenda, JACS 73 (1951) 373.
rpn
2
(rkn  tn )
capillary evaporation

n 1
2
 tn  
j 1
rpj  t j
rpj
Apj
rp
For each desorption step the average diameter of the pore, which
undergoes capillary evaporation is calculated from the Kelvin
equation and the t-plot equation: rp = rk + t
log


© IFP
2V p
correction term
p
13.99
log
 0.034  2
p0
t

Ap 
p  4.14

p0
rk
t is the change in layer thickness in each desorption step
Vn is the volume desorbed in each step
A plot of pore volume vs. pore radius is obtained.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Limits of the BJH-model

BJH underestimates the pore size below 7.5 nm –
WHY?



Model separates the adsorbed film and the capillary
condensate – not a realistic picture
Fluid-wall interactions are neglected
Kelvin equation may not be valid in very narrow pores

One should speak about a BJH-value rather than pore
diameter.
© IFP

Surface tension might increase with curvature
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Adsorption vs. desorption isotherm

Adsorption isotherm – Pros and cons



Desorption branch – Pros and cons


© IFP


Condensation can be delayed – not in thermodynamic
equilibrium
Cylindrical meniscus not stable – not advisable to use
adsorption branch for cylindrical pores
Evaporation not delayed, in thermodynamic equilibrium –
generally preferred
Ink-bottle type pores: smallest openening determines the
desorption
In a network of interconnected pores: percolation (transport)
effects determine the desorption
Conclusion: look at both and compare them
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Ink-bottle pores
rdes


In desorption smallest pore openening is determining.
When pressure of capillary evaporation is reached for
the smallest pore opening, whole pore is suddenly
emptied.
Sudden drop in the desorption isotherm – type H2
© IFP

rdes
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Pore network effects
N2 can only desorb from
pore A and C if pore B has
already been emptied.
Seaton, CES, 1991.


Desorption branch of isotherm not in thermodynamic
equilibrium any more, due to pore blocking.
Desorption is controlled by a percolation process.
probability that the pore is connected to the outer surface.
© IFP

Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Ink-bottle type hysteresis loop
Type H2
© IFP
Adsorbent:
xerogel and
alcogel
• If pore size distribution is calculated from desorption branch, an
artificially narrow pore size distribution is obtained
• The adsorption branch has to be used to calculate the pore size
distribution.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Ink-bottle type hysteresis loop
Isotherm
Pore size distribution
1000
900
6
Desorption
700
dV/dD
3
Vol ads (cm /g)
800
600
3
500
Adsorption
0
400
300
0
200
2
4
6
8
Pore size (nm)
100
0.0
0.2
0.4
0.6
0.8
1.0
p/p0
© IFP

Narrow peak in the pore size distribution of the adsorption branch
is an artefact caused by the forced closure of the isotherm at p/p 0
= 0.43
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Isotherm without hysteresis loop
700
3
Vol ads (cm /g)
600
500
400
300
200
100
0.0
0.2
0.4
0.6
0.8
1.0
© IFP
p/p0
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Closure point of the isotherm






© IFP

Critical temperature inside a pore lower than in bulk:
Tc,pore < Tc, bulk
Tc,pore decreases with decreasing pore diameter1
Above Tc no condensate-vapour meniscus
For narrow pores Tc < 77 K
No hysteresis for filling and emptying of these pores
Tc,pore < 77 K for pores, which show capillary
condensation at p/p0 = 0.4  closure point of isotherm
Tc(Ar) > Tc(N2)  Ar isotherms show hysteresis when
N2 isotherms don‘t2
[1] R. Evans, J. Phys. Condensed Matter 2 (1990) 8989.
[2] M. Thommes, R. Köhn, M. Fröba, J.Phys.Chem. B 104 (2000) 7932.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Summary – Pore size analysis

BJH relies on Kelvin equation and the universal tcurve for determining pore size distribution


both concepts have weaknesses
Alternative methods exist



Density Functional Theory
Broekhoff de Boer – improvement of BJH
Derjaguin – concept of disjoining pressure

Every model assumes a certain pore geometry
(cylindrical, spherical, slit-shaped) – influences the
results!!
 The adsorption and the desorption branch contain
Characterization
of porous solids
- Characterization of Catalysts
and Surfaces
- G Pirngruber 25 october 2016
different
information
– look
at both.

© IFP
surface tension concept that takes interaction with solid into
account
Literature

Textbooks on adsorption



Review articles discussing a more moderne view of
hysteresis effects and pore size analysis


© IFP
D.A. Ruthven, Principles of Adsorption and Adsorption
Processes, Wiley
D.D. Do, Adsorption Analysis: Equilibria and Kinetics,
Imperial College Press
P.A. Monson, Understanding adsorption/desorption
hysteresis for fluids in mesoporous materials using simple
molecular models and classical density functional theory,
Microporous Mesoporous Materials 160 (2012) 47.
B. Coasne et al., Adsorption, intrusion and freezing in porous
silica: the view from the nanoscale, Chem. Soc. Rev. 42
(2013) 4141.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
CO2 maîtrisé | Carburants diversifiés | Véhicules économes | Raffinage propre | Réserves prolongées
© IFP
Alternative explanations of the
Kelvin equation
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Physical principles of capillary
condensation

Vapour pressure of a liquid under external pressure
External pressure

H2O
H2O
H2O
g
H2O
© IFP
H2O
l
pg = new vapour pressure of H2O
pg0 = orignial vapour pressure of H2O
pl = external pressure on liquid (H2O),
which induces the change in vapour
pressure
vl = molar volume of liquid (H2O)

Equlibrium gas-liquid g = l
vapour pressure pg,0
Equilibrium disturbed by external
pressure
dg = dl  vl dpl = vg dpg
vl dpl = RT/pg * dpg
vlpl = RT ln (pg/pg0)
vl
pg  pg ,0  exp(
pl )
RT
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Vapour pressure of a drop of liquid
pin  pout
2

r
pout
 = surface tension
r = radius of curvature
© IFP
pl  pout
2

r
2
pl  p 
r
pin - pout
overpressure
inside the drop
holds against
pin
surface tension
tries to contract
the drop
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Vapour pressure of a liquid void
pin  pout
2

r
2
pin  pl 
r
© IFP
2
pl  p 
r
2vl
pg  pg ,0  exp(
)
RT  r
pout
pin - pout
overpressure
in gas bubble
holds against
pin
surface tension
tries to collapse
the void
Kelvin equation: vapour pressure
inside a void is lowered
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016
Kelvin equation

Capillary forces in the pore lower the vapor pressure
of the condensed liquid
cap  0  RT ln
pcap
p
liq  0  RT ln sat
p0
p0
Evaporate n moles from capillary
and condense on a flat surface.
You have to work against the
surface tension γ.
n
© IFP
Δl
The work done to overcome the surface
tension is equal to change in chemical
potential.
θ
capillary
psat
V
n   
RT ln
Vm
pcap
unconfined liquid
Explains why vapor pressure of the
unconfined liquid is higher than in the
capillary.
Characterization of porous solids - Characterization of Catalysts and Surfaces - G Pirngruber 25 october 2016