porous

1. Dept. Inorganic Chemistry, Fritz Haber Institute
Characterization of porous solids and powders:
Density, surface area and pore size
Christian Hess
0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

2. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

3. Catalysis: Nanostructured materials as support
• High(er) specific surface area of nanostructured materials
’ increased activity
’ formation of isolated, catalytic sites
…an estimate based on a material with spherical particles
using (1) S = 4 P r2 / m
(2) m = 4/3 r P r3
with r ~ 3 g/cm3 (typ. oxide value)
shows that for r = 1 cm _ S = 10-2 m2/g
r = 1 mm _ S = 1 m2/g
r = 1 nm _ S = 1000 m2/g
_ S = 3/rr
_ Creation of high surface requires nanoscale structuring

4. Novel mesoporous silica molecular sieve: SBA-15
TEM characterization
100 nm
in a section of hexagonal pores
along a longitudinal section
_ Silica SBA-15 is a very ordered large-pore (7 nm) material

5. Model catalysts based on mesoporous SBA-15
• Support properties are well defined (structural homogeneity)
and tunable (pore diameter)
’ control over support properties
’ well-structured silica platform for vanadia catalysts
_ 3-D model catalyst for partial oxidation reactions
50 nm
20 nm VO x
/SBA-15
silica
OH OH
O
O
V
O
O
O
H2O
silica
O
O
V
O
O

6. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

7. Real surfaces - Factors affecting surface area
• Idealization: cube → S = 6 l2
sphere → S = 4 π r2
Reality: Surface roughness due to voids, pores, steps, other imperfections
and surface atomic and molecular orbitals
’ real surface area > corresponding theoretical area
• Particle size: e.g. cube l = 1 m → S = 6 m2
l = 1 μm → S = 6 10-12 m2
N = 1018 → S = 6 106 m2
• Particle shape: e.g. two particles with same composition and mass
Mcube = Msphere
(Vδ)cube = (Vδ)sphere
l3
cube = 4/3 π r3
sphere
(Scube lcube)/6 = Ssphere rsphere/3
Scube/Ssphere = 2 rsphere / lcube

8. Factors affecting surface area
• Porosity: can overwhelm size and shape factors
e.g. powder consisting of spherical particles
St = 4 π (r1
2 N1 + r2
2 N2 + … + ri
2 Ni) = 4 π Σ ri
2 Ni
V = 4/3 π (r1
3 N1 + r2
3 N2 + … + ri
3 Ni) = 4/3 π Σ ri
3 Ni
S = St/M = 3 Σ ri
2 Ni/δ Σ ri
3 Ni
S = 3/δ r
for spheres of uniform radius:
with r ~ 3 g/cm3 (typ. oxide value):
r = 1 cm → S = 10-2 m2/g
r = 1 mm → S = 1 m2/g
r = 1 nm → S = 1000 m2/g

9. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

10. Introduction
• most popular method for surface and pore size characterization (0.35-100nm)
other methods include: - small angle X-ray scattering (SAXS)
- neutron scattering (SANS)
- electron microscopy (scanning and transmission)
- NMR methods
- mercury porosimetry
• adsorption: enrichment of components in interfacial layer
gas adsorption → gas/solid interface
W = f (P,T,E)
W: weight of gas adsorbed (per unit weigth adsorbent)
E: interaction potential between adsorbate and adsorbent
T = const. → W = f (P,E)
plot of W vs. P referred to as sorption isotherm

11. Physical adsorption forces
• Physical adsorption forces
van der Waals’ forces: dispersion forces, ion-dipole, ion-induced dipole
dipole-dipole, quadrupole
• physisorption on a planar surface:
London-van der Waals’ interaction energy:
C1, C2: constants
Us(z) = C1z-9 – C2z-3
Thermodynamics: ΔG = ΔH - TΔS
→ ΔH always negative for adsorption
→ at low T: δg << δads ’ nσ ~ nads
(e.g. N2 adsorption at 77K)
→ surface excess: nσ = ng - δgVg
ng = nads + nbulk
Vg = Vads + Vbulk
nσ = nads - δgVads = (δads- δg)Vads

12. Experimental
• Experimental
measure nads vs P under static or quasi-equilibrium conditions using
- volumetric (manometric) methods
- gravimetric methods
Gravimetric method:
based on sensitive microbalance and pressure gauge
→ adsorbed amount measured directly (sensitivity ~ 10-8 g)
→ adsorbent not in direct contact with thermostat (→ T control difficult!)
Volumetric method:
based on calibrated volumes
and pressure measurements
→ known amounts of gas are
admitted stepwise into cell
→ Vads difference between gas
admitted and gas filling void volume
→ requires manifold (Vm) and void
volume (Vv) to be accurately known

13. Experimental
→ Vm determination via expansion of He into
calibrated reference volume VR
→ Vv determination via expansion of He into
sample cell volume occupied by adsorbent
m
R
m
2
m
m
1 )
(
T
V
V
P
T
V
P +
=
m
v
m
4
m
m
3 )
(
T
V
V
P
T
V
P +
=
→ total volume of adsorptive dosed:
P
T
T
PV
T
V
P
V ´
÷
÷
ø
ö
ç
ç
è
æ
-
=
me
m
m
m
m
d
v
d
S V
V
V -
=
→ adsorbed volume:
)
( v
v
d
S a
pV
V
V
V +
-
=
→ correction for non-ideality of real gases:
α: non-ideality factor
e.g. N2: α = 6.6×10-5 torr-1(77K)

14. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

15. Introduction
→ contribution by pores/internal voids must be subtracted
• Density measurements
e.g. for permeametry
volume specific surface area
true density =
mass
volume occupied by mass
No porosity: true density measured by displacement of fluid
→ accuracy: fluid volume determination
Porosity: fluid displaced by powder does not penetrate (all) pores
→ apparent density < true density
→ true density via gas displacement (pycnometer)
using e.g. He (inert, small size)

16. Experimental
• sample cell:
a
a
p
c
a RT
n
V
V
P =
- )
(
a
p
c RT
n
V
V
P 1
1 )
( =
-
• open valve 2:
a
R
a
R
p
c RT
n
RT
n
V
V
V
P +
=
+
- 1
2 )
(
R
aV
P
(2) in (3): R
a
p
c
R
p
c V
P
V
V
P
V
V
V
P +
-
=
+
- )
(
)
( 1
2
(4)
(1)
(2)
(3)
)
/
(
1
)
/
(
1
2
1
2
P
P
P
P
V
V
V a
R
c
p
-
-
+
=

17. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

18. Pore size and adsorption potential
nearly flat
surface
• classification: macropore mesopore micropore
d > 50nm 2nm < d < 50nm d < 2nm
fluid-wall and
fluid-fluid i.a.
(→ capillary
condensation)
fluid-wall i.a.,
overlapping
ads. potential
internal pore width

19. Classification of adsorption isotherms
• classification → IUPAC 1985
• type I: ads. limited to a few layers
(chemisorption, micropores)
• type II: unrestricted mono/multilayer
ads. → at point B monolayer
(non-porous, macropores)
• type III: unrestricted multilayer ads.
• type IV: monolayer/multilayer ads.,
limited uptake, hysterisis due
to pore condens. (mesopores)
• type V: multilayer ads. (→ type III),
pore condensation (hysterisis)
• type VI: stepwise multilayer ads. on a
uniform, non-porous surface

20. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

21. Langmuir isotherm
• Langmuir isotherm: → applicable to chemisorption and type I physisorption
- identical adsorption sites
- no interactions between adsorbed molecules
ΔHads ≠ f (θ)
)
1
(
a
a q
-
= P
k
v
q
a
a -
- = k
v
adsorption of A:
desorption of A:
equilibrium: q
q a
a )
1
( -
=
- k
P
k
KP
KP
+
=
1
q
a
a
-
=
k
k
K
m
V
V
=
q
substitute/rearrange:
KP
V
V
V m
m
1
1
1
+
= plot 1/V vs 1/P → straight line:
intercept 1/Vm → nads
x
A
ads
t A
N
n
S = Ax: cross-sectional area
surface area:

22. Brunauer, Emmett and Teller (BET) theory
• BET isotherm:
- first layer serves as substrate for further adsorption (physisorption)
- almost universally used → accommodates isotherm types I-V
)
/
(
1
1
]
1
)
/
[(
1
0
m
m
0
P
P
C
V
C
C
V
P
P
V
-
+
=
-
→ derivation BET
→
Vm → nads
x
A
ads
t A
N
n
S =
surface area:
BET isotherm:
W
M
n =
ads
C: BET constant

23. Single point BET method
plot 1/W[(P0/P)-1] vs P/P0
→ usually straight line
within 0.05 < P/P0 < 0.35
)
/
(
1
1
]
1
)
/
[(
1
0
m
m
0
P
P
C
W
C
C
W
P
P
W
-
+
=
-
intercept:
slope:
C
W
b
m
1
=
C
W
C
m
m
1
-
=
,
1
m
m
b
W
+
= 1
+
=
b
m
C
• Single point BET method: → simplicity and speed with little loss of accuracy
1
-
= C
b
m
for large values of C → m >> b
→ b ≈ 0
)
/
(
1
]
1
)
/
[(
1
0
m
0
P
P
C
W
C
P
P
W
-
=
-
→

24. Comparison of single point and multipoint methods
0
0
mp
m
sp
m
mp
m
/
)
1
(
1
/
1
)
(
)
(
)
(
P
P
C
P
P
W
W
W
-
+
-
=
- mp: multipoint
sp: single point
→
BET isotherm: W = Wm
1
1
m
0 -
-
=
÷
÷
ø
ö
ç
ç
è
æ
C
C
P
P
→
m
0
mp
m
sp
m
mp
m
1
1
)
(
)
(
)
(
÷
÷
ø
ö
ç
ç
è
æ
=
-
-
=
-
P
P
C
C
W
W
W

25. Applicabilitiy of the BET theory
- accommodates isotherm types I-V
- good agreement between theory and exp.
within 0.05 < P/P0 < 0.30
why?
- porosity: micropores/mesopores ~2-4nm
pore filling ↔ mono/multilayer
- variation from linearity at P/P0 > 0.30
→ ΔHads varies throughout the layers

26. Cross-sectional area
x
a
ads
t A
N
n
S =
surface area: Ax: cross-sectional area (temperature, ads.-ads. i.a.,
ads.-surface i.a., texture)
• approximation of Ax of adsorbate molecules
→ spherical, exhibiting bulk liquid properties
3
/
2
A
x 091
.
1 ÷
÷
ø
ö
ç
ç
è
æ
´
=
N
V
A
• surfaces of high polarity (such as silica) may show significant deviation…
→ N2 is standard BET adsorptive
e.g. N2 (77.35 K): Ax = 13.5 Å2

27. Permeametry
known amount of air is drawn through compacted bed of powder
→ resistance to air flow is a function of the surface area
• ambient p and T (λ < d): gas collisions → viscous flow
→ ignores blind pores
→ `envelope` area of particle
• reduced p (λ ≈ d): → no flow retardation near wall
• small p (λ > d): gas-wall collisions (→ diffusion)
→ senses blind pores and gives
good agreement with BET
÷
÷
ø
ö
ç
ç
è
æ
ú
û
ù
ê
ë
é
-
÷
ø
ö
ç
è
æ D
=
v
f
p
p
l
P
S
h
d 2
2
3
2 1
)
1
(
viscous flow through cylindrical pores → Poiseuille
p: porosity, v: velocity
f = 2(l/lp)2
aspect ratio
→ aspect ratio highly dependent upon particle size and shape (typically f = 3-6)

28. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.

29. Adsorption in mesopores - multilayer adsorption
depends on fluid-wall and attractive fluid-fluid interactions
→ multilayer adsorption and pore condensation
)
/
ln( 0
0 P
P
RT
a
a -
=
-
=
D m
m
m
• Adsorption in mesopores:
• Multilayer adsorption:
- novel ordered mesoporous materials (e.g. MCM, SBA)
- Microscopic methods (e.g. NLDFT)
Fluid in contact with planar surface
→ thickness l: l → ∞ for P/P0 → 1
m
-
´
-
= l
a
(m typ. 2.5-2.7)
α: fluid-wall interaction parameter
μa: chem. potential of adsorbed liquid-like film
μ0: chem. potential of bulk fluid at gas-liquid coexistence

30. Pore condensation – modified Kelvin equation
• in pore l can not grow unlimited
c
a m
m
m D
+
D
=
D
for small l:
m
-
´
-
=
D
»
D l
a a
m
m
for large l: r
q
g
m
m D
-
=
D
»
D a
c /
cos
2
a: core radius
(a = rp – l; rp: pore radius)
γ: surface tension
Δρ = ρl - ρg
• at critical thickness lc:
pore condensation (i.e. gas-like state
→ liquid-like state at μ < μ0)
Kelvin equation:
aRT
V
P
P
g
2
)
/
ln( 0
-
=
for complete
wetting, i.e. θ = 0

31. Methodes based on modified Kelvin equation
→
- from isotherm: Vads/g catalyst as function of P/P0
t
)
/
( m
a W
W
l =
• Calculation of the pore size distribution:
- Kelvin equ.: calulate a for θ=0 as function of P/P0
- calculate statistical film thickness l: l
a
r -
=
p
- calculate a and rp in each decrement from successive entries
- calculate Δl, the change in film thickness in successive steps
- calculate ΔVads, the change ads. volume in successive steps
- calculate ΔVliq, corresponding to ΔVads
- calculate ΔlΣS, the volume change of film remaining adsorbed
- calculate the pore volume Vp:
- calculate the surface area S:
p
p
2
r
V
S =
p
2
p
p l
r
V ×
×
= p
[ ]
)
(
liq
2
p
p S
l
V
a
r
V S
D
-
D
÷
÷
ø
ö
ç
ç
è
æ
=
S
l
l
a
V S
D
+
×
×
=
D p
liq p

32. Testing the modified Kelvin equation
• direct exp. test of Kelvin eq. (MCM, SBA)
and comparison with XRD, TEM results
→ Kelvin eq. underestimates pore size
• accurate pore size analysis by applying
microscopic models based on stat. mech.
→ local fluid structure near curved wall
→ capture properties of confined fluid

33. Adsorption vs desorption branch of hysteretic isotherm
• ordered mesoporous materials
→ desorption branch of hysterisis loop reflects equilibrium phase transition
• disporered mesoporous materials
→ desorption branch not necessarily correlated with the pore size
e.g. tensile strength effect → artifact!

34. 0. Motivation
1. Introduction
2. Gas adsorption
3. Density measurements
4. Adsorption isotherms
5. Surface area determination
6. Mesopore analysis
Literature: Characterization of porous solids and powders,
Lowell, Shields, Thomas, Thommes, Kluwer, Dordrecht, 2004.