European Master
Adsorption
Modeling of physisorption in porous materials
Part 2
Bogdan Kuchta
Laboratoire MADIREL
Université Aix-Marseille
Typical hysteresis of adsorption-desorption cycle
H1
H2
H3
H4
n / ms
p / p0
Hysteresis loops H1 and H2, are characteristic for isotherms of type IV
(nanoporous materials). Loop of hysteresis H1 shows nearly vertical and
parallel branches of the loop : it indicates a very narrow distribution of
pore sizes. Loop of hysteresis H2 is observed if there are many
interconnections between the pores.
Typical hysteresis of adsorption-desorption cycle
H1
H2
H3
H4
n / ms
p / p0
Loops of hysteresis H3 et H4, appear on isotherms of type II where
there is no saturation. They are not always reproducible. Loop of
hysteresis H3, is observed in porous materials formed from agregats,
where the capillary condensation happens in a non-rigid framework
and porosity not definitly defined. Loop of hysteresis H4 are often
observed in structures built from planes that are not rigidly
n
p/p0
n
p/p0
n
p/p0
n
p/p0
n
p/p0
Theories of adsorption
Frundlich model
Langmuir model
BET
Theory of adsorption by Freundlich:
x – adsorbed mass
x/m =  c1/n
m – mass of adsorbent
c – concentration
, n – experimental constants
x/m
lg(x/m)
c
Conclusion: adsorption is better at
higher pressure
lg(c)
Langmuir theory
bp


n  nm
1  bp

n
 
nm
- 1 one type of » adsorption sites"
1.2
- No lateral interactions
1
- 1 site of adsorption allows 1 particle to
be there:
adsorption is limited to one layer
0.8
0.6
N s = number of adsorption sites
N a = number of adsorbed molecules
0.4
 = fraction of the surface covered
0.2
0
Pression
Langmuir theory
 Isotherm of Chemisorption
 at low pressure bp << 1, so
 Henry’s law

n  n bp
 at high pressure, bp >> 1, si


n n

m

m
bp
n n
1  bp

b  K exp

m
E / RT
Langmuir isotherm :
influence of the coefficient ‘b’
1
n/ n m
0.8
0.6
b=0.01
b=0.01
b
= 0.1
b=0.01
b = 0.1
b=1
0.1
b=1
b = 10
0.4
0.2
0
0
bp
n n
1  bp


m
100
200
300
Pression
400
500
Variations on Langmuir and Henry
Henry
n  kH p
Freundlich
n  k H p1/ m
Langmuir
Sips (Langmuir-Freundlich)
Toth
Jensen & Seaton
n
bp

nL 1  bp
n
(k H p)1/ m

nL 1  (k H p)1/ m
n
p

nL (b  p m )1/ m
  k p 
H

n  k H p 1  
  a(1  p) 
m 1/ m



Variations of Langmuir and Henry
n
p

nL (b  p m )1/ m
1
n
bp

nL 1  bp
0.8
n  k H p1 / m
n / nm
n
(k H p )1/ m

nL 1  (k H p )1/ m
0.6
  k p 
H

n  k H p 1  
  a (1  p ) 
0.4
0.2
Henry
Freundlich
Langmuir
Toth
Sips
Seaton
100
200
300
Pression
400



n  kH p
0
0
m
500
1 / m
Methode « BET »
Théorie de Brunauer Emmett et
Teller (BET)
•
Hypothèses
–
- 1 one type of » adsorption sites"
- No lateral interactions
}
E1=energy of adsorption of
the first layer
– Starting from the second layer E1EL energy of
molecules in liquid state
Basic hypothesis of the BET theory

EL
1
B
E1
Energy of adsorption Relative pressure p/p°
E1 = Energy of adsorption for the first layer
El = Energy of liquid state
Basic hypothesis of the BET theory
so
s1
s2
s3
A
surface so covered with 0 adsorbed layers
... s1
...
1
...
...
...
... si
...
i
Accessible surface A = so + s1 + … + si + ...
For s0
Derivation of the BET formula
Rate of condensation of an empty
surface
=
for s1
Rate of evaporation from the surface
covered with two layers
=
Rate of evaporation from a surface
covered with one layer
Rate of condensation on the surface
covered with one layer
General, in the case of si
Rate of condensation on a surface
=
covered with i layers
Rate of evaporation from a surface
covered with i+1 layers
ki si-1 p = k-i si
Derivation of the BET formula
ki si-1 p = k-i si
Total surface of adsorbent A 

s
i

~ nm
i 0
n

nm
Total quantity of adsorbed gas
 is
i
i 0

s
i
i 0
Asuming, that the layer properties are all the same
k 
p
si   i  si 1 p  xi si 1 ou xi 
gCi (T )
 k i 
si  xi si 1
k i
 gCi (T ) ; i  1
ki
C1(T)=exp(-E1/kT)
Ci(T)=exp(-EL/kT)
p
 EL 
 E1  E L 
 Cx s0 ; x  exp
 ; C  exp

g
kT
kT




i
Derivation of the BET formula

n

nm
 is
C
i
i 0


i 0
si





ix i
i 1
i 1

1 C

ixi 


xi
i 1
xi  x
i 1
n
Cx

nm (1  x)(1  x  Cx )
À p° :
Cx

(1  x)(1  x  Cx)
donc :
x
p
p0
x
(1  x) 2
1
1 x
Theory of Brunauer Emmett and Teller
(BET)
•
Equations
– N= number of layers
 E1  E L 
C  exp 

 RT 
n
Cx 1  ( N  1) x N  Nx N 1


nm 1  x 1  (C  1) x  Cx N 1
x = p/p0 = relative equilibrium
pressure
– if N  
n
Cx

nm (1  x)[1  x(C  1)]
– Transformed equation BET
x
1 C 1


x
n(1  x) nmC nmC
Influence of number of layers N on the shape of
isotherms of adsorption (BET)
N=7
4
N = 25 à 
N=6
N=5
3
N=4

2
n
Cx 1  ( N  1) x N  Nx N 1


nm 1  x 1  (C  1) x  Cx N 1
1
p / po
0
0
0.2
0.4
0.6
0.8
1
Influence of the constant ‘C’ on the shape of
isotherms of adsorption (BET)
5
4.5
4
C = 0.1
C = 0.1
CC==10.1
C
C == 10.1
CC==10
1
C
1
C == 10
CC==100
10
C = 100
C = 1000
3
2.5
a
n /n
a
m
3.5
2
1.5
1
0.5
0
0
0.2
0.6
0.4
p / p°
0.8
1
n
Cx

nm (1  x)[1  x(C  1)]
Application for calculation of the adsorption surface
example : alumin NPL / N2 / 77 K
350.08
200
35
y = 89.95x + 1.31
y = 93.59x + 0.94
x
1 C 1
 a  a x
a
n (1  x) nmC nmC
a
a.(1-x) -1
xxa///nnmmol
n
.(1-x)g
180
0.07
3030
160
0.06
2525
140
0.05
120
2020
0.04
100
1515
80
0.03
60
10
100.02
40
5
50.01
20
0
000
000
0
C 1
nmC
1
nma C
Pente
: + 1.16
y = 91.52x
a
Ordonnée :
0.2 0.1
0.05 0.2
0.05
0.1
0.4
0.4
0.15
0.15
0.20.6
0.6
0.2
0
p
/
p°
p°
p
/
p
p / p°
0.25 0.8
0.80.3
0.25
0.3
1
n 
 0.011
PO
P
C   1  70
O
a
m
11
0.35
0.35
Verifications of BET results
example : alumin NPL / N2 / 77 K
nma  0.011
0.016
0.014
-1
a
n / mmol g
C  70
nma
0.012
0.01
1
( p / p) n a 
 0.107
m
C 1
0.008
0.006
0.004
0.002
0
0
0.05
0.1
0.15
0.2
p / p0
0.25
0.3
0.35
Lateral interactions
Normal interactions
Simulation of adsorption
1. Calculation of energy of adsorption
2. Simulation of isothermes (with different strength of interaction)
3. Analyse the results
1. Simulation Monte Carlo grand canonique (GCMC)
2. Tool: program GCMC (Fortran)
Numerical challenge:
1. Simulations of equilibrium between gas
and adsorbed phase
2. Modeling of interaction between pore
walls and adsorbed particles
Working case: MC simulation of adsorption in a pore
Problem: Fluid adsorption in cylindrical pores.
Grand Canonical Monte Carlo
VT- constant
(gas) = (adsorbate)
(gas, ideal) = 0(gas) + kBT ln(P)
VT
PVT - constant
External ideal
gas pressure P
Working case: MC simulation of adsorption in a pore
4,0
P1 and T fixed
3,5
Density
3,0
R (radius)
2,5
2,0
1,5
1,0
0,5
0,0
20
6
10
5
0
P2 and T fixed
5
R (radius)
4
Density
15
3
2
1
0
20
15
10
5
0
Working case: MC simulation of adsorption in a pore
1400
1200
T = const
1000
<N>
800
600
400
p
0.05
0.1
0.2
0.3
0.4
0.5
0.7
0.8
0.9
<N>
234.7
362.8
385.8
401.9
421.2
448.3
558.3
691.6
1259.1

16.2
7.6
5.8
6.9
9.7
13.2
31.6
26.7
8.3
200
0
0.0
0.2
0.4
0.6
Pressure
0.8
1.0
Directory Run
program (compiled)
input files
gcmc_H2.dat
gcmc_H2_par.dat
pos_inp.dat
spline*
execute
Results files
ene.ini
ene.fin
mc.pos
mc_ene.dat
mc_ent.dat
pos_inp.res
- initial molecular energies
- final molecular energies
- molecular position after each bin
- energies after each bin (wall and total)
- energies
-
analysis of results
Rename :
pos_inp.res
OK
STOP
NO
 pos_inp.dat
mc.pos
20
15
N
y
1
154
10.886520 -14.887360
10.898990 14.983010
14.028510 11.913710
-14.459990 11.605350
1.908520 18.285780
1.256716 -18.238170
13.606980 -12.499060
-15.536920 -9.600965
-15.764630 -9.760927
-4.318254 -17.841070
15.933630 -9.115001
………….
……..
10
z
9.244424
21.000650
2.990251
.722188
22.916110
4.253842
7.607756
16.492660
24.275380
23.939530
17.567660
Nbin = 1
N = 154
Y Axis Title
Nbin
x
5
0
-5
-10
-15
-20
-20 -15 -10 -5
0
5
10 15 20
X Axis Title
20
15
10
Y Axis Title
20
15
Nbin = 2000
N = 615
Y Axis Title
10
5
5
-5
-10
0
-15
-5
-20
-10
Nbin = 1000
N = 258
0
-20 -15 -10 -5
0
5
X Axis Title
-15
-20
-20 -15 -10 -5
0
5
X Axis Title
10 15 20
10 15 20
Equilibrium situation
-200
1400
-400
1200
T = const
-600
Energy
1000
p
0.05
0.1
0.2
0.3
0.4
0.5
0.7
0.8
0.9
<N>
234.7
362.8
385.8
401.9
421.2
448.3
558.3
691.6
1259.1

16.2
7.6
5.8
6.9
9.7
13.2
31.6
26.7
8.3
600
-800
-1000
400
-1200
200
-1400
0
0.0
0.2
0.4
0.6
0.8
1.0
-1600
0
2000
Pressure
4000
6000
8000
10000
6
MC steps/10
500
Mean values
Variation
-337.7
-1160.2
-1497.9
13.5
15.3
10.2
400
300
N
<N>
800
200
100
234.7
16.2
0
0
2000
4000
6000
8000
6
MC steps/10
10000
Experimental results of adsorption
Milestones results
1. Isotherms
2. Energy of adsorption
3. Hysteresis properties
Approach thermodynamic – energie of adsorption
 = g
u +pv -Ts = ug +pvg - Tsg
u-Ts = ug +RT - Tsg (v = 0, pvg =RT)
sg=sg,0 – R ln(p/p0)
adsh = u- ug - RT  const.
adss0 = s - sg,0  const.
 ads h
 

ln p   

RT 2
 T

Adsorption is a
phenomenon
exothermic !!!

ln


ln

p  u σ  u g  RT s σ  s g,0


0 
RT
R
p 
p   ads h  ads s 0


0 
RT
R
p 







ln
p

  ads h  R
1 
   
 T  
RT1T2  p2 

 ads h  
ln
T2  T1  p1 
Approach thermodynamic – energie of adsorption
n
ms
T1
T2 > T1
 Isosteric enthalpy
 p2 
T1.T2

 adsh   R.
. ln  
T2  T1  p1  n / m s
p1
p2
p
Basic types of adsorption energy curves
Curves 3 and 4 correspond delocalized and localized
adsorption on a homogeneous surface, with lateral
interactions between molecules.
adsh
4
Curve 2 appears in
homogeneous systems with
no lateral interaction.
3
2
5
1
Curve 5 shows an existence of well
defined fomains.
Curve 1 is characteristic for
heterogeneous surfaces.
p/p0
Example:
mesoporous system: MCM-41 et 77K
CO & CH4
20
CO
 Typical for
heterogeneous surface
16
-adsh / kJ.mol
-1
Kr
  ads
. h (2 kJ.mol-1)
during the capillary
condensation
12
Kr
CH4
  adsh (5 kJ.mol-1)
. the capillary
during
condensation
solidification ?
8
4
0
0.2
0.4
0.6
na/nap/p°=0.4
0.8
1
Milestone properties
 Capillary condensation is accompnied with histeresis of variable form
40
Ar
35
900
30
vol.adsorbé /cm .g (STP)
800
na / mmol.g -1
60
100
600
3
-1
700
500
400
300
200
25
20
N2
15
10
100
5
0
0
0,2
0,4
0,6
p/p0
0,8
1
0
0
0.2
0.4
0.6
0
lichrospher
p/p
CPG
0.8
1
Milestone properties
 Hysteresis disappears at some
high temperature
Argon / MCM41
(Morishige et al)
Milestone properties
 For each temperature, there is a size of pore (and/or equilibrium
pressure), that the hysteresis disappears below this value.
Argon
Nitrogen
1.2
1
1
0.8
0.8
a
0.9
n /n
0.4
2.5 nm
4.0 nm
4.6 nm
0.6
a
2.5 nm
4.0 nm
4.6 nm
0.6
a
n /n
a
0.9
1.2
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
p / p°
0.8
1
0
0.2
0.4
Llewellyn et al., Micro. Mater. 3 (1994) 345.
0.6
p / p°
0.8
1
Adsorption - Desorption Isotherms :
 MCM41 à 77K
20
Ar
N2
CO
10
a
n / mmol.g
-1
15
5
0
0
0.2
0.4
p / p0
Llewellyn et al., Surf. Sci., 352 (1996) 468.
0.6
0.8
1
Nitrogen / black of de carbon (Carbopack)
0.2
0.1
a
n / mmol.g
-1
0.15
0.05
0
0.00001
0.0001
0.001
0.01
0.1
1
p / p°
M. Kruk, Z. Li, M. Jaroniec, W. B. Betz, Langmuir 15 (1999) 1435-1441.
Adsorption on precipitated silica
Isotherms : N2 & Ar à 77K
P. J. M. Carrott & K. S. W. Sing, Ads. Sci. Tech., 1 (1984) 31.
na
/ mmol
g-1
7
300°C
6
200°C
110°C
25°C
5
4
3
2
1
p / p0
0
0
•
0.2
0.4
0.6
0.8
1
The conditions of the sample preparation are very important!!!!