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Structural characterization of porous carbonaceous materials using high-pressure adsorption measurements
Studies in Surface Science and Catalysis 160
113
P.L. Llewellyn, F. Rodriquez-Reinoso, J. Rouqerol and N. Seaton (Editors)
92007 Elsevier B.V. All rights reserved
Structural characterization of porous carbonaceous materials using high-pressure adsorption measurements Y. Belmabkhout, M. Fr6re (*), G. De Weireld. Facult6 polytechnique de Mons - Thermodynamics department, 31 bd Dolez 7000 M o n s Belgium The aim of this work is to present a new method for porous carbonaceous solids characterization: the pore size distribution function (PSDF) of the adsorbent is determined by a theoretical treatment applied to adsorption data. The measurements are performed with different adsorbates at different temperatures in a wide pressure range. The theoretical model is based on the concept of integral adsorption equation (IAE). The main assumptions of the model are: (1) The PSDF is considered as an intrinsic property of the adsorbent. (2) The slit shaped pore model is used to describe the geometric configuration of the porous structure. (3) The adsorbent-adsorbate interactions are described by a Lennard-Jones potential model. (4) The pore wall surface is considered to be energetically homogeneous and the adsorbed phase is monolayer. (5) Both the gas phase and the adsorbed phase are supposed to be efficiently described by a Redlich-Kwong type equation of state. (6) We assumed a priori the analytical form of the pore size distribution function. We used adsorption isotherms of nitrogen, oxygen, argon and methane on different carbonaceous solids (four activated carbons and one molecular sieve) at 283 K, 303 K and 323 K and for pressures up to 2200 kPa. 1. INTRODUCTION Characterization of heterogeneous micropourous carbonaceous solids from theoretical treatments of adsorption data is a complex issue. The large number of existing characterization methods is an evidence of the constant evolution in the understanding of the way adsorption properties and sorbents structural characteristics are related [ 1-13]. During the last decades, the disposal of efficient calculation means [11] has led to significant improvements in this matter and the major challenges which were clearly identified a few tens years ago are about to be overcome. These challenges may be classified as follows: 9 the analytical form of the pore size distribution function should not be assumed a priori; 9 the geometrical structure of the pores should be well defined; 9 the interaction models (adsorbate-adsorbate and adsorbate-adsorbent) should be accurate; 9 the local adsorption model should be efficient in the whole pore range (micro, meso and macropores); 9 The energetic heterogeneity of the pore walls should be taken into account. All these aspects have been partially or thoroughly treated in previous works so that their relative importance for the assessment of a reliable pore size distribution function can be evaluated [ 1,10]. On such a basis, it should be interesting to develop an efficient model which would be able to simulate adsorption data in a wide range of temperature and pressure
Y. Belmabkhout, M. Frbre and G. De Weireld
114
conditions and for any adsorbate. Such a model should remain as simple as possible so that it could be used for adsorption process simulation. The purpose of this paper is to present such a model which can be regarded as a synthesis of fundamental considerations resulting from the activities in porous sorbents characterization and classical thermodynamics developments. This model is able to represent in a correct way adsorption data of oxygen, argon, nitrogen and methane on a given carbonaceous adsorbent using a unique pore size distribution function whatever the adsorbate. The procedure was applied on four different activated carbons and on a carbon molecular sieve. The adsorption isotherms were measured at 283 K, 303 K and 323 K and for pressures up to 2200 kPa. 2. T H E O R I T I C A L DEVELOPMENT
2.1. General approach Our theoretical approach is based on the Integral Adsorption Equation (IAE) concept. The IAE allows the generalization of statistic models initially developed for homogeneous surfaces to heterogeneous ones. Such surfaces are characterized by an adsorption energy distribution function F(e') for a given adsorbate. Considering porous carbonaceous adsorbents, it is generally accepted that the apparent energetic heterogeneity results from pore size heterogeneity. The concept of Integral Adsorption Equation may then be used for such adsorbents if FOz) may be correlated to the pore size distribution function Fz(H). The basic expression of the Integral Adsorption Equation applied to geometrically heterogeneous adsorbents is: Bs
m(T,P) = [MI'(T,P,H)F~(H)dH
(1)
Bi
In which: 9 m(T, P) is the adsorbed mass at temperature T and pressure P; 9 Mis the molar mass of the adsorbate; 9 1-'(T, P, H) is the molar adsorbate surface concentration on the walls of the micropores of diameter H at temperature T and pressure P; 9 FA(H) is the pore size distribution function defined on a surface basis; 9 Bi and Bs are the integration limits. Using Eq. (1) for the calculation of adsorbed masses requires the knowledge of both/-'(T, P, H) and FA(H). FA(H) may be related to the pore size distribution function defined on a volume basis Fz(H) for a given pore shape. 1-'(T, P, H) is called the local adsorption isotherm model. Considering the pore walls as energetically homogeneous surfaces, classical models derived from statistical thermodynamics developments may be used once the correlation between the adsorption energy c and the pore size H is known. The analytical form of such a correlation depends on the pore shape on the one hand and on the adsorbate-pore wall interaction energy function on the other hand.
2.2. Geometrical and energetic modelling of the porous structure of carbonaceous adsorbents Classical carbonaceous adsorbents (activated carbons and carbon molecular sieves) are composed of amorphous and graphitic carbon. Such materials exhibit a very important microporous structure due to missing graphitic layers in the graphite crystallites. The disordered arrangement of these crystallites is responsible for the mesoporous and macroporous structure. The presence of heteroatoms in the raw materials used for the
115
Structural characterization of porous carbonaceous materials
synthesis of such adsorbents leads to a chemical heterogeneity of the pore wall surface. As far as gas adsorption at supercritical temperatures is concerned, it is generally assumed that the adsorption process occurs mainly in micropores of molecular dimensions of which the walls may be considered as energetically homogeneous [13]. As a consequence, we chose the simplified slit-shaped model to describe the porous structure of our adsorbents. The pore diameter H is defined as the distance between the parallel graphite walls. The interactions between the adsorbate molecules and the pore wall are assumed to be well described by the Lennard-Jones model. The adsorption energy in a pore of diameter H is calculated by summing the contribution of each wall of the pore. The resulting correlation between the adsorption energy and the pore diameter e(H)is used to develop the local model I"(T, P, H) on the basis of classical existing models defined on a energetic basis 1-(T, P, ~). In this work, we set the mathematical form of the pore size distribution function Fv(H); we opted for a bimodal distribution: F v ( H ) = .f~O.inpore ~ exp
2 cr 1Hpores
2
! ,.ores +
~ ( 7
2 Hpores
exp
2 (7 2Hpores
2
!
(2)
In which: 0 V1 pores and V2 pores are the micropores volumes; o mlHpores and m2Hpores are the averages pore diameters; 0 O'lHpores and O'2Hpores are the standard deviations. Given the slit-like shape of the pores, the pore size distribution function defined on a surface basis FA(H) is related the corresponding function defined on a volume basis Fv(H) by Eq. (3): FA(H)_ X Fv(H) (3) H
In which x is equal to 1 or 2 according to the number of monomolecular layer formed within the pore. 2.3. Local adsorption isotherm model I'(T, P, 0 and hence F(T, P, t ~ are obtained by expressing the equality of the chemical potentials of the adsorbate in both the gas and adsorbed phases. The chemical potentials are derived from statistical thermodynamics developments [ 14]. 2.3.1. Gas phase modeling The canonical partition function of a system composed of Ng molecules in a volume Vg at temperature T is given by: 1 u N ~ ~ ( u~.,) (4) Qg = - - qgtrans s qgrot * qgv, b, ~ qge ~ expL) 2-~ Ng !
In which: is the translational contribution to the partition function (3 degrees of freedom of the centre of mass of the molecule); 9q g rot is the rotational contribution to the partition function; 9qg vibi is the internal vibrational contribution to the partition function; 9q g e is the electronic contribution to the partition function;
9q g trans
116
Y. Belmabkhout, M. Frkre and G. De Weireld
9 UiPnt is the potential energy of interaction between any molecule and all the others in the
system; 9 kis the Boltzmann constant. qg tr,,,,smay be expressed by: qg ....
Vg - Ngb A3
(5)
[2xM.,kT] 5
(6)
=
with: X'
=L~J
In which: 9 Ais the thermal de Broglie wavelength; 9 his the Planck constant; 9Mini is the mass of the molecule; 9 bis the volume of the molecule calculated using the Redlich-Kwong approach [ 15]. The molecular chemical potential of the gas phase is obtained by differentiating lnQg (given by Eq. (4) with respect to Ng at constant temperature and volume:
It, = -k< OlnQ" ON, )r.v,
(7/
Using Eq. (4) and Eq. (7) and the Redlich-Kwong approach for the calculation of Ui',t [16], we obtain:
IlnkTq,~o,~_lnP+ln x
lXg=
-k, l N,b +7
a
[ Ve - Ngb L b k r ~
PVg.-In
krN~
Vg
v~--N~b
In(V'+Ngbil [ ~
V,
(8) a
)J +Lv, + Ngb bkrx/-T
In which: a is the molecular parameter of the Redlich-Kwong eos[ 15] for the adsorbate in the gas phase. Eq. (8) may also be written: /.tg =/zg~ + kr lnP + kTlncb' (9) With:
(1 O)
/~0 = -krln kr qg,o,q~b,q,e A3 PVg+In
ln*'=-lnkTNg
Vg
N,b
a[~V,+N,b]+
Vg-N,b+g,-N,b bkT4-T 1
N,b ]
gg ) V +NgbJ
(11)
9 is ' the fugacity coefficient which takes into account the non-ideality of the gas phase.
2.3.2. Adsorbed phase modeling We consider a system of Ns molecules adsorbed on a surface A; the molecules interact with each other. The same kind of developments as the ones presented for the gas phase may be achieved for the adsorbed phase. They lead to the expression of the chemical potential of the adsorbate in the adsorbed phase by using the corresponding two dimensional equation Redlich-Kwong equation of state:
117
Structural characterization of porous carbonaceous materials
In k T q s v i b q . . . . q ~ q ....b, Az
- U----~~- In FI + In ~ 1 _ ~ _ In 1 / F r o / - kT k T Fro1 1/F - b s
(12)
/d s = - k T
b,
+
b, kr 4 Y
In -
-
+
1/r~,
~/r~, + b, In which: * U0 is the energy of adsorption. It can be calculated from the Lennard-Jones expression of the potential energy of interaction of an adsorbate molecule in a pore; 9H i s the spreading pressure; 9Fml is the surface molecular concentration F,, = N, ; A
9a~ and b~ are the Redlich-Kwong molecular parameters of the adsorbate in the adsorbed phase; 9q~ vib is the contribution to the partition function due to the vibration of the molecule perpendicularly to the adsorption surface. qs trans and qs vibmay be calculated respectively by: A-U~b, (13)
q.....
--
A2 - -
and
exp-2kT -
q.v,b =
(14)
1 oxp( )
In which: 9Vz is the vibration frequency of the molecule perpendicularly to the surface. It can be calculated from the Lennard-Jones expression of the potential energy (adorbate-pore). Eq. (12) may also be written: P, = p o + kTlnI-I + k T l n * (15) With: o _ k T I l n k T q S v , oq . . . . q~eq~v,o,I /~s = A2 + Uo
(16)
And In ~ = - l n - - - - + n 1 In 1~Fret + b. k r Fro, 1/rm,-b. l/Fret-b,
a. [ln(1/Fmt+b'/ b. 1 krvC-Tb. L ~. 1)rmZ-) + 1/rmt +bs
(17)
.J
is the fugacity coefficient of the adsorbate in the adsorbed phase; it takes into account of the non-ideality of the adsorbed phase. 2.3.3. Expression of equilibrium The equality of the chemical potentials of the adsorbate in both phases calculated respectively by eq. (8)and Eq. (12) lead to the final expression of the local adsorption isotherm model. The electronic, the internal vibrations and the rotational behaviour of the molecules are supposed to undergo no change when the molecules pass from the gas to the adsorbed phase. Using molar quantities instead of molecular quantities leads to: (
r" i'r'bS.o,1
exp
exp
1-rb..,Vb"~[l+Fbs]-~r~exp
( - NoU o ) p -RT- ~ v exp b•~ RT Pv v-b.o , V - bmo,
In which:
RT#-T
v + b.o I - ~ v
N~
= RT42xMRT-1-~_(-N--~](18) exp
Noa~~ RT ~t-f V + bmo,
118
Y. Belmabkhout, M. FrOre and G. De Weireld
9 vis the molar volume of the adsorbate in the gas phase; it is calculated as a function of temperature and pressure using the Redlich-Kwong equation of state; 9 Ris the ideal gas constant; 9 No is the Avogadro number 9 amol, bmol, asmol, bsmol, [" are the molar values corresponding respectively to a, b, as, bs, Fret. 2.4. Global model The use of Eq. (1), (2), (3) and eq. (18) allows the calculation of the adsorbed mass as a function of temperature and pressure for a given adsorbate and a given carbonaceous adsorbent. The unknown parameters are: 9 asmotand bsmot (for each adsorbate). 9 V1 pores and V2 pores, mlHpores and m2Hpores a n d O'lHpores and O'2Hpores, the parameters of the bimodal pore size distribution function
3. RESULTS AND DISCUSSION The theoretical developments presented in the previous section were applied to experimental adsorption isotherms. We studied four activated carbons (Centaur, BPL, F30/470, WS42) and one molecular sieve (CMS 1). High pressure adsorption isotherm measurements of N2, Ar, CH4, and 02 (at three different temperatures) were carried out with a volumetric device described earlier by Belmabkhout et al [17]. These experimental data have also been published [18]. By minimizing the discrepancies between the experimental and calculated results, it was possible to determine the adjustable unknown parameters of the model. Fig. 1 shows an example of the experimental and theoretical isotherms on Centaur (AC).
Fig. 1. Experimental and theoretical adsorption isotherms of N2, Ar, CH4 and 02 on Centaur For all the studied systems, a perfect fitting is noticed between experiment and theory (discrepancies are within the experimental error on the adsorbed mass: 5 " 1 0 -3 g/g)
119
Structural characterization of porous carbonaceous materials
Tables 1 and 2 present the values of the fitted parameters for all the studied systems. Table 1 Fitted parameters of the pore size distribution function for each adsorbent Sample m lHpores/A OrlHpores//~ Vlpores/ m eHporeJ~ O'2HporeJ~ Vepores/ m3/kg m3/kg Centaur BPL F30/470 WS42 CMS1
6.98 6. 73 6.27 6. 74 6.00
0.49 0.47 0.66 O.5 7 0.87
9. 75E-05 1.74E-04 2.26E-04 1.65E-04 3.13E-04
10.46 10. 77 8.51 9. 62 13.04
2.49 2.35 3.28 1.52 2.53
1.98E-04 1. lOE-04 1.70E-04 1.29E-04 3.32E-09
Vtotalmicr/ "
m3/kg
2.95E-04 2.84E-04 3.97E-04 2.93E-04 3.13E-04 =,
Table 2 Fitted energetic and geometric bidimensional Redlich-Kwong parameters for each adsorbentadsorbate s~,stem (asmol / J m 2 mole -2 , bsmol/m 2 mole -l) Sample asrnolOV2) bsmol (N2) asmol(Ar) bsmot (Ar) asmol(Cff4) bsmoI asmol(02) bsmot (02) .....
Centaur BPL
8.10E+09 9.43E+09
7.88E+04 7.59E+04
9.80E+08 7.15E+08
3.42E+04 3.24E+04
4.18E+09 5.83E+09
(CH4) 4.45E+04 4.27E+04
2.18E+09" 1.29E+09
4.25E+04 3.69E+04
F30/4 70 WS42
8.14E+09 6.52E+09
6. 74E+04 5.76E+04
2.50E+09 9.91E+08
4.64E+04 2.88E+04
4.93E+09 2.96E+09
3.96E+04 3.13E+04
1.09E+09 1.53E+09
3.67E+04 3.32E+04
CMS1
8.66E+09
5.94E+04
2.26E+09
3.87E+04
4.20E+09
2.89E+04
2.23E+09
4.19E+04
In Fig. 2 we plot the derived PSDF representative for each adsorbent. The PSDF of CMS1 seems to reduce to a Gaussian which points out that the range of pore size of CMS 1 is very small. The values of the standard deviations (YlHpore, (Y2Hporeand the volumes Vlpore and V2pore for F30/470 point out that the degree of the structural heterogeneity of this adsorbent is the highest as far as micropores are concerned. Centaur, WS42, and BPL seem to have a relatively intermediate degree of microporous structural heterogeneity. 3.00E+05
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
/
E 2.00E+05
l= !-SOE+0S
l .OOE+05
0.00E+00 0,00E+00
$.OOE+00
!.00E+0t
1.50E+01
2.00E,01
2.50E+01
Diameter I Angstrom
Fig. 2. PSDF of the studied adsorbents.
From Table 2, it appears that the value of the fitted energetic and geometric bidimensional Redlich-Kwong parameters asmol and bsmol seem to be physically realistic; they vary weakly
120
Y. Belmabkhout, M. Frkre and G. De Weireld
from one adsorbent to another. The variation of these bidimensional parameters can be explained by the difference in the surface chemistry for the different adsorbents which is not taken into account in our model. 4. SUMMARY AND CONCLUSIONS In this paper, we have presented and tested a model which allows the calculation of adsorption isotherms for carbonaceous sorbents. The model is largely inspired of the characterization methods based on the Integration Adsorption Equation concept. The parameters which characterize the adsorbent structure are the same whatever the adsorbate. In comparison with the most powerful characterization methods, some reasonable hypothesis were made: the pore walls of the adsorbent are assumed to be energetically homogenous; the pores are supposed to be slit-like shaped and a simple Lennard-Jones model is used to describe the interactions between the adsorbate molecule and the pore wall; the local model is obtained considering both the three-dimension gas phase and the two-dimension adsorbed phase (considered as monolayer) described by the Redlich-Kwong equation of state; the pore size distribution function is bimodal. All these hypotheses make the model simple to use for the calculation of equilibrium data in adsorption process simulation. Despites the announced simplifications, it was possible to represent in an efficient way adsorption isotherms of four different compounds at three different temperatures on a set of carbonaceous sorbents using a unique pore size distribution function per adsorbent.
REFERENCES 1. M. Kruk, M. Jaroniec, andJ. Choma, Carbon., 36 (1998)1447. 2. K. Kaneko, C. Ishii and t. Rybolt. J.Rouquerol,F.Rodrigues-Reinoso, K.S.W.Sing and K.K. Unger (eds.), Proceeding of Characterisation of porous solids III.. Elsevier, Amsterdam, (1994), 583. 3. L.S. Cheng and R.T Yang., Chemical engineering science., 49 (1994) 2599. 4. P.I. Ravikovitch, A. Vishnyakov, R. Russo, and A.V.Neimark, Langmuir., 16 (2000) 2311. 5. K. Wang and Do .D.D, Langmuir., 13 (1997) 6226. 6. C. Lastoskie, and al, J.phys.Chem., 97 (1993) 4786. 7. C. Dapeng and al, Carbon., 40 (2002) 2359. 8. J.P. Olivier, Carbon., 36 (1998)1469. 9. L.P. Ding, and S. K. Bathia, Carbon., 39 (2001) 2215 10. P. I. Ravikovitch, and A. V. Neimark, J.Phys.Chem.B., 102 (2001) 6817. 11. P. Kowalczyk, and al., Carbon., 41 (2003) 1113. 12. Z. Tan, and K. E. Gubbins, J.Phys.Chem., 94 (1990) 6061. 13. J. Jagiello, T. Bandosz, K. Putyera, and J. Schwarz. Rouquerol J, Rodriguez-Reinoso F., Sing K.S.W.and K.K.Unger (eds.), Proceeding of Characterisation of porous solids III., Elsevier Science,Amsterdam, 1994, 679. 14. T. L. Hill, An introduction to statistical thermodynamics, Addison-Wesley Publishing company, London, 1962. 15. J. M. Pruasnitz, R. N. Lichtenthaler, E.G de Azevedo, Second edition, Molecular thermodynamics of fluid-phase equilibria, Prentice-Hall, Inc, New Jersey, 1986. 16. J. H. Vera, J. M. prausnitz, The chemical engineering journal., 3 (1972) 1. 17. Y Belmabkhout, M Frrre and Guy De Weireld., Meas.Sci.Technol., 15 (2004) 848. 18. Y Belmabkhout, M Frrre and Guy De Weireld., J.Chem.eng.data., 49 (2004) 1379.
113
P.L. Llewellyn, F. Rodriquez-Reinoso, J. Rouqerol and N. Seaton (Editors)
92007 Elsevier B.V. All rights reserved
Structural characterization of porous carbonaceous materials using high-pressure adsorption measurements Y. Belmabkhout, M. Fr6re (*), G. De Weireld. Facult6 polytechnique de Mons - Thermodynamics department, 31 bd Dolez 7000 M o n s Belgium The aim of this work is to present a new method for porous carbonaceous solids characterization: the pore size distribution function (PSDF) of the adsorbent is determined by a theoretical treatment applied to adsorption data. The measurements are performed with different adsorbates at different temperatures in a wide pressure range. The theoretical model is based on the concept of integral adsorption equation (IAE). The main assumptions of the model are: (1) The PSDF is considered as an intrinsic property of the adsorbent. (2) The slit shaped pore model is used to describe the geometric configuration of the porous structure. (3) The adsorbent-adsorbate interactions are described by a Lennard-Jones potential model. (4) The pore wall surface is considered to be energetically homogeneous and the adsorbed phase is monolayer. (5) Both the gas phase and the adsorbed phase are supposed to be efficiently described by a Redlich-Kwong type equation of state. (6) We assumed a priori the analytical form of the pore size distribution function. We used adsorption isotherms of nitrogen, oxygen, argon and methane on different carbonaceous solids (four activated carbons and one molecular sieve) at 283 K, 303 K and 323 K and for pressures up to 2200 kPa. 1. INTRODUCTION Characterization of heterogeneous micropourous carbonaceous solids from theoretical treatments of adsorption data is a complex issue. The large number of existing characterization methods is an evidence of the constant evolution in the understanding of the way adsorption properties and sorbents structural characteristics are related [ 1-13]. During the last decades, the disposal of efficient calculation means [11] has led to significant improvements in this matter and the major challenges which were clearly identified a few tens years ago are about to be overcome. These challenges may be classified as follows: 9 the analytical form of the pore size distribution function should not be assumed a priori; 9 the geometrical structure of the pores should be well defined; 9 the interaction models (adsorbate-adsorbate and adsorbate-adsorbent) should be accurate; 9 the local adsorption model should be efficient in the whole pore range (micro, meso and macropores); 9 The energetic heterogeneity of the pore walls should be taken into account. All these aspects have been partially or thoroughly treated in previous works so that their relative importance for the assessment of a reliable pore size distribution function can be evaluated [ 1,10]. On such a basis, it should be interesting to develop an efficient model which would be able to simulate adsorption data in a wide range of temperature and pressure
Y. Belmabkhout, M. Frbre and G. De Weireld
114
conditions and for any adsorbate. Such a model should remain as simple as possible so that it could be used for adsorption process simulation. The purpose of this paper is to present such a model which can be regarded as a synthesis of fundamental considerations resulting from the activities in porous sorbents characterization and classical thermodynamics developments. This model is able to represent in a correct way adsorption data of oxygen, argon, nitrogen and methane on a given carbonaceous adsorbent using a unique pore size distribution function whatever the adsorbate. The procedure was applied on four different activated carbons and on a carbon molecular sieve. The adsorption isotherms were measured at 283 K, 303 K and 323 K and for pressures up to 2200 kPa. 2. T H E O R I T I C A L DEVELOPMENT
2.1. General approach Our theoretical approach is based on the Integral Adsorption Equation (IAE) concept. The IAE allows the generalization of statistic models initially developed for homogeneous surfaces to heterogeneous ones. Such surfaces are characterized by an adsorption energy distribution function F(e') for a given adsorbate. Considering porous carbonaceous adsorbents, it is generally accepted that the apparent energetic heterogeneity results from pore size heterogeneity. The concept of Integral Adsorption Equation may then be used for such adsorbents if FOz) may be correlated to the pore size distribution function Fz(H). The basic expression of the Integral Adsorption Equation applied to geometrically heterogeneous adsorbents is: Bs
m(T,P) = [MI'(T,P,H)F~(H)dH
(1)
Bi
In which: 9 m(T, P) is the adsorbed mass at temperature T and pressure P; 9 Mis the molar mass of the adsorbate; 9 1-'(T, P, H) is the molar adsorbate surface concentration on the walls of the micropores of diameter H at temperature T and pressure P; 9 FA(H) is the pore size distribution function defined on a surface basis; 9 Bi and Bs are the integration limits. Using Eq. (1) for the calculation of adsorbed masses requires the knowledge of both/-'(T, P, H) and FA(H). FA(H) may be related to the pore size distribution function defined on a volume basis Fz(H) for a given pore shape. 1-'(T, P, H) is called the local adsorption isotherm model. Considering the pore walls as energetically homogeneous surfaces, classical models derived from statistical thermodynamics developments may be used once the correlation between the adsorption energy c and the pore size H is known. The analytical form of such a correlation depends on the pore shape on the one hand and on the adsorbate-pore wall interaction energy function on the other hand.
2.2. Geometrical and energetic modelling of the porous structure of carbonaceous adsorbents Classical carbonaceous adsorbents (activated carbons and carbon molecular sieves) are composed of amorphous and graphitic carbon. Such materials exhibit a very important microporous structure due to missing graphitic layers in the graphite crystallites. The disordered arrangement of these crystallites is responsible for the mesoporous and macroporous structure. The presence of heteroatoms in the raw materials used for the
115
Structural characterization of porous carbonaceous materials
synthesis of such adsorbents leads to a chemical heterogeneity of the pore wall surface. As far as gas adsorption at supercritical temperatures is concerned, it is generally assumed that the adsorption process occurs mainly in micropores of molecular dimensions of which the walls may be considered as energetically homogeneous [13]. As a consequence, we chose the simplified slit-shaped model to describe the porous structure of our adsorbents. The pore diameter H is defined as the distance between the parallel graphite walls. The interactions between the adsorbate molecules and the pore wall are assumed to be well described by the Lennard-Jones model. The adsorption energy in a pore of diameter H is calculated by summing the contribution of each wall of the pore. The resulting correlation between the adsorption energy and the pore diameter e(H)is used to develop the local model I"(T, P, H) on the basis of classical existing models defined on a energetic basis 1-(T, P, ~). In this work, we set the mathematical form of the pore size distribution function Fv(H); we opted for a bimodal distribution: F v ( H ) = .f~O.inpore ~ exp
2 cr 1Hpores
2
! ,.ores +
~ ( 7
2 Hpores
exp
2 (7 2Hpores
2
!
(2)
In which: 0 V1 pores and V2 pores are the micropores volumes; o mlHpores and m2Hpores are the averages pore diameters; 0 O'lHpores and O'2Hpores are the standard deviations. Given the slit-like shape of the pores, the pore size distribution function defined on a surface basis FA(H) is related the corresponding function defined on a volume basis Fv(H) by Eq. (3): FA(H)_ X Fv(H) (3) H
In which x is equal to 1 or 2 according to the number of monomolecular layer formed within the pore. 2.3. Local adsorption isotherm model I'(T, P, 0 and hence F(T, P, t ~ are obtained by expressing the equality of the chemical potentials of the adsorbate in both the gas and adsorbed phases. The chemical potentials are derived from statistical thermodynamics developments [ 14]. 2.3.1. Gas phase modeling The canonical partition function of a system composed of Ng molecules in a volume Vg at temperature T is given by: 1 u N ~ ~ ( u~.,) (4) Qg = - - qgtrans s qgrot * qgv, b, ~ qge ~ expL) 2-~ Ng !
In which: is the translational contribution to the partition function (3 degrees of freedom of the centre of mass of the molecule); 9q g rot is the rotational contribution to the partition function; 9qg vibi is the internal vibrational contribution to the partition function; 9q g e is the electronic contribution to the partition function;
9q g trans
116
Y. Belmabkhout, M. Frkre and G. De Weireld
9 UiPnt is the potential energy of interaction between any molecule and all the others in the
system; 9 kis the Boltzmann constant. qg tr,,,,smay be expressed by: qg ....
Vg - Ngb A3
(5)
[2xM.,kT] 5
(6)
=
with: X'
=L~J
In which: 9 Ais the thermal de Broglie wavelength; 9 his the Planck constant; 9Mini is the mass of the molecule; 9 bis the volume of the molecule calculated using the Redlich-Kwong approach [ 15]. The molecular chemical potential of the gas phase is obtained by differentiating lnQg (given by Eq. (4) with respect to Ng at constant temperature and volume:
It, = -k< OlnQ" ON, )r.v,
(7/
Using Eq. (4) and Eq. (7) and the Redlich-Kwong approach for the calculation of Ui',t [16], we obtain:
IlnkTq,~o,~_lnP+ln x
lXg=
-k, l N,b +7
a
[ Ve - Ngb L b k r ~
PVg.-In
krN~
Vg
v~--N~b
In(V'+Ngbil [ ~
V,
(8) a
)J +Lv, + Ngb bkrx/-T
In which: a is the molecular parameter of the Redlich-Kwong eos[ 15] for the adsorbate in the gas phase. Eq. (8) may also be written: /.tg =/zg~ + kr lnP + kTlncb' (9) With:
(1 O)
/~0 = -krln kr qg,o,q~b,q,e A3 PVg+In
ln*'=-lnkTNg
Vg
N,b
a[~V,+N,b]+
Vg-N,b+g,-N,b bkT4-T 1
N,b ]
gg ) V +NgbJ
(11)
9 is ' the fugacity coefficient which takes into account the non-ideality of the gas phase.
2.3.2. Adsorbed phase modeling We consider a system of Ns molecules adsorbed on a surface A; the molecules interact with each other. The same kind of developments as the ones presented for the gas phase may be achieved for the adsorbed phase. They lead to the expression of the chemical potential of the adsorbate in the adsorbed phase by using the corresponding two dimensional equation Redlich-Kwong equation of state:
117
Structural characterization of porous carbonaceous materials
In k T q s v i b q . . . . q ~ q ....b, Az
- U----~~- In FI + In ~ 1 _ ~ _ In 1 / F r o / - kT k T Fro1 1/F - b s
(12)
/d s = - k T
b,
+
b, kr 4 Y
In -
-
+
1/r~,
~/r~, + b, In which: * U0 is the energy of adsorption. It can be calculated from the Lennard-Jones expression of the potential energy of interaction of an adsorbate molecule in a pore; 9H i s the spreading pressure; 9Fml is the surface molecular concentration F,, = N, ; A
9a~ and b~ are the Redlich-Kwong molecular parameters of the adsorbate in the adsorbed phase; 9q~ vib is the contribution to the partition function due to the vibration of the molecule perpendicularly to the adsorption surface. qs trans and qs vibmay be calculated respectively by: A-U~b, (13)
q.....
--
A2 - -
and
exp-2kT -
q.v,b =
(14)
1 oxp( )
In which: 9Vz is the vibration frequency of the molecule perpendicularly to the surface. It can be calculated from the Lennard-Jones expression of the potential energy (adorbate-pore). Eq. (12) may also be written: P, = p o + kTlnI-I + k T l n * (15) With: o _ k T I l n k T q S v , oq . . . . q~eq~v,o,I /~s = A2 + Uo
(16)
And In ~ = - l n - - - - + n 1 In 1~Fret + b. k r Fro, 1/rm,-b. l/Fret-b,
a. [ln(1/Fmt+b'/ b. 1 krvC-Tb. L ~. 1)rmZ-) + 1/rmt +bs
(17)
.J
is the fugacity coefficient of the adsorbate in the adsorbed phase; it takes into account of the non-ideality of the adsorbed phase. 2.3.3. Expression of equilibrium The equality of the chemical potentials of the adsorbate in both phases calculated respectively by eq. (8)and Eq. (12) lead to the final expression of the local adsorption isotherm model. The electronic, the internal vibrations and the rotational behaviour of the molecules are supposed to undergo no change when the molecules pass from the gas to the adsorbed phase. Using molar quantities instead of molecular quantities leads to: (
r" i'r'bS.o,1
exp
exp
1-rb..,Vb"~[l+Fbs]-~r~exp
( - NoU o ) p -RT- ~ v exp b•~ RT Pv v-b.o , V - bmo,
In which:
RT#-T
v + b.o I - ~ v
N~
= RT42xMRT-1-~_(-N--~](18) exp
Noa~~ RT ~t-f V + bmo,
118
Y. Belmabkhout, M. FrOre and G. De Weireld
9 vis the molar volume of the adsorbate in the gas phase; it is calculated as a function of temperature and pressure using the Redlich-Kwong equation of state; 9 Ris the ideal gas constant; 9 No is the Avogadro number 9 amol, bmol, asmol, bsmol, [" are the molar values corresponding respectively to a, b, as, bs, Fret. 2.4. Global model The use of Eq. (1), (2), (3) and eq. (18) allows the calculation of the adsorbed mass as a function of temperature and pressure for a given adsorbate and a given carbonaceous adsorbent. The unknown parameters are: 9 asmotand bsmot (for each adsorbate). 9 V1 pores and V2 pores, mlHpores and m2Hpores a n d O'lHpores and O'2Hpores, the parameters of the bimodal pore size distribution function
3. RESULTS AND DISCUSSION The theoretical developments presented in the previous section were applied to experimental adsorption isotherms. We studied four activated carbons (Centaur, BPL, F30/470, WS42) and one molecular sieve (CMS 1). High pressure adsorption isotherm measurements of N2, Ar, CH4, and 02 (at three different temperatures) were carried out with a volumetric device described earlier by Belmabkhout et al [17]. These experimental data have also been published [18]. By minimizing the discrepancies between the experimental and calculated results, it was possible to determine the adjustable unknown parameters of the model. Fig. 1 shows an example of the experimental and theoretical isotherms on Centaur (AC).
Fig. 1. Experimental and theoretical adsorption isotherms of N2, Ar, CH4 and 02 on Centaur For all the studied systems, a perfect fitting is noticed between experiment and theory (discrepancies are within the experimental error on the adsorbed mass: 5 " 1 0 -3 g/g)
119
Structural characterization of porous carbonaceous materials
Tables 1 and 2 present the values of the fitted parameters for all the studied systems. Table 1 Fitted parameters of the pore size distribution function for each adsorbent Sample m lHpores/A OrlHpores//~ Vlpores/ m eHporeJ~ O'2HporeJ~ Vepores/ m3/kg m3/kg Centaur BPL F30/470 WS42 CMS1
6.98 6. 73 6.27 6. 74 6.00
0.49 0.47 0.66 O.5 7 0.87
9. 75E-05 1.74E-04 2.26E-04 1.65E-04 3.13E-04
10.46 10. 77 8.51 9. 62 13.04
2.49 2.35 3.28 1.52 2.53
1.98E-04 1. lOE-04 1.70E-04 1.29E-04 3.32E-09
Vtotalmicr/ "
m3/kg
2.95E-04 2.84E-04 3.97E-04 2.93E-04 3.13E-04 =,
Table 2 Fitted energetic and geometric bidimensional Redlich-Kwong parameters for each adsorbentadsorbate s~,stem (asmol / J m 2 mole -2 , bsmol/m 2 mole -l) Sample asrnolOV2) bsmol (N2) asmol(Ar) bsmot (Ar) asmol(Cff4) bsmoI asmol(02) bsmot (02) .....
Centaur BPL
8.10E+09 9.43E+09
7.88E+04 7.59E+04
9.80E+08 7.15E+08
3.42E+04 3.24E+04
4.18E+09 5.83E+09
(CH4) 4.45E+04 4.27E+04
2.18E+09" 1.29E+09
4.25E+04 3.69E+04
F30/4 70 WS42
8.14E+09 6.52E+09
6. 74E+04 5.76E+04
2.50E+09 9.91E+08
4.64E+04 2.88E+04
4.93E+09 2.96E+09
3.96E+04 3.13E+04
1.09E+09 1.53E+09
3.67E+04 3.32E+04
CMS1
8.66E+09
5.94E+04
2.26E+09
3.87E+04
4.20E+09
2.89E+04
2.23E+09
4.19E+04
In Fig. 2 we plot the derived PSDF representative for each adsorbent. The PSDF of CMS1 seems to reduce to a Gaussian which points out that the range of pore size of CMS 1 is very small. The values of the standard deviations (YlHpore, (Y2Hporeand the volumes Vlpore and V2pore for F30/470 point out that the degree of the structural heterogeneity of this adsorbent is the highest as far as micropores are concerned. Centaur, WS42, and BPL seem to have a relatively intermediate degree of microporous structural heterogeneity. 3.00E+05
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
/
E 2.00E+05
l= !-SOE+0S
l .OOE+05
0.00E+00 0,00E+00
$.OOE+00
!.00E+0t
1.50E+01
2.00E,01
2.50E+01
Diameter I Angstrom
Fig. 2. PSDF of the studied adsorbents.
From Table 2, it appears that the value of the fitted energetic and geometric bidimensional Redlich-Kwong parameters asmol and bsmol seem to be physically realistic; they vary weakly
120
Y. Belmabkhout, M. Frkre and G. De Weireld
from one adsorbent to another. The variation of these bidimensional parameters can be explained by the difference in the surface chemistry for the different adsorbents which is not taken into account in our model. 4. SUMMARY AND CONCLUSIONS In this paper, we have presented and tested a model which allows the calculation of adsorption isotherms for carbonaceous sorbents. The model is largely inspired of the characterization methods based on the Integration Adsorption Equation concept. The parameters which characterize the adsorbent structure are the same whatever the adsorbate. In comparison with the most powerful characterization methods, some reasonable hypothesis were made: the pore walls of the adsorbent are assumed to be energetically homogenous; the pores are supposed to be slit-like shaped and a simple Lennard-Jones model is used to describe the interactions between the adsorbate molecule and the pore wall; the local model is obtained considering both the three-dimension gas phase and the two-dimension adsorbed phase (considered as monolayer) described by the Redlich-Kwong equation of state; the pore size distribution function is bimodal. All these hypotheses make the model simple to use for the calculation of equilibrium data in adsorption process simulation. Despites the announced simplifications, it was possible to represent in an efficient way adsorption isotherms of four different compounds at three different temperatures on a set of carbonaceous sorbents using a unique pore size distribution function per adsorbent.
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