Langmuir 1992,8, 1789-1795
1789
Texture and Surface Energetic Heterogeneity of Solids from Modeling of Low Pressure Gas Adsorption Isotherms F. Villibras, J. M. Cases, M. FranCois, L. J. Michot, and F. Thomas* Centre de Recherche sur la Valorisation des Minerais, UA 235, BP 40, 54501 Vandoeuvre cedex, France Received September 23,1991. I n Final Form: April 1, 1992 A quick method of surface heterogeneity analysis based on the summation of local derivative isotherms is presented. Each local isotherm is described by three parameters: (i) the normal interaction between the adsorbate and the adsorbent; (ii) the lateral interactions between molecules; (iii) the quantity of gas adsorbed on each homogeneous domain. The position of the maximum of a local derivative isotherm on the In (PlP,)axis depends on the normal interactions and the intensity of lateral interactions. Lateral interactions are easily detected through the shape of the derived isotherms. The relationship between these parameters and the position of a maximum is obtained through the use of second-order derivative equations of the fundamental equation of gas adsorption on a homogeneous surface. The height of the maximum yields the monolayer capacity for each local isotherm. For each homogeneous domain, experimentalparameters are determined and adjusted by the operator. Lateral interactions are introduced if necessary. The isotherm and derivative isotherm are computed by summation of all the local models and compared to the experimental isotherms. In order to study the textural properties of finely divided solids, low-pressure isotherms were obtained through a quasi-equilibrium continuous gas adsorption procedure. The adjustment method was tested with a kaolinite and a sepiolite. Energetic parameters are consistent with those obtained by other methods, such as low temperature adsorption microcalorimetry. In the case of kaolinite, broken-edge and basal faces are identified. In the case of sepiolite, three domains are easily distinguished. They correspond to the filling of the two different types of micropores (intraand intercrystalline) and to the adsorption on external faces.
Introduction Solid surfaces are energetically heterogeneous. In the case of powdered minerals, heterogeneity originates from (i) crystal faces with different surface energies, (ii) the presence of pores with various sizes and shapes, and (iii) the presence of impurities or local crystalline disorder. It is well established that surface heterogeneity influences physical adsorption proce~ses.~-~ Until very recently, volumetric measurements were the main source of experimental information and surface heterogeneity was mainly studied by the isosteric method which measures the adsorption isotherm a t two different temperatures. Unfortunately, this method lacks accuracy in the low pressure range as it is based on the assumption, unlikely for clays, that the state of the adsorbed phase is the same at the two temperatures. Calorimetric methods such as low temperature gas adsorption microcalorimetry (LTAM)allow a direct measurement of energetic heterogeneity and textural properties by measuring the differential molar enthalpy of adsorption IAad,h1* which is equal to qst + R T where qat is the isosteric heat of adsorption. The plot of IAadshlversus surface coverage 0 is related to the distribution of domains with different energies. This method has been successfully applied in the qualitative and quantitative description of heterogeneous adsorbents." For instance, in the case of (1) Ashworth,F. Advances in Electronics; Academic Press: New York, 1951; Vol. 3, p 1. (2) Beebe, R. A. Handbuch der Katalyse; Springer: Vienna, 1943; Vol. 4, p 473. (3) Branauer, S. The Adsorption of Gases and Vapors; Princeton University Press: Princeton, NJ, 1943. (4) Rouquerol,J. Calorimhtrie &adsorption aux basses temperatures, CNRS Publ.: Paris, 1972; 538 p. (5) Grillet, Y.; Cases, J. M.; Francois, M.; Rouquerol,J.; Poirier, J. E. Clays Clay Miner. 1988,36, 233. (6) Cases, J. M.; Grillet, Y.; FranCois, M.; Michot, L.; Villibras, F., Yvon, J. Clays Clay Miner. 1991,39, 191. (7) Cases, J. M.; Cunin, P.; Grillet, Y.; Poinsignon, C.; Yvon, J. Clay Miner. 1986,21, 55.
microporous minerals the distribution of micropore volumes can be calculated.516 In the case of kaolinite, Cases et aL7calculated the area of basal and lateral surfaces, i.e. the aspect ratio or shape factor. Another powerful method of assessing surface heterogeneity is based on the adsorption of ionic surfactants from aqueous solution.g13 The adsorption mechanism is described by taking into account (i) the energy of the normal adsorbate-adsorbent bond, (ii) the energy of the lateral adsorbateadsorbate bonds, which controls the twodimensional condensation through a cooperative process, and (iii) the distribution of homogeneous domains characterized by different adsorption energies. Cases and co-workers+l3 demonstrated that the use of the 0-1 approximation (condensation approximation, CA) (firstorder change if lateral interactions are greater than 4kT) for the two-dimensional condensation yields a direct representation of the energetic surface distribution when the derivative of adsorption isotherm is used. Thus, surfactants can be considered as reliable tracers of surface heterogeneity. For instance, adsorption isotherms of alkyldodecylammonium ions were used to calculate the aspect ratio of kaolins with various crystalline properties.' Theories based on similar considerations have been developed by several authors in order to describe adsorption on heterogeneous surfaces at the solid-gas and solidliquid interface.14J5 In the case of gas adsorption, lateral interactions are generally lower than 4kT, and the CA (8) Cases, J. M.; Mutaftschiev, B. Surf. Sci. 1968, 9, 57. (9) Cases, J. M. Bull. Mineral. 1979, 102, 684-707. (10) Cases, J. M.; Canet, D.; Doerler, N.; Poirier, J. E. In Adsorption at the Gas-Solid and Solid-Liquid Interface;Rouquerol, J., Sing, K. S. W., Eds.; Elsevier Publisher: Amsterdam, 1982; p 21. (11) Cases, J. M.; Poirier, J. E.; Canet, D. In Solid-liquid Interactions in Porous Media; Cases, J. M., Ed.; Technip Publisher: Paris, 1985; p 335. (12) Cases,J.M.;Levitz, P.;Poirier,J. E.;VanDa"e,H. In Advances in Mineral Processing; Somasundaran, P., Ed.; SME Publisher: Littleton, CO, 1986; p 171. (13) Cases, J. M.; Villibras, F. Langmuir 1992, 8, 1251.
0 1992 American Chemical Society
Villidras et al.
1790 Langmuir, Vol. 8, No. 7, 1992
cannot be used. Nevertheless, it is frequently used in the literature even for Langmuir isotherm.l4J5 In the present paper, a new method of a quantitative estimation of surface heterogeneity from low pressure gas adsorption isotherms is presented. This method is based on fitting the derivative of an experimental isotherm with respect to In (PIP01by a linear combination of derivative isotherm equationsdescribingadsorption on homogeneous surfaces.
Theory Equations for the Local and Total Adsorption Isotherms. A heterogeneous surface is commonly described by the differential distribution of the number of adsorption sites among corresponding values of adsorption energy. Then, two types of topographical distribution modela have been proposed: (i) the random model which assumes a random distribution of sites;lS (ii) the patchwise modelSJ7 which assumes the surface to be composed of independent homogeneous domains, or patches. The experimentally measured isotherm, et, has to be related to the following average
where 8j is the surface coverage on the homogeneous domain i and f(-qoa,i) the distribution function of normal adsorbate-adsorbent interaction (-q0a,i) in the energetic domain qoa,i - Aqoa,i to qoa,i + Aqo,i(dqo,i). AS n approaches infiiity, the summation in eq 1can be replaced by an integral. Numerous methods using this equation have been developed in order to obtain the energetic distribution function.14J5 In the case of CA,eq 1can be reduced to
where Si is the surface area of the homogeneous domain i and S the total surface area. In order to calculate f(-qO,i) in eq 1, the adsorption equation on each homogeneous class has to be defined. Now, let us consider the Langmuir isotherm first, as well as its extension for weak lateral bonds, the BraggWilliams-Temkin isothermls
kT In P =
Da e - kT In + kT In kT 1-8
(2)
where P is the equilibrium pressure, k the Boltzmann constant, T the absolute temperature, -qa the differential energy of adsorption defined as =
+ we
b PRO
Figure 1. Theoretical Langmuir adsorption isotherm and first derivative versus In P/Powith C = loo0calculated from eqs 7and 8.
Another form of eq 2 is Ap
= kT In P - kT In Po= -'poll
kT In Po+ kT In 1e 8 (4) where POis the saturation pressure of the adsorbate and Ap the undersaturation of the actual adsorbed layer with respect to the reference phase defined as the condensed phase at saturation. While denoting K = q o a + kT In (DJkT) + kT In POas a new constant, for a given adsorbata-adsorbent system at constant temperature, eq 4 takes the form
P e kT In - = -K - we + kT In -= Ap P O 1-8
(14) Jaroniec, M.; Madey, R. In Physical Adsorption on Heterogenous Solids; Elsevier Publisher: New York, 1988; 351 p. (15)Rudzinski, W.; Everett, 0. H. In Adsorption of Gases on Heterogeneous Surfacea; Academic Press: New York, 1991. (16) Hill, T. L. J. Chem. Phys. 1949, 17,762. (17) Row, 5.;Oliver,J. P. On Physical Adsorption; Wiley Interscience: New York, 1964. (18) Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, 1960.
(5)
Equation 5 can be rewritten as follows:
Experimental isotherms give the pairs (PIP0 - 8). In order to fit these isotherms by 8 defined in eq 6, the parameters K and w have to be determined. In the following section, a simple method to get these parameters is presented. Derivatives of Adsorption Isotherms. Adsorption Limited to One Layer. For the most simple case, adsorption without lateral interactions, i.e. when w = 0, the Langmuir isotherm, plotted as B vs In (PIPo),exhibits, an inflection point that can be characterized by the second isotherm derivative (Figure 1). Introducing the notation u = In (P/Po) = Ap/kT and C = exp(K/kT) we have Ce"
e=-
(3)
where we is the average force field exerted on an adsorbed molecule interacting with nearest neighbor adsorbed molecules at surface coverage0, and On the mean vibrational volume of the molecules in the adsorbed state.
0, - we - kT In kT
1
+ CeU
(7)
and
The inflection point of the isotherm (d28/du2 = 0) is given by the relation CeU = 1, i.e. 0 = 0.5. Provided that our experimentalisotherm obey eq 2, the position of this point
Langmuir, Vol. 8, No. 7,1992 1791
Surface Heterogeneity Analysis
0.0
I
I
-12
.IO
8
4
4
2
0
In PPo In PIP0
Figure 2. Theoretical derivative of Temkin adsorptionisotherms versus In PIP, with C = lo00 and various values of w calculated from eq 13.
can be determined experimentally on the experimental first derived curve (arrow in Figure 1): u* or P, the value of the C constant is
Figure 3. Theoretical derivative of BET adsorption isotherm versus In P/P,with C = lo00 calculated from eq 18.
Multilayer Adsorption. When the affinity of the adsorbate for the surface is low (C < lOOO), multilayer formation starts before the first layer is completed. This influences the shape of the derivative isotherm, and a multilayer correction has to be applied. The BET theory is then used in the same manner as previously. In the In (PlPo)- 8 plot, the BET equation and its derivatives read
and the value of the maximum of the first-order derivative
e= dB --
Thus, from the determination of the couple (u*, (dVad du),,,.) on the experimental curve we obtain the constant C = e-"' and the volume of gas required for the monolayer formation V, = 4*(dVaddu),,,*. Taking into account the lateral interactions adds some complexity to the problem. With a = w/kT and C = K, eq 6 can be written as follows:
e=
du
[l - aO(1- @I3
(14)
The coordinates of the maximum of eq 13 depend on the value of o (Figure 2). deldu presents a maximum when d2Bldu2 = 0, then when 0 = 0.5. Thus, from eq 12, the relation between C and u* is
The height of the maximum can be evaluated by eq 13 1 de (zL* 4-a =:-
(1 + (C - l)eh)Ceu (1 - eU)'(1+ (C - l)eU)'
(18)
d20 --
du2 [Ce"(l - (C - 2)e" + 6(C - l)e& + (C- 1)(C- 2)ek + (C - ~ ) ~ e $ l / [ ( l -e')3(1 + (C - I ) ~ " ) ~(19) I
1
The derivatives are
du2
(17)
With B = C - 1, the relation between C and u* is given by the equation
CeaBeu 1 + CeaBeU
d2e - e (1 - e)(i - 28) --
Ce" (1 - e")(l + (C - 1)e")
(16)
From the determination of the position u* of the maximum and a correct adjustment of the parameter "a" with respect to the shape of an experimental derived isotherm, the C constant and the theoretical height of the derivative are calculated. Comparisonbetween theoretical and experimental heights yields directly the monolayer capacity.
+ e" + B(-e" + 6e2"- e3")+ @(e3"+ e&) = 0
(20)
The derivative of BET isotherm for C = 10oO (Figure 3) exhibits both a maximum and a minimum. A real solution of eq 20 exists if PIP0 I0.0718 or u* I-2.63. The C constant corresponding to the maximum position can be evaluated by calculating the lower solution of eq 20, eq 21 (arrow noted u*mb, Figure 3). The C constant B z c - 1 ~ e"(1- 6e" + e'")
- (e2"(1- e")'(1 - 14e" + e2"))'/2 2e3U(1+ e")
(21)
corresponding to the minimum position can be evaluated by calculating the higher solution of eq 20 (arrow noted U*mm, Figure 3). Maxima are easier to determine and energetic constants should then be calculated with the use of eq 18. The value (dOldu),,,* is then calculated by substituting the value of C obtained from eq 21 in eq 16. The final treatments are the same as for the Langmuir model. Nevertheless, for -2.90 < u* < -2.63, the values obtained by this procedure are wrong because there is no maximum in the first derivative but an inflection point (Figure 4). The values of C have been tabulated taking into account this last point and are presented in Table I. In the case of nonnegligible lateral interactions, a multilayer contribution can be introduced by using the formula
1792 Langmuir, Vol. 8,No.7, 1992
[ Plotof
-c=35
-g
o,3
-
0.2
-
0.1
-
___
C=l4
...........C=17
'ads versus lnP/Po dVads / dlnPPo
Maximum identification
. .u
m
.u
-
Villibrae et al.
0.0 -12
-10
8
6
I
1
4
2
C
Plot of experimentaland modelled isotherms and derived isotherms
1
riot accepted
O=O+AW
-2.90 -2.85 -2.80
35 31 27
-2.75 -2.70 -2.65
24 21 17
A
Plot of experimental and modelled isotherms and derived isotherms
first proposed by Hill19
coordinates of the maximum
Experimental values of a and C can be determined step by step: a and C are fixed, 8 is computed from eq 22. dB/du is then calculated from eq 23 and the position of the maximum is determined. If this position is different from u*, C is corrected and tested in the same way. If the shape of the theoretical isotherm obtained, after C adjustment, differs from the shape of the experimental isotherm, a is modified and C recalculated. Isosteric Heat of Adsorption Derivation. Knowing the thermodynamical parameters of the different local isotherms, it is possible to derive the theoretical isosteric heat of adsorption curves qat. The expression of qatis
where h, is the molecular enthalpy of a molecule in the gas and ha the molecular differential enthalpy of a molecule in the adsorbed phase. Traditionally qat is taken in a positive form that is+,&. After differentiation of B versus 1/T, one obtains the following equations for qst
or
for monolayer limited and multilayer adsorption, respectively. (19)Hill, T. L. In Statistical Mechanics; McGraw-Hilk New York, 1966.
.L Edition of final results
Figure 5. Algorithm of the fitting procedure. Fitting Procedure The fitting procedure is based on the definition and summation of local derivative isotherms corresponding to a homogeneous domain of the surface and has been named derivative isotherm summation (DIS).The results of this summation are obtained by successive adjustments through an interactive step by step graphic display. The fitting algorithm is presented in Figure 5. The experimental isotherm (V,,) and the isotherm derivative (dVadd In (PlPo))are plotted versus ln (PlPo).The user first defines the best resolved maximum on the experimental derivative isotherm. Depending on his knowledge of the adsorbent, he then chooses the model isotherm describing this maximum: a multilayer model is used for adsorption on external surfaces whereas a Langmuir or Temkin model is preferred when adsorption is likely to be space-limited (for instance in micropores). The pair (ln (PIPo), (dVadd In (P/Po))p=p.) corresponding to this maximum is then determined. The result of this first step is displayed on the screen together with the experimental curves. On the basis of a comparison between the shapes of the experimental and calculated derived isotherms, lateral interactions are added to the model until a satisfactory shape is obtained. This first local model is then subtracted from the experimental curves. This allows the definition of a second maximum treated in the same way. When all the local isotherms have been defined, the user checks the fit. He can then "tune" it finely by either adding a new domain or modifying a previously defined model. In this latter case, he can modify the coordinates of the maximum or its lateral interactions. The final results are then edited. They yield for each local isotherm i, the type of model used, the coordinates
Surface Heterogeneity Analysis
Langmuir, Vol. 8, NO. 7, 1992 1793 Table 11. Adsomtion Isotherm of Araon at 77 K on FU7. ~~
domain A
B C a
position of the maxima P/Pa In P/Pa 1.5 x -11.1 6.3 X 1 W 2.2 x 10-2
-7.4 -3.8
energetic constant
lateral interaction
adsorbed volume, cm3/g NTP
67770 1604 41
0 0 0.4 kT
0.25 1.07 9.71
specific surface area, m2/g 0.9 4.0 36.0
Parameters obtained by the DIS procedure.
-13
12 C I
experimental curve
.
I
I
6 1
0
---._ __....---’ __ 4
2
Figure 6. Derivative of the experimental adsorption isotherm and fitted first derivatives of argon at 77 K on FU7.
of the maximum, the C energetic constant, the intensity of lateral interactions, and the monolayer capacity ( Vmi). It is then possible to calculate the proportion of the different energetic domains as well as the total specific surface area, which can be compared to the surface area obtained by a BET treatment. The evolution of the isosteric heat of adsorption with surface coverage can then be simulated. The values of q a t obtained on each domain using eqs 25 and/or 26 are then taken into account according to the following equation20:
Experimental Section Apparatus and Procedures. Low-pressure isotherms were obtained through a quasi-equilibrium continuous gas adsorption procedure,21.22as described by Michot et al.2392‘ The procedure was tested with two previously studied adsorbents, a kaolinite and a sepiolite. Pressure measurement were performed using two Datametrics differential pressure gauges: 0-10 Torr and 0-1000 Torr ((0-1.3) x lo2and (0-1.3) X 106 Pa) ranges and 0.15% accuracy. A dynamic vacuum of about lo4 Torr was obtained through the use of a turbomolecular vacuum pump provided by Balzers. Pressure data were collected and stored with a microcomputer. The frequency of measurements was always adjusted Thus, to get around 200 experimental points per unit of In (PlPo). 2000 to 3500 points were collected for relative pressures lower than 0.15. The specific surface areas were calculated using the classical BET procedure. Materials. Two adsorbents were used to test the method a kaolin from the Charentes (France) deposit (FU7) supplied by AGS (France) and a sepiolite from Vallecas (Spain) supplied by Tolsa S.A. (20) Patrykiejew A. Quoted in ref 17, p 117. (21) Rouquerol, J.; Rouquerol, F.; Grillet, Y.; Ward, R. J. in Characterhation of Porous solids; Unger, K. K., Rouquerol, J., Sing, K. S. W., Krd, H., Eds.; Elsevier Publisher: Amsterdam, 1988; Vol. 39, p 67. (22) Weeeon, S. P. In Fundamentals ofddsorption;Myers, A. L., Belfort, G., Eds.; Engineering Fondation Publieher: New York, 1984, p 693. (23) Michot, L.; Franpis, M.; Cases, J. M.Langmuir 1990, 6, 697. (24) Michot, L. Propri6tes physicochimiques superficiellesdu talc et de la chlorite. Thbe INPL Nancy, 1990; 285 p.
0.2
0.4
0.6
0.8
I
1.2
Surface Covernge
o
In PIP0
I
Figure 7. Experimental and simulated curves of isosteric heat of argon adsorption on FU7 at 77 K. Table 111. Determination of Textural Parameters of FU7 by Various Methods quasiequilibrium LTAMa procedure surfactantaa BET energetic constant 57.0 43.8 BET surface area, m2/g 47.3 DIS surface area, m2/g 40.9 41.6 basal surface, m2/g 36.0 lateral surface, mYg 5.7 4.9 lateral surface/total surface 0.12 0.12 0.124 0 Cases et al.’ The kaolin is poorly crystallized. Ita specific surface area, as measured with the BET-argon method, is 47 m2gl and ita lateral faces account for about 12% of the total surface.’ The sepiolite is made up of talc-like layers arranged in long ribbons stuck together to form fibers. This microporous mineral bears different kinds of sites: (i) external surface, with a specific area of 120 m2gl,as measured by the Harkins and Jura method, and of 120 and 105 m2gl,respectively, as measured by nitrogen and argon adsorption at 77 K (ii) intrafiber and (iii) interfiber microporosities, both volumes being derived from LTAM experimenta.5 The volume of gas adsorbed in the intra- and interfiber microporosity depends on the outgassing procedure (temperature and v a c ~ u m ) . ~ The kaolin was outgassed during 14 h a t 100 “C. The sepiolite was outgassed during 24 h at room temperature. The adsorbate was high purity grade (>99.996)argon.
Results and Discussion Kaolin. The adsorption isotherm was split into three local BET-type isotherms (Figure 6 and Table 11). The less energetic domain (domain C)corresponds to 88% of the total area and is representative of the basal surface of the kaolin. Thus, domains A and B can be assigned to the adsorption on broken-edgefaces. The specificsurface area obtained with the DIS method, 40.9 m2/g, is lower than that obtained with the conventional BET procedure, 43.8 m2/g. The difference may be explained by capillary condensation in slit-shaped pores between the edges of these platy particules, as shown by Delon et alS25 The simulated evolution of qst with surface coverage is presented in Figure 7 together with the experimental curve (25) Delon, J. F.; Lietard, 0.;Cases, J. M.; Yvon, J. Clay Miner. 1986,
21, 361.
1794 Langmuir, Vol. 8, No. 7, 1992
Villibras et al.
25
........
Experimental derived isotherm
"
20
Adjusted derived isotherm
____________
Sites A, B and C derived isotherms
15
d . 4
10
P
B P
5
0 -15
-12
-9
-6
-3
0
In P/Po
Figure 8. Derivative of the experimental adsorption isotherm and fitted first derivatives of argon at 77 K on sepiolite.
domain A B
C
Table IV. Adsorption Isotherm of Aruon at 77 K on Sedolite* position of the maxima energetic lateral adsorbed volume, P/Po P/Po constant interaction cm3/g NTP -13.8 527030 1.0 x 1o-g 1.2 kT 24.7 1.1 x 10-4 -9.1 7976 0.3 kT 12.0 -5.9 313 2.8 x 10-3 28.6 0.3kT
equivalent specific surface area, mZ/g 91.6 44.5 106.1
Parameters obtained by the DIS procedure. Table V. Determination of Textural Parameters of Sepiolite by Various Methods quasi-equilibrium LTAM la LTAM 2 Procedure BET energetic constant 809 160 772 BET surface area, m2/g 301 227 239 DIS surface area 242 intramicroporosity, cm3/g 0.041 0.024 0.031 intermicroporosity, cm3/g 0.025 0.015 0.015 external surface, m2/g 105 110 106 a Grillet et al.6
obtained with low temperature adsorption microcalorimetry. The simulated curve is strictly parallel to the experimental one. The difference between the two curves can be explained as follows: The simulated curves are based on the BET assumption, i.e. the heat of adsorption of the second layer is equal to the heat of liquefaction of the adsorbate (-6.8 kJ/mol in the case of argon) that is not really observed on the experimental curves. The three methods (LTAM, SA, and DIS) (Table 111) gave the same value of lateral specific surface area: 12% of the total surface. The energetic parameters obtained by the summation method yield a qBtcurve close to those obtained by the LTAM method. Sepiolite. The argon equivalent specific surface area of sepiolite, as measured by the BET procedure, was 239 mz/g. This value is much lower than that obtained by Grillet et aL5 and might be due to an incomplete outgassing. Resulta obtained by using LTAM with weak outgassing are given Table V (LTAM2). The experimental isotherm was split into three local isotherms (Figure 8 and Tables IV and V). According to LTAM,S class C was assigned to sites on the external surface and was modeled with a BET isotherm. The evaluated external surface area, 106 m2/g, corresponded to those obtained by LTAM. The two most energetic classes, A and B, were assigned
to the filling of intra- and intermicroporosity,as shown by LTAM, and were modeled with Temkin isotherms. At the very beginning of the adsorption the derivative isotherm shows a vertical slope (Figure 8), which can not be strictly modeled with a Temkin derivative isotherm. Thereforethe parameters of classA (Table IV) were chosen in order to obtain the same integral area as the experimental one. The filling of the microporosity of sepiolite depends on both the temperature and vacuum during the outgassing procedure.5 The quantity of gas adsorbed inside the microporosity, expressed in liquid volumes (Table V), is always considerably smaller5 than the total micropore capacity as measured using COZadsorption treated by the Dubinin26procedure, 0.13 cm3/g at 293 K5, or calculated from the theoretical dimension of the channels,0.248 cm3/ g.27 Such deviations have been observed on porous carbons28 and may be explained by the small size of these micropores that act as molecular sieves with regards to the molecular size of argon or other gases. The time required to reach equilibrium is large and depends on the diffusion rate and on the relative pressure at the beginning of the isotherms. Then, the filling of small micropores is not observed at the true equilibrium pressure. The simulated qatcurve (Figure 9) shows some discrepancies with the experimental one (LTAMB). At very low pressure, the difference may be explained by the uncertainty in the measured pressure due to the gas flow rate and to an overestimation of the relative pressure of intramicroporosity filling as explained above. The differences obtained in the external surface domain may be explained by a very heterogeneous energetic distribution on the external surface of the solid. (26) Dubinin, M. M. Pure Appl. Chem. 1966, 10, 309. (27) Rautureau, M.; Tchoubar, C. Clays Clay Miner. 1976,24,43. (28) Rodriguez-Reinoeo,F.; Torregroea, R.; Venero, A. F. Langmuir 1991, 7, 350.
Surface Heterogeneity Analysis
Langmuir, Vol. 8, No. 7, 1992 1795
IR I
16
I4
f
- - - - - _ _ _ _ _ ,_ . - 1
e x p r i m n h l curve simulated curve
-0'2 . IO
.
R -
0
0.2
0.4
0.6
0.8
I
1.2
Surface coverage
Figure 9. Experimental and simulated curves of isosteric heat of argon adsorption on sepiolite at 77 K.
Finally, the different microporous amounts and the external surface estimated with the DIS method are in good agreement with those obtained with LTAM.S
Conclusions The measurement of adsorption isotherms by a quasiequilibrium procedure allows calculation of an experimental derivative curve. The two examples have shown that the proposed method based on a thermodynamical formulation (eq 11) yields a good adjustment of experimental adsorption isotherms. Each domain is described by three parameters: the normal interaction, the lateral interaction, and the quantity of gas adsorbed on this site.
Normal interactions depend on (i) the position of the maxima on the In (PIPo) axis and (ii)the intensity of lateral interactions. Simulated qat curves obtained by using these parameters are consistent with those obtained by microcalorimetric measurements. Lateral interactions are easily detected through the shape of the derived isotherms and must be taken into account to fit correctly the experimental curves. Multilayer adsorption isotherms with lateral interactions can be fitted if a BET-like correction is introduced in eq 11. The energetic domains determined by this method are representative of textural sites of the surface. The quantity of gas adsorbed in these domains allows a quantitative estimation of the textural parameters of phyllosilicates, for instance basal and lateral surfaces,and of microporous adsorbenta, for instance external and internal surfaces. Finally, this method must be considered as a global method of surface heterogeneity analysis because the energetic distribution is made up of a limited number of classes, each class having its own energetic distribution.
Acknowledgment. We acknowledge the Ministere de la Recherche of France (Phygis program) for ita financial support, the Maurice Letord Research Center (Nancy, CNRS) for help in the design and realization of the quasiequilibrium volumetry apparatus, and Dr. J. E. Poirier for helpful discussions. Registry No. Ar,7440-37-1; sepiolite,15501-743; kaolin, 1318 74-7.
