- Email: [email protected]
Physical adsorption of gases on heterogeneous solid surfaces: Evaluation of the adsorption energy distribution from adsorption isotherms and heats of adsorption
Physical Adsorption of Gases on Heterogeneous Solid Surfaces: Evaluation of the Adsorption Energy Distribution from Adsorption Isotherms and Heats of Adsorption W. RUDZIlqSKI, *'1 J. JAGIELLO,* AND Y. GRILLETt *Department of Theoretical Chemistry, Institute of Chemistry UMCS, 20-031 Lublin, Nowotki 12, Poland, and tCentre de Recherches de Microcalorimetrie et de Thermochimie du C.N.R.S., 26 rue du 141e, R.1.A., F-13003 Marseille, France Received April 27, 1981; accepted September 4, 1981 A new simple method is proposed for evaluating the adsorption energy distribution, based on a simultaneous use of adsorption isotherm and heat of adsorption as the sources of experimental information. Both the interactions between the adsorbed molecules and the effects of surface topography are taken into account. The method is illustrated by a model calculation, as well as by its practical use to two adsorption systems exhibiting very different types of the energy distribution function. INTRODUCTION
The generally used, quantitative description of the energetic heterogeneity of surfaces is the differential distribution of the adsorption sites ×(C), among various values of the adsorption e n e r g y e'. This function is usually called the "adsorption energy distribution," and is used in its form normalized to unity. The numerical evaluation of this function from experimental adsorption data was one of the main problems of adsorption science in the last 30 years. Several numerical methods have been developed (1-22); most of them are advanced, but also time-consuming routines, usually designed for the Langmuir model of adsorption. When the interactions between the adsorbed molecules cannot be neglected, then the spatial distribution (correlation) of the adsorption sites having the same adsorption energy becomes a very important factor to be taken into account in this calculation. This spatial correlation is usually called 1 To whom correspondence should be addreSsed.
the "topographical distribution of adsorption sites," or simply the "topography of surface." The importance of this factor in relation to the evaluated energy distribution was emphasized first by Adamson and coworkers (2), and quite recently by Rudzifiski et al. (23, 24). Meanwhile, almost all the numerical methods for evaluating the adsorption energy distribution have been designed for only one extreme model of the surface topography, namely, for the "patchwise" model of surface topography, introduced to adsorption theory by Ross and co-workers (6-8). It is assumed in this model that the adsorption sites having the same adsorption energy are grouped into large patches. These patches are large enough, so that the statistical thermodynamics could be applied to every one of them, and the states of the whole adsorption system could be neglected, in which the interacting molecules are adsorbed on two different patches. In other words, the whole adsorption system can be considered to be a collection of independent subsystems.
478
0021-9797/82/060478-14502.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, VoL 87, No. 2, June 1982
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES
The second extreme model of the surface topography assumes that the adsorption sites having the same value as the adsorption energy are distributed completely at random on the heterogeneous surface. In other words, the whole adsorption system must be considered as one thermodynamic entity. This adsorption model has been considered first by Hill (25) and Tompkins (26), and quite recently by Rudzifiski and Lajtar (27), and by Stoeckli and co-workers (28). However, the problem of a right choice of a surface topography model, when evaluating the adsorption energy distribution, is far from being solved. Meanwhile, it has been shown by Rudzifiski et al. (23, 24) that this choice very strongly affects the form of the evaluated energy distribution function. It is also surprising that almost all the numerical methods for evaluating the adsorption energy distribution are based on applying the adsorption isotherm as the source of experimental information. Meanwhile, for some basic thermodynamic reasons, the heat of adsorption should be much more sensitive to the effects of surface heterogeneity than the experimental adsorption isotherms. However, except for the works by Drain and Morrison (20) and by Dormant and Everett (4), no one has tried to apply the calorimetric adsorption data as the source of experimental information, when evaluating the energy distribution function. It is the purpose of this paper to show that simultaneous knowledge of the experimental adsorption isotherm and the isosteric heat of adsorption offers a new possibility for a proper and exact evaluation of this important function. THEORY
The experimentally measured adsorption isotherm N(p, T) has the following relation to the adsorption energy distribution
N(p, T) = No
O(e',p, T)X(e')dE'
479
[1]
t 1
where 0(E', p, T) is the relative surface coverage of the adsorption sites having the adsorption energy e', at the pressure p, and temperature T. This function is traditionally called the "local adsorption isotherm." No is the total number of the adsorption sites on the investigated heterogeneous surface (surface capacity), and e~ and Em are the lowest and the maximum values of the adsorption energy, e', on the heterogeneous surface. The adsorption energy e' is defined to be equal to the sum - ( U 0 + Ev), of a local minimum U0 of the gas surface adsorption potential U(x, y, z), and the vibrational energy Ev of the adsorbed molecules. The surface heterogeneity is caused mainly by the variation of U0 on these local minima adsorption sites. Following many authors, we shall accept Em = +~, for the purpose of mathematical convenience. The nonphysical part of the integral [1] from E" to infinity plays an essential role only in the region of very low surface coverages. For the problems considered here, much more essential is the lower energy limit e~, since it should affect the adsorption at moderate and higher surface coverages. We shall still take it into account, and write Eq. [1] in the following form,
v(p, T) = I~ O(E,p, T)x(e)d~
[2]
where e = (e' - E~), and v = N/No. Let us now consider the form of the local adsorption isotherm 0(e, p, T). We shall accept here the model of monolayer localized adsorption, with nearest neighbor interactions between the adsorbed molecules. Furthermore, we will accept Bragg-Williams approximation when taking these interactions into account. With these assumptions the local adsorption isotherm O(e,p, T) can be written in the Journal o f Colloid and Interface Science, Vol. 87, No. 2, June 1982
480
RUDZIlqSKI, JAGIELLO, AND GRILLET
following form, 0 ( • , p , T ) = [1 + exp • c - e]-x kT J
[3]
where ec(P, T) is some function which depends on the model of the surface topography (25), • c = - k T In p
- z u f - •~.
K
[4]
In Eq. [4] f is equal to 0(e, p, T) or v ( p , T ) for the patchwise and the random topography, respectively; K is the Langmuir temperature-dependent constant, z is the number of the nearest neighbors adsorption sites, and u is the interaction energy between two molecules adsorbed on two neighboring sites. The simplest way to solve the integral equation [2] with respect to the energy distribution function X(E), is to replace the true kernel [3] by a step function 0CA(•,P, T), OCA(• , p ,
T) =
01 for for
• < •~(p, T) • >/ e~(p, T)
[5]
which is the essential idea of the condensation approximation. The choice of the function e~(p, T) is governed by Cerofolini' s variational princip,e (29),
from which one can easily evaluate the approximate solution for the energy distribution function XCA(•)
Ov(•~)
XCA(•~) -
[101
0ee~
Another simple solution can be obtained when the true kernel 0 is replaced by a combination of the step function 0CAand a Henry isotherm. This is the essential idea of the so-called assymptoticaUy correct condensation approximation. The approximate solution for X(•), denoted later by X A C C A ( • ) , then takes the following form (30):
Ov(•D
XACCA(•s) =
0Ec~
02v(e~)
[11]
kT ~ 0(•s) ~
However, this solution does not represent a big improvement over the CA solution. Recently, Rudzifiski and Jagie~o have found another simple solution for X(•) (31), which for typical experimental conditions does not diverge by more than 5% from an appropriate exact solution, which will be discussed in the next section. Now, let us consider our approximate solution. Let us consider to this purpose the derivative (Ov/O•~) 0V
[12]
1
0(•, •c = •~) = - • [61 2 Thus, for the patchwise topography we have, • ~ . •~.
-.k T In . p
K
1 zu - •~ 2
[71
whereas for the random topography we obtain,
The derivative (00/0e s) is a bell-shaped function of e, centered about • = •g. For the case of a patchwise surface topography, this function has been investigated by Cerofolini [32]
fd 00
1
-
Es
~s = •~ = •c(f = V).
X(e)d•
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
_
~?
2
[81
Replacing the true kernel 0 by the step function 0CAin Eq. [2] leads to the simple integral equation, v ( p , T) =
E~
exp/~)T
[9]
1 -
- -
kT
0(1
-
0)y
[13]
where y is some function which reflects the difference between the patchwise and the
PHYSICALADSORPTIONOF GASES ON SOLID SURFACES random surface topography. In the case of the random topography (and also for the Langmuir model of adsorption neglecting the interactions between admolecules), y is equal to unity. In the case of the patchwise surface topography 3/ takes the following form: 7=
1 ---0(1
kT
- 0)
[14]
The derivative (00/0~) is a symmetrical function of E (for both the patchwise and the random topography), which becomes more and more narrow as the temperature decreases. In the limit T ---> 0, this function degenerates finally into the Dirac delta distribution 6(~ - E). In this limit, the CA approximate solution becomes identical with the exact solution. Now, we are approaching the idea on which the Rudzifiski- Jagietlo (RJ) approximate solution has been based. Namely, the derivative (Ov/Oe~) is evaluated by expanding the distribution function X(e) into Taylor series around ~ = e~. Introducing the new variable x = (e - e~)/kT, we obtain,
Ov
(kT) ~ O"X -
Oe~
Cn
[15]
kTIOOtxndx.
[16]
~,
,~o
- -
n!
O(eg)"
where,
C , = I~
Except for the region of very high temperatures, where the model of localized adsorption does not apply, e~ >> kT. Furthermore, even for the Langmuir model of adsorption (and also for the random surface topography), where the spread of the weighting function (O0/Oe~) is the largest one, the width at its half in maximum is equal to 1.76 kT. Therefore, we can take safely (-oo) as the lower integration limit in Eq. [16]. The differences in the surface topography will result in the different values of the coefficients C,'s, since the function (00/0~) is different for these two kinds of surface topography. For the random surface topography
481
we have,
C~=
,1
for n = 0
0
for
n =2m + 1
[17]
-2(2 - 1)~-~Bn for n = 2 m . We will be interested further only in the first Co, and the third C2 coefficient, since cutting the expansion [15] leads to a satisfactory approximation, as we will soon show. Thus, according to the above equation, the coefficient C~ takes the value -3.29, and is also temperature-independent. In the case of the patchwise surface topography, C~ cannot be evaluated analytically, but we have found that its value can well be approximated by the following formula, C~ = -3.29 exp bl\ kT / J
[18]
where bl = - 1 . 6 4 and b2 = 1.24. The values of Cf given by Eq. [18] do not differ more than 3% from appropriate exact values found numerically. Because of the symmetry of the function (Ov/Oeg) in the case of the patchwise surface topography, all the noneven coefficients C2m+i's disappear. Neglecting in Eq, [15] all the terms of order higher than O(kT) 2, we arrive at the following second-order differential equation for the energy distribution function,
Ov _ ae~
X(e s) - ~
C~
(kT)Z~'(es).
[19]
For the case of the patchwise topography, this equation has already been discussed by Rudzifiski et al. (33~ 34), who have attempted to solve it numerically. Meanwhile, our recent numerical investigation has shown that the second term in Eq. [19] can be treated as a correction term. As such, it can be evaluated approximately using the CA solution for X(¢~). Doing so, we arrive at the following equation,
es
C~
O2Xc------~A
Xru(~) = Xca(¢~) - -~- (kT) ~ O(e~),
[20]
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
482
RUDZIIqSKI, JAGIELLO, AND GRILLET
Equation [20] can be applied to both the patchwise and the random surface topography. Below, we are going to discuss in detail the interrelations of the solutions obtained with the assumption of a patchwise and a random surface topography. Let us consider for this purpose the relation, er = e°~ - z u ( v - ~) .
[21]
It can be deduced easily from this relation that the energy distribution function evaluated with the assumption of a random surface topography will be wider, by the value zu, than the distribution function evaluated with the assumption of a patchwise surface topography. According to our model, the maximum value of e~ should correspond to the limit v ---> 0, whereas the minimum value ofe~ will be reached in the limit v ~ I. Thus, 1 el,max = ~.max + -- • 2 Ec,min
=
P
Ee,min
1 --
_
2
.
[22] [23]
This result has already been reported by Rudziriski et al. (23, 24) together with the interrelation of the CA solutions, obtained for these two kinds of surface topography. X~A = X~A[1 q- ZUX~A]-1.
[241
From this equation, and the general relation XCA~ cJ = XSA(E~(E~))~0e~ /
[25]
one can deduce the interrelation of the second derivatives of the function X, which is necessary to establish the relation between the ILl solutions obtained for the patchwise and the random surface topography. o(e~) ~
LO(E~) 2
k O~ x
J oe~J, [Oe~] [ - ~ j 4. [26]
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
Having the X~A solution evaluated, from Eqs. [21]-[26] one can easily find the relation between the RJ solutions, obtained for the patchwise and the random surface topography. And the X~a(e~) function can be evaluated from the experimental adsorption isotherm, according to the relation, Ov 0 In p 0 In p 0e~
X~A(e~) = 1
Ov
k T O In p
[271
However, after evaluating the X~A solution, we still have to choose properly the model of surface topography, as well as the values of the parameters K , zu, ~ to be used in our further calculation, for the adsorption system under consideration. As to the parameters K, E~, their choice causes only some shift of the final solution on the energy scale without changing the form of the solution itself. On the other hand, the choice of the model of surface topography, and of the interaction parameter zu influences very strongly the form (shape) of the final solution. To our knowledge, no one has yet proposed how to choose these important factors when evaluating the energy distribution function. Below we are going to show that a simultaneous analysis of the experimental adsorption isotherm and the experimental heat of adsorption creates some hopes for a simple solution of this problem. We shall write the isosteric heat of adsorption Qst(V, T) in the following form:
Q~t = k
P
[28]
The derivative (Ov/O In P)T is just the ~(~A solution multiplied by kT. Now, let us consider the derivative (Ov/O1/T)p in more detail:
PHYSICALADSORPTIONOF GASES ON SOLID SURFACES
t o0) p
483
Using in Eq. [29] the same expansion as in Eq. [15] leads to the following expressions for the derivative (Ov/O1/T)p: For the patchwise topography we have,
p
Depending on the surface topography, the function (O0/O1/T)p takes the following explicit form. For the patchwise topography we have
(00)
= T[(E ° + Q°)X~A + ,-.2~:'P:kTa2"":d :t ~ c,
: 0P(1- 0P) k [ e + z u 0
+
[30]
p+Q°]
where the superscript p in 0p denotes the "local" adsorption isotherm, corresponding to the patchwise model of the surface topography, and Q0 is defined as follows:
x
0~(1
_
. [32]
T
p
[33]
x X~cA + C[(kT)2x'(E~)
+ C~
3! ( k T ) 4 x ' ( e ~ ) + "'"
0~)~
e + zuv + QO +
"
For the random surface topography, we obtain,
0 InK QO = - k ~ [31] 1 0-T For the random surface topography, we obtain =
3[ (kT)4x"(eD + "
1
[34]
where e ° = - k T In ( p / K ) - e'x. Consequently, we obtain the following two equations for the isosteric heat of adsorption, for the surfaces exhibiting a patchwise and a random surface topography:
(co+ Q0)X~A + C~(kT) , zX , (ec) p + ~C~ ( k T ) 4 x " ( e D QPst -
[35] (e ° + Q°)X~A +
Cr4 (kT)4x,,,(e~) C~(kT)2x'(e r) + -~.
Qrt = [1
- - Z l l 'Ar C A J1. / ~ CpA ' ~i f eP: l
Our numerical investigation has shown that C~ can be approximated by a formula, similar to the formula [18], C~ = -45.5.exp bl
.
[37]
[36]
We shall treat further the term C J3 !(kT)4x " in Eqs. [35]-[36] as a correction one, and evaluate it approximately replacing the third derivative X" by the second derivative of XCA- We will soon see that this approximation is consistent with that in Eq. [20], when Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
484
RUDZII~SKI, JAGIELLO, AND GRILLET
evaluating the adsorption energy distribution from the experimental heat ~of adsorption Qst. With this approximation the solution of Eq, [35] with respect to X'(¢P), and its further integration yields, x~(e~) - C~(kT)------5
o
X~A[Qs, - ~ - Q°]de~
C~(kT) 2 -
-
c~3~
fCA(Eep) + X0
[38]
where e0 corresponds to the highest investigated adsorbate pressure, and X0is the value ofx at this point. The ratio (C~/C~3 !) is equal to 2.30, and is practically temperature-independent. Thus, when comparing Eq. [20] and [38] we can see that the second term in Eq. [38] represents a correction of the same kind, as the second term in Eq. [20]. It proves the consistency of the approximation accepted in Eqs. [33]-[36], with that leading to Eq. [20]. Now, let us suppose that we have to deal with an adsorption system exhibiting a typical patchwise topography of surface. We mean, we have measured the adsorption isotherm and the isosteric heat of adsorption for this system. Having these experimental data, we can calculate two solutions for the energy distribution function: one of them given by Eq. [20], and the other one from Eq. [38]. Then, a proper choice of the parameters K , QO, zu, xo should cause the solutions to match each other. In other words, the best possible convergence of these solutions could serve as the criterion for the numerical (best-fit) choice of these parameters. After a short algebra, one may arrive at the counterpart of Eq. [38], related to the adsorption systems exhibiting a random topography of surface:
1 -
I~ ~1
-
Xr(e~)- C[(kT) 2 j~o
C[3!
NUMERICAL RESULTS AND DISCUSSION We begin our illustrative numerical calculation with the illustration of the effectiveness of our approximation, accepted in Eq. [20]. Let us consider for this purpose the very typical case of the adsorption systems, characterized by the right-hand widened, gaussian distribution of adsorption energy. The overall adsorption isotherm v ( p , T) then takes the following form: v(p, T) = exp{-B(~e)2}.
[40]
Let us consider further the subclass of these systems, in which the lateral interactions between the adsorbed molecules can be neglected. Then, according to our theoretical consideration, we can have at our disposal the following simple approximate solutions for the adsorption energy distribution: (1) The crude CA solution, XcA(e) = 2BE exp{-B~ 2} ;
[41]
(2) the more accurate ACCA solution, X~A[Qst- E° - Q°]dE~
C~4(k T ) 2
- -
Now, let us assume a situation in which we cannot judge a priori which kind of surface topography has to be assumed when calculating the adsorption energy distribution. Then, we could try the following procedure: we could perform the calculation of the adsorption energy distribution twice: once with the assumption of a patchwise surface topography and the second time assuming a random surface topography. Doing so, we should obtain two pairs of solutions, (from the experimental adsorption isotherm, and the heat of adsorption), corresponding to the two kinds of surface topography. Then, a better convergence of the solutions in one of the pairs could eventually serve as the check as to which kind of surface topography has to be accepted.
~ A ( 4 ) + X0.
[39]
Journal o f Colloid and Interface Science, Vol. 87, No. 2, June 1982
XACCA(E) = {2Be + k T [ 2 B - (2Be)Z]} x exp{-Be2};
[42]
(3) the RJ solution, given by Eq. [20],
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES
-¢" 1.0 ,. P< A~'L~f o
//'~
1.1
",::X "~'~,,X,
,
''/., 0.5
I/
0.0
485
Q5
tO
t,~
RELATIVE ADGORPION ENER6Y
2,0
F.. [KCAL/MOLE]
FIG. l. Comparison of the approximate solutions CA, ACCA, RJ, with the exact solution ST for the adsorption systems obeying the Dubinin-Radushkevich Eq. [40]. The calculation is done for the Langmuir model of adsorption, and the parameter B = 1.4 × 10-6 moles ca1-2.
XRJ(E)= [ 2Be
(TrkT)2 [-12BZe + × exp{-Be~}.
[43]
H o w e v e r , for this particular case of the Langmuir local isotherm, a simple accurate solution of the integral equation [2] can be obtained, by means o f the method of Stieltjes transform (ST) (35) ×ST(e) =
exp{B(~kT) 2} kT × sin{2~'BkT}.cxp{-Be2}.
[44]
For the purpose o f illustration, we have taken the v a l u e B = 1.4 x l0 -6 mole 2 ca1-2, which characterizes for example the heterogeneity of the surface in the adsorption system a r g o n - a e r o s i l at 75°K (36, 37). This value lies in the range of typical values of the parameter B, as shown by Rudzihski et al. (38). Figure 1 shows the comparison o f all the approximate solutions, CA, ACCA, RJ, with the exact solution ST obtained by the method o f Stieltjes transform.
It can be seen from this figure that for such a typical case o f surface heterogeneity, the approximate solution RJ does not diverge more than by 5% from the exact solution S T . Note that the accuracy of the RJ solution will be the same in the case o f the adsorption systems possessing a random topography o f surface (also when the interactions between the adsorbed molecules are taken into account). This is because the spread o f the function (O0/Oe~) and the coefficient C2 will be the same as in the case of the Langmuir local adsorption. In the case of the adsorption systems characterized by the patchwise surface topography, the approximate RJ solution will be even closer to the exact solution ST. This is because the spread of the function (O0/Oe~) and the coefficient C~ will be smaller, compared to the Langmuir model o f adsorption. (We assume at this moment the more probable case of the attractive forces between the adsorbed molecules.) Now, let us come to the essential problem of the simultaneous calculation of the adsorption energy distribution from the exJournal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
486
RUDZIlqSKI,JAGIELLO,AND GRILLET
,-¢ -g
2.6. (3
as
2,2. 0,4.
ao:
-.I
:~ "I.8
&
,
d4
~
&
n
ag
i
!
,o
ADSO~B,~T£ P~£SSU~£ p(mrnH 9)
FI6. 2. Comparison of the adsorption isotherms and the isosteric heats of adsorption for oxygen adsorbed on boron nitride (& A) and argon adsorbed on Gasil I, outgassed at 150°C • o). perimental adsorption isotherm and the isosteric heat of adsorption. These two functions are the two adsorption observables which are most often investigated. However, papers in which they are reported simultaneously are very rare, especially those where the heats of adsorption were measured directly by a calorimetric method. The so-called "isosteric method" involves some uncertainties, as discussed by Grillet et al. (39) and by Koubek et al. (40). The C.N.R.S. Laboratory was the place where several experimental methods have been developed for a simultaneous independent measurement of adsorption isotherm and the isosteric heat of adsorption. Using these methods, Rouqueroll and coworkers have investigated a variety of adsorption systems. Among the experimental data they have published, we have selected two adsorption systems for the purpose of our numerical investigation. One of them is the system argon-Gasil I, investigated experimentally by Rouqueroll et al. (41), in the scope of the SCI/IUPAC/ Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
NPL Project on Surface Area Standards, described by Everett et al. (42). The adsorption data for two samples of Gasil I, outgassed at 150 and 400°C, have been taken into consideration by us. The second system considered by us was oxygen adsorbed on boron nitride at 77.5°K, investigated experimentally in the C.N.R.S. Laboratory in Marseille. Figure 2 shows the unpublished experimental data for this system. There the experimental data for argon on Gasil I have been shown for the purpose of comparison. The first general conclusion which can be drawn from Fig. 2 is that the adsorption isotherms themselves do not seem to indicate serious differences between the properties of these two adsorption systems. At the same time, the isosteric heats of adsorption seem to reveal deeper differences, which will be confirmed soon by our calculation of the energy distribution functions for these systems. Figures 3 and 4 show the related energy distributions, evaluated for these systems
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES
487
OAStL I / ~O0*C
C
Z 0
~2o, b.
o ~o.
10o
Q
-~jOO ~
-I0o
"foO
FIG. 3. The energy distribution functions for argon adsorbed on Gasil I. The solid lines denote the energy distribution functions, evaluated from adsorption isotherms, whereas the dotted lines denote the distribution functions evaluated from isosteric heats of adsorption. by means o f the method described in the previous section. We can see there that, while the distribution o f the adsorption energy on the silica (Gasil I) samples is of an "exponential"
~.. []
~.
/ ~
type, the boron nitride sample can be well characterized by the " s k e w - g a u s s i a n " energy distribution function. It has been k n o w n for a long time that the exponential energy distribution is responsible for the
RANrlOPI
PATOHWt.5£
TOPOGRAPHY
" r o P o O R A laPRY
'\ 200
- 600
-
- ~0o
O
Rf.LATIVE A D , 5 0 ~ P T I O H EHEROY F~C(.o (co,L/rnoLo~ FIG. 4. The energy distribution functions for oxygen adsorbed on boron nitride, evaluated with the assumption of the patchwise and random surface topography. Journal o f Colloid and Interface Science, Vol. 87, No. 2, June 1982
488
RUDZIlqSKI, JAGIELLO,AND GRILLET
"Freundlich behavior" of adsorption systems, whereas the skew-gaussian distribution causes the adsorption systems to exhibit the "Dubinin-Radushkevich behavior." This, of course, must be considered as a crude historical classification. We are now at the stage of an accurate determination of the form of the adsorption energy distribution. Regarding the details of our calculation, we would like to draw the reader's attention to the parameters to be fitted best numerically. In fact, there are only the following two parameters: (1) the parameter P1 = - k T In K - Q0 + e~; and (2) the parameter P~ = z u . This is because the parameter P0 = k T In K - E~ causes only some shift of the evaluated energy distributions on the energy scale, and does not affect the form of the distribution function. Consequently, this parameter has no effect on the obtained best fit, and cannot, therefore, be estimated numerically. This was the reason why we used in Figs. 3 and 4 the relative energy scale esc0~ ~(o~ = es~ - k T
In K
+ e'l= -kTlnp
-zuf.
[45]
A proper choice of the parameters P1 and P2 (and of the surface topography model), should cause the two energy distributions obtained from adsorption isotherm and heat of adsorption to match each other. They are marked respectively by the solid and dotted lines in Figs. 3 and 4. In the case of argon adsorbed on the silica (Gasil I) samples, the best possible convergence of the two distribution functions is obtained v¢ith the assumption z u = 0: in other words, by accepting the Langmuir model of adsorption. The surface topography does not, therefore, play any role in this case. A slightly different situation has been discovered in the oxygen on boron nitride system. Here the assumption of a patchwise surface topography seems to lead to a better convergence of the two distribution funcJournal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
tions calculated from adsorption isotherm and heat of adsorption. The numerically estimated parameter z u takes the value 175 cal/mole, which corresponds to the critical temperature 22°K (the critical temperature of the 2D phase transition, lattice gas dense ordered phase). The assumption of a random topography for this system yields the parameter z u = 0, and the convergence which is slightly worse than previously. It seems that the difference between the two convergences are not great enough to determine without doubt that the patchwise topography is the proper model for the oxygen on boron nitride system. Nonetheless, in view of the present results, we shall accept it as a working hypothesis. Both in the case of the patchwise and the random surface topography, the distribution functions obtained from adsorption isotherm show, at the highest investigated energies (lowest pressures), some slightly different features than the distribution functions evaluated from heat of adsorption. There may be various explanations for this: We feel that either experimental errors are responsible for that, or the existence of a maximum value of the adsorption energy e', neglected in our hitherto theoretical consideration. There is still another reason why the oxygen on boron nitride system is very interesting: the skew-gaussian form of its distribution function makes possible the estimation of the value of the relative energy ect0)°,corresponding to the minimum adsorption energy E~. At this point we have, Ecco~° = P~
= - k T In K + el. '
[46]
The use of CA approximation also yields a simple physical meaning for the lowest heats of adsorption Qst,1, observed at the highest surface coverages. For the patchwise surface topography to be accepted for this system, we have, 1 ¢ Qst,1 = P 4 = QO + _ z u + el. 2
[47]
489
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES TABLE I The Experimentally Estimated, and Numerically Calculated Values of the Parameters Appearing in the Proposed Method System
Oxygen on boron nitride (patchwise topography) Argon on Gasil I/ 150°C (Langmuir model) Argon on Gasil I/ 400°C (Langmuir model)
P, (¢al/mole)
~st,i (cal/mole)
P3 (cal/mole)
{~}o (cal/mole)
~dt (¢al/mote)
K (rnm Hg)
2230
1820
-600
1630
100
109.9
2690
2160
-550
2140
20
44.7
2680
2150
-550
2130
20
44.7
Knowing the numerically determined parameters P1, P~, and the experimentally estimated parameters P3 and P4, one can evaluate Q0, K, and E;. For the considered oxygen-boron nitride system, Table I reports the values of the parameters Pi's, together with the values of Q0, K, el, calculated in this way. A similar calculation has been done for the adsorption systems formed by argon adsorbed on the two silica (Gasil I) samples. Here, an assumption has been made that the lowest investigated energies E~t0~correspond to the value el on the true energy scale. This assumption is justified by the fact that these lowest relative energies correspond to the experimental coverages, which are very near unity. Then, according to CA approximation, the adsorption sites are covered, possessing the lowest adsorption energy E~. The results of this calculation are reported in Table I. Table I reveals several interesting features of our adsorption systems such as the low values of the lowest adsorption energy el. It suggests thai the cancellation takes place between the values of U0 and Ev, on the adsorption sites exhibiting the lowest adsorption energy E~. This result should not surprise us. Rather it helps us to realize the relative sense of the "monolayer capacity" value, which is in fact temperature dependent.
Namely, in the model of the localized adsorption accepted here, not all the local minima of the adsorption potential U(x, y, z) can serve as "adsorption sites." These are only the minima where the adsorption potential U0 prevails over the vibrational energy Ev. (It is generally assumed that Ev is the same for all adsorption sites.) When the experimental temperature increases, the vibrational energy Ev also increases, which causes the effective number of the adsorption sites to decrease. This effect will result in an apparent decreasing of the estimated monolayer capacity value, as the experimental temperature increases. Observations of this kind have already been reported by Rudzifiski and Wojeiechowski (15). Now, let us consider at last theproblem of estimating the surface capacity of the heterogeneous surfaces. It seems that the only fully justified procedure is ttie. a posteriori normalization (to unity)' of the evaluated energy distribution function. We have applied this procedure to the oxygen-boron nitride system, and in this way obtained the value 5.27 cm3/g. This value has been used by us to make the plots in Fig. 2 (for oxygen on boron nitride). The form of the distribution function in Fig. 4 suggests that our experiment covers the essential range of the adsorption energies Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
490
RUDZIlqSKI, JAGIELLO, AND GRILLET
~
4
2
(19
"1
0
~bo rH£
~0o rmu£
~ o v
~,oa
4oo
"
SCALE £ ' C(x~L/moLe)
Fxo. 5. Comparison of the energy distribution functions, using the true energy scale c'. (1) The distribution function for oxygen on boron nitride, (2) the distribution function for argon on Gasil I outgassed at 150°C, and (3) the distribution function for argon on Gasil I outgassed at 400°C.
for this system, and the normalization is therefore possible. On the other hand, the f o r m of the distribution functions in Fig. 3 cannot m a k e us certain that the evaluated energy distributions c o v e r the whole essential range o f the adsorption energies for argon adsorbed on these silica samples. F o r this reason, we h a v e accepted the B E T estimation for these systems, reported in the previous p a p e r by GriUet et al. (41). D o r m a n t and A d a m s o n (3) have suggested that some special compensation for interaction and heterogeneity effects at high relative coverages m a y decrease the errors ha the B E T method, when applied to adsorption systems with heterogeneous surfaces. An a d v a n c e d theoretical analyze of this p r o b l e m can be found in the recent w o r k s by Oh and K i m (42). W e conclude our experimental section with the observation, which can be drawn f r o m Fig. 3, that the outgassing o f the silica samples in higher t e m p e r a t u r e s yields m o r e heterogeneous surfaces. In Fig. 3 the distribution function for the silica sample outgassed at 400°C is more flattened, c o m p a r e d Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
with that outgassed at 150°C. This can be seen e v e n more clearly in Fig. 5, where all three distribution functions have b e e n shown simultaneously, using the true energy scale e'. The energy distribution functions shown in Fig. 5 r e p r e s e n t the arithmetic average from the two energy distribution functions, calculated f r o m the adsorption isotherm and heat o f adsorption. REFERENCES 1. Adamson, A. W., and Ling, I., Adv. Chem. Ser. 33, 51 (1961). 2. Adamson, A. W., Ling, I., Dormant, L., and Orem, M., J. Colloid Interface Sci. 21, 445 (1966).
3. Dormant, L. M., and Adamson, A. W., J. Colloid Interface Sci. 38, 285 (1972). 4. Dormant, L. E., and Everett, D. H., unpublished results, work done in 1970 at the University of Bristol. 5. Dormant, L. M., and Adamson, A. W., Surface Sci. 62, 337 (1977). 6. Ross, S., and Olivier, J. P., "On Physical Adsorption." Interscience, London, 1964. 7. Ross, S., and Morrison, D., Surface Sci. 52, 103 (1975). 8. Morrison, I. D., and Ross, S., Surface Sci. 62, 331 (1977).
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES 9. Rudzifiski, W., and Jaroniec, M., Surface Sci. 42, 552 (1974). 10. Rudzifiski, W., Jaroniec, M., Soko~owski, S., and Cerofolini, G. F., Czech. J. Phys. B25, 891 (1975). 11. Jaroniec, M., and Rudzifiski, W., Colloid Polym. Sci. 253, 683 (1975). 12. Jaroniec, M., Surface Sci. 50, 553 (1975). 13. Rudzifiski, W., Waksmundzki, A., Jaroniec, M., and Sokotowski, S., Polish J. Chem. 48, 1985 (1974). 14. Jaroniec, M., and Rudzifiski, W., Acta Chim. Acad. Sci. Hungaricae 88, 351 (1976). 15. Rudzifiski, W., and Wojciechowski, B. W., Colloid Polym. Sci. 255, 869 (1977), 255, 1086 (1977). 16. House, W. A., and Jaycock, M. J., J. Colloid Interface Sci. 59, 252 (1977). 17. House, W. A., J. Colloid Interface Sci. 67, 166 (1978). 18. House, W. A., and Jaycock, M. J., J. Colloid Polym. Sci. 256, 52 (1978). 19. House, W. A., J. Chem. Soc. Faraday Trans. I 74, 1045 (1978). 20. Drain, L. E., and Morrison, J. A., Trans. Faraday Soc. 48, 316 (1952), 48, 840 (1952). 21. Jackson, D. J., and Dawis, B. W., J. Colloid Interface Sci. 47, 499 (1974). 22. van Dongen, R. H., Surface Sci. 39, 341 (1973). 23. Rudzifiski, W., Lajtar, L., and Patrykiejew, A., Surface Sci. 67, 195 (1977). 24. Rudzifiski, W., Patrykiejew, A., and Lajtar, L., Surface Sci. 77, L655 (1978). 25. Hill, T. L., J. Chem. Phys. 17, 762 (1949). 26. Tompkins, F. C., Trans. Faraday Soc. 46, 569 (1950).
49 1
27. Rudzifiski, W., and Lajtar, L., J. Chem. Soc., Faraday Trans. H 77, 153 (1981), 28. Stoeckli, F., Ph. Thesis by Dr. Morel, University of Neuchatel, 1980. 29. Cerofolini, G. F., Surface Sci. 24, 391 (1971). 30. Hobson, J. P., Canad. J. Phys. 43, 1934 (1965). 31. Jagiek~o, J., Ph. thesis, UMCS Lublin, 1981. 32. Cerofolini, G. F., Thin Solid Films 26, 53 (1975). 33. Rudzifiski, W., Narkiewicz, J., Wojciechowski, B. W., and Hsn, C. C., J. Colloid Interface Sci. 67, 292 (1978). 34. Rudzifiski, W., Narkiewicz, J., and Patrykiejew, A., Z. Phys. Chem. 260, 1097 (1979). 35. Landman, U., and Montroll, E. W., J. Chem. Phys. 64, 1762 (1976). 36. Nicolaon, G., and Teichner, S., J. Colloid Interface Sci. 38, 172 (1972). 37. Hsu, C. C., Rudzifiski, W., and Wojciechowski, B. W., Phys. Lett. 54A, 365 (1975). 38. Rudzifiski, W., Jaroniec, M., and Lajtar, L., Wiadomogci Chemiczne 30, 305 (1976). 39. Grillet, Y., Rouqueroll, F., and Rouqueroll, J., Rev. G~n. Therm. Fr. 171,237 (1976). 40. Koubek, J., Pasek, J., and Vole, J., J. Colloid Interface Sci. 51, 491 (1975). 41. RouqueroU, J., Rouqueroll, F., P6r6s, C., Grillet, Y., and Boudellal, M., in "'Characterization of Porous Solids" (S. J. Gregg, K. S. W. Sing, and H. F. Stoeckli, Eds.). London Soc., 1979. 42. Everett, D. H., Parfitt, Go D., Sing, K. S. W., and Wilson, R., J. Appl. Chem. Biotechnol. 24, 199 (1974). 43. Oh, B. K., and Kim, S. K., J. Chem. Phys. 61, 1797 (1974), 67, 3416 (1977).
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
The generally used, quantitative description of the energetic heterogeneity of surfaces is the differential distribution of the adsorption sites ×(C), among various values of the adsorption e n e r g y e'. This function is usually called the "adsorption energy distribution," and is used in its form normalized to unity. The numerical evaluation of this function from experimental adsorption data was one of the main problems of adsorption science in the last 30 years. Several numerical methods have been developed (1-22); most of them are advanced, but also time-consuming routines, usually designed for the Langmuir model of adsorption. When the interactions between the adsorbed molecules cannot be neglected, then the spatial distribution (correlation) of the adsorption sites having the same adsorption energy becomes a very important factor to be taken into account in this calculation. This spatial correlation is usually called 1 To whom correspondence should be addreSsed.
the "topographical distribution of adsorption sites," or simply the "topography of surface." The importance of this factor in relation to the evaluated energy distribution was emphasized first by Adamson and coworkers (2), and quite recently by Rudzifiski et al. (23, 24). Meanwhile, almost all the numerical methods for evaluating the adsorption energy distribution have been designed for only one extreme model of the surface topography, namely, for the "patchwise" model of surface topography, introduced to adsorption theory by Ross and co-workers (6-8). It is assumed in this model that the adsorption sites having the same adsorption energy are grouped into large patches. These patches are large enough, so that the statistical thermodynamics could be applied to every one of them, and the states of the whole adsorption system could be neglected, in which the interacting molecules are adsorbed on two different patches. In other words, the whole adsorption system can be considered to be a collection of independent subsystems.
478
0021-9797/82/060478-14502.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, VoL 87, No. 2, June 1982
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES
The second extreme model of the surface topography assumes that the adsorption sites having the same value as the adsorption energy are distributed completely at random on the heterogeneous surface. In other words, the whole adsorption system must be considered as one thermodynamic entity. This adsorption model has been considered first by Hill (25) and Tompkins (26), and quite recently by Rudzifiski and Lajtar (27), and by Stoeckli and co-workers (28). However, the problem of a right choice of a surface topography model, when evaluating the adsorption energy distribution, is far from being solved. Meanwhile, it has been shown by Rudzifiski et al. (23, 24) that this choice very strongly affects the form of the evaluated energy distribution function. It is also surprising that almost all the numerical methods for evaluating the adsorption energy distribution are based on applying the adsorption isotherm as the source of experimental information. Meanwhile, for some basic thermodynamic reasons, the heat of adsorption should be much more sensitive to the effects of surface heterogeneity than the experimental adsorption isotherms. However, except for the works by Drain and Morrison (20) and by Dormant and Everett (4), no one has tried to apply the calorimetric adsorption data as the source of experimental information, when evaluating the energy distribution function. It is the purpose of this paper to show that simultaneous knowledge of the experimental adsorption isotherm and the isosteric heat of adsorption offers a new possibility for a proper and exact evaluation of this important function. THEORY
The experimentally measured adsorption isotherm N(p, T) has the following relation to the adsorption energy distribution
N(p, T) = No
O(e',p, T)X(e')dE'
479
[1]
t 1
where 0(E', p, T) is the relative surface coverage of the adsorption sites having the adsorption energy e', at the pressure p, and temperature T. This function is traditionally called the "local adsorption isotherm." No is the total number of the adsorption sites on the investigated heterogeneous surface (surface capacity), and e~ and Em are the lowest and the maximum values of the adsorption energy, e', on the heterogeneous surface. The adsorption energy e' is defined to be equal to the sum - ( U 0 + Ev), of a local minimum U0 of the gas surface adsorption potential U(x, y, z), and the vibrational energy Ev of the adsorbed molecules. The surface heterogeneity is caused mainly by the variation of U0 on these local minima adsorption sites. Following many authors, we shall accept Em = +~, for the purpose of mathematical convenience. The nonphysical part of the integral [1] from E" to infinity plays an essential role only in the region of very low surface coverages. For the problems considered here, much more essential is the lower energy limit e~, since it should affect the adsorption at moderate and higher surface coverages. We shall still take it into account, and write Eq. [1] in the following form,
v(p, T) = I~ O(E,p, T)x(e)d~
[2]
where e = (e' - E~), and v = N/No. Let us now consider the form of the local adsorption isotherm 0(e, p, T). We shall accept here the model of monolayer localized adsorption, with nearest neighbor interactions between the adsorbed molecules. Furthermore, we will accept Bragg-Williams approximation when taking these interactions into account. With these assumptions the local adsorption isotherm O(e,p, T) can be written in the Journal o f Colloid and Interface Science, Vol. 87, No. 2, June 1982
480
RUDZIlqSKI, JAGIELLO, AND GRILLET
following form, 0 ( • , p , T ) = [1 + exp • c - e]-x kT J
[3]
where ec(P, T) is some function which depends on the model of the surface topography (25), • c = - k T In p
- z u f - •~.
K
[4]
In Eq. [4] f is equal to 0(e, p, T) or v ( p , T ) for the patchwise and the random topography, respectively; K is the Langmuir temperature-dependent constant, z is the number of the nearest neighbors adsorption sites, and u is the interaction energy between two molecules adsorbed on two neighboring sites. The simplest way to solve the integral equation [2] with respect to the energy distribution function X(E), is to replace the true kernel [3] by a step function 0CA(•,P, T), OCA(• , p ,
T) =
01 for for
• < •~(p, T) • >/ e~(p, T)
[5]
which is the essential idea of the condensation approximation. The choice of the function e~(p, T) is governed by Cerofolini' s variational princip,e (29),
from which one can easily evaluate the approximate solution for the energy distribution function XCA(•)
Ov(•~)
XCA(•~) -
[101
0ee~
Another simple solution can be obtained when the true kernel 0 is replaced by a combination of the step function 0CAand a Henry isotherm. This is the essential idea of the so-called assymptoticaUy correct condensation approximation. The approximate solution for X(•), denoted later by X A C C A ( • ) , then takes the following form (30):
Ov(•D
XACCA(•s) =
0Ec~
02v(e~)
[11]
kT ~ 0(•s) ~
However, this solution does not represent a big improvement over the CA solution. Recently, Rudzifiski and Jagie~o have found another simple solution for X(•) (31), which for typical experimental conditions does not diverge by more than 5% from an appropriate exact solution, which will be discussed in the next section. Now, let us consider our approximate solution. Let us consider to this purpose the derivative (Ov/O•~) 0V
[12]
1
0(•, •c = •~) = - • [61 2 Thus, for the patchwise topography we have, • ~ . •~.
-.k T In . p
K
1 zu - •~ 2
[71
whereas for the random topography we obtain,
The derivative (00/0e s) is a bell-shaped function of e, centered about • = •g. For the case of a patchwise surface topography, this function has been investigated by Cerofolini [32]
fd 00
1
-
Es
~s = •~ = •c(f = V).
X(e)d•
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
_
~?
2
[81
Replacing the true kernel 0 by the step function 0CAin Eq. [2] leads to the simple integral equation, v ( p , T) =
E~
exp/~)T
[9]
1 -
- -
kT
0(1
-
0)y
[13]
where y is some function which reflects the difference between the patchwise and the
PHYSICALADSORPTIONOF GASES ON SOLID SURFACES random surface topography. In the case of the random topography (and also for the Langmuir model of adsorption neglecting the interactions between admolecules), y is equal to unity. In the case of the patchwise surface topography 3/ takes the following form: 7=
1 ---0(1
kT
- 0)
[14]
The derivative (00/0~) is a symmetrical function of E (for both the patchwise and the random topography), which becomes more and more narrow as the temperature decreases. In the limit T ---> 0, this function degenerates finally into the Dirac delta distribution 6(~ - E). In this limit, the CA approximate solution becomes identical with the exact solution. Now, we are approaching the idea on which the Rudzifiski- Jagietlo (RJ) approximate solution has been based. Namely, the derivative (Ov/Oe~) is evaluated by expanding the distribution function X(e) into Taylor series around ~ = e~. Introducing the new variable x = (e - e~)/kT, we obtain,
Ov
(kT) ~ O"X -
Oe~
Cn
[15]
kTIOOtxndx.
[16]
~,
,~o
- -
n!
O(eg)"
where,
C , = I~
Except for the region of very high temperatures, where the model of localized adsorption does not apply, e~ >> kT. Furthermore, even for the Langmuir model of adsorption (and also for the random surface topography), where the spread of the weighting function (O0/Oe~) is the largest one, the width at its half in maximum is equal to 1.76 kT. Therefore, we can take safely (-oo) as the lower integration limit in Eq. [16]. The differences in the surface topography will result in the different values of the coefficients C,'s, since the function (00/0~) is different for these two kinds of surface topography. For the random surface topography
481
we have,
C~=
,1
for n = 0
0
for
n =2m + 1
[17]
-2(2 - 1)~-~Bn for n = 2 m . We will be interested further only in the first Co, and the third C2 coefficient, since cutting the expansion [15] leads to a satisfactory approximation, as we will soon show. Thus, according to the above equation, the coefficient C~ takes the value -3.29, and is also temperature-independent. In the case of the patchwise surface topography, C~ cannot be evaluated analytically, but we have found that its value can well be approximated by the following formula, C~ = -3.29 exp bl\ kT / J
[18]
where bl = - 1 . 6 4 and b2 = 1.24. The values of Cf given by Eq. [18] do not differ more than 3% from appropriate exact values found numerically. Because of the symmetry of the function (Ov/Oeg) in the case of the patchwise surface topography, all the noneven coefficients C2m+i's disappear. Neglecting in Eq, [15] all the terms of order higher than O(kT) 2, we arrive at the following second-order differential equation for the energy distribution function,
Ov _ ae~
X(e s) - ~
C~
(kT)Z~'(es).
[19]
For the case of the patchwise topography, this equation has already been discussed by Rudzifiski et al. (33~ 34), who have attempted to solve it numerically. Meanwhile, our recent numerical investigation has shown that the second term in Eq. [19] can be treated as a correction term. As such, it can be evaluated approximately using the CA solution for X(¢~). Doing so, we arrive at the following equation,
es
C~
O2Xc------~A
Xru(~) = Xca(¢~) - -~- (kT) ~ O(e~),
[20]
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
482
RUDZIIqSKI, JAGIELLO, AND GRILLET
Equation [20] can be applied to both the patchwise and the random surface topography. Below, we are going to discuss in detail the interrelations of the solutions obtained with the assumption of a patchwise and a random surface topography. Let us consider for this purpose the relation, er = e°~ - z u ( v - ~) .
[21]
It can be deduced easily from this relation that the energy distribution function evaluated with the assumption of a random surface topography will be wider, by the value zu, than the distribution function evaluated with the assumption of a patchwise surface topography. According to our model, the maximum value of e~ should correspond to the limit v ---> 0, whereas the minimum value ofe~ will be reached in the limit v ~ I. Thus, 1 el,max = ~.max + -- • 2 Ec,min
=
P
Ee,min
1 --
_
2
.
[22] [23]
This result has already been reported by Rudziriski et al. (23, 24) together with the interrelation of the CA solutions, obtained for these two kinds of surface topography. X~A = X~A[1 q- ZUX~A]-1.
[241
From this equation, and the general relation XCA~ cJ = XSA(E~(E~))~0e~ /
[25]
one can deduce the interrelation of the second derivatives of the function X, which is necessary to establish the relation between the ILl solutions obtained for the patchwise and the random surface topography. o(e~) ~
LO(E~) 2
k O~ x
J oe~J, [Oe~] [ - ~ j 4. [26]
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
Having the X~A solution evaluated, from Eqs. [21]-[26] one can easily find the relation between the RJ solutions, obtained for the patchwise and the random surface topography. And the X~a(e~) function can be evaluated from the experimental adsorption isotherm, according to the relation, Ov 0 In p 0 In p 0e~
X~A(e~) = 1
Ov
k T O In p
[271
However, after evaluating the X~A solution, we still have to choose properly the model of surface topography, as well as the values of the parameters K , zu, ~ to be used in our further calculation, for the adsorption system under consideration. As to the parameters K, E~, their choice causes only some shift of the final solution on the energy scale without changing the form of the solution itself. On the other hand, the choice of the model of surface topography, and of the interaction parameter zu influences very strongly the form (shape) of the final solution. To our knowledge, no one has yet proposed how to choose these important factors when evaluating the energy distribution function. Below we are going to show that a simultaneous analysis of the experimental adsorption isotherm and the experimental heat of adsorption creates some hopes for a simple solution of this problem. We shall write the isosteric heat of adsorption Qst(V, T) in the following form:
Q~t = k
P
[28]
The derivative (Ov/O In P)T is just the ~(~A solution multiplied by kT. Now, let us consider the derivative (Ov/O1/T)p in more detail:
PHYSICALADSORPTIONOF GASES ON SOLID SURFACES
t o0) p
483
Using in Eq. [29] the same expansion as in Eq. [15] leads to the following expressions for the derivative (Ov/O1/T)p: For the patchwise topography we have,
p
Depending on the surface topography, the function (O0/O1/T)p takes the following explicit form. For the patchwise topography we have
(00)
= T[(E ° + Q°)X~A + ,-.2~:'P:kTa2"":d :t ~ c,
: 0P(1- 0P) k [ e + z u 0
+
[30]
p+Q°]
where the superscript p in 0p denotes the "local" adsorption isotherm, corresponding to the patchwise model of the surface topography, and Q0 is defined as follows:
x
0~(1
_
. [32]
T
p
[33]
x X~cA + C[(kT)2x'(E~)
+ C~
3! ( k T ) 4 x ' ( e ~ ) + "'"
0~)~
e + zuv + QO +
"
For the random surface topography, we obtain,
0 InK QO = - k ~ [31] 1 0-T For the random surface topography, we obtain =
3[ (kT)4x"(eD + "
1
[34]
where e ° = - k T In ( p / K ) - e'x. Consequently, we obtain the following two equations for the isosteric heat of adsorption, for the surfaces exhibiting a patchwise and a random surface topography:
(co+ Q0)X~A + C~(kT) , zX , (ec) p + ~C~ ( k T ) 4 x " ( e D QPst -
[35] (e ° + Q°)X~A +
Cr4 (kT)4x,,,(e~) C~(kT)2x'(e r) + -~.
Qrt = [1
- - Z l l 'Ar C A J1. / ~ CpA ' ~i f eP: l
Our numerical investigation has shown that C~ can be approximated by a formula, similar to the formula [18], C~ = -45.5.exp bl
.
[37]
[36]
We shall treat further the term C J3 !(kT)4x " in Eqs. [35]-[36] as a correction one, and evaluate it approximately replacing the third derivative X" by the second derivative of XCA- We will soon see that this approximation is consistent with that in Eq. [20], when Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
484
RUDZII~SKI, JAGIELLO, AND GRILLET
evaluating the adsorption energy distribution from the experimental heat ~of adsorption Qst. With this approximation the solution of Eq, [35] with respect to X'(¢P), and its further integration yields, x~(e~) - C~(kT)------5
o
X~A[Qs, - ~ - Q°]de~
C~(kT) 2 -
-
c~3~
fCA(Eep) + X0
[38]
where e0 corresponds to the highest investigated adsorbate pressure, and X0is the value ofx at this point. The ratio (C~/C~3 !) is equal to 2.30, and is practically temperature-independent. Thus, when comparing Eq. [20] and [38] we can see that the second term in Eq. [38] represents a correction of the same kind, as the second term in Eq. [20]. It proves the consistency of the approximation accepted in Eqs. [33]-[36], with that leading to Eq. [20]. Now, let us suppose that we have to deal with an adsorption system exhibiting a typical patchwise topography of surface. We mean, we have measured the adsorption isotherm and the isosteric heat of adsorption for this system. Having these experimental data, we can calculate two solutions for the energy distribution function: one of them given by Eq. [20], and the other one from Eq. [38]. Then, a proper choice of the parameters K , QO, zu, xo should cause the solutions to match each other. In other words, the best possible convergence of these solutions could serve as the criterion for the numerical (best-fit) choice of these parameters. After a short algebra, one may arrive at the counterpart of Eq. [38], related to the adsorption systems exhibiting a random topography of surface:
1 -
I~ ~1
-
Xr(e~)- C[(kT) 2 j~o
C[3!
NUMERICAL RESULTS AND DISCUSSION We begin our illustrative numerical calculation with the illustration of the effectiveness of our approximation, accepted in Eq. [20]. Let us consider for this purpose the very typical case of the adsorption systems, characterized by the right-hand widened, gaussian distribution of adsorption energy. The overall adsorption isotherm v ( p , T) then takes the following form: v(p, T) = exp{-B(~e)2}.
[40]
Let us consider further the subclass of these systems, in which the lateral interactions between the adsorbed molecules can be neglected. Then, according to our theoretical consideration, we can have at our disposal the following simple approximate solutions for the adsorption energy distribution: (1) The crude CA solution, XcA(e) = 2BE exp{-B~ 2} ;
[41]
(2) the more accurate ACCA solution, X~A[Qst- E° - Q°]dE~
C~4(k T ) 2
- -
Now, let us assume a situation in which we cannot judge a priori which kind of surface topography has to be assumed when calculating the adsorption energy distribution. Then, we could try the following procedure: we could perform the calculation of the adsorption energy distribution twice: once with the assumption of a patchwise surface topography and the second time assuming a random surface topography. Doing so, we should obtain two pairs of solutions, (from the experimental adsorption isotherm, and the heat of adsorption), corresponding to the two kinds of surface topography. Then, a better convergence of the solutions in one of the pairs could eventually serve as the check as to which kind of surface topography has to be accepted.
~ A ( 4 ) + X0.
[39]
Journal o f Colloid and Interface Science, Vol. 87, No. 2, June 1982
XACCA(E) = {2Be + k T [ 2 B - (2Be)Z]} x exp{-Be2};
[42]
(3) the RJ solution, given by Eq. [20],
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES
-¢" 1.0 ,. P< A~'L~f o
//'~
1.1
",::X "~'~,,X,
,
''/., 0.5
I/
0.0
485
Q5
tO
t,~
RELATIVE ADGORPION ENER6Y
2,0
F.. [KCAL/MOLE]
FIG. l. Comparison of the approximate solutions CA, ACCA, RJ, with the exact solution ST for the adsorption systems obeying the Dubinin-Radushkevich Eq. [40]. The calculation is done for the Langmuir model of adsorption, and the parameter B = 1.4 × 10-6 moles ca1-2.
XRJ(E)= [ 2Be
(TrkT)2 [-12BZe + × exp{-Be~}.
[43]
H o w e v e r , for this particular case of the Langmuir local isotherm, a simple accurate solution of the integral equation [2] can be obtained, by means o f the method of Stieltjes transform (ST) (35) ×ST(e) =
exp{B(~kT) 2} kT × sin{2~'BkT}.cxp{-Be2}.
[44]
For the purpose o f illustration, we have taken the v a l u e B = 1.4 x l0 -6 mole 2 ca1-2, which characterizes for example the heterogeneity of the surface in the adsorption system a r g o n - a e r o s i l at 75°K (36, 37). This value lies in the range of typical values of the parameter B, as shown by Rudzihski et al. (38). Figure 1 shows the comparison o f all the approximate solutions, CA, ACCA, RJ, with the exact solution ST obtained by the method o f Stieltjes transform.
It can be seen from this figure that for such a typical case o f surface heterogeneity, the approximate solution RJ does not diverge more than by 5% from the exact solution S T . Note that the accuracy of the RJ solution will be the same in the case o f the adsorption systems possessing a random topography o f surface (also when the interactions between the adsorbed molecules are taken into account). This is because the spread o f the function (O0/Oe~) and the coefficient C2 will be the same as in the case of the Langmuir local adsorption. In the case of the adsorption systems characterized by the patchwise surface topography, the approximate RJ solution will be even closer to the exact solution ST. This is because the spread of the function (O0/Oe~) and the coefficient C~ will be smaller, compared to the Langmuir model o f adsorption. (We assume at this moment the more probable case of the attractive forces between the adsorbed molecules.) Now, let us come to the essential problem of the simultaneous calculation of the adsorption energy distribution from the exJournal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
486
RUDZIlqSKI,JAGIELLO,AND GRILLET
,-¢ -g
2.6. (3
as
2,2. 0,4.
ao:
-.I
:~ "I.8
&
,
d4
~
&
n
ag
i
!
,o
ADSO~B,~T£ P~£SSU~£ p(mrnH 9)
FI6. 2. Comparison of the adsorption isotherms and the isosteric heats of adsorption for oxygen adsorbed on boron nitride (& A) and argon adsorbed on Gasil I, outgassed at 150°C • o). perimental adsorption isotherm and the isosteric heat of adsorption. These two functions are the two adsorption observables which are most often investigated. However, papers in which they are reported simultaneously are very rare, especially those where the heats of adsorption were measured directly by a calorimetric method. The so-called "isosteric method" involves some uncertainties, as discussed by Grillet et al. (39) and by Koubek et al. (40). The C.N.R.S. Laboratory was the place where several experimental methods have been developed for a simultaneous independent measurement of adsorption isotherm and the isosteric heat of adsorption. Using these methods, Rouqueroll and coworkers have investigated a variety of adsorption systems. Among the experimental data they have published, we have selected two adsorption systems for the purpose of our numerical investigation. One of them is the system argon-Gasil I, investigated experimentally by Rouqueroll et al. (41), in the scope of the SCI/IUPAC/ Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
NPL Project on Surface Area Standards, described by Everett et al. (42). The adsorption data for two samples of Gasil I, outgassed at 150 and 400°C, have been taken into consideration by us. The second system considered by us was oxygen adsorbed on boron nitride at 77.5°K, investigated experimentally in the C.N.R.S. Laboratory in Marseille. Figure 2 shows the unpublished experimental data for this system. There the experimental data for argon on Gasil I have been shown for the purpose of comparison. The first general conclusion which can be drawn from Fig. 2 is that the adsorption isotherms themselves do not seem to indicate serious differences between the properties of these two adsorption systems. At the same time, the isosteric heats of adsorption seem to reveal deeper differences, which will be confirmed soon by our calculation of the energy distribution functions for these systems. Figures 3 and 4 show the related energy distributions, evaluated for these systems
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES
487
OAStL I / ~O0*C
C
Z 0
~2o, b.
o ~o.
10o
Q
-~jOO ~
-I0o
"foO
FIG. 3. The energy distribution functions for argon adsorbed on Gasil I. The solid lines denote the energy distribution functions, evaluated from adsorption isotherms, whereas the dotted lines denote the distribution functions evaluated from isosteric heats of adsorption. by means o f the method described in the previous section. We can see there that, while the distribution o f the adsorption energy on the silica (Gasil I) samples is of an "exponential"
~.. []
~.
/ ~
type, the boron nitride sample can be well characterized by the " s k e w - g a u s s i a n " energy distribution function. It has been k n o w n for a long time that the exponential energy distribution is responsible for the
RANrlOPI
PATOHWt.5£
TOPOGRAPHY
" r o P o O R A laPRY
'\ 200
- 600
-
- ~0o
O
Rf.LATIVE A D , 5 0 ~ P T I O H EHEROY F~C(.o (co,L/rnoLo~ FIG. 4. The energy distribution functions for oxygen adsorbed on boron nitride, evaluated with the assumption of the patchwise and random surface topography. Journal o f Colloid and Interface Science, Vol. 87, No. 2, June 1982
488
RUDZIlqSKI, JAGIELLO,AND GRILLET
"Freundlich behavior" of adsorption systems, whereas the skew-gaussian distribution causes the adsorption systems to exhibit the "Dubinin-Radushkevich behavior." This, of course, must be considered as a crude historical classification. We are now at the stage of an accurate determination of the form of the adsorption energy distribution. Regarding the details of our calculation, we would like to draw the reader's attention to the parameters to be fitted best numerically. In fact, there are only the following two parameters: (1) the parameter P1 = - k T In K - Q0 + e~; and (2) the parameter P~ = z u . This is because the parameter P0 = k T In K - E~ causes only some shift of the evaluated energy distributions on the energy scale, and does not affect the form of the distribution function. Consequently, this parameter has no effect on the obtained best fit, and cannot, therefore, be estimated numerically. This was the reason why we used in Figs. 3 and 4 the relative energy scale esc0~ ~(o~ = es~ - k T
In K
+ e'l= -kTlnp
-zuf.
[45]
A proper choice of the parameters P1 and P2 (and of the surface topography model), should cause the two energy distributions obtained from adsorption isotherm and heat of adsorption to match each other. They are marked respectively by the solid and dotted lines in Figs. 3 and 4. In the case of argon adsorbed on the silica (Gasil I) samples, the best possible convergence of the two distribution functions is obtained v¢ith the assumption z u = 0: in other words, by accepting the Langmuir model of adsorption. The surface topography does not, therefore, play any role in this case. A slightly different situation has been discovered in the oxygen on boron nitride system. Here the assumption of a patchwise surface topography seems to lead to a better convergence of the two distribution funcJournal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
tions calculated from adsorption isotherm and heat of adsorption. The numerically estimated parameter z u takes the value 175 cal/mole, which corresponds to the critical temperature 22°K (the critical temperature of the 2D phase transition, lattice gas dense ordered phase). The assumption of a random topography for this system yields the parameter z u = 0, and the convergence which is slightly worse than previously. It seems that the difference between the two convergences are not great enough to determine without doubt that the patchwise topography is the proper model for the oxygen on boron nitride system. Nonetheless, in view of the present results, we shall accept it as a working hypothesis. Both in the case of the patchwise and the random surface topography, the distribution functions obtained from adsorption isotherm show, at the highest investigated energies (lowest pressures), some slightly different features than the distribution functions evaluated from heat of adsorption. There may be various explanations for this: We feel that either experimental errors are responsible for that, or the existence of a maximum value of the adsorption energy e', neglected in our hitherto theoretical consideration. There is still another reason why the oxygen on boron nitride system is very interesting: the skew-gaussian form of its distribution function makes possible the estimation of the value of the relative energy ect0)°,corresponding to the minimum adsorption energy E~. At this point we have, Ecco~° = P~
= - k T In K + el. '
[46]
The use of CA approximation also yields a simple physical meaning for the lowest heats of adsorption Qst,1, observed at the highest surface coverages. For the patchwise surface topography to be accepted for this system, we have, 1 ¢ Qst,1 = P 4 = QO + _ z u + el. 2
[47]
489
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES TABLE I The Experimentally Estimated, and Numerically Calculated Values of the Parameters Appearing in the Proposed Method System
Oxygen on boron nitride (patchwise topography) Argon on Gasil I/ 150°C (Langmuir model) Argon on Gasil I/ 400°C (Langmuir model)
P, (¢al/mole)
~st,i (cal/mole)
P3 (cal/mole)
{~}o (cal/mole)
~dt (¢al/mote)
K (rnm Hg)
2230
1820
-600
1630
100
109.9
2690
2160
-550
2140
20
44.7
2680
2150
-550
2130
20
44.7
Knowing the numerically determined parameters P1, P~, and the experimentally estimated parameters P3 and P4, one can evaluate Q0, K, and E;. For the considered oxygen-boron nitride system, Table I reports the values of the parameters Pi's, together with the values of Q0, K, el, calculated in this way. A similar calculation has been done for the adsorption systems formed by argon adsorbed on the two silica (Gasil I) samples. Here, an assumption has been made that the lowest investigated energies E~t0~correspond to the value el on the true energy scale. This assumption is justified by the fact that these lowest relative energies correspond to the experimental coverages, which are very near unity. Then, according to CA approximation, the adsorption sites are covered, possessing the lowest adsorption energy E~. The results of this calculation are reported in Table I. Table I reveals several interesting features of our adsorption systems such as the low values of the lowest adsorption energy el. It suggests thai the cancellation takes place between the values of U0 and Ev, on the adsorption sites exhibiting the lowest adsorption energy E~. This result should not surprise us. Rather it helps us to realize the relative sense of the "monolayer capacity" value, which is in fact temperature dependent.
Namely, in the model of the localized adsorption accepted here, not all the local minima of the adsorption potential U(x, y, z) can serve as "adsorption sites." These are only the minima where the adsorption potential U0 prevails over the vibrational energy Ev. (It is generally assumed that Ev is the same for all adsorption sites.) When the experimental temperature increases, the vibrational energy Ev also increases, which causes the effective number of the adsorption sites to decrease. This effect will result in an apparent decreasing of the estimated monolayer capacity value, as the experimental temperature increases. Observations of this kind have already been reported by Rudzifiski and Wojeiechowski (15). Now, let us consider at last theproblem of estimating the surface capacity of the heterogeneous surfaces. It seems that the only fully justified procedure is ttie. a posteriori normalization (to unity)' of the evaluated energy distribution function. We have applied this procedure to the oxygen-boron nitride system, and in this way obtained the value 5.27 cm3/g. This value has been used by us to make the plots in Fig. 2 (for oxygen on boron nitride). The form of the distribution function in Fig. 4 suggests that our experiment covers the essential range of the adsorption energies Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
490
RUDZIlqSKI, JAGIELLO, AND GRILLET
~
4
2
(19
"1
0
~bo rH£
~0o rmu£
~ o v
~,oa
4oo
"
SCALE £ ' C(x~L/moLe)
Fxo. 5. Comparison of the energy distribution functions, using the true energy scale c'. (1) The distribution function for oxygen on boron nitride, (2) the distribution function for argon on Gasil I outgassed at 150°C, and (3) the distribution function for argon on Gasil I outgassed at 400°C.
for this system, and the normalization is therefore possible. On the other hand, the f o r m of the distribution functions in Fig. 3 cannot m a k e us certain that the evaluated energy distributions c o v e r the whole essential range o f the adsorption energies for argon adsorbed on these silica samples. F o r this reason, we h a v e accepted the B E T estimation for these systems, reported in the previous p a p e r by GriUet et al. (41). D o r m a n t and A d a m s o n (3) have suggested that some special compensation for interaction and heterogeneity effects at high relative coverages m a y decrease the errors ha the B E T method, when applied to adsorption systems with heterogeneous surfaces. An a d v a n c e d theoretical analyze of this p r o b l e m can be found in the recent w o r k s by Oh and K i m (42). W e conclude our experimental section with the observation, which can be drawn f r o m Fig. 3, that the outgassing o f the silica samples in higher t e m p e r a t u r e s yields m o r e heterogeneous surfaces. In Fig. 3 the distribution function for the silica sample outgassed at 400°C is more flattened, c o m p a r e d Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982
with that outgassed at 150°C. This can be seen e v e n more clearly in Fig. 5, where all three distribution functions have b e e n shown simultaneously, using the true energy scale e'. The energy distribution functions shown in Fig. 5 r e p r e s e n t the arithmetic average from the two energy distribution functions, calculated f r o m the adsorption isotherm and heat o f adsorption. REFERENCES 1. Adamson, A. W., and Ling, I., Adv. Chem. Ser. 33, 51 (1961). 2. Adamson, A. W., Ling, I., Dormant, L., and Orem, M., J. Colloid Interface Sci. 21, 445 (1966).
3. Dormant, L. M., and Adamson, A. W., J. Colloid Interface Sci. 38, 285 (1972). 4. Dormant, L. E., and Everett, D. H., unpublished results, work done in 1970 at the University of Bristol. 5. Dormant, L. M., and Adamson, A. W., Surface Sci. 62, 337 (1977). 6. Ross, S., and Olivier, J. P., "On Physical Adsorption." Interscience, London, 1964. 7. Ross, S., and Morrison, D., Surface Sci. 52, 103 (1975). 8. Morrison, I. D., and Ross, S., Surface Sci. 62, 331 (1977).
PHYSICAL ADSORPTION OF GASES ON SOLID SURFACES 9. Rudzifiski, W., and Jaroniec, M., Surface Sci. 42, 552 (1974). 10. Rudzifiski, W., Jaroniec, M., Soko~owski, S., and Cerofolini, G. F., Czech. J. Phys. B25, 891 (1975). 11. Jaroniec, M., and Rudzifiski, W., Colloid Polym. Sci. 253, 683 (1975). 12. Jaroniec, M., Surface Sci. 50, 553 (1975). 13. Rudzifiski, W., Waksmundzki, A., Jaroniec, M., and Sokotowski, S., Polish J. Chem. 48, 1985 (1974). 14. Jaroniec, M., and Rudzifiski, W., Acta Chim. Acad. Sci. Hungaricae 88, 351 (1976). 15. Rudzifiski, W., and Wojciechowski, B. W., Colloid Polym. Sci. 255, 869 (1977), 255, 1086 (1977). 16. House, W. A., and Jaycock, M. J., J. Colloid Interface Sci. 59, 252 (1977). 17. House, W. A., J. Colloid Interface Sci. 67, 166 (1978). 18. House, W. A., and Jaycock, M. J., J. Colloid Polym. Sci. 256, 52 (1978). 19. House, W. A., J. Chem. Soc. Faraday Trans. I 74, 1045 (1978). 20. Drain, L. E., and Morrison, J. A., Trans. Faraday Soc. 48, 316 (1952), 48, 840 (1952). 21. Jackson, D. J., and Dawis, B. W., J. Colloid Interface Sci. 47, 499 (1974). 22. van Dongen, R. H., Surface Sci. 39, 341 (1973). 23. Rudzifiski, W., Lajtar, L., and Patrykiejew, A., Surface Sci. 67, 195 (1977). 24. Rudzifiski, W., Patrykiejew, A., and Lajtar, L., Surface Sci. 77, L655 (1978). 25. Hill, T. L., J. Chem. Phys. 17, 762 (1949). 26. Tompkins, F. C., Trans. Faraday Soc. 46, 569 (1950).
49 1
27. Rudzifiski, W., and Lajtar, L., J. Chem. Soc., Faraday Trans. H 77, 153 (1981), 28. Stoeckli, F., Ph. Thesis by Dr. Morel, University of Neuchatel, 1980. 29. Cerofolini, G. F., Surface Sci. 24, 391 (1971). 30. Hobson, J. P., Canad. J. Phys. 43, 1934 (1965). 31. Jagiek~o, J., Ph. thesis, UMCS Lublin, 1981. 32. Cerofolini, G. F., Thin Solid Films 26, 53 (1975). 33. Rudzifiski, W., Narkiewicz, J., Wojciechowski, B. W., and Hsn, C. C., J. Colloid Interface Sci. 67, 292 (1978). 34. Rudzifiski, W., Narkiewicz, J., and Patrykiejew, A., Z. Phys. Chem. 260, 1097 (1979). 35. Landman, U., and Montroll, E. W., J. Chem. Phys. 64, 1762 (1976). 36. Nicolaon, G., and Teichner, S., J. Colloid Interface Sci. 38, 172 (1972). 37. Hsu, C. C., Rudzifiski, W., and Wojciechowski, B. W., Phys. Lett. 54A, 365 (1975). 38. Rudzifiski, W., Jaroniec, M., and Lajtar, L., Wiadomogci Chemiczne 30, 305 (1976). 39. Grillet, Y., Rouqueroll, F., and Rouqueroll, J., Rev. G~n. Therm. Fr. 171,237 (1976). 40. Koubek, J., Pasek, J., and Vole, J., J. Colloid Interface Sci. 51, 491 (1975). 41. RouqueroU, J., Rouqueroll, F., P6r6s, C., Grillet, Y., and Boudellal, M., in "'Characterization of Porous Solids" (S. J. Gregg, K. S. W. Sing, and H. F. Stoeckli, Eds.). London Soc., 1979. 42. Everett, D. H., Parfitt, Go D., Sing, K. S. W., and Wilson, R., J. Appl. Chem. Biotechnol. 24, 199 (1974). 43. Oh, B. K., and Kim, S. K., J. Chem. Phys. 61, 1797 (1974), 67, 3416 (1977).
Journal of Colloid and Interface Science, Vol. 87, No. 2, June 1982