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Langmuir 1997, 13, 990-994
Kinetics of Desorption from Heterogeneous Surfaces† G. F. Cerofolini* EniChemsIstituto Guido Donegani, 28100 Novara NO, Italy
N. Re Dipartimento di Chimica, Universita` di Perugia, 06100 Perugia PG, Italy Received September 29, 1995X The desorption kinetics from heterogeneous surfaces contain information on energy heterogeneity. Mathematical methods for extracting the energy distribution function from desorption kinetics are developed. The overall desorption kinetics are described as weighted averages of the local desorption kinetics from each energetically homogeneous zone forming the surface. Whatever the local kinetics, the energy distribution is determined from the overall kinetics by using the condensation approximation. The method is applied to two situations: the time-logarithm law observed in a lot of systems of practical interest in catalysis, and the time-power law observed in many relaxation kinetics of macromolecules.
1. Introduction Information on the energetic structure of a surface is contained in the equilibrium adsorption isotherm θ(p), i.e., the functional relationship linking the average surface coverage θ to the partial pressure p of a given gas in equilibrium conditions. Roughly speaking, the most energetic sites are preferentially filled at low pressure, while a high pressure is necessary for the occupation of low-energy sites. These naive considerations have been developed to the stage of a formal theorysthe mathematical theory of adsorption on heterogeneous surfaces.1-3 On the same intuitive basis it is clear that information on the energetic structure of a surface is contained in the desorption kinetics too. In fact, it is expected that the initial stages of desorption involve the least bonded molecules, while the final stages are related to the desorption of the most bonded molecules. In general, because of the thermally activated nature of the desorption phenomenon, the characteristic desorption time τ depends on the desorption energy E as
τ ) τ0 exp(E/kBT)
(1)
where τ0 is a suitable pre-exponential factor, kB is the Boltmann constant, and T is the absolute temperature. On one hand, the exponential dependence (1) is such that the desorption from different zones takes place at very different timessthat allows the energy distribution function of the surface to be determined, at least in principle. On another hand, if the difference EM - Em between the maximum and minimum desorption energy is of several kBT, the determination of the energy distribution function requires the observation of the desorption isotherm over a time duration covering many orders of magnitudesthat may be unpractical. This difficulty may be overcome either by determining portions of the complete isothermal desorption kinetics * Corresponding author. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, September 15, 1996. (1) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (2) Rudzin´ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (3) Cerofolini, G. F.; Re, N. Riv. Nuovo Cimento 1993, 16, No. 7, 1.
S0743-7463(95)00813-4 CCC: $14.00
at different temperatures or by determining the desorption spectrum when a temperature ramp is applied to the system (as it happens for thermal programmed desorption). Both of these situations can however be considered as special cases of the conceptually simpler case of isothermal desorption. The theory of isothermal desorption kinetics from heterogeneous surfaces can be established by describing the overall desorption kinetics as weighted averages of the local desorption kinetics from each energetically homogeneous zone forming the surface. In general, the problem of extracting the energy distribution function φ(E) from the overall kinetics δν(t) (where ν is the supposedly known kinetic order) is an ill-posed problem and originates unstable solutions. Whatever is the order the local desorption kinetics, the instability can however be removed, thus allowing the energy distribution to be determined from the overall kinetics, via approximate methods like the condensation approximation and the asymptotically correct approximations.4 In this work the first and most simple of these methods is applied to two situations: the Elovich equation observed in a lot of systems of practical interest in catalysis, and the nonexponential relaxation isothermal kinetics observed in protein molecules. 2. Desorption Kinetics from Homogeneous Surfaces The rate equation giving the time derivative dΘ/dt of the surface coverage Θ of a homogeneous surface with desorption time τ is assumed to be of the form
-
dΘ 1 ν ) Θ dt τ
(2)
where ν is the reaction order. The analysis will be limited to kinetics of the first order (ν ) 1) and second order (ν ) 2) and to kinetic coefficients τ independent of Θ. If Θ0 is the local coverage at time t ) 0 the rate equation (2) can be solved for all ν by separation of variables
-
t ) ∫ΘΘ(τ) dΘ Θν τ 0
(3)
thus giving a function Θ ) Θν(t,τ). Equation 3 can be solved in closed form for all ν. (4) Cerofolini, G. F.; Re, N. J. Colloid Interface Sci. 1995, 174, 428.
© 1997 American Chemical Society
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Defining ∆ν(t,τ) :) Θ0(τ) - Θν(t,τ), one has
∆1(t,τ) ) Θ0(τ)[1 - exp(-t/τ)]
δ1(t) ) (4)
Θ02(τ)t Θ0(τ)t + τ
(5)
for second-order kinetics. 3. Desorption Kinetics from Heterogeneous Surfaces In general the desorption kinetics from patchwise heterogeneous surfaces are given by
δν(t) )
∫D
τ
∆ν(t,τ)φ(τ) dτ
(6)
where φ(τ) dτ is the fraction of surface with desorption time between τ and τ + dτ and Dτ is the support of φ(τ). Equation 6 originates the problem of determining the distribution function φ(τ) when the overall kinetics δν(t) are experimentally known. The overall amount δν(t) desorbed at time t depends on the lifetime distribution function φ(τ) and on the set of initial conditions Θ0(τ). Clearly enough, if one wants to extract the distribution function φ(τ) from the experimental function δν(t), one must operate in conditions for which Θ0(τ) is known. In the forthcoming development the attention will be limited to the case of Θ0 constant with τ. In this case one may, without loss of generality, take Θ0 ) 1, so that eqs 4 and 5 become
∆1(t,τ) ) 1 exp(-t/τ)
(7)
and
∆2(t,τ) )
t t+τ
(8)
respectively. The theory proceeds more easily specifying the desorption time τ as in (1). Since in most cases τ0 ≈ 10-13 s, E is the unique random variable characterizing the surface in desorption experiments. The situation does not change even when τ0 depends on E, and the forthcoming considerations can be extended with minor modifications provided that τ0(E) exp(E/kBT) is a monotonically increasing function of E; an analysis of the case in which τ0 is not constant is given in ref 5. Rather than describing the surface with the distribution function φ(τ), it is convenient to describe the surface with the distribution function φ(E), where φ(E) dE is the fraction of surface with desorption energy between E and E + dE. Since in the desorption kinetics of physisorbed molecules one may reasonably assume that E coincides with the adsorption energy q, and since φ(q) can be determined from adsorption equilibrium isotherms, there is the possibility to compare the results of the theory with those of a well established theory. Defining ∆ ˆ ν(t,E) :) ∆ν(t,τ(E)), where τ(E) is given by eq 1, eq 6 becomes
δν(t) )
[
(
1 - exp -
(
))]
t E exp τ0 kBT
φ(E) dE (10)
for first-order kinetics, and
for first-order kinetics and
∆2(t,τ) )
∫0+∞
∫0+∞ ∆ˆ ν(t,E)φ(E) dE
Equation 9 is specialized to the form (5) Bogillo, V. I.; Shkilev, V. P. Kinet. Katal. 1995, 36, 849.
(9)
δ2(t) )
∫0+∞ t + τ
t φ(E) dE exp(E/k 0 BT)
(11)
for second-order kinetics. It is noted that though the integration interval has been taken (0,+∞), in practice the support DE of φ(E) is contained in a finite interval (Em, EM). The integral representations (10) and (11) of δ1(t) and δ2(t) are the basic equations of the theory. 4. The Condensation Approximation A close inspection of kernels (7) and (8) with τ given by eq 1 shows that they resemble strictly the Langmuir isotherm and the Jovanovic isotherm, respectively, of equilibrium adsorption.1,2 The analogy, however, is not complete: in fact, assuming the one-to-one correspondences p T t and q T E, the equilibrium equations do not depend on p and q separately but rather on the function p exp(q/kBT), while the desorption kinetics do not depend on t and E separately, but rather on the function t exp(-E/kBT)snote the different signs of the exponents. In spite of this, as first suggested by Jaroniec6 many of the methods developed for the theory of adsorption equilibrium on heterogeneous surfaces can be extended to desorption kinetics from heterogeneous surfaces. For both the Langmuir and Jovanovic isotherms several exact methods exist for the determination of the energy distribution function φ(q) when the overall adsorption isotherm is precisely known. Rather than trying to modify these exact methods for desorption kinetics, this work will use the condensation approximation, which gives a simple analytical expression for the distribution function in terms of the derivative of the overall isotherm and which does not produce unstable solutions.3 We mention that the analysis of the overall desorption kinetics can be extended by using the methods developed for equilibrium adsorption by Hobson8 and Cerofolini9 (asymptotically correct approximation), Nederlof et al. (logarithmic symmetrical local isotherm approximation),7 Rudzin´ski and co-workers (third order approximation),10-13 and Re (nth order approximation).14 Details are given in ref 4. The condensation approximation (CA) was introduced in the study of equilibrium adsorption on heterogeneous surfaces by Roginsky in the 1940s.15,16 However, it was only after Harris’ rediscovery17 and Cerofolini’s systematic application of CA to realistic model isotherms18 that this (6) Jaroniec, M.; Garbacz, J. K. Thin Solid Films 1979, 62, 237. (7) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. J. Colloid Interface Sci. 1990, 135, 410. (8) Hobson, J. P. Can. J. Phys. 1965, 43, 1934, 1941. (9) Cerofolini, G. F. Thin Solid Films 1974, 23, 129. (10) Hsu, C. C.; Wojciechowski, B. W.; Rudzin´ski, W.; Narkiewicz, J. J. Colloid Interface Sci. 1978, 67, 292. (11) Rudzin´ski, W.; Narkiewicz, J.; Patrykiejew, A. Z. Phys. Chem. (Leipzig) 1979, 260, 1097. (12) Rudzin´ski, W.; Jagiełło, J. J. Low Temp. Phys. 1981, 45, 1. (13) Rudzin´ski, W.; Jagiełło, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478. (14) Re, N. J. Colloid Interface Sci. 1994, 166, 191. (15) Roginsky, S. Z. C. R. Acad. Sci. USSR 1944, 45, 61, 194. (16) Roginsky, S. Z. Adsorption and Catalysis on Heterogeneous Surfaces; Akad. Nauk USSR: Moscow, 1949. (17) Harris, L. B. Surf. Sci. 1968, 10, 129; 1969, 13, 377; 1969, 15, 182. (18) Cerofolini, G. F. Surf. Sci. 1971, 24, 391; J. Low Temp. Phys. 1972, 6, 473.
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method has become of wide use in the analysis of real adsorption systems. The CA consists in replacing the kernel ∆ ˆ ν(t,E) by the step function
∆νc(t,E) )
{
1 for t g tνc(E) 0 for t < tνc(E)
(12)
where the function tνc(E) is obtained by minimizing the distance d between the local kinetics ∆ ˆ ν(t,E) and their approximant (12)
d [∆ ˆ ν(t,E), ∆νc(t,E)] ) min
(13)
It is immediately verified that condition 13 is satisfied for the Lagrangian distance and for the L2 distance by taking
tνc(E) ) [ν - 1 + ln(3 - ν)]τ0 exp(E/kBT)
(14)
which applies to both first-order and second-order kinetics. Defining the inverse function of (14), Eν :) kBT ln(t/[ν - 1 + ln(3 - ν)]τ0), and putting δˆ ν(Eν) Z δν(t(Eν)), one has
δˆ ν(Eν) )
∫-∞E φνc(E) dE ν
(15)
Of course, the solution of eq 15 will in general differ from the solutions of eq 10 or 11, and for this reason the distribution function φνc(E) in eq 15 has been denoted with the index “c”. Equation 15 is immediately solved by a differentiation with respect to Eν
φνc(Eν) ) ∂δˆ ν(Eν)/∂Eν
(17)
δˆ 2mM(E2) )
{
(
rE ln 1 + δM
( ))
E2 τ0 exp tm kBT
for -∞ < E2 e E2M
where E2M :) E2m + kBT ln(exp(δM/rE) - 1), with E2m :) kBT ln(tm/τ0). In the sequel the index “2” to E and φc will be omitted to have more manageable expressions. The application of (16) to (20) gives
φc(E) )
{
exp
(
rE/δM × kBT
)[
E - Em kBT
0
(
1 + exp
)]
E - Em kBT
-1
for -∞ < E e EM for EM < E < +∞ (21)
Of the many isothermal kinetics observed on real surfaces for different processes like oxidation, chemisorption, and desorption, one has deserved special attention: the time-logarithm law usually known as the Elovich equation.19 An extended list of systems obeying the Elovich equation in chemisorption is reported in ref 20. 5.1. The Elovich Equation. The Elovich equation is usually written
{
rE ln(1 + t/tm) for 0 e t < tM δM for tM e t
(20)
for E2M < E2 < +∞
5. The Energy Distribution Function Originating the Time-Logarithm Law
δmM(t) )
(19)
in some situations tM (and hence δM) may be allowed to go to +∞. The macroscopic meaning of the Elovich parameters is straightforward: tm is a characteristic time below which the logarithmic behavior is no longer observed, tM is the time required to complete the process, while rE/tm is the initial desorption rate. The first attempts to understand the time-logarithm law started immediately after its experimental discovery.21 However, the first systematic search of the reasons explaining the Elovich equation is probably due to Porter and Tompkins.22 These authors assumed that the timelogarithm law (18) is observed during activated chemisorption and interpreted the resulting equation in terms either of pre-existing surface heterogeneity or of variation of the activation energy with coverage during the process (induced heterogeneity). As discussed in ref 23, the time-logarithm law in desorption experiments can hardly be understood in terms of induced heterogeneity, so that in this part we shall describe how the Elovich equation is explained in terms of fixed heterogeneity by discussing eq 18 in light of the CA method. 5.2. The Time-Logarithm Law in the Condensation Approximation. Let us first consider second-order kinetics. Expressed in terms of E2 the Elovich equation (18) reads
(16)
Since E2 ) E1 + kBT ln(ln 2) and dE2 ) dE1 the distribution function calculated in the hypothesis of firstorder kinetics coincides with that calculated in the assumption of second-order kinetics provided that the energy axis is shifted by an amount -kBT ln(ln 2) ) 0.367kBT. The approximation involved in this method can be evaluated with the methods developed by Harris17 and Cerofolini;18 it is mentioned without further discussion that eq 16 gives an adequate description of the distribution function φ(E) provided that the energy spectrum DE is much wider than kBT, EM - Em . kBT, and φ(E) varies smoothly with E
φ′(E)/φ(E) , 1/kBT
tM :) tm[exp(δM/rE) - 1]
(18)
where δ(t) is the amount of matter involved in the kinetics at time t and rE, tm, tM, and δM are four characteristic parameters mutually related by the relationship (19) Elovich, S. Y.; Zhabrova, G. M. Zhr. Fiz. Khim. 1939, 13, 1761. (20) Aharoni, C.; Tompkins, F. C. Adv. Catal. 1970, 21, 1.
whatever the value of δM. A study of the distribution function (21) shows the following limiting behaviors:
φc(E) =
{
rE/δM × kBT
(
exp 1 0
)
E - Em kBT
for -∞ < E j Em - Ο(kBT) for Em + Ο(kBT) j E e EM for EM < E < +∞
(22)
The calculated distribution function cannot be an arbitrary (21) Characorin, F.; Elowitz, S. Acta Physicochim. URSS 1936, 5, 325. (22) Porter, A. S.; Tompkins, F. C. Proc. R. Soc. London 1953, A217, 529. (23) Cerofolini, G. F. In Adsorption on New and Modified Inorganic Sorbents; Dabrowski, A., Teztykh, V. A., Eds.; Elsevier: Amsterdam, 1995; p 435.
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Langmuir, Vol. 13, No. 5, 1997 993
function of E and must satisfy the mathematical conditions of non-negativity and of normalization. Physical conditions too have to be satisfied; for instance, φc(E) must be identically null for E < 0 and must be temperature independent. There is no choice of Em and EM for which function 21 is temperature independent; however, the asymptotic expansion (22) shows that φc(E) varies with T only in a region centered on Em of width Ο(kBT), while the calculated distribution function is approximately constant in the whole interval (Em, EM) provided that the extremes Em and EM, and therefore the difference EM - Em, are constant with T. Equation 22 states that EM must be interpreted as the maximum adsorption energy, EM ) EM, while the distribution function vanishes exponentially with E - Em for E j Em. Though strictly speaking DE does not admit a minimum, Em can in a way be interpreted as an estimate of the minimum adsorption energy, Em = Em. Since in general φ(E) * φc(E), even the maximum desorption energy EMc determined within the CA is in principle different from the true desorption energy EM. The normalization condition for φc(E)
∫-∞E φ(E)c dE ) 1 c
M
gives the following equation for Emc
(
(
))
EMc - Em rE ln 1 + exp kBT
Figure 1. Energy distribution function calculated in the condensation approximation from time-logarithm law for three values of Ω. The distribution function is substantially independent of T for Ω/kBT . 1.
) δM
whose solution is
( ( ) )
EMc - Em ) kBT ln exp
δM δM - 1 = kBT (23) rE rE
Combining this relationship with the physical meaning of Em and EM, one eventually gets the meaning and temperature dependence of the parameters tm and rE: tm ≈ τ0 exp(Em/kBT), with Em = Em; and rE = δMkBT/Ω, with Ω :) EM - Em = EM - Em (thus Ω is the width of the energy spectrum). When first-order kinetics are considered, the Elovich equation (18) reads
δˆ 1mM(E1) )
{
(
rE ln 1 + δM
( ))
E1 τ0 ln 2 exp tm kBT
for -∞ < E1 e E1M for E1M < E1 < +∞ (24)
where
E1M Z E1m + kBT ln(exp(δM/rE) - 1)
(25)
with
is fulfilled only for τE/δM , 1, which therefore is the condition for the validity of the above argument. Note that when this condition is satisfied, the difference between the calculated distribution functions for first- and secondorder kinetics, consisting of a shift of the order of kBT on the energy axis, is negligible. Figure 1 confirms these statements. 6. The Energy Distribution Originating the Time-Power Law The time-power law is another important isothermal equation, observed in adsorption-desorption kinetics of light molecules on macromolecules like proteins. 6.1. The Time-Power Law. The time-power law reads
δp(t) ) 1 -
( )
(26)
The application of eq 16 to eq 24 gives a distribution function with the same shape as (21) in which E2, E2m, and E2M are replaced by E1, E1m, and E1M, respectively. All the considerations on minimum and maximum adsorption energies made above for second-order kinetics therefore apply to first-order kinetics provided that E2m and E2M are replaced by E1m and E1M, respectively, as calculated from eqs 25 and 26. Irrespective of the kinetic order, the CA is valid when the energy spectrum is much wider than kBT; this condition
s
(27)
where t0 and s are temperature-dependent parameters, t0 ) t0(T) and s ) s(T).24,25 As far as no crossing of electronic levels is involved in a process, first-order kinetics should be observed
(
∆ ˆ 1(t,E) ) 1 - exp E1m Z kBT ln(tm/τ0) - kBT ln(ln 2)
t0 t0 + t
(
))
t E exp τ0 kBT
The deviations of this relationship from the experimental behavior (27) can then be ascribed to a distribution of activation energies. Denoting with φ(E) the barrier-height distribution function, one gets eq 10. 6.2. The Time-Power Law in the Condensation Approximation. Substituting (27) for δ1(t) in eq 10, one (24) Iben, I. E. T.; Braunstein, D.; Doster, W.; Frauenfelder, H.; Hong, M. K.; Johnson, J. B.; Luck, S.; Ormos, P.; Schulte, A.; Steinbach, P. J.; Xie, A. H.; Young, R. D. Phys. Rev. Lett. 1989, 62, 1916. (25) Young, R. D.; Frauenfelder, H.; Johnson, J. B.; Lamb, D. C.; Nienhaus, G. U.; Philipp, R.; Scholl, R. Chem. Phys. 1991, 158, 315.
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Figure 2. Energy distribution function calculated in the condensation approximation from the time-power law for three temperatures at assigned TF. The distribution function is substantially independent of T for E > E0 + Ο(kBT).
gets an inverse problem of the same type as the one considered for the time-logarithm law. Particularly straightforward is the application of the condensation approximation. Defining E :) kBT ln(t/τ0 ln 2), the time-power law (27) expressed in terms of E reads
[
δˆ p(E) ) 1 - 1 +
( )]
τ0 ln 2 E exp t0 kBT
-s
(28)
for -∞ < E < +∞. The application of eq 16 to eq 28 gives
φc(E) )
[
(
)]
E - E0 s 1 + exp kBT kBT
(
-s-1
exp
)
E - E0 kBT
E0 , E w φc(E) =
(
)
E - E0 s exp kBT kBT
(
)
Of course, the above analysis is based on the assumption that the energy landscape of the adsorbent has a sharp minimum, so that the adsorption-energy distribution function can be considered temperature independent. When this condition is not satisfied (that is expected to occur in many situations of biological interest, as in the case of low-temperature CO readsorption on myoglobin (Mb) after photocleavage of Mb-CO26,27), the above considerations cannot be applied. 7. Conclusions
(30)
E - E0 s exp -s kBT kBT
is therefore reminescent of the one which originates the Freundlich isotherm in equilibrium adsorption.
(29)
where E0 :) kBT ln(t0/τ0 ln 2). A study of (29) shows that φc(E) has a maximum in E0 - kBT ln s and that
E , E0 w φc(E) =
Figure 3. Energy distribution function calculated in the condensation approximation from the time-power law for three values of T/TF. The distribution function is substantially independent of T for T/TF , 1.
(31)
The distribution function φc(E) satisfies the conditions of non-negativity and normalization but is temperature dependent. However, if t0(T) depends on T as t0(T) ) τ0 ln 2 exp(E0/kBT) (where E0 is an assigned energy) and s depends on T as s ) T/TF (where TF is a suitable temperature much greater than T), then (a) the most probable energy, E0 + kBT ln(TF/T) ) E0 + Ο(kBT), is almost independent of T, (b) for E > E0 the distribution φc(E) is asymptotic to a temperature-independent distribution function (as described by (31)), and (c) for E < E0 the distribution function depends strongly on T, but because of (31) is appreciably different from 0 only in a narrow region of width Ο(kBT). Figure 2 shows that for T , TF the calculated energy distribution function varies with T only in a region of width Ο(kBT) centered on E0. Figure 3 shows that the width of the temperature-dependent region lowers with T/TF. The distribution function (29)
The rate equation δ˙ ) δ˙ 0(1 + t/th)-R, with δ˙ 0, ht, and R suitable constants (R > 0), describes the most frequently observed kinetic equations for adsorption-desorption on real surfaces: the time-logarithm law for R ) 1, and the time-power law for R > 1. If these behaviors are ascribed to surface heterogeneity, one runs into the problem of finding the energy distribution functions which account for them. This problem can easily be solved with approximate analytical methods. This condensation approximation, originally proposed for adsorption equilibrium, has been extended to desorption kinetics from heterogeneous surfaces with local kinetics of the first or second order. The application of this method to the time-logarithm law and to the time-power law shows that these equations play in kinetics a similar role to the Temkin and Freundlich isotherms in adsorption equilibrium. LA950813E
(26) Austin, R. H.; Beeson, K. W.; Eisenstein, L.; Frauenfelder, H.; Gunsalus, I. C. Biochemistry 1975, 14, 5355. (27) Chu, K.; Ernst, R. M.; Frauenfelder, H.; Morant, J. R.; Nienhaus, G. U.; Philipp, R. Phys. Rev. Lett. 1995, 74, 2607.
