6398
J. Phys. Chem. 1992, 96, 6398-6405
(MgdOs), respectively, which are in excellent agreement with the experimentally observed value. However, higher level calculations, which included configurational interactions and a correction for higher excitations, produced barrier heights of 4.8 and 6.3 kcal/mol for LiO and LiO(Mg40S),respectively. If the latter values are indeed correct, then another model must be sought for the activating center.
Conclusions Under conventional catalytic reaction conditions, the activation energy for the oxidative coupling of CH4 over Li/MgO catalysts is influenced by the presence of C 0 2 in the gas phase. As an adsorbent, C 0 2poisons the basic oxygen ions that are responsible for the activation of CH4. The intrinsic activation energy for the generation of CH3' radicals at the surface appears to be about 25 kcal/mol, but when COz is present, the activation energy approaches 50 kcal/mol. Qualitatively, more of the active sites are poisoned at low temperatures than at high temperatures; therefore, there is an increase in the apparent activation energy when C02is present. Acknowledgment. We gratefully acknowledge financial support
of this research by the U.S. Department of Energy, Division of Basic Energy Sciences.
References and Notes (1) Campbell, K. D.; Morales, E.; Lunsford, J. H. J. Am. Chem. Soc. 1987, 109, 7900. (2) Feng, Y.; Gutman, D. J . Phys. Chem. 1991, 95, 6558. Feng, Y.; Niiranen, J.; Gutman, D. J. Phys. Chem. 1991, 95, 6564. (3) Driscoll, D. J.; Campbell, K. D.; Lunsford, J. H. Adv. Catal. 1987, 35, 139. (4) Lunsford, J. H. funnmuir 1989. 5. 12. (5) Campbell, K. D.; Linsford, J. H.'J. Phys. Chem. 1988, 92, 5792. (6) Ito, T.; Wang, J.-X.; Lin, C.-H.; Lunsford, J. H. J. Am. Chem. SOC. 1985. 107., -5062. ~. --(f)Amorebieta, V. T.; Colussi, A. J. J . Phys. Chem. 1988, 92, 4576. (8) DeBoy, J. M.; Hicks, R. F. J . Catal 1988, 113, 517. (9) Korf, S. J.; Roos, J. A.; de Bruijn, N. A. van Ommen, J. G.; Ross, J. R. H. J . Chem. SOC.,Chem. Commun. 1987, 1433. (10) Martir, W.; Lunsford, J. H. J . Am. Chem. SOC.1981, 103, 3728. (11) Lin, C.-H.; Wang, J.-X.; Lunsford, J. H. J. Catal 1988, 111, 302. Tong, Y.; Lunsford, J. H. J. Chem. Soc., Chem. Commun. 1990, 792. (12) Roos, J. A.; Korf, S. J.; Veehof, R. H. J.; Van Ommen, J. G.; Ross, J. R. H. Appl. Catal. 1989, 52, 131. (13) Shi, C.; Hatano, M.; Lunsford, J. H. Catal. Today 1992, 13, 191. (14) Shi, C.; Xu, M.; Rosynek, M. P.;Lunsford, J. H. To be published. (15) Bolrve, K. J.; Pettersson, L. G. M. J . Phys. Chem. 1991, 95, 7401. ~
Diffusion-Controlled Reactions on Porous Silicas: Mechanisms, Surface Diffusion Coefficients, and Effects of Geometry Joshua Samuel, Michael Ottolenghi,* and David Avnir* Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel (Received: April 3, 1992)
The luminescence quenching reaction of R ~ ( b p y ) by ~ ~molecular + oxygen on a porous silica and on controlled porous glass was studied in the 88-353 K temperature range. Several distinct reaction mechanism ranges were observed. In the lowest temperature range the reaction is exclusively Langmuir-Hinshelwood (LH) and is controlled by the surface diffusion of O2 The theoretical model of Freeman and Doll is applied to the experimental data, yielding, for the first time, surface diffusion coefficients in a gas-phase porous solid interface. At higher temperatures the reaction remains LH but crosses into a non-diffusion-controlledrange. At still higher temperatures a modified Eley-Rideal (ER) reaction is observed, where although O2surface diffusion is minimal, surface residence times influence the reaction rate. The effects of surface geometry (average pore size and fractal dimension) are analyzed for each of the above mechanisms. We find that surface geometry plays a role only in the LH domain. The geometry of the surface affects the diffusion coefficient of the mobile O2 reactant with respect to both the pre-exponential factors and the activation energies.
Introduction Bimolecular reactions at the interface of amorphous solids are of considerable practical and theoretical interest.' A primary question that pertains to reactions at the gas-solid interface is their basic reaction mechanism. Such procaw are usually defined as Eley-Rideal (ER)z for a target annihilation reaction between a gas-phase molecule and a surface-bound molecule or Langmuir-Hinshelwood (LH)3 for a reaction between two adsorbed molecules. A clear discrimination between the two mechanisms is usually unavailable. This issue was treated by Gafney and co-workers4for a quenching reaction on porous Vycor, where an LH mechanism was proposed. More recently Thomas and coworkerss studied the oxygen quenching of aromatic molecules on nonporous silica,describing the reaction as a modified LH process. Once the reaction mechanism has been established, one can address the question of the effect of surface geometry on the rates of reaction at the interface. Of particular interest are the effects of geometry on the transport of reactants toward each other. Theoretical interest has been generated particularly by the advent of fractal geometry and its use in formulating kinetic laws on disordered structures? While theoretical understanding has increased, most experimental work has concentrated on percolation' and other low dimensional systems.a 0022-3654/92/2096-6398$03.00/0
In the present work we have carried out a systematic study of the diffusion-influenced reaction between R ~ ( b p y ) , ~and + O2on SiOz surfaces (porous silica and controlled porous glasses) over a wide temperature range. Our objectives were as follows: (a) To establish the relative weights of the ER and LH mechanisms at various temperatures, clarifying the factors determining the transition between the two mechanisms. For the LH mechanism our goal was to determine the conditions under which the reaction crosses from a diffusion-controlled regime to a reaction-controlled regime. (b) To determine surface diffusion coefficients by using a surface-modified Smoluchowski relationship at the diffusioncontrolled regime of the LH reaction. Another approach has recently been applied by Wong and Harrisg for the diffusioncontrolled reaction between excited pyrene and iodine at the interface of a silica surface and liquid alcohol. (c) To study the effects of surface geometry on the rates of reaction at both high and low temperatures. Experimental Details S i c a Samples and Chemicals. The following S O z materials
were used in which the numbers refer to the nominal average pore diameter: Porous silica-40 (Si40 Merk Lichroprep); controlled0 1992 American Chemical Society
Diffusion-Controlled Reactions on Porous Silicas
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6399
pore glass-75 (CPG75) and controlled pore glass-lOOO (CPGlOOO), both supplied by C.P.G., Inc. The N2 BET surface areas of the glasses as supplied by the producer are, respectively, 700, 182, and 26 m2/g, the pore diameters are 33, 72, and 962 A, respectively. Ruthenium tris(bipyridy1) chloride, Ru(bpy)?+, was used as received from Aldrich. Oxygen 99.5% was from Merkaz Hahamtsan, Israel. Sample Preparation. R ~ ( b p y ) , ~was + adsorbed by overnight equilibration with an aqueous solution followed by sample filtration and drying. The amount of Ru(bpy)?+ adsorbed was determined by spectrophotometricanalysis of the filtrate. The surface coverage corresponding to the experimentally determined amount adsorbed was calculated as follows: For the relatively smooth surface C.P.G.s, we used the simple relation: coverage = (moles adsorbed X molecular cross section)/N -BET area.I0 With the cross section of Ru(bpy)?+ taken as 172 A2,4the surface merage was 10% for both CPG75 and CPG1000. For Si-40, which has a very irregular surface, one can estimate the upper limit of the coverage'' from the cross-sectional areas of R ~ ( b p y ) , ~and + of N2 (16.2 A2) and from the highest reported value of the fractal dimension of the molecularly accessible surface, which according to Huppert et al.12 is close to 3. One then gets a maximum coverage of 6% for Si-40. Samples were placed in a long stemmed 1-mL cuvette under a vacuum of Torr and dried at 100 OC until stable pressure was achieved with the vacuum cut off. Oxygen was added at room temperature while pressure was measured using an MKS Baratron transducer (Model 122AA) with a 10 or 1000 Torr full scale. Luminescence Measurement. Samples measured at specific oxygen doses had a total volume of 46 mL. Samples measured at constant pressure were equilibrated with a room temperature gas reservoir with a total volume of 1300 mL. Temperature was controlled by placing the samples in a Dn-704 Oxford Instrument cryostat cooled by a flow of liquid nitrogen and equipped with a Eurotherm adjusted resistive heater. Temperatures studied ranged from 88 to 353 K. Pressures were measured in situ with an MKS Baratron transducer. R ~ ( b p y ) , ~was + excited by a P.R.A. LN 102 dye laser pumped by an LN lo00 pulsed nitrogen laser at 445 nm. Fluorescence was collected through a monochromator onto a multichannel plate (Hamamatsu R15644). The signal was amplified and digitized using a Tektronix 7912 AD oscilloscope for fast signals and a Tektronix 2440 digital oscilloscope for longer times, followed by averaging and storing with an Olivetti PC. The fwhm of the system is 1 ns. Oxygen Surface Coverage Meammments. O2surface coverage was determined by measuring the pressure of O2in a calibrated total volume during the luminescence experiments. Measurements at upper temperatures are bounded by small amounts adsorbed on the surface and at low temperatures by measurement of low pressures, when most of the O2is adsorbed on the surface. The correction factor for the pressure change not caused by adsorption was determined by replacing oxygen by helium under identical conditions (Appendix A).
-
Results I. Analysis of the Quenching Reaction. Both the self-decay and the quenching reaction of Ru(bpy)?+ adsorbed on the porous silica and on the controlled porous glasses cannot be fit to a single exponent. The reactions are pseudo first order in R~(bpy),~+*, as witnessed by the fact that the intensity of the exciting light does not influence the rate of decay, Le., there is no bimolecular annihilation of the excited state. The nonexponentiality stems from the inherent heterogeneity of the porous solids. In previous studies, data have been analyzed as bi- or triexponential processes with three or five independent variables,13implying the arbitrary assumption of two or three distinct chromophore populations. In the present work we have chosen to analyze our regults using a Gaussian model developed by Albery et al.14 which has recently been applied to porous silica by Thomas and co-worker~.~ The basic assumption of the model is that the distribution of decay rate constants, k, is caused by a normal distribution of free energies, leading to a Gaussian. The distribution of energies leads
I
0
I
I
I
50
100 TIME (nsec)
150
200
b
0
I
I
I
50
100 TIME (nsec)
150
20
Figure 1. Luminescence decay of adsorbed Ru(bpy),2+* quenched by gaseous O2and fit to a Gaussian distribution of In (k): (a) CPG75,0.57 Torr 02,173 K (b) Si40,0.42Torr 02,173 K. to a distribution of log (k)around some log (kav),where k,, is the average rate constant of the reaction. The full derivation of the model can be found in ref 14. The two fitting parameters are k,, and y, the dispersion of the Gaussian in In (k). When the distribution falls to 1/ e of its maximum value, the rate coefficient is given by k = k,, exp(fy). This gives a physical meaning to the two fitting parameters. An advantage of the model is that y enables a quantitative assessment of the extent of the heterogeneity of the system.'$ Figure 1 shows the fit of this model to the decay of excited R~(bpy),~+ on CPG75 and Si40. The agreement between the model and the experimental decay is remarkable. Except for very low temperatures, the selfdecay of Ru(bpy):+ on silica is nonexponential, as mentioned above. The rate constant of the process is markedly temperature dependent, an effect that has also been observed in solution.16 Whenever the rate constant of self-decay was over 5% of the observed decay rate constant (particularly at high temperatures), the reported rate constant, k,is given by k = kaV- kava, where kava is the average rate constant of self-decay. This is an approximation because the Gaussian model, while fitting the self-decay better than a single exponential, clearly is not exact. II. Quenching at Controkd O2Coverage. An LH mechanism is expected to prevail at low temperatures when O2 adsorption is substantial. Temperature will affect such a surface reaction not only by controlling the surface coverage but also by determining the intrinsic bimolecular rate constant. In order to separate these two effects, experiments were carried out under conditions of controlled O2coverage. These were carried out below 170 K. At higher temperatures the amount of O2 adsorbed is not measurable in our experimental setup. Experiments were carried out by dosing CPG75 and Si40 with amounts of 02,that upon total adsorption would yield an equal surface coverage of 1 f 0.1%. Due to the respective differences in surface areas, this necessitated an O2pressure (at room temperature) which was approximately
-
6400 The Journal of Physical Chemistry, Vol. 96, No. IS, 1992
Samuel et al. 34
+ +
+ h
A
+
33
+
A
t
A
32
3 31
I +
I
4
30
A
10'
29
80
140
110
200
170
T-ratuei'K
230
'
I
0.07
0.09
0.08
0.10
0.1 1
~fterrperattrd'
)
Figure 2. Quenching of R ~ ( b p y ) , ~ +luminescence * by O2as a function of temperature: (+) si40; (A)cPG75. At full adsorption, coverages are -1%.
0.12 (E-11
~ 7 Figure 4. Arrhenius plot (In (k3 versus 1/T) of the linear, low temperature range of the reaction: (+) Si40; (A) CPG75.
10" 1
2.0
I
1.7
+ + + +
0
+ 10" 7
&
A
+
1.4
-
1.1
-
+ + +
+ +++ A
A
+
t
0.8 -
A
+
0.5 50
IV
80
100
120
160
140
90
130
170
250
210
180 renweratue ( 'K 1
Tenperatud'K )
Figure 3. k'as a function of temperature: (+) Si40; (A) cPG75.
Figure 5. Width, y, of the Gaussian, used in the fitting procedure as seen in Figure 1, as a function of temperature: (+) Si40; (A) CPG75.
TABLE I: Residence Times rod Surface Coverage of O2Molecules on the Surfaces of Si40 and CPG75 7,
T,K 173 163 153 143 133 123 113 103
n, nmol n i 2
ns
Si40 0.37
CPG75
0.66
0.87 1.30 3.20 5.70 14.00 47.00 220.00
1.18 2.80 7.00 22.00 84.00 445.00
Si40 21 34 51 76 100 116 124 125
CPG75 10 14 24 47 12 100 113
4 times higher on Si40 than on CPG75. The experimental results, presented in Figure 2, show that in the high temperature range the rate of reaction on Si40 is approximately 4 times that on CFG75, mirroringthe difference in pressure. As the temperature is reduced, the rate constants on the two materials approach each other. At very low temperatures, the reaction rate constant on CFG75 slightly exceeds that on Si40. In both materials, the reaction rate constant increases as the temperature is reduced, as a result of increased oxygen surface coverage (Table I), until a maximum is reached at about 120 K. The sharp drop in the observed rate of reaction below 120 K stems from the fact that the decrcaw in the intrinsic rate of the surface reaction is no longer offset by an increase in the amount of adsorbed O2 (at these temperatures most of the O2 present in the system is already adsorbed). As will be shown in section I of the Discussion, the low temperature range may be attributed to an LH reaction between two adsorbed reactants. For an LH type reaction the rate constant should be expressed in terms of a second-order rate constant defined as k' = kob/ [0210b, where [O2Iobis the surface concentration of 0,expressad in units of mol/m2. As shown in Figure 3, k' increases with temperature, reaching a plateau at about 170 K,above which [OJd values are experimentally unavailable. The same data are plotted in Figure 4 in the form of Arrhenius type plots which are
1.OE+07
I
250
270
290
310
330
350
370
TenpOratue ( ' K )
Figure 6. Quenching of the luminescence of Ru(bpy),*+* adsorbed on CPGlOOO at constant concentration of O2as a function of temperature.
linear in the 88-1 13 K range. The activation energies found for the quenching reaction are 10.4 kJ/mol for ( 2 x 7 5 and 9.4 kJ/mol for Si40. The dispersion of the Gaussian distributions, y, shown in Figure 5 is larger for Si40 than for CPG75. III. Qlleacbhg in the High Temperature Range. At temperatures between 353 and 173 K, for which the amount of adsorbed Ozwas not established, two types of experiments were canid out: (a) ExperimenQ at Constant O2Concentration. Experiments at constant O2concentration were directed toward establishing the relative contributions of ER and LH mechanisms. Since at relatively high tempratures the amount of O2adsorption is small, an ER mechanism is expected to prevail. When the incoming gaseous reactant is kept at constant concentration by adjusting the pressure, the rate of a simple ER reaction should be proportional to the collision rate per unit area, Z,of the gas phase molecule with the surface, given by
z = ([02]kbT/2Tm)'/2
(1)
where kb is the Boltzmann constant and m is the mass of O2(cf.
The Journal of Physical Chemistry, Vol. 96, NO. 15, 1992 6401
Diffusion-Controlled Reactions on Porous Silicas
-
I
I
\\
b
Q
10' 150
180
210
240
tewratue ( ' K
8 I
270
300
)
& 1 Torr.
1
100
150
200
tenperatue
Figure 7. Quenching of Ru(bpy)32+luminescence by O2 at constant pressure: (+) SiM; (A)CPG1000; (V)cPG75. Pressure of o2was 50
-16
50
250 (
300
350
K )
Figure 9. Distance traversed by an adsorbed O2 molecule during its residence time, 7,on the surface of (---) CFG75 and (-) function of temperature.
Si40 as a
agreement with the literature estimate of 10-14-10-1s s. T~ is smaller than the period of a vibration by 1 to 2 orders of magnitude because of entropic effects: the unbound molecule has rotational and vibrational degrees of freedom unavailable to the bound m0le~ule.l~ Table I gives the residence times, T , as calculated from I'/Z at different temperatures.
/+*'I
Discussion
-22
I
0.50
0.60
0.70
0.80
0.90
1.00
(E-2) 1IT ( K-')
Figure 8. Plot of In ( 7 ) versus 1/T: (+) Si40; (A) CPG75. I is the resitence time of an O2 molecule on the surface. The slope yields the heat of adsorption.
eq 6). Experiments were carried out in the 173-353 K temperature range at a constant O2concentration of 5.5 X lW3 M. As shown in Figure 6, the average reaction rate for CPGlOOO shows a decrease rather than an increase with temperature up to 353 K, indicating that the mechanism is not simply ER in nature. (b) Experh"ets at Constant O2 Pressure. Experiments at constant O2pressure were carried out to determine the role of the bulk, Knudsen type, diffusion of O2 in the porous solids. Previous publications1' have reported that the room temperature quenching rates of triplet benzophenone on porous silica surfaces, by gaseous 02,scale with the average pore diameter when the latter is varied in the 68-572 A range. The data were interpreted in terms of an ER mechanism controlled by the bulk (Knudsentype) diffusion of 02. To test the validity of this approach, experiments were carried out at an O2pressure of 50 Torr,which was kept constant by using a large gas reservoir, kept at room temperature. As shown in Figure 7, k shows essentially identical values for Si40, CPG75, and CPGlOOO over a relatively wide temperature range, namely, no scaling wirh the pore size is observed. IV. Estimation of Heats of Adsorptionuldof Residence Times. Molar heats of adsorption, Q, on surfaces at low coverages can be determined from the temperature dependence of the average surface residence time, 7 , of the adsorbed molecule, given by1* 7 = 70 exp(Q/RT) (2) T , can be determined from = r/z (3) where r is the fraction of surface covered and Z is given by eq 1. A plot of In (I'/Z) versus 1 / T (Figure 8 ) yields Q = 13.5 kJ/mol for CPG75 and Q = 15 W/mol for Si40 (Le., an opposite trend to that found for the reaction activation energies). The preexponential factors, rO,calculated from the intercept, so = 4.2 X s and T~ = 9.4 X s, respectively, are in good
As mentioned in the Introduction, we shall address three main points dealing with the quenching of a fluorophore adsorbed on a porous solid by a gaseous reactant. The first concerns the reaction mechanism. On the one hand the reaction may be target annihilation in nature, in our case,between gas phase oxygen and adsorbed Ru(bpy)32+,via a basically Eley-Rideal (ER) type mechanism. At the other extreme the reaction may take place between an adsorbed O2molecule and an adsorbed Ru(bpy)?+, via a Langmuir-Hinshelwood (LH) type mechanism. Furthermore, each of the two mechanisms may be either diffusion controlled or reaction controlled. Having discriminated between the mechanisms, and determined the range in which the reaction is purely LH and essentially diffusion controlled, we proceed with calculating the surface diffusion coefficients from the observed reaction rate constants. The third point to be addressed is the effect of surface geometry on the reaction rates and on the rates of surface diffusion for both LH and ER processes. I. The Reaction Mechanism. One indication of a basic surface reaction mechanism is the average distance, r, traversed by an adsorbed O2molecule before desorption. By use of the diffusion coefficients, D, calculated in the next section, r may be estimated by applying the expression r = ( ~ D T ) ~The / ~ .temperature dependence of r is shown in Figure 9: At room temperature r((2x75) is about 16 A, while r(Si40) is, at most, 4 A. The short distance traversed by an O2molecule on the surface before desorption indicates that the mechanism may be of an ER character. This point can be further assessed by using simple kinetic theory: for a simple pseudo-first-order process, assuming a gas-phase target-annihilation reaction, the rate of reaction may be described by dA/dt = -kA (4) in which A is the concentration of the excited-state molecules and
k
Zce,rf
(5)
where uecnis the effective cross section of the reaction. From eqs 5 and 1
k
[02](kbT/2~m)'/~~,,r~
(6)
Equation 6 predicts that at constant quencher concentration, the rate constant of a nonactivated process, will be proportional to T1I2.This,as mentioned above, is in variance with the data shown in Figure 6, where the reaction rate constant falls continuously with the increase in temperature up to 333 K. It is again evident
6402 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
Samuel et al. To verify this model the results have been fit to a scheme that describes “diffusion influenced” reactions, covering the complete range between a distinctly diffusion-controlled process and a reaction-controlled process which is independent of the rate of diffusion. The rate of diffusion-influenced reactions is given by the classical expression2’
‘
29 0.06
I
0.07
0.08
0.10
0.1 1
0.12 (E-1)
lltenperatue (‘K-’l
Figure 10. Full Arrhenius plot of In (k3 versus 1/T: (+) Si40; (A) CFG75. The experimental points are fit to a diffusion influenced reaction scheme. The dashed line is a fit assuming an Arrhenius form for the reaction rate constant and assuming that the diffusion rate coefficient is proportional to the reaction rate. The solid line is a fit assuming an Arrhenius form for the diffusion cocfficient followed by calculation of the reaction rate.
that a simple ER mechanism cannot be inferred. We therefore suggest that the short residence times of the O2molecules on the surface (4 ps on Si-40 and 8 ps on CPG-75at 300 K), point to a reaction that is surface mediated. The short residence time on the surface increases the reaction probability to the degree that the T112behavior is overshadowed. A similar conclusion, based on different arguments, was recently reached by Thomas and c o - w o r h in a study of oxygen quenching of fluorescent aromatic molecules adsorbed on colloidal sili~a.~ Accordingly, they defined this type of process as a modified LH reaction. At lower temperatures the reaction mechanism is obviously far from ER in nature. This can be seen by evaluating the reaction probability P, defined as uar/u, where u is the reaction cross d o n given by a hard sphere model. Owing to the relatively small cross section of O2(14 A2) compared to that of Ru(bpy),2+ ( 1 72 A2), we may approximate the overall hard sphere reaction cross section as u 5~ 170 A*. Around room temperature, at 213 K, and under an O2pressure of 1.56 Torr, ueff,as calculated from eq 6, is 14 A2. This leads to a reaction probability of P z 0.1, which is in keeping with a modified ER model. On the other hand at a temperature of 113 K and a pressure of 0.22 Torr 02, is 2000 A2,leading to a reaction probability of P z 10, which considerably exceeds unity. It is thus obvious that at low temperatures the reaction is far from ER in character. We now consider in more detail the low temperature range for which the anomalous P value and the longer (nanosecond) residence times of O2molecules on the surface are in keeping with an LH mechanism. The important feature of the temperature dependence of the second-order rate constant, k’, is that the respective Arrhenius plots, Figure 10, show a linear portion only at lower temperatures, while leveling off at higher temperatures. When defining &’as k/[O2Iadsthe assumption is made that the contribution of the gas phase O2to the quenching is negligible. (If an appreciable component of gas-phase quenching existed at these temperatures, the value of k’would be artificially inflated. This is because the gas-phase component would increase the observed rate of reaction without a concomitant increase in the amount of O2adsorbed. The overestimation in k’ would then manifest itself as an upward trend in the Arrhenius plot.) Our explanation for the leveling-off in the Arrhenius graph is based on the temperature dependence of the ratelimiting step: At lower temperatures the rate of surface diffusion is small and the reaction is diffusion limited, at higher temperatures the rate of diffusion ex& the intrinsic rate of reaction and the total rate observed is reaction limited. The upper part of the Arrhenius curve shows, asymptotically, a very small slope, meaning that the reaction is nearly activationless. In fact the reaction rate constant in homogeneous solution, k = 3.7 x 109 s-1 (mol/L)-l 2o implies an essentially diffusion-controlled mechanism.
k ’ = kdckrc/(kdc + krc) (7) where kdcis the rate constant of the purely diffusion-controlled reaction and k , is the reaction rate constant once the reactants have reached the critical reaction distance, or in effect the intrinsic rate of reaction. In solution kdcis given by the expression kdc = ~ R A B D (8) where RAB is the critical reaction distance and D is the sum of the diffusion coefficients for molecules A and B. Freeman and Doll have carried out an analysis of diffusion-influenced reactions on surfaces using models similar to those used in solution kinetics.u They have shown that eq 7 also applies to bimolecular reactions taking place on surfaces. The diffusion coefficient, in all dimensions, is given by D = Do exp[-E&], where /3 = l/kbT,Em is the activation energy for diffusion, and Do is the preexponential factor. However, The linear relationship between D and kd,found in three dimensions (eq 8) may only serve as a first approximation in two dimensions, and a more rigorous approach is required. We shall pursue this approach in the next section but will first look at a qualitative picture by assuming that the linear relationship that exists between kd,and D in homogeneous solution also applies to surface reactions. Only under this approximation, the diffusion-controlled rate constant may then be described by the expression kdc = (9) where Adcis the pre-exponential factor and Edcis the activation energy for the diffusion-controlled reaction. As discussed above, in relation to Figure 10, the intrinsic reaction rate is essentially activationless; namely k, is practically independent of temperature. The fitting of the experimental data to eq 7 is then done as follows: k, is taken from the asymptotic intercept in Figure 10, which is in the high-temperature regime where krc prevails. In the low-temperature regime k’ = kdcand eq 9 is obeyed giving the linear part of the Arrhenius plot in Figure 10 from which 4 and Edcare calculated. These values are then used to obtain kdc for the range in which k, cannot be neglected. The fit is shown in Figure 10. It is evident that this simplified approach, in which there are no adjustable parameters, fits the results quite well. The analysis confirms our interpretation of the temperature effects in terms of a crossover of the reaction from a purely diffusioncontrolled reaction at low temperatures to a non-diffusion-controlled regime at higher temperatures when surface mobility becomes sufficiently high. IL C!dadntioo of M a c e Diffusion Coefficie~~ls.The classical relationship between the diffusion coefficient, D, and the diffusion-controlled rate constant, kk, for a reaction in homogeneous liquids, is given by eq 8. The Smoluchowski type approach2’ that leads to this equation has been adapted to surfaces by a number of authors.24 Freeman and argued that the Smoluchowski equation has no solution in two dimensions. They proceeded by deriving an expression for a surface-diffusion controlled reaction between an adsorbed molecule and an adsorbed reactant, which is in equilibrium with the gas phase, by adding a source term accounting for the limited residence time of the gaseous reactant (in our case the O2quencher). The twedimensional expression for kdc which takes into account the added source is given by kdc = ~ X R A B ~ (10) in which the parameter X is given by = ~KI(~RAB)/Ko(~RAB) (11) K , and KOare the modified Bessel functions of the first and zero order respectively, where Y = (D7)-’/2 (12)
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6403
Diffusion-Controlled Reactions on Porous Silicas
C = 0.1159 -21.4
1
'
-27.0 0.08
AE
I 0.10
0.10
0.11
I l T ( K.'l
0.12 (E-1)
Figure 11. Arrhenius plot of the diffusion coefficient calculated from eq 13: (+) Si-40;(A) CPG75. TABLE II: Arrhenius Parameters for Surface Quenching Reaction of Ru(bpy)f+ by O2 on Si40 and CPG79
Si40 Edc, kJ mol-'
ko,
(mol/m2)-' kJ mol-'
DO,m2/s Q,kJ mol-' 709
S
(Edc- E,jifi)cxp, kJ mol-' (Edc - E,jiff)calc, kJ mol-'
9.4 4.7 x 1018 7.9 3.1 X 15.0 9.4 x 10-15 1.5 2.3
CPG75 10.4 2.1 x 1019 9.9 4.1 X lo4 13.1 4.2 x 10-14 0.5 1.3
OEdCand Ediff are the activation energies for reaction and diffusion respectively. Q is the heat of adsorption. ko,Do,and T~ are the Arrhenius prefactors for reaction, diffusion, and adsorption, respectively.
On a flat surface v is the inverse of half the average distance traveled by a gas molecule before desorption ( 2 / r ) . Since the increase in the residence time, T , exceeds the drop in the diffusion coefficient, Y decreases as the temperature is lowered. Using eqs 10 and 11 one arrives at
Equation 13 enables one to calculate the surface diffusion coefficient of a reactant adsorbed on the surface from the gas phase. This calculation was carried out as follows: kdc was obtained experimentally for the range in which k' = kdc (Figure 10); RAB was taken as the sum of the radii of the two reactants, 9.4 A; T , the residence time, was calculated from eqs 1-3. One then obtains D by iteration and numerical calculation of the Bessel functions K, and KO for the argument uRAB. For the temperature region for which k' = kdcwe obtain an Arrhenius plot of the diffusion coefficients (Figure 11). The resultant energies of activation for diffusion and the corresponding prefactors are shown in Table 11. The diffusion coefficient at room temperature for CPG75 is 8 X m2 s-l. For Si40, the calculated value of the diffusion coefficient is 1.3 X m2 s-I which for reasons discussed in the following section, is an upper limit. Another important outcome of this treatment is that in contradistinction to homogeneous solutions, the activation energy for diffusion, Ediff,and the activation energy for the diffusion controlled reaction, Edcrare not equal. Our results for surface diffusion show that EM, as calculated from Figure 11, is appreciably smaller than Edcras calculated from the linear portion of Figure 4 (see Table 11). This effect stems mainly from the temperature dependence of the surface residence time. The relationship between Edcand EM can be derived for the low temperature range, for which the argument of the Bessel function, YRAB,is small. Then, as will be shown in Appendix B, the relationship is given by the expression Edc
Ediff+ m / ( c+ mfl)
where C is a temperature-independent constant
~ )-In / 2RAB
(15)
and AE is given by
I
0.09
+ (In Do+ In ~
(14)
(Q- Ediff)/2
(16) Edois seen to be temperature dependent through the denominator in eq 14, particularly for small values of C. This temperature dependence may cause an error in the assessment of the diffusion-influenced reaction rate as approximated in the previous section. Finally we reanalyze the data using the diffusion coefficients calculated from eq 13 and extrapolated to higher temperatures by means of the Arrhenius relationship. By introducing these calculated diffusion coefficients in eq 10, we arrive at the correct value of kdc. We complete our treatment by analyzing these corrected data by means of eq 7, thus arriving, as shown in Figure 10, at a closer and more correct fit between theory and experiment. III. Effects of Surface Geometry. (a) Gas-Phase Transport at High Temperatures. As shown previously, the predominant mechanism of O2transport toward the excited photosensitizer in the high temperature range takes place via the gas phase. The effect of pore geometry on this type of reaction has been addressed both theoretically and experimentally by Drake et al." who studied the O2quenching of triplet benzophenone on a series of porous silicas with varying pore diameter. The observed quenching rate constants were found to be proportional to the average pore diameter (apd) of the material. The data were interpreted in terms of an ER diffusion controlled (Smoluchowsky-type) mechanism, where the effective (bulk) diffusion coefficient is determined by "Knudsen's diffusion", i.e., under conditions in which the mean free path of O2in the gas phase is much larger than the apd. In a recent detailed analysis25we have shown that under any feasible experimental conditions ER target-annihilation reactions in porous solids cannot be controlled by the Knudsen diffusion rate and consequently should be independent of pore size. This prediction is now confirmed experimentally by our present data (Figure 3). Thus, at constant pressure, the R ~ ( b p y ) ~ ~ +quenching /O~ rate is independent of the pore size over an apd range of 40 to 1000 A for which the Knudsen diffusion coefficient varies by a factor of 25. (b) SIlrfaCe Dhsitm in the Low Temperahw LH Domain. We have shown that at low temperatures the mechanism of the reaction is LH in character and the transport of the O2molecules takes place mainly on the surface. Under these conditions,effects of surface geometry may be expected to show up. Surface diffusion of an adsorbate may be influenced both by energetic factors (activation of a hopping movement) and by geometric factors, these two factors being intertwined. Porous solids have been traditionally characterized by surface areas and average pore diameters.'O Recently the concept of a surface fractal dimension has been used to asses the roughness of highly disordered surfaces." Fractal theory shows that reaction rate laws and rates of diffusion are dictated both by the fractal dimension, df,and by the spectral dimension dS5,6s2'which is, in essence, the dimension of connectivity. On Euclidean objects df and d, are equal. On fractal objects the spectral dimension is bounded from above by d f and from below by the topological dimension, which in our case is most probably equal to 2. The strict application of eq 10 is limited to the case in which d, = dtop = 2. CFG75 fulfills, to a large degree, this condition and the diffusion coefficient on CPG75 can be therefore calculated directly from eq 10. For Si40, d, is unknown and may exceed 2. To the best of our knowledge an explicit relation between D and kd, for the case in which d, # d has not been derived. We can, however, say that the diffusion coeficient calculated under the assumption that d, = 2 may serve as an upper limit and that if d, > 2, the actual diffusion coefficient is smaller than D calculated by using eq 10. This may be seen qualitatively in the following way: The relationship between kdc and D is dictated by the mean number of distinct sites that a random walker visits in a given period of time.6-28For a reaction taking place in a system in which d, = 2, diffusion is barely recurrent, meaning that a random walker tends to revisit sites. On the other hand, on a d, > 2 structure,
6404 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
diffusion is nonrecurrent, Le., the random walker is more efficient in visiting new sites in the same period of time, compared to the d, = 2 random walker. Since we measured kdcand calculated D assuming the less efficient case, i.e. d, = 2, then if, in reality, d, > 2, and the reaction is more efficient than we assumed, it is sufficient to have a smaller diffusion coefficient than the calculated one, in order to get the same kdc. Furthermore, using eq 10, we found that D(Si40) < D(CPG(75), and since D(Si40) calculated in this way is an upper limit of the true diffusion coefficient, the difference between the D values for the two materials is a minimal estimation. The activation energies for diffusion, calculated in the previous section, are 9.9 kJ mol-' for CPG75 and 7.9 kJ mol-' for Si40. Both values are smaller than the respective heats of adsorption, 13.5 and 15 kJ mol-', respectively. This observation may be rationalized as follows: Diffusion on the surface entails hopping from site to site and would be expected to have an activation energy of less than total bond breakage, in keeping with the above observations. A similar trend is also observed on metallic surfaces: Activation energies are of the order of 30% of heats of adsorption.B It is interesting to note the trend of the heats of adsorption for the two systems, Q(Si40) > Q(CPG75), is reversed for the corresponding activation energies for diffusion, namely, EdidSi40) < EdCPG75). We approach this observation by suggesting that both the lower diffusion activation energy and the higher heat of adsorption found for Si40 may be explained by factors of surface geometry. The high degree of surface roughness of Si40, as witnessed by the high fractal dimension, gives rise to many nooks and crannies in which O2can adsorb with multiple surface interactions. These types of interactions are consistent with the higher heats of adsorption as compared with those found on the relatively smooth surface of CPG75. However, in spite of the higher heats of adsorption found on Si40, the barrier for diffusion is less than that found on CPG75. We suggest that the multiple surface interactions possible on a rough surface, while possibly constraining diffusional pathways, may enable the molecule to diffuse through energetic troughs with lower barriers than those found on relatively flat surfaces. The purely geometric effects, i.e. the constraints on diffusion pathways, are expressed in the prefactor, Do,of the Arrhenius expression for D. These effects are witnessed by the fact that in spite of the higher activation energy for diffusion the overall diffusion rate on CPG75 exceeds that found on Si40. This is due to the pre-exponential factor, Do,which for CPG75 is 13 times larger than for Si40. The pre-exponential factor, formulated in the framework of transition state theory is a function of the average jump distance and of an entropic factor.30 The purely geometric effects as seen in the different magnitude of the prefactor may be understood in terms of hindered or constrained diffusion. Although the form of the rate equation is not influenced by the fractal dimension when d, L 2, the rate of diffusion, and thus the rate of the diffusion controlled reaction, is affected. On fractal objects the diffusion constant has been theoretically derived as3I
Samuel et al. fluenced by the intrinsic rate of reaction: A distinct higher temperature range is finally reached in which the (essentially activationless) intrinsic reactivity becomes the rate limiting factor and the reaction is no longer affected by diffusion. These observations provide the first experimental example of a system in which the whole range of a diffusion influenced reaction may be spanned by varying the temperature. At still higher temperatures the gas-phase transport of the mobile reactant predominates and the reaction is no longer LH in character. It may now be defined as a "modified ER" or "modified LH" reaction in the sence that the residence time of the gase~usmolecules on the surface influences the rate of reaction. At still higher temperatures the reaction is expected to become strictly target annihilation in nature. Due to intrinsic limitations (mainly thermal stability) this range could not be attained in the present system. A second major implication of the present work concerns the effects of surface geometries of irregular porous solids on the various reaction mechanisms. Effects of surface geometry are present only at the lowest temperature range when the reaction is both LH and diffusion controlled. At higher temperatures, when the mechanism is LH but not diffusion controlled, or when it enters an essentially ER range, the effects of surface geometry are negligible: In these cases transport of the reactants is fast and no longer constitutes the rate-limiting step. Finally, by applying diffusion controlled reaction theory adapted to two-dimensional surfaces to the range in which the reaction is LH and strictly diffusion controlled, we were able to calculate microscopic surface diffusion coefficients. In the case of flat surfaces, this approach is rigorous, while in the case of surfaces with a high fractal dimension this treatment gives an upper limit for the rate of diffusion. The strategy employed in this work constitutes a novel tool for the determination of surface diffusion coefficients based on chemical kinetic data.
Acknowledgment. We acknowledge support by the Krupp Foundation, by the US-Israel Binational Foundation, and by the Farkas Center for Light Induced Processes. The Farkas center is supported by the Bundesministerium fur Forschung and Technologie and by the Minerva Gesellschaft fur die Forschung. D.A. is a member of the F. Haber Research Center for Molecular Dynamics. Appendix A. Measurement of the Amount of Adsorbed O2 The amount of gas adsrobed on the surface was measured in a long-stemmed cell in which the lower part of the cell held the adsorbent at the set temperature, T,, and the upper part was at an indeterminate temperature. Measurements of the pressures of the adsorptive gas, 02,and of a nonadsorptive gas, He, were taken under identical conditions. The amount of O2adsorbed on the surface, assuming ideal gas behavior, is described by I
where Dmacr0 is the observed diffusion constant, Dviltis a virtual diffusion coefficient on an otherwise identical but smooth surface, and r,, and r,, are the inner and outer cutoffs of fractal behavior. We therefore tentatively suggest that the smaller diffusion coefficient on Si40 indicates that, for that material, d, < df.
Conclusion In the present work we have shown that a bimolecular reaction of a molecule immobilized by adsorption on the surface, with a gas-phase reactant present in excess, is controlled by a complex superposition of several mechanisms whose relative contribution varies with temperature. In the low temperature range, where the reaction is LH, several stages may be observed: At the lowest temperature limit, when the surface diffusion of the gaseous reactant is slow, the reaction is strictly diffusion controlled. As the temperature is raised and the rate of the surface diffusion increases, the reaction remains LH in character but is also in-
where 6 and Ti are the volumes and temperatures of the various parts of the cell, p(02) is the pressure of O2measured at T,, and Oh, and OZtotare the adsorbed and total amount of oxygen, respectively. For the helium measurements, under the assumption that no adsorption takes place, the amount of helium in the system is given by Hetot = p(He)/RXV/Ti i
(19)
where p(He) is the pressure of helium at T,, From eq 18 and eq 19 we have at temperature T , All quantities on the right side of eq 20 are measurable. B. Derivation of the Difference between EM and E&. Starting from eq 13, the full expression for the diffusion coefficient in two dimensions, we use the expansion of the Bessel functions around zero for the low temperature range for which the argument of
J . Phys. Chem. 1992,96,6405-6410 the function vRAB is small. The expansions of the two Bessel functions are KO(YRAB)= -In (vRAB) - y m In (2) ...
+
K~(YRAB) =
(vRAB)-'
...
where ymis the Euler Mascharoni constant. Using these two expressions in eq 13, we obtain for the diffusion coefficient kdc
D = -(0.1159 - In (YRAB)) (21) 217 Using the Arrhenius expression for the diffusion coefficient and for the diffusion-controlled rate coefficient and the definition of Y , given in eq 12, we arrive at
Using the definition of T given in eq 2, we expand the rightmost expression in eq 22 2
In
(5)$ (%) + Q =
-0 - Ediff
In
We now define C = 0.1159
+
In
2
(23)
(s)
Using these two definitions and taking the logarithm of both sides of eq 23, we arrive at
Finally under the assumption that the prefactors Do and ko are similarly temperature dependent and differentiating eq 25 results in
6405
Registry No. Ru(bpy)32+, 15158-62-0; 02,7782-44-7; silica, 763186-9.
References and Notes (1) Photochemistry on Solid Surfaces; Anpo, M., Matsuura, T., Eds.; Elsevier, Amsterdam, 1989. Photochemistry in Constrained and Organized Media; Ramamurthy, V., Ed.; VCH: New York, 1991. (2) Rideal, E. K. Proc. Cambridge Philos. SOC.1939, 35, 130. (3) Hinshelwood, C. W. Kinetics of Chemical Change; Claredon, Oxford, UK, 1940. (4) Wolfgang, S.;Gafney, H. D. J . Phys. Chem. 1983, 87, 5395. (5) Karansky, R.; Koike, K.; Thomas, J. K. J. Phys. Chem. 1990,94,4521. (6) Havlin, S. In The Fractal Approach To Heterogeneous Chemistry: Polymers, Colloids and Surfaces; Avnir, D., Ed.; Wiley: Chichester, 1989; Chapter 4.1.1. (7) Evesque, P. J . Phys. 1983, 44, 1217. (8) Kopelman, R. J . Stat. Phys. 1986, 42, 185. (9) Wong, A. L.; Harris, J. M. J . Phys. Chem. 1991, 95, 5895. (10) Gregg, S.J.; Sing, K. S.W. Adsorption Surface Area and Porosity; Academic Press: London, 1982. (1 1) Avnir, D. J . Am. Chem. SOC.1987, 109, 2931. (12) Pines,D.; Huppert, D. Isr. J . Chem. 1989,29,473, and earlier reports. (13) Lochmiiller, C. H.; Hunnicut, M. L. J. Phys. Chem. 1986, 90,4318. Avnir, D.; Busse, R.; Ottolenghi, M.; Wellner, E.; Zacchariasse, K. J . Phys. Chem. 1985,89, 3521. (14) Albery, W. J.; Bartlett, P. N.; Wilde, C. P.; Darwent, J. R. J . Am. Chem. SOC.1985, 107, 1854. (15) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (16) Gafney, H. D. Coord. Chem. Reu. 1990, 104, 113. (17) Drake, J. M.; Levitz, P.; Klafter, J.; Turro, N. J.; Nitsche, K. S.; Cassidy, K. F. Phys. Rev. Lett. 1988.61, 865. Drake, J. M.; Levitz, P.; Turro, N. J.; Nitsche, K. S.;Cassidy, K. F. J . Phys. Chem. 1988, 92, 4680. (18) Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, 1990; p 594. (19) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics and Chemical Reactivity; Oxford University Press: New York, 1987; p 463. (20) Demas, J. W.; Harris, E. W.; Mcbride, R. P. J . Am. Chem. Soc. 1977, 99, 3547. (21) Noyes, R. M. Prog. React. Kinet. 1961, 1 , 129. (22) Freeman, D. L.; Doll, J. D. J. Chem. Phys. 1983, 78, 6002. (23) Smoluchowski, M. V. Z . Phys. Chem. 1917, 92, 129. (24) Cukier, J. L. J . Stat. Phys. 1986, 42, 69. (25) Samuel, J.; Ottolenghi, M.; Avnir, D. J . Phys. Chem. 1991, 95, 8440. (26) Avnir, D.; Pfeifer, P. Nouu. J . Chem. 1983, 70, 144. Sato, M.; Sukeyawa, T.; Harigawa, S.; Kaneko, K. Chem. Phys. Lett. 1991,181,526. (27) Alexander, S.;Orbach, R. J . Phys. Lett. 1982, 43, 626. (28) Pfeifer, P. In Preparutiue Chemistry Using Supported Reagents; Laszlo, P., Ed.; Academic Press: San Diego, CA, 1987, Chapter 2. Pfeifer, P. In ref 6, Chapter 1.2. (29) Zdahnov, V. P. Surf. Sci. Rep. 1991, 12, 185. (30) King, D. A. J . Vac. Technol. 1980, 17, 241. (31) Dozier, W. D.; Drake, J. M.; Klafter, J. Phys. Rev. Lett. 1986, 56, 197.
Structural Role of H20 in Sodium Silicate Glasses: Results from 2sSiand 'H NMR Spectroscopy Jorg Kiimmerlen, Lawrence H. Merwin: Angelika &bald,* and Hans Keppler Bayerisches Geoinstitut, Universitiit Bayreuth, Postfach I O 12 51, W-8580 Bayreuth, Germany (Received: December 18, 1991; In Final Form: March 26, 1992)
The structural role of H 2 0 in Na2Si409glasses has been investigated by high-resolution solid-state NMR methods (29SiMAS, 29SiCP MAS, ' H MAS, 'H CRAMPS). As a result of this study it is shown that (i) H 2 0 does depolymerize the silicate network, and (ii) both Si-OH and molecular water are present in hydrous sodium silicate glasses.
htroduction The structural role of water in silicate and aluminosilicate glasses has been, for a long time now, an issue of controversial
* Corresponding author. t R M n t address: Naval Air Warfare Center, codc 3851, China Lake, CA 93555.
0022-3654/92/2096-6405$03.00/0
discussion in the literature.'-l3 Various spectroscopic techniques have been used to characterizethe species involved; the conclusions drawn have to be considered as strongly divergent. o n e remining point of controversy is the question of whether H 2 0 does or does not depolymerize the Al, Si, or Si network in silicate and aluminosilicate glasses. In other words, it is still debated whether there is molecular water, Al+H/Si+H species, or both, present 0 1992 American Chemical Society
