J. Phys. Chem. C 2010, 114, 3991–3997
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Entropic Contributions to the Atomic-Scale Charge-Carrier/Surface Interactions That Govern Macroscopic Surface Conductance Shuhui Cai,†,‡ Monica Caldararu,§ and Karl Sohlberg*,‡ Department of Physics, Fujian Key Laboratory of Plasma and Magnetic Resonance, State Key Laboratory of Physical Chemistry of Solid Surfaces, Xiamen UniVersity, Xiamen 361005, PR China, Department of Chemistry, Drexel UniVersity, Philadelphia, PennsylVania 19104, and Ilie Murgulescu Institute of Physical Chemistry of the Romanian Academy, Spl. Independentei 202, 060021 Bucharest, Romania ReceiVed: October 24, 2009
In this paper it is demonstrated how to use quantum-chemical electronic structure calculations to estimate the values of the entropic parameters in a mathematical model of surface conductance. Entropy changes involved in both charge-carrier-hopping and desorption processes are considered. The model is then fit to experimental data for the temperature dependence of surface conductance of γ-alumina and the best-fit values of the entropic parameters are compared to those obtained by quantum-chemical calculations. 1. Introduction The surfaces of many metal oxides present excellent binding sites for the adsorption of molecules from the gas phase, making them attractive for application in gas sensors and as heterogeneous catalysts. The surfaces of SnO2,1 ZnO,2,3 and Al2O3,4,5 for example, have been studied extensively for these applications. Most commercial gas detectors available at present make use of porous SnO2 sensing elements, mainly because they offer high sensitivity at low operating temperature, but disadvantages such as lack of reproducibility in the presence of moisture are frequently observed,6 as their surfaces are usually covered with water/OH groups. Surface hydroxyl groups also play important roles in adsorption and catalysis on oxides, since they promote water adsorption through hydrogen bonding. For example, it is known that the catalytic behavior of γ-alumina (when used as a catalyst or as a support for catalysts) depends very much on the extent of its dehydration.7,8 Its catalytic activity is mainly attributed to Lewis acidity, created on the surface as a result of removal of the OH groups on heating above 400 °C. γ-Alumina easily adsorbs water, however, even at room temperature, and water adsorption would change part of the Lewis type acidity to Bro¨nsted acidity.9 Surface hydroxyl groups can generally be removed by condensation dehydration at higher temperature in vacuo, but the removal temperature varies considerably depending on the nature of the solid substrate.10 Surface conductance measurements are a valuable tool to investigate adsorption behavior of these materials because adsorption/desorption of gases occurs frequently with charge transfer, which plays a central role in governing the catalytic activity. Surface conductance is also often the transduction metric in gas sensors. It is frequently influenced by the presence of moisture in the atmosphere.11 In particular, the effect of water on the conductivity of porous materials and ceramics has been investigated with the intent of identifying various reliable humidity sensors. Besides its crucial role in controlling surface * Author for correspondence. E-mail:
[email protected]. Fax: 1-215895-1265. † Xiamen University. E-mail:
[email protected]. ‡ Drexel University. § Ilie Murgulescu Institute of Physical Chemistry of the Romanian Academy. E-mail:
[email protected].
acidity, water adsorption would generate parasite reactions, would be involved in spillover of species12 and, as the main atmospheric contaminant, would compete with other gases for adsorption on the same adsorption sites.13 Humidity-dependent conductivity in such systems arises from protonic transport in water molecules adsorbed. Protons are among the most frequently encountered mobile cations on the metal oxide surfaces. Proton conduction occurs either facilitated by other molecules (as water or ammonia through so-called Vehicle mechanism) or by Grotthuss chain (hops between neighboring OH groups).14-16 A comprehensive examination of the characteristics of a wide range of different humidity sensors concluded that all operate by the same physical mechanism.17 Although surface conductance depends on the atomic-scale structures present on the surface of a material, its measurement is a macroscopic technique. The dependence of surface conductance on surface morphology is well documented,18 but studies that relate surface conductance to the atomic-scale surface structure are more sparse. In a recent paper we presented a mathematical model that makes a direct connection between atomic-scale surface structures and macroscopic surface conductance when there are two independent types of charge carriers.19 Herein we review the mathematical model and present it in a form generalized to describe an arbitrary number of atomic or molecular charge carrier species. We then demonstrate how to estimate the entropic parameters in this model with quantum chemical electronic structure calculations using γ-alumina as an example. 2. Theory Since the mathematical description of the thermal dependence of surface conductance G for the multiple-charge-carrier model was derived previously in ref 19, we present only a summary here, starting with the definitions used in our earlier work and generalizing to an arbitrary number of types of charge carriers. The main parameter definitions are G ) surface conductance q ) charge on charge carrier, assumed to be + qe d ) hopping distance m ) charge-carrier mass n ) concentration of charge carriers (per unit area)
10.1021/jp910169n 2010 American Chemical Society Published on Web 02/16/2010
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N ) concentration of charge-carrier binding sites (per unit area) E ) electric field µ ) mobility ) ratio of drift velocity to applied field ) υ/E υ ) drift velocity ) µE ∆F ) free energy change on adsorption ∆E ) internal energy change on adsorption ∆S ) entropy change on adsorption ∆f ) free energy barrier to hopping ∆ε ) internal energy barrier to hopping ∆ξ ) entropic contribution to hopping barrier If there are J independent and noninteracting types of charge carriers, we may write J
G)B
∑ υiniqi
(1)
i)1
where the index of summation runs over all types of charge carrier species. Since the overall magnitude of conductance is governed by external factors such as contact resistance that are not intrinsic to the material, the constant B in expression 1 is taken as an adjustable parameter. The overall drift velocity υ of a charge carrier is given by19
(
)
∆fi - Eqidi /2 υi ) νidi exp - νidi × kT ∆fi + Eqidi /2 exp kT
(
Pibi 1 + Pibi
(
)
)
(
)
(
{
[
(
νidi · exp -
∆fi + Eqidi /2 exp kT
)]
)
∆fi - Eqidi /2 - νidi · kT
(
)
}
∆Fi kT × Niqi × ∆Fi 1 + Pi · exp kT (5) Pi · exp -
(
)
Expression 5 has the same form as the usual empirical expression for surface conductance with a factor containing the hopping barrier in the exponent,21,22 and a factor dependent on the partial pressure. In (5), there is one term in the summation for each type of charge carrier species. Assuming negligible PV work during the adsorption and hopping processes, we may write
G)B
∑ i)1
(3)
∆Fi kT ni ) Ni × ∆Fi 1 + Pi exp kT
i)1
J
(2)
Here bi is an equilibrium constant and is therefore related to the free energy for the adsorption process by bi ) exp(-∆Fi/ kT), from which it follows that
Pi exp -
∑
∆Fi ) ∆Ei - T · ∆Si
(6a)
∆fi ) ∆εi - T · ∆ξi
(6b)
Substitution of (6a) into (5) yields
where νi is the frequency with which a charge carrier of type i attacks the potential barrier between adjacent rest sites on the surface, di is the distance it hops if it does surmount the barrier, and the exponential term is a Boltzmann factor describing the probability of surmounting the barrier given energy kT. Assuming Langmuir-type gas adsorption on the surface, for Ni total adsorption sites (which are also the type i charge-carrier sites), the number of carriers on the surface ni may be described by20
ni ) Ni ×
J
G)B
Cai et al.
(4)
Here Pi is the ratio of the pressure of the gas-phase species (Pi′) to standard pressure, Pi ) Pi′/(1 atm). Substituting the expression for carrier velocity (2) and the expression for carrier density (4) into the expression for conductance in the multiplecharge-carrier model (1) yields
{( ) [
νidi · exp
exp
( ) (
)
∆εi - Eqidi /2 ∆ξi · exp - νidi · k kT
(
)]
∆εi + Eqidi /2 ∆ξi · exp × Niqi × k kT ∆Ei ∆Si · exp Pi · exp k kT ∆Ei ∆Si 1 + Pi · exp · exp k kT
( ) ( ) ( ) ( )
}
(7)
Expression 7 is the fully expanded model of surface conductance by multiple-independent-charge-carriers. It gives the thermal dependence of the macroscopic surface conductance in terms of the microscopic energy, entropy and positional changes involved in the charge transport. 3. Entropy Calculations The multiple-charge-carrier model has been applied to the surface conductance of γ-alumina, a case with two types of charge carriers. The thermal dependence of surface conductance of γ-alumina is shown in Figure 1.23 Upon heating from room temperature (RT) to 673 K (solid points), the surface conductance displays two distinct regimes. After a brief initial rise (we denote this regime 0), the surface conductance falls with increasing temperature (regime I), but after reaching a minimum at some intermediate temperature (ca. 450 K), the surface conductance rises again (regime II). This phenomenon arises from the interplay of two independent types of atomic or molecular charge carriers.19 At low temperatures hydrogenbonded water carries charge as H3O+ and at high temperatures charge is transported by proton hopping. Starting at RT, the brief initial rise in conductance in regime 0 is due to enhanced surface mobility of the molecular charge carriers with temperature, but upon further increase in temperature, this water is lost from the surface, leading to decreasing surface conductance
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Figure 1. Thermal dependence of surface conductance of γ-alumina. Points are experimental data from ref 23. Solid curve is best fit based on eq 8.
in regime I. The conductance only increases again when protons in the material become mobile at high temperature in regime II. For this system with water (H3O+) and hydrogen (protons) as two charge carrier types, expression 7 can be rewritten as
G ) B×
{[
(8a)
( ) ( ( ) (
)
∆εw - Eqwdw /2 ∆ξw · exp k kT ∆εw + Eqwdw /2 ∆ξw νwdw exp · exp k kT
νwdw exp
)]
( ) ( ) ( ) ( )
× (8b)
∆Ew ∆Sw · exp k kT Nw × + ∆Ew ∆Sw 1 + Pw · exp · exp k kT Pw · exp
[
( ) ( ( ) (
)
∆εH - EqHdH /2 ∆ξH · exp k kT ∆εH + EqHdH /2 ∆ξH νHdH exp · exp k kT
νHdH exp
NH}
)]
(8c)
× (8d)
(8e)
where subscripts w and H represent hydrogen-bonded water and proton on the alumina surface, respectively. Since the energy cost of removing an H atom from an oxide surface is quite large (see Table 1), we expect the H atoms to remain bound to the alumina throughout the experimentally sampled temperature range 300 K < T < 673 K, i.e., the number of H atoms on the surface nH ≈ NH. In our previous work,19 we estimated the energetic parameters in the model with first-principles calculations following standard practice.24 Here we demonstrate how to use electronic structure calculations to estimate the entropic factors in the model. We have also refit the model with an improved optimization scheme. The best-fit energetic and entropic parameters are listed in Table 1 alongside the values estimated with electronic structure calculations. Note that in our previous work we were unable to compute the value of ∆εw, instead suggesting that it should be comparable to the strength of the hydrogen bond in the water dimer. We have now carried out a linear synchronous transit optimization to locate the transition state (TS) for a hydrogen atom hopping between the alumina surface and physisorbed water. The calculation was based on density functional theory (DFT)25 with the generalized gradient approximation (GGA) to
the exchange-correlation energy,26,27 as described in the review by Payne et al.28 and coded in CASTEP. The electron-ion interactions were described by ultrasoft pseudopotentials29 and a plane wave basis set with a cutoff energy of 340 eV to construct the (valence) electronic wave functions. In the dual-charge-carrier model there are four entropy changes: 1. ∆S during water adsorption/desorption (∆Sw) 2. ∆S during H3O+ hopping (∆ξw) 3. ∆S during H adsorption/desorption (∆SH) 4. ∆S during H hopping (∆ξH) The entropy change during a reaction can be related to the partition function Q of the reaction system by
(
∆S ) -kN0 ln
Qproducts Qreactants
)
(9)
where N0 is the Avogadro number and k is the Boltzmann constant. Computing the entropy changes thus reduces to estimating the relevant partition functions for each step in the charge transport process. Of the above entropy changes, ∆SH is not relevant in the present case because H is bound so strongly to the surface that it is not desorbed within the temperature range that is sampled in the experiments. The remaining three entropy changes are computed as follows: 3.1. Entropy Change upon Water Adsorption. The water adsorption/desorption is represented by the process
H2O(g) + slab.H+ f slab.H3O+
(10)
Here the process is defined as an adsorption with the sign determining the actual direction of the reaction. For purposes of calculation, the alumina material is represented by a finite slab as shown in Figure 2. The entropy of the reactants consists of vibrational entropy of gas-phase water, translational entropy of gas-phase water, and vibrational entropy of H+ bound to the alumina slab. (Here the rotational entropy of gas-phase water is neglected.) The entropy of the products consists of vibrational entropy of H3O+ bound to the alumina slab. In the reactants, there are three vibrational modes for the gas-phase water, three translational degrees of freedom for the gas-phase water, and three vibrational modes for the H+ relative to the alumina slab. In the products the nine degrees of freedom are all vibrational modes of the H3O+ relative to the alumina slab. Given these degrees of freedom, the entropy change upon water adsorption (∆Sw) is given by
(
∆Sw ) N0k ln
H 3O Qvib
+.slab
H.slab H2O Qvib Qvib
)
- Strans
(11)
Here QHvib2O is the vibrational partition function for gas-phase water, QH.slab is the vibrational partition function for H+ bound vib + to the alumina slab, QHvib3O .slab is the vibrational partition function for H3O+ bound to the alumina slab, and Strans is the translational entropy of gas-phase water. Using the harmonic oscillator approximation, the vibrational partition function is given by24,30
Qvib ) Πi
exp(-hνi /2kT) 1 - exp(-hνi /kT)
(12)
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TABLE 1: Values Relevant to the Dual-Charge-Carrier Model of γ-Alumina Surface Conductance from Theoretical Computation and Fitting Experimental Dataa ∆Ew (eV) fitting values computed values
-0.81 -0.58c
∆εw (eV) 0.34 0.51
∆EH (eV) -∞ -4.05c b
∆εH (eV)
∆Sw [J/(K · mol)]
∆ξw [J/(K · mol)]
∆SH [J/(K · mol)]
∆ξH [J/(K · mol)]
0.41 0.78c
-178 -191
71 31
NA NA
10 9
a Entropy changes are computed at 302.8 K. Electric field E ) 667 V, hopping distance d ) 0.265 nm, vibrational frequency υ ) 284 cm-1 and υ ) 413 cm-1 for W and H, respectively. b Assumed to be effectively infinite within the temperature range of interest. c From ref 19.
proximation the translational entropy of an ideal gas is given by
(( ( )) )
Strans ) N0k ln
V 4πmU N0 3N h2 0
3/2
+
5 2
(13)
Here h is the Planck constant, m is the molecular mass of the gas and U ) 3N0kT/2. For an ideal gas, V ) N0kT/P, where P is the pressure. It follows that
(( )
Strans ) N0k ln
Figure 2. (Top) oxygen terminated surface of γ-alumina (110C) for hydrogen-bonded H3O+ frequency calculations. (Bottom) Al-O terminated surface of γ-alumina (110C) for H atom frequency calculations. The H atom moves along an exposed edge of oxygen atoms. Key: red ) oxygen, purple ) aluminum, yellow ) hydrogen.
where the νi are the normal mode vibrational frequencies of the species, k is the Boltzmann constant, and T is the Kelvin temperature. The index on the product runs over all vibrational modes in the species. The vibrational frequencies (listed in Table 2) are obtained from quantum-chemical electronic structure calculations. For the entropy calculations, herein the vibrational frequency calculations were based on the PM3 Hamiltonian using the slab models shown in Figure 2. The computational efficiency of semiempirical calculations allows for modeling a cluster that includes all atoms up to and including second-nearest neighbors of the adsorption site, larger than is currently practical with firstprinciples methods. All atoms were frozen during these calculations except for the inspected H, H2O, or H3O+. The resulting vibrational frequencies were scaled by a factor of 0.9761, as is recommended for PM3 calculations.31 We have used this computational protocol with good success in modeling the interactions of various small molecules with alumina including alcohols,32 alkanes,33 and hexene,34 and the predictions support conclusions consistent with the results of benchmark QM/MM32 and ab initio calculations.35-37 As a further validation of our choice of the PM3 Hamiltonian, vibrational frequencies for the subset of species necessary to compute the entropy change upon water adsorption (∆Sw) were recomputed at the B3LYP/631G(d,p) level of theory. For computational efficiency, the smaller slab model shown in Figure 3 was employed for these DFT calculations. The results, which are listed in Table 2, are in very good agreement with those obtained with the PM3 Hamiltonian. As reported below, the computed value of ∆Sw varies little when computed on the basis of DFT frequencies instead of PM3 frequencies. To estimate the translational entropy of gas-phase water, we apply the Sackur-Tetrode equation.24,30 In this ap-
(
) )
kT 2πmkT 3 5 + ln + P 2 2 h2
(14)
The pressure of water vapor corresponds to 1% relative humidity at 22 °C, 2.6 × 10-4 atm. On the basis of these data, the change in entropy upon water adsorption is predicted to be -191 J · K-1 · mol-1, which differs by only 10% from the value of -171 J · K-1 · mol-1 determined by fitting the dual-charge-carrier model to experimental data. When recomputed on the basis of the B3LYP/6-31G(d,p) frequencies, ∆Sw ) -199 J · K-1 · mol-1, very similar to the value obtained on the basis of the PM3 frequencies. 3.2. Entropy Change during H3O+ Hopping. In the “H3O+ hopping” process, charge is carried by the shuttling of H+ from one water molecule hydrogen-bonded on the surface to an adjacent one. This is the well-known Grotthuss process.14,38 Protons move easily among the water molecules hydrogenbonded to the alumina surface. Effectively, H3O+ is moving along the surface, but only the proton is actually mobile.39,40 The mechanism is shown schematically in Figure 4. The “reactant” state consists of H3O+ bonded to the alumina surface. The “transition” state corresponds to a proton in transit between the oxygen atom of a hydrogen-bonded water molecule and an adjacent surface oxygen atom. In this transition state one of the vibrational degrees of freedom that is present in the reactant state becomes the reaction coordinate, so the transition state has one fewer vibrational degree of freedom than the reactant. To compute the entropy change during the H3O+ hopping process, we compute the vibrational partition function for H3O+ + bound to the alumina slab (QHvib3O .slab) and the vibrational partition function for the corresponding transition state ( + QHvib3O .slab.ts) using the harmonic approximation. The entropy change for H3O+ hopping (∆ξw) is then given by
(
∆ξw ) N0k ln
+.slab.ts
H3O Qvib
+.slab
H 3O Qvib
)
(15)
The computed vibrational frequencies for H3O+ bonded to the alumina surface and the corresponding transition state are shown in Table 2. The computed entropy cost of the hopping process is ∆ξw ) 31 J · K-1 · mol-1, which is lower than the value determined by fitting experimental data by about a factor of 2.
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TABLE 2: Frequencies Used in Estimating the Vibrational Partition Functions (Values in cm-1) H+.slab H+.slab (DFT) H+ hop ts gas-phase H2O gas-phase H2O (DFT) H3O+.slab H3O+.slab (DFT) H3O+ hop ts
3863 2989 3863 3992 3919 3648 3708 3736
1021 1001 1021 3871 3801 3536 3264 3473
476 790 rxn coord 1737 1660 2251 3073 1693
3.3. Entropy Change during H+ Hopping. In the H+ hopping process, the “reactant” state consists of H+ bonded to a surface oxygen atom. The “transition” state corresponds to a proton in transit between two adjacent surface oxygen atoms. In this transition state one of the vibrational degrees of freedom that is present in the reactant state becomes the reaction coordinate, so the transition state has one fewer vibrational degree of freedom than the reactant. To compute the entropy change during the H+ hopping process, we compute the vibrational partition function for H+
1751 1429 1560
757 768 1295
651 625 486
607 503 241
496 295 156
279 178 rxn coord
+
bound to the alumina slab (QHvib.slab) and the vibrational partition H+.slab.ts ) using function for the corresponding transition state (Qvib the harmonic approximation. The entropy change for H+ hopping (∆ξH) is then given by
( ) +
∆ξH ) N0k ln
H .slab.ts Qvib +
H .slab Qvib
(16)
The computed vibrational frequencies for H+ bonded to the alumina surface and the corresponding transition state are shown in Table 2. The potential energy surface in the vicinity of the transition state appears to be rather flat rendering the TS difficult to isolate. If we suppose that the high frequency vibrations of the reactant state remain essentially unchanged and that the lowest frequency vibration corresponds to the reaction coordinate, the computed entropy cost of the hopping process is predicted to be ∆ξH ) 9 J · K-1 · mol-1. This is in good agreement with the value of 10 J · K-1 · mol-1 determined by fitting experimental data. 4. Discussion
Figure 3. Cluster model used in B3LYP/6-31G(d,p) calculations, shown with H3O+ adsorbed. Color scheme is the same as in Figure 2.
Figure 4. Schematic of Grotthuss mechanism by which H3O+ moves across a surface through physical movement of only H.
To obtain the values of the energy and entropy parameters, we have fit expression 8 to experimental data. Since the experiments were carried out in “dry” gases, we assume 1% relative humidity at room temperature corresponding to Pw ) 0.000 26 atm. We use the calculated values of νw, νH, dw, and dH from ref 19, and vary the energy and entropy parameters to identify the best-fit curve. The best-fit parameters are collected in Table 1. The optimization procedure finds two minima with comparable total residual errors, i.e., the sum-of-squares of the residual errors at the 75 data points. The minimum that corresponds to the more physically sensible energy and entropy parameters is used here. (The reported entropy changes correspond to 302.8 K, although in the fitting all 75 different experimental temperature values were used.) Note that the energy parameters are in generally good agreement with those determined by first-principles calculations. The fitted value for the adsorption entropy of water is consistent with measured adsorption entropies for other small molecules.41 We note from expression 8, however, that increasing the partial pressure of the adsorbate mimics an increase (more negative) in the entropy of adsorption, exp(-∆S/k). Fitted values for adsorption entropy are therefore no more reliable than the uncertainty in the estimate of partial pressure. To gauge the reliability of the fitted parameters, we preformed a sensitivity analysis. With the fitted parameters at their optimized values we computed the sum-of-squares of the residual errors at each of the 75 data points. For each fitted parameter in turn we then increased the parameter by ca. 2% and recomputed the sum-of-squares of the residual errors and did the same with the parameter decreased by ca. 2%. These two points plus the optimum were then fit to a quadratic polynomial. The coefficient on the squared term gives an
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TABLE 3: Sensitivity Coefficients for Fitting the Energy and Entropy Parameters in the Dual-Charge-Carrier Modela ∆Ew (eV-2)
∆εw (eV-2)
∆εH (eV-2)
17 602 28 124 18 616
∆Sw (K2 · mol2 · J-2)
∆ξw (K2 · mol2 · J-2)
∆ξH (K2 · mol2 · J-2)
0.262
0.372
0.604
a
The values for the energy and entropy parameters are not directly comparable because the units differ, but the analysis shows that of the entropy parameters, ∆Sw, is the least tightly constrained by the experimental data.
indication of how tightly the experimental data determine the parameter. A large value indicates that a small change in the fitted value leads to a large increase in the sum-of-squares of the residual errors, hence the parameter is tightly defined by the experimental data. These “sensitivity coefficients” for each of the fitted parameters are collected in Table 3. The values for the energy and entropy parameters are not directly comparable because the units differ, but the analysis does show that of the entropy parameters, ∆Sw, is the least tightly constrained by the experimental data. The poorest agreement between the fitted and computed results is for ∆ξw, the entropy parameter is associated with the enhanced mobility of the low-temperature charge carrier with increasing temperature. Note that this is the dominant process in regime 0 in Figure 1 where there are very few experimental data points so the fitting is effectively based on less data than for the other values. This parameter is also the most challenging to compute with electronic structure calculations because it depends on the vibrational partition function of the transition state in the Grotthuss process, the structure that is the most difficult to isolate and contains the largest number of vibrational frequencies of all structures considered. A possible source of error in the predicted entropy changes (as opposed to the fitted values) is the use of the ideal gas model (Sackur-Tetrode equation) to estimate the translational entropy of gas-phase water. Since the vapor pressure of water in the experiments is very low, this approximation is probably quite good, but the equation is known to break down at sufficiently low temperatures so if the conductance measurements were conducted in cryogenic conditions (which would be useful to more accurately fit the parameters associated with the waterhopping process), an improved estimate of the translational entropy might be needed. Another possible source of error in the predicted entropy changes is the use of the harmonic approximation in determining the vibrational partition functions. This approximation is widely used in transition state theory and generally produces errors less significant than the errors introduced in the determination of the molecular potential energy surface by the electronic structure calculations. In general, we note that the entropy changes are most sensitively dependent on the lowest frequency vibrations, which are the most difficult to compute with accuracy. With the level of electronic structure theory employed here, calculations of entropy changes best serve as a “sanity check” on the values obtained by fitting experimental data. That is, the computed values can be used to verify that the values from fitting experimental data are physically sensible. Precise prediction of entropy changes would require much higher-level electronic structure calculations and more accurate determination of transition-state structures, both very demanding computational tasks. 5. Conclusions Previously, we cast a dual-charge-carrier model of surface conductance on γ-alumina in mathematical form and showed
how to estimate the values of the energy parameters in the model with first-principles calculations. Here we have shown that the extension of the analytic form of the model to include more types of charge carriers is straightforward. We have fit the model to experimental data for the thermal dependence of the surface conductance of γ-alumina. We have then gone on to demonstrate how to estimate the values of the entropic parameters in the model using quantum-chemical electronic structure calculations. This analysis includes entropy changes involved in both chargecarrier hopping and desorption processes. Sensitivity analysis of the fitting gives an indication of how tightly the energy and entropy parameters are determined by the experimental data. By comparing the best-fit values of the energetic and entropic parameters in the model to those obtained from quantumchemical electronic structure calculations, one can dramatically reduce the number of candidates for surface charge carriers and their modes of interaction with the surface. The model therefore provides a new way to use macroscopic conductance measurements to probe the atomic-scale surface structure of a material. Acknowledgment. This work was supported in part by DuPont, by the USDOE under contract number DE-FC0201CH11085 and by NATO-PST-CLG-980354. Computations were partially supported by the National Center for Supercomputing Applications (NCSA) and utilized the SGI Origin2000 at NCSA, University of Illinois at Urbana-Champaign. References and Notes (1) Batzill, M.; Diebold, U. Prog. Surf. Sci. 2006, 79, 47. (2) Woll, C. Prog. Surf. Sci. 2007, 82, 55. (3) Schmidt, O.; Kiesel, P.; Ehrentraut, D.; Fukuda, T.; Johnson, N. M. Appl. Phys. A: Mater. Sci. Process. 2007, 88, 71–75. (4) Sohlberg, K.; Pennycook, S. J.; Pantelides, S. T. Chem. Eng. Commun. 2000, 181, 107. (5) Ballarini, A. D.; Ricci, C. G.; de Miguel, S. R.; Scelza, O. A. Catal. Today 2008, 28, 133. (6) Ghiotti, G.; Chiorino, A.; Martinelli, G.; Carotta, M. C. Sens. Actuators B 1995, 24/25, 520. (7) Satterfield, C. N. Heterogeneous Catalysis in Practice; McGraw Hill: New York, 1980. (8) Knozinger, H.; Ratnasamy, P. Catal. ReV. - Sci. Eng. 1978, 17, 31. (9) Liu, X. J. Phys. Chem. C 2008, 112, 5066–5073. (10) Morimoto, T.; Nagao, M.; Tokuda, F. J. Phys. Chem. 1969, 73, 243–248. (11) Chakrapani, V.; Eaton, S. C.; Anderson, A. B.; Tabib-Azar, M.; Angus, J. C. Electrochem. Solid-State Lett. 2005, 8, E4. (12) Boudart, M.; Benson, J. E.; Vannice, M. A. Z. Phys. Chem. Neue Folge 1969, 64, 171. (13) Caldararu, M.; Popa, V. T.; Sprinceana, D.; Ionescu, N. I. Sens. Actuators B 1996, 30, 35. (14) Kreuer, K.-D. In Proton Conductors; Colomban, P., Ed.; Cambridge University Press: Cambridge, 1992; p 409. (15) Ikawa, H. In Proton Conductors; Columban, P., Ed.; Cambridge University Press: Cambridge, 1992; p 190. (16) Colomban P. Novak, A. In Proton Conductors; Colomban, P., Ed.; Cambridge University Press, Cambridge, 1992; p 38. (17) Kulwicki, M. B. J. Am. Ceram. Soc. 1991, 74, 697–708. (18) Tudose, I. V.; Horvath, P.; Suchea, M.; Christoulakis, S.; Kitsopoulos, T.; Kiriakidis, G. Appl. Phys. A: Mater. Sci. Process. 2007, 89, 57–61. (19) Cai, S.; Caldararu, M.; Chihaia, V.; Munteanu, C.; Hornoiu, C.; Sohlberg, K. J. Phys. Chem. C 2007, 111, 5506. (20) Aroutiounian, V. M.; Aghababian, G. S. Sens. Actuators B 1998, 50, 80–84. (21) Antipov, V. V.; Dorofeev, V. P.; Malkov, A. A.; Malygin, A. A.; Mikhailovsky, S. V.; Sosnov, E. A. Glass Phys. Chem. 2008, 34, 534–542. (22) Sberveglieri, G. Sens. Actuators 1995, B23, 103. (23) Caldararu, M.; Postole, G.; Hornoiu, C.; Bratan, V.; Dragan, M.; Ionescu, N. I. Appl. Surf. Sci. 2001, 181, 255. (24) Loffreda, D. Surf. Sci. 2006, 600, 2103. (25) Kohn, W.; Sham, L. J. Phys. ReV. A 1965, 140, 1133. (26) Perdew, J. P. Phys. ReV. B 1986, 33, 8822. (27) Perdew, J. P.; Wang, Y. Phys. ReV. B 1992, 45, 13244. (28) Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.; Joannopoulos, J. D. ReV. Mod. Phys. 1992, 64, 1045.
Macroscopic Surface Conductance (29) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892. (30) McQuarrie, D. A., Statistical Thermodynamics; Harper and Row: New York, 1973. (31) Scott, A. P.; Radom, L. J. Phys. Chem. 1996, 100, 16502. (32) Cai, S.-H.; Sohlberg, K. J. Molec. Catal. A 2003, 193, 157. (33) Cai, S.; Chihaia, V.; Sohlberg, K. J. Mol. Catal. A 2007, 275, 63. (34) Cai, S.-H.; Sohlberg, K. J. Mol. Catal. A 2006, 248, 76. (35) Smirnov, E. P.; Tsyganenko, A. A. React. Kinet. Catal. Lett. 1977, 7, 425. (36) Smirnov, E. P.; Tsyganenko, A. A. React. Kinet. Catal. Lett. 1984, 26, 405.
J. Phys. Chem. C, Vol. 114, No. 9, 2010 3997 (37) Fleisher, M. B.; Golender, L. O.; Shimanskqya, M. V. React. Kinet. Catal. Lett. 1992, 46, 173. (38) Kreuer, K. D. Chem. Mater. 1996, 8, 610–641. (39) Knapp, E.-W.; Schulten, K.; Schulten, Z. Chem. Phys. 1980, 46, 215. (40) Hayashi, K.; Muramatsu, H.; Matsuishi, S.; Kamiy, T.; Hosono, H. Electrochem. Solid-State Lett. 2009, 12, J11. (41) Bartels-Rausch, T.; Eichler, B.; Zimmermann, P.; Gaggeler, H. W.; Ammann, M. Atmos. Chem. Phys. 2002, 2, 235–247.
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