Distinguishing Bulk Conduction from Band Bending Transduction

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C: Physical Processes in Nanomaterials and Nanostructures

Distinguishing Bulk Conduction from Band Bending Transduction Mechanisms in Chemiresistive Metal Oxide Gas Sensors Aravind Reghu, Jay LeGore, John F. Vetelino, Robert J. Lad, and Brian G. Frederick J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01446 • Publication Date (Web): 24 Apr 2018 Downloaded from http://pubs.acs.org on April 24, 2018

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The Journal of Physical Chemistry

Distinguishing Bulk Conduction from Band Bending Transduction Mechanisms in Chemiresistive Metal Oxide Gas Sensors Aravind Reghu,1 L. Jay LeGore,1 John F. Vetelino,1,2 Robert J. Lad,1,3 and Brian G. Frederick1,4,5* 1

Laboratory for Surface Science and Technology (LASST), 2Dept. of Electrical and Computer

Engineering, 3Dept. of Physics and Astronomy, 4Forest Bioproducts Research Institute (FBRI), 5

Dept. of Chemistry, University of Maine, Orono, ME 04469.

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ABSTRACT: Chemiresistive metal oxide gas sensors based on materials including SnO2, ZnO, TiO2, and WO3 have been investigated extensively for a wide range of applications. The band bending model, based on the surface chemistry of highly reactive ionosorbed species ( or ) and the semiconducting material properties of SnO2, TiO2 and ZnO, adequately predicts the dependence of steady state response on target gas pressure and temperature for these materials. However, the assumptions associated with the band bending model are not valid for sensors based on reducible oxides such as WO3, MoO3 and V2O5, in which lattice oxygen reacts with adsorbed target gases creating oxygen vacancies which diffuse rapidly into the bulk and modulate the conductivity. Here, we develop a model that includes surface reactions and vacancy diffusion for the bulk conduction mechanism and describe characteristics of the sensor behavior that allow bulk conduction to be distinguished from the band bending transduction mechanism. We illustrate the predictions of the model regarding how the change in conductivity, , and

response time, , depend on target gas pressure, temperature, and film thickness using well characterized WO3 sensors, including epitaxially oriented polycrystalline thin films and nano-rod structured glancing angle deposition (GLAD) films. The physical/chemical parameters of the model are determined in independent measurements. Expressions for the limiting cases in which is determined either by surface reactions or by bulk diffusion provide design criteria to predict the theoretical performance limits of sensors which operate via this transduction mechanism.


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1. INTRODUCTION

Semiconducting metal oxide (SMO) sensors, commonly based on SnO2,1 ZnO,2-5 MoO3,6 and WO37 deposited on an electrically insulating substrate, are widely studied for gas sensing8-9 and are commercially available.10 Their material properties11 and nanostructure10 have been evaluated for detecting a wide range of gases for industrial, military, environmental, and civilian applications.12-13 SMO sensors detect gases by undergoing a change in electrical resistance as a target gas interacts with the SMO film. A widely accepted mechanism, known as the band bending model has been developed over the past 50 years by many workers, including Wolkenstein,14 Heiland,

15

Morrison,16-17 Geistlinger,18 Göpel,19 Semancik and Cavicchi,1,

20-21

Bârsan and Weimar,22-25 and Sberveglieri26 to describe the interaction of the so-called “ionosorbed” molecular ( ) and atomic ( ) oxygen species with target gases on SnO2, TiO2, and ZnO materials and establish the fundamental solid state physics and surface chemistry involved. The mechanism assumes that 1) ionosorbed species are stable on the surface at sensor operating temperatures; 2) the electrons in the near-surface region of the semiconductor are repelled by the localized charge of the ions at the surface causing upward band bending of the electrostatic potential and a depletion of electron carrier density near the surface; and 3) there are no other mobile charged species (e.g. dopants, point defects) that can neutralize the surface charge. Whether these assumptions are satisfied or not depends on the oxide material properties. Electron spin resonance (ESR),27-29 infrared (IR),30 and thermal desorption (TDS)31 data indicate that ionosorbed atomic oxygen species ( ) are stable at temperatures up to 500 °C on SnO2 and TiO2.27-28, 31 The presence of a surface potential has been measured directly between ZnO101 0 single crystals when exposed to oxygen,32-33 consistent with the predictions of Weisz.34 For pure SnO2, equilibrium oxygen vacancy concentrations are of order 1014 cm-3, but


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the diffusion constant for oxygen vacancies is negligible, increasing from 10-100 cm2/s at room temperature to 10-35 at 500 °C, as illustrated in Figure 1.35 Comparison of diffusivities for single crystal TiO2 and ZnO (dashed curves, S) with values for the polycrystalline materials (solid curves, P) illustrates that defect mobility can be dramatically higher in polycrystalline materials, and may reach values of order 10-20 cm2/s for polycrystalline TiO2. The presence of impurities may cause assumption 3 to break down. In fact, Hernandez-Ramirez, et al., have shown that oxygen vacancy diffusion accounts for long-term drift in SnO2 nanowires, even at room temperature.36 Nevertheless, there is strong evidence that the band bending model applies to SnO2, ZnO, and TiO2. Effects of parameters including reaction rate constants,22-25 crystallite size,37-38 film surface area,39-41 and target gas concentration22-25, 42 on the sensor response have been studied through experimental work and modeling of surface kinetics and gas phase diffusion through pores.43-44

Figure 1. Temperature dependence of diffusion coefficients for single crystal (---S) and polycrystalline (__ P) oxides showing that oxygen vacancy diffusion in WO3, V2O5, and MoO3 is many orders of magnitude faster than in pure SnO2, ZnO, and TiO2, but diffusivities in polycrystalline materials can be much higher than single crystals. Based on data by Freer.35


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The band bending model has been applied to sensors based on WO3, V2O5, and MoO3, assuming explicitly45 or implicitly46-53 that the model is applicable, although defect concentrations are higher and drastically more mobile. The intrinsic equilibrium vacancy concentration in reducible oxides like WO3 is approximately 100 times greater than in oxides like SnO2, as estimated from defect creation energies.54-56 As illustrated in Fig. 1, diffusion constants for WO3 are 20 – 60 orders of magnitude greater than SnO2. Furthermore, little ESR evidence exists regarding the presence of or on these oxides, because oxygen is thought to react directly with the reduced oxide surface to form lattice oxygen.57-58 In cases where ESR has detected ionosorbed oxygen, oxygen desorption is not observed and is reduced to surface

lattice oxygen ( ) and migrates into the film lattice around 350 °C.27-28, 31, 59-60 Analysis of impedance spectroscopy measurements during oxidation of tungsten metal within the point defect model show that the oxygen vacancy concentration varies with depth below the oxide surface and is compensated by the local electron density.61 The large oxygen vacancy

concentration and the rapid vacancy diffusion rates in these materials, as well as the lack of evidence for ionosorbed species implies that the band bending model is not appropriate for these oxides. Instead, reactions with lattice oxygen create oxygen vacancies and excess electrons that modulate the bulk conductivity of these type of chemiresistive metal oxide film sensors. 1.1 WO3 Material Properties and Surface Chemistry. We summarize the experimental basis for an alternative description which we refer to as the bulk conduction transduction mechanism. The solid state properties of WO3 are well understood. Substoichiometric WO3-x single crystals were studied by Berak and Sienko, showing that oxygen vacancies act as electron traps 0.18 eV below the conduction band edge,54 which are fully ionized above room temperature62 and


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certainly at the higher temperatures where sensors typically operate. Band structure calculations show that in substoichiometric WO3-x, for x > 0.2, the Fermi level moves into the conduction band and the material transitions to a conductor.56, 63-64 We have fabricated chemiresistive sensors with well-defined materials that allow the mechanism of transduction to be investigated systematically. We have previously grown and studied amorphous, polycrystalline, and epitaxially oriented WO3 films on r–cut sapphire substrates, as characterized by RHEED and other techniques.65-67 As illustrated in Figure 2a, scanning electron microscopy shows that epitaxially oriented WO3 films grown on r-cut sapphire are dense with no evidence of porosity. The transmission electron microscopy (TEM) image in Figure 2b and the selected area diffraction pattern in Figure 2c reveals well defined crystal structure. Crystallographic shear planes are clearly observed,65, 68 which enhance oxygen vacancy diffusion in the bulk as well as at grain boundaries. The electrical conductivity after annealing in air varies between 10-3 and 10-4 Ω ,66 corresponding to a bulk oxygen vacancy concentration of 1014 – 1015 cm-3 based on a mobility of 2.7 cm2V-1s-1 measured by the Hall effect.69 Therefore, the structure and morphology of these WO3 chemiresistive sensors are ideal for comparison with the model illustrated in Figure 2d in which a dense oxide film, which may be polycrystalline with grain boundaries, is grown with a thickness, , and width, , between

electrodes with spacing, . The black dots illustrate oxygen vacancies at the surface and the shading on the cross-sectional face illustrates a transient vacancy concentration gradient into the film.


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a

500 nm

WO3

c

b

20 nm

d

Figure 2. a) SEM image of a cleaved epitaxially oriented planar WO3 film on r-cut sapphire substrate showing the dense film structure with no porosity. b) Cross sectional TEM image showing shear planes which can enhance oxygen vacancy diffusion. c) TEM selected area diffraction (SAD) pattern through film in part b showing well defined crystal structure. d) Schematic drawing of a sensor described by the bulk conduction transduction mechanism consisting of a continuous, polycrystalline planar reducible metal oxide film, deposited between metal electrodes, as specified by geometric parameters; the intrinsic oxidation/reduction reactions and target gas reaction creates oxygen vacancies (speckled surface) creating a transient concentration gradient in the z direction.

The surface chemistry of reducible oxides has also been investigated extensively. Surface oxygen vacancies on WO3(100) crystals have been observed directly with STM70-71 under UHV conditions, leading to reduced W5+ ions as characterized by photoemission.72-73 The vacancy


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creation energy at the surface is different from that in the bulk of the film due to differences in coordination between the surface and bulk oxygen species. DFT cluster model studies find the energy of vacancy creation at the surface to be less than in the bulk for reducible oxides.63, 74 Calculation of surface vacancy creation energies on MoO3 (Ed = 4.85 eV)75-77 and WO3 (Ed = 2.98 eV)78 shows that the energy required to directly remove an oxygen atom is very large. However, adsorption of hydrogen and dehydroxylation to produce water is far less costly (Δ = 0.07 .77-78

Under sensor operating conditions vacancy creation occurs by

dehydroxylation of surface hydroxyls, 2 → + !∙∙ + 2 # ,

(1)

where hydroxyls can be generated from H2, C-H, or N-H bond breaking, !∙∙ is a doubly ionized vacancy on an oxygen site, and Kröger-Vink notation79 is used to specify conservation of mass, charge and sites. The relative stability of oxygen vacancies at the surface and in the lattice has been calculated for MoO3 and WO3 showing that vacancy diffusion into the bulk is thermodynamically favorable.64, 80-81 The kinetics for oxidation of WO3 show that the rate law57 is ⁄

$%& = '(! ) ,

(2)

indicating that molecular oxygen reacts directly with coordinatively unsaturated metal ions (a.k.a. Lewis acid sites or surface oxygen vacancies) to produce lattice oxygen, according to, + )

+ !∙∙ + 2 # → !& .

(3)

Therefore, the oxidation and reduction reactions occur via different paths. Direct reaction of hydrocarbons with lattice oxygen has been studied extensively, particularly with isotope labeling studies of the selective oxidation of olefins, in which feeding 18

O2 and the olefin yields

16

O labeled products. The rate of the oxidation reaction depends


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directly on the metal-oxygen bond energy, with MO3, V2O5, and WO3 near the maximum in socalled “volcano plots”.82-84

The overall oxidation/reduction (Redox) reaction, that creates

vacancies and determines the magnitude of change in sensor conductivity can be generally described as, ,-& . + /0, +

1.

2 !& → 0,-1 + 34 + /0, + )

1.

2 !∙∙ + 2 /0, +

1.

2 # , (4)

where R may represent C, or heteroatoms such as N or S. The rate of the reaction may be controlled by mechanistic steps that are not necessarily the redox steps, such as the vacancy creation or proton transfer steps. For specific reactions, such as the hydrodeoxygenation of acrolein to propene, propanol, and allyl alcohol on MoO3 and WO3, we have constructed microkinetic models based on quantum potential energy surfaces to predict adsorption, desorption, and reaction rates, associated with Brønsted and Lewis acid catalyzed elementary steps.78 Deposition of noble metal particles, such as Pt and Au, on catalyst and sensor surfaces can lower the dissociation energy barrier for breaking hydrocarbon bonds and increase the rate of the chemical reactions that reduce the metal oxide. Unlike the band bending mechanism, a thorough understanding of the bulk conduction mechanism is lacking. Sensor response descriptions based on reaction of lattice oxygen and vacancy diffusion in SrTiO3,85 WO360,

86

and V2O559 have been suggested before, but are not

widely recognized. Schilling and Colbow59 showed clearly that the time-dependent response of V2O5 to hydrogen gas fits the solution of Fick’s law for oxygen vacancies diffusing into a finite thickness film and they reviewed catalysis literature that shows hydrogen reacts with lattice oxygen. Williams interpreted the increase in resistance of WO3 sensors during ozone exposure to reactions between surface oxygen vacancies and O3.86 LeGore60 developed a model in which surface reactions were coupled to bulk diffusion to explain the oxidation and reduction of WO3


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thin film sensors and their response to H2S. A general framework that predicts the characteristics of the bulk conduction transduction mechanism is needed to distinguish it from the band bending mechanism. 1.2 Qualitative Comparison of Bulk Conduction and Band Bending Mechanisms. In Sect. 2 we develop a simplified model for the bulk conduction transduction mechanism. We first describe the predictions of the model and some qualitative ways to distinguish it from band bending. Quantitative predictions of the magnitude of sensor response and response time are given in Section 2. As summarized in Table 1, the magnitude of sensor response in the band bending mechanism depends on the target gas partial pressure through the rates of the surface reactions between ionosorbed species and the target gas. The ratio of the conductivity, , in the

presence of a target gas with partial pressure, (, to the conductivity in air, 5 , depends on the

specific ionosorbed species ( or ) and the specific target gas as ⁄5 ∝ ( 7 . The partial

pressure exponent, z, has been determined to have values of 0.3-0.6 for CO and H2, but depends on depth of the depletion layer relative to the sensor film thickness or grain structure.22,

42, 87

Modulation of the conductance leads to sub-linear behavior because adsorption rates are dependent on the surface potential which limits the coverage of ionosorbed species, known as the Weisz limit.34 The bulk conduction mechanism predicts that the magnitude of the response, ∆, at steady state conditions is also determined by the rate of surface reactions. When the rate of the target gas reaction is slow compared to the intrinsic rate of reduction of the oxide ∆ ∝ (9 ,

where b is the partial order of the target gas in the reaction rate law. At higher pressure, ∆ ∝

(:9/ . While predicting the pressure dependence is a critical test for any transduction model, the

experimental pressure dependence does not allow the two mechanisms to be distinguished a priori.


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By contrast, the dependence on film thickness differs for the two mechanisms. In band bending, as the film thickness increases relative to the depletion layer depth, the magnitude of ⁄5 decreases.

For the bulk conduction model, when steady state conditions have been

achieved, the vacancy concentration is uniform through the material and therefore ∆ is independent of film thickness. The time dependence of the sensor response in principle contains information regarding the surface kinetics and transduction mechanism.

For the band bending model, the transient

response is determined by the rates of the competing adsorption, desorption, and surface reactions, but band bending will affect the rate of adsorption of oxygen (and other species which trap electrons to form ions) and has generally not been treated theoretically in literature. We expect that the electron distribution responds rapidly to changes in the surface potential associated with the coverage of ionosorbed species and is therefore essentially dependent only on the rates of the surface reactions (and potentially gas phase diffusion in the case of porous sensors). The response time, , which we define as the time to achieve 90% of the steady state value, should be independent of film thickness for band bending. Under isothermal testing conditions with a step change in target gas pressure, an exponential dependence is generally observed. We have examined two limiting cases for the step response within the bulk conduction mechanism: 1) if surface reaction rates are slow compared to bulk vacancy diffusion, then the transient response is determined by the surface reaction kinetics and becomes a pseudo first order reaction in lattice oxygen.

In this case, the exponential time dependence is not easily

distinguished experimentally from band bending; however, 2) if diffusion limits the rate of sensor response, the shape will have t1/2 behavior characteristic of diffusion and should be strong


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evidence of bulk conduction. Our model predicts that increases proportional to the volume/area ratio (i.e. the film thickness, L, or rod diameter, R, for nanowires) for the surface reaction limited case and proportional to L2 for the diffusion limited case.

Table 1. Characteristic Behavior of the Band Bending vs. Bulk Conduction models. Characteristic Characteristic

Band Bending Band Bending

Magnitude of Mag nitude of response vs. concentration of target gas. of target gas.

Depends on analyte;X⁄XY ∝ Z[ , where z = 0.3-0.6 have been shown.

Magnitude of response vs. film thickness thickness film thickness

X⁄XY decreases

with increasing thickness if depletion layer width is too thin to modulate entire film.

Functional dependence of step response to target gas vs. time at constant temperature temperature

Transient response of surface reactions and effect of surface potential barrier on rate of adsorption of ionic species gives

Response time, l, vs. film thickness film thickness

l is independent of

gX ∝ h − jk⁄l

film thickness

Bulk Conduction Bulk Conduction

Surface Reaction Limited Limited

Diffusion Limited Limited

Depends on analyte; ∆X ∝ Zc, at low pressure; or ∆X ∝ Zdc/e at high pressure, where b is order of the rate in target gas Magnitude of ∆X is independent of film thickness.

Psuedo-first order reaction in lattice oxygen gives exponential behavior

gX ∝ h − jk⁄l

l increases proportional to volume/area ratio e.g. film thickness.

Determined by solution to diffusion in finite film. ∆X ∝ kh⁄e q for k ≤. d e, r then saturates.

Increases proportional to square of film thickness.


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We present the model with quantitative predictions in Section 2, and then in Section 3 show that the bulk conduction model describes sensor performance for well-defined planar and nanorod structured gold (Au) doped WO3 sensors exposed to NH3 and H2S, demonstrating both reaction- and diffusion limited cases, and compare best-fit model parameters to independently measured physical and chemical properties. We also illustrate the use of a microkinetic model to predict theoretical performance limits and trade-offs between response and response time, given temperature and pressure dependence of reaction rates and the diffusion coefficient. 2. BULK CONDUCTION MODEL

2.1 Magnitude of Sensor Response. For simplicity, we consider the conductivity through a dense planar film of thickness, L, width, w, and distance between electrodes, , which in general may be single crystal, polycrystalline, or amorphous, with the surface at z = L exposed to the gas phase, as illustrated in Figure 2d. When the sensor is exposed to pure air, we simplify the oxidation and reduction reactions as if reaction (3) is reversible, describing the change in the surface vacancy coverage, uv , as wxy wz

= −' (!{) uv + ' u|} − uv

(5)

where a is the order in oxygen pressure and u|} is the number of lattice oxygen sites per monolayer. This approximation is valid if there is a rate limiting step in the generally multipleelementary step oxidation and reduction pathways, where ' ~ and ' ~ are the temperature

dependent rate constants. The order in O2, = +), has been determined for WO3.57 Within this

approximation, we associate an effective equilibrium constant, K, =

=

)

+ )

(6)

that characterizes the ratio of the vacancy creation and annihilation rates. K is dependent on temperature and O2 pressure. It decreases with increasing metal-oxygen bond strength and


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therefore describes the reducibility of the oxide. K increases with temperature. As an illustration of how reducibility affects sensor response for the reaction of NH3 on WO3 (see Sect. 3.4), the symbols in Figure 3 shift upward with temperature crossing curves of constant K. We use dimensionality60 to approximately relate the surface and bulk oxygen vacancy concentrations, !9 ≅ uv :⁄ . We note that the Gibbs free energy for vacancy creation precisely determines

!9 while uv will vary depending on the surface facet. For doubly ionized vacancies, the

carrier concentration, = 2!9 , and therefore the conductivity at steady state conditions with a

uniform distribution of vacancies in the film,

= = 2uv :⁄ = , }

(7)

where is the electron mobility and R is the measured resistance of the film at steady state. Thus, the steady state conductivity in air, { = 2 u|}

:⁄

,

(8)

obtained from Eqs. 5 and 6, is a function of the rates of oxidation and reduction. Because u|}

and are material dependent parameters, we define a dimensionless bulk vacancy concentration,

= !9 u|} :⁄, which is a function only of kinetic constants, and thus values of Δ plotted in Figure 3 are proportional to the change in conductivity when exposed to the target gas, where Δ = 2u|} :⁄ Δ.

(9)

In the presence of a reducing target gas of pressure, Pt, we include an additional term in Eqn. 5 representing the rate determining step for the reaction with surface lattice oxygen atoms of coverage, u|} − uv , rate constant, k3, and partial order b, wxy wz

= −' (!{) uv + ' u|} − uv + ': u|} − uv (z9 −

wv w7

7}

,

(10)

and allow for diffusion of surface vacancies into the bulk, where the last term on the r.h.s. is the flux at the surface, z = L, which depends on the bulk vacancy concentration gradient and


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diffusion constant, D, appropriate to the material. For arbitrary variation in the target gas pressure with time, simultaneous solution of Eqn. 10 and Fick’s 2nd law of diffusion, wv wz

=

w) v w7 )

,

(11)

over 0 ≤ z ≤ L yields the time dependent coverage of surface vacancies and depth distribution of bulk vacancies.

A transient spatial vacancy distribution requires a non-uniform carrier

concentration to achieve local charge neutrality, analogous to the distributions determined from impedance spectroscopy and the point defect model describing electrochemical passivation of tungsten films.61 In general, the time-dependent response of the system may be influenced by the rate of surface reactions, bulk diffusion, or both as well as the initial vacancy depth distribution. Under typical sensor testing conditions, a step function in target gas pressure is applied at constant temperature, ideally with an initially uniform vacancy depth distribution corresponding to { , for which closed form solutions are available when the rate is diffusion limited. After the sensor has achieved steady state conditions, the change in conductivity is determined only by the rates of the surface reactions, Δ = 2u|}

:⁄

:⁄

¡

−/

2

:⁄

¢,

(12)

z where £¤w = ': (z9 . We analyze the response of the system using an effective equilibrium

constant, z =

= ¥ ,

(13)

+ )

z representing the relative ability of the target gas to reduce the oxide. Note that £¤w incorporates

both the temperature dependence of the rate constant, ': ~, as well as the target gas pressure and its power law dependence, b.

Therefore, at steady state conditions in the target gas

compared to that in air, Δ, depends on the rates of reduction by target gas and intrinsic oxide


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reduction, according to Eqn. 12. Using Eqns. 6 and 13 to express the rates relative to the rate of oxidation by air and transforming to dimensionless vacancy concentrations ( simplifies analysis of the behavior of Eqn. 12.

Figure 3. Behavior of the dimensionless change in bulk vacancy concentration, Δ (proportional

to Δ), as a function of the relative ability of the target gas to reduce the oxide, Kt, for a range of oxide reducibilities, K. Symbols illustrate the variation of K and Kt for a microkinetic model of

NH3 reactions in air on WO3 at indicated concentrations from 250 to 600 °C (see Sect. 3.4).

Figure 3 shows the dimensionless change in vacancy concentration, Δ, as a function of z

for a wide range of oxide reducibility, . This simplified kinetic model reveals three regimes.

When the rate of target gas reaction is less than the intrinsic oxide reduction rate (z < ; i.e.)

the response, Δ~z ⁄ . For constant sensor temperature ( is constant) Δ is proportional

to ': (z9 . Because the value of increases with temperature, the response increases with

temperature. More reducible oxides (larger K) give greater response. The value of t typically increases with temperature due to the temperature dependence of ': . To illustrate these effects,


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the diamonds in Figure 3 labeled 250 °C through 600 °C show the parametric dependence of ~ and z ~ on temperature for a microkinetic model for 10 ppb NH3 detection on Au/WO3

z in air, described in Section 3, where £¤w < £¤w . For other reactions, the response may go

through a maximum if the reaction rate becomes adsorption limited (decreasing the surface coverage) or due to thermodynamic limitations if the reaction is exothermic. ⁄

In the regime where z > for £¤w and asymptotically approach the common z:

curve

below 400 °C. When ≥ 1, the oxide will be completely reduced to metal. Therefore, the magnitude of Δ is independent of target gas pressure: it is simply the difference between the

conductivity of the oxide in air and the conductivity of the metal. The material could be used in this regime as a dosimeter, where the time for reduction would depend on both the target gas (via ': ) and its partial pressure. 2.2 Time Dependence of Response. Although the time dependent response of the coupled surface reactions and diffusion of bulk vacancies can be solved with finite element methods,60 the limiting cases of surface reaction or diffusion controlled processes reveal qualitative behavior that is useful to identify the transduction mechanism and provide a means to verify numerical solutions of the partial differential equations. We consider a step function in target gas pressure applied at constant temperature, starting from a uniform vacancy depth distribution corresponding to { . Experimentally, multiple processes may affect the functional form of the


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conductivity vs. time curve, so we adopt a practical approach to quantify response time, , as the time to reach 90% of the steady state value. 2.2.1 Surface Reaction Limited Case. For the surface reaction limited case, we assume bulk diffusion is fast and the vacancies are uniformly spatially distributed, so that there is a negligible concentration gradient in the film, similar to observed behavior in

18

O2 isotope exchange

experiments on WO3, MoO3, and V2O5, where 18O was uniformly distributed through the bulk.88 The rate of vacancy creation on a surface of area © is given by ©

wxy wz

concentration of bulk vacancies for a material of volume is given by is given by the first three terms in Eqn 10. Transforming uv into

and the rate of change in wv wz

%9 ⁄:

ª wxy

=v

wz

, where

wxy wz

and dividing by u|}

gives, w« wz

+

)

)

)

= u|}) ¬ −£%& ¥ + £¤w /1 − ¥ 2 + ®5 £¤w /1 − ¥ 2¯, ª

which is a linear differential equation in

)

¥

z

with constant coefficients, where ®5 =

(14) 0, ° < 0 is 1, ° ≥ 1

the unit step function. Because the differential equation is pseudo-first order in surface oxygen vacancies, the solution is closely approximated by ° = { + Δσ 1 − e³⁄´ , independent of the orders (a and b) in O2 or target gas pressure where, within an accuracy of 10%, +

v

) = µ u|} ¯ ª

¶·· ¥⁄) s, ··

(15)

where τ is the exponential time constant if µ = 1, and τ corresponds to 90% of steady state if

µ ≈ 3.44. Therefore, the functional form of the time-dependent response contains no information regarding the orders of the surface reactions. Because the vacancy concentration as

a function of depth is uniform, the local electron carrier concentration is uniform to maintain


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charge neutrality. We assume that the conductivity through a dense film is not depth dependent, and the conductivity can be calculated directly from Eqn 7. Figure 4 shows the response time as a function of and z . Equation 14 was solved

z numerically for a wide range of values of £¤w , £¤w , and £%& to determine τ(at 90% of the

steady state value known exactly from Eqn. 12). Plotting the product τ£%& simplifies the behavior such that the response time curves collapse onto the common set of curves dependent only on and z , consistent with the approximate expression of Eqn 15.

In the low

+

temperature/pressure regime, where z < and < 1, (where ∆σ ∝ K ³ ) ), the limiting +

behavior of Eqn. 15 predicts that τ~ ) /£%& , independent of target gas pressure (z ). Since both +

∆σ and τ depend on oxide reducibility as ) , for sensors operating in the low pressure regime there is a tradeoff between magnitude and speed of response when choosing the oxide and operating temperature for the sensor. For faster detection of toxic gases in this regime, a less reducible oxide should be better. On the other hand, if the objective is to detect low concentration gases for environmental monitoring, where longer response times are acceptable, more reducible oxides can be used to increase ∆. The diamonds in Figure 4 for 10 ppb exposure to NH3 show the increasing response time as the temperature is raised from 250 to 500 °C, increasing by several orders of magnitude.


19


Page 20 of 45

Figure 4. Dependence of the time to reach 90% of steady state, , following a step function change in target gas pressure as a function of the relative ability of the target gas to reduce the oxide, z , for a range of oxide reducibilities, , with

v

ª

= 1 and u|} = 1 . Symbols

illustrate the variation of and z for a microkinetic model of NH3 reactions in air for the same conditions indicated in Figure 3. ¥

For the intermediate regime, where z > and ∆σ ∝ K )³ , the limiting behavior of Eqn. 15 + )

gives τ~z /£%& , which predicts increasing response time with target gas pressure. Both ∆σ and τ increase with target gas partial pressure. There is still a tradeoff between magnitude and speed

of response, but the reducibility of the oxide, , is not a factor in this pressure regime. The +

triangles in Figure 4 for 10,000 ppb at T ≤ 350 °C NH3 illustrate the z) dependence of τ. If

≥ 1, then the oxide is completely reduced to a metal, and the sensor response time becomes

faster with increasing target gas pressure. The response time, as given by Eqn. 15, shows the direct dependence on the volume to surface area ratio. These equations are applicable to planar films of thickness, L, where ⁄© = ; cylindrical rods of radius, r, where ⁄© = $⁄2; nanostructured materials such as nanorods

and nanosheets,89-94 and GLAD films95 (see Sect. 3); and porous materials such as MCM-48 and SBA-15 which achieve rather low ⁄© ratios. For SBA-15, ⁄© ratios of 2.3 nm and 1.8 nm have been achieved.96 Fabricating continuous WO3 oxide films in the L< 20 nm range can be challenging65 but growth of WO3 nanorods97 of radius r < 2 nm offers an order of magnitude improvement in response time with no direct affect on ∆σ.


20

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2.2.2 Diffusion Limited Case. When the rate of reaction is sufficiently fast to maintain a steady state surface vacancy concentration, the time dependent response of the sensor is controlled by diffusion of oxygen vacancies (and charge compensating electron carriers) into the film. Closed form solutions for arbitrary initial depth distributions exist for finite thickness films and rods,98 but for simplicity and consistency with the surface reaction limited solutions of Sect. 2.2.1, we consider a uniform vacancy depth distribution prior to a step function change in target gas pressure at constant temperature. The solution to the 1-D finite thickness diffusion problem (Eqn. 11), i.e. the vacancy concentration vs. z, is plotted for a dimensionless time of 0.024

}) ¼

as the

solid blue curve in Figure 5 a, where ½ = is the film-gas interface and D is a concentration independent diffusivity. The surface vacancy concentration is greater than the initial bulk concentration due to the reduction reaction. The carrier concentration, shown schematically in Figure 5b, is higher at the surface, since the local electron carrier concentration, ¤ = 2! to maintain charge neutrality. The red curves in Figure 5 a and b illustrate a later dimensionless time of 0.24, where vacancies and electrons have accumulated deeper into the film. Note that the behavior contrasts with the band bending model, which predicts a depletion of charge carriers at the surface, relative to the bulk. The conductivity therefore varies with depth, and we calculate the film conductivity as the average of the carrier concentration over the depth of the film, ̅ ,

from the dimensionless vacancy concentration. The time dependence of ̅ is shown as a function ¼z

of time in Figure 5c, replotted in the inset as ¿̅ ,À. ¿ / ) 2 to show the characteristic t1/2 } })

behavior for ° ≤ 0.05 . ¼


21


Page 22 of 45

Figure 5. a) Vacancy concentration vs. z, after a step function increase in target gas pressure establishing a steady state vacancy concentration at the surface, !9 zÂÂ > !9 { ÂÂ , under diffusion controlled conditions, for dimensionless times indicated. Dashed curve illustrates effect of grain boundary diffusion on the depth profile, which is most pronounced at later times. b) Illustration of the corresponding carrier concentration vs. depth required to maintain charge neutrality and the accumulation of carriers at the surface. c) Depth averaged bulk vacancy concentration, ̅ , vs. dimensionless time, with inset showing that the conductivity increases with })

t1/2 for ° < 0.05 , irrespective of lattice or grain boundary diffusion. d) Comparison of the 90% ¼

response times for films of WO3, MoO3, V2O5, TiO2 and ZnO, based on data of Freer35 shown in Figure 1, illustrating the strong temperature dependence, where curves of thickness, L, are marked as follows: () 50 nm, () 125 nm, () 250 nm, () 500 nm and () 1000 nm.


22

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The effects of grain boundary diffusion have been modelled theoretically by a number of workers.99-103 Although grain boundary diffusion accelerates the motion of vacancies into the bulk, neither the depth profile nor the average conductivity vs. time are predicted to differ significantly in functional form from simple lattice diffusion. The dashed curve in Figure 5a illustrates a condition revealing the greatest effect of grain boundaries on the depth dependent vacancy concentration using a model by Gilmer and Farrell104 in which the grain boundary diffusion constant, Dgb, is greater than the lattice diffusion constant, D, by a factor of ∆= 9 ⁄

and the width of grain boundaries running perpendicular to the surface occupies a fraction, Ã, of the surface area. Despite rapid diffusion through the grain boundaries and subsequent lateral diffusion into the grains, the shape of the depth distributions (Figure 5a) and depth averaged vacancy concentration vs. time (Figure 5c) do not differ significantly from the characteristic t1/2 dependence expected for diffusion controlled processes.

Several models100-104 have been

proposed for grain boundary diffusion and yield effective diffusivities that can be employed in the solutions to the 1-D diffusion equation.98 Therefore, we conclude that for the diffusion limited case, the shape of the conductivity vs. time curve should follow the standard solutions to the diffusion equation, whether the diffusivity is due to lattice diffusion or grain boundary diffusion. The response time for the average conductivity (∝ ̅ ) to achieve 90% of steady state value is given by = 0.85

}) ¼

for a planar film, or = 0.4

(16) ) ¼

for a cylindrical rod, where - is the radius of the cylindrical

nanorod. Figure 5d compares the dependence of response time, if diffusion limited, as a function


23


Page 24 of 45

of temperature for planar films of WO3 to other oxides. The strong temperature dependence of D can give diffusion limited behavior at lower temperature for thicker films, as has been clearly demonstrated by Schilling59 for planar films of V2O5 during H2 detection. For a WO3 film with D = 3x10-12 cm2/s at 300 °C (Figure 1) to achieve a response time of 1s, requires L = 19 nm. However, growth of continuous epitaxial films is strongly influenced by interfacial surface energies and poses a practical limitation on response time.65 By contrast, for radial diffusion into WO3 nanorods of diameter 4 nm, diffusion would only limit the response time to 5 ms. 3. APPLICATION TO WO3 SENSORS

3.1 Experimental. The fabrication and characterization of the WO3 sensing films66-67 and the single crystal sapphire platforms with platinum interdigitated electrodes, RTD, and heater elements67 have been described previously. Briefly, 120 nm, 250 nm and 600 nm WO3 films were deposited using magnetron sputtering in an O2/Ar environment. Enhanced surface diffusion during deposition at 650 °C generates dense, epitaxially oriented polycrystalline monoclinic WO3 films, with a majority of crystallites with the (002) orientation parallel to the r-cut sapphire surface, with some (020), and (200) oriented crystallites present.66-67 Film composition as a function of annealing temperature in air was measured using XPS.66 Film dimensions were defined by a shadow mask and film thicknesses controlled with a quartz crystal oscillator, calibrated against profilometer measurements.66 Therefore, the intrinsic conductivity was calculated directly from measured resistances and film geometry. Carrier concentrations were calculated using mobilities measured with the Hall effect and van der Pauw methods.69 Separately, 125 nm and 250 nm WO3 films were sputter deposited at room temperature in an O2/Ar environment. The films were subjected to post-deposition annealing treatments at 500 °C in air for 6 h. Films with higher surface area to volume ratios were fabricated using the glancing angle deposition (GLAD) technique, and SEM and conductivity measurements indicate the


24

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presence of a thin, continuous base layer, with columnar rods extending 100 nm perpendicular to the surface.95 Gold (1.5 nm equivalent film thickness) was deposited onto the planar and GLAD films using electron beam evaporation, which led to Au nanoparticles on the surface. Sensor testing methods were described previously.60, 105 Planar films were exposed to 10 ppm H2S at 350 °C and 100-600 ppb of NH3 at 450 °C while the GLAD film was exposed to 100-600 ppb of NH3 at 450 °C. To independently determine the products and rate constant, k3, for the reaction of NH3 in air on Au supported WO3 surfaces, we carried out experiments using a continuous flow packed bed microreactor with an integrated GC/MS105-106 at reactor temperatures up to 500 °C. The Au/WO3 powder catalyst was synthesized by wet impregnation of WO3 (Aldrich, 20 µm) using gold(III) chloride trihydrate (HAuCl4.3H2O) at a 0.1 % mass loading to achieve a similar gold loading as on the GLAD sensor surface, estimated to be 50 atoms/nm2. X-ray diffraction (Panalytical X’pert Pro) analysis of the catalyst after reduction at 200 °C for 2 hrs in 10% H2/Ar and catalytic reaction measurements confirmed the presence of gold nanoparticles supported on monoclinic WO3; the BET surface area (Micromeritics ASAP 2020) was 0.9 m2/g. The reactant and product gas mixtures were sampled and injected into the GC (ThermoScientific Trace GC) with a split ratio of 100 onto an Agilent Plot Q column and analyzed with a quadrupole mass spectrometer (ThermoScientific ISQ). 3.2 Determination of Transduction Mechanism. The qualitative characteristics predicted by the model described in Sect. 2, and summarized in Table 1, provide factors to differentiate experimentally between the bulk conduction and band bending transduction mechanisms. The magnitude of the response to ammonia in air at 450 °C as a function of NH3 concentration is shown for planar and GLAD WO3 sensors in Figure 6a. The dependence on NH3 concentration,


25


Page 26 of 45

∆ = Å(9 , was determined from a weighted non-linear least squares fit to be b = 1, within 95% confidence intervals based on the individual sensor uncertainties in ∆ and (u: , for both

planar and GLAD films. The sensitivity, S, is the slope, Æ∆⁄Æ(, which is not significantly dependent of the V/A ratio. For these conditions, where z < (see below), b corresponds to a

rate law that is 1st order in NH3 within the bulk conduction model. Within the band bending model,22 the exponent typically varies between 0.3-0.6 so a value of Ç = 1 suggests that the band bending mechanism is not likely. Note that ½ order dependence on target gases, such as dissociative adsorption of H2, could result in Ç = 0.5 within the bulk conduction model and therefore a fractional order could be consistent with both mechanisms. The data is re-plotted in the inset of Figure 6a vs. the volume to area ratio, where the error bars represent an estimate of the sensor to sensor reproducibility. Within the uncertainty, there is no dependence of ∆ on V/A, which is consistent with the bulk conduction model. For the band bending model, the magnitude of ∆ depends on the Debye length (estimated to be 50 nm) relative to the film thickness. The variation in film thickness from the GLAD film, in which the base layer is 6.4 nm, to the 250 nm planar film indicates that there should be a substantial decrease in ∆ over this range if band bending were important.


26

Page 27 of 45

-3

(a) ∆∆ (1/ ∆ cm)

1.6x10

∆∆ (1/∆ cm)

1.4 1.2 1.0

0.0020 0.0015

600 ppb 500 ppb

0.0010

200 ppb 100 ppb

0.0005 0.0000

100 150 200 250

0.8

V/A (nm)

Response to NH 3 at 450 °C Fit: ∆∆ = S PNH3b

0.6 0.4

250nm planar film b = 0.91± 0.08 125nm planar film b = 0.96 ± 0.16 100 nm GLAD film b = 1.1± 0.5

0.2 0.0 0 3.5x10

100

200

300

400

500

600

700

Concentration (ppb)

-6

(b)

∆∆ (1/∆ cm)

3.0 2.5 2.0 1.5

Response to 10 ppm H 2S at 300 °C 120 nm Epitaxial Au/WO 3 Rxn Limited: tau = 1.47 ± 0.05 min 2 Diffusion Limited: D = 36 nm /s

1.0 0.5 0.0 0

2

4

6

8

Time/(min) -4.5

(c)

c

-5.0

log(∆∆)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-5.5 -6.0 Exposure to 10 ppm H 2S @ 300 °C 120 nm slope = 1.11 ± 0.01 250nm slope = 0.98 ± 0.03 600nm slope = 0.764 ± 0.009

-6.5

-1.0

-0.5

0.0

0.5

1.0

Log(Time/min)

Figure 6. a) Power law fit of response of Au/WO3 films vs. NH3 concentration showing 1st order dependence. Inset: Data replotted as ∆ vs. volume/surface area ratio showing that, within

sensor-to-sensor reproducibility, ∆ is independent of V/A. b) Comparison of time-dependent

response of ∆ for exposure of 120 nm planar WO3 film to H2S, showing that the surface

reaction limited response fits significantly better than diffusion controlled response (È¼ÉÉ =

25È&Ê ). c) Log-log plots of sensor response illustrating surface reaction limited behavior (slope

=1) decreasing with film thickness towards diffusion controlled behavior (slope = 0.5).


27


Page 28 of 45

The functional behavior of ° is found to vary with film thickness. For thin films, as illustrated in Figure 6b for exposure of a 120 nm WO3 sensor to 10 ppm H2S, the data fits ° = { + Δσ 1 − e³⁄´ , which is consistent with both the band bending and the surface reaction limit of the bulk conduction model. However, for thicker films the diffusion timescale becomes significant. For surface reaction limited behavior, the initial slope of a log ∆ vs. log t plot should be 1, while in the diffusion controlled limit, the slope should be 0.5 (as illustrated in Figure 5c). The behavior illustrated in the log-log plot in Figure 6c, shows a decrease in slope with increasing film thickness. For the 600 nm sensor, the slope of 0.76 is intermediate between the surface reaction and bulk diffusion limits. When both surface reaction and diffusion occur on similar timescales, the time dependent response requires solution of the coupled partial differential equations,60 which can be performed using finite element methods.107 The film thickness dependence provides additional evidence that the transduction mechanism is likely to be bulk conduction rather than band bending.

3.3 Analysis of Sensor Response. We now demonstrate that the response of the Au/WO3 sensors to NH3 can be predicted semi-quantitatively from the surface reaction limit solution, Eqn. 14, for a range of pressures, using independent experimental measurements of the mobility, V/A ratio, and rate constant for the reaction of NH3 in air on Au/WO3 catalysts. The model parameters are given in Table 2 and the time-dependent solution to Eqn. 14 (obtained numerically) is compared to data for a planar Au/WO3 film and a GLAD Au/WO3 film in Figure 7.


28

Page 29 of 45

1.6x10

-3

(a)

Response of 125 nm planar Au/WO 3 to NH3 Experiment Model with test cell effect Intrinsic sensor response NH3 test cell conc.

1.4

∆∆ (1/ ∆ cm)

1.2 1.0 0.8

600 ppb

0.6

500 ppb

0.4

200 ppb

0.2

100 ppb

-3 0.0 1.6x10

Response of 100 nm GLAD Au/WO 3 to NH3 Experiment Model with test cell effect Intrinsic sensor response NH3 test cell conc.

(b) 1.4 1.2

∆∆ (1/ ∆ cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0 0.8

600 ppb

0.6

500 ppb 0.4

200 ppb

0.2

100 ppb

0.0 0

50

100

150

200

Time (min)

(c)

100 nm

Figure 7. Comparison of experimentally measured and predicted ∆σ as a function of time for exposure to NH3 in air at 450 °C for (a) 125 nm planar Au/WO3 film and (b) 100 nm GLAD Au/WO3 film, using the surface reaction-limited model. (c) Cross sectional SEM image of the GLAD film showing 100 nm nanorods on top of a continuous base layer of small grains resulting in a smaller V/A ratio.


29


Page 30 of 45

The mobility was determined previously69 as a function of temperature using the Hall effect. The number of adsorption sites, u|} , taken to be the number of terminal oxygens (or W ions) per unit area on the WO3 (100) surface was based on the monoclinic x-ray crystal structure.108 The thickness of the films was determined by quartz crystal oscillator measurements during film deposition, calibrated against profilometry for the planar films. The baseline conductivity in air was relatively constant at 450 °C for film thicknesses greater than 20 nm. For the GLAD film, SEM measurements,95 shown in Figure 7c, indicate that there is a continuous base layer of small WO3 grains that determines the baseline conductivity, with rods extending 100 nm. The estimated base layer thickness of 6.4 nm, calculated from the known conductivity at 450 °C was consistent with SEM images. The V/A ratio was estimated from SEM images of the rod height and diameter and the base layer thickness. The conversion of NH3 in air was measured in a packed bed microreactor as a function of temperature. The only reaction product identified by mass spectrometry was N2O. The overall rate constant for the reaction, 'xË¥ = 8.1 atm–1 nm-2 s-1, was calculated from 35% conversion at 450 °C at a weight hourly space velocity, WHSV = 0.66 hr-1. The ratio of

was constrained by the baseline conductivity at 450 °C, using Eqn. 8 and

the parameters u|} and μ given above. The best fit values of k3 were identical for the planar and GLAD films, but about 20% of the value estimated from

ÍÎ¥ xÏÐ

, which we consider well within

experimental uncertainties of determining rate constants.


30

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Table 2. Surface Reaction Model Parameters for NH3-Au/WO3. Parameter Parameter

100nm GLAD GLAD 100nm GLAD

Planar Planar

×Ør ÙÔe Öh

7.716

7.716

μÓÔe Õh Öh

1.2

1.2

Method Method

Hall Effect XRD

L/nm L/nm

6.4

125

SEM

Õ⁄Ú ÙÔ

90

125

SEM

à. há × hYã

h. Yã × hYä

ÛÞß ÜjÝ

h ÛÞßÜ ÜjÝ Ö h ÛÞßÜ åæ Ö

Y. Yhe

Y. Yhá

ÞßÜ Ûåæ

= 7010ç

{ = 5010 è

from

:

éd ÞkÔh Öh

Y. hêd

Y. hêd

'xË¥ = 1.05 u|}

from microreactor

Figure 7a and 7b show that the simplified surface reaction model (solid black curve) accounts for both the magnitude of the conductivity change and the response time for step increases and decreases in NH3 concentration (red curve). The ability of the model to account for the magnitude of the response for films with thickness ranging from 6.4 to 125 nm demonstrates that conduction occurs uniformly through the bulk of the film. The small value of L in the GLAD film results in a higher resistance, and consequently poorer signal to noise ratio. The response time depends on V/A (Eqn. 15). The GLAD film is shown as an example of using nanofabrication to decrease V/A and improve sensor response time. The cross sectional SEM image (Figure 7c) shows that the rods are not perfect cylinders. The diameter at the top of the


31


Page 32 of 45

rods is greater than at the base due to atomic diffusion during growth. Annealing can also affect the morphology of the GLAD films which in turn can also affect the V/A ratio.95 The V/A ratio, estimated to be 90 nm, is about 70% that of the 125 nm planar film, accounting for the slightly faster response but still within the surface reaction limited regime. The red curves include the response time of the test cell, which has been measured to be approximately twice the volume/flow rate ratio for NO.105 Because NH3 is quite sticky, the test cell response times required to fit the sensor data (after the initial dose) were approximately 200 s and accounts for the slower measured response during decreases in NH3 concentration. The blue curves predict the theoretically achievable sensor response time, that would be observed with a gas delivery system with instantaneous step function changes in concentration. 3.4 Sensor Optimization. In sensor development, the primary performance specifications are typically sensitivity and response time. We illustrate how the bulk conduction model can be used to optimize sensor performance, which requires specific temperature and pressure dependence of reaction rates and temperature dependence of the diffusion coefficient. We developed a microkinetic model to predict the temperature and pressure dependence of the reaction rates for NH3 in air, shown in Figure 8. The reactions given in Table 3 allow for both adsorption and desorption of NH3 and a surface reaction stoichiometry consistent with the observed N2O product, resulting in the coupled set of differential equations in surface vacancies, NV, and ammonia coverage, ëxË¥ . From the reaction rates we calculated and z , and hence

the response, ∆, which was plotted on Figure 3. For the reaction limited case, the response time was plotted in Figure 4, while in the diffusion limit, the behavior depends on temperature and material properties, as illustrated in Figure 5d.


32

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Figure 8. Bulk conduction model illustrating temperature dependence of a) NH3 surface reaction rate in air on WO3, b) bulk vacancy concentration in air (red) and 10,000 ppb NH3 (dashed blue), c) response for indicated NH3 concentrations, and d) contributions to response time from diffusion (black) and surface reaction (colors for indicated NH3 concentrations) for planar films of 1000 nm (solid curves) and 100 nm (dashed curves), based on a microkinetic model of adsorption and reaction rates. Note minimum in response time depends on temperature and V/A ratio. The magnitude of response increases with temperature for this reaction because NH3 adsorption is strong.


33


Page 34 of 45

Table 3. Parameters in Microkinetic Model.

+

!∙∙

+ 2

u: ¥ :

í

+ #→

← )

!&

îïð u ñïò : Â í

u: Â + 2!& → u + 2!∙∙

Æuv = −' (!{) uv + ' u|} − uv Æ° + 2': u|} − uv ëxË¥ ÆëxË¥ = '{wÂ óu|} − ëxË¥ ô(xË¥ − 'w¤Â ëxË¥ Æ° − 2': u|} − uv ëxË¥ Vacancy creation

A2 = 3.8x102 s-1 Evac=120 kJ/mol

Oxidation

s = 2.6x10-10

NH3 Ades = 2x1011 s- Edes = 1 desorption kJ/mol NH3 reaction

A3 = 390 s-1

Erxn = kJ/mol

100 115

Values of the kinetic parameters, chosen to give measured values of and z at 450 °C and 600 ppb NH3, are given in Table 3. The temperature dependence of the rate of vacancy creation in Eqn. 5 was described by ' = © exp− 1{õ ⁄-~, where the vacancy creation activation

energy, 1{õ , is the dehydroxylation barrier and © is a constant. Thibodeau, et al.,106 found apparent activation energies of 120 kJ/mol decreasing as low as 40 kJ/mol with greater substoichiometry, WO3-x. The temperature dependence of the oxidation rate is weak, as specified by ' =

Â

¶ö÷) ø ù

, where s is the sticking coefficient, !) is the molecular mass, and 'ú is


34

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Boltzmann’s constant. The steady state solution for the vacancy concentration in air constrains A2 and s, given a value for Evac. The NH3 reaction rate constant, ': , in Eqn. 10 is actually an effective rate constant that may in general be controlled by the adsorption rate or the surface reaction rate. The rate constants for adsorption, '{wÂ =

¶ö÷ÍÎ¥ ø ù

,

and

desorption,

'w¤Â = ©w¤Â exp− w¤Â ⁄-~,

were

implemented in a Langmuir adsorption model, where the NH3 coverage was constrained to the monolayer coverage, u|} . The NH3 desorption energy depends on the adsorption site. Microcalorimetry data find strongly bound species on WO3 with heats of adsorption, ∆{wÂ , of 175 and 145 'û/ü¿, with more weakly bound species at 75 kJ/mol.109 Our DFT calculations for

NH3 on Lewis acid sites (i.e. surface oxygen vacancies) on W9O42 clusters give ∆{wÂ = 142 'û/ü¿, while interaction of NH3 with surface terminal hydroxyl groups (i.e. Brønsted acid sites) give ∆{wÂ = 78 'û/ü¿.110 The sticking coefficient was taken as 2.6x10-10. The surface

reaction rate constant, ': = ©: exp− &Ê ⁄-~ was implemented in a second order process,

where the reaction rate, $&Ê = ': ëxË¥ ë! , depends on both NH3 and lattice oxygen coverage, but

for small surface oxygen vacancy coverages, ë! ≈ u|} , and the rate becomes pseudo first order in the NH3 coverage. The microkinetic model predicts that the surface reaction rate increases with temperature, as

shown in Figure 8a. Using Campbell’s criterion for rate control,111 we find that the surface reaction is rate controlling, consistent with relatively high NH3 coverages.

The predicted

response, ∆, increases continuously with temperature, shown in Figure 8c, similar to experimental measurements.112 This behavior can be understood qualitatively from the bulk vacancy concentrations in air and 10 ppm NH3, shown in Figure 8b, in that both the surface reaction and intrinsic vacancy creation reaction increase with temperature while the oxidation


35


Page 36 of 45

rate is relatively constant. We calculated ~ from the predicted temperature dependence of the intrinsic vacancy creation and oxidation reactions, while we determined z ~ from the ratio of the surface reaction to oxidation rates.

These are the values plotted parametrically with

temperature in Figure 3. The value of ∆, for several NH3 concentrations was calculated from Eqn. 12 and plotted in Figure 8c. Note that the response increases proportional to NH3 pressure, but does not depend on the V/A ratio. The response time depends on film thickness and temperature, as shown in Figure 8d. At lower temperatures for thicker films, the response time is diffusion controlled, given by Eqn. 16. Note the L2 dependence in the low temperature regime. At higher temperatures, the response time increases with surface reaction rates, but because the rate of reduction by NH3 is less than the intrinsic reduction rate (i.e. z < ), the response time is not strongly dependent on NH3 concentration. However, the response time increases linearly with L. The ability to fabricate nanowires could dramatically decrease response times in the diffusion limited regime, and significantly reduce response time in the surface reaction limit. In summary for this reaction, the general predictions of the model are that the magnitude of the response increases with temperature, but there is a minimum in response time with increasing temperature that depends on the film thickness. For target gas reactions that decrease at higher temperatures, for example due to decreasing coverage of less strongly bound intermediates or due to endothermic reactions,113 the response may go through a maximum. 4. CONCLUSIONS

We have analyzed the effects of oxide reducibility and oxygen vacancy diffusion rates in WO3 and related oxides and shown that the assumptions of the band bending model are not valid. We developed a framework to predict the magnitude of chemiresistive sensor response, , for materials in which the bulk conductivity is controlled by steady state rates of surface reactions.


36

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The response, , is independent of film thickness, which contrasts with the predictions for the

band bending mechanism. The model predicts that increases with target gas partial pressure,

revealing a power law dependence that is determined by the order in the surface reaction rate. The time dependence of the response was analyzed for both surface reaction-limited and bulk vacancy diffusion-limited conditions. In the reaction-limited regime, an approximate analytical solution to a step change in target gas pressure under isothermal conditions was obtained; unfortunately, the functional form of the time-dependent solution may not distinguish the bulk conduction from band bending mechanisms. In the diffusion controlled regime, the functional dependence is t1/2, which is a good indication of the bulk conduction mechanism. The model predicts that response time increases with film thickness for both surface reaction limited ( ∝ ) and bulk diffusion limited cases ∝ ). By contrast, response time should be independent of

film thickness for the band bending mechanism. We illustrated the qualitative behavior using examples of experimental data from well characterized planar and GLAD Au/WO3 sensors exposed to H2S and NH3 gases. In every aspect, these sensors display behavior characteristic of the bulk conduction mechanism.

The simplified model predicts the sensor response semi-

quantitatively and the best fit parameters are in good agreement with independently measured chemical and physical values for the reaction of NH3 in air. We illustrated how to use the model for sensor optimization. A microkinetic model was developed to predict sensor performance as a function of temperature and NH3 pressure. For this reaction, both reaction rate and increased with temperature. Response time decreased with temperature initially in the diffusion limited regime, passing through a minimum, and increasing at higher temperatures in the surface reaction limited regime. The minimum in response time is


37


Page 38 of 45

determined by the film thickness, reaction rates, and diffusion constant. In general, fabricating thinner nanowires, nanorods or planar films will result in faster sensor response.

AUTHOR INFORMATION

Corresponding Author *B.G. Frederick. Email: [email protected] ACKNOWLEDGMENT

We would like to thank the W.M. Keck foundation for the funding experimental work and support of AR. Microstructural characterization was performed at the Oak Ridge National Laboratory ShaRE User Facility, under U.S. Dept. of Energy contract DE-AC05-96OR22464 and DE-AC05-76OR00033. We would also like to thank Dr. Paul Mayewski, Dr. David Frankel, Dr. Derya Deniz, Dr. George Bernhardt, Rahim Stennett and Michael Call for experimental assistance.


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Table of Contents Graphic:


45

Distinguishing Bulk Conduction from Band Bending Transduction

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