Dynamic structural disorder in supported nanoscale catalysts J. J. Rehr and F. D. Vila Citation: The Journal of Chemical Physics 140, 134701 (2014); doi: 10.1063/1.4869178 View online: http://dx.doi.org/10.1063/1.4869178 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Critical review: Effects of complex interactions on structure and dynamics of supported metal catalysts J. Vac. Sci. Technol. A 32, 020801 (2014); 10.1116/1.4820493 A classical trajectory study of the intramolecular dynamics, isomerization, and unimolecular dissociation of HO2 J. Chem. Phys. 139, 084319 (2013); 10.1063/1.4818879 The dynamic effects on dissociation probability of H2–Pt(111) system by embedded atom method J. Appl. Phys. 109, 063509 (2011); 10.1063/1.3554690 Charge redistribution in core-shell nanoparticles to promote oxygen reduction J. Chem. Phys. 130, 194504 (2009); 10.1063/1.3134684 Influence of initial oxygen coverage and magnetic moment on the NO decomposition on rhodium (111) J. Chem. Phys. 122, 154702 (2005); 10.1063/1.1878692 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 67.183.65.130 On: Thu, 10 Apr 2014 02:26:01
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Dynamic structural disorder in supported nanoscale catalystsJ. J. Rehr and F. D. Vila
Citation: The Journal of Chemical Physics 140, 134701 (2014); doi: 10.1063/1.4869178 View online: http://dx.doi.org/10.1063/1.4869178 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Critical review: Effects of complex interactions on structure and dynamics of supported metal catalysts J. Vac. Sci. Technol. A 32, 020801 (2014); 10.1116/1.4820493 A classical trajectory study of the intramolecular dynamics, isomerization, and unimolecular dissociation of HO2 J. Chem. Phys. 139, 084319 (2013); 10.1063/1.4818879 The dynamic effects on dissociation probability of H2–Pt(111) system by embedded atom method J. Appl. Phys. 109, 063509 (2011); 10.1063/1.3554690 Charge redistribution in core-shell nanoparticles to promote oxygen reduction J. Chem. Phys. 130, 194504 (2009); 10.1063/1.3134684 Influence of initial oxygen coverage and magnetic moment on the NO decomposition on rhodium (111) J. Chem. Phys. 122, 154702 (2005); 10.1063/1.1878692
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THE JOURNAL OF CHEMICAL PHYSICS 140, 134701 (2014)
Dynamic structural disorder in supported nanoscale catalystsJ. J. Rehr and F. D. VilaDepartment of Physics, University of Washington, Seattle, Washington 98195, USA
(Received 9 September 2013; accepted 4 March 2014; published online 1 April 2014)
The structure and behavior of materials at the nanoscaleare both of fundamental and technological importance. Theyare particularly relevant to the understanding of supportednanoscale catalysts. This challenging problem is complicatedby the fact that nano-scale properties differ from those of bulkcondensed systems, especially at high temperatures. Their un-usual thermodynamic properties have been widely recognizedat structural,1 spectroscopic,2 and catalytic3 levels. Theseproperties differ from the bulk since the thermodynamic limitdoes not apply for small N, where N refers to the number ofatoms in the nanocluster. For example, cluster surface effectsof O(N2/3) in extensive physical quantities (i.e., quantities pro-portional to N like total cluster energy) are crucial, and thusnano-scale systems are necessarily inhomogeneous. In addi-tion supported nano-scale catalysts exhibit an intrinsic fluctu-ating, inhomogeneous structure, referred to here as dynamicstructural disorder (DSD).4 Moreover, statistical physics ar-guments imply that nano-scale systems exhibit substantialO(1/
√N ) fluctuations in local temperature. Thus, in contrast
to bulk solids, these systems have no well-defined static “me-chanical equilibrium structure,” particularly at high temper-atures. The presence of these intrinsic fluctuations suggestthat it is useful to examine their behavior from a dynamicalperspective. This approach is in contrast to conventional sur-face science methodology typically based on static structuralproperties. Indeed, there has been a growing interest in suchreal-time approaches in recent years.4–7 The possible impli-cations of DSD on catalytic properties at the nanoscale arealso briefly explored. Dynamic effects have been postulated toplay a role in other treatments of catalysis, ranging from het-
erogeneous catalysis8, 9 at surfaces and in the catalytic activ-ity in biostructures.10, 11 The difference in the treatment hereis the emphasis on the effects of nano-scale dynamics on cat-alytic properties, which are investigated using nudged-elastic-band (NEB) transition state theory (TST) calculations.
Our approach for modelling the effects of DSD is basedon a concept dubbed picturesquely as “Shake-Rattle-and-Roll” (SRR). This approach was inspired by observations ofunusual behavior in nanoscale supported Pt and Pt-alloy cat-alysts: For example, x-ray absorption spectra (XAS) studiesshowed that small Pt nano-clusters exhibit negative ther-mal expansion, anomalously large disorder, and tempera-ture dependent shifts in XAS threshold energies.5 Theseanomalous properties were subsequently explained by finite-temperature density functional theory/molecular dynamics(DFT/MD) calculations, which showed that they are dynam-ical in origin, involving multiple time-scales.6 They include(1) fast bond vibrations (i.e., shaking), (2) soft anharmonic orfluxional modes, particularly involving the nano-particle sur-face atoms; and (3) librational motion of the center of mass(CM), which is due to the thermal contact of the nano-clusterstethered to the support bonding sites. As discussed below,these CM dynamics are analogous to hindered 2-dimensionalBrownian motion.6 These longer-time whole-cluster dynam-ics have been previously shown to be important in the char-acterization of epitaxial nanoclusters.12–15 When combinedwith the stochastic thermal coupling to the support, these low-frequency modes induce a rattle-like motion in the system,typically at sub-THz frequencies. At longer time scales (tensof ps for supported Pt clusters on γ -alumina at high tem-peratures), bonding to the support is transient. When thosebonds break, the nano-clusters can then roll or slide to new
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134701-2 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
positions on the support. This behavior differs from that inprevious studies of diffusion,12–15 in that the characteristictime-scale is set by the librational motion of the cluster withrespect to the support, rather than the frequency of individ-ual atomistic events. According to the crystallite model ofsintering, the combined motion of many such clusters typi-cally leads to reduced catalytic activity.16 The rattle-like mo-tion has been recently4 interpreted as dynamic structural dis-order, in contrast to static bond distortions or substitutionaldisorder typical in condensed systems. Instead, the DSD is“non-equilibrium” in character, in the sense that the systemexhibits fluctuations from mechanical and thermal equilib-rium, even though, of course, the full system (cluster plus sup-port) is in overall “thermal-equilibrium,” with a well definedglobal temperature T. Thus the mean geometric structure ofnano-clusters is amorphous and fluctuating, rather like that ina partly-melted solid. Indeed, DSD affects the nano-clusterbehavior through stochastic fluctuations in structure, clustercharge, thermal fluctuations, and center of mass position.
These remarkable observations have motivated us toreinvestigate the structure and catalytic behavior of supportednanoscale catalysts in terms of their dynamic behavior.To this end, we investigate the SRR concept based on acombination of real-time DFT/MD simulations, transientcoupled-oscillator models, and statistical mechanics. Giventhe computational demands of DFT/MD simulations, wefocus primarily on the few ps DSD regime. This regime issufficient to capture the “shake” and “rattle” dynamics, andthe faster components of the “roll” dynamics. However, thenanosecond dynamics involved in longer-range nanoparticledrift over the support are taken into account statistically, bysampling results for a number of independent trajectories.
As discussed below, the SRR model can explain theanomalous properties observed for supported nano-clusters,including both DSD and charge fluctuations. As an illustra-tion we have applied this approach to calculations of both theinteraction between the supported nanocatalysts and prototyp-ical molecules, and their dissociation reaction paths. Theseshow that the DSD provides a mechanism for increasing re-action rates. Although still rather preliminary, the SRR modeloffers tantalizing insights into dynamic mechanisms that con-tribute to catalytic processes.
II. DYNAMIC STRUCTURAL DISORDER
A. Structure and disorder in nanoclusters
In order to investigate DSD within the framework ofSRR, and to differentiate DSD from other types of disor-der, we consider the effect of large fluctuations on the dy-namic structure and disorder of the cluster. This structurecan be described formally as a point in 6N-dimensionalphase space defining the nuclear coordinates and momenta{ �Ri(t), �Pi(t), i = 1 . . . N}. Due to the fluctuations in the struc-ture, e.g., the change in shape due to the librational motionand fluxional surface bonds, the cluster potential energy sur-face V [{ �Ri}, t] will also fluctuate on a time-scale comparableto the librational motion, which is much slower than typicalbond vibrations. During that time the motion of the cluster is
vibrational with respect to the instantaneous minima on itsPES, i.e., fluctuating equilibrium positions, denoted by Ri .The fluctuating structure of the clusters can be characterizedin terms of the pair-distribution function (PDF) g(R), which isdefined as the average distance between all bond pairs (i, j) inthe cluster,
g(R) = 1
N (N − 1)
∑i �=j
〈δ(R − | �Ri(t) − �Rj (t)|)〉. (1)
This PDF can be measured in experiment. For example, theXAFS signal
χ (k) =∫
dR g(R)feff(k)
kR2sin(2kR + �)e−2R/λ (2)
is closely related to the Fourier transform of g(R). In con-ventional XAFS analysis, however, one usually restricts con-sideration to the near neighbor bonds, e.g., by Fourier filter-ing over the first coordination shell, which is described bythe near-neighbor distribution function g(R). Physical quan-tities of interest are then obtained from the cumulant mo-ments of g(R), e.g., the mean near-neighbor distance R = 〈R〉= ∫
dR R g(R), and the mean-square radial disorder (MSRD)by σ 2 = 〈(R − R)2〉.
We emphasize that the quantities R and σ 2 refer to av-erages over the entire cluster and thus can give a mislead-ing picture of the structure of inhomogeneous systems. Thisdifference is important in the Pt nano-clusters, for whichthe mean Pt-Pt bond lengths and fluctuations depend on thelocal environment, and thus their locations inside the clus-ter. Simulations of σ 2 can be carried out in terms of thedistribution of (partial) PDFs at each site g(R) = ∑
igi(R).The local near-neighbor distributions gi(R) have net weightsgi = ∫
dR gi(R), mean near neighbor bond lengths Ri
= ∫dR R gi(R), with fluctuations σ 2
i = ∫dR (R − Ri)2gi
(R). Consequently, the global average near-neighbor bonddistance R and MSRD σ 2 corresponding to experimentalmeasurements are
R =∑
i
Ri gi , σ 2 = σ 2D + σ 2
V , (3)
where the local bond vibrations (V) are σ 2V = σ 2
i gi , and themean-squared disorder (D) due to cluster inhomogeneity isσ 2
D = ∑i(Ri − R)2gi . Interestingly, both of these contribu-
tions have DSD contributions from low frequency fluctua-tions.
B. DFT/MD calculations
In order to understand the fluctuations in structure quanti-tatively we have performed VASP20 ab initio DFT/MD simu-lations of nanoscale Pt10Sn10 and Pt15Sn5 alloy clusters on γ -Al2O3 at 298 K and 598 K. All simulations were performedat the � point, using the PBE functional and a plane wavecutoff of 396 eV. Initial structures for the clusters were ob-tained by randomly substituting Sn atoms into the optimal gasphase Pt20 cluster, and then placing them onto the “d” layer ofthe dehydroxylated [110] surface of γ -Al2O3. This surfaceis represented by 4 layers (2 fully relaxed and 2 frozen) ina supercell of 19.4 Å × 13.7 Å with an effective vacuum
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134701-3 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
2.2 2.4 2.6 2.8 3.0 3.2 3.4RPtPt (Å)
0.00
0.02
0.04
0.06
0.08
0.10P
(RP
tPt)
R < 2.7 Å2.7 Å < R < 2.8 Å2.8 Å < R < 2.9 Å2.9 Å < R < 3.0 Å3.0 Å < R < 3.1 ÅTotal × 0.04
2.2 2.4 2.6 2.8 3.0 3.2 3.4RPtPt (Å)
0.00
0.02
0.04
0.06
0.08
0.10
P(R
PtP
t)
R < 2.7 Å2.7 Å < R < 2.8 Å2.8 Å < R < 2.9 Å2.9 Å < R < 3.0 Å3.0 Å < R < 3.1 ÅTotal × 0.04
FIG. 1. Decomposition of the total pair distribution function of Pt15Sn5 onγ -Al2O3 at 298 K (top) and 598 K (bottom) into individual pair components.The different colors label the mean distance for the pair and the height of thedistributions indicate their relative weight. Much of the width and asymmetryof the total near-neighbor distribution arises from cluster inhomogeneity.
separation of 16 Å. Four clusters were then relaxed to theiroptimal ground state conformations which were subsequentlyused to start the 298 K simulations. The initial structures forthe 598 K runs were obtained by randomly sampling andquenching four conformations from the 298 K trajectories. Allsimulations used a 3 fs time step, which is adequate to capturethe “fast” bond vibrations of the system. After an initial 3 psthermalization period, statistical averages were acquired for8 ps. These simulations provide a wealth of information on theelectronic and structural properties of these systems, whichhave been described in detailed in Refs. 4 and 6. Briefly,we find that the particles are approximately hemi-spherical.Their volume increases with temperature for low Pt concen-tration while it decreases at high concentrations, in agreementwith experimental observations5 of negative thermal expan-sion in pure Pt clusters. Analysis of the short-range order pa-rameter indicates that the Sn atoms preferentially segregateto the particle surface, likely due to their markedly positivenet charge. Pt atoms, which are mostly neutral, tend to formthe core of the clusters. Finally, the effective contact betweenthe Pt atoms and the support decreases with increasing Snconcentration.
2.6 2.7 2.8 2.9 3.0
RPtPt (Å)
0
5
10
15
20
25
30
35
σ2 (
10-3
Å2 )
298 K598 K
FIG. 2. Correlation between the mean Pt-Pt bond distances and their associ-ated mean MSRDs for Pt10Sn10 on γ -Al2O3 at 298 K and 598 K. Both tem-peratures show similar behavior with weaker/longer bonds associated withlarger MSRDs.
The decomposition of the g(R) obtained from these MDsimulations is shown in Fig. 1 vs. mean bond length. Inparticular, much of the width of the first neighbor peak ing(R) comes from cluster inhomogeneity, and thus explains theanomalously large disorder observed for the nanoclusters.5
Moreover, the individual PDFs in Fig. 1 are not randomlydistributed, suggesting a correlation between the mean Pt-Ptbond length and the associated MSRD of the bond. This cor-relation is clearly shown in Fig. 2, with an increase in theMSRD as the Pt-Pt bonds get longer/weaker. Another exam-ple of structural correlations is shown in Fig. 3, which depictsthe mean Pt-Pt bond distance RPtPt as a function of its dis-tance to the center of the nanoparticle (RCC). Bonds near thesurface of the cluster (RCC � 2.5 Å) are shorter, as expectedfrom their reduced number of near neighbors. This thresh-old is in good agreement with previous results4 showing thatthe interior-surface transition zone occurs between about 2.5and 3 Å.
1.0 1.5 2.0 2.5 3.0 3.5
RCC (Å)
2.5
2.6
2.7
2.8
2.9
3.0
3.1
RP
tPt (
Å)
298 K598 K
FIG. 3. Correlation between the Pt-Pt mean bond distances and their distanceto the center of the nanoparticle (RCC) for Pt10Sn10 on γ -Al2O3 at 298 K and598 K. Both temperatures show similar behavior with shorter bonds near thesurface of the particle.
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134701-4 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
C. Transient coupled oscillator model
While the DFT/MD simulations provide a quantitativeaccount of DSD, they do not explicitly expose the underly-ing physical mechanisms that drive the dynamical disorderor explain the thermal behavior of tethered Pt nano-clusters.In order to understand their thermodynamic behavior withinthe SRR picture, we introduce a “transient coupled oscillatormodel.” That is, we consider the internal vibrations with re-spect to fluctuating instantaneous-equilibrium structures. Thismodel is presumably valid over a ps time-scale which is longcompared to typical bond vibrational periods. The dominantmodes of the system coupled to the support thus consist of:(i) vibrational modes; (ii) soft and fluxional modes, partic-ularly close to the cluster surface; and (iii) CM librationalmodes, i.e., soft modes about which the cluster center ofmass fluctuates. For Pt, the vibrational modes are of THzscale, while the soft and librational modes are sub-THz. Themodel is transient, since anharmonicity and bond-breakingcan be substantial, and hence the characteristic modes of os-cillation fluctuate. The thermodynamic quantities of inter-est here can be obtained from the free energy, e.g., 〈F(T)〉averaged over several such models, each of which can becalculated over sufficiently short (e.g., ps time-scale) time-intervals within a quasi-harmonic approximation using therelation,19
F (T ) = E0(R) + kT
∫dω ρ(ω) ln[2 sinh(β¯ω/2)]. (4)
Here ρ(ω) is the total density of modes per unit frequencyand E0(R) is the internal energy at the transient mean atomicpositions. The thermodynamic average 〈F(T)〉 is given by thesame expression but with E0(R) replaced by 〈E0(R)〉 andρ(ω) by 〈ρ(ω)〉. This approach is efficient, since despite thefluctuations in structure, the average spectrum 〈ρ(ω)〉 is rel-atively stable and well defined. The term E0(R) is importantsince it contains the transient effects of charge fluctuationsand affects changes in potential energy that dominate the re-action paths. The support provides both a heat bath and chargereservoir at fixed T which give rise to stochastic CM motionand DSD. Since we are primarily interested in catalytic be-havior at high-temperatures, e.g., 300–600 ◦C, we focus ourdiscussion here to the classical limit with mean phonon occu-pations n(T) ≈ kT/¯ω 1. At 600 K this limit is a reason-able approximation for Pt, which has a Debye-temperature of240 K and an equivalent Debye frequency of 5.2 THz. In thisclassical limit
F (T ) = E0(R) + kT
∫dω ρ(ω) ln(β¯ω). (5)
This statistical approach can also be directly compared to ex-periment, since observed physical properties are usually de-fined as averages over a statistical ensemble, and are equiv-alent to time-averages over a long time interval sufficientto cover the accessible phase space. Low frequency modesare important since n(T) varies inversely with ω. However,the effect of breathing modes is expected to be small sincefor a 3-d bulk system ρ(ω) ≈ ω2 at long-wavelengths.From the identity E = ∂βF/∂β, we obtain a mean energy E
= 3NkT at high temperatures, consistent with the equipar-
2.4
2.6
2.8
3.0
3.2
RP
tPt (
Å)
TotalDisorder
0 2 4 6 8 10Time (ps)
-0.4
-0.2
0.0
0.2
0.4
ΔR
PtP
t (Å
) Vibrational
0 1 2 3 4 5 6Frequency (THz)
RP
tPt P
DO
S ρ
(ω)
298 K598 K
Disorder
Vibrational
FIG. 4. (Top) Typical trajectory for a bound Pt-Pt pair in Pt15Sn5 on γ -Al2O3 at 298 K (black) decomposed into DSD (red) and vibrational (blue)components by applying a 0.5 THz low-pass filter. (Bottom) Average pro-jected density of states for the vibrational and dynamic structural disordercomponents of bound Pt-Pt pairs in Pt15Sn5 as a function of temperature.
tition theorem. This energy includes contributions from the2-d librational motion in the x − y plane, which can be ap-proximated by low frequency (sub THz) vibrational modes ofthe CM (Fig. 4). Thus the mean stochastic librational kineticenergy (1/2)Mv2
lib = kT . This result is the analog of Brown-ian motion, the difference being that such motion is hinderedand 2-dimensional in character in nanoparticles tethered to thesupport. This stochastic CM motion is crucial to the origin ofDSD and drives non-equilibrium fluctuations in the internalstructure.
The low frequency DSD modes in Eq. (3) are particularlyimportant, since they give large contributions to the MSRD
σ 2 = ¯
2μ
∫dω
ρ(ω)
ωcoth(β¯ω/2) ≈ kT
μ
∫dω
ρ(ω)
ω2,
(6)where the expression on the right is the high temperaturelimit. Here μ refers to the reduced mass of a given bondpair, e.g, MPt/2 for Pt-Pt bonds. Thus it should be possibleto distinguish DSD from conventional vibrations by filteringthe density of modes ρ(ω). The result is shown in Fig. 4.Similarly, the contribution from each site σ 2
i can be obtainedby replacing ρ(ω) with the projected density of vibrationalmodes ρ i(ω). In this way, one can analyze the net MSRD into
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134701-5 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
contributions from different positions in the cluster and theirvibrational and DSD components. A similar expression canbe used to calculate the librational fluctuations of the CM,with the reduced mass μ replaced by MCM and ρ by the pro-jected density of librational modes. Note that this expressionimplies that σ 2 for the Pt-Pt bonds is linear in T at hightemperatures, with a slope depending on the inverse secondmoment of ρ(ω). Estimates of ρ(ω) can be obtained in var-ious ways. For example, one approach is to use DFT/MDcalculations of 〈σ 2〉, in real time over a sufficiently longinterval τ ,
〈σ 2〉 = 1
τ
∫ τ
0dt σ 2(t) =
∫dω σ 2(ω), (7)
where σ 2(ω) = ρ(ω)/μω2. Such real-time calculations natu-rally account for the transient coupled-oscillator motion andeffects of charge fluctuations on the dynamics of the cluster,as illustrated in Fig. 4 (bottom). Detailed information on thevarious times time-scales can be obtained from the Fourier-transform of real-time DFT/MD trajectories or from equationof motion techniques,21 which give for the density of modesρR(ω),
ρR(ω) =∫ ∞
0〈R(0) · R(t)〉 cos(ωt)e−εt2
, (8)
where 〈R(0) · R(t)〉 is the autocorrelation function of the co-ordinate of interest R and ε is a broadening parameter addedfor convenience.
D. Thermal fluctuations at the nanoscale
Part of the driving force behind DSD is due to the largethermal fluctuations in nano-scale clusters, which is brieflyexplored here. As noted in the Introduction, thermal prop-erties of nano-structures with relatively small N differ sub-stantially from those in macroscopic condensed matter due tofinite-size effects enhanced by their poor thermal coupling tothe support. As in bulk systems, their total thermal energiesat a given substrate temperature T are not sharply defined,but exhibit substantial energy fluctuations17 of order kT
√N .
Moreover, these variations lead to substantial long-time fluc-tuations in the local temperature (or energy per particle) of or-der T/
√N . This latter effect is not immediately obvious since
the global temperature T is a constant throughout the sys-tem, as fixed by the substrate. To understand these results, werecount the classical arguments of statistical thermodynam-ics. We consider a nano-cluster with N atoms weakly boundto a support at fixed temperature T, which serves as a heat-bath. For simplicity of argument the cluster volume V , chargeQ, and composition can be regarded as fixed, but these con-straints are not essential and will be relaxed below. Due tocontact with the support, which is in continuous thermal andvibrational motion, energy will fluctuate between the clusterand the support, with a probability distribution
P (E, T ) ≈ �(E)e−βE = e−βF(E,T ), (9)
where F(E, T ) = E − T S(E) and kln �(E) = S(E) is thenet cluster entropy due to distinguishable configurations�(E) of the nano-cluster at energy E. True thermal equi-
librium corresponds to the maximum probability or, equiva-lently, minF(E, T ) = F (T ), which in bulk thermal equilib-rium would correspond to the Helmholtz free energy. Themean total cluster energy E is then fixed by the relation∂S(E)/∂E|E = 1/T , and hence both the nano-cluster and thesupport will have the same average temperature T. The clus-ter entropy S(E) is additive in terms of the independent 6Ndegrees of freedom in the cluster dynamics. In addition tothe (3N − 3) internal vibrational modes (including both po-tential and kinetic degrees of freedom), the SRR dynamicsinclude four additional degrees of freedom from the 2-d libra-tional modes of the CM parallel to the support, and additionalmodes binding the cluster to it. The distribution P(E, T) issharply peaked for large N and approximately Gaussian nearthe peak at E. However, due to fluctuations the most probablevalue may differ from the thermal equilibrium value.18 Themean square energy fluctuations σ 2
E are obtained from the 2ndderivative of the entropy σ 2
E = k/[∂2S(E)/∂E2]. The quantityσ 2
E is clearly O(N) since both the total energy E and entropyS(E) of the cluster are extensive. At high temperatures, whereequipartition is valid, E = 3NkT and S(E) ≈ 3Nkln E. Thusthe energy fluctuations are given by σE ∝ kT
√3N and simi-
larly since T = E/3Nk, fluctuations in the internal tempera-ture of the cluster are σT = T/
√3N . The effect is rather like
that of a fluctuating thermostat for which any nonlinear effectson temperature (e.g., reaction rates) do not cancel. This effectshows up in DFT/MD calculations of the total energy per par-ticle, and we have found that the effect persists in substantiallong-time fluctuations of ps time-scales. Consequently, onemay expect substantial DSD effects on reaction rates. For ex-ample, for N = 20 at 600 K, E is distributed within a fewtenths of an eV of the mean E = 3NkT ≈ 3 eV, and σ T
≈ 75K. In contrast, the thermal fluctuations of the semi-infinite substrate itself are negligible. This statistical argumentsuggests why finite temperature DFT/MD approaches may bemore efficient than full canonical ensemble sampling for cal-culating the physical properties of nanoclusters, since theynaturally probe the range of accessible phase space within afew σ E of E over a time-scale comparable to the periods ofvibrational and librational motion.
The above arguments can be generalized for other con-served physical quantities, for example total volume Vtot
= V + VS , net electronic charge Qtot = Q + QS, chemi-cal composition, etc., where the subscript S refers to thesupport. To this end one must generalize the calculationsof accessible states in terms of S(E,V,Q,Ni) with vary-ing V , Q, etc. These additional physical quantities are sta-bilized by those from the support bath, so that F → F − pV
+ μQ/e + �iμiNi where p is the pressure, μ the chemicalpotential, Q/e is the number of electrons in the cluster, andNi is the number of atoms of species i. Since quantities likeV and Q are extensive, one also expects their fluctuationsto be of order N1/2. The charge (and hence chemical poten-tial) fluctuations, are important to explain the variation of theFermi energy with temperature in the XAS studies.6 Due tothe strength of Coulomb forces and the importance of oxida-tion states in many catalytic reactions, these charge fluctua-tions may play a key role in controlling nano-scale propertiesand chemical reaction rates.
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134701-6 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
III. EFFECTS OF DSD ON NANOSCALE REACTIONS
A. Effects of fluctuations in nanoscale reaction rates
Formally, calculations of catalytic properties such as re-action rates depend on free energies from equilibrated statis-tical ensembles that contain all accessible regions of phasespace. Some effects of disorder22, 23 and the presence of avariety of possible transition paths24, 25 have been addressedin the past. Due to its simplicity, however, transition statetheory (TST) is still the most widely used approach forrate calculations in molecular and nanoscale systems.26 Thisapproach is usually based on time-independent nanoparti-cle potential energy surfaces and the assumption of quasi-equilibrium between reactants and transition state. In contrast,our SRR model based on finite-temperature DFT/MD calcu-lations avoids many of these difficulties. We stress that ab ini-tio DFT/MD is crucial for many properties, as classical MDsimulations with model potentials do not adequately capturecharge fluctuations, bond-breaking, diffusion, and other non-equilibrium effects.27 Moreover DFT/MD is a good approachfor non-equilibrium structural and physical properties at hightemperatures, since it builds in anharmonic and structuraldisorder.
Statistical thermodynamic arguments similar to those inSec. II D can be applied to nano-scale reaction rates. Our anal-ysis below suggests that it may be important to treat the be-havior of reactant molecules on the dynamically fluctuatingsurface of a nano-cluster, e.g., with our SRR model, ratherthan on selected static structures. Thus, the slow fluctuatingmodes of both the cluster and the reactant molecules can nowcouple, resulting in an statistical ensemble of reaction barriersand reaction rates.
The SRR catalysis model thus consists of a system ofreactant molecules in contact with a fluctuating cluster thatcan exchange energy to and from it. Our treatment below fo-cuses on the rate limiting step of crossing the transition bar-rier. More detailed approximations should take into accountthe diffusion rates of the molecules on the surface of thenano-clusters to the reaction site. Given that diffusion pro-cesses are dominated by low frequency modes, such ratesare likely influenced by the slow nature of the DSD. Usingarguments analogous to those of Sec. II D, the probabilitydistribution P(ε) of reactant energies ε is given by a statis-tical ensemble with a fluctuating cluster energy distribution Erelative to the mean E: �(E − ε) ∝ exp[S(E) − βε] whereβ = ∂S(E)/∂E = 1/kT is the fluctuating inverse local clus-ter temperature at energy E. Thus we obtain
P (ε) ≈ 〈�R(ε)e−βε〉 = 〈e−βf (ε;T )〉, (10)
where �R(ε) is the configurational entropy of the reactantmolecules on the cluster, and the average is carried out overthe distribution of cluster energies E and hence values of β.This result shows why the average over terms with β is notthe same as that with fixed β when temperature fluctuationsare large.
Consequently, calculations of reaction rates κ for thetransition [A] + [B] → [AB∗] → [P] require a generalization
of transition state theory to the case with fluctuating barriers,
κ = kT
h
⟨Z∗
AB
ZAZB
e−βεb
⟩. (11)
Here h is Planck’s constant, and the averages in the partitionfunctions are carried out over all nano-cluster configurations,Z∗
AB is the partition function of the activated complex, andZA and ZB the partition functions of the reactants A and B, andagain the average is carried out over the range of local temper-atures β. This result is analogous to the usual Arrhenius law,but with a dynamically averaged “attempt frequency” pref-actor and fluctuating reaction barriers 〈e−βεb〉. Both of thesefactors are affected by DSD. It is easy to see, for example,using a first order cumulant expansion, why the latter averageis always greater than the reactivity at the mean barrier 〈εb〉
〈eβεb〉 = e−β〈εb〉e(+1/2)β2σ 2εb ≥ e−β〈εb〉. (12)
The average leads to a reduced, temperature dependent “effec-tive barrier” ε′
b = εb − (1/2)βσ 2εb, which is comparable to the
lowest barriers typically encountered during reactions. Note,however, that the cumulant estimate assumes small fluctionsin εb, more generally the inequality is always valid and thelowest barriers dominate.
Approximate calculations of the molecular partitionfunctions in these fluctuating TST models can be carried outin terms of the local free energies of the molecules bonded tothe fluctuating cluster or at the transition state using a general-ization of the coupled-oscillator model discussed above. Oneexpects that the low frequency modes of the reacting moleculewill be strongly coupled to the cluster and depend both on theinstantaneous orientation and local geometry of the moleculeon the cluster surface at a given time. This coupling is ex-pected to be transient, and it is plausible that cluster fluctu-ations will assist a molecule in probing a wider range of de-grees of freedom than on a static support. Within this model,the local free energy of an adsorbed molecule with position �ris given by
f (T ) = ε0(�r) + kT
∫dω ρR(ω) ln[2 sinh(β¯ω/2)], (13)
where ρR(ω) is the projected local density of vibrational statesat a given reactant molecule at position �r on the surface of thecluster and T is the effective temperature of the cluster. Aswith the total free energy of the cluster, the local free energyof the molecule f (T ) contains terms coupling to the fast vi-brational density of states of the cluster, and to the relativelyslow librational motion of the CM and other contributions toDSD. Since reactant molecular vibration frequencies are of-ten large compared to thermal motion (¯ω kT), their vibra-tional states are typically frozen in their ground state, and theclassical high temperature limit is inappropriate. On the otherhand, relatively low energy bending and rotational modes maybe thermally active and hence contribute strongly. As withDSD, one expects that the MSRD σ 2 of the reactant moleculewill contain contributions from the low frequency librationalmodes, even for weak couplings as contributions to σ 2 areproportional to 1/ω2 (cf. Fig. 5 of Ref. 19). Thus it is pos-sible that these low frequency fluctuations can significantlyincrease the entropy fluctuations and hence the catalytic
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134701-7 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
FIG. 5. Structural snapshots of the dynamics of CO on Pt10Sn10 on γ -Al2O3at 598 K. Here the sphere colors represent: red, oxygen; blue, carbon; green,aluminium; yellow, tin; brown, platinum. These snapshots show that the COmolecule remains tethered to a Pt atom through its C-end for most of thesimulation time, despite the high mobility on the surface.
activity of reactants, though quantitative calculations areneeded to assess the net effect.
B. DSD and adsorbate dynamics
As an illustration of the concepts discussed above, wepresent calculations and sample reaction paths for prototypi-cal molecules adsorbed on nano-clusters. In order to addressthe effects of DSD on their catalytic activity we first addressthe “simpler” problem of adsorbate dynamics on the surfaceof the particles. This approach serves two main purposes:First, it probes the coupling between the adsorbate dynam-ics and the dynamical disorder, which is key to understandingreactivity, in particular in the pre-transition state regime. Sec-ond, given that vibrational spectra of adsorbates are readilyavailable, it provides a good benchmark for the DFT/MD sim-ulations showing whether they are capable of reproducing thedynamics of adsorbed molecules. Thus, we have carried outDFT/MD simulations similar to those described in Secs. II Band II C, but with the addition of an adsorbate molecule (CO)on the surface of the nanoparticle. Fig. 5 shows a series ofsnapshots illustrating the high mobility of the molecule overthe surface.
Qualitatively, while the CO molecule position is stronglycoupled to the DSD of the nanoparticle, its internal structureis not expected to depend strongly on disorder. The MDsimulations support this statement: First, the CO moleculeis tethered to a Pt atom through its carbon-end for most ofthe simulation, so its position is modulated by the slow Ptmotion. Nevertheless, brief incursions into bridging sites arealso observed. These noticeably affect the internal structureof the adsorbate by elongating the CO bond. Second, the lowfrequency dynamics associated with DSD are likely to couplebetter with the low frequency C–Pt bonds, rather than withthe high frequency adsorbate covalent bonds.
To investigate this coupling between the adsorbate dy-namics and the nanoparticle DSD more quantitatively, wehave have filtered out the slow, stochastic component from theintra-adsorbate (RCO) and adsorbate-particle (RCPt) dynamics,similar to that in Fig. 4. The fast transition between Pt anchor-ing points described above can be clearly seen in a typicaltrajectory (Fig. 6). Here the region between ∼2–4 ps showsthe brief incursions into bridging sites. These transient stateshave a clear effect on the internal structure of the adsorbateas seen in Fig. 7, which shows the decomposition of the RCO
dynamics into vibrational and disorder components. The tran-sient mean bond distance (represented by the red “Disorder”curve) is increased in the bridging site, while the amplitude of
0 2 4 6 8 10 12Time (ps)
2.0
3.0
4.0
5.0
6.0
RC
Pt (
An
g)
Pt1Pt2Pt3
FIG. 6. Dynamics of the three shortest RCPt bonds in a typical trajectoryfor CO adsorbed on Pt10Sn10 on γ -Al2O3 at 598 K. The vertical green lineindicates the end of the thermalization stage. The region between ∼2-4 psshows fast transitions between Pt anchoring points, with brief incursions intobridging sites. These states have a clear effect on the RCO distance (Fig. 7).
the vibrational component (blue “Vibrational” curve) is en-hanced due to the weakened bond.
The projected densities of states ρR(ω) for each compo-nent of these trajectories can be calculated using Eq. (8), andare shown in Fig. 8 (top). The most prominent features cor-respond to the CO stretch for both on-top (between 60 and64 THz, or 2001 to 2135 cm−1) and on-bridge (small peaksbetween 54 and 58 THz, or 1800 to 1935 cm−1). These fre-quency ranges are in good agreement with the experimentalvalues.28–31 Also noticeable is a small feature between 12 and16 THz (400 and 535 cm−1) associated with the molecule-particle interaction, as discussed below in further detail. Asexpected, the negligible weight of the disorder confirms thatthe intramolecular dynamics of CO are largely independent ofDSD.
The disorder has a much larger role in the molecule-nanoparticle dynamics, as seen in Fig. 8 (middle), where theRCPt modes below 5 THz (170 cm−1) are highly coupled tothe DSD. The vibrational component shows a prominentfeature between 13 and 15 THz (430 and 500 cm−1), in
1.10
1.15
1.20
1.25
RC
O (
Å)
TotalDisorder
0 2 4 6 8 10 12Time (ps)
-0.05
0.00
0.05
ΔR
CO (
Å)
Vibrational
FIG. 7. Decomposition of the RCO dynamics (black) into vibrational (blue)and DSD (red) components for CO adsorbed on Pt10Sn10 on γ -Al2O3 at598 K. The vertical green line indicates the end of the thermalization stage.
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134701-8 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
FIG. 8. Average projected density of states for the vibrational and DSD com-ponents of the RCO (top), RCPt (middle), and P tCO (αOCPt, bottom) trajecto-ries in Pt10Sn10 on γ -Al2O3 at 598 K. The experimental CO stretch frequencyfor on-top adsorption sites ranges from 61.5 to 63 THz (2050 to 2100 cm−1),while for on-bridge sites it ranges from 54 to 55 THz (1800 to 1835 cm−1).For the C–Pt stretch, the frequency ranges from 13 to 14 THz (434 to467 cm−1).
reasonable agreement with the experimental range of 13–14 THz (434 to 467 cm−1) for the C-Pt stretch mode.32 Inaddition, the RCPt dynamics show a broad feature between 10and 13 THz (334 to 434 cm−1) which results from strong cou-pling to the P tCO bending, as seen in Fig. 8 (bottom). Thelarge component of disorder in the RCPt and P tCO coordi-nates highlights the importance of nanoparticle fluctuationson the pre-dissociation regime where the reacting molecule
0 2 4 6 8 10
Reaction Path (Å)
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
En
erg
y (e
V)
Path 1Path 2Path 3
FIG. 9. Typical NEB reaction paths for the dissociation of O2 on fully re-laxed Pt10Sn10 supported on γ -Al2O3. Path energies have been shifted tomatch at the reactant state. The reaction path coordinate corresponds to theintegrated displacement of all atoms in the system.
both explores a variety of surface sites, and the physical prop-erties of the sites themselves vary dynamically. Moreover, itpoints to an enhanced role of disorder during the reactionregime, where the breaking bonds become weaker. In thisregime we expect the frequencies associated with the reac-tion path to be lowered into the disorder-dominant region andthus become more coupled with DSD.
C. Effects of DSD on reaction barriers
As discussed in Sec. III A, the DSD creates an effectiveactivation barrier ε′
b ≤ εb due to barrier fluctuations. To studythese fluctuations, we performed nudged elastic band (NEB)calculations of the dissociation of O2 on Pt10Sn10 supportedon γ -Al2O3. Fig. 9 shows results for three sample reactionpaths, generated for nanoparticle conformations with differentDSD, sampled from DFT/MD trajectories. In principle, thedifferent nanoparticle conformations should be sampled fromMD trajectories including the adsorbate. In practice, however,the effect of the latter is local and largely does not affect theoverall dynamics of the nanoparticle. Nevertheless, given thepoor sampling implied by the use of just three paths, thesesimulations should be regarded as proof-of-principle. Fig. 10shows the structures of the initial, transition, and final statesfor these reaction paths. These paths represent rather differ-ent behaviors, including exothermic and endothermic thermo-dynamics, and the presence of a high energy intermediate inpath 2. We find that reactions paths starting in perpendicu-larly absorbed molecules go through a very stable interme-diate state where the O2 molecule is oriented parallel to thesurface. Thus, we only show those portions of the paths. De-spite their similar orientations, the initial states correspond toadsorption on sites with different local environments. For in-stance, while path 1 starts with absorption on top of a sin-gle Pt atoms, paths 2 and 3 start with absorption over a Pt-Sn bond. Although the activation barriers occur close to theinitial state, and thus the initial and transition states (TS)have similar structural characteristics, the activation energy of
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134701-9 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
FIG. 10. Initial, transition and final states for the NEB reaction paths (1-3, asin Fig. 9) for the dissociation of O2 on fully relaxed Pt10Sn10 supported on γ -Al2O3. The sphere colors represent: red, oxygen; green, aluminium; yellow,tin; brown, platinum.
path 1 is significantly lower to those of the other paths. Thisis due to the relative stabilization of the TS in 1 through theincipient formation of a strong O–Sn bond. The difference inrelative final state energies are also related to the strength ofthe bonds O–Sn bonds formed: path 3 has the O atoms form-ing only single bonds to Pt and Sn atoms, and thus results in aless stable structure. Paths 1 and 2 have final states where theO atoms form multiple anchors to the nanoparticle that makethem very stable. From these results we can roughly estimatethe mean barrier (εb � 0.8 eV), and the barrier fluctuations(σεb
� 0.5 eV). In addition, we can estimate the nanoparticledistortions induced during the dissociation process by analyz-ing the atomic displacements along the reaction path. For thepaths shown in Fig. 9, the metal atoms directly bound to theO2 molecule experience displacements of the order of 0.2 Åalong the reaction path. Such nanoparticle distortions are ofsimilar magnitude to those observed due to DSD (0.2–0.4 Å).The remainder of the nanoparticle also participates in the re-action, but with smaller displacements of order 0.1 Å.
The effects of DSD on the reaction barriers are twofold:First, direct effects change the chemical nature of the re-actants, transition states, and products. These direct effectsare usually visible as large energy changes, such as the shiftfrom endothermic to exothermic behavior in paths 3 and 2,respectively, and are driven by major rearrangements of thestructure of the nanoparticle and its interaction with the ad-sorbate. Second, smaller fluctuations in structure that do notchange the chemical nature of the different states yet still in-duce changes in the activation barrier. To quantify these indi-rect effects we have also computed reaction barriers where allatoms, except those in the dissociating O2 molecule, are fixedat the transition state structure of the paths in Fig. 9. Theserigid nanoparticle paths are shown in Fig. 11. Overall, thesepaths are smoother than the fully relaxed ones. For example,the nanoparticle fluctuations induce a variety of minima andshoulders in the reaction path. Given that both the nanoparti-cle and surface are fixed at the structure of the fully relaxedtransition state, by definition the rigid reaction paths map to
-2.0
-1.0
0.0 Path 1
-1.0
0.0
1.0
En
erg
y (e
V)
Path 2
0 2 4 6
Reaction Path (Å)
0.0
1.0 Path 3
FIG. 11. Comparison of the fully relaxed NEB reaction paths for the disso-ciation of O2 on Pt10Sn10 supported on γ -Al2O3 (full lines) with the reactionpaths obtained by fixing the surface and nanoparticle structure at the tran-sition state conformation (dashed lines). The energies for the fully relaxedpaths have been shifted to match at the reactant state. The reaction path coor-dinate corresponds to the integrated displacement of all relaxed atoms in thesystem.
higher energies in the potential energy surface, resulting insmaller barriers. This does not preclude the fact that, on av-erage, the dynamic disorder tends to reduce the reaction bar-riers. The difference is only 0.1–0.2 eV, but clearly the effecton the exponential character of reaction rates is neverthelesssubstantial. This result demonstrates our premise that DSD innano-structures can significantly affect reaction rates.
IV. CONCLUSIONS
We have investigated the effects of dynamic structuraldisorder on the properties of supported metal nanoparticles,from both their morphological and reactivity perspectives.Our primary focus is on aspects of DSD that have not beenadequately described previously. Our results suggest that areal-time approach that accounts for their fluctuating bondingand electronic structure provides a powerful method to bet-ter understand their catalytic activity. In particular, we havefound that the DSD in nanoclusters is an important conceptfor understanding structural disorder. This can be probed bytechniques like EXAFS, where disorder has typically beenaccounted for in terms of globally averaged properties. Inaddition, our real time approach can be especially impor-tant in interpreting time-resolved x-ray spectroscopy experi-ments, such as those performed using x-ray free electron lasersources. Moreover, the possible effects of DSD on reactionpaths and reaction rates in supported nano-scale catalysts hasnot, to our knowledge, been carefully elucidated previously.DSD leads to a larger statistical ensemble of configurationsthan that from thermal vibrations plus static disorder on solidsurfaces and overlayers. The SRR concept makes it possi-ble to simulate the statistical ensemble of possible reactionsites efficiently, compared to the effort needed for enumerat-ing structures and Boltzmann factors. It can reveal both thesurface structure and dynamics, which can be more importantthan the global average morphology typically measured in
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134701-10 J. J. Rehr and F. D. Vila J. Chem. Phys. 140, 134701 (2014)
experiment. Since experimental probes of nano-scale systemsmeasure their global average properties, it is important in theanalysis to differentiate between surface and internal struc-ture. Our results highlight the strong coupling between thesurface disorder and the adsorbate dynamics which is likelyto have significant consequences in reactivity. In addition, thelocal electronic structure of the binding sites is highly het-erogeneous. For example, the oxidation state of a Pt atomvaries significantly depending on its local environment. Thus,the DSD in these systems effectively induces additional ac-tive sites for reaction pathways, which likely account for theincreased reaction rates in nano-scale systems. Our resultsemphasize the importance of large fluctuations driven by thestochastic motion of the center of mass of the nanoparticlesand their transient bonding to the support. These fluctuationsaffect the internal energy and temperature distribution as wellas the structure of the nanoparticles, thus having a measur-able effect on both the properties observed by experimental.They also affect both the prefactors and distribution of energybarriers, thus tending to increase the observed reaction rates.Consequently, we believe the SRR model of dynamic struc-tural disorder constitutes a timely integration of concepts thatprovides a compelling understanding of disorder phenomenain nano-catalysts with potentially wide ranging implications.
ACKNOWLEDGMENTS
The authors wish to thank S. R. Bare, J. J. Kas, S. D.Kelly, M. Tromp, and especially A. Frenkel for advice anduseful references. This work was supported by DOE GrantNo. DE-FG02-03ER15476 with substantial computer supportfrom DOE-NERSC.
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