principles of dynamic heterogeneous catalysis: …heterogeneous catalysis is an integral component...

doi.org/10.26434/chemrxiv.7790009.v1

Principles of Dynamic Heterogeneous Catalysis: Surface Resonance andTurnover Frequency ResponseM. Alexander Ardagh, Omar A. Abdelrahman, Paul Dauenhauer

Submitted date: 01/03/2019 • Posted date: 01/03/2019Licence: CC BY 4.0Citation information: Ardagh, M. Alexander; Abdelrahman, Omar A.; Dauenhauer, Paul (2019): Principles ofDynamic Heterogeneous Catalysis: Surface Resonance and Turnover Frequency Response. ChemRxiv.Preprint.

Acceleration of catalytic transformation of molecules via heterogeneous materials occurs through design ofactive binding sites to optimally balance the requirements of all steps in a catalytic cycle. In accordance withthe Sabatier principle, the characteristics of a single binding site are balanced between at least two transientphenomena, leading to maximum possible catalytic activity at a single, static condition (i.e., a ‘volcano curve’peak). In this work, a dynamic heterogeneous catalyst oscillating between two electronic states was evaluatedto demonstrate catalytic activity as much as three-to-four orders of magnitude (1,000-10,000x) above theSabatier maximum. Surface substrate binding energies were varied by a given amplitude (0.1 < ΔU < 3.0 eV)over a broad range of frequencies (10-4 < f < 1011 s-1) in square, sinusoidal, sawtooth, and triangularwaveforms to characterize surface dynamics impact on average catalytic turnover frequency. Catalyticsystems were shown to exhibit order-of-magnitude dynamic rate enhancement at ‘surface resonance’ definedas the band of frequencies (e.g., 103-107 s-1) where the applied surface waveform kinetics were comparableto kinetics of individual microkinetic chemical reaction steps. Key dynamic performance parameters arediscussed regarding industrial catalytic chemistries and implementation in physical dynamic systemsoperating above kilohertz frequencies.

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____________________________________________________________________________ Ardagh, et al. Page 1

Principles of Dynamic Heterogeneous Catalysis: Surface Resonance and Turnover Frequency Response

M. Alexander Ardagh1,2, Omar A. Abdelrahman2,3, Paul J. Dauenhauer1,2* 1University of Minnesota, Department of Chemical Engineering and Materials Science, 421 Washington

Ave SE, Minneapolis, MN 55455 2Catalysis Center for Energy Innovation, University of Delaware, 150 Academy Street, Newark, DE

19716 3University of Massachusetts Amherst, Department of Chemical Engineering, 159 Goessmann

Laboratory, 686 North Pleasant Street, Amherst, MA 01003.

*Corresponding author: [email protected]

Introduction. Heterogeneous catalysis is an

integral component of many industrial processes

that manufacture food, materials, and energy for a

high quality of life[1]. Catalytic surfaces, pores, and

single-atom sites are responsible for accelerating

the rates of reactions that protect and provide the

most important resources including fuels[2],

fertilizers[3], medicines[4], basic chemicals[5,6], CO2

feedstock[7], clean air[8], and polymers[9,10] that

would be inaccessible absent catalytic acceleration.

In the past century, enhancement of these processes

has progressed via increased catalyst selectivity and

stability along with increased overall activity for

smaller reactors that operate at higher efficiency

and lower temperature[11].

Catalytic rate enhancement occurs primarily

through catalyst design to tune the binding

characteristics of surface species and transition

states for maximum catalytic turnover

frequency[12,13,14]. In the past two decades, advances

in nanostructured materials led to detailed synthesis

of atomic-scale active sites that precisely balance

the surface substrate binding energies[15]. The limit

of this approach is characterized by the Sabatier

principle, which states that the binding of substrates

must be neither too strong nor too weak[16,17].

Quantitative description of the Sabatier principle

was captured in Balandin-Sabatier volcano-shaped

curves (“volcano curves” for the remainder of the

manuscript), which depicted a metric of catalyst

activity relative to a descriptor of substrate

binding[18,19]. Balandin depicted volcano-shaped

curves in 1960 and 1964 for the dehydration and

dehydrogenation of alcohols, with the catalytic

Abstract. Acceleration of catalytic transformation of molecules via heterogeneous materials occurs

through design of active binding sites to optimally balance the requirements of all steps in a catalytic

cycle. In accordance with the Sabatier principle, the characteristics of a single binding site are balanced

between at least two transient phenomena, leading to maximum possible catalytic activity at a single,

static condition (i.e., a ‘volcano curve’ peak). In this work, a dynamic heterogeneous catalyst oscillating

between two electronic states was evaluated to demonstrate catalytic activity as much as three-to-four

orders of magnitude (1,000-10,000x) above the Sabatier maximum. Surface substrate binding energies

were varied by a given amplitude (0.1 < ΔU < 3.0 eV) over a broad range of frequencies (10-4 < f < 1011

s-1) in square, sinusoidal, sawtooth, and triangular waveforms to characterize the impact of surface

dynamics on average catalytic turnover frequency. Catalytic systems were shown to exhibit order-of-

magnitude dynamic rate enhancement at ‘surface resonance’ defined as the band of frequencies (e.g.,

103-107 s-1) where the applied surface waveform kinetics were comparable to kinetics of individual

microkinetic chemical reaction steps. Key dynamic performance parameters are discussed regarding

industrial catalytic chemistries and implementation in physical dynamic systems operating above

kilohertz frequencies.

____________________________________________________________________________ Ardagh, et al. Page 2

activity dependent on the bond energies between

the alcohol oxygen and the metal oxide

catalysts[18,19]. Since that time, volcano curves have

been generated for numerous catalytic chemistries

including NOx decomposition[20], propylene

oxidation[21], hydrodesulfurization[22,23], ammonia

synthesis[24,25], CO oxidation[26], and oxygenate

decomposition[27], among many other reactions[28].

The simplest surface catalytic mechanism of

species A reacting to species B depicted in Figure

1A obeys the Sabatier principle regarding

adsorption enthalpy (ΔHA, ΔHB) and surface

reaction activation energy, (Ea). Reactant molecule

A adsorbs to the surface as A*, undergoes surface

reaction to B*, and then desorbs to gas-phase

product B; the overall turnover rate can potentially

be limited by any one of these three steps. Reactant

adsorption is a fast, barrierless step unless it is

combined with surface reactions; a combined step

of dissociative adsorption (e.g., N2) is commonly

rate limiting on some catalytic materials[29]. The

volcano curve therefore results from the sequential

kinetics of surface reaction(s) and product

desorption, as presented originally by Balandin and

depicted in Figure 1B, the transition between

surface reaction- and desorption-control exhibits

the characteristic ‘volcano’ two-kinetic-regime

plot. As depicted in Figure 1C, surface adsorbates

desorb slowly on strong-binding materials, while

surface reactions occur slowly on weak-binding

materials; the kinetic balance of these two steps

forms the optimum turnover frequency of the

system (i.e., volcano peak) characteristic to

materials only exhibiting the optimal binding

energy[30].

The asymmetry of some volcano curves

depicted in Figure 1B arise from the relationship

between the surface binding energy and the surface

reaction activation energy. As described in the

Brønsted-Evans-Polyani (BEP) principle[31,32], the

activation energy of a catalytic reaction linearly

correlates with the surface reaction enthalpy by a

linear-scaling parameter, α, and offset of E0

associated with a reaction class[33].

Ea = ∝∗ ∆Hsr + E0 (1)

As depicted in Figure 1B, α ~ 0 (purple) indicates

negligible relationship between the enthalpy of

surface reaction and the surface activation energy

resulting in a ‘flat’ volcano, while a completely

proportional relationship, α ~ 1.0 (red), forms a

more symmetric volcano curve; values between

zero and one form the interspersed curves: blue of

α ~ 0.8, green of α ~ 0.6, yellow of α ~ 0.4, and

orange of α ~ 0.2 all with an offset E0 of 102

kJ/mole. The linear scaling relationships can be

universal and consistent across different

molecules[34,35,36,37], while other chemicals such as

formic acid exhibit different linear scaling

relationships across different transition metal

groups of the periodic table[38]. The Figure 1B set of

volcano curves are also defined by the condition

that the surface energy of B* changes at twice the

rate of the surface energy of A*, which is a ratio

A

A*

B

B*

ΔHA ΔHB

EA

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1

1E+2

1E+3

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Tu

rno

ve

r F

req

ue

ncy [s

-1]

Relative Binding Energy of B* [eV]

Surface

Reaction

ControlledDesorption

Controlled

10-5

10-4

10-3

10-2

10-1

1

101

102

103

-160

-120

-80

-40

0

40

En

thalp

y [

kJ m

ol-1

]

A

B

ΔHBEAA*

B*

ΔHA

Tu

rno

ver

Fre

qu

en

cy

[s-1

]

En

thalp

y [k

J m

ol-1

]

A B C

Figure 1. Catalyst Optimization via the Sabatier Principle. A. Surface reaction of A converting to B via transition

state with forward activation energy, EA. B. Volcano plots for the turnover frequency of A-to-B with variable binding

energy of A* and B* and variable Brønsted-Evans-Polyani relationships of Ea to ΔHsr (0 < α < 1.0: 0.2 increments,

purple to red). Conditions: perfectly mixed reactor at 150 °C, YB ~ 1%. C. Three conditions of surface intermediate

binding energy: black (+0.4 eV), red (-0.1 eV, α = 1.0), blue (-0.5 eV, α = 0.2).

____________________________________________________________________________ Ardagh, et al. Page 3

that can vary between surface chemistries and

materials.

Catalyst activity optimization within the

context of volcano curves has focused on catalyst

design to achieve optimal turnover at the volcano

curve apex. Of the existing catalysts and multi-

metal combinations, computational screening of the

relevant surface-binding descriptors aims to

identify single- or multi-descriptor optima from

databases of catalytic materials[39,40,41]. Other

strategies have aimed to create and tune the

properties of new materials including physical and

electronic descriptors such as metal spacing and

coordination, d-band center and fermi level, and

electronic interaction with supports, solvents, and

co-adsorbents via multi-metal mixing and/or

nanostructured synthesis[42,43,44,45,46,47]; all of these

approaches have achieved success in creating new

materials near the maximum theoretical turnover

frequency of a static catalyst.

The limitation of the Balandin-Sabatier

maximum arises from the multi-purpose catalyst,

which must balance the kinetics of competing

reaction steps (activation, desorption, etc.). One

strategy to exceed this maximum is to decouple the

reaction phenomena and physically disassociate

sequential chemistries. The physical reaction

dissociation approach was recently demonstrated

with ammonia synthesis, whereby gas-phase

activation of N2 occurred via plasma, and metals

catalyzed hydrogenation of nitrogen to ammonia

with rapid desorption[48,49]. Deconvoluted control of

individual reactions (activation with plasma and

hydrogenation with a metal catalyst) permits

independent system tuning to yield an overall rate

potentially in excess of the Balandin-Sabatier

maximum in conventional catalytic reactors.

In this work, we will focus on the kinetics of

temporally decoupling surface reaction steps via

oscillation of the catalytic surface binding energy.

As depicted in Figure 2A, the volcano curve can be

depicted with its independent slopes extended

above the apex (dashed black lines); these represent

the potential rates of surface reaction and

desorption absent other limitations. In a dynamic

system, the surface energy could oscillate between

two or more binding energy states, with the

oscillation amplitude identified as the total distance

in traversed binding energy (ΔU [=] eV) at the

frequency of oscillation (f [=] s-1); in Figure 2A, the

volcano plot has a BEP of moderate slope, α ~ 0.8,

and amplitude (-0.10 to +0.50 eV, ΔU = 0.60 eV).

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1

1E+2

1E+3

1E+4

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Tu

rno

ve

r F

requ

en

cy [

s-1

]

Relative Binding Energy of B* [eV]

Optimum Dynamics at

Resonant Frequency

Amplitude

Static

Optimum

Minimum

Dynamics far

From Resonant

Frequency

Dyn

am

ic P

erfo

rma

nce

Ra

ng

e

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

A B

-0.3

-0.1

0.1

0.3

0.5

0.7

0

5

10

15

20

25

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6

Rela

tive B

indin

g

Energ

y o

f B

[eV

]

Turn

over

Fre

quency,

TO

FB

[s-1

]

Surf

ace C

overa

ge

of A

, θ

A[-

]

Time [s]

C

D

Figure 2. Dynamic Catalysis – Conditions and Surface Response. A. Transient variation of the catalyst surface (α

= 0.8) binding between a maximum and minimum binding energy comprises the overall surface amplitude resulting

in dynamic performance with optimum (purple) turnover frequency at the reaction resonance frequency. B. The

catalyst binding energy changes as a square wave below the resonant frequency (f = 10 Hz), resulting in maximum

and minimum surface coverage of surface intermediate B* and A*; loading and unloading of B from the surface

produces transient B production rates.

____________________________________________________________________________ Ardagh, et al. Page 4

The optimal turnover frequencies are depicted for

each state at the given amplitude as purple points,

while the minimum turnover frequencies below the

static optimum are identified as green points.

The response of the substrate on the catalyst

surface depends on the relative dynamics of the

system to the kinetics of the surface steps (i.e.,

reactions, desorption). For a catalyst oscillating

between two states as a square waveform with

amplitude of ΔU and frequency (f ~ τ-1), the

optimum time-averaged turnover frequency will

occur when the time scale of each state is

approximately the same as the time scale of the

individual surface steps. Referred to here as

“surface resonance”, the resonant frequencies

depicted in Figure 2A permit the surface coverage

of B* to vary (θmin < θB < θmax) without stabilization

before switching surface states.

In this work, the concept of ‘dynamic catalysis’

illustrated in Figure 2A is explored for a broad

range of catalyst and dynamic applied conditions to

understand the connection between catalyst-system

design combinations and catalytic turnover

frequency. For BEP relations identified in Figure

1B (0 < α < 1.0), we present the averaged turnover

frequency for a broad range of conditions including

applied frequency (f), surface energy amplitude

(ΔU), and wave shape (e.g. square versus

sinusoidal). Optimal performance is then identified

with the constraints of practical implementation in

mind.

Results and Discussion. The following results

were generated computationally using Matlab (see

Methods section for more details). Turnover

frequencies (TOFs) were calculated at 1% yield of

product (B) in all cases for a CSTR reactor setup.

TOFB can be defined as follows,

TOFB = [B]q̇

# of active sites [=] s−1 (2)

where q̇ is the volumetric flow rate through the

reactor and [B] is the concentration of component

B in the reactor effluent.

The impact of oscillating the surface binding

energy of B* with time is depicted in Figures 2B-

2D for a square waveform of amplitude ΔU ~ 0.6

eV and frequency of f ~ 10 Hz. The square

waveform of surface binding energies of B*

depicted in Fig. 2B was simulated for a perfectly

mixed reactor operating at 1% yield of B, 150 °C,

and 100 bar A inlet. These conditions produce the

instantaneous turnover frequency depicted in Fig.

2C which ranges from 3-19 s-1 in a complex

oscillating form; the TOFB achieves a maximum of

19 s-1 soon after the binding energy of B* switches

to relatively low energy (BEB ~ 0.9 eV), while the

minimum TOFB of 3 s-1 occurs just before BEB

switches from 1.5 to 0.9 eV. These turnover

frequencies of B are below the predicted resonance

frequencies identified in purple in Fig. 2A (~100 s-

1). An explanation for the lower-than-expected

TOFB is provided by the surface coverages of Fig.

2C. At 10 Hz, the surface coverage of A* achieves

complete oscillation between θA ~ 0 and θA ~ 1;

moreover, the surface coverage of A* stabilizes for

a significant fraction of the period of oscillation,

indicating a period (~ 0.02 s) with negligible change

in the surface composition of the catalyst. In other

words, faster oscillation above 10 Hz of the surface

binding energy of B* should more efficiently utilize

the catalyst.

The TOFB of Fig. 2C indicate that the highest

rates occur when surface states flip from high to

low binding energy of B*. The energetic path

leading to this unloading of the surface of B* is

depicted in Figure 3A. In the initial strong binding

state 1, A adsorbs to the surface as A* and forms a

thermodynamic distribution with state B*. When

the surface flips to weaker-binding state 2, B*

readily desorbs with lower activation energy to

form product B. By these two states, the complete

cycle can be interpreted as filling of the surface

sites (state 1) followed by forced desorption (state

2), the overall rate of which is determined by the

surface frequency and amplitude associated with

the surface binding energies of the two states.

The impact of the surface state-flipping

frequency on the time-averaged turnover frequency

is depicted in Figure 3B for fixed square waveform

amplitude (ΔU = 0.6 eV). At low frequencies (10-4

< f < 10-2 Hz), the average TOFB is an average of

the static conditions of the two states (i.e., a slow

catalyst). At the corner frequency (fC1) of ~0.02 Hz,

the average turnover frequency begins to increase

until the dynamic system eventually matches the

optimal turnover frequency of the static system at

the volcano apex (depicted in red). Further

increasing the surface waveform frequency

increases the average turnover frequency until

maximizing over a range of dynamic resonance

(~103 < f < ~107), identified in Figure 3B in purple.

Above the resonance frequency band, the average

____________________________________________________________________________ Ardagh, et al. Page 5

turnover frequency decreases before stabilizing at

2.6•10-1 s-1 at a waveform frequency of ~1011 Hz,

the TOFB associated with optimal conditions of the

static system at the volcano curve optimum.

For the volcano curve system depicted in Fig.

2B with amplitude of 0.6 eV square waveform, the

instantaneous TOFB is depicted in Figure 3C for

four frequencies: 0.001 Hz in green, 0.25 Hz in red,

10 Hz in blue, and 1,000 Hz in purple. At low

frequency (0.001 Hz), the surface coverages of A*

and B* (in SI Fig. S2) rapidly respond to the change

in surface state, with static operation occurring in

either of the two states. Low frequency below fC1

results in TOFB response comparable to a mix of the

two low activity states identified in green in Fig.

2A. The unique behavior to the general TOFB

response exists only at the condition of flipping

surface states from strong to weak binding of B*;

as noted in the highlighted region of Fig. 3C for

0.001 Hz, the TOFB overshoots, resulting from the

unloading of surface B* species into the gas phase

as product B. As the waveform frequency increases

to 0.25 and 10 Hz, the unloading of B* species from

the surface becomes the dominant mechanism

leading to catalyst activity. For these two

frequencies, the TOFB and the surface coverages of

A* and B* (Fig. S3-S4) are transient for most of the

waveform period. At 1,000 Hz in Fig. 3C, the

0.00001

0.0001

0.001

0.01

0.1

1

10

100

Insta

nta

neo

us T

urn

ove

r F

req

uen

cy

to B

[s

-1]

102

101

1

10-1

10-2

10-310-3

10-4

10-5

0.001 Hz

0.25 Hz

10 Hz

1000 Hz

1000

sec

4.0 s 0.1 s

Time (Different Scales)

Insta

nta

neous T

urn

over

Fre

quency to B

[s

-1]

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

En

tha

lpy [

kJ

mo

l-1]

A

B

A*

State 2

B*

State 1

Osc

illa

tio

n

Fre

qu

en

cy

B*

TS

En

thalp

y [k

J m

ol-1

]A B

D C

0.01

0.10

1.00

10.00

100.00

-4.00 -2.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00

Ave

rage T

urn

ove

r F

requ

en

cy,

B [

s-1

]

Frequency [s-1]

Activity

Response

Static

Optimum

10-2 1 102 104 106 108 1010 101210-2

10-4

10-1

1

101

102

fC1 fC2

Ave

rag

e T

urn

ove

r F

req

uen

cy,

B [s

-1]

Waveform Amplitude [eV]

Wavefo

rm F

req

uen

cy [

s-1

]

0 0.2 0.4 0.6 0.8 1.0

10-2 10-1 100 101 102 103

Average Turnover Frequency, B [s-1]

1010

108

106

104

102

1

10-2

10-4

Figure 3. Activity response of applied oscillating surface binding energy - square waveform. A. Oscillating state

energy diagram for A reacting on a catalytic surface to B product (-0.1 to 0.5 eV of B*). B. Average catalytic turnover

frequency to product B at waveform amplitude of 0.6 eV at 150 °C and 100 bar; resonance frequencies identified in

purple. C. Instantaneous turnover frequency to B for four frequencies at ΔU of 0.6 eV at 150 °C and 100 bar. D.

Average turnover frequency to B at 150 °C and 100 bar for variable square waveform amplitude and frequency.

____________________________________________________________________________ Ardagh, et al. Page 6

TOFB and surface coverages of A* and B* (Fig. S5)

are always transient; under these conditions TOFB

and surface coverages only minimally oscillate in

value (e.g., 0.29 < θB < 0.32).

An interpretation of catalytic surface resonance

comes from evaluating the TOFB response of each

condition independently, as shown in Fig. 3C. The

rate of production for surface species B* is defined

by the forward surface reaction rate constant and

surface coverage of A* (which is in equilibrium

with gas phase A). Similarly, the rate of desorbing

B* to gas product B is defined by the desorption

rate constant and surface coverage of B*. In this

case, the time scales of these two processes sum to

the total time scale, which is comparable to the

applied square waveform time scale at resonance.

This concept is visually observed in Fig. 2A, where

TOFB for the two purple points predict 60 s-1 for

each independent process, while the actual TOFB

predicted by simulation is exactly half of that value,

29 s-1 (Fig. 3B and 3C). To put it simply, catalytic

surface resonance occurs when the frequency of the

applied surface state-switching waveform matches

the natural frequency of the catalytic kinetics.

Variation of the surface square waveform

amplitude will change the kinetics of the surface

chemistry, resulting in a shift of the resonance

frequency band. As depicted in the heat map of Fig.

3D, a range of amplitudes (0 < ΔU < 1.0 eV) was

evaluated for the volcano curve of Fig. 2A for

frequencies varying over 15 orders of magnitude

(10-4 < f < 101 s-1) to determine the average steady

state turnover frequency to B, TOFB. The steady

state system response is oscillatory, so the average

TOF was calculated via integration over a range of

time after the system reached stable oscillation. For

each value of the oscillation amplitude ΔU, the two

extreme values of U [eV] corresponding to the two

states of the square surface waveform were selected

to yield two conditions of equal rate; put simply,

each value of ΔU should produce a horizontal line

connecting two purple points as in Figure 2A. The

variation in surface kinetics with square waveform

frequency and amplitude is visually apparent in

Figure 3D, where low frequencies below 0.1 Hz are

1.00E-14

1.00E-13

1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

1.00E+09

1.00E+10

-1.0 -0.5 0.0 0.5 1.0 1.5

Relative Binding Energy, B [eV]

Turn

over

Fre

quency,

B [s

-1]

α = 1.0

100

10-3

10-6

10-9

10-12

103

106

109

0.6 eV1.0 eV

1.5 eV

A

Avera

ge T

urn

over

Fre

quency,

B [s

-1]

0

10

20

30

40

50

60

70

80

90

100

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10

0.6 eV

Avera

ge T

urn

over

Fre

quency,

B [s

-1]

Square Wave Frequency [s-1]

10-4 10-2 100 102 104 106 108 1010

α = 1.0

1.00E+001.00E+021.00E+041.00E+06

1

2

31.5 eV

1.0 eV

0.6 eV

106104102100

Resonance Turnover Frequency [s-1]

B

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

-0.5 0.0 0.5 1.0 1.5


100

10-2

10-4

10-6

102

104

106

Turn

over

Fre

quency,

B [s

-1]

0.6 eV1.0 eV

1.5 eV

α = 0.4C

0

2

4

6

8

10

12

1E-04 1E-02 1E+00 1E+02 1E+04 1E+06 1E+08 1E+10


α = 0.4

1010108106104102110-210-4

0.6 eV

1.00E+001.00E+021.00E+041.00E+06

1

2

31.5 eV

1.0 eV

0.6 eV

106104102100

Resonance Turnover Frequency [s-1]

D

1.00E-15

1.00E-14

1.00E-13

1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-1.0 -0.5 0.0 0.5 1.0 1.5

Turn

over

Fre

quency,

B [s

-1]


α = 0

0.6 eV1.5 eV

108

104

1

10-4

10-8

10-12

E

1.00E+001.00E+021.00E+041.00E+06

1

2

3-0.43 to +0.17 eV

-0.43 to +1.07 eV

-0.75 to +0.75 eV

106104102100

Maximum Turnover Frequency [s-1]

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

1E-04 1E-01 1E+02 1E+05 1E+08 1E+11

α = 0

Avera

ge T

urn

over

Fre

quency,

B [s

-1]


0.6 eV

10-4 10-1 10+2 10+5 10+8 10+11

F

Figure 4. Dynamic Catalytic Response of Three Linear-Scaling Relationships (α of 1.0, 0.4, 0.0). All panels

comprised of an A-to-B reaction at 150 °C, 100 bar, perfectly mixed system operating at 1 % yield of product B with

BEP relationship parameter α as the sole variable. A. Volcano plot for α of 1.0 system. B. Catalytic turnover frequency

response of α of 1.0 system. C. Volcano plot for the α of 0.4 system. D. Catalytic turnover frequency response of the

α of 0.4 system. E. Volcano plot for the α of zero system. F. Catalytic turnover frequency response of the α of zero

system.

____________________________________________________________________________ Ardagh, et al. Page 7

slower (dark blue) than static catalysis for

amplitudes greater than 0.1 eV. Alternatively,

above ~1 Hz, the average turnover frequency

increases dramatically to 10 and 1,000 s-1 per

catalytic site for amplitudes above 0.3 eV.

The ability to dynamically accelerate catalytic

turnover depends on the energetics of the

obtainable states defined by the shape of the

volcano curve. Of the many parameters that define

the volcano shape, the linear-scaling relationship

parameter, α, relating the surface reaction enthalpy

to the surface reaction activation energy can

dramatically shift the slope of the volcano plot.

While Figures 2 and 3 describe a system with α of

0.8, three volcano plots for α of 1.0, 0.4. and zero

are shown in Figure 4. For steep volcano plots such

as Fig. 4A, extension of the slopes as dashed lines

above the volcano apex indicate rapid increase in

the turnover frequency for amplitudes of 0.6, 1.0

and 1.5 eV at resonance conditions. This is

supported by the catalytic reactor simulation

kinetics of Fig. 4B, which considered the catalyst

system of Figure 4A at variable applied square

waveform frequency (10-4 < f < 1011). For a square

waveform at an amplitude of 0.6 eV, the resonance

frequencies of 103 to 107 s-1 yield an average

turnover frequency to B of about 52 s-1 per catalytic

site. At higher amplitudes of 1.0 and 1.5 eV, the

average turnover frequency per catalytic site at

resonance achieves 2,074 and 2.0•105 s-1.

A broader volcano of α of 0.4 in Fig. 4C limits

the overall speed achievable for a given amplitude;

the purple points above the curve are further apart

and at lower turnover frequencies. This translates to

lower overall reaction rates at resonance conditions

as shown in Fig. 4D. At the extreme case where the

activation energy of the surface reaction does not

change with the binding energy of an adsorbate

such as the volcano curve of Fig. 4E, the potential

of dynamic operation is limited as shown in Fig. 4F.

There exist at least two cases where the slope of the

volcano curve is horizontal on one side as drawn:

(1) catalytic systems where the surface reaction

enthalpy does not change with the binding energies

of the descriptor component (e.g., B*), thus leading

to constant surface activation energy of reaction,

and (2) systems with α of zero. In these rare cases,

the rate of the surface reaction can never be

accelerated to match a fast rate of desorption, and

the average overall turnover frequency is limited to

the rate of the surface reaction.

Applying dynamic operation to heterogeneous

catalytic applications will require identifying the

conditions of optimal performance in addition to

new design variables such as surface waveform

shape that can be implemented in real reactor

technology. As depicted in Fig. 5A, the sinusoidal

surface binding waveform varying from -0.1 to

+0.5 eV at frequency of 10 Hz applied to the

catalyst system characterized by the volcano plot of

Fig. 2A yields oscillatory turnover frequency (Fig.

5B) and surface coverage of A* (Fig. 5C) at 100 bar

A inlet, 150 °C, and 1% yield of B. Similar to the

case with the square waveform, the turnover

frequency of B increases when the applied

waveform changes from strong binding of B* to

weak binding. At the same time, the surface

coverage of A* increases to take the place of the B*

that was removed from the surface as desorbed

product B. Other considered waveform types

including triangle and sawtooth are depicted in Fig.

5D. For all conditions, the square waveform

exhibits superior activity at surface resonance

conditions. At higher frequencies of 10 and 1,000

Hz, the sinusoidal waveform outperforms the

triangle and sawtooth shapes.

Implementation of dynamic operation of

heterogeneous catalysts requires the capability to

modify the binding energy of surface intermediates

with time. Based on the simulations of Figs. 2-5,

catalyst and system parameters should be selected

to achieve average turnover frequencies above the

optimum of static conditions and preferably as high

as resonance conditions. This implies that a

physical catalyst system must achieve surface

waveform amplitudes of at least 0.3 eV (and

preferably above 0.5 eV) and operating frequencies

above 10 Hz (and preferably 100-1,000 Hz). These

performance targets change with the selected

surface chemistry, which will likely have more than

two surface intermediates exhibiting linear scaling

relationships over a broad range (0.2 < α < 0.8). The

complexity of each catalytic chemistry combined

with the large number of dynamic catalysis

parameters indicates that each system can be guided

by the principles proposed here but will require

detailed microkinetic modeling for design and

optimization.

Device construction for tuning of the surface

intermediate adsorbate binding energy can be

interpreted via the electronic state of the catalyst

material; surface intermediates such as adsorbed

____________________________________________________________________________ Ardagh, et al. Page 8

nitrogen, N*, correlate linearly with the d-band

edge/center when compared across a broad range of

metals[17,50,51,52]. Temporal variation of metal d-

bands exists in at least two categories including

electronic and physical (and even electro-

mechanical) manipulation. Straining of surfaces

has been shown to shift the d-band centers of

metals, metal alloys, and other 2D materials[53],

which alters the binding energy of adsorbates such

as carbon monoxide[54]. When combined with

dynamic approaches such as sound waves or

piezoelectrics capable of 1% strain oscillation

exceeding kilohertz frequencies, this approach can

potentially achieve the frequencies and amplitudes

required for resonant dynamic catalytic

acceleration. Other methods electronically

manipulate a catalyst surface including field effect

modulation[55,56,57] or non-Faradaic electrochemical

modification[58,59,60,61,62], both of which can

potentially achieve the frequency and amplitude

targets necessary for surface catalytic resonance.

Future work will expand the principles developed

here to more complex surface chemistries in

parallel with implementation in physical catalytic

reactors.

Methods. Continuously-stirred tank reactor

(CSTR – perfect mixing assumed) models were

implemented in Matlab 2017b and Matlab 2018b,

using GPU computing resources at the Minnesota

Supercomputing Institute (MSI) at the University

of Minnesota. See SI section S1 for the full shell

and reactor codes, which can be used and modified

to reproduce these simulations. The shell code set

reactor parameters included the temperature (T),

inlet volumetric flow rate (q̇), catalyst weight (w),

and active site loading. Reactor time-on-stream

data was generated using the Matlab ODE15s

differential equation solver. This solver was

selected based on its performance; see SI section S2

for a comparison with Matlab ODE45, ODE23s,

ODE23t, and ODE23tb. The set of differential

equations consisted of forward and reverse rates for

the consumption of gas phase (A, B) and surface

species (A*, B*). This general reaction system, A

↔ B, was modeled using three reversible

elementary steps: (i) adsorption of A, (ii)

conversion of A* to B*, and (iii) desorption of B

with a site balance involving *, A*, and B*.

A ⇌ A* (3)

A* ⇌ B* (4)

B* ⇌ B (5)

0.01

0.1

1

10

100

0.001 0.25 10 1000

Ave

rag

e T

urn

ove

r F

req

ue

ncy,

B [s

-1]

Waveform Frequency [s-1]

A

B

C

D

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0

2

4

6

8

10

12

14

16

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Rela

tive B

ind

ing

En

erg

y o

f B

[eV

]

Tu

rno

ver

Fre

qu

en

cy

B [s

-1]

Su

rface C

ove

rag

e

of A

* [-

]

Time [s]

Figure 5. Surface B* Binding Energy Waveform. Volcano curves comprised of ΔU ~ 0.6 eV, 150 °C, perfectly

mixed reactor at 1% yield of B product. A. Relative binding energy of B* varying in sinusoidal waveform at f~10 Hz.

B. Turnover frequency to B response to a 10 Hz sinusoidal waveform. C. Surface coverage of A* response to a 10

Hz sinusoidal waveform. D. Comparison of average turnover frequency to B for an applied B* surface binding energy

oscillation of four waveform types: square (blue), sinusoidal (orange), triangle (green), and sawtooth (red).

____________________________________________________________________________ Ardagh, et al. Page 9

[Sites] = [∗] + [A]∗ + [B]∗ (6)

Generalized forms of the differential equation used

for each gas phase and surface species are:

d[A]

dt=

q̇

V([𝐴]feed − [A]) − r1,forward + r1,reverse

(7)

d[A]∗

dt= r1,forward − r1,reverse + r2,forward −

r2,reverse (8)

Reaction rate equations consisted of rate constants

and species concentrations, with each elementary

step assumed to be 1st order in all participating

reactants. Since this was modeled as a gas phase

reaction, adsorption steps were expressed in terms

of A and B pressures (bar).

𝑟𝑎𝑑𝑠,𝑖 = 𝑘𝑎𝑑𝑠,𝑖𝑃𝑖[∗] (9)

𝑟𝑑𝑒𝑠,𝑖 = 𝑘𝑑𝑒𝑠,𝑖𝜃𝑖[𝑠𝑖𝑡𝑒𝑠] (10)

𝑟𝑠𝑢𝑟𝑓 𝑟𝑥𝑛 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 = 𝑘𝑠𝑢𝑟𝑓 𝑓𝑥𝑛 𝑓𝑜𝑟𝑤𝑎𝑟𝑑𝜃𝐴[𝑠𝑖𝑡𝑒𝑠]

(11)

Rate constants were constructed as Arrhenius

expressions using pre-exponential factors and

activation energies for adsorption, desorption, and

surface reactions[63]. Pre-exponential factors were

set to 106 (bar-s)-1 for adsorption steps and 1013 s-1

for surface reaction and desorption steps. These

values are typical starting points when fitting

microkinetic models to experimental data.

Activation energy was set to 0 kJ/mol for

adsorption and to the binding energies (BEs) of A

and B for their respective desorption steps. The

binding energies for A and B, the surface reaction

activation energy (Ea), and the surface enthalpy of

reaction were selected; the base conditions were

BEA = 1.3 eV, BEB = 1.0 eV, Ea = 102 kJ mol-1, ΔHsr

= 0 kJ/mol, and ΔHovr = -20 kJ/mol.

Brønsted-Evans-Polyani relationships between

Ea and BEs were held at a constant offset (E0) of

102 kJ/mol and the slope of the relationship, α, was

varied (0 < α < 1.0). These values fall within

previously observed ranges for experimentally

derived BEP parameters and resulted in volcano

peak TOFs between 0.1-10 s-1. Thus, the activation

energy was expressed as a linear function of the

surface enthalpy of reaction, ΔHsr (i.e., the

difference in binding energies between A* and B*

plus the overall enthalpy of reaction from A B):

Ea = α*ΔHsr + E0 (12)

Volcano plots were generated by varying ΔHsr and

measuring the time-averaged turnover frequency

(TOF) at 1.0 % overall yield of B. Turnover

frequency was defined as 𝑇𝑂𝐹 = [𝐵]q̇

# 𝑜𝑓 𝑠𝑖𝑡𝑒𝑠 for

the CSTR design equation, so in practice q̇ (the gas

flowrate [=] L/s) was adjusted until the outlet yield

of component B was 1.0 %. Variation in the BEP

slope (0 < α < 1.0) resulted in surface reaction

activation energies (25 < Ea < 170 kJ/mol) between

binding energies of 0.5 and 2.0 eV, which are

comparable with a broad range of reactions

described in the literature.

Dynamic catalysis was simulated by running

ODE15s for a system where BEs varied with time

on stream as square, sinusoidal, triangle, or

sawtooth waves. Binding energy changes for A*

and B* were specified in the shell code, and these

shifts resulted in varying ΔHsr and Ea. Oscillation

period/frequency was set by specifying the time

duration spent at each condition. Reported TOFs

were calculated when the system oscillation was

centered on 1.0 % yield of B and after the reactor

had achieved oscillatory steady state, defined as a

steady time-averaged turnover frequency. Example

time on stream data for dynamic catalysis can be

found in the supporting information section S4.

Plots of the average turnover frequency as a

function of surface binding energy oscillation

amplitude and frequency (i.e., heat maps) were

generated in Matlab 2018b using the jet color

scheme to indicate low and high TOF, ranging from

dark blue to dark red. The heat map data consists of

a modified Akima cubic Hermite fit through

discrete data points calculated at 0-1.0 eV ΔBE.

Data points are tabulated in supporting information

section S5. This data was obtained for symmetric

dynamic catalysis starting at the volcano peak (ΔBE

~ -0.05 to 0.05 eV) and oscillating the same

amplitude in each direction (from 0-0.75 eV). Data

was also obtained for asymmetric dynamic catalysis

where the end points were chosen based on

extrapolated linear fits of each side of the volcano

curve. These lines were set equal with a specified

____________________________________________________________________________ Ardagh, et al. Page 10

oscillation amplitude between 0-1.5 eV, and the end

points were chosen by drawing a vertical line down

to the volcano plot. Frequency response figures

were generated for scenarios with varying BEP

relationships where the BEP slope ranged from zero

to one.

Acknowledgements. We acknowledge financial

support of the Catalysis Center for Energy

Innovation, a U.S. Department of Energy - Energy

Frontier Research Center under Grant DE-

SC0001004. The authors acknowledge the

Minnesota Supercomputing Institute (MSI) at the

University of Minnesota for providing resources

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download fileview on ChemRxivManuscript_Dauenhauer.pdf (2.03 MiB)



Page S1

SUPPORTING INFORMATION

Principles of Dynamic Heterogeneous Catalysis:

Surface Resonance and Turnover Frequency Response

M. Alexander Ardagh1,2, Omar A. Abdelrahman2,3, Paul J. Dauenhauer1,2*

1University of Minnesota, Department of Chemical Engineering and Materials Science, 421 Washington

Ave SE, Minneapolis, MN 55455 2Catalysis Center for Energy Innovation, University of Delaware, 150 Academy Street, Newark, DE

19716 3University of Massachusetts Amherst, Department of Chemical Engineering, 159 Goessmann

Laboratory, 686 North Pleasant Street, Amherst, MA 01003.

*Corresponding author: [email protected]

# of Figures: 9

# of Tables: 1

# of Equations: 0

Table of Contents:

Section S1. Matlab 2017b and 2018b Code

CSTR Model

Volcano Plot Generator

Dynamic Catalysis Shell Code for Square, Triangle, and Sawtooth Waveforms

Dynamic Catalysis Shell Code for Sinusoidal Waveform

Section S2. Matlab ODE Solver Performance

ODE45

ODE15s

ODE23s

ODE23t

ODE23tb

Section S3. Static Catalysis Time on Stream Data

Example at 1 % Yield of B

Section S4. Dynamic Catalysis Time on Stream Data

Square Waveform:

o 0.001 Hz

o 0.25 Hz

o 10 Hz

o 1000 Hz

10 Hz Oscillation Frequency:

o Square Waveform

o Sinusoidal Waveform

o Triangle Waveform

o Sawtooth Waveform

Section S5. Figure 3 Heatmap Data

Page S2

Section S1. Matlab 2017b and 2018b Code

CSTR Model

% CSTR model

% Description:

% Continuously Stirred Tank Reactor with overall reaction A -> B.

% The number of catalytic active sites is the control.

function xdot=cstr1(t,x)

global T Rg Caf Cbf V u deltE0 q

% Volcano parameters

% Binding energies

BEAeV0 = 1.3; % eV

BEA0 = BEAeV0*96.485e3; % J/gmol

BEBeV0 = 1.0; % eV

BEB0 = BEBeV0*96.485e3; % J/gmol

% Correlation between BE A and BE B

dCorrA = 0.5;

dCorrB = 1;

% Overall reaction enthalpy

dHovr = -20e3; % J/gmol

% BEP relationship parameters

alpha = 0.4;

beta = 102e3; % J/gmol

% Input (1):

% Number of catalytic active sites (gmol)

N = u;

% States (2):

% Concentration of A in CSTR (M)

Ca = x(1,1);

% Concentration of B in CSTR (M)

Cb = x(2,1);

% Number of A* in CSTR (gmol)

Ca_star = x(3,1);

% Number of B* in CSTR (gmol)

Cb_star = x(4,1);

% Parameters:

% Binding energy (J/gmol)

BEa = BEA0 + dCorrA*deltE0;

BEb = BEB0 + dCorrB*deltE0;

% Heat of reaction (J/gmol)

delH1 = -BEa; % A --> A*

delH2 = dHovr + BEa - BEb; % A* --> B*

delH3 = BEb; % B* --> B

% E - Activation energy in the Arrhenius Equation (J/gmol)

R = 8.314459848; % J/gmol-K

Page S3

Eaf1 = 0e3;

Eaf2 = beta + alpha*delH2;

Eaf3 = delH3;

% Pre-exponential factor

Af1 = 1e6; % 1/bar-sec

Af2 = 1e13; % 1/sec

Af3 = 1e13; % 1/sec

% Equilibrium constants

K1 = 1e-7*exp(-delH1/(R*T)); % 1/bar

K2 = 1*exp(-delH2/(R*T)); % unitless

K3 = 1e7*exp(-delH3/(R*T)); % bar

% Rate constants

kf1 = Af1*exp(-Eaf1/(R*T)); % 1/bar-sec

kr1 = kf1/K1; % 1/sec

kf2 = Af2*exp(-Eaf2/(R*T)); % 1/sec

kr2 = kf2/K2; % 1/sec

kf3 = Af3*exp(-Eaf3/(R*T)); % 1/sec

kr3 = kf3/K3; % 1/bar-sec

% Reactant pressure (bar)

Pa = Ca*Rg*T;

Pb = Cb*Rg*T;

% Site balance (gmol)

star = N - Ca_star - Cb_star;

% Surface coverages (unitless)

Theta_a_star = Ca_star/N;

Theta_b_star = Cb_star/N;

Theta_star = star/N;

% Compute xdot:

xdot(1,1) = (q/V*(Caf - Ca) - kf1*Pa*Theta_star*(N/V) + kr1*Theta_a_star*(N/V));

xdot(2,1) = (q/V*(Cbf - Cb) + kf3*Theta_b_star*(N/V) - kr3*Pb*Theta_star*(N/V));

xdot(3,1) = kf1*Pa*Theta_star*N - kr1*Theta_a_star*N - kf2*Theta_a_star*N + kr2*Theta_b_star*N;

xdot(4,1) = - kf3*Theta_b_star*N + kr3*Pb*Theta_star*N + kf2*Theta_a_star*N - kr2*Theta_b_star*N;

Volcano Plot Generator

% Remove size discrepancies

clear

% Step test for Model 1 - CSTR

global T Rg Caf Cbf V u BEA0 BEB0 deltE0 dCorrA dCorrB dHovr alpha beta q

% Input (1):

% Temperature

Tc = 150; % deg C

T = Tc + 273.15; % K

Page S4

% Feed Pressure (bar)

Paf = 100;

Pbf = 0;

% Ideal Gas Constant (L-bar/K-gmol)

Rg = 8.314459848*10^-2;

% Feed Concentration (M)

Caf = Paf/(Rg*T);

Cbf = Pbf/(Rg*T);

% CSTR Volume

w = 200; % mg-cat

rho = 3.578; % g-cat/mL-cat

V = (w/1000)*(1/rho)*(1/1000); % L-reactor

% Volumetric Flowrate

qs = 160; % mL/min

% Steady State Initial Condition for the Control

u_ss = 0; % gmol sites

% Open Loop Step Change

srho = 20e-6; % gmol sites/g-cat

u = (w/1000)*srho; % gmol sites

% Steady State Initial Conditions for the States

Ca_ss = Caf; % M

Cb_ss = Cbf; % M

Ca_star_ss = 0; % gmol

Cb_star_ss = 0; % gmol

x_ss = [Ca_ss;Cb_ss;Ca_star_ss;Cb_star_ss];

% Startup Time (sec)

ts = 5e13;

% Create TOF matrix

storx = zeros(201,1);

story = zeros(201,1);

% Volcano parameters

% Binding energies

BEAeV0 = 1.3; % eV

BEA0 = BEAeV0*96.485e3; % J/gmol

BEBeV0 = 1.0; % eV

BEB0 = BEBeV0*96.485e3; % J/gmol

% Correlation between BE A and BE B

dCorrA = 0.5;

dCorrB = 1;

% Overall reaction enthalpy

dHovr = -20e3; % J/gmol

% BEP relationship parameters

alpha = 0.4;

beta = 102e3; % J/gmol

Page S5

% Iterate i

for i = -1:0.01:1

% Binding energy shift

delteV0 = i; % eV

deltE0 = delteV0*96.485e3; % J/gmol

% Iterate at i

for n = 1:inf


q = qs*1.66667e-5; % L/sec

% Spacetime (sec)

taur = V/q;

% Run ODE solver

[t,x] = ode15s('cstr1',[0 ts],x_ss);

% Parse out the state values

Ca = x(:,1); % M

Cb = x(:,2); % M

% Calculate rates

TOF = Cb*q/u; % 1/sec

Effluent = Cb./(Ca + Cb)*100; % Mole percent

% Constrain to 1 percent yield

if abs(1-Effluent(end)) > 0.01

qs = qs*Effluent(end);

clear Ca Cb TOF Effluent

else

j = (i+1)/0.01 + 1;

storx(round(j)) = i;

story(round(j)) = TOF(end);

clear Ca Cb TOF Effluent

break

end

end

end

% Plot volcano

figure(1)

semilogy(storx,story)

Dynamic Catalysis with Square, Triangle, or Sawtooth Waveforms


clear

% Step test for Models 1, 2, and 3 - CSTRs

Page S6

global T Rg Caf Cbf V u deltE0 deltE1 deltE2 q

% Temperature

Tc = 150; % deg C

T = Tc + 273.15; % K


Paf = 100;

Pbf = 0;


Rg = 8.314459848*10^-2;


Caf = Paf/(Rg*T);

Cbf = Pbf/(Rg*T);

% CSTR Volume

w = 200; % mg-cat


V = (w/1000)*(1/rho)*(1/1000); % L-reactor







ts = 5e1;

% Dynamic catalysis parameters

delteV0 = 0.04; % eV


delteV1 = delteV0 + 1.04; % eV


delteV2 = delteV0 - 0.46; % eV


tau1 = 5000; % sec

tau2 = tau1; % sec

chkr = tau1/tau2;

Nosc = 11; % Number of oscillations

fosc = 1/(tau1 + tau2); % 1/sec


qs = 160; % mL/min

q = qs*1.66667e-5; % L/sec

% Create empty matrices

x_ss = zeros(4,Nosc);

Ca_ss = zeros(1,Nosc);

Cb_ss = zeros(1,Nosc);

Page S7

Ca_star_ss = zeros(1,Nosc);

Cb_star_ss = zeros(1,Nosc);


Ca_ss(1) = Caf; % M

Cb_ss(1) = Cbf; % M

Ca_star_ss(1) = 0; % gmol

Cb_star_ss(1) = 0; % gmol

x_ss(:,1) = [Ca_ss(1);Cb_ss(1);Ca_star_ss(1);Cb_star_ss(1)];

% Run ODE Solver

[t,x] = ode15s('cstr1',[0 ts],x_ss(:,1));


Ca(:,1) = x(:,1); % M

Cb(:,1) = x(:,2); % M

Ca_star(:,1) = x(:,3); % gmol

Cb_star(:,1) = x(:,4); % gmol

star = u - Ca_star(:,1) - Cb_star(:,1); % gmol vacant sites

% Calculate rates

TOF = zeros(size(Cb,1),Nosc);

TOF(:,1) = Cb(:,1)*q/u; % 1/sec

Effluent = zeros(size(Cb,1),Nosc);

Effluent(:,1) = Cb(:,1)./(Ca(:,1) + Cb(:,1))*100; % Mole percent

% Surface coverage (unitless)

Theta_a_star(:,1) = Ca_star(:,1)/u;

Theta_b_star(:,1) = Cb_star(:,1)/u;

Theta_star(:,1) = star(:,1)/u;

% Create time matrix

tshift = zeros(size(t,1),Nosc);

tshift(:,1) = t;

% Create temporary storage

store = zeros(Nosc,5);

store(1,:) = [x(end,:),t(end)];

% Begin dynamic catalysis

for k = 2:Nosc


Ca_ss(k) = store(k-1,1); % M

Cb_ss(k) = store(k-1,2); % M

Ca_star_ss(k) = store(k-1,3); % gmol

Cb_star_ss(k) = store(k-1,4); % gmol

x_ss(:,k) = [Ca_ss(k);Cb_ss(k);Ca_star_ss(k);Cb_star_ss(k)];

% CSTR Selection

Page S8

if mod(k,2) == 0 % even step #

% Run ODE solver

[t,x] = ode15s('cstr2',[0 tau1],x_ss(:,k));

% Shift time span

t = t + store(k-1,5);

store(k,5) = t(end);

else % odd step #

% Run ODE solver

[t,x] = ode15s('cstr3',[0 tau2],x_ss(:,k));

% Shift time span

t = t + store(k-1,5);

store(k,5) = t(end);

end

% Resolving size discrepancies 1

if size(Ca,1) > size(x,1)

store(k,1:4) = x(end,:);

x = [x;repmat(zeros,size(Ca,1)-size(x,1),size(x,2))];

else

store(k,1:4) = x(end,:);

Ca = [Ca;repmat(zeros,size(x,1)-size(Ca,1),size(Ca,2))];

Cb = [Cb;repmat(zeros,size(x,1)-size(Cb,1),size(Cb,2))];

Ca_star = [Ca_star;repmat(zeros,size(x,1)-size(Ca_star,1),size(Ca_star,2))];

Cb_star = [Cb_star;repmat(zeros,size(x,1)-size(Cb_star,1),size(Cb_star,2))];

star = [star;repmat(zeros,size(x,1)-size(star,1),size(star,2))];

end


Ca(:,k) = x(:,1); % M

Cb(:,k) = x(:,2); % M

Ca_star(:,k) = x(:,3); % gmol

Cb_star(:,k) = x(:,4); % gmol

star(:,k) = u - Ca_star(:,k) - Cb_star(:,k); % gmol vacant sites


if size(TOF,1) > size(Cb,1)

Ca = [Ca;repmat(zeros,size(TOF,1)-size(Ca,1),size(Ca,2))];

Cb = [Cb;repmat(zeros,size(TOF,1)-size(Cb,1),size(Cb,2))];

else

Page S9

TOF = [TOF;repmat(zeros,size(Cb,1)-size(TOF,1),size(TOF,2))];

Effluent = [Effluent;repmat(zeros,size(Cb,1)-size(Effluent,1),size(Effluent,2))];

end

% Calculate rates

TOF(:,k) = Cb(:,k)*q/u; % 1/sec

Effluent(:,k) = Cb(:,k)./(Ca(:,k) + Cb(:,k))*100; % Mole percent


if size(Theta_a_star,1) > size(Ca_star,1)

Ca_star = [Ca_star;repmat(zeros,size(Theta_a_star,1)-size(Ca_star,1),size(Ca_star,2))];

Cb_star = [Cb_star;repmat(zeros,size(Theta_a_star,1)-size(Cb_star,1),size(Cb_star,2))];

star = [star;repmat(zeros,size(Theta_a_star,1)-size(star,1),size(star,2))];

else

Theta_a_star = [Theta_a_star;repmat(zeros,size(Ca_star,1)-size(Theta_a_star,1),size(Theta_a_star,2))];

Theta_b_star = [Theta_b_star;repmat(zeros,size(Ca_star,1)-size(Theta_b_star,1),size(Theta_b_star,2))];

Theta_star = [Theta_star;repmat(zeros,size(Ca_star,1)-size(Theta_star,1),size(Theta_star,2))];

end


Theta_a_star(:,k) = Ca_star(:,k)/u;

Theta_b_star(:,k) = Cb_star(:,k)/u;

Theta_star(:,k) = star(:,k)/u;

% Correcting time matrix


if size(tshift,1) > size(t,1)

t = [t;repmat(zeros,size(tshift,1)-size(t,1),size(t,2))];

else

tshift = [tshift;repmat(zeros,size(t,1)-size(tshift,1),size(tshift,2))];

end

tshift(:,k) = t;

end

% End dynamic catalysis

% Clean up data

Page S10

txls = nonzeros(tshift);

TOFxls = nonzeros(TOF);

Theta_A_starxls = nonzeros(Theta_a_star);

Theta_B_starxls = nonzeros(Theta_b_star);

for p = 1:size(tshift,1)

for r = 1:size(tshift,2)

if tshift(p,r) == 0

Theta_star(p,r) = 0;

else

if tshift(p,r) > 0 && Theta_star(p,r) == 0

Theta_star(p,r) = 1;

else

continue

end

end

end

end

Theta_starxls = nonzeros(Theta_star);

Theta_starxls(Theta_starxls > 0.99) = 0;

% Generate Time Matrix

tsh = tshift - tshift(1,:);

tsh(tsh < 0) = 0;

tshxls = nonzeros(tsh);

% Generate BE shift wave

BEshift = zeros(size(TOF));

% Cleanup Errors

for v = 1:size(TOF,1)

if tshift(v,1)>0

BEshift(v,1) = delteV0;

else

BEshift(v,1) = 0;

end

end

for m = 2:size(TOF,2)

if mod(m,2) == 0

for n = 1:size(TOF,1)

if tshift(n,m)>0

BEshift(n,m) = delteV1;

else

BEshift(n,m) = 0;

end

end

else

for n = 1:size(TOF,1)

if tshift(n,m)>0

BEshift(n,m) = delteV2;

else

BEshift(n,m) = 0;

end

Page S11

end

end

end

% Clean up wave

BEshiftxls = nonzeros(BEshift);

Dynamic Catalysis with Sinusoidal Waveform


clear

% Step test for Models 1 and 2 - CSTRs

global T Rg Caf Cbf V u deltE0 deltE1 deltE2 tau q

% Temperature

Tc = 150; % deg C

T = Tc + 273.15; % K


Paf = 100;

Pbf = 0;


Rg = 8.314459848*10^-2;


Caf = Paf/(Rg*T);

Cbf = Pbf/(Rg*T);

% CSTR Volume

w = 200; % mg-cat


V = (w/1000)*(1/rho)*(1/1000); % L-reactor







ts = 5e1;

% Dynamic catalysis parameters

delteV0 = 0.02; % eV


delteV1 = delteV0 + 0.48; % eV


delteV2 = delteV0 - 0.12; % eV


tau = 5000; % sec

Page S12

fosc = 1/tau; % 1/sec

tspan = 10*tau; % sec

Nosc = tspan/tau; % Unitless


qs = 160; % mL/min

q = qs*1.66667e-5; % L/sec


Ca_ss = Caf; % M

Cb_ss = Cbf; % M

Ca_star_ss = 0; % gmol

Cb_star_ss = 0; % gmol

x_ss = [Ca_ss;Cb_ss;Ca_star_ss;Cb_star_ss];

% Run ODE Solver

[t,x] = ode15s('cstr1',[0 ts],x_ss);


Ca = x(:,1); % M

Cb = x(:,2); % M

Ca_star = x(:,3); % gmol

Cb_star = x(:,4); % gmol

star = u - Ca_star - Cb_star; % gmol vacant sites

% Calculate rates

TOF = Cb*q/u; % 1/sec

Effluent = Cb./(Ca + Cb)*100; % Mole percent


Theta_a_star = Ca_star/u;

Theta_b_star = Cb_star/u;

Theta_star = star/u;

% Begin dynamic catalysis


Ca_ss2 = Ca(end); % M

Cb_ss2 = Cb(end); % M

Ca_star_ss2 = Ca_star(end); % gmol

Cb_star_ss2 = Cb_star(end); % gmol

x_ss2 = [Ca_ss2;Cb_ss2;Ca_star_ss2;Cb_star_ss2];

% Run ODE solver

[t2,x2] = ode15s('sin_cstr2',[0 tspan],x_ss2);


Ca2 = x2(:,1); % M

Cb2 = x2(:,2); % M

Ca_star2 = x2(:,3); % gmol

Cb_star2 = x2(:,4); % gmol

Page S13

star2 = u - Ca_star2 - Cb_star2; % gmol vacant sites

% Calculate rates

TOF2 = Cb2*q/u; % 1/sec

Effluent2 = Cb2./(Ca2 + Cb2)*100; % Mole percent


Theta_a_star2 = Ca_star2/u;

Theta_b_star2 = Cb_star2/u;

Theta_star2 = star2/u;

% Shift time matrix

t2s = t2 + t(end);

% End dynamic catalysis

% Clean up data

txls = [nonzeros(t);nonzeros(t2s)];

TOFxls = [nonzeros(TOF);nonzeros(TOF2)];

Theta_A_starxls = [nonzeros(Theta_a_star);nonzeros(Theta_a_star2)];

Theta_B_starxls = [nonzeros(Theta_b_star);nonzeros(Theta_b_star2)];

Theta_starxls = [nonzeros(Theta_star);nonzeros(Theta_star2)];

% Generate BE shift wave

BEshift1 = zeros(size(t));

BEshift1(2:end,1) = delteV0;

BEshift2 = ((delteV2-delteV1)/2)*cos(2*pi*t2/tau) + mean([delteV1,delteV2]);

BEshiftxls = [nonzeros(BEshift1);nonzeros(BEshift2)];

Page S14

Section S2. ODE Solver Selection and Justification

Matlab ODE solvers were screened using a sinusoidal dynamic catalysis waveform. ODE45 is the

recommended solver for general use in Matlab, however, this solver failed to generate a solution after 8 h

of continuous computation.

Stats for ODE15s:

3179 successful steps

312 failed attempts

4727 function evaluations

1 partial derivatives

631 LU decompositions

4721 solutions of linear systems

Elapsed time is 1.449973 seconds.

Stats for ODE23s:


44 failed attempts






Stats for ODE23t:


125 failed attempts






Stats for ODE23tb:


92 failed attempts






ODE23s has the least number of failed attempts, but ODE15s performed most efficiently in terms of

number of steps and amount of time to generate the solution. Therefore, ODE15s was used throughout

this manuscript to solve CSTR equations under static and dynamic catalysis conditions.

Page S15

Section S3. Time on Stream Data for Static Catalysis

Figure S1. Example of time on stream data generated with ODE15s for a CSTR operating at 1 % yield of

B. Reaction conditions: 150 oC, 100 bar A inlet, and a volumetric spacetime of 1 s.

Page S16

Section S4. Time on Stream Data for Dynamic Catalysis

Section S4.1. Varying Oscillation Frequency with a Square Waveform

Figure S2. Dynamics of 0.001 Hz System. A. Waveform amplitude of -0.1 to +0.5 eV. B. Catalytic

turnover frequency of B with time. C. Surface coverage of A* with time. D. Surface coverage of B* with

time. E. Surface coverage of open sites with time. Conditions: 150 °C, 100 bar A inlet, ΔU = 0.6 eV, 1%

yield of B, CSTR reactor, oscillation frequency 0.001 Hz, Volcano Curve Parameters: α of 0.8 and E0

of 102 kJ/mol, B* shifts twice as much as A*

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

0 500 1000 1500 2000

Wa

vefo

rm A

mp

litu

de

[e

V]

Time [Seconds]

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 500 1000 1500 2000Tu

rno

ver

Fre

qu

en

cy o

f B

[s

-1]

Time [Seconds]

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 500 1000 1500 2000Surf

ace

Cove

rage

of A

, θ

A[-

]

Time [Seconds]

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 500 1000 1500 2000Surf

ace C

overa

ge o

f B

, θ

B[-

]

Time [Seconds]

0.00E+00

5.00E-07

1.00E-06

1.50E-06

2.00E-06

2.50E-06

3.00E-06

0 500 1000 1500 2000

Surf

ace

Cove

rage

of O

pen

, θ

*[-

]

Time [Seconds]

A B

C D

E

Page S17

Figure S3. Dynamics of 0.25 Hz System. A. Waveform amplitude of -0.1 to +0.5 eV. B. Catalytic



yield of B, CSTR reactor, oscillation frequency 0.25 Hz, Volcano Curve Parameters: α of 0.8 and E0 of

102 kJ/mol, B* shifts twice as much as A*

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 5 10 15 20

Wa

vefo

rm A

mp

litu

de

[e

V]

Time [Seconds]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20

Tu

rno

ve

r F

req

ue

ncy,

B

[s-1

]

Time [Seconds]

-0.5

0.0

0.5

1.0

1.5

0 5 10 15 20Su

rfa

ce

Co

ve

rag

e o

f A

, θ

A[s

-1]

Time [Seconds]

-0.5

0.0

0.5

1.0

1.5

0 5 10 15 20Su

rfa

ce

Co

ve

rag

e o

f B

, θ

B[s

-1]

Time [Seconds]

0.0E+00

5.0E-07

1.0E-06

1.5E-06

2.0E-06

2.5E-06

3.0E-06

0 5 10 15 20Surf

ace C

ove

rage o

f O

pen, θ

*[s

-1]

Time [Seconds]

A B

C D

E

Page S18

Figure S4. Dynamics of 10 Hz System. A. Waveform amplitude of -0.1 to +0.5 eV. B. Catalytic



yield of B, CSTR reactor, oscillation frequency 10 Hz, Volcano Curve Parameters: α of 0.8 and E0 of

102 kJ/mol, B* shifts twice as much as A*

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6

Tu

rno

ve

r F

req

ue

ncy,

B

[s-1

]

Time [seconds]

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6

Wa

vefo

rm A

mp

litu

de

[e

V]

Time [seconds]

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6Surf

ace

Cove

rage

of A

, θ

A[-

]

Time [seconds]

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6Surf

ace

Cove

rage

of B

, θ

B[-

]

Time [seconds]

0.0E+00

5.0E-07

1.0E-06

1.5E-06

2.0E-06

2.5E-06

3.0E-06

0 0.2 0.4 0.6

Surf

ace C

ove

rage o

f O

pen S

ites, θ

*[-

]

Time [seconds]

A B

C D

E

Page S19

Figure S5. Dynamics of 1,000 Hz System. A. Waveform amplitude of -0.1 to +0.5 eV. B. Catalytic


time. E. Surface coverage of open sites with time. Conditions: 150 °C, 100 bar inlet A, ΔU = 0.6 eV,

1% yield of B, CSTR reactor, oscillation frequency 1,000 Hz, Volcano Curve Parameters: α of 0.8 and

E0 of 102 kJ/mol, B* shifts twice as much as A*

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

Wavefo

rm A

mplit

ude

[eV

]

Time [Milliseconds]

28.0

28.5

29.0

29.5

30.0

30.5

31.0

0 1 2 3 4 5

Turn

over

Fre

quency,

B

[s-1

]

Time [Milliseconds]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5Surf

ace C

overa

ge o

f A

, θ

A[-

]

Time [Milliseconds]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5Surf

ace C

overa

ge o

f B

, θ

B[-

]Time [Milliseconds]

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

0 1 2 3 4 5

Surf

ace C

overa

ge o

f O

pen, θ

*[-

]

Time [Milliseconds]

A B

C D

E

Page S20

Section S4.2. 10 Hz Dynamic Catalysis with Varying Waveforms

Figure S6. Square waveform at 10 Hz. Conditions: A CSTR reactor for A converting to B operating at

1% yield, 150 oC, and 100 bar A inlet. Volcano plot: ΔU ~ 0.6 eV, α ~ 0.8, E0 of 102 kJ/mol, and B*

moves twice as much as A*

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.40 0.50 0.60 0.70 0.80 0.90 1.00

Rela

tive

Bin

din

g E

nerg

y, B

[eV

]

Time [seconds]

0

2

4

6

8

10

12

14

16

18

20

0.40 0.50 0.60 0.70 0.80 0.90 1.00

Tu

rnove

r F

req

uen

cy

to B

[s

-1]

Time [seconds]

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0.40 0.50 0.60 0.70 0.80 0.90 1.00

Su

rface c

ove

rag

e A

[-]

Time [seconds]

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0.40 0.50 0.60 0.70 0.80 0.90 1.00

Su

rface c

ove

rag

e B

[-]

Time [seconds]

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

0.40 0.50 0.60 0.70 0.80 0.90 1.00

Su

rface c

ove

rag

e:

Op

en

Sites

[-]

Time [seconds]

Page S21

Figure S7. Triangle waveform at 10 Hz. Conditions: A CSTR reactor for A converting to B operating at

1% yield, 150 oC, and 100 bar A inlet. Volcano plot: ΔU ~ 0.6 eV, α ~ 0.8, E0 of 102 kJ/mol, and B*


-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

51.50 51.60 51.70 51.80 51.90 52.00Bin

din

g E

ne

rgy o

f B

(e

V)

Time (seconds)

0

1

2

3

4

5

6

7

8

9

10

51.50 51.60 51.70 51.80 51.90 52.00

Turn

ove

r F

requency t

o B

[s

-1)

Time (seconds)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

51.50 51.60 51.70 51.80 51.90 52.00

Su

rfa

ce

Co

vera

ge

A [-]

Time (seconds)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

51.50 51.60 51.70 51.80 51.90 52.00

Su

rfa

ce

Co

vera

ge

B [

-]

Time (seconds)

A B

C D

Page S22

Figure S8. Sawtooth waveform at 10 Hz. Conditions: A CSTR reactor for A converting to B operating

at 1% yield, 150 oC, and 100 bar inlet A. Volcano plot: ΔU ~ 0.6 eV, α ~ 0.8, E0 of 102 kJ/mol, and B*


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.40 0.50 0.60 0.70

Rela

tive

Bin

din

g E

nerg

y, B

[eV

]

Time [seconds]

0

1

2

3

4

5

6

7

8

9

10

0.40 0.50 0.60 0.70

Tu

rnove

r F

req

uen

cy

to B

[s

-1]

Time [seconds]

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0.40 0.50 0.60 0.70

Su

rface c

ove

rag

e A

[-]

Time [seconds]

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0.40 0.50 0.60 0.70

Su

rface c

ove

rag

e B

[-]

Time [seconds]

Page S23

Figure S9. Sinusoidal waveform at 10 Hz. Conditions: A CSTR reactor for A converting to B operating

at 1% yield, 150 oC, and 100 bar inlet A. Volcano plot: ΔU ~ 0.6 eV, α ~ 0.8, E0 of 102 kJ/mol, and B*


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.2 0.4 0.6 0.8 1.0

Bin

din

g E

nerg

y B

[eV

]

Time (s)

0

2

4

6

8

10

12

14

16

0.0 0.2 0.4 0.6 0.8 1.0

Tu

rnover

Fre

qu

en

cy B

[s-1

]

Time (s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Su

rface C

overa

ge A

[-]

Time (s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Su

rface C

overa

ge A

[-]

Time (s)

Page S24

Section S5. Figure 3 Heatmap Data

Table 1. Heatmap data (TOFs [=] 1/s) for oscillation amplitudes between 0.0-1.0 eV and frequencies

between 10-4-1010 Hz. Volcano parameters: α of 0.8, E0 of 102 kJ/mol, 150 oC, 100 bar, 1 % yield of

product B, and B* shifts twice as much as A*

0.0 eV 0.1 eV 0.2 eV 0.3 eV 0.4 eV 0.5 eV 0.6 eV 0.7 eV 0.8 eV 0.9 eV 1.0 eV

10-4 Hz 0.2609 0.2106 0.1397 0.1219 0.1021 0.0779 0.0621 0.0465 0.0289 0.0216 0.0174

10-3 Hz 0.2609 0.2106 0.1442 0.1219 0.1021 0.0808 0.0621 0.0403 0.0289 0.0216 0.0181

10-2 Hz 0.2609 0.2056 0.1407 0.1219 0.1021 0.0808 0.0617 0.0556 0.0364 0.0325 0.0293

10-1 Hz 0.2609 0.2263 0.1989 0.1926 0.1907 0.1820 0.1695 0.1501 0.1334 0.1232 0.1113

100 Hz 0.2609 0.2972 0.4505 0.7468 0.8541 0.9143 1.001 1.139 1.288 1.541 2.366

101 Hz 0.2609 0.4149 0.9343 2.231 4.321 7.425 8.030 8.721 9.325 13.26 28.00

102 Hz 0.2609 0.4202 0.9620 2.371 5.581 12.81 26.88 49.68 90.94 120.1 152.9

103 Hz 0.2609 0.4203 0.9632 2.385 5.611 13.04 29.08 65.08 142.4 296.3 542.5

104 Hz 0.2609 0.4203 0.9634 2.383 5.618 13.05 29.43 65.38 144.0 315.9 690.5

105 Hz 0.2609 0.3990 0.9550 2.383 5.616 13.05 29.43 65.39 144.2 316.6 693.1

106 Hz 0.2609 0.2750 0.3364 0.8856 4.947 13.01 29.43 65.39 144.2 316.7 693.9

107 Hz 0.2609 0.2609 0.2613 0.2667 0.2833 0.3799 3.078 59.92 144.0 316.7 694.0

108 Hz 0.2609 0.2609 0.2609 0.2611 0.2617 0.2637 0.2700 0.2917 0.4478 17.81 40.14

109 Hz 0.2609 0.2609 0.2609 0.2609 0.2609 0.2611 0.2614 0.2622 0.2645 0.2719 0.3063

1010 Hz 0.2609 0.2609 0.2609 0.2609 0.2609 0.2609 0.2609 0.2610 0.2611 0.2615 0.2625

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