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Catalytic Resonance Theory: SuperVolcanoes, Catalytic MolecularPumps, and Oscillatory Steady StateM. Alexander Ardagh, Turan Birol, Qi Zhang, Omar Abdelrahman, Paul Dauenhauer
Submitted date: 12/07/2019 • Posted date: 15/07/2019Licence: CC BY 4.0Citation information: Ardagh, M. Alexander; Birol, Turan; Zhang, Qi; Abdelrahman, Omar; Dauenhauer, Paul(2019): Catalytic Resonance Theory: SuperVolcanoes, Catalytic Molecular Pumps, and Oscillatory SteadyState. ChemRxiv. Preprint.
Catalytic reactions on surfaces with forced oscillations in physical or electronic properties undergo controlledacceleration consistent with the selected parameters of frequency, amplitude, and external stimuluswaveform. In this work, the general reaction of reversible A-to-B chemistry is simulated by varying the catalytic(heat of reaction, transition state and intermediate energies) and oscillation parameters (frequency, amplitude,endpoints, and waveform) to evaluate the influence on the overall catalytic turnover frequency and steadystate extent of conversion. Variations of catalytic cycle energies are shown to comprise a superVolcano ofsuperimposed individual Balandin-Sabatier volcano plots, with variations in linear scaling relationshipsleading to unique turnover frequency response to forced oscillation of the catalyst surface. Optimization ofcatalytic conditions identified a band of forced oscillation frequencies leading to resonance and rateenhancement as high as 10,000x above the static Sabatier maximum. Dynamic catalytic reactions conductedat long times achieved oscillatory steady state differing from equilibrium consistent with the imposed surfaceoscillation amplitude acting as a ‘catalytic pump’ relative to the Gibbs free energy of reaction.
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____________________________________________________________________________ Ardagh, et al. Page 1
Catalytic Resonance Theory: SuperVolcanoes, Catalytic Molecular Pumps,
and Oscillatory Steady State
M. Alexander Ardagh1,2, Turan Birol1, Qi Zhang1, Omar 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]
1.0 Introduction. The efficient catalytic
transformation of feedstocks to chemicals and
materials for society remains at the forefront of
technological needs in the 21st century.[1] Grand
challenges in catalysis aim for the sustainable
manufacture of monomers[2,3,4], the elimination of
carbon dioxide[5,6], the fixing of nitrogen[7,8,9,10], and
the environmental remediation of pollutants[11,12].
New catalysts should be synthesized from low-cost,
earth-abundant materials[13] that operate with long-
term stability and negligible environmental impact
within a process designed to optimally benefit from
catalytic rate enhancement. Catalyst development
also aims for maximum catalytic rate within the
limitations of the Sabatier principle[14] for multi-
step chemistry which includes adsorption, surface
catalytic reactions, and desorption (Figure 1A).
Catalyst design has historically focused on
selecting the optimum catalytic active site within a
material which provides a balance of strong and
weak adsorbate binding energies[15]. Strong
binding is rate limiting in desorption resulting in
rate inhibition from the product, while weak
binding is limiting in dissociative adsorption or
surface reaction[16,17,18]. The original strategy in
these systems tuned catalyst characteristics to exist
at the maximum possible turnover frequency (i.e.,
the Sabatier volcano peak)[19]. More recent
strategies aim to shift the volcano peak via
alteration of the linear scaling relations, which
describe the predicted surface activation energy as
a function of surface reaction enthalpy[20,21,22,23].
Varying of the linear scaling relations of surface
reaction activation energies has the potential to
Abstract. Catalytic reactions on surfaces with forced oscillations in physical or electronic properties
undergo controlled acceleration consistent with the selected parameters of frequency, amplitude, and
external stimulus waveform. In this work, the general reaction of reversible A-to-B chemistry is
simulated by varying the catalytic (heat of reaction, transition state and intermediate energies) and
oscillation parameters (frequency, amplitude, endpoints, and waveform) to evaluate the influence on the
overall catalytic turnover frequency and steady state extent of conversion. Variations of catalytic cycle
energies are shown to comprise a superVolcano of superimposed individual Balandin-Sabatier volcano
plots, with variations in linear scaling relationships leading to unique turnover frequency response to
forced oscillation of the catalyst surface. Optimization of catalytic conditions identified a band of forced
oscillation frequencies leading to resonance and rate enhancement as high as 10,000x above the static
Sabatier maximum. Dynamic catalytic reactions conducted at long times achieved oscillatory steady
state differing from equilibrium consistent with the imposed surface oscillation amplitude acting as a
‘catalytic pump’ relative to the Gibbs free energy of reaction.
____________________________________________________________________________ Ardagh, et al. Page 2
change the shape of Sabatier volcano plots and shift
the maximum possible turnover frequency.
We have recently described an alternative
concept for catalyst enhancement in the form of
surface adsorbate binding oscillation[24]. By this
approach depicted in Figure 1B, the catalyst surface
undergoes periodic external stimulation (e.g.
electric field, strain) such that reaction surface
intermediates experience oscillating binding energy
with time. The amplitude of the oscillating
stimulus generates variation in surface species
binding energy, with the endpoints of the oscillation
amplitude dictated by the combination of the
selected catalyst surface and the condition of the
external stimulus. Stimulus waveforms can also
vary in frequency, type (e.g. square, sinusoidal),
and periodicity (e.g. combinations of waveforms).
For the reversible A to B reaction, the highest
catalytic turnover frequency occurs over a band of
imposed frequencies which resonate with the
natural frequencies of the catalytic surface reaction.
The turnover frequency response of the
catalytic system to external oscillating stimuli
depends on the kinetics of the surface mechanism
and the resulting shape of the Balandin-Sabatier
volcano curve[25,26]. A catalytic system may exhibit
a linear Brønsted-Evans Polanyi (BEP) relationship
with slope, α, and intercept, β, relating the surface
reaction thermodynamics to the surface reaction
transition state. However, another consideration in
volcano plot architecture is the dynamic response
of individual surface species (e.g., A*, B*) to
external stimulus. For variation of any descriptor
(e.g. binding energy of an intermediate or single
atom), surface species exhibit differing binding
energies with the surface. In our previous work on
surface resonance of the reversible A to B
reaction[24], the considered catalytic system
exhibited two-fold variation in the binding energy
of B* relative to A* for the same extent of external
stimulus. A broader understanding of this ratio
(referred to here as gamma, γ) on the structure of
volcano plots is required to predict the potential of
imposed external dynamic stimulus across a broad
range of catalytic chemistries and materials.
In addition to rate, chemical equilibrium
between the reactants and products limits many
catalytic systems. As shown in Figure 1C,
industrially important reactions including ammonia
synthesis, water-gas shift chemistry, dry reforming
of methane, methanol synthesis, and alkane
dehydrogenation all exhibit Gibbs free energy of
reaction (and associated equilibrium constants, Ki)
at standard conditions limiting high overall
conversion[27,28,29,30,31]. To overcome these inherent
limitations, equilibrium between reactants and
products can be manipulated via reaction
conditions; ammonia synthesis is operated at high
pressure[32,33], while water-gas-shift chemistry is
conducted in staged reactors of varying
temperature[34,35]. Other strategies for overcoming
equilibrium limitations combine a second reaction
or separation phenomenon to drive a reaction. For
example, alkane dehydrogenation is combined with
oxidation[36], ammonia synthesis is combined with
absorption[37,38], and carbon-carbon coupling
chemistries such as cycloaddition can be combined
with dehydration[39,40].
1
10
100
50 160 270 380 490 600
Equili
brium
Convers
ion [
%]
Temperature [ C]
100 atm
N2 + 3H2 ↔ 2NH3
CO2 + H2 ↔ CO + H2O
1.0 atm
1.0 atm: CO2 + CH2 ↔ 2CO + 2H2
A
A*
B
B*
ΔHA ΔHB
EA
CA
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
-0.50 -0.25 0.00 0.25 0.50 0.75 1.00
Tu
rno
ver
Fre
qu
en
cy
to
B,
TO
FB
[s-1
]
Relative Binding Energy of B [eV]
Amplitude
Accessib
le
Reaction
Rate
s
Desorption Rate
Surface Reaction
Rate
XA~1%105
104
103
102
101
1
10-1
10-2
10-3
10-4
10-5
B
Figure 1. Kinetic and Thermodynamic Challenges of Catalytic Reactions. (A) Gas-phase chemical species A
and B react on catalyst surfaces through adsorbed surface species A* and B* with forward activation energy, Ea. (B)
Conversion of A-to-B general chemicals volcano curve operating at 1% conversion with oscillating binding energy
of B* with resonance rate in purple. (C) Reaction equilibrium of ammonia synthesis (red), reverse water-gas-shift
(blue), and dry reforming of methane (green).
____________________________________________________________________________ Ardagh, et al. Page 3
The driving force for chemical reaction is the
difference in Gibbs free energy between reactant
and product, with a reaction system stabilizing at
equilibrium defined as the two states existing at
equal Gibbs free energy. The two strategies for
manipulating reactions are based on this definition;
change the conditions and equilibrium, or deprive
the system of one of the two components (reactant
or product) to unbalance the free energy
distribution (i.e., Le Chatelier’s principle)[41,42]. A
third strategy for driving physical systems away
from equilibrium is the application of work; added
energy can perturb a physical system to a steady-
state condition different from equilibrium such as
the case of electrocatalytic water splitting[43,44] or
electrocatalytic ammonia synthesis[45].
Alternatively, a catalyst surface with dynamically-
oscillating adsorbate binding energy provides work
to a catalytic reaction to raise the energy of
adsorbed surface species[46]. The imposed
amplitude of surface binding energy oscillation (0.1
< ΔU < 1.5 eV) serves as a ‘catalytic pump’ to raise
a surface adsorbate to a higher energy state. This
surface energy input manifests as a rate
enhancement and a deviation from equilibrium of
the surface reaction, with the extent of variation in
reaction rate and conversion depending on the type
of catalytic chemistry as well as the parameters
associated with the imposed oscillating surface
stimulus.
Here we will map out the different structures of
Sabatier volcano plots based on definitions of the
five key parameters of single-reaction (A-to-B)
reversible catalytic systems. Example systems
representative of the different kinetic regimes
associated with different kinetic parameter
combinations are simulated as catalytic reactors to
explore the ability of the dynamically oscillating
catalyst surface to preferentially promote catalytic
turnover frequency. Selected systems are evaluated
within the context of the forced oscillator
parameters of imposed surface binding energy
frequency, amplitude, amplitude position, and
oscillating waveform. Specific conditions are
identified leading to surface resonance and
enhanced overall catalytic rate relative to the
Sabatier maximum, while other conditions
preferentially promote a steady state condition
differing from equilibrium. The relationship
between the applied oscillation energy (i.e., surface
work) and the resulting oscillatory steady state is
then evaluated to understand the conditions leading
to tunable reaction conversion with broad
application.
2.0 Computational Methods. Computational
simulations were conducted using Matlab 2017b
and 2019a, as well as the supercomputing resources
at the Minnesota Supercomputing Institute (MSI).
Reactor and dynamic catalysis codes are provided
in SI section S1. A model reversible reaction (A ↔
B) was implemented as a gas-phase catalytic
reaction system with three reversible elementary
steps: (i) reversible adsorption of A, (ii) surface
forward and reverse reaction of A* ↔ B*, and (iii)
reversible desorption of B. Pre-exponential factors
were set to constant values typical for each type of
elementary step. For adsorption steps, a pre-
exponential of 106 (bar-s)-1 was selected, while a
pre-exponential of 1013 s-1 was used for surface
reaction and desorption steps. Surface reaction
activation energies (Ea,sr) were calculated based on
the specified Brønsted-Evans-Polanyi (BEP)
relationship parameters and the heat of reaction for
the surface reaction (ΔHsr)[47], as shown in Equation
1. The activation energy of adsorption was set to 0
kJ mol-1, and the activation energy of desorption
was set to the binding energy for each species,
𝐸𝑎,𝑠𝑟 = 𝛼𝛥𝐻𝑠𝑟 + 𝛽 (1)
where α is the BEP proportionality constant, and β
is the BEP offset.
The reaction chemistry was specified by the
overall gas-phase heat of reaction (ΔHovr), species
binding energies (BEA and BEB), and the BEP
relationship parameters (α and β). Vapor-phase
flow reactors (CSTR) and batch reactors were
modeled by systems of ordinary differential
equations (ODEs) with two gas phase species, A
and B, and two surface species, A*, B*, with open
site *. Differential equations for the CSTR were as
follows:
𝑑[𝐴]
𝑑𝑡=�̇�
𝑉([𝐴]𝑓 − [𝐴]) − 𝑘1,𝑓[𝐴]𝑅𝑇𝜃
∗ 𝑁𝑠𝑖𝑡𝑒𝑠
𝑉+
𝑘1,𝑟𝜃𝐴∗ 𝑁𝑠𝑖𝑡𝑒𝑠
𝑉 (2)
𝑑[𝐵]
𝑑𝑡=�̇�
𝑉([𝐵]𝑓 − [𝐵]) − 𝑘3,𝑟[𝐵]𝑅𝑇𝜃
∗ 𝑁𝑠𝑖𝑡𝑒𝑠
𝑉+
𝑘3,𝑓𝜃𝐵∗ 𝑁𝑠𝑖𝑡𝑒𝑠
𝑉 (3)
____________________________________________________________________________ Ardagh, et al. Page 4
𝑑𝜃𝐴∗
𝑑𝑡= 𝑘1,𝑓[𝐴]𝑅𝑇𝜃
∗ − 𝑘1,𝑟𝜃𝐴∗ − 𝑘2,𝑓𝜃𝐴
∗ + 𝑘2,𝑟𝜃𝐵∗
(4)
𝑑𝜃𝐵∗
𝑑𝑡= 𝑘3,𝑟[𝐵]𝑅𝑇𝜃
∗ − 𝑘3,𝑓𝜃𝐵∗ + 𝑘2,𝑓𝜃𝐴
∗ − 𝑘2,𝑟𝜃𝐵∗
(5)
where [A] and [B] are gas phase concentrations in
M, �̇� is the volumetric flow rate in liters/s, V is the
reactor bed volume in liters, k’s are rate constants
in (bar-s)-1 or s-1, θ’s are surface coverages, and
Nsites is the number of catalytic active sites. In
addition, surface coverage is constrained by the site
balance shown in Equation 6.
1.0 = 𝜃∗ + 𝜃𝐴∗ + 𝜃𝐵
∗ (6)
For batch reactors, the flowrate terms from
Equations 2 and 3 are removed.
Differential equations were solved using built-
in Matlab ODE solvers ODE15s and ODE23tb
along with the “Radau” 3rd party Matlab ODE
solver. Tight solver tolerances of 10-6-10-8 relative
tolerance and 10-9 absolute tolerance were used due
to the ultra-stiff nature of the static and dynamic
catalysis systems. Performance comparison of the
various Matlab ODE solvers can be found in SI
section S2. Computational times were measured
using the ‘tic’ and ‘toc’ commands which act as a
stopwatch timer. The stopwatch was initiated right
before dynamic catalysis was applied in a
simulation and stopped after the last dynamic
catalysis iteration.
Dynamic catalysis was implemented by
varying the binding energies of A and B using
square, sinusoidal, triangle, and sawtooth shaped
waveforms. Waveforms were defined by their
oscillation amplitude ΔU in electron volts (eV),
oscillation frequency f in hertz (Hz), and oscillation
endpoints in eV. Oscillation amplitude is noted
relative to either surface intermediate as ΔUA or
ΔUB. The binding energies of A and B were varied
using a defined proportionality constant, γ,
according to Equation 7,
𝛥𝐵𝐸𝐵 = 𝛾𝛥𝐵𝐸𝐴 (7)
where the change in the binding energy of B was
specified using one of the four different waveform
equations (e.g. sinusoidal). A second parameter, δ,
was defined for the relationship between binding
energies. At any binding energy of B, the binding
energy of A was calculated using Equation 8:
𝐵𝐸𝐴 = (𝐵𝐸𝐵 − (1 − 𝛾)𝛿 − 𝛥𝐻𝑜𝑣𝑟)/𝛾 (8)
Heat maps of catalytic turnover frequency to B
(TOFB) were created in Matlab 2019a by
interpolating over discrete data sets obtained at
varying oscillation amplitudes, oscillation
frequencies, and oscillation endpoints. The
modified Akima cubic Hermite or spline
interpolation method was used to fit data sets and
the ‘smoothdata’ function was employed with the
moving average or Savitzky-Golay filter method to
average out overshoots due to interpolation.
Complete tabulated data sets of all heat maps are
provided in the supporting information.
3.0 Results and Discussion. As shown in Figure
2A, a general reaction enthalpy diagram for
chemical species A reversibly converting to B
forms intermediates A* and B* with transition
ΔHrxn
ΔHBEa = f(α,β)
ΔHsurf = f(γ,δ)
δ
A A* TS B* B
State 3
State 2
State 1
Heat of Adsorption of A, ΔHA
He
at o
f Adso
rptio
n o
f B, Δ
HB
(δ - ΔHrxn)
δSlope, γ
= 𝐻𝐵 𝐻𝐴
1.0
Figure 2. Parameters of Dynamic Heterogeneous Catalysis. Left. State-energy diagram of oscillating
heterogeneous catalyst. Right. Variation of the binding energy of B* linearly scales with the binding energy of A*
with slope, γB-A, and common point, δ.
____________________________________________________________________________ Ardagh, et al. Page 5
state, TS, in between. The overall heat of reaction,
ΔHovr, remains fixed for all conditions of catalyst
surface dynamics. Two more variables are required
to define the transition state (TS) energy, Ea, which
will vary with the surface intermediate energies. As
stated in Equation 1, the existing BEP relationships
with slope, α, and intercept, β, fully define all
possible transition state energies for catalytic
systems exhibiting this linear relationship.
Two new variables are required to define the
variation of surface intermediate energies. Due to
the differences in electronic and steric interactions
with the surface, variation of surface adsorbate
binding energies with external stimuli will not
affect all surface adsorbates equally. As depicted in
Figure 2A, the binding energy of A* changes less
than B* between each of the three potential states
resulting from external stimulus. To describe this
change differential, the extent of change is defined
to be gamma equal to the difference in binding
energy of B* divided by the difference in binding
energy of A*,
𝛾𝐵−𝐴 = 𝛥𝛥𝐻𝐵
𝛥𝛥𝐻𝐴 (9)
Additionally, there exists one condition (purple in
Figure 2A) leading to A* and B* existing at the
same surface energy; that energy is identified in
Figure 2A as delta, δ [eV].
If the external stimulus of a catalytic surface
imposes a linear response in the binding energy of
the surface adsorbates, then potential dynamic
catalysis scenarios can be depicted in the right panel
of Figure 2B. As shown, the binding enthalpies of
A* and B* are related by a line with slope, gamma
γB-A. It is important to note that gamma is defined,
and consideration of the reaction in reverse will
produce a gamma of inverse value. The point of
common surface energy, delta δ, exists for the
enthalpy of adsorption of B offset by the heat of
reaction. More complicated relationships (i.e., non-
linear) between the surface energies of intermediate
species require more than two variables, depending
on the applied model.
3.1 External Catalytic Stimuli. The ability to
tune adsorption enthalpy (also referred to as
binding energy, BE = -ΔHads), requires external
stimulus to modify the catalyst, adsorbate, or
interacting surface bond. One method of catalyst
stimulus is surface strain, which has been shown in
static systems (e.g., films on lattice-mismatched
substrates) to shift the metal d-band center and alter
the adsorption enthalpy of surface
adsorbates[48,49,50,51]. Electric fields applied to
adsorbed species can also tune the binding energy
of adsorbates with benefits for controlling and
accelerating catalytic surface chemistry, including
externally imposed fields or internal fields arising
in electrocatalytic reactors[52,53,54]. Other strategies
aim to alter the electronic structure of surface metal
atoms, such as field-effect modulation with a
dielectric and imposed back gate-voltage; this
method moves charge carriers into the surface
catalyst layer (e.g. metal or metal oxide) to shift the
d-band center and ultimately the interaction with
surface adsorbates[55,56, 57,58]. The impact of these
techniques on the relative changes in surface
intermediate energies and variation in the gamma-
delta plot (e.g., Figure 2B) of two surface
adsorbates depends with each combination of
surface intermediates, surface material/facet, and
type of imposed stimulus which so far has been
primarily explored via computation.
An initial comparison across surface
intermediate binding energies can be made by
comparing the energies between metals and metal
surface planes. For the case of ammonia, the
surface adsorption enthalpy of N*, NH*, NH2*, and
NH3* have been calculated by Mavrikakis and co-
workers on Re(0001)[59], Ru(0001)[60], Pt(111)[61],
Pd(111)[62], and Au(111)[63]. As shown in Figure
3A and 3B, the adsorption enthalpies of the various
intermediates are plotted relative to one another
with the gamma values of the intermediates
apparent in the fitted linear slopes. In the
comparison of NH* and NH2*, the binding energy
of NH* changes almost twice as much as the
binding energy of NH2* (slope ~ γNH2-NH ~ 0.53).
Similarly, in the comparison of NH* and NH3*, the
binding energy of NH* changes almost five times
as much as the binding energy of NH3* (γNH3-NH ~
0.21). These comparisons exist across different
metals, which is not a realistic scenario for
implementing dynamic catalysis (i.e., metals
cannot be periodically interchanged). Only
external stimulus on a single catalyst surface is
likely to physically achieve the conditions
necessary for dynamic catalytic resonance and rate
promotion.
Comparison of the potential for tuning
adsorbate binding energy via external stimuli are
presented for imposed electric field (Figure 3C) and
____________________________________________________________________________ Ardagh, et al. Page 6
surface strain (Figure 3D). As calculated by
McEwen and co-workers[64], electric fields ranging
from -1.0 to +1.0 V/Å applied to a Nickel surface
resulted in significant variation of the binding
energy of methane (blue in Figure 3C), hydrogen
(black), and formaldehyde (red) relative to
methanol. From the slopes of a linear best fit,
methane varied less with changes in methanol
(slope ~ γCH4-CH3OH ~ 0.18) than hydrogen (γH2-CH3OH
~ 0.51), while formaldehyde actually decreased in
binding energy as methanol increased in binding
energy (γCH2O-CH3OH ~ -1.31). Another method of
varying surface binding energy of surface strain is
presented in Figure 3D using the calculated binding
energies of Shengchun Yang and co-workers[65]. As
depicted in Figure 3D, the binding energy of carbon
monoxide and atomic oxygen were calculated for
the lowest energy adsorption geometry on the
M(111) surface of five different metals: Cu (black),
Rh (green), Ir (purple), Pd (orange), and Pt (blue).
Binding energies varied due to imposed surface
strain of -3.0% to +3.0%; while this large surface
strain is likely not physically feasible, it does
provide insight into the relative impact of strain on
oxygen relative to carbon monoxide. From these
calculations, there exists some initial evidence that
a two-parameter linear model can effectively
describe the relative change in binding energy
between two surface adsorbates.
3.2 Catalytic Dynamics - Definition. With the
definition of parameters in Figure 2 and the
calculations presented in Figure 3, it is possible to
completely define the surface catalytic kinetics for
a dynamic chemistry with five parameters: ΔHovr,
y = 0.53x - 0.12
-6
-5
-4
-3
-2
-1
0
-6 -5 -4 -3 -2 -1 0
He
at
of
Ad
so
rpti
on
, N
H2*
[eV
]
Heat of Adsorption, NH* [eV]
-1-2-3-4-5
-1
-2
-3
-4
-5Au(111)
Pd(111)
Pt(111)
Ru(0001)
Re(0001)
y = 0.21x + 0.34
-6
-5
-4
-3
-2
-1
0
-6 -5 -4 -3 -2 -1 0
Heat of Adsorption, NH* [eV]
He
at
of
Ad
so
rpti
on
, N
H3*
[eV
]
-1-2-3-4-5
-1
-2
-3
-4
-5
NH2* vs. NH* NH3* vs. NH*
A B
Au(111)
Pd(111)
Pt(111)
Ru(0001)
Re(0001)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
Ad
so
rpti
on
En
erg
y:
H2,
CH
2O
, C
H4
[eV
]
Adsorption Energy Methanol [eV]
CH2O
H2
CH4
E: -2.0 < V/Å < 2.0
C
-2.5
-2.0
-1.5
-1.0
-0.5-2.5 -2.0 -1.5 -1.0 -0.5
Ca
rbo
n M
on
ox
ide
Bin
din
g E
ne
rgy [
eV
]
Oxygen Binding Energy [eV]D
Pt
Cu
Rh
Ir
Pd
Figure 3. Gamma Parameters of Ammonia Synthesis. (A) The heat of adsorption of NH* and NH2* vary by γ ~
0.53. (B) The heat of adsorption of NH* and NH3* vary by γ ~ 0.21. Data for panels A and B calculated by Mavrikakis
and co-workers[59-63]. (C) The heat of adsorption of methanol varies differently than methane (CH4), hydrogen (H2),
and formaldehyde (CH2O) on Ni metal in the presence of an electric field varying between -1.0 to +1.0 V/Å calculated
by McEwen and co-workers[64]. (D) The binding energy of oxygen, O*, and carbon monoxide, CO*, on Copper
(black), Rhodium (green), Iridium (purple), Palladium (orange), and Platinum (blue) with strain of -3.0% to +3.0%[65].
____________________________________________________________________________ Ardagh, et al. Page 7
α and β to define the activation energy linear scaling
relationship, and γ and δ to define the linear
adsorption energy scaling relationship. Four
additional parameters exist related to the applied
dynamics of the catalyst binding energy oscillation
including: (1) surface oscillation frequency, f, (2)
surface oscillation amplitude, ΔU, (3) surface
oscillation endpoint, UE, and (4) surface oscillation
waveform shape (e.g., square, sinusoidal, etc.).
Additional parameters are required if the applied
surface stimulus is a composite of imposed
oscillations each with their own frequency and
amplitude. Finally, simulation of the dynamic
catalyst system requires all of the parameters
associated with the reactor including temperature,
pressure, reactant composition, and space time.
The introduction of dynamics for the simple
system of A reversibly reacting to B requires nine
variables for definition, which is more than double
the four parameters required to define the static
catalytic reaction (two adsorption enthalpies, one
activation energy, and overall reaction enthalpy).
As depicted in Figure 4A, as the complexity of
catalytic systems expands by the number of
catalytic surface reactions, the number of
parameters required to define the oscillating
systems significantly expands. Dynamic systems
require at least 2n + 1 parameters to define all
energetic states, where n is the number of
parameters required to define the static catalytic
reaction. If the scaling relationships to predict
changes in the transition state energy or the relative
binding energies of surface intermediates become
nonlinear, then the number of parameters further
expands accordingly.
The time required to achieve steady state in a
simulated continuous-flow stirred-tank reactor with
a dynamically operating catalyst increases orders of
magnitude from the simulation with a static
catalyst. As depicted in Figure 4B, the considered
A-to-B reversible catalytic reaction was simulated
using Matlab to achieve steady-state reaction
conditions in less than one second. However,
introduction of dynamics immediately increased
computational time to greater than a second for all
conditions. In particular, computational time
increased logarithmically above about 100 Hz such
that simulations above megahertz-imposed catalyst
binding energy waveforms required more than 104
seconds (~ 3.0 hrs).
Increased computational time is associated
with the large number of oscillations required to
achieve steady-state surface and reactor conditions.
As depicted in Figure 5A, a catalytic batch reactor
0
5
10
15
20
25
30
1 2 3 4 5
Num
ber
of
Reaction
Para
mete
rs [-]
Number of Surface Reactions [-]
Dynamic
Static
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1.E+10
1.E+11
1.E+12
1E-06 1E-03 1E+00 1E+03 1E+06 1E+09 1E+12
Co
mp
ute
r T
ime
[s]
Oscillation Frequency [Hz]
10-6 10-3 1 103 106 109 1012
10-2
1
10+2
10+4
10+6
10+8
10+10
10+12
Static
Dynamic, γ < 1
Dynamic, γ > 1
A B
Figure 4. Reaction Parameters of Static and Dynamic Surface Mechanisms. (A) The number of parameters
required to define all energies of all states in a catalytic surface reaction mechanism differ between a static, steady-
state reaction and a dynamic catalytic reaction with oscillating states. Reactor parameters (e.g., temperature, pressure,
composition, space time) are not included in this figure. (B) Computational time for static (black), dynamic (γB-A <
1, red), and dynamic (γB-A > 1, blue) simulations; filled data points are determined from simulation, while open circles
are determined by extrapolation. Conditions: CSTR, P of 100 bar, α ~ 0.6, β ~ 102 kJ mol-1, δ ~ 1.4 eV, Yield of B
of 1.0 %, fosc varies, and ΔUB ~ 0.6 eV. For γB-A < 1: T of 100 °C, ΔHovr of 0 kJ mol-1, and asymmetric oscillation
endpoints. For γB-A > 1: T of 150 °C, ΔHovr of -20 kJ mol-1, and symmetric oscillation endpoints.
____________________________________________________________________________ Ardagh, et al. Page 8
operating at long times with initially high
concentrations of B (95 mol%) slowly decreases in
concentration until achieving the steady state limit
cycle (red). When starting from a low
concentration of B (10 mol%), the composition
slowly increases until achieving the same limit
cycle (Figure 5B). This steady state solution
(Figure 5C) varies in time in surface coverage and
reactor composition, but the average composition is
fixed in time. As is visually apparent, the
composition approaches the limit cycle only
minimally for each imposed binding energy cycle,
requiring significant computational time to identify
the solution for each parameter set of catalyst and
oscillation conditions.
3.3 Surface Parameters and SuperVolcanoes.
The large number of catalytic parameter
combinations indicates the importance of
categorizing general behaviors of Sabatier-
principle-controlling catalytic reactions. One
strongly-determining parameter is γ, the ratio of the
extent of change of one surface species relative to
another as defined in Equation 9. To understand the
impact of γ on the catalytic reaction, all possible
volcano plots are generated in Figure 6A for 17
variations (0.3 < γB-A < 9.0), while the surface
coverages of A* and B* are depicted for low γB-A
(Figure 6B-6C) and high γB-A (Figure 6D-6E). As
shown, the change in γB-A from above and below
unity dramatically alters the surface coverages and
overall kinetics of the catalytic reaction system.
However, by superimposing all rate volcano plots
on the same panel (Figure 6A), it is apparent that
there exists global behavior; the entire
superimposed set of possible volcano curves are
bounded by a ‘superVolcano’ of extreme limits.
All potential volcano curves are bounded at low
binding energy of B* by the rate of adsorption,
while high bending energy binds the overall rate of
desorption. All variations from the superVolcano
bounds result only from limitations arising from the
surface reaction.
For catalytic systems with high γB-A (>1.0,
blue), the surface product B* lowers in energy
faster than A*, resulting in a decreasing transition
state energy and activation energy by the linear
scaling relationship; for this reason, high gamma
systems exhibit an increasing turnover frequency
with increased binding energy of B* consistent with
a sharp volcano peak depicted in Figure 6A. In
contrast, the low γB-A catalytic systems (γ <1.0, red)
exhibit three regimes. At low binding energy (< -
0.4 eV), the surface is bare, and the overall rate is
controlled by the rate of adsorption; at moderate
binding energies (-0.4 eV < BEB < 0.2 eV), the rate
is controlled by desorption. Even higher binding
energies lead to surface reaction control. The
common point of all curves (0.2 eV relative binding
energy of B*) is equal to the linear scaling
parameter, δ ~ 1.4 eV (offset relative to a binding
energy of B of 1.0 eV).
The surface coverages of the volcano curves of
Figure 6 shift with the binding energy of B*
consistent with the rate-controlling phenomenon.
At low BEB, the surface is bare. For the high γB-A
systems (blue, γ > 1.0), the surface becomes
89.10
89.15
89.20
89.25
89.30
89.35
89.40
89.45
89.50
89.55
89.60
0.4 0.5 0.6 0.7 0.8 0.9 1.0
XB,
Reacto
r C
om
positio
n [
mol
% B
]
θA, Surface Coverage of A [-]
88.10
88.30
88.50
88.70
88.90
89.10
89.30
89.50
0.4 0.5 0.6 0.7 0.8 0.9 1.0
XB,
Reacto
r C
om
positio
n [
mol
% B
]
θA, Surface Coverage of A [-]
Limit
Cycle
Limit
Cycle
Reacto
r C
om
positio
n [
mol%
B]
A B C
Figure 5. Dynamic Forced Oscillation of Surface Binding Energy – Limit Cycles. A simulated catalytic batch
reactor reversibly converting A to B with oscillating binding energy of B* exhibits dynamic variation in surface
coverage and gas-phase composition approaching a limit cycle. (A) Initial conditions above the limit cycle reactor
composition, 95 mol% B. (B) Initial conditions below the limit cycle reactor composition, 10 mol% B. (C) The limit
cycle consists of a three-dimensional stable loop with non-overlapping variations of binding energy of B*, surface
coverage of A, and reactor composition. Catalytic conditions: α ~ 0.6, β ~ 130 kJ mol-1, γB-A ~ 2.0, δ ~ 1.4 eV, f ~
1.0 Hz, ΔUB ~ 0.6 eV, and ΔHrxn ~ 0 kJ/mol.
____________________________________________________________________________ Ardagh, et al. Page 9
covered in A* as binding energy increases only to
be completely replaced by B* at the transition point
of +0.2 eV. The opposite behavior exists for low
γB-A systems (red, γ < 1.0); the surface is covered in
A* at high binding energy. The only variation is
the transition observed between high and low
surface coverage of A* and B* at moderate binding
energies.
Four superVolcano plots were created to
compare the impact of changing α, β, and δ on the
superVolcano shapes and kinetic behaviors as
shown in Figure 6A and Figure 7A-7C. The
independent-axis for Figure 6A is zero when the
binding energy of B is 1.0 eV, and the independent-
axis zeros for Figure 7A, 7B and 7C are the
corresponding delta values (0.3, 1.2, and 0.3 eV,
respectively). In the first superVolcano in Figure
6A, low γB-A (< 1.0) systems had higher overall
performance with a maximum TOFB of ~104 s-1 at
relatively weak binding of A and B (0.6 eV for the
binding energy of B). Low γB-A volcano plots
exhibited three distinct regions with varying rate-
limiting steps: (i) adsorption, (ii) surface
reaction/adsorption, and (iii) desorption.
Conversely, high gamma (γ > 1.0) volcano plots
had two primary regions: (i) surface reaction
limited and (ii) desorption-limited regimes. All
volcano plots intersected at a single point,
corresponding to the equivalence point, where the
binding energy of B is (δ + ΔHovr).
Figure 7A demonstrates the changes due to
lowering the value of δ from 1.4 eV to 0.5 eV in
comparison with Figure 6A. At low δ, the high γB-A
curves have higher TOF performance across a wide
span of binding energies, but the maximum TOFB
is lowered to ~102 s-1. In addition, the intermediate
regime of the low γB-A plots moves to the right of
the volcano plot peak at absolute binding energies
of 0.5-1.3 eV, so there now exists a surface
reaction/desorption limited regime. The high γB-A
curves coalesce on the desorption limited line at
~1.3 eV and onwards in Figure 7A, meaning that
these systems are nearly saturated with B* and the
TOFB is solely dependent on the binding energy of
B past this point.
Figure 7B depicts the superVolcano trend for
lower β values as compared with Figure 6A; β was
lowered to 65 kJ/gmol versus 102 kJ/gmol in Figure
6A. The high γB-A curves gain two new rate-limiting
kinetic regimes, with adsorption limited behavior
until an absolute binding energy of 0.4 eV, surface
reaction/adsorption until 0.9 eV, surface
reaction/adsorption from 0.9 eV to 1.4 eV, and
desorption limited behavior from 1.4 eV onwards.
As expected, the overall maximum TOFB increases
to ~105 s-1 since a lower β allows for higher surface
1.0E-08
1.0E-06
1.0E-04
1.0E-02
1.0E+00
1.0E+02
1.0E+04
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Relative Binding Energy of B [eV]
Tu
rnover
Fre
qu
en
cy t
o B
, T
OF
B[s
-1] 102
104
1.0
10-2
10-4
10-6
10-8
0.0
0.2
0.4
0.6
0.8
1.0
-1.0 -0.5 0.0 0.5 1.0
Relative Binding Energy of B [eV]
Su
rface C
ove
rag
e o
f A
[-]
0.0
0.2
0.4
0.6
0.8
1.0
-1.0 -0.5 0.0 0.5 1.0
Relative Binding Energy of B [eV]
Su
rface C
ove
rag
e o
f B
[-]
0.0
0.2
0.4
0.6
0.8
1.0
-1.0 -0.5 0.0 0.5 1.0
Relative Binding Energy of B [eV]
Su
rface C
ove
rag
e o
f A
[-]
0.0
0.2
0.4
0.6
0.8
1.0
-1.0 -0.5 0.0 0.5 1.0
Su
rface C
ove
rag
e o
f B
[-]
Relative Binding Energy of B [eV]
γ > 1.0
γ < 1.0
1.00.90.80.70.60.50.40.3 1.1 1.2 1.3 1.5 1.75 2.0 2.5 4.5 9.0γB
C
D
E
A
Figure 6. Balandin-Sabatier Volcano Curves for Varying γ-Parameter. (A) Turnover frequency to B product.
(B). Surface coverage of A in low γB-A systems. (C) Surface coverage of B in low γB-A systems. (D) Surface coverage
of A in high γB-A systems. €. Surface coverage of B in high γB-A systems. All panels – conditions: YB~1%, P ~ 100
bar, T ~ 150 °C, α ~ 0.6, β ~ 102 kJ mol-1, and δ ~ 1.4 eV.
____________________________________________________________________________ Ardagh, et al. Page 10
reaction rates. High γB-A curves coalesce on an
adsorption line from 0.4 eV and lower values of
relative binding energy of B.
Finally, α was raised from the intermediate
value of 0.6 to its maximum value of 1.0, such that
Figure 7C can be readily compared with Figure 7A
since they share all other parameters in common.
The maximum TOFB was relatively unchanged, but
all volcano curves in Figure 7C exhibit steeper
slopes due to the influence of α. The tighter
coalescence of all curves below 0.5 eV shows that
these systems are largely adsorption limited due to
a low δ of 0.5 eV. The desorption lines from 1.3 eV
onwards for low γ systems are shifted to higher
binding energies as compared with Figure 7A. This
is due to the high coverage of A* at binding
energies > 1.3 eV, and this limits the desorption
TOF. In summary, the superVolcano shape and
kinetic behavior is highly sensitive to both the
transition state BEP relationship parameters and the
binding energy equivalence point, δ. Since these
parameters are a strong function of the reaction
chemistry and metal crystal face/structure, practical
application of dynamic catalysis will be guided by
a combination of catalyst design and external
stimulus selection to achieve beneficial catalytic
reaction control.
3.4 Low Gamma Catalytic Kinetics. The
dynamic kinetics of high gamma (γB-A ~ 2.0)
catalytic systems were previously evaluated and
shown to exhibit significant rate enhancement
(1,000-10,000x above the Sabatier maximum) in
the resonant frequency range[24]. Under optimal
conditions, the surface coverage of high γ
chemistries oscillate between high coverage in A*
and high coverage in B*, ultimately yielding a
turnover frequency about equal to the imposed
surface oscillation frequency. As these dynamic
catalytic systems modulated into the strong binding
state, the heat of the surface reaction was at its most
negative condition leading to the lowest energy
transition state; this permitted A* to readily convert
to B*. When the surface binding energy then
converted to weaker overall binding, the binding
energy of B* decreased thereby decreasing the
desorption barrier. Additionally, the high energy
binding state inhibited the reverse surface reaction
(B* to A*) due to a high transition state energy.
This behavior held for multiple waveform types
(e.g. square, sinusoidal) over a range of linear
transition state scaling relationships (0 < α < 1.0),
making these types of reactions amenable to
dynamic rate enhancement.
Low gamma (γB-A <1.0) catalytic systems
exhibit similar behavior with the variation of
surface intermediate binding, but conditions
leading to surface resonance remain to be
identified. As depicted in Figure 8A, dynamic
catalysis can lead to performance above the
Sabatier maximum rate for low gamma systems. In
this example, catalytic conversion of A-to-B occurs
in a continuous-flow catalytic reactor at fixed
temperature and pressure, and the system switches
from static operation to five different dynamic
frequencies for comparison (f ~ 0.001, 0.035, 1.0,
10, and 1000 Hz, ΔUB ~ 0.93 eV). Oscillation
frequencies above the Sabatier maximum rate
(~0.036 s-1) lead to high average turnover frequency
1.E-09
1.E-07
1.E-05
1.E-03
1.E-01
1.E+01
1.E+03
-1.0 -0.5 0.0 0.5 1.0
Turn
ove
r F
reque
ncy [
s-1
]
Relative Binding Energy of B [eV]
1.E-13
1.E-10
1.E-07
1.E-04
1.E-01
1.E+02
1.E+05
-1.5 -1.0 -0.5 0.0 0.5 1.0
Turn
ove
r F
reque
ncy [
s-1
]
Relative Binding Energy of B [eV]
1.E-11
1.E-09
1.E-07
1.E-05
1.E-03
1.E-01
1.E+01
1.E+03
1.E+05
-1.0 -0.5 0.0 0.5 1.0
Turn
ove
r F
reque
ncy [
s-1
]
Relative Binding Energy of B [eV]
1.00.90.80.70.60.50.40.3 1.1 1.2 1.3 1.5 1.75 2.0 2.5 4.5 9.0γ
10-7
10-9
10-5
10-3
10-1
101
103
10-7
10-10
10-13
10-4
10-1
102
105
10-9
10-7
10-5
10-3
10-1
101
103
105
10-11
B CA α ~ 0.6
β ~ 102 kJ/mol
δ ~ 0.5 eV
α ~ 0.6
β ~ 65 kJ/mol
δ ~ 1.4 eV
α ~ 1.0
β ~ 102 kJ/mol
δ ~ 0.5 eV
Figure 7. Parameter Variations Catalytic Reactions Depicted as Balandin-Sabatier SuperVolcanoes. A
catalytic flow reactor reversibly converts A to B at 150 °C, 100 bar of pure feed A, and 1% yield of B. In each case,
the heat of reaction was exothermic, ΔHrxn ~ -20 kJ/mol with varying gamma, 0.3 < γB-A < 9.0. Surface parameters
varying between cases included: (A) Alpha of 0.6, beta of 102 kJ mol-1, and delta of 0.5 eV. (B) Alpha of 0.6, beta
of 65 kJ mol-1, and delta of 1.4 eV. (C) Alpha of 1.0, beta of 102 kJ mol-1, and delta of 0.5 eV.
____________________________________________________________________________ Ardagh, et al. Page 11
up to 25 s-1. The depicted instantaneous TOFB for
each frequency shows flipping between high TOFB
and low TOFB states due to the large change in
surface coverages of A* and B*.
The low gamma system (γB-A ~ 0.5) of Figure
8B exhibits a broad range of resonance frequencies.
For an oscillation amplitude of 0.93 eV, the TOFB
frequency response is shown in Figure 8B for a
range of imposed square wave oscillation
frequencies (10-6 < f < 1012) in the binding energy
of B*. From this plot, four corner frequencies can
be determined as the transitions in slope; the
resonant frequency range extends from about 102 to
107 Hz. Frequencies below 10-4 Hz yield a TOF of
~0.02 s-1, which is the average of the steady state
TOF at two oscillation endpoints, while frequencies
above 1011 Hz yield the TOF from the starting
condition (i.e. the volcano plot peak).
Figures 8A and 8B can be directly compared to
the performance of the high gamma system (γB-A ~
2.0) depicted in our previous publication[24]. At
0.001 Hz, the TOFB response in Figure 8A exhibits
an overshoot consistent with the flipping of the
surface binding energies and off-loading of the
surface. At intermediate frequencies (0.035, 1.0,
and 10 Hz) in Figure 8A, the TOFB response has
sharp features due to rapid uptake of gas-phase
species. In the resonant region (1000 Hz), a
significant number of oscillations are required for
the system to achieve steady state consistent with a
gradual change in the surface coverages; for this
condition, there exists an initial overshoot of the
average TOFB above the final steady state value.
This example indicates that the catalytic response
of both low and high gamma systems, in this case
γB-A ~ 0.5 and γB-A ~ 2.0, can exhibit similar general
reaction behavior, with the caveat that any
particular dynamic catalytic system must be
evaluated to identify the conditions that permit
resonance and rate enhancement. This key
observation is relevant to general applicability of
dynamics, since forward and reverse directions of
any reaction will exhibit inverse values of gamma;
this indicates that at least some catalytic systems
could be dynamically promoted in either direction
(forward or reverse) depending on the selected
parameters of the imposed surface oscillation.
While both the forward and reverse directions
of a reaction can be promoted dynamically, the
strategies for selecting the parameters of the
imposed surface binding energy oscillation
between systems are indeed different. For the low
gamma system (γB-A ~ 0.5) depicted in Figure 9A,
the Sabatier curve is depicted in black with
extended dashed lines above the Sabatier peak. The
oscillation amplitude is depicted as a red bar with
two endpoints: (i) the strong binding energy
maximum is located at the position of the volcano
peak, and (ii) the weak binding energy minimum is
permitted to vary. The average turnover frequency
to B is depicted in the heatmap of Figure 9B with
variable square wave surface frequency and tunable
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Tu
rno
ve
r F
req
ue
nc
y t
o B
[s
-1]
Time on Stream [Arbitrary Units]
10-3
10-1
1
10+1
10-2
10+2A
0.001
0.01
0.1
1
10
100
1E-06 1E-02 1E+02 1E+06 1E+10
Tim
e A
ve
rag
ed
TO
FB
[s-1
]
Oscillation Frequency [Hz]
B
1000 Hz
10 Hz
1.0 Hz
0.001 Hz
0.035 Hz
Figure 8. Kinetics of Low Gamma (γ ~ 0.5) Continuous Flow Catalytic Reactor. (A) A continuous flow reactor
operating under static conditions exhibits TOFB ~ 0.036 s-1; implementation of catalyst dynamics at 0.001 Hz (red),
0.035 Hz (green), 1.0 Hz (light blue), 10 Hz (dark blue) and 1000 Hz (purple) varies the instantaneous TOFB. (B)
Continuous variation of the catalyst binding energy over varying frequencies (10-6 < f < 1011 s-1) reveals a band of
resonance frequencies highlighted in purple. Conditions: T ~ 100 °C, P ~ 100 bar feed of A, YB ~ 1%, ΔHrxn ~ 0 kJ
mol-1, α ~ 0.6, β ~ 102 kJ mol-1, γB-A ~ 0.5, δ ~ 1.4 eV, ΔUB ~ 0.93 eV, [0.10 eV < UB < 1.03 eV]
____________________________________________________________________________ Ardagh, et al. Page 12
minimum binding energy of B. Until the oscillation
amplitude low binding energy endpoint (left) is less
than -0.4 eV relative binding energy of B, the time
averaged turnover frequency is negligible, and the
A → B reaction is inhibited by dynamic catalysis.
Above the -0.4-eV transition, the average turnover
frequency to B is about constant and oscillation
frequencies above 103 Hz lead to a high rate
performance of 25 s-1 for a total rate enhancement
of ~715x.
The importance of the transition at about -0.4
eV relative binding energy of B is indicative of the
mechanism leading to dynamic enhancement. As
shown in the surface coverage plot of Figure 6C,
this transition is associated with the shift from a
clean surface (θ* ~ 1) to a surface covered in B* (θB
~ 1). When the oscillation amplitude extends into
the low binding energy range associated with a
clean surface, then B* is removed from the surface
producing a dramatic reaction rate enhancement.
The significance of transitioning between a
surface covered in reaction surface species and a
clean surface for low gamma systems under
dynamic conditions has implications for selection
of catalyst materials. In general, the binding
energies of A* and B* need to be sufficiently weak
that all chemical species can be desorbed during the
weaker of the surface oscillation states; in this case
δ ~ 1.4 eV exhibited this capability. However, as
the binding energy of B* varied with the imposed
surface oscillation, a new limitation exists; the heat
of adsorption of both B* and A* must be less than
or equal to zero. For low gamma systems, small
variations in the binding energy of B* produce large
variations in the binding energy of A* such that it
readily approaches zero enthalpy of adsorption.
This scenario can be interpreted by the gamma-
delta plot of Figure 2B; as binding energies weaken;
one surface species will achieve negligible
adsorption enthalpy before the other one. In this
scenario, the surface species with negligible
adsorption enthalpy will remain in that state as the
other surface species continues to change with the
imposed oscillation.
The physical restriction of non-repulsive
surfaces for adsorption alters the enthalpy of the
surface reaction and the surface transition state
energy by association. The implication is apparent
in the TOFB heat map of Figure 9C, where the TOFB
for the surface reaction elementary step is presented
(color) as a function of the surface coverage of A
(θA) and the relative binding energy of B [eV]. At
low relative binding energies of B, the system
exhibits a gradual change in the TOFB before a
sharp transition above -0.7 eV; this transition is the
condition where the binding energy of A
approaches zero. Figure 9C also reveals part of the
origin for fast dynamic TOFB; at high surface
coverage of A, the surface reaction can achieve
TOFB of ~102 s-1. These unique behaviors indicate
that three-order-of-magnitude enhancement in rate
is achievable for low gamma systems, but selection
1.E-08
1.E-06
1.E-04
1.E-02
1.E+00
1.E+02
1.E+04
1.E+06
-1.0 -0.5 0.0 0.5
Turn
ove
r fr
equ
ency
(1
/s)
Relative binding energy of B (eV)Relative Binding Energy of B [eV]
Tu
rnover
Fre
qu
en
cy [
s-1
]
1
106
104
102
10-2
10-4
10-6
10-8 1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
Osc
illat
ion
fre
qu
ency
(H
z)
Relative binding energy of B (eV)Relative Binding Energy of B, Left Endpoint [eV]
Oscill
ation
Fre
qu
en
cy [
s-1
]
FAST SLOW
Turnover Frequency to B, TOFB [s-1]
105
104
103
102
101
1
10-1
10-2
10+1 1 10-1 10-2
-1.40
-1.18
-0.96
-0.74
-0.52
-0.30
-0.08
1.E-10 1.E-08 1.E-06 1.E-04 1.E-02 1.E+00
Re
lati
ve
Bin
din
g E
ne
rgy o
f B
[e
V]
Surface Coverage of A, θA [-]Surface Coverage of A, θA [-]
Rela
tive B
ind
ing
En
erg
y o
f B
[eV
]
10-10 10-8 10-6 10-4 10-2 1
Turnover Frequency to B, TOFB [s-1]
10+2 10+1 1 10-1 10-2 10-3A B C
Figure 9. Catalytic Dynamics in a Flowing Stirred Tank Reactor at Differential Conversion with Variable
Amplitude at Low Gamma. (A) Volcano kinetics of reversible A to B with variable amplitude. (B) Average
turnover frequency to B (color distribution) as a function of square wave oscillation frequency and the amplitude low
energy endpoint; the high energy endpoint is fixed at +1.03 eV. (C) Maximum possible surface reaction rate (color
distribution) for A* reversibly converting to B* as a function of variable surface coverage and binding energies;
negligible reverse reaction occurred under these conditions. Conditions: T~ 100 °C, P~100 bar A feed, YB ~ 1%,
ΔHrxn ~ 0 kJ mol-1, α ~ 0.6, β ~ 102 kJ mol-1, γB-A ~ 0.5, δ ~ 1.4 eV, and ΔUB ~ variable.
____________________________________________________________________________ Ardagh, et al. Page 13
of the dynamic oscillation of surface binding
energies must be carefully selected unique to each
system.
3.5 Dynamic Steady State. As stated in the
introduction, a grand challenge in catalysis and
reaction engineering targets high yields of a desired
product in the constraint of thermodynamic
equilibrium and the mass action of Le Chatelier’s
principle. As shown in Figure 10A, reaction
systems can be thought of as a thermodynamic
trajectory between the reactant and product. Under
thermodynamic control, the reaction system
performance will approach the product composition
where the overall free energy, or chemical
potential, is minimized. Since each molecule has
significantly different free energy due to varying
chemical potentials, bonding, and functional
groups, this free energy landscape is a strong
function of the reaction chemistry.
The importance of controlling extent of
reaction has led to the strategy depicted in Figure
10B; application of external work to one or both of
the two thermodynamic states perturbs the free
energy landscape in favor of a new equilibrium
extent of conversion. For example, work via
compression of reactant gases in chemistries such
as hydrogenation of hydrocarbons alter the free
energy landscape in favor of saturated bonds,
requiring high pressure hydrogen as a co-reactant.
Dynamic oscillation of surface binding energies
potentially permits work to be imparted directly to
the surface reaction (Figure 10C). Since work is
applied directly to the surface reaction, the free
energy landscape between A and B is modified
without changing the free energy of the gas phase
species. A system of this type is depicted in Figure
10D, with the minimum energy offset from
equilibrium by ΔΔG. Thermodynamic work as a
‘catalytic molecular pump’ moving surface
adsorbates from strong to weak binding could
therefore permit reactor systems that achieve high
catalytic conversion at thermodynamically
unfavorable gas-phase conditions.
To assess catalytic reactor performance and
extent of reaction of dynamic catalytic systems,
simulations were conducted using a batch reactor
charged with 100 bar of initial gas at varying
composition of A and B with a fixed amount of
catalyst with dynamically varying surface binding
energies. Figure 11 depicts the time-resolved gas-
phase composition within the catalytic batch reactor
(100 bar, 250 °C) with seven different initial
compositions of A and B. Dynamic surface binding
energy of the catalyst was applied for 200 seconds
of square wave of amplitude 0.5 eV and 1000 Hz,
after which the system was converted to static
catalysis (in this case at the volcano plot peak
binding energy). Regardless of the initial gas-
phase composition of the reactor, the system
approached a dynamic steady state of 70 mol% B in
the gas phase. While the small oscillatory behavior
is difficult to observe in Figure 11, oscillation in the
instantaneous TOFB was observed upon zooming in
on any of the data sets. Once the catalyst was
converted from dynamic to static operation at 200
s, the system returned to thermodynamic
equilibrium at 40 mol % B (ΔHovr of +2 kJ/gmol),
indicating that deviation from equilibrium resulted
from dynamic ‘catalytic pumping’ on the catalyst
surface.
Gib
bs F
ree E
nerg
y, G
Conversion
ΔGEQ
ΔΔ
G
XEQ
XSS
AG
ibbs F
ree E
nerg
y, G
Conversion
ΔGEQ
XEQ
XEQ
ΔΔ
G
D
State A State B
Reaction
Path
State A State BReaction
Path
WORKWORK
B
C
Figure 10. (A) Gibbs free energy landscape for a reaction with differing overall heats and entropy of reaction in state
B, ΔΔG. (B) Work applied to reactant state, B, alters the overall Gibbs free energy of reaction and the extent of
conversion at equilibrium. (C) Work applied in the reaction path between reactant A and product B alters the Gibbs
free energy minimum and extent of conversion at steady state. (D) General reaction system Gibbs free energy with
respect to extent of conversion with added work, ΔΔG, and minima identified as equilibrium points.
____________________________________________________________________________ Ardagh, et al. Page 14
Figure 11. Dynamic Catalytic Reaction to Steady
State Different from Equilibrium - High γB-A
Condition. The reversible reaction of A to B undergoes
dynamic catalytic conversion in a batch reactor at 250
°C, 100 bar total pressure, and a square waveform at
1000 Hz and 0.5 eV amplitude. Variable initial
concentrations of component A. Surface chemistry
parameters, α ~ 0.6, β ~ 135 kJ mol-1, γB-A ~ 2.0, δ ~ 1.0
eV, and ΔHrxn ~ +2.0 kJ mol-1.
3.6 Oscillatory Steady State of High γ
Catalytic Systems. The steady state performance
of a representative high gamma catalytic system
(γB-A ~ 2.0, δ ~ 1.4 eV) was assessed using a batch
reactor model. Sabatier volcano plots for this
system (α ~ 0.6, β ~ 102 kJ/mol) are shown in
Figure 12A, with reactor temperatures varying
between 150-250 oC. Qualitatively, temperature
does not change the shape and the rate-limiting
kinetic regimes of the volcano plots. Similarly, the
associated surface coverages for the considered
system was plotted in Figure 12B for both A* and
B*. B* dominated the surface at high relative
binding energies while A* dominated at weaker
binding energies. Each set of curves crossed over at
the delta point, which is the equivalence point
where A* and B* have the same energy. As
temperature increased, the onset of surface vacancy
shifted to stronger binding energies, and the
remaining coverage of A* decreased faster with
higher temperature (not shown, BE < -1.0 eV).
The extent of conversion at oscillatory steady
state of the considered system of Figure 12 was
evaluated as a function of applied square wave
frequency f, waveform amplitude, and waveform
amplitude position. As depicted in Figure 12C at
150 °C, dynamic oscillation of the surface binding
energy of B at a fixed amplitude can occur over a
range of oscillation endpoints. The position of the
strong binding energy of B endpoint was the
independent variable in the heat maps of Figures
12D (ΔUA ~ 0.5 eV), 12E (ΔUA ~ 1.0 eV), and 12F
(ΔUA ~ 1.5 eV), where the oscillatory steady state
TOFB was determined via simulation.
In the simulations of Figure 12D-12F, the heat
and entropy of reaction were zero, indicating that
the system should achieve only 50% composition
of both A and B in the gas phase (green) at
thermodynamic equilibrium. However, the broad
color range associated with the concentration at
oscillatory steady state indicates that there exist
catalytic conditions yielding substantial shift away
from equilibrium as high as >99% yield of B for all
oscillation frequencies above 1.0 Hz and oscillation
endpoints above about 0.2 eV. This region of high
conversion to B at steady state extends over a broad
range that ends at lower oscillation endpoints
before transitioning into a region favorable to
forming the reactant species. Frequencies this slow
and even as high as 0.1 Hz are achievable for all
conceivable experimental methods of imposing
oscillatory surface binding energy of adsorbates
including oscillating electric fields and surface
stress/strain.
The higher amplitude heat map plots of TOFB
at ΔUA ~ 1.0 eV (Figure 12E) and ΔUA ~ 1.5 eV
(Figure 12F) show enhanced performance at lower
frequency (about 10-2 s-1), because the amplitude
oscillation endpoints move further and further away
from the volcano plot peak. Moreover, there exists
a sharp transition between the forward and
backwards reaction at 0.4 eV relative binding
energy, which exactly matches the absolute value
of delta, δ ~ 1.4 eV. High conversion to either A or
B (>99%) is achievable at these higher amplitudes
requiring only the selective application of surface
binding oscillation at the relevant frequency and
oscillation amplitude endpoints.
3.7 Oscillatory Steady State of Low γB-A
Systems. Low gamma catalytic systems introduced
in section 3.4 exhibited dramatic rate enhancement
under dynamic operation (Figure 8) but with more
complex kinetic behaviors as compared with high
gamma systems in CSTR simulations. To assess the
potential for controlling the extent of reaction of
low gamma catalysis, a simulation with parameters
____________________________________________________________________________ Ardagh, et al. Page 15
identical to the volcano plot of Figure 12A (except
for a γB-A of 0.5) was conducted in a batch reactor
with a square wave in surface binding energy
oscillation. As depicted in Figure 13A-13C, three
oscillation amplitudes of ΔUB of 0.5, 1.0, and 1.5
eV were evaluated as a function of oscillation
endpoints and frequency. Again, the simulated
system reaction equilibrium was 50% composition
(green in Figure 13) of both A and B.
For an oscillation amplitude of ΔUB ~ 0.5 eV in
Figure 13A, there exists a small region (yellow)
where the forward A → B reaction is promoted
between relative binding energies of B of 0-0.2 eV
and about 102 Hz. However, above about 0.15 eV
of relative binding energy of B exists a large region
(dark blue) overwhelmingly favorable to the
reverse reaction (formation of A). Higher
oscillation amplitudes in low gamma γB-A catalytic
systems again lead to high conversion at oscillatory
steady state, in this case for both reaction directions
as yields > 90 % B are observed while other
oscillation endpoints yield > 99 % A. The two
regions are split by the delta point (δ ~ 1.4 eV, 0.4
relative binding energy of B). The implication is
that this system can be tuned to produce an outlet
stream with nearly completely controllable
composition at high reactor residence times, just by
varying the oscillation endpoints at frequencies
between 1-10 Hz. The high and low conversion
regimes exist as low as 0.001 Hz making them
accessible to slowly oscillating systems. This
behavior is nearly identical to the high gamma
system in Figure 12; as the only difference between
Figure 12 and 13 being the inverse values of gamma
(γB-A of 2.0 and 0.5, respectively), this inverse
mirrored behavior is expected. Both low and high
gamma systems provide versatility in conversion as
nearly pure product streams of A or B can be
produced by merely changing the dynamic catalysis
oscillation endpoints relative to the catalytic
chemistry delta point.
3.8 Mechanism of Catalytic Molecular
Pumping. The tunable directionality (forward
versus reverse) of catalytic molecular pumping
observed in Figures 12 and 13 derives from the
mechanism of molecular movement through the
oscillating energy profiles on the catalytic surface.
As shown earlier, catalytic systems can be driven in
the forward or reverse directions by selection of the
catalytic material or stimulus method (affecting α,
0.0
0.2
0.4
0.6
0.8
1.0
-1.0 -0.5 0.0 0.5 1.0
Surf
ace
cove
rage
(u
nit
less
)
Relative binding energy of B (eV)
1.E-08
1.E-06
1.E-04
1.E-02
1.E+00
1.E+02
-1.0 -0.5 0.0 0.5 1.0
Turn
ove
r fr
equ
ency
(1/s
)
Relative binding energy of B (eV)
Binding Energy of B* [eV]
-1.0 0.5 1.00-0.5
Tu
rnover
Fre
qu
en
cy t
o B
[s
-1]
10-2
Binding Energy of B* [eV]
-1.0 0.5 1.00-0.5
Su
rface C
overa
ge [
-]
1.00
0.80
0.60
0.40
0.20
0
250 C
200 C
150 C
θA
θB
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
-1.00 -0.50 0.00 0.50 1.00
Turn
ove
r Fr
equ
ency
to
B [s
-1]
Binding energy shift of B* (eV)
Binding Energy of B* [eV]
-1.00 0.50 1.000-0.50
Tu
rnover
Fre
qu
en
cy t
o B
[s
-1]
P1
P2P3
P4P5
106
104
102
100
10-2
10-4
A B C
1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
0.0 0.2 0.4 0.6 0.8 1.0
Osc
illat
ion
freq
uen
cy (H
z)
Oscillation endpoint (eV)
102
100
10-1
10-2
10-3
10-4
10-5
10-6
Oscill
ation
Fre
qu
en
cy [
s-1
]
Oscillation Endpoint, B [eV]
0 0.40 1.60 2.001.200.80
E
101
ΔUA = 1.0 eV1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Osc
illa
tio
n fr
eq
ue
ncy
(Hz)
Oscillation endpoint (eV)
Con
vers
ion
of A
[%]
100
90
80
70
60
50
40
30
20
10
0
F 102
100
10-1
10-2
10-3
10-4
10-5
10-6
Oscill
ation
Fre
qu
en
cy [
s-1
] 101
Oscillation Endpoint, B [eV]
0 0.5 2.5 3.01.51.0 2.0
ΔUA = 1.5 eV1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
0.0 0.2 0.4 0.6 0.8 1.0
Turn
ove
r fre
qu
ency
(1/s
)
Relative binding energy of B (eV)
101
100
10-1
10-2
10-3
10-4
10-5
10-6
Oscill
ation
Fre
qu
en
cy [
s-1
]
Oscillation Endpoint, B [eV]
-0.2 0 0.6 0.80.40.2
D102
ΔUA = 0.5 eV
10-4
10-6
10-8
1
102
Figure 12. Dynamic Catalytic Conversion of A to B for High Gamma (γ ~ 2.0, δ ~ 1.4 eV). (A) Turnover
frequency to product chemical B from reactant A for α ~ 0.6, β ~ 102 kJ/mol, and YB~1% for 150, 200, and 250
°C. (B) Surface coverages of A* (θA) and B* (θB) for α ~ 0.6 and YB~1% for 150, 200, and 250 °C. (C) Varying
oscillation endpoints for ΔUB ~ 0.8 eV. (D/E/F) Oscillatory steady state conversion of A (0 < XA < 100%) at 150 °C
and 100 bar for varying applied surface oscillation frequency and oscillation endpoint with oscillation amplitudes
ΔUA of 0.5, 1.0, and 1.5 eV
____________________________________________________________________________ Ardagh, et al. Page 16
β, γ, and δ) but also by the selection of imposed
surface oscillation including the frequency,
amplitude, and amplitude endpoints. The
relationship between selected parameters and
catalytic pumping directionality is depicted in
Figure 14 for the system previously described in
Figure 13 with a low gamma (ΔHovr ~ 0 kJ gmol-1, α
~ 0.6, β ~ 102 kJ gmol-1, γB-A ~ 0.5, and δ ~ 1.4 eV).
Two conditions were selected; Figure 14A-B
correspond to an oscillation amplitude promoting
the reverse reaction, while Figure 14C-D has an
amplitude promoting the forward reaction. In each
energy diagram (Figure 14A, 14C), the amplitude
minimum (Umin, blue) and maximum (Umax, red) are
depicted along with the relative binding energy of
B* corresponding to the Sabatier peak under static
conditions (purple, dashed) and the binding energy
associated with delta (black, δ ~ 1.4 eV).
The directionality of the dynamic catalytic
system is visually apparent from the values of the
enthalpy of A* and B* at the minimum amplitude
(Umin). The reverse reaction (B-to-A) is favored
when A* is lower in energy than B* at Umin in
Figure 14A, while the forward reaction (A-to-B) is
favored when B* is lower in energy than A* at Umin
in Figure 14C. By this interpretation, the lower
energy state Umin serves as the condition whereby
the surface reaction proceeds to accumulate A* or
B* on the surface; the higher energy state Umax then
serves to push this surface reaction product into the
gas phase. In the low energy state Umin, the
transition state enthalpy must be lower than the
desorption enthalpy to permit the surface reaction
to proceed. This general behavior is depicted in the
schemes for the reverse reaction (Figure 14B) and
forward reaction (Figure 14D).
Interpreting general reaction systems for
potential catalytic molecular pumping relies on the
determination of the reaction energy profile under
different oscillation conditions with respect to the
parameter delta, δ. These intermediate and
transition state energies can now be determined
computationally for almost any catalytic system[66].
As defined earlier, the quantity delta δ identifies the
adsorption enthalpy whereby both A* and B* have
equal binding energy with the surface. Delta
therefore serves as a separation point between the
forward and reverse directionality of catalytic
molecular pumping; for any system with a non-
unity gamma, the surface species of lowest energy
(A* or B*) will differ for systems operating on
either side of delta. This precise transition in
directionality was observed in Figures 12 and 13,
where the shift from high to low conversion was
demarcated by the relative binding energy of B*
equaling delta.
3.9 Efficiency of Dynamic Catalysis at
Oscillatory Steady State. Returning to the concept
of imparting work during catalysis as described in
Figure 10C, the observed free energy change for the
reaction (ΔΔG) is related to the applied oscillation
amplitude (ΔUi). In mechanical applications, this
relationship is typically expressed as an efficiency,
which compares the applied work (i.e. energy units
after integrating over time) to the resulting change
in system enthalpy, internal energy, or another
indicator such as temperature. Here, the efficiency
of the dynamic catalytic system is defined as,
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
0 0.1 0.2 0.3 0.4 0.5
Oscill
atio
n F
reque
ncy [
Hz]
Relative Binding Energy of B [eV]
Ste
ady-S
tate
Co
nve
rsio
n [%
]
10-1
10-2
10-3
10-4
10-5
10-6
1
101
102
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
0.0 0.2 0.4 0.6 0.8 1.0
Oscill
atio
n F
reque
ncy [
Hz]
Relative Binding Energy of B [eV]
10-5
10-4
10-3
10-6
10-2
10-1
1
101
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
0.00 0.25 0.50 0.75 1.00 1.25 1.50
Oscill
atio
n F
reque
ncy [
Hz]
Relative Binding Energy of B [eV]
10-1
1
101
10-2
10-3
10-4
10-5
10-6
A B C
ΔUB = 0.5 eV ΔUB = 1.0 eV ΔUB = 1.5 eV
Figure 13. Dynamic Catalytic Conversion of A to B for Low Gamma (γB-A ~ 0.5, δ ~ 1.4 eV). Steady-state
average conversion of A-to-B in a batch reactor (equilibrium at 50% - green) for varying square wave amplitude high
binding energy state [eV] and oscillation frequency [Hz] for fixed amplitudes ΔUB of: (A) 0.5 eV, (B) 1.0 eV, and
(C) 1.5 eV. Batch reactor conditions: 150 °C, initial reactor composition of 100 bar pure A, ΔHrxn ~ 0 kJ mol-1, α ~
0.6, β ~ 102 kJ mol-1, γB-A ~ 0.5, and δ ~ 1.4 eV.
____________________________________________________________________________ Ardagh, et al. Page 17
휀 =|𝛥𝛥𝐺|
𝛥𝑈𝑖=
𝐹𝑟𝑒𝑒 𝐸𝑛𝑒𝑟𝑔𝑦 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝑂𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
(10)
The efficiency is the absolute value of the free
energy deviation from equilibrium to account for
both the forward and reverse reaction promotion at
oscillatory steady state relative to the applied
oscillation amplitude. For consistency with the
mechanism of dynamic catalysis of Figure 14, the
amplitude in the denominator of the efficiency
(ΔUA or ΔUB) was selected based on the
directionality of the promoted reaction; ΔUB was
selected for the forward reaction, and ΔUA was
selected for the reverse reaction.
Representative high and low gamma systems
were analyzed to assess the efficiency as a function
of oscillation amplitude, waveform, and endpoint
selection. As demonstrated in a previous
publication for a continuous flow reactor under
dynamic catalysis operation[24], oscillation
waveform shape has a significant effect on dynamic
catalysis rate promotion. In the CSTR for a high
gamma system (γB-A ~ 2.0), the square waveform
exhibited the highest TOFB followed by sinusoidal
(2-3x slower) and then triangle and sawtooth
waveforms (4-6x slower). As shown in Figure 15,
these waveform types were applied to the batch
reactor system, and the resulting steady state
composition was compared with the
thermodynamic equilibrium composition (50% A
and B in all cases). The ΔΔG was computed as the
difference in the apparent free energy of the
-250
-200
-150
-100
-50
0
50
100
En
tha
lpy [
kJ
mo
l-1]
Umin
Umax
δ
Static
Peak
A* B*
Reverse
A
B
-175
-125
-75
-25
25
75
125
En
tha
lpy [
kJ
mo
l-1]
A* B*
δ
Umin
Umax
Static
Peak
Forward
C
D
Umax
Umin
Umax
Umin
Reverse Forward
Figure 14. Mechanism of Catalytic Molecular Pumping and Rate Enhancement. Energy diagrams for catalytic
unimolecular reaction system with reaction chemistry defined as: ΔHovr ~ 0 kJ gmol-1, α ~ 0.6, β ~ 102 kJ gmol-1, γB-
A ~ 0.5, and δ ~ 1.4 eV. (A) Oscillation amplitude of 1.0 eV with oscillation endpoints of B* of Umin ~ 0.6 eV and
Umax ~ 1.6 eV. (B) For the selected oscillation amplitude, the catalytic molecular pump moves molecules of B in
green through adsorption to B* and then reaction to A*; subsequent oscillation desorbs A* to A. (C) Oscillation
amplitude of 1.0 eV with oscillation endpoints of B* of Umin ~ 0.2 eV and Umax ~ 1.2 eV. (D) For the selected
oscillation amplitude, the catalytic molecular pump moves molecules of A in orange through adsorption to A* and
then reaction to B*; subsequent oscillation moves molecules of B* to product B in the gas phase.
____________________________________________________________________________ Ardagh, et al. Page 18
reaction at oscillatory steady state from the actual
equilibrium (i.e., 50% A and 50% B).
Figure 15A shows the efficiency of the applied
waveforms as a function of imposed amplitude at a
constant frequency of 10 Hz. Initially, near zero
amplitude, the efficiency is low for all waveforms
as shown by the shallow slope of the plot. Once the
oscillation amplitude achieves 10-20 kJ mol-1 (i.e.
0.1-0.2 eV), the relationship between ΔΔG and ΔU
is effectively described by a straight line. The slope
of this line is the system efficiency during dynamic
catalysis since it is the ratio of two energy units. The
maximum efficiency observed was ~40 % for the
square wave with the sinusoidal and triangular
wave being slightly less effective at 34% and 30%,
respectively.
Due to the importance of oscillation amplitude
endpoints in low gamma γB-A catalytic systems, the
efficiency of low gamma systems was evaluated in
Figure 15B as a function of both oscillation
amplitude and oscillation amplitude endpoint in a
flow reactor. As observed in the low gamma batch
reactor results in Figure 13A-13C, the forward A →
B reaction (yellow-to-red) was only enhanced at
oscillation endpoints up to the delta point. Once
stronger binding energies were imposed on the
system, the B → A reaction (blue) was highly
favorable due to the dominant surface coverage of
A*. The highest observed efficiency here was ~16
%, specifically for the B → A reaction.
A parity region was observed in Figure 15B
that runs diagonally along the blue color-coded B
→ A region with a slope of about unity. This region
traces points where the difference between the
oscillation endpoint and amplitude equals delta.
Once this difference was less than delta (i.e. the
lower right quadrant of the heat map), the system
performance rapidly declined to 0% efficiency
(green). This inefficiency resulted from the surface
coverage which needs to be turned over
significantly to enhance reactor performance, and
the delta point determines the crossover between
A* and B* covered surfaces. In summary, the
system efficiency for high gamma and low gamma
cases is a strong function of oscillation amplitude,
frequency, and endpoint selection but waveform
shape also has a minor effect.
3.10 Catalytic Molecular Pump and
Molecular Machines. The design of molecules
with dynamic functionality has been extensively
described as ‘molecular machines’[67] with both
synthetic and biological examples[68,69]. While
‘switchable catalysts’ are capable of turning on and
off as needed[70], other molecular machines and
molecular devices are capable of implementing
motion including molecular motors and
pumps[68,71,72]. For example, an artificial molecular
pump is capable of moving molecules such as
charged rings up a concentration gradient[73],
conceptually similar to nature’s capability for
0.0
0.3
0.6
0.9
1.2
1.5
0.0 0.3 0.6 0.9 1.2 1.5
Oscill
atio
n A
mplit
ude
[e
V]
Oscillation Endpoint [eV]
Surf
ace
Oscill
atio
n E
ffic
iency,
ε[%
]
Fo
rwa
rdR
eve
rse
20
15
10
5
0
5
10
15
20-40
-35
-30
-25
-20
-15
-10
-5
0
0 20 40 60 80 100
Amplitude [kJ mol-1]
De
lta D
elta
Gib
bs F
ree
Ene
rgy,
ΔΔ
G [
kJ m
ol-1
]
Square (40.1%)
Sine (34.1%)
Triangle (30.2%)
A B
Oscillation Endpoint [eV]
Figure 15. Conversion Efficiency of A-to-B Reaction on Dynamic Catalytic Surfaces. (A) Square, sinusoidal,
or triangle wave with varying amplitude operating at 10 Hz (α ~ 0.6, β ~ 135 kJ mol-1, γ ~ 2.0, δ ~ 1.0 kJ mol-1, and
ΔHovr ~ 0 kJ mol-1) on a catalyst in a batch reactor at oscillatory steady state offset from equilibrium by delta delta
Gibbs free energy, ΔΔG. The efficiency is defined as the absolute value of ΔΔG relative to the oscillation amplitude,
ΔU. (B) The surface oscillation efficiency (ε, color map) of a square wave operating at 1.0 Hz with varying amplitude
and oscillation amplitude endpoint exhibits two bands of increased efficiency (α ~ 0.6, β ~ 102 kJ mol-1, γ ~ 0.5, δ ~
1.4 kJ mol-1, and ΔHovr ~ 0 kJ mol-1).
____________________________________________________________________________ Ardagh, et al. Page 19
controlling the mobility of metal ions across
membranes against equilibrium[74]. These
molecular machines and pumps are relevant to the
dynamic oscillating catalyst surface concept
addressed in this work due to a common
mechanism. Of the many methods of manipulating
molecules, many molecular machines and pumps
utilize energy to implement a dynamic ‘pulsating’
ratchet energy profile[75,76,77]. Similar to Figure 14,
progression along a pulsating ratchet permits
molecules to spontaneously access the current low
energy states; the input of energy into the ratchet
system alters the free energy landscape permitting
molecules to access new low energy states that
constitute progress in position (i.e., molecular
pumps). The evolution of changing energy profiles
biases progress in one direction based on local peak
heights, consistent with its ‘ratchet’ name[73].
Molecular pumps and ‘catalytic pumps’ share a
common purpose; move molecules against
thermodynamics. Molecular pumps move
molecules against the free energy gradient
associated with concentration or pressure, while the
catalytic pumps described here aim to advance the
extent of reaction against the chemical potential
associated with chemical reaction equilibrium.
Molecular pumps utilize the ratchet mechanism,
while the catalytic pump utilizes the ratchet
mechanism plus the addition of a surface reaction
as one of the free energy profile transition states.
Similarly, molecular pumps utilize external energy
sources (e.g. ATP in biological membrane
transport), while catalytic pumps oscillate binding
energy by a variety of proposed methods as
described in Figure 4 (e.g., oscillating surface strain
or electric field).
4.0 Conclusions. The dynamic catalytic promotion
of A-to-B surface reactions permits order-of-
magnitude rate enhancement and control of extent
of conversion for a broad set of surface
mechanisms. Classification of surface mechanisms
and their associated kinetic behavior requires
detailed description of the key parameters defined
here as gamma (γ, the ratio of variation in surface
adsorbate binding energy) and delta (δ, the surface
adsorbate binding energy common to two
intermediates). High gamma catalytic systems (γ >
1) achieve surface resonance between an oscillating
external stimulus and an oscillating surface reaction
by selecting a catalyst and dynamic conditions that
maintain two oscillating states of A* and B*,
respectively. Alternatively, low gamma catalytic
systems (γ < 1) require one of the two oscillating
states to consist of a completely bare surface (open
sites only). The two different dynamic catalytic
behaviors are visually apparent in their Sabatier
volcano shapes; superposition of numerous
Sabatier volcanoes with varying γ comprise a
‘supervolcano’ where the regime of surface-
reaction rate control is visible as deviation from the
superVolcano border. Dynamic catalysis was also
show to serve as a ‘catalytic molecular pump’ by
altering the binding energy of surface adsorbate
species via external work. The extent of work
converted into catalytic conversion deviating from
equilibrium was defined as the dynamic catalytic
efficiency. For both low and high γ catalytic
systems, conditions of efficient catalytic promotion
as high as 30-40% were identified as a strong
function of the selected dynamic oscillation
conditions of frequency, amplitude, and waveform
type.
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
that contributed to the research results reported
within this paper. URL: http://www.msi.umn.edu/
We acknowledge helpful discussions with
Professors Dan Frisbie, Michael Tsapatsis, and
Dionisios Vlachos.
Keywords. Catalysis, Sabatier, Dynamics,
Frequency, Resonance, Volcano, Ammonia
Supporting Information. Additional information
including computer code, time-on-stream data, and
simulation methods are included in the supporting
information.
____________________________________________________________________________ Ardagh, et al. Page 20
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Ardagh, et al. Supplementary Information Page S1
Electronic Supplementary Information
Catalytic Resonance Theory:
SuperVolcanoes, Catalytic Molecular Pumps, and
Oscillatory Steady State
M. Alexander Ardagh1,2, Turan Birol1, Qi Zhang1,
Omar A. Abdelrahman3, Paul J. Dauenhauer1,2*
1
University 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, Colburn
Laboratory, Newark, DE 19716.
3University of Massachusetts Amherst, Department of Chemical Engineering, 686 N. Pleasant Street,
Amherst, MA 01003.
*Corresponding Author: [email protected]
# of Figures: 2
# of Tables: 8
# of Equations: 6
Table of Contents:
Section S1. Matlab 2017b and 2019a Code
● CSTR Model
● Batch Reactor Model
● Dynamic Catalysis CSTR Shell Code with Square Waveform
● Dynamic Catalysis Batch Reactor Shell Code with Square or Triangle Waveform
● Dynamic Catalysis Batch Reactor Shell Code with Sinusoidal Waveform
Section S2. Matlab ODE Solver Performance
● Static Catalysis
● Dynamic Catalysis
● Solvers:
○ ODE45
○ ODE15s
○ ODE23s
○ ODE23t
○ ODE23tb
○ Radau
Section S3. Static Catalysis Time on Stream Data
● CSTR at 1 % Yield of B
● Batch reactor at 50 % Yield of B
Section S4. Data from Heatmap Figures
Ardagh, et al. Supplementary Information Page S2
Section S5. Binding Energy Derivation
Section S1. Matlab 2017b and 2019a Code
CSTR Model
% CSTR model
% Description:
% Continuously Stirred Tank Reactor (CSTR) with reaction A -> B.
% The binding energies of A and B are the controls.
function xdot = cstr1(t,x)
global Rg R TK Caf Cbf qdot w u delHovr alpha beta gamma delta BEb0 delBEb
% Input (1):
% Number of catalytic active sites (gmol)
Ns = u;
% States (2):
% Concentration of A in CSTR (M)
Ca = x(1,1);
% Concentration of B in CSTR (M)
Cb = x(2,1);
% Amount of adsorbed A in CSTR (gmol)
Ca_star = x(3,1);
% Amount of adsorbed B in CSTR (gmol)
Cb_star = x(4,1);
% Number of vacant sites in CSTR (gmol)
C_star = Ns - Ca_star - Cb_star;
% Convert units
% Pressure of A in CSTR (bar)
Pa = Ca*Rg*TK;
% Pressure of B in CSTR (bar)
Pb = Cb*Rg*TK;
% Surface coverage of A in CSTR (unitless)
Theta_A_star = Ca_star/Ns;
% Surface coverage of B in CSTR (unitless)
Theta_B_star = Cb_star/Ns;
% Surface coverage of * in CSTR (unitless)
Theta_star = C_star/Ns;
% Reactor parameters:
% Volume of CSTR
% Catalyst density (g/mL)
rho = 3.58;
% Bed void fraction (unitless)
epsilon = 0.375;
% Volume calculation (L)
V = (w/1000)*(1/rho)*(1/1000)*(1/(1-epsilon));
Ardagh, et al. Supplementary Information Page S3
% Reaction parameters:
% Binding energies
% Volcano x-axis zero point (eV)
BEa0 = (BEb0 - ((1-gamma)*delta) - (delHovr/96.485e3))/gamma;
% Binding energy of B (J/gmol)
BEb = (BEb0 + delBEb)*96.485e3;
% Restrict binding energy of B to positive values
if BEb < 0
BEb = 0;
End
% Binding energy of A (J/gmol)
BEa = (BEa0 + ((1/gamma)*delBEb))*96.485e3;
% Restrict binding energy of A to positive values
if BEa < 0
BEa = 0;
End
% Heats of reaction (J/gmol)
delH1 = -BEa; % A -> A*
delH2 = delHovr + BEa - BEb; % A* -> B*
delH3 = BEb; % B* -> B
% Ea - Activation energies in the Arrhenius equation (J/gmol)
Ea1f = 0e3; % A -> A*
Ea2f = alpha*delH2 + beta; % A* -> B*
% Restrict activation energy to positive values
if Ea2f < 0
Ea2f = 0;
end
Ea3f = delH3; % B* -> B
% Pre-exponential factors
A1f = 1e6; % 1/bar-sec
A2f = 1e13; % 1/sec
A3f = 1e13; % 1/sec
% Equilibrium constants
K1 = 1e-7*exp(-delH1/(R*TK)); % 1/bar
K2 = 1*exp(-delH2/(R*TK)); % unitless
K3 = 1e7*exp(-delH3/(R*TK)); % bar
% Rate constants
k1f = A1f*exp(-Ea1f/(R*TK)); % 1/bar-sec
k1r = k1f/K1; % 1/sec
k2f = A2f*exp(-Ea2f/(R*TK)); % 1/sec
k2r = k2f/K2; % 1/sec
k3f = A3f*exp(-Ea3f/(R*TK)); % 1/sec
k3r = k3f/K3; % 1/bar-sec
Ardagh, et al. Supplementary Information Page S4
% Compute xdot:
xdot(1,1) = (qdot/V*(Caf - Ca) - k1f*Pa*Theta_star*(Ns/V) + k1r*Theta_A_star*(Ns/V)); % M/sec
xdot(2,1) = (qdot/V*(Cbf - Cb) + k3f*Theta_B_star*(Ns/V) - k3r*Pb*Theta_star*(Ns/V)); % M/sec
xdot(3,1) = k1f*Pa*Theta_star*Ns - k1r*Theta_A_star*Ns - k2f*Theta_A_star*Ns +
k2r*Theta_B_star*Ns; % gmol/sec
xdot(4,1) = - k3f*Theta_B_star*Ns + k3r*Pb*Theta_star*Ns + k2f*Theta_A_star*Ns -
k2r*Theta_B_star*Ns; % gmol/sec
Batch Reactor Model
% Batch reactor model
% Description:
% Batch reactor with reaction A -> B.
% The binding energies of A and B are the controls.
function xdot = batch1(t,x)
global Rg R TK w u delHovr alpha beta gamma delta BEb0 delBEb
% Input (1):
% Number of catalytic active sites (gmol)
Ns = u;
% States (2):
% Concentration of A in batch reactor (M)
Ca = x(1,1);
% Concentration of B in batch reactor (M)
Cb = x(2,1);
% Amount of adsorbed A in batch reactor (gmol)
Ca_star = x(3,1);
% Amount of adsorbed B in batch reactor (gmol)
Cb_star = x(4,1);
% Number of vacant sites in batch reactor (gmol)
C_star = Ns - Ca_star - Cb_star;
% Convert units
% Pressure of A in batch reactor (bar)
Pa = Ca*Rg*TK;
% Pressure of B in batch reactor (bar)
Pb = Cb*Rg*TK;
% Surface coverage of A in batch reactor (unitless)
Theta_A_star = Ca_star/Ns;
% Surface coverage of B in batch reactor (unitless)
Theta_B_star = Cb_star/Ns;
% Surface coverage of * in batch reactor (unitless)
Theta_star = C_star/Ns;
% Reactor parameters:
% Volume of batch reactor
% Catalyst density (g/mL)
rho = 3.58;
Ardagh, et al. Supplementary Information Page S5
% Bed void fraction (unitless)
epsilon = 0.375;
% Volume calculation (L)
V = (w/1000)*(1/rho)*(1/1000)*(1/(1-epsilon));
% Reaction parameters:
% Binding energies
% Volcano x-axis zero point (eV)
BEa0 = (BEb0 - ((1-gamma)*delta) - (delHovr/96.485e3))/gamma;
% Binding energy of B (J/gmol)
BEb = (BEb0 + delBEb)*96.485e3;
% Restrict binding energy of B to positive values
if BEb < 0
BEb = 0;
End
% Binding energy of A (J/gmol)
BEa = (BEa0 + ((1/gamma)*delBEb))*96.485e3;
% Restrict binding energy of A to positive values
if BEa < 0
BEa = 0;
End
% Heats of reaction (J/gmol)
delH1 = -BEa; % A -> A*
delH2 = delHovr + BEa - BEb; % A* -> B*
delH3 = BEb; % B* -> B
% Ea - Activation energies in the Arrhenius equation (J/gmol)
Ea1f = 0e3; % A -> A*
Ea2f = alpha*delH2 + beta; % A* -> B*
% Restrict activation energy to positive values
if Ea2f < 0
Ea2f = 0;
end
Ea3f = delH3; % B* -> B
% Pre-exponential factors
A1f = 1e6; % 1/bar-sec
A2f = 1e13; % 1/sec
A3f = 1e13; % 1/sec
% Equilibrium constants
K1 = 1e-7*exp(-delH1/(R*TK)); % 1/bar
K2 = 1*exp(-delH2/(R*TK)); % unitless
K3 = 1e7*exp(-delH3/(R*TK)); % bar
% Rate constants
k1f = A1f*exp(-Ea1f/(R*TK)); % 1/bar-sec
k1r = k1f/K1; % 1/sec
Ardagh, et al. Supplementary Information Page S6
k2f = A2f*exp(-Ea2f/(R*TK)); % 1/sec
k2r = k2f/K2; % 1/sec
k3f = A3f*exp(-Ea3f/(R*TK)); % 1/sec
k3r = k3f/K3; % 1/bar-sec
% Compute xdot:
xdot(1,1) = -k1f*Pa*Theta_star*(Ns/V) + k1r*Theta_A_star*(Ns/V); % M/sec
xdot(2,1) = k3f*Theta_B_star*(Ns/V) - k3r*Pb*Theta_star*(Ns/V); % M/sec
xdot(3,1) = k1f*Pa*Theta_star*Ns - k1r*Theta_A_star*Ns - k2f*Theta_A_star*Ns +
k2r*Theta_B_star*Ns; % gmol/sec
xdot(4,1) = - k3f*Theta_B_star*Ns + k3r*Pb*Theta_star*Ns + k2f*Theta_A_star*Ns -
k2r*Theta_B_star*Ns; % gmol/sec
Dynamic Catalysis CSTR Shell Code with Square Waveform
% Remove prior data and runs
clear
clc
% Step test for Model 1 - CSTR
global Rg R TK Caf Cbf qdot w u delHovr alpha beta gamma delta BEb0 delBEb
% Constants:
% Gas constant
Rg = 8.314459848e-2; % L-bar/K-gmol
R = 8.314459848; % J/K-gmol
% Reactor parameters:
% Temperature
Tc = 150; % deg C
TK = Tc + 273.15; % K
% Feed pressure (bar)
Paf = 100;
Pbf = 0;
% Feed concentration (M)
Caf = Paf/(Rg*TK);
Cbf = Pbf/(Rg*TK);
% Volumetric flowrate
qsdot = 50; % mL/min
qdot = qsdot/60000; % L/sec
% 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];
Ardagh, et al. Supplementary Information Page S7
% Steady state initial condition for the control
u_ss = 0; % gmol active sites
% Open loop step change
% Catalyst weight (mg)
w = 138;
% Active site density (gmol/g catalyst)
srho = 20e-6;
% Active site calculation (gmol)
u = (w/1000)*srho;
% Reaction parameters:
% Overall heat of reaction for A -> B (J/gmol)
delHovr = 0e3;
% Brønsted-Evans-Polanyi relationship parameters
alpha = 0.6; % unitless
beta = 102e3; % J/gmol
% Binding energy relationship parameters
gamma = 0.5; % unitless
delta = 1.4; % eV
% Volcano plot parameter:
% x-axis zero point (eV)
BEb0 = delta + delHovr/96.485e3;
% Volcano plot peak calculation
% Iterate binding energies of A and B (eV)
is = -delta;
ii = 0.01;
ie = delta;
% Preallocate BEb, TOF, and surface coverage matrices
je = ((ie - is)/ii + 1); % final index
delBEbv = zeros(round(je),1); % eV
TOFbvf = zeros(round(je),1); % 1/sec
Theta_A_starvf = zeros(round(je),1); % unitless
Theta_B_starvf = zeros(round(je),1); % unitless
% Begin iteration (eV)
for i = is:ii:ie
% Set binding energy of B (eV)
delBEb = i; % eV
% Iterate at each binding energy
for n = 1:inf % index
% Final Time (sec)
tfv = 5e100;
% Generate CSTR data
opts = odeset('RelTol’,1e-8,’AbsTol’,1e-9);
[tv,xv] = ode15s('cstr1',[0 tfv],x_ss,opts); % Time and state outputs
% Parse out the state values (M)
Ardagh, et al. Supplementary Information Page S8
Cav = xv(:,1);
Cbv = xv(:,2);
% Measure reactor performance
Ybv = (Cbv./(Cav + Cbv))*100; % mol %
% Converge on C yield of B (mol %)
C = 1;
if abs(Ybv(end) - C) > 0.01 % Convergence criterion
qdot = qdot*Ybv(end)/C; % L/sec
nv = n; % Indicates that volcano criterion was not met
nv; % Display iteration count
clear tv xv Cav Cbv Ybv
else
% Compute matrix index
j = ((i - is)/ii) + 1; % unitless
% Parse out state values (gmol)
Ca_starv = xv(:,3);
Cb_starv = xv(:,4);
% Measure reactor performance
TOFbv = Cbv*qdot/u; % 1/sec
% Convert units
Theta_A_starv = Ca_starv/u; % unitless
Theta_B_starv = Cb_starv/u; % unitless
% Store volcano plot results
delBEbv(round(j)) = i; % eV
TOFbvf(round(j)) = TOFbv(end); % 1/sec
Theta_A_starvf(round(j)) = Theta_A_starv(end); % unitless
Theta_B_starvf(round(j)) = Theta_B_starv(end); % unitless
clear tv xv Cav Cbv Ybv Ca_starv Cb_starv TOFbv Theta_A_starv Theta_B_starv
break
end
end
end
% Extract volcano plot peak information
peakTOF = max(TOFbvf); % 1/sec
% Preallocate relative binding energy matrix (eV)
peakdelBEb = zeros(round(je),1);
% Iterate relative binding energy of B (eV)
for k = 1:je % index
if abs(TOFbvf(k) - peakTOF) < peakTOF/100 % Convergence criterion
peakdelBEb(k) = delBEbv(k); % Store relative binding energy
end
end
Ardagh, et al. Supplementary Information Page S9
% Calculate volcano plot peak position (eV)
peakdelBEb = mean(nonzeros(peakdelBEb));
% Display volcano plot peak results
peakdelBEb; % eV
peakTOF; % 1/sec
% Transition to dynamic catalysis
% Dynamic catalysis parameters
% Oscillation frequency
tau2 = 5e-4; % sec
taur = 1.0; % unitless
tau1 = taur*tau2; % sec
fosc = 1/(tau1 + tau2); % Hz
% Number of oscillations (unitless)
Nosc = round(fosc) + 1;
% Set minimum number of oscillations
if Nosc < 11
Nosc = 11;
end
% Oscillation endpoints (eV)
delBEb0 = peakdelBEb;
delU = 0.6;
UR = delBEb0 + delU/2.0;
UL = UR - delU;
% Static catalysis calculation
% Starting time
ts = 0;
% Final time
te = 5e3;
% Generate CSTR data
% Reset volumetric flowrate (L/sec)
qdot = qsdot/60000;
% Static catalysis at volcano plot peak
delBEb = delBEb0; % eV
% Start static catalysis timer (sec)
tic
% Iterate until convergence
for n = 1:inf % index
% Generate time span matrix (sec)
% Time interval
ti = te/40;
% Time span matrix
tspan = ts:ti:te;
[t,x] = ode15s('cstr1',tspan,x_ss,opts); % Time and state outputs
Ardagh, et al. Supplementary Information Page S10
% Parse out the state values
Ca(:,1) = x(:,1); % M
Cb(:,1) = x(:,2); % M
% Measure reactor performance
Yb = (Cb./(Ca + Cb))*100; % mol %
% Converge on C yield of B (mol %)
if abs(Yb(end) - C) > 0.01 % Convergence criterion
qdot = qdot*Yb(end)/C; % L/sec
ns = n; % Indicates that static catalysis criterion was not met
ns; % Display iteration count
clear t x Ca Cb Yb
else
% Stop static catalysis timer (sec)
toc
% Store static catalysis results
tsv(:,1) = t; % sec
Ca_star(:,1) = x(:,3); % gmol
Cb_star(:,1) = x(:,4); % gmol
TOFb = Cb*qdot/u; % 1/sec
% Remove static catalysis data and run
clear t x
% Reset volumetric flowrate (L/sec)
qdot = qsdot/60000;
break
end
end
% Begin dynamic catalysis
% Initial conditions for dynamic catalysis
x_ss(:,2) = [Ca(end,1);Cb(end,1);Ca_star(end,1);Cb_star(end,1)];
% Start dynamic catalysis timer (sec)
tic
% Iterate until convergence
for n = 1:inf % index
% Begin dynamic catalysis
for l = 2:Nosc % index
% Even numbered oscillations
if mod(l,2) == 0
% Operate at strong binding (eV)
delBEb = UR;
Ardagh, et al. Supplementary Information Page S11
% Construct new time span matrix (sec)
% Final time
te = tau1;
% Time interval
ti = te/40;
% Time span matrix
tspan = ts:ti:te;
% Generate CSTR data
[t,x] = ode15s('cstr1',tspan,x_ss(:,l),opts); % Time and state outputs
% Parse out the state values
tsv(:,l) = t + tsv(end,l-1); % sec
Ca(:,l) = x(:,1); % M
Cb(:,l) = x(:,2); % M
Ca_star(:,l) = x(:,3); % gmol
Cb_star(:,l) = x(:,4); % gmol
% Initial conditions for the next oscillation
x_ss(:,l+1) = [Ca(end,l);Cb(end,l);Ca_star(end,l);Cb_star(end,l)];
clear t x
% Odd numbered oscillations
else
% Operate at weak binding (eV)
delBEb = UL;
% Construct new time span matrix (sec)
% Final time
te = tau2;
% Time interval
ti = te/40;
% Time span matrix
tspan = ts:ti:te;
% Generate CSTR data
[t,x] = ode15s('cstr1',tspan,x_ss(:,l),opts); % Time and state outputs
% Parse out the state values
tsv(:,l) = t + tsv(end,l-1); % sec
Ca(:,l) = x(:,1); % M
Cb(:,l) = x(:,2); % M
Ca_star(:,l) = x(:,3); % gmol
Cb_star(:,l) = x(:,4); % gmol
% Initial conditions for the next oscillation
x_ss(:,l+1) = [Ca(end,l);Cb(end,l);Ca_star(end,l);Cb_star(end,l)];
clear t x
end
end
% End dynamic catalysis
% Calculate state values (gmol)
Ardagh, et al. Supplementary Information Page S12
C_star = u - Ca_star - Cb_star;
% Convert units
Pa = Ca*Rg*TK; % bar
Pb = Cb*Rg*TK; % bar
Theta_A_star = Ca_star/u; % unitless
Theta_B_star = Cb_star/u; % unitless
Theta_star = C_star/u; % unitless
% Measure reactor performance
Yb = (Cb./(Ca + Cb))*100; % mol %
% Preallocate TOF matrix
TOFb = [TOFb(:,1),zeros((te/ti)+1,Nosc-1)]; % 1/sec
TOFb(:,2:end) = Cb(:,2:end)*qdot/u; % 1/sec
% Converge on C yield of B (mol %)
% Data at the end of the run
% Time data (sec)
tdiff = [tsv(:,end-1);tsv(:,end)];
% Yield data (mol %)
Ybdiff = [Yb(:,end-1);Yb(:,end)];
% Data in the middle of the run
% Time data (sec)
tdiff2 = [tsv(:,(end-1)/2);tsv(:,(end-1)/2+1)];
% Yield data (mol %)
Ybdiff2 = [Yb(:,(end-1)/2);Yb(:,(end-1)/2+1)];
% Integrate yield of B over 1 oscillation period
% Preallocate integral matrix (sec*mol %)
Ybint = zeros(size(Ybdiff));
% Integration with midpoint Riemann sum
for m = 2:size(tdiff,1)
Ybint(m) = (tdiff(m) - tdiff(m-1))*mean([Ybdiff(m-1);Ybdiff(m)]);
End
% Preallocate integral matrix (sec*mol %)
Ybint2 = zeros(size(Ybdiff2));
% Integration with midpoint Riemann sum
for p = 2:size(tdiff2,1)
Ybint2(p) = (tdiff2(p) - tdiff2(p-1))*mean([Ybdiff2(p-1);Ybdiff2(p)]);
end
% Calculate time-averaged yield of B (mol %)
Ybavg = sum(Ybint)*fosc;
Ybavg2 = sum(Ybint2)*fosc;
% Check for convergence
if abs(Ybavg - Ybavg2) > min([Ybavg,Ybavg2])/100 % Convergence criterion 1
Nosc = round(2*Nosc) + 1; % Adjust number of oscillations
Ardagh, et al. Supplementary Information Page S13
nn = n; % Indicates that criterion 1 was not met
nn; % Display iteration number
clear tsv(:,2:end) Ca(:,2:end) Cb(:,2:end) Ca_star(:,2:end) Cb_star(:,2:end) x_ss(:,3:end)
else
if abs(Ybavg - C) > 0.01 % Convergence criterion 2
qdot = qdot*Ybavg/C; % Adjust volumetric flowrate
nq = n; % Indicates that criterion 2 was not met
nq; % Display iteration number
clear tsv(:,2:end) Ca(:,2:end) Cb(:,2:end) Ca_star(:,2:end) Cb_star(:,2:end) x_ss(:,3:end)
else
% Stop dynamic catalysis timer (sec)
toc
break
end
end
end
% Integrate TOF over 1 oscillation period
TOFbdiff = [TOFb(:,end-1);TOFb(:,end)];
% Preallocate TOF integral matrix (unitless)
TOFbint = zeros(size(TOFbdiff));
% Integration with midpoint Riemann sum
for m = 2:size(tdiff,1)
TOFbint(m) = (tdiff(m) - tdiff(m-1))*mean([TOFbdiff(m-1);TOFbdiff(m)]);
end
% Calculate the time-averaged TOF (1/sec)
TOFbavg = sum(TOFbint)*fosc;
% Display final results
plot(tsv,Yb) % Visually check for convergence
TOFbavg; % 1/sec
Dynamic Catalysis Batch Reactor Shell Code with Square or Triangle Waveform
% Remove prior data and runs
clear
clc
% Step test for models - CSTR and Batch Reactor
global Rg R TK Caf Cbf qdot w u delHovr alpha beta gamma delta BEb0 delBEb
% Constants:
% Gas constant
Rg = 8.314459848e-2; % L-bar/K-gmol
R = 8.314459848; % J/K-gmol
% Reactor parameters:
% Temperature
Tc = 150; % deg C
Ardagh, et al. Supplementary Information Page S14
TK = Tc + 273.15; % K
% Feed pressure (bar)
Paf = 100;
Pbf = 0;
% Feed concentration (M)
Caf = Paf/(Rg*TK);
Cbf = Pbf/(Rg*TK);
% Volumetric flowrate
qsdot = 50; % mL/min
qdot = qsdot/60000; % L/sec
% 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];
% Steady state initial condition for the control
u_ss = 0; % gmol active sites
% Open loop step change
% Catalyst weight (mg)
w = 138;
% Active site density (gmol/g catalyst)
srho = 20e-6;
% Active site calculation (gmol)
u = (w/1000)*srho;
% Reaction parameters:
% Overall heat of reaction for A -> B (J/gmol)
delHovr = 0e3;
% Brønsted-Evans-Polanyi relationship parameters
alpha = 0.6; % unitless
beta = 102e3; % J/gmol
% Binding energy relationship parameters
gamma = 0.5; % unitless
delta = 1.4; % eV
% Volcano plot parameter:
% x-axis zero point (eV)
BEb0 = delta + delHovr/96.485e3;
% Volcano plot peak calculation
% Iterate binding energies of A and B (eV)
is = -delta;
ii = 0.01;
ie = delta;
% Preallocate BEb, TOF, and surface coverage matrices
je = ((ie - is)/ii + 1); % final index
Ardagh, et al. Supplementary Information Page S15
delBEbv = zeros(round(je),1); % eV
TOFbvf = zeros(round(je),1); % 1/sec
Theta_A_starvf = zeros(round(je),1); % unitless
Theta_B_starvf = zeros(round(je),1); % unitless
% Begin iteration (eV)
for i = is:ii:ie
% Set binding energy of B (eV)
delBEb = i; % eV
% Iterate at each binding energy
for n = 1:inf % index
% Final Time (sec)
tfv = 5e100;
% Generate CSTR data
opts = odeset('RelTol',1e-8,'AbsTol',1e-9);
[tv,xv] = ode15s('cstr1',[0 tfv],x_ss,opts); % Time and state outputs
% Parse out the state values (M)
Cav = xv(:,1);
Cbv = xv(:,2);
% Measure reactor performance
Ybv = (Cbv./(Cav + Cbv))*100; % mol %
% Converge on C yield of B (mol %)
C = 1;
if abs(Ybv(end) - C) > 0.01 % Convergence criterion
qdot = qdot*Ybv(end)/C; % L/sec
nv = n; % Indicates that volcano criterion was not met
nv; % Display iteration count
clear tv xv Cav Cbv Ybv
else
% Compute matrix index
j = ((i - is)/ii) + 1; % unitless
% Parse out state values (gmol)
Ca_starv = xv(:,3);
Cb_starv = xv(:,4);
% Measure reactor performance
TOFbv = Cbv*qdot/u; % 1/sec
% Convert units
Theta_A_starv = Ca_starv/u; % unitless
Theta_B_starv = Cb_starv/u; % unitless
% Store volcano plot results
delBEbv(round(j)) = i; % eV
TOFbvf(round(j)) = TOFbv(end); % 1/sec
Theta_A_starvf(round(j)) = Theta_A_starv(end); % unitless
Ardagh, et al. Supplementary Information Page S16
Theta_B_starvf(round(j)) = Theta_B_starv(end); % unitless
clear tv xv Cav Cbv Ybv Ca_starv Cb_starv TOFbv Theta_A_starv Theta_B_starv
break
end
end
end
% Extract volcano plot peak information
peakTOF = max(TOFbvf); % 1/sec
% Preallocate relative binding energy matrix (eV)
peakdelBEb = zeros(round(je),1);
% Iterate relative binding energy of B (eV)
for k = 1:je % index
if abs(TOFbvf(k) - peakTOF) < peakTOF/100 % Convergence criterion
peakdelBEb(k) = delBEbv(k); % Store relative binding energy
end
end
% Calculate volcano plot peak position (eV)
peakdelBEb = mean(nonzeros(peakdelBEb));
% Display volcano plot peak results
peakdelBEb; % eV
peakTOF; % 1/sec
% Transition to dynamic catalysis
% Dynamic catalysis parameters
% Oscillation frequency
tau2 = 5e0; % sec
taur = 1.0; % unitless
tau1 = taur*tau2; % sec
fosc = 1/(tau1 + tau2); % Hz
% Number of oscillations (unitless)
Nosc = round(fosc) + 1;
% Set minimum number of oscillations
if Nosc < 11
Nosc = 11;
end
% Oscillation endpoints (eV)
delBEb0 = peakdelBEb;
delU = 1.50;
UR = delBEb0 + delU/2.0;
UL = UR - delU;
% Static catalysis calculation
% Starting time (sec)
ts = 0;
% Final time (sec)
te = 5e3;
Ardagh, et al. Supplementary Information Page S17
% Generate batch reactor data
% Static catalysis at volcano plot peak
delBEb = delBEb0; % eV
% Start static catalysis timer (sec)
tic
% Iterate until convergence
for n = 1:inf % index
% Generate time span matrix (sec)
% Time interval
ti = te/40;
% Time span matrix
tspan = ts:ti:te;
[t,x] = ode15s('batch1',tspan,x_ss,opts); % Time and state outputs
% Parse out the state values
Ca(:,1) = x(:,1); % M
Cb(:,1) = x(:,2); % M
% Measure reactor performance
Yb = (Cb./(Ca + Cb))*100; % mol %
% Equilibrium yield calculation (mol %)
Yeq = exp(-delHovr/(R*TK))/(1+exp(-delHovr/(R*TK)))*100;
% Converge on equilibrium yield of B (mol %)
if abs(Yb(end) - Yeq) > 0.01 % Convergence criterion
te = te*2; % sec
ns = n; % Indicates that static catalysis criterion was not met
ns; % Display iteration count
clear t x Ca Cb Yb
else
% Stop static catalysis timer (sec)
toc
% Store static catalysis results
tsv(:,1) = t; % sec
Ca_star(:,1) = x(:,3); % gmol
Cb_star(:,1) = x(:,4); % gmol
% Remove static catalysis data and run
clear t x
break
end
end
% Begin dynamic catalysis
Ardagh, et al. Supplementary Information Page S18
% Initial conditions for dynamic catalysis
% Feed pressure (bar)
Pafb = rand*(Paf + Pbf);
Pbfb = rand*(Paf + Pbf);
if (Pafb + Pbfb) > (Paf + Pbf)
Pbfb = Paf + Pbf - Pafb;
end
% Feed concentration (M)
Cafb = Pafb/(Rg*TK);
Cbfb = Pbfb/(Rg*TK);
% Surface coverage (unitless)
Ca_star_b = rand*u;
Cb_star_b = rand*u;
if (Ca_star_b + Cb_star_b) > u
Cb_star_b = u - Ca_star_b;
end
x_ss(:,2) = [Cafb;Cbfb;Ca_star_b;Cb_star_b];
% Start dynamic catalysis timer (sec)
tic
% Iterate until convergence
for n = 1:inf % index
% Begin dynamic catalysis
for l = 2:Nosc % index
% Even numbered oscillations
if mod(l,2) == 0
% Operate at strong binding (eV)
delBEb = UR;
% Construct new time span matrix (sec)
% Final time
te = tau1;
% Time interval
ti = te/40;
% Time span matrix
tspan = ts:ti:te;
% Generate batch reactor data
[t,x] = ode15s('batch1',tspan,x_ss(:,l),opts); % Time and state outputs
% Parse out the state values
tsv(:,l) = t + tsv(end,l-1); % sec
Ca(:,l) = x(:,1); % M
Cb(:,l) = x(:,2); % M
Ca_star(:,l) = x(:,3); % gmol
Cb_star(:,l) = x(:,4); % gmol
% Initial conditions for the next oscillation
Ardagh, et al. Supplementary Information Page S19
x_ss(:,l+1) = [Ca(end,l);Cb(end,l);Ca_star(end,l);Cb_star(end,l)];
clear t x
% Odd numbered oscillations
else
% Operate at weak binding (eV)
delBEb = UL;
% Construct new time span matrix (sec)
% Final time
te = tau2;
% Time interval
ti = te/40;
% Time span matrix
tspan = ts:ti:te;
% Generate batch reactor data
[t,x] = ode15s('batch1',tspan,x_ss(:,l),opts); % Time and state outputs
% Parse out the state values
tsv(:,l) = t + tsv(end,l-1); % sec
Ca(:,l) = x(:,1); % M
Cb(:,l) = x(:,2); % M
Ca_star(:,l) = x(:,3); % gmol
Cb_star(:,l) = x(:,4); % gmol
% Initial conditions for the next oscillation
x_ss(:,l+1) = [Ca(end,l);Cb(end,l);Ca_star(end,l);Cb_star(end,l)];
clear t x
end
end
% End dynamic catalysis
% Calculate state values (gmol)
C_star = u - Ca_star - Cb_star;
% Convert units
Pa = Ca*Rg*TK; % bar
Pb = Cb*Rg*TK; % bar
Theta_A_star = Ca_star/u; % unitless
Theta_B_star = Cb_star/u; % unitless
Theta_star = C_star/u; % unitless
% Measure reactor performance
Yb = (Cb./(Ca + Cb))*100; % mol %
% Converge on steady state yield of B (mol %)
% Data at the end of the run
% Time data (sec)
tdiff = [tsv(:,end-1);tsv(:,end)];
% Yield data (mol %)
Ybdiff = [Yb(:,end-1);Yb(:,end)];
Ardagh, et al. Supplementary Information Page S20
% Data in the middle of the run
% Time data (sec)
tdiff2 = [tsv(:,(end-1)/2);tsv(:,(end-1)/2+1)];
% Yield data (mol %)
Ybdiff2 = [Yb(:,(end-1)/2);Yb(:,(end-1)/2+1)];
% Integrate yield of B over 1 oscillation period
% Preallocate integral matrix (sec*mol %)
Ybint = zeros(size(Ybdiff));
% Integration with midpoint Riemann sum
for m = 2:size(tdiff,1)
Ybint(m) = (tdiff(m) - tdiff(m-1))*mean([Ybdiff(m-1);Ybdiff(m)]);
End
% Preallocate integral matrix (sec*mol %)
Ybint2 = zeros(size(Ybdiff2));
% Integration with midpoint Riemann sum
for p = 2:size(tdiff2,1)
Ybint2(p) = (tdiff2(p) - tdiff2(p-1))*mean([Ybdiff2(p-1);Ybdiff2(p)]);
end
% Calculate time-averaged yield of B (mol %)
Ybavg = sum(Ybint)*fosc;
Ybavg2 = sum(Ybint2)*fosc;
% Check for convergence
if abs(Ybavg - Ybavg2) > 0.01 % Convergence criterion
Nosc = round(2*Nosc) + 1; % Adjust number of oscillations
nn = n; % Indicates that dynamic catalysis criterion was not met
nn; % Display iteration number
clear tsv(:,2:end) Ca(:,2:end) Cb(:,2:end) Ca_star(:,2:end) Cb_star(:,2:end) x_ss(:,3:end)
else
% Stop dynamic catalysis timer (sec)
toc
break
end
end
% Calculate ddG (kJ/gmol)
ddG = -(log(Ybavg/(100-Ybavg)) - log(Yeq/(100-Yeq)))*R*TK/1000;
% Calculate efficiency (%)
eff = abs(ddG/(delU*96.485))*100;
% Display final results
plot(tsv,Yb) % Visually check for convergence
Ybavg; % mol %
ddG; % kJ/gmol
eff; % %
Ardagh, et al. Supplementary Information Page S21
Dynamic Catalysis Batch Reactor Shell Code with Sinusoidal Waveform
% Remove prior data and runs
clear
clc
% Step test for models - CSTR and batch reactor
global Rg R TK Caf Cbf qdot w u delHovr alpha beta gamma delta BEb0 delBEb tau UR UL tdel
% Constants:
% Gas constant
Rg = 8.314459848e-2; % L-bar/K-gmol
R = 8.314459848; % J/K-gmol
% Reactor parameters:
% Temperature
Tc = 250; % deg C
TK = Tc + 273.15; % K
% Feed pressure (bar)
Paf = 100;
Pbf = 0;
% Feed concentration (M)
Caf = Paf/(Rg*TK);
Cbf = Pbf/(Rg*TK);
% Volumetric flowrate
qsdot = 50; % mL/min
qdot = qsdot/60000; % L/sec
% 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];
% Steady state initial condition for the control
u_ss = 0; % gmol active sites
% Open loop step change
% Catalyst weight (mg)
w = 138;
% Active site density (gmol/g catalyst)
srho = 20e-6;
% Active site calculation (gmol)
u = (w/1000)*srho;
% Reaction parameters:
% Overall heat of reaction for A -> B (J/gmol)
delHovr = 0e3;
Ardagh, et al. Supplementary Information Page S22
% Brønsted-Evans-Polanyi relationship parameters
alpha = 0.6; % unitless
beta = 135e3; % J/gmol
% Binding energy relationship parameters
gamma = 2.0; % unitless
delta = 1.4; % eV
% Volcano plot parameter:
% x-axis zero point (eV)
BEb0 = delta + (delHovr/96.485e3);
% Volcano plot peak calculation
% Iterate binding energies of A and B (eV)
is = -delta;
ii = 0.01;
ie = delta;
% Preallocate BEb, TOF, and surface coverage matrices
je = ((ie - is)/ii + 1); % final index
delBEbv = zeros(round(je),1); % eV
TOFbvf = zeros(round(je),1); % 1/sec
Theta_A_starvf = zeros(round(je),1); % unitless
Theta_B_starvf = zeros(round(je),1); % unitless
% Begin iteration (eV)
for i = is:ii:ie
% Set binding energy of B (eV)
delBEb = i; % eV
% Iterate at each binding energy
for n = 1:inf % index
% Final Time (sec)
tfv = 5e100;
% Generate CSTR data
options = odeset('RelTol',1e-8,'AbsTol',1e-9);
[tv,xv] = ode15s('cstr1',[0 tfv],x_ss,options); % Time and state outputs
% Parse out the state values (M)
Cav = xv(:,1);
Cbv = xv(:,2);
% Measure reactor performance
Ybv = (Cbv./(Cav + Cbv))*100; % mol %
% Converge on C yield of B (mol %)
C = 1;
if abs(Ybv(end) - C) > 0.01 % Convergence criterion
qdot = qdot*Ybv(end)/C; % L/sec
nv = n; % Indicates that volcano criterion was not met
nv; % Display iteration count
clear tv xv Cav Cbv Ybv
else
% Compute matrix index
Ardagh, et al. Supplementary Information Page S23
j = ((i - is)/ii) + 1; % unitless
% Parse out state values (gmol)
Ca_starv = xv(:,3);
Cb_starv = xv(:,4);
% Measure reactor performance
TOFbv = Cbv*qdot/u; % 1/sec
% Convert units
Theta_A_starv = Ca_starv/u; % unitless
Theta_B_starv = Cb_starv/u; % unitless
% Store volcano plot results
delBEbv(round(j)) = i; % eV
TOFbvf(round(j)) = TOFbv(end); % 1/sec
Theta_A_starvf(round(j)) = Theta_A_starv(end); % unitless
Theta_B_starvf(round(j)) = Theta_B_starv(end); % unitless
clear tv xv Cav Cbv Ybv Ca_starv Cb_starv TOFbv Theta_A_starv Theta_B_starv
break
end
end
end
% Extract volcano plot peak information
peakTOF = max(TOFbvf); % 1/sec
% Preallocate relative binding energy matrix (eV)
peakdelBEb = zeros(round(je),1);
% Iterate relative binding energy of B (eV)
for k = 1:je % index
if abs(TOFbvf(k) - peakTOF) < peakTOF/100 % Convergence criterion
peakdelBEb(k) = delBEbv(k); % Store relative binding energy
end
end
% Calculate volcano plot peak position (eV)
peakdelBEb = mean(nonzeros(peakdelBEb));
% Display volcano plot peak results
peakdelBEb; % eV
peakTOF; % 1/sec
% Transition to dynamic catalysis
% Dynamic catalysis parameters
% Oscillation frequency
tau = 1e0; % sec
fosc = 1/tau; % Hz
% Number of oscillations (unitless)
Nosc = fosc;
Ardagh, et al. Supplementary Information Page S24
if Nosc < 11
Nosc = 11;
end
% Dynamic time span (sec)
tdyn = tau*Nosc*10;
% Oscillation endpoints (eV)
delBEb0 = peakdelBEb;
delU = 0.6;
UR = delBEb0 + delU/2.0;
UL = UR - delU;
% Equilibrium yield calculation (mol %)
Yeq = exp(-delHovr/(R*TK))/(1+exp(-delHovr/(R*TK)))*100;
% Begin dynamic catalysis
% Start dynamic catalysis timer (sec)
tic
% Batch reactor initial conditions
% Feed pressure (bar)
Pbfb = rand*(Paf + Pbf);
Pafb = Paf + Pbf - Pbfb;
% Feed concentration (M)
Cafb = Pafb/(Rg*TK);
Cbfb = Pbfb/(Rg*TK);
% Surface coverage (gmol)
Ca_star_ssb = rand*u;
Cb_star_ssb = rand*u;
if (Ca_star_ssb + Cb_star_ssb) > u
Cb_star_ssb = u - Ca_star_ssb;
End
% Initial condition array
x_ssb = [Cafb;Cbfb;Ca_star_ssb;Cb_star_ssb];
% Generate batch reactor data
tdel = rand*tau;
% Iterate until convergence
for n = 1:inf
[t,x] = ode15s('sin_batch1',[0 tdyn],x_ssb,options); % Time and state outputs
% Parse out the state values
Ca = x(:,1); % M
Cb = x(:,2); % M
% Measure reactor performance
Yb = (Cb./(Ca + Cb))*100; % mol %
Ardagh, et al. Supplementary Information Page S25
% Find convergence point
index = zeros(size(Yb));
for l = 1:size(Yb,1)
if abs(Yb(end)-Yb(l)) > 0.5
index(l) = l;
end
end
index = max(index);
if index > size(Yb,1)/2
tdyn = 2*tdyn;
else
break
end
end
% Stop dynamic catalysis timer (sec)
toc
% Parse out the state values
Ca_star = x(:,3); % gmol
Cb_star = x(:,4); % gmol
C_star = u - Ca_star - Cb_star; % gmol
% Convert units
Pa = Ca*Rg*TK; % bar
Pb = Cb*Rg*TK; % bar
Theta_A_star = Ca_star/u; % unitless
Theta_B_star = Cb_star/u; % unitless
Theta_star = C_star/u; % unitless
% Track binding energies
delBEbf = (((UR - UL)/2)*(cos(2*pi()*(t-tdel)/tau))) + ((UR + UL)/2);
% Display final results
plot(t,Yb) % Visually check for convergence
Ardagh, et al. Supplementary Information Page S26
Section S2. ODE Solver Selection and Justification
Matlab ODE solvers were screened for several different reactor types and catalytic systems under static
conditions or using a sinusoidal dynamic catalysis waveform. ODE45 is the recommended solver for
general use in Matlab, however, this solver failed to generate solutions for static and dynamic catalysis.
Static Catalysis
Stats for ode15s:
452 successful steps
58 failed attempts
1047 function evaluations
39 partial derivatives
131 LU decompositions
851 solutions of linear systems
Elapsed time is 0.159894 seconds.
Stats for ode23s:
2546 successful steps
1197 failed attempts
20218 function evaluations
2546 partial derivatives
3743 LU decompositions
11229 solutions of linear systems
Elapsed time is 1.949530 seconds.
Stats for ode23t:
1143 successful steps
8 failed attempts
2207 function evaluations
8 partial derivatives
106 LU decompositions
2166 solutions of linear systems
Elapsed time is 0.347212 seconds.
Stats for ode23tb:
1054 successful steps
8 failed attempts
2902 function evaluations
5 partial derivatives
99 LU decompositions
3934 solutions of linear systems
Elapsed time is 0.276370 seconds.
Stats for radau:
72 successful steps
72 failed attempts
1377 function evaluations
Ardagh, et al. Supplementary Information Page S27
147 LU decompositions
397 solutions of linear systems
Elapsed time is 0.210526 seconds.
ODE23t and ODE23tb have the least number of failed attempts, but Radau performed most efficiently in
terms of number of steps and ODE15s in terms of time to generate the solution. Therefore, ODE15s was
used throughout this manuscript to solve CSTR and Batch Reactor equations under static catalysis
conditions.
CSTR with Gamma < 1.0 and Sinusoidal Waveform
ODE23s, ODE23tb, and Radau did not converge on a stable sinusoidal CSTR solution after 8 h of
computational time.
Stats for ode15s:
18819 successful steps
3682 failed attempts
46463 function evaluations
1291 partial derivatives
5906 LU decompositions
40007 solutions of linear systems
Elapsed time is 7.945276 seconds.
Stats for ode23t:
51424 successful steps
2625 failed attempts
102764 function evaluations
1671 partial derivatives
8781 LU decompositions
94408 solutions of linear systems
Elapsed time is 17.107914 seconds.
ODE23t and has the least number of failed attempts, but ODE15s performed most efficiently in terms of
number of steps and time to generate the solution. Therefore, ODE15s was used throughout this
manuscript to solve CSTR and Batch Reactor equations under dynamic catalysis conditions.
Ardagh, et al. Supplementary Information Page S28
Section S3. Static Catalysis Time on Stream Data
CSTR at 1 % Yield of B
Figure S1. Example CSTR time on stream data with A) gas phase concentration of A and B and
B) amount of adsorbed species A* and B*. Conditions: 150 oC, 100 bar, and 1 % yield of B.
Batch Reactor at 50 % Yield of B
Figure S2. Example Batch Reactor time on stream data with A) gas phase concentration of A
and B and B) amount of adsorbed species A* and B*. Conditions: 150 oC, 100 bar, and
equilibrium yield of B.
Ardagh, et al. Supplementary Information Page S29
Section S4. Data from Heatmap Figures
Table S1. CSTR TOF (1/s) heatmap data for gamma of 0.5 with varying oscillation endpoints
and frequency. Conditions: 100 oC, 100 bar, and 1 % yield of B. Reaction parameters: ΔHovr of 0
kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 0.5, and delta of 1.4 eV. Oscillation
amplitude of 1.03 eV.
0 -0.10 -0.21 -0.31 -0.41 -0.52 -0.62 -0.72 -0.82 -0.93 -1.03
0.01 0.0364 0.0256 0.0192 0.0176 0.0188 0.0252 0.0282 0.0281 0.0280 0.0280 0.0280
0.1 0.0364 0.0257 0.0200 0.0187 0.0209 0.0418 0.0644 0.0669 0.0669 0.0669 0.0669
1 0.0364 0 0.0199 0.0191 0.0315 0.1817 0.3434 0.3620 0.3629 0.3629 0.3629
10 0.0364 0 0 0 0.1385 1.629 3.294 3.486 3.495 3.495 3.495
100 0.0364 0 0 0 0.7139 9.540 16.98 17.65 17.68 17.68 17.68
1000 0.0364 0 0 0 1.020 13.88 24.02 24.84 24.88 24.88 24.88
10000 0.0364 0 0 0 0 14.51 25.00 25.83 25.87 25.87 25.87
100000 0.0364 0 0 0 0 14.55 25.06 25.89 25.93 25.93 25.93
Table S2. Batch reactor composition (mol % B) heatmap data for gamma of 2.0 with varying
oscillation endpoints and frequency. Conditions: 150 oC, 100 bar, and steady state yield of B.
Reaction parameters: ΔHovr of 0 kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 2.0, and
delta of 1.4 eV. Oscillation amplitude of 0.5 eV.
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.000001 50.00 50.00 50.00 50.00 50.01 50.01 50.02 50.01 50.00 50.00 50.00
0.00001 49.98 50.00 50.00 50.01 50.02 50.09 50.22 50.14 50.07 50.04 50.01
0.0001 49.99 50.00 50.00 50.01 50.06 50.66 52.61 52.73 51.24 50.44 50.21
0.001 49.99 50.00 50.00 50.03 50.49 55.91 69.00 69.14 63.25 57.27 53.31
0.01 49.98 49.95 50.01 50.31 54.79 87.24 93.69 92.80 88.42 81.79 71.86
0.1 49.99 49.92 50.10 53.06 91.80 99.02 99.18 98.79 98.29 96.97 94.09
1 49.17 49.26 50.44 77.73 99.65 99.84 99.79 99.61 99.52 99.34 98.61
10 26.00 35.62 59.77 96.98 99.78 99.95 99.94 99.91 99.88 99.88 99.90
100 18.43 27.60 66.32 97.21 99.79 99.96 99.97 99.97 99.97 99.97 99.97
Table S3. Batch reactor composition (mol % B) heatmap data for gamma of 2.0 with varying
oscillation endpoints and frequency. Conditions: 150 oC, 100 bar, and steady state yield of B.
Reaction parameters: ΔHovr of 0 kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 2.0, and
delta of 1.4 eV. Oscillation amplitude of 1.0 eV.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.000001 50.16 52.38 50.89 50.72 49.90 50.94 50.32 50.04 50.00 50.00 50.00
0.00001 49.34 49.79 50.03 49.40 51.90 59.85 54.01 50.76 50.10 50.01 50.00
0.0001 49.50 49.17 49.49 53.91 66.34 85.70 75.96 58.62 51.66 50.21 50.03
0.001 49.23 43.99 44.88 74.52 91.44 97.69 95.55 83.97 64.31 52.56 50.62
0.01 45.34 22.38 37.31 92.37 99.05 99.71 98.85 97.06 90.06 72.12 56.66
0.1 25.29 6.21 39.68 97.33 99.90 99.98 99.81 99.30 98.36 94.20 80.59
1 4.72 3.47 39.53 97.73 99.98 100.00 100.00 99.80 99.52 99.20 97.19
10 0.60 3.22 39.52 97.76 99.99 100.00 100.00 99.98 99.89 99.93 99.80
100 0.24 3.21 39.53 97.76 99.99 100.00 100.00 100.00 100.00 100.00 99.98
Ardagh, et al. Supplementary Information Page S30
Table S4. Batch reactor composition (mol % B) heatmap data for gamma of 2.0 with varying
oscillation endpoints and frequency. Conditions: 150 oC, 100 bar, and steady state yield of B.
Reaction parameters: ΔHovr of 0 kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 2.0, and
delta of 1.4 eV. Oscillation amplitude of 1.5 eV.
0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0
0.000001 49.69 49.65 50.96 50.12 47.07 51.05 50.16 50.94 50.03 50.00 50.00
0.00001 49.62 49.56 59.79 53.98 50.37 47.82 52.05 59.91 50.75 50.04 50.00
0.0001 49.59 47.26 83.50 84.69 57.65 52.96 66.47 84.29 58.40 50.58 50.04
0.001 48.85 34.51 87.83 95.17 82.83 71.89 91.50 97.66 83.91 57.41 50.60
0.01 45.46 16.37 88.55 99.29 97.63 94.39 99.11 99.53 97.37 82.14 56.66
0.1 25.45 11.89 88.65 99.71 99.74 99.38 99.91 99.98 99.20 97.01 80.59
1 4.70 11.38 88.66 99.79 99.98 99.96 99.99 100.00 99.78 99.29 97.18
10 0.60 11.34 88.66 99.79 99.99 99.99 100.00 100.00 99.99 99.88 99.80
100 0.24 11.34 88.66 99.79 100.00 100.00 100.00 100.00 100.00 100.00 99.98
Table S5. Batch reactor composition (mol % B) heatmap data for gamma of 0.5 with varying
oscillation endpoints and frequency. Conditions: 150 oC, 100 bar, and steady state yield of B.
Reaction parameters: ΔHovr of 0 kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 0.5, and
delta of 1.4 eV. Oscillation amplitude of 0.5 eV.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.000001 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00
0.00001 50.00 50.00 50.00 50.00 49.99 50.00 50.00 49.99 49.99 50.00 50.00
0.0001 50.02 50.01 50.00 50.00 50.00 49.98 49.93 49.89 49.88 49.91 49.93
0.001 50.03 50.01 50.00 50.00 49.98 49.82 49.18 48.31 47.95 48.42 49.07
0.01 50.05 50.02 50.01 50.00 49.84 48.21 42.55 36.49 34.06 36.21 40.22
0.1 50.05 50.02 50.01 50.00 48.39 34.44 16.94 10.54 8.884 10.47 14.46
1 50.06 50.02 50.01 50.00 34.60 5.573 2.340 1.447 1.363 1.663 2.249
10 50.06 50.02 50.01 50.00 6.778 1.030 0.5733 0.5133 0.5231 0.5665 0.5999
100 73.78 75.94 74.63 49.99 5.182 0.7511 0.4453 0.4232 0.4231 0.4270 0.4275
Table S6. Batch reactor composition (mol % B) heatmap data for gamma of 0.5 with varying
oscillation endpoints and frequency. Conditions: 150 oC, 100 bar, and steady state yield of B.
Reaction parameters: ΔHovr of 0 kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 0.5, and
delta of 1.4 eV. Oscillation amplitude of 1.0 eV.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.000001 50.23 50.09 50.01 50.00 50.00 50.00 50.00 49.99 50.00 50.00 50.00
0.00001 50.38 50.34 50.18 50.04 50.00 49.98 49.93 49.83 49.95 49.99 50.00
0.0001 50.39 50.39 50.42 50.53 49.92 49.67 48.74 47.04 49.16 49.80 49.94
0.001 50.40 50.45 51.57 55.31 48.64 45.23 37.92 29.56 40.77 46.69 49.03
0.01 50.45 51.25 61.15 74.50 40.67 23.71 11.94 6.554 15.18 28.23 39.89
0.1 50.98 58.15 86.67 88.18 29.33 4.267 1.654 1.140 2.470 6.000 14.06
1 55.70 82.71 97.77 90.43 26.52 0.7109 0.1376 0.1313 0.6546 1.166 2.177
10 77.49 97.48 99.25 90.64 26.29 0.4056 0.0190 0.0060 0.1439 0.3561 0.3556
Ardagh, et al. Supplementary Information Page S31
Table S7. Batch reactor composition (mol % B) heatmap data for gamma of 0.5 with varying
oscillation endpoints and frequency. Conditions: 150 oC, 100 bar, and steady state yield of B.
Reaction parameters: ΔHovr of 0 kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 0.5, and
delta of 1.4 eV. Oscillation amplitude of 1.5 eV.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.000001 50.23 50.09 50.01 50.00 50.00 50.00 50.00 49.99 50.00 50.00 50.00
0.00001 50.38 50.34 50.18 50.04 50.00 49.98 49.93 49.83 49.95 49.99 50.00
0.0001 50.39 50.39 50.42 50.53 49.92 49.67 48.74 47.04 49.16 49.80 49.94
0.001 50.40 50.45 51.57 55.31 48.64 45.23 37.92 29.56 40.77 46.69 49.03
0.01 50.45 51.25 61.15 74.50 40.67 23.71 11.94 6.554 15.18 28.23 39.89
0.1 50.98 58.15 86.67 88.18 29.33 4.267 1.654 1.140 2.470 6.000 14.06
1 55.70 82.71 97.77 90.43 26.52 0.7109 0.1376 0.1313 0.6546 1.166 2.177
10 77.49 97.48 99.25 90.64 26.29 0.4056 0.0190 0.0060 0.1439 0.3561 0.3556
Table S8. Batch reactor efficiency (%) heatmap data for gamma of 0.5 with varying oscillation
endpoints and amplitudes. Conditions: 150 oC, 100 bar, and steady state yield of B. Reaction
parameters: ΔHovr of 0 kJ/mol, alpha of 0.6, beta of 102 kJ/mol, gamma of 0.5, and delta of 1.4
eV. Oscillation frequency of 1 Hz.
0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50
0.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.15 -0.0002 -0.3224 -7.612 -12.61 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.30 0.0063 -0.2547 -10.52 -13.65 -7.409 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.45 0.0957 -0.1519 -13.02 -14.85 -9.846 -4.949 0.0000 0.0000 0.0000 0.0000 0.0000
0.60 0.7986 1.962 -12.27 -13.79 -10.44 -7.394 -3.712 0.0000 0.0000 0.0000 0.0000
0.75 1.102 10.88 -3.757 -15.71 -11.01 -9.258 -5.915 -2.970 0.0000 0.0000 0.0000
0.90 0.9411 11.04 6.624 -10.38 -13.63 -9.247 -7.573 -4.930 -2.482 0.0000 0.0000
1.05 0.7954 9.554 7.996 -3.502 -10.48 -11.81 -8.199 -6.612 -4.215 -2.121 0.0000
1.20 0.6960 8.368 7.056 -2.714 -7.385 -9.223 -10.07 -7.027 -5.786 -3.695 -1.856
1.35 0.6186 7.438 6.273 -2.407 -7.558 -6.842 -8.202 -8.949 -5.629 -5.143 -3.283
1.50 0.5568 6.694 5.646 -2.167 -7.130 -8.343 -6.162 -7.360 -8.183 -5.066 -4.629
Ardagh, et al. Supplementary Information Page S32
Section S5. Binding Energy Derivation
In this section, we demonstrate that the binding energies of A and B can be calculated at any
point by using the definitions of ΔHovr, gamma, and delta.
Definitions for gamma and delta are summarized in Equations S1 and S2:
𝛾 ≡∆𝐵𝐸𝐵
∆𝐵𝐸𝐴 (S1)
𝑤ℎ𝑒𝑛 𝐵𝐸𝐴 = 𝛿, 𝐵𝐸𝐵 = 𝛿 + ∆𝐻𝑜𝑣𝑟 (S2)
Equation S1 can be integrated with an indefinite integral to obtain Equation S3:
𝐵𝐸𝐵 = 𝛾𝐵𝐸𝐴 + 𝐶 (S3)
After plugging Equation S2 into Equation S3, the constant of integration is obtained:
𝐶 = (1 − 𝛾)𝛿 + ∆𝐻𝑜𝑣𝑟 (S4)
Equation S4 can therefore be substituted into Equation S3 to define a relationship between the
binding energies of A and B at any point.
𝐵𝐸𝐵 = 𝛾𝐵𝐸𝐴 + (1 − 𝛾)𝛿 + ∆𝐻𝑜𝑣𝑟 (S5)
Alternatively, Equation S5 may be rearranged to calculate the binding energy of A from a known
binding energy of B.
𝐵𝐸𝐴 = (𝐵𝐸𝐵 − (1 − 𝛾)𝛿 − ∆𝐻𝑜𝑣𝑟)/𝛾 (S6)
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![[Supporting Information] Polymers for Metal-free Catalytic ... filemeasured at 77 K under a Gemini 2375 surface area analyzer. 1 H nuclear magnetic resonance (NMR) spectra were recorded](https://img.dokumen.tips/doc/110x75/5d4efe9388c993790d8b5146/supporting-information-polymers-for-metal-free-catalytic-at-77-k-under-a-gemini.jpg)
