This is the peer reviewed version of the following article: Nature Catalysis 2, 971-976(2019), which
has been published in final form at https://www.nature.com/articles/s41929-019-0376-6. This article
may be used for non-commercial purposes in accordance with Macmillan Terms and Conditions for
Self-Archiving.
https://doi.org/10.1038/s41929-019-0376-6

PERSPECTIVE

Strategies to break linear scaling relationships
Javier Pérez-Ramírez

1

and Núria López

2

The search for new catalytic materials has relied on highly time- and human- resource-consuming procedures. The appearance
of theoretical methods that employ density functional theory coupled to kinetic models has allowed the rational understanding of activity volcano plots and selectivity abrupt profiles that resemble cliffs. However, these methodologies present several
drawbacks as the optimization is confined to a family of materials, typically metals, and not applied to the overall phase and
compositional space, therefore the maximum activity might not be sufficient for practical applications. Volcanos emerge from
the symmetry between the adsorption energies of different intermediates on the catalyst, and thus circumventing these dependencies is crucial to identify better catalytic materials. Here we present a revision of the advances in the field that indicate that
complexity in the materials is key to identifying alternative paths and thus overcome the drawbacks of scaling relationships.

T

he not-too-strong, not-too-weak bonding of key intermediates marks the sweet spot fingerprint of optimal catalytic
materials. This principle was introduced first by Sabatier, who
showed that the light-off temperature of catalysed reactions displayed a volcano shape as a function of formation energies1,2. Since
then, theoretical approaches to rational catalytic design have been
based on identifying the activity of volcanoes and how they depend
on a few, descriptors, typically the binding energies of (one or two)
intermediates3,4. With the appearance of the density functional theory (DFT) and wide-spread access to computers, simulations have
been established as one of the pillars in understanding and predicting catalysis. Nowadays, it is possible to implement robust DFT
techniques in computational codes, providing detailed electronic
structure, thermodynamic and kinetic parameters5 of increasingly
complex reaction networks. Therefore, computationally derived
volcano plots have become the gold standard in catalyst design6.
Volcanos provide a systematic way for catalytic optimization by
finely tuning the adsorption energies or through dynamic operation7, and have proven to be a powerful tool for assessing alloy composition8, size9, metal/oxide interfaces10, pH dependence11 and have
even been reported in homogeneous catalysts12. The same principles
can be used in electrocatalysis once the potential and pH terms are
introduced, and it is likely to appear in photochemistry, although
this has not been demonstrated yet.
However, there is an increasing pressure to go beyond the top of
these volcanoes13–15. There are several reasons behind this need. It is
well known that the maximum of the observed rate provides the limit
for the activity, and this might be insufficient for the practical application of catalytic technologies16. In addition, descriptors are typically energies, and thermodynamics for intermediates cumbersome
to obtain experimentally. Moreover, addressing selective processes
through volcanoes is not straightforward as several paths need to
be investigated with similar accuracy, and the respective volcanoes
need to be superimposed17. As for the practical aspects, the identified maximum does not provide a synthetic route18 towards the
desired composition. Finally, some false negatives (outliers) might
appear as the real phase space is much bigger than a single catalyst
family. In summary, the concept of volcano is practical for activity
although addresses selectivity and stability aspects poorly. The latter
have been solved by applying filters, for instance to assess that the
active phase can exist under reaction conditions. Volcanos are not

so useful for full-phase space discovery in their present formulation, but they are very convenient for catalyst optimization within
a family of compounds, for example, alloys and bimetallics. In the
following, we address the different approximations that have been
employed in the literature to alleviate the limitations of the volcanoes by increasing the complexity of materials and the chemical
variability. To this end, we first describe the appearance of the linear
scaling relationships (LSRs) and how these are transferred into the
volcano and cliff shapes once microkinetic balances are considered
and then provide a perspective on the strategies in the literature to
alter LSRs by increasing complexity.

The origin of volcanos and cliffs

To improve catalytic performance beyond the volcanos it is necessary to understand why they appear. The procedure is exemplified
in Fig. 1, where the particular model employed comprises a selective
process (A+1/2B→P) and a side path (A→Q), the models for which
can be accessed from ref. 19. The standard procedure is as follows: the
detailed reaction network is mapped carefully for at least one catalytic compound and the corresponding energies are calculated by
DFT, then a microkinetic model that wraps all the elementary steps
is developed. In a complex reaction with multiple intermediates, it
turns out that the bond order conservation theory and the electron
counting rules imply that the energies of intermediates are linked one
to another in a linear manner (tLSRs20, Fig. 1a). This is particularly
applicable to metals for which the smoothness of the electron cloud
helps address density fluctuations better. For metals, the local coordination number can also generate LSR and cLSR21. Brønsted–Evans–
Polanyi, as well as several others, found that the kinetic terms depend
linearly on thermodynamic parameters, and thus the activation barriers can be traced back to the energy of one or more intermediates
in a linear form (kinetic linear scaling relationships, kLSRs), Fig. 1b22.
These kLSRs are also known in the literature as linear free energy
relationships (LFER) or Bronsted–Evans–Polanyi (BEP) relationships. When the dependencies encoded in the LSRs are introduced in
the rate expression, activity shows the volcano shape as a function of
specific adsorption energies for key intermediates, as seen in Fig. 1c,
with the highest activity marked by an asterisk (see Fig. 1e for a
particular cut in a monodimensional space). Therefore, a strong
adsorption implies high coverage and surface poisoning, and a weak
adsorption entails very low coverage and thus low rate; in both cases

Institute for Chemical and Bioengineering, ETH Zurich, Zurich, Switzerland. 2Institute of Chemical Research of Catalonia, The Barcelona Institute of
Science and Technology, Tarragona, Spain. e-mail: jpr@chem.ethz.ch; nlopez@iciq.es

1

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971

PERSPECTIVE

NATURE CATALYSIS

a

1

A*

A(g)

Q*

B(g)

B*

b

0.0

β E I*
E B* = B
α+
E A* = A

2

P*

P(g)

1.5

0.5

–1.0

2

0.0
E a,3

–1.5

= α3

–1.0

EP* = αP eV

I*

+E 2

=α

2

E TS

– 0.5

–2.0
–2.0

3

Q*+I*

2I*

1.0

β A E Q*
ETS / eV

–0.5
E / eV

Q(g)

+ β3

∆E 3

1

=α

1

+E

A*

E TS

–1.5
–1.5

–1.0
E / eV

–0.5

0.0

c

–2.0

–1.5

–1.0
E / eV

– 0.5

0.0

d
–0.6

<10–5s–1

<1%

–0.6
107 s–1

*
0%

>9

0%

>99%

%

–0.9

<1

–0.9

50

EI / eV

EI / eV

*

%

<1

–1.2

–1.2
–0.9

–0.6
EQ / eV

–0.3

e

–0.9

f

*

5

–0.6
EQ / eV

–0.3

100
80

*

sP / %

log(r/ s–1)

0
–5

60
40

–10
20
–15
0
–0.9

–0.6
EQ / eV

–0.3

–0.9

–0.6
EQ / eV

–0.3

Fig. 1 | Competitive reaction network to convert A and B into P and Q as desired and undesired products, respectively. a,b, Thermodynamic and kinetic
linear scaling relationships (tLSRs and kLSRs, respectively. c,d, Activity and selectivity volcanos as a function of the adsorption energy of Q and I. An
asterisk indicates the condition that maximizes the production of P although with a moderate selectivity (~50%). An optimization strategy is presented as
a dashed white line. e,f, One-dimensional activity volcano and selectivity cliff (the cuts were done along the dotted lines marked in c,d. The selectivity has
a cliff where Q binds more strongly than I and thus poisons the surface. Models are from ref. 19, Zenodo.

the reaction is hindered. Contrarily to the smoothness found for
activity volcanoes, selectivity relies on the difference between two
competing steps, and these very small energy differences generate discontinuities in the behaviour, acting as an on/off switch and
resulting in much more abrupt profiles, as seen in Fig. 1f. The reason
behind this is that very small energy differences, as small as <0.10 eV
at normal conditions, might turn a selective process into an unselective one23, thus approaching the limit of DFT accuracy energy. If the
kLSRs for the desired and undesired products have different slopes
when written as a function of the energy of the key intermediate, as
seen in Fig. 1b, then cliffs emerge in the volcano plots (see Fig. 1d, f),
and actually the point with the highest selectivity might depart from
the point with the best activity.
Therefore, reaching activities that are higher than those marked
by the volcano limitations implies breaking the symmetry rules
972

(dependencies) imposed by the LSRs. This can only be done by
including alternative energy forms or increasing the complexity of
the catalyst.

Circumventing linear scaling relationships

In 2015, Vojvodic and Nørskov listed several potential ways of
breaking or circumventing LSRs24. Following the traditional analysis, geometric and electronic terms, as well as intrinsic or extrinsic effects, were taken as orthogonal directions for development.
Disentangling the geometric and electronic effects is difficult, some
suggestions have been put forward25. The activity has been very
large in the field and in Fig. 2 we present the already identified ways
to elude the LSRs for which examples exist.
The simplest way to circumvent LSRs is through a pure geometrical effect. Strain has been used in this context26, as the binding
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PERSPECTIVE

NATURE CATALYSIS
Pd Cu Pb Pt

Ir Rh Ru

O Ag S

Ti

C

N Au

Ce

P Cl

Ni

Pd +
mechanic force

NiO +
magnetic field

Lindlar

Au - N ligand
Ru - IL

Pd3S

Ru - HHDMA

Pd - C3N4
Pt - CeO2
Ag - Cu

IrPdPtRhRu

External field
assisted catalysts,
Mechanically
tuned catalysts,
Magneticactive
phase locking...

CuS

PdPb

Br - TiO2

Metals

Ge

om

etr

ic

Alloys

nic

tro

c
Ele

Metals,
NPs, strained films,
coreshell NPs ...

Heteroatom
doped catalysts,
‘poisoned’ metals,
diluted alloys,
high entropy alloys,
p - block doped
catalysts...

Dynamically assisted catalysts,
Ligand modified surfaces,
dynamic single cluster coordination,
polaron assisted catalysts...

The singularity of catalytic activity

Fig. 2 | The cone timeline of breaking linear-scaling relationships strategies. Structures are placed according to their mostly electronic or mostly
geometric contributions. The identified rules can be described in terms of complexity (diagonal timeline). The lowest design level corresponds to metalonly systems and simple alloys and the changes induced by strain26 or coordination. The second level is formed by doped systems: metal-in-metal single
atoms27, high entropy alloys39, self-doped semiconductors41, p-doped systems53. The third complexity level encompasses ligand modified systems42,
organometallic systems59, single atoms in carbon matrixes30 and their dynamic electronic and geometric effects56. The highest level corresponds to the
introduction of external forces62–64, such as light, mechanic or magnetic forces.

energy of different species may depend on the geometric perturbation in different directions. Alternatively, ensemble control requires
the identification of the minimum site to perform the desired
reaction and space, and to resolve it in a controlled manner. Actually,
this effect is really acting on the selectivity control, particularly
as some reactions might require larger ensembles than others.
The way to access this control is by alloying with inert species
(Au, ref. 27); adsorbing molecular fragments (selectivity modulators), as in the Lindlar catalyst28 or NanoSelect29. This method has
been used recently in single-atom catalysts30,31, and p-block modified phases like PdSx active materials32. All these changes also modify
the electronic structure (and even bifunctional mechanisms appear
with reactants adsorbed in different sites, such as PdSx) but the
ensemble contribution is the most obvious by far. The best example
can be found for the hydrogenation of alkynes: while the desired
path requires very small ensembles (3–4 atoms), the parasitic reaction C–C bond formation demands more than six atoms. Therefore,
limiting the ensemble is a selectivity enhancer, and this is achieved
by breaking the LSRs for the C–C bond formation. This has been
NATURE CATA LYSIS | VOL 2 | NOVEMBER 2019 | 971–976 | www.nature.com/natcatal

found by sulfiding supported Pd nanoparticles under mild conditions, which results in facet control by kinetic terms33. Therefore,
the nuclearity control when using small clusters can fine-tune the
selectivity34–36. Similar concepts have been employed in metal-only
single-atom catalysts, where the resistance towards coking37,38 and
the long-term stability of the catalyst are due to the suppression of
the side decomposition path that does not scale any longer with the
intermediates. High-entropy alloys are an alternative, as the continuous nature of adsorption is an effective way to modulate reactivity39. Notice, however, that metal in metal systems (without a phase
change) can only provide solutions to the selectivity cliffs (and in
doing so improve the yield and the long-term stability) but not to
the underlying activity, as the density anisotropies are very small in
these materials.
A fully electronic effect was found in the Br version of the Deacon
reaction ðHBr þ 1 O2 ! Br2 þ H2 OÞ on oxides of the rutile family,
2
most of them with a metallic character. The reaction encompasses
I
surface oxidation, the adsorption of the acid, the halogen evolution, hydroxyl recombination and, finally, water elimination. When
973

PERSPECTIVE
considered in this order, the reaction is found to be unfeasible
on TiO2 rutile, a semiconductor that cannot adsorb O2 (ref. 40).
However, the order of the competing adsorbing reactions is arbitrary. For example, when starting the reaction by HBr adsorption,
the system self-dopes and leads to an effective catalyst by breaking
the rules between O and Br adsorption on rutiles41. An important
conclusion is that, besides geometry, the particular electronic structure for the material is crucial, and results from metal-like materials
cannot be extrapolated to semiconductors.
Electronic and geometric terms are generally not so easy to
separate. The addition of external elements, like ligands to metal
surfaces, can also provide other electronic effects. For instance, electrostatics and hydrogen bonds are found to rearrange intermediates
in the direct hydrogenation of O2 (towards H2O2) or for nitrogen
to ammonia from planarly bonded OOH or NNH (prone to dissociation) towards standing species where the lack of bonds to the
surface prevent effective dissociation42,43. The LSR for the planar and
vertical configurations is thus different when electric fields, ligands
and ionic liquids exist42,43. Again, in the first case, the selectivity
switch goes on top of the activity control.
In all of the previous cases, the changes mostly affected the
viability of the reaction network and not in the particular nature
of the species involved. However, this is one of the most effective
ways to circumvent LSRs. For instance, hydrogenation on metal surfaces takes place via the Horiuti–Polanyi mechanism44, that is, by
splitting H2 into two H atoms and transferring them sequentially,
although other ways are also possible. On gold-decorated nanoparticles with ligands containing H2 bonds, the ligands play an active
role and induce the dissociation as hydride-proton45. On the other
hand, if the ligand contains C=O terminations, then hydrogenation
occurs through a concerted step46 transfer hydrogenation. In both
cases, the kLSR breaks because the transition states are different
from those of the regular surface. This is also the case of the single
Pd atoms stabilized on carbon nitride in which the scaffold acts as
a ligand30. Moreover, the presence of polarity between cations and
anions in oxides leads to a heterolytic dissociation path47.
Hydrogen molecules can still be separated as two protons and
two electrons, which gives rise to two alternative hydrogenation
mechanisms. For instance, catalyst acidity is fundamental to the
conversion of acidic biomass molecules. Ru nanoparticles decorated
by surfactants (HHDMA) split H2 into protons and electrons. The
metal–ligand interface is so strongly acidic48 that it can protonate
levulate, the most common form in a water-based solution of levulinic acid, and thus enable the transformation to γ-valerolactone,
one of the twelve most demanded chemicals derived from biomass.
This protonation overrules the reaction mechanism of levulate on
Ru, which is rather inefficient. pH effects can also tune the catalytic route and change the active volcano49. Very small decorated
Pt nanoparticles can also store electrons at high potentials, which
can be used for the selective reduction of nitrobenzene to aniline.
For such systems, the interface electric field, coupled to the size of
the nanoparticles (around 1.2–1.5 nm), creates the driving force for
changing the active species from H atoms on the surface into high
potential electrons50. The decoration with chiral substrates can also
alter the reactivity and displace the rate-determining reaction to be
spin controlled, as has been found for the oxygen evolution reaction
on chiral-decorated iron oxides. This appears to be consequence of
the spin-polarized nature of O2 that can be only achieved when the
surface is ferromagnetically aligned51.
Cooperative effects can also help reducing inefficiencies in reaction paths. Simple coverage effects can be deemed responsible for
better performance due to rearrangements that make competing
kLSRs more chemoselective towards the desired product at higher
coverages52. Alternatively, impurities such as sulfur can preferentially trap CO2 on Cu in electrochemical environments. The tethering53 of the key species allows blocking of the unselective path for
974

NATURE CATALYSIS
geometric reasons. The ultimate consequence is a complete change
in selectivity as the decorated system produces formate instead
of CO, as would occur on Cu, producing formate instead. Active
ligands can be employed as reactant relays54 in homogeneous electrocatalysis; this effect could be equivalent to Mars-van-Krevelen
processes in heterogeneous catalysis, and thus does not follow surface LSRs. Bifunctional mechanisms also belong to this class, since
a part of the path can occur at one centre while a second conversion
takes place on a different area of the catalyst55.
Dynamics can offer a way to overcome LSRs, however, the control at the nanoscale is not evident from our present tools. For
instance, the fluxionality of small nanoparticles can interconnect
different geometric configurations with different properties for
the adsorption of intermediates, thus breaking the rules by adapting the coordination in each case56. Ligands can also dynamically
cover parts of the surface, leading to short-lived highly selective
active sites57. This effect can be amplified and modified when a support exists58. The effect is particularly important in homogeneous
catalysis, and in this way in molecular systems LSR can be disrupted
through geometric strain, which influences the electronic properties and decouples reaction steps59. Alternatively, the electronic
dynamics can also contribute to differential reactivity, as has been
found for CO oxidation on dynamic charge transfer systems, such
as single-atom platinum on CeO2 where more than one electronic
state is available, allowing a low-temperature reaction path60.
Finally, the use of other external forces can break LSRs61. Recently,
mechanochemistry has emerged as a way to enhance the catalytic
response, and in this case reactions are carried out with the minimum solvent and by grinding the solid reactants62. Plasmonic metal
nanoparticles can also drive new transformations effectively. The
incoming electromagnetic radiation in the form of light concentrates
in the form of plasmons and therefore results new activity rules63.
Similarly, the oxygen evolution reaction on antiferromagnets at high
pH has been found to be enhanced by magnetic effects due to spincontrol in the reaction64. All these processes differ from standard
temperature control and different linear scaling can be envisaged.

Conclusions

In summary, much progress has been made in alternative ways to
circumvent linear scaling relationships, particularly those affecting
selectivity, as blocking the lateral paths improves both activity and
stability. However, a general overall optimization including unchartered territories is not yet possible, partially because of the lack of
fundamental understanding of the properties of complex electronic
structure materials (https://materialsproject.org/) and the very limited atomistic investigations on complex multiphase materials or
interfaces. Understanding complexity will require the combination
of the descriptor strategies with high-throughput experimentation
approaches, as well as detailed synthetic control and characterization tools to unveil alternative catalytic routes that circumvent the
present linear-scaling relationships.

Code A vailability

The model systems for Fig. 1 can be retrieved from ref. 19.
Received: 25 July 2019; Accepted: 27 September 2019;
Published online: 28 October 2019

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PERSPECTIVE
A cknowledgements

We would like to thank E. Fako and R. García-Muelas for their assistance with Figs.,
microkinetic modelling and for critically reading the manuscript.

Competing Interests

The authors declare no competing interests.

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