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REPORT◥
NANOMATERIALS
Entropy-driven stability of chiralsingle-walled carbon
nanotubesYann Magnin1*, Hakim Amara2, François Ducastelle2,Annick
Loiseau2, Christophe Bichara1†
Single-walled carbon nanotubes are hollow cylinders that can
grow centimeters longvia carbon incorporation at the interface with
a catalyst. They display semiconductingor metallic characteristics,
depending on their helicity, which is determined duringtheir
growth. To support the quest for a selective synthesis, we develop
a thermodynamicmodel that relates the tube-catalyst interfacial
energies, temperature, and the resultingtube chirality. We show
that nanotubes can grow chiral because of the
configurationalentropy of their nanometer-sized edge, thus
explaining experimentally observedtemperature evolutions of chiral
distributions. Taking the chemical nature of the catalystinto
account through interfacial energies, we derive structural maps and
phasediagrams that will guide a rational choice of a catalyst and
growth parameters towarda better selectivity.
Electronic properties of single-walled car-bon nanotubes (SWNTs)
depend on theirchirality—i.e., the way the SWNTs are rolledalong
their axis— which is characterizedby two indices (n, m).
Controlling chiral-
ity during the tube’s synthesis would enable usto avoid costly
sorting and trigger the imple-mentation of promising applications
[such asthe use of SWNT yarns as strong, light, and con-ductive
wires (1) or the development of SWNT-based electronics (2)], with
the ultimate goal ofovercoming the limitations of silicon.
Notablebreakthroughs have been reported (3, 4), andprogress toward
carbon nanotube computers(5, 6) has been very rapid. However,
selectivesynthesis still appears to be the weak link,though new
studies using solid-state catalysts(7–9) have reported a
chiral-specific growth ofSWNTs. Detailed mechanisms underlying
thisselective growth are still being debated, thusunderlining the
need for realistic growth mod-els explicitly including the role of
the catalyst.Existing models focus on kinetics (10), neg-lecting
the role of the catalyst (11, 12), but failto calculate chiral
distributions in line withexperiments. Atomistic computer
simulationsemphasize chemical accuracy (13, 14) but needto be
complemented with a model so as toprovide a global understanding of
the process.In this study, we developed a thermodynamicmodeling of
the interface between the tube
and the catalyst to relate its properties tothe resulting chiral
distribution obtained dur-ing chemical vapor deposition (CVD)
synthesisexperiments.Vapor-liquid-solid and vapor-solid-solid
CVD
processes have both been used to grow SWNTs(8), the latter
leading to a (n, m) selectivity.Growth can proceed through
tangential or per-pendicular modes (15), and ways to control
thesemodes have been proposed recently (16). Forspecific catalysts
and growth conditions favor-ing the perpendicular mode, a
pronounced near-armchair selectivity can be observed (16). In sucha
mode, the interface between the tube and thecatalyst nanoparticle
(NP) is limited to a line,and a simple model describing the
thermo-dynamic stability of the tube-NP system canbe developed. We
thus considered an ensembleof configurations of a catalyst NP,
possibly ametal or a carbide, in perpendicular contact witha (n,m)
SWNT, as in Fig. 1. The total numbers ofcarbon and catalyst atoms
are constant. Config-urations differ by the structure of the
NP-tubeinterface, defined by (n, m), for which we have(n +m)
SWNT-NP bonds, with typically 10 < n +m < 50. On the tube
edge, 2m among the bondsare armchair, and (n − m) are zigzag (17).
In afirst approximation, the atomic structure of theNP is
neglected, and the catalyst appears as asmooth flat surface, in a
jellium-like approxi-mation. The interface is then a simple
closedloop with two kinds of species: armchair andzigzag contact
atoms. Under these conditions,the total energy of the system can be
separatedinto three terms
E(n, m) = E0 + ECurv(n, m) + EInt(n, m) (1)
where E0 includes all terms independent of(n, m), such as the
energy of the threefold
coordinated carbon atoms in the tube walland the atoms forming
the NP. The surfaceenergy of the NP and the very weak surfaceenergy
of the tube are also included in E0,because these surfaces are kept
constant. Ad-ditionally, ECurv is the curvature energy, andEInt is
the interfacial energy. Note that thismodel could possibly also
apply in tangentialmode, if the lateral tube-catalyst
interactiondoes not depend on (n, m).The (n, m)-dependent energy
terms con-
cern the tube curvature and its interface withthe NP. Using
density functional theory (DFT)calculations, Gülseren et al. (18)
evaluatedthe curvature energy of the isolated tube asECurv ¼ 4 a
D�2CNT , where DCNT is the tubediameter and a = 2.14 eV·Å2 per C
atom. Weassume that the interfacial energy for a (n, m)tube in
contact with the NP surface dependsonly on the number of its 2m
armchair and (n,m) zigzag contacts
Eðn;mÞInt ¼ 2mEAInt þ ðn�mÞEZInt ð2Þ
where the armchair ðEAIntÞ and zigzag ðEZIntÞinterfacial
energies are given by EXInt ¼ gXG þEXAdh , with X standing for A or
Z. The edgeenergy per dangling bond, gXG , is positive be-cause it
is the energy cost of cutting a tubeor a graphene ribbon and
depends on thetype of edge created. The adhesion energy ofthe tube
in contact with the NP, EXAdh, is neg-ative because energy is
gained by reconnect-ing a cut tube to the NP. EXInt, the sum of
thesetwo terms, has to be positive to create a driv-ing force for
SWNT formation. DFT calcula-tions of the edge energies of different
edgeconfigurations of a (8, 4) tube (Fig. 2) show nopreferential
ordering. We thus assume that alltube-catalyst interfaces with the
same numberof armchair and zigzag contacts have the sameenergy.This
leads us to introduce the edge config-
urational entropy as a central piece of themodel. We assume that
the tube is cut almostperpendicular to its axis, forming the
short-est possible interface, for a given (n, m). Weneglect
vibrational entropy contributions, whichare essentially the same
for all tubes, exceptfor radial breathing modes. Armchair
twofoldcoordinated C atoms always come as a pair;thus, this entropy
(S) that relates the numberof ways of putting (n − m) zigzag C
atoms andm pairs of armchair atoms on n sites (degen-eracy) is
Sðn;mÞkB
¼ ln n !m!ðn�mÞ! ð3Þ
where kB is Boltzmann’s constant. Interfacialenergies can be
evaluated using DFT calcula-tions, described in the materials and
methods.In agreement with previous studies (17, 19), wefind gAG ¼
2:06 eV per bond and gZG ¼ 3:17 eVper bond for graphene, and 1.99
and 3.12 eV
RESEARCH
Magnin et al., Science 362, 212–215 (2018) 12 October 2018 1 of
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1Aix Marseille Université, CNRS, Centre Interdisciplinaire
deNanoscience de Marseille, Campus de Luminy, Case 913,F-13288
Marseille, France. 2Laboratoire d’Etude desMicrostructures,
ONERA-CNRS, UMR104, UniversitéParis-Saclay, BP 72, 92322 Châtillon
Cedex, France.*Present address: MultiScale Material Science for
Energy andEnvironment, MIT-CNRS Joint Laboratory at MIT, Cambridge,
MA02139, USA.†Corresponding author. Email:
[email protected]
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per bond, respectively, for cutting (6, 6) and(12, 0) tubes. The
lower value of gAG is due to therelaxation (shortening) of the C–C
bonds ofthe armchair edge that stabilizes it. Adhesionenergies of
(10, 0) and (5, 5) tubes on icosa-hedral clusters of various
metals, including Fe,Co, Ni, Cu, Pd, and Au, were calculated in
(20, 21).Thus, orders of magnitude for interface ener-gies, EXInt,
of armchair and zigzag terminationsin contact with typical
catalysts can be esti-mated: They lie between 0.0 and 0.5 eV
perbond, with EAInt < E
ZInt for these metals. An
example of free energy and corresponding prob-ability
distribution is plotted as a function of(n, m) in fig. S1.Instead
of focusing on a specific catalytic
system, it is more relevant at this stage tostudy the general
properties of the model thatlinks the (n, m) indexes of a SWNT to
threeparameters characterizing its CVD growth
conditions—namely, temperature and the in-terfacial energies of
armchair ðEAIntÞ and zig-zag ðEZIntÞ tube-catalyst contacts. For
each setof parameters, a free energy can be calculated,and its
minimization yields the stable (n, m)value. This model displays
similarities with asimple alloy model on a linear chair, but
thecurvature term, dominant for small diame-ters, and the small and
discrete values of nand m prevent it from being analytically
sol-vable, except for ground states (i.e., stablestructures at zero
kelvin), for which a solu-tion is provided in the materials and
methods.We thus define a three-dimensional (3D) spaceof stable
configurations in the ðT ; EAInt;EZIntÞcoordinates.Setting T and,
hence, the entropy contribu-
tion to zero, the ground states are readily cal-culated and
displayed in Fig. 3A. Only armchairor zigzag tubes are found to be
stable, sepa-
rated by a line EZInt ¼ 43 EAInt. With increasingtemperature,
they become unstable, and atransition toward chiral tubes takes
place.Figure 3B is a contour plot of the surface de-fined by the
transition temperatures. Abovethis surface, for each set of ðT ;
EAInt;EZIntÞ pa-rameters, a chiral (n, m) tube is found sta-ble,
defining “volumes” of stability for eachchirality. To explore it,
we can cut slices atconstant temperature to obtain an isother-mal
stability map (Fig. 3C). In such maps, onlythe most stable (n, m)
tube structures areshown, whereas the model yields a distribu-tion
of chiralities for each ðT ; EAInt;EZIntÞ point.Within a (n, m)
domain, this distribution isnot constant, especially close to the
bounda-ries, which are calculated by searching forpoints where the
free energies and hence theprobabilities of two competing
structures areequal. As illustrated in fig. S1B, around the
chi-rality that displays a maximal probability setto 1, neighboring
chiralities have non-negligiblecontributions that depend on ðT ;
EAInt;EZIntÞ.We can also fix either EAInt or E
ZInt to obtain
temperature-dependent phase diagrams, asin Fig. 4A (for EAInt ¼
0:15 eV per bond) andFig. 4B (for EZInt ¼ 0:25 eV per bond). As
anexample, we can follow the temperature sta-bility of a (6, 6)
tube. Figure 4A shows a largestability range with a maximal
stability temper-ature rising from 200 to 800 K by increasingEZInt
from 0.20 to 0.30 eV per bond, whereas thesecond map, orthogonal to
the first one in the3D configuration space, shows an upper
temper-ature limit varying from 500 to 700 K, within anarrowerEAInt
range. Above the armchair tubes,chiral (n, n − m) tubes become
stable start-ing with (n, n − 1) and then with increasing(n − m)
values such as (6, 5), (7, 5), etc. Chiraltubes—i.e., tubes
different from armchair orzigzag tubes—are stabilized at finite
temper-ature by the configurational entropy of thetube edge.An
isothermal map calculated at 1000 K is
plotted in Fig. 3C. Chiral tubes are spread alongthe EZInt ¼
43EAInt diagonal, between armchairand zigzag ones. Small-diameter
tubes are sta-bilized for larger values of ðEAInt;EZIntÞ and
hencefor weaker adhesion energies of the tube onthe catalyst.
Larger-diameter tubes are obtainedfor small values of ðEAInt;EZIntÞ
because the en-tropy cannot counterbalance the energy cost ofthe
interface, proportional to n + m. A com-parison of maps at 1000 and
1400 K is providedin fig. S3. As shown in movie S1, the effect
ofincreasing temperature is to expand and shiftthe stability domain
of chiral tubes along andon both sides of the EZInt ¼ 43EAInt
diagonal,with a larger spread on the armchair side. Thestability
domain of chiralities between central(2n, n) and near-armchair (n,
n − 1) expandssubstantially at high temperature. However,the
free-energy differences become smaller,leading to broader chiral
distributions and thusexplaining the lack of selectivity reported
fortubes grown at very high temperature by elec-tric arc or laser
ablation methods (22).
Magnin et al., Science 362, 212–215 (2018) 12 October 2018 2 of
4
Fig. 1. From experiments to a model. (A and B) (Top)
Postsynthesis transmission electronmicroscopy (TEM) images of a
SWNT attached to the NP from which it grew at 1073 K, using
eitherCH4 (A) or CO (B) feedstocks, leading to a tangential or
perpendicular growth mode, illustrated atthe atomic scale (bottom).
TEM images are reproduced from (16) with permission from the
RoyalSociety of Chemistry. The experiments and our analysis of the
growth modes are described in (16).(C) Sketch of the model, with a
SWNT in perpendicular contact with a structureless
catalyst.Armchair edge atoms are in red, zigzag ones in blue.
Fig. 2. Key elements of the model.(Top) Different ways of
cutting a (8, 4)tube, leading to the formation of zigzag(blue) and
armchair (red) undercoordi-nated atoms. For a (8, 4) tube, there
are70 different edge configurations withalmost the same energy.
(Bottom) For-mation energies of all possible (8, 4)edges, from DFT
calculations describedin the materials and methods. Theenergy
levels lie within 25 meV per bondand can thus be considered
degenerate.
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This very simple model displays a fair agree-ment with
literature data, as illustrated in thefollowing examples. Figure 3B
suggests a wayto grow either zigzag or armchair tubes, thelatter
being metallic for any diameter. For both,growth kinetics is slow,
because each new ringof carbon atoms has to nucleate once the
pre-vious one has been completed (11, 12). To over-come this
nucleation barrier, one should seekregions in the map where such
tubes remainstable at high temperature. For armchair spe-cies, this
corresponds to the lower right cornerof the map in Fig. 3B, where
the adhesion energyof armchair edges is strong and that of
zigzagones is weak, and the opposite is true for theinterfacial
energies. Such requirements havepossibly been met in
high-temperature (1473 K)CVD experiments (23) that also used
thiophenein the feedstock. Those experiments might in-dicate that
the presence of sulfur at the interfacecould modify the relative
interaction strength ofzigzag and armchair edges with the Fe NP.The
temperature dependence of the chiral
distributions, measured by photoluminescence(24–26) or Raman and
transmission electronspectroscopies (27) in previous studies,
seemsmore robust. The maps presented in Fig. 4, Aand B, are
consistent with these experiments,showing that armchair or
near-armchair chiral-ities [(6, 6) and (6, 5)] are grown at low
tem-perature (873 K) and that the chiral distributiongradually
shifts toward larger chiral angles [(7, 5),(7, 6), and (8, 4)...]
at higher temperatures. Re-ferring to our model, this suggests that
Co- andFe-based catalysts used in these experimentscorrespond to
interfacial energy values aroundEAInt ¼ 0:15 eV per bond and EZInt
¼ 0:24 eVper bond, as indicated by the dashed boxes inthe maps. A
quantitative comparison with fourdifferent sets of experimental
data is providedin fig. S2, showing a slight tendency to
overes-timate the width of the distributions. This over-
estimation partly results from the fact that weuse a
two-parameter thermodynamic model toaccount for experiments that
include the var-iability in the catalyst size and chemical
com-position and the growth kinetics. Our resultsalso confirm that
overlooking metallic tubes inphotoluminescence experiments
introduces aserious bias in the resulting chiral
distribution.Further, the dependence of the quantum yieldon chiral
angles of semiconducting tubes mayalso contribute to
underestimating the width ofthe experimental distributions.The
present model thus sets a framework for
understanding why a number of experiments,using metallic
catalysts in perpendicular growthconditions as discussed in (16),
report a near-armchair selectivity. For such catalysts, EAInt
isgenerally lower than EZInt (20). At low temper-ature, zigzag or
armchair tubes are thermody-namically favored butmay not always be
obtained,
owing to kinetic reasons.On the armchair side, ourmodel
indicates that near-armchair helicities arethen favored by a
temperature increase, becausetheir stability domain is large and
they are lesskinetically impaired (11). At even higher
temper-atures, tube chiralities tending toward (2n, n)indexes
should be stabilized by their larger edgeconfigurational entropy,
but their stability do-mains turn out to be narrower in the
presentmodel. Taking the atomic structure of the cat-alyst into
account in our model could rule outsome neighboring structures and
contribute toopen up these domains.Concerning the practical use of
these maps,
a first issue is to select the appropriate lo-cation for a
catalyst in the ðT ; EAInt;EZIntÞ co-ordinates, so as to favor the
desired tube helicity.Looking at Fig. 3C, one can see that the
largestand most interesting parameter ranges corre-spond to either
metallic armchair tubes or to
Magnin et al., Science 362, 212–215 (2018) 12 October 2018 3 of
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Fig. 3. Structural maps. (A) Map of the ground states, with
armchairtubes in the lower right corner and zigzag ones in the
upper left corner,separated by a line EZInt ¼ 43 EAInt.
Small-diameter tubes [e.g., (5, 5) and(8, 0)] are obtained for
large values of the interfacial energies EInt, whereasstability
domains of large-diameter tubes are narrower, with a widthdecaying
as 1nðnþ1Þ, and are obtained for small values of EInt. (B) Contour
plot ofthe highest temperatures of stability of the ground-state
structures,armchair or zigzag. Chiral tubes are found only above
this surface,
stabilized by the configurational entropy of the tube’s edge.
Armchairand zigzag tubes can remain stable at high temperatures, in
thebottom right and upper left corners, respectively. (C)
Chiralitymap at 1000 K. Iso-n (iso-m) values are delimited by solid
black (dashedblue) lines. Metallic tubes, for which (n − m) is a
multiple of 3, areshown in red, and semiconducting ones are flesh
colored. Theparameter space for armchair (metallic) and (n, n − 1)
and (n, n − 2)(semiconducting) tubes is larger than for other
chiralities.
Fig. 4. Chirality phase diagrams. Phase diagrams calculated for
constant values of EAInt (A)and EZInt (B). These diagrams would be
orthogonal in a 3D plot. The blue dashed boxesindicate possible
parameter ranges corresponding to the analysis of growth productsby
He et al. (26), based on a photoluminescence assignment of tubes
grown using a FeCucatalyst. (6, 5) tubes are reported stable up to
1023 K, (7, 5) and (8, 4) become dominantat 1023 K, and (7, 6) at
1073 K.
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(n, n − 1) and (n, n − 2) semiconducting tubes.A second, more
difficult issue is to design acatalyst that would display
appropriate EAIntand EZInt values. DFT-based calculations, inthe
same spirit as those used in other studies(7, 9, 20, 21, 28),
should probably be helpful.However, the evidence of the important
roleof the edge configurational entropy calls intoquestion the
possibility of explaining the highselectivity reported in (7, 9) on
the basis of astructural or symmetry matching. The intrin-sic
disorder at the edge could be taken intoaccount by averaging over
various atomic con-figurations and using molecular dynamics
atfinite temperature.The present model reevaluates the role of
thermodynamics in the understanding of SWNTgrowth mechanisms. It
accounts for experimen-tal evidence, such as the near-armchair
prefer-ential selectivity, hitherto attributed to kinetics(11), and
the temperature-dependent trends inchiralities. It also provides a
guide to designbetter, more selective catalysts. However, onemust
also consider the importance of kineticsin a global understanding
of the SWNT growthprocess. An attempt to combine thermodynamicand
kinetic aspects of the growth has beenproposed in (12), but,
overlooking the role of theedge configurational entropy, it led to
unrealisticchiral distributions. Those resulting from thepresent
thermodynamic analysis are slightly
broader than the experimental distributions(fig. S2) but should
be narrower if the reportedchirality dependence of the growth
kinetics(11) is taken into account. Owing to the highsynthesis
temperatures and the very small sizeof the interface, there may be
SWNT growthregimes where the atomic mobility and theresidence time
of atoms close to the interfaceare large enough to achieve a local
thermody-namic equilibrium.
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ACKNOWLEDGMENTS
Funding: Support from the French research funding agency
(ANR)under grant 13-BS10-0015-01 (SYNAPSE) is
gratefullyacknowledged. C.B. thanks P. Müller for stimulating
discussions.Author contributions: C.B. designed the project; Y.M.,
H.A.,F.D., and C.B. developed the model; and all authors
contributed tothe data analysis and manuscript preparation.
Competinginterests: The authors declare no competing interests.
Data andmaterials availability: All data are available in the main
text orthe supplementary materials.
SUPPLEMENTARY MATERIALS
www.sciencemag.org/content/362/6411/212/suppl/DC1Materials and
MethodsFigs. S1 to S3References (29–35)Movie S1Data S1 and S2
21 March 2018; accepted 8 August 201810.1126/science.aat6228
Magnin et al., Science 362, 212–215 (2018) 12 October 2018 4 of
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Entropy-driven stability of chiral single-walled carbon
nanotubesYann Magnin, Hakim Amara, François Ducastelle, Annick
Loiseau and Christophe Bichara
DOI: 10.1126/science.aat6228 (6411), 212-215.362Science
, this issue p. 212Sciencenanotube edge. The model should be
useful in helping to guide nanotube growth parameters to enhance
selectivity.carbon nanotubes. The model explains the origin of
nanotube chirality in terms of the configurational entropy of
the
developed a thermodynamic model for the growth of
single-walledet al.their growth are still not fully known. Magnin
Despite progress in growing single-walled carbon nanotubes of
specific size and chirality, the factors that control
The twisted carbon nanotube story
ARTICLE TOOLS
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REFERENCES
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