Download Importance of supersaturated carbon concentrations in catalytic

Importance of supersaturated carbon concentrations in catalytic metal particles
for single-walled carbon nanotube nucleation
Feng Ding1∗ and Kim Bolton1,2
1) Department of Physics, Göteborg University, SE-412 96, Gothenburg, Sweden
2) School of Engineering, University College of Borås, SE-501 90, Borås, Sweden
Abstract: The carbon concentration in catalytic metal particles during initial stages of single-walled
carbon nanotube (SWNT) nucleation is studied by thermodynamics and molecular dynamics
modeling. Highly supersaturated carbon concentrations are necessary to nucleate the initial small
carbon structures (e.g., islands) on the particle surface.
A lower, but still supersaturated,
concentration is required to enlarge the carbon island, with the required carbon concentration
decreasing with increasing island size. The lower carbon concentration at the end of the nucleation
process prevents the formation of a second island (which requires highly supersaturated conditions)
and poisoning of the metal particle.
PACS: 65.80. +n, 61.46. +w, 68.55. -k
∗
Corresponding author, Email: [email protected]
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The immense interest in single-walled carbon nanotubes (SWNTs) stems, in part, from their
potential use in nanoscale electronics.
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Depending on their chirality, SWNTs may be
semiconducting or metallic1 and can thus be used as transistors and interconnects in electronic
circuitry. However, current production methods yield mixtures of semiconducting and metallic
SWNTs1 that need to be separated before they can be used in circuits. Alternatively, new or
improved production methods that produce only metallic or semiconducting nanotubes, preferably
with a specific chirality, must be identified. A step towards achieving this goal is to improve our
understanding of the SWNT growth process.
The vapor-liquid-solid (VLS) model2,3 is often invoked to describe SWNT growth. Here the
carbon atoms in the liquid metal-carbide clusters arise from vapor-phase carbon feedstock. When the
liquid clusters are supersaturated in carbon, and/or when they cool, solid state SWNTs are formed.
A critical aspect in SWNT nucleation is the initial formation of carbon structures on the catalyst
particle surface that eventually grow into a cap (according to the VLS model) and then the SWNT.
The microscopic mechanisms of these initial stages are still not well understood3,4. It is believed that
the cap is nucleated when carbon atoms, which have precipitated on the metal surface, condense
(coalesce) into larger carbon strings and islands. This can be viewed as a two dimensional phase
transition on the catalyst particle surface. In this contribution we present a thermodynamic model for
the nucleation of the graphitic island on the catalyst particle surface, and support this analysis with
molecular dynamics (MD) simulations.
2
Figure1. Illustration of the three kinds of carbon (C) atoms used in
the thermodynamic model of SWNT nucleation on the catalyst
particle surface: Carbon atoms that have condensed (coalesced) to
form the graphitic island, CC; carbon atoms that have precipitated
on the particle surface but do not form part of the island, CP; and
carbon atoms that are dissolved in the catalyst particle, CD. The
large spheres are metal atoms and the carbon atoms are shown as
ball-and-stick.
The thermodynamic model discussed here distinguishes three types of carbon (C) atoms during the
initial stages of SWNT nucleation, and these are shown in Fig. 1. Previous MD simulations5, 6, 7 have
shown that, for small particles that are molten under experimental growth conditions, all three types
of C atoms exist during SWNT nucleation. The dissolved C atoms are denoted CD, the atoms that
have precipitated on the particle surface but that do not form part of the graphitic island are denoted
CP, and the precipitated atoms that have condensed into islands are denoted CC. Both the C-C bondsaturated atoms in the middle of the island and the bond-unsaturated atoms at the edge of the island
are included as condensed atoms CC. During the two-dimensional phase transition (i.e., coalescence
of precipitated atoms into the island), the change in Gibbs free energy is:
dG = − SdT + Vdp + µ D dN D + µ P dN P + µ C dN C + σdL( N C )
(1)
where G, S, V and µ are the Gibbs free energy, entropy, volume and chemical potential, respectively8.
T and p are the temperature and pressure, and N D , N P and N C are the number of dissolved,
precipitated and condensed carbon atoms. σ is the edge tension of the graphitic cap, and L( N C ) is
the edge length around the cap which contains N C carbon atoms. In the thermodynamic limit of
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infinitely slow nucleation there is thermal equilibrium between dissolved, precipitated and condensed
C atoms, i.e.,
µD = µP ,
µ D = µC + σ
(2)
dL( N C )
.
dN C
(3)
Hence, at thermal equilibrium the chemical potential of CD (and CP) is higher than that of CC. Eq. 3
also shows that the relationship between µ D and µ C depends on
dL( N C )
, i.e., the rate of change of
dN C
the island edge length with change in number of condense atoms.
dL( N C )
decreases with increasing
dN C
island size, and in the limit of an infinitely large island
dL( N C )
→ 0 9, i.e., µ D = µ P = µ C . This
dN C
limiting condition is expected since the effect of the bond-unsaturated C atoms (or rate of change of
island edge length) becomes negligible for very large islands. This relative dependence of µ D and
Chemical potential
µ C on the graphite island size is illustrated in Fig. 2.
µD
Figure 2. Dependence of the dissolved and condensed C atom
µC
chemical potentials, µ D and µ C , on the graphitic cap size.
Size of the graphitic cap
Since, at constant temperature and pressure, the chemical potential for a single thermodynamic
phase is larger at higher concentrations, it is clear from Fig. 2 that higher concentrations of CD are
required for thermal equilibrium with smaller graphitic islands than for larger islands. This is in
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agreement with previous thermal dynamical calculations10. Also, assuming that the island growth
occurs at thermal equilibrium, the concentration of CD decreases over time (i.e., as the island
increases in size).
In addition, in accordance with classical nucleation theory,11 the dissolved carbon concentration
must be (highly) supersaturated in order to initiate the condensation of a graphitic island. The highly
supersaturated carbon concentration in the metal particle during the initial stages of graphite island
formation is similar to the oversaturated vapor required for the nucleation of a liquid drop,11 which is
a three dimensional phase transition. One important feature of liquid drop nucleation is that smaller
drops are less stable than larger ones because of their larger surface tensions. Hence, the sustained
growth of smaller drops requires higher vapor concentrations (or pressures) than that of larger drops.
This requires a highly oversaturated vapor for nucleation and growth of small drops.
We have recently performed molecular dynamics (MD) studies to understand the growth
mechanism of SWNTs.5 Based on experimental and electronic theory data for different carbon-iron
systems,12, 13 an empirical potential energy surface (PES) was developed to study features of the ironcarbide phase diagram, the change in cluster melting point with cluster size and the nucleation of
SWNTs on catalyst particles. This PES correctly predicts the eutectic point in the iron-carbide phasediagram and the decrease in melting point with decreasing particle size.14 MD simulations based on
this PES also show that SWNTs grow on iron catalyst particles between 800K and 1400K, which is
the same temperature range used in CCVD experiments (at lower temperatures the particle becomes
encapsulated in a graphene sheet and at higher temperatures soot is formed). The detailed SWNT
growth mechanism that was revealed by the simulations appears elsewhere.5, 6, 7
5
Figure 3. Typical SWNT nucleation and growth as revealed by MD simulations. The iron cluster
contained 50 atoms (diameter 1 nm) and the temperature is 1000 K. The graph shows the change in
the number of dissolved carbon atoms (CD) during nucleation and growth. The big spheres are metal
atoms and the carbon atoms are shown as ball-and-stick. The horizontal line is shown to guide the
eye.
Fig. 3 shows a typical scenario of SWNT nucleation and growth on a catalyst particle (Fe50) as
revealed by MD simulations, as well as the change in the number of dissolved C atoms, CD, over time.
The simulation is initiated with a pure iron cluster, Fe50 (Fig. 3a), that is thermalized at 1000 K before
carbon atoms are added (one every 50 ps). Hence, in the simulations we do not consider the catalytic
role of the metal particle, but focus solely on its role as a solvent for the C atoms. Initially all of the
C atoms dissolve in the Fe particle (seen as the linear increase in CD over time). When the iron is
supersaturated in C, some of the C atoms precipitate. However, these are not stable on the surface
and redissolve into the cluster. Only when the cluster is highly supersaturated are there sufficient
precipitated C atoms (Fig. 3b) that are sufficiently close to coalesce (condense) to form strings (Fig.
3c), polygons (Fig. 3d), and islands (Fig. 3e).
Under certain conditions (i.e., temperature and
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pressure) the island lifts off the surface to form a graphitic cap (Figs. 3f and 3g) and a SWNT (Fig.
3h).
Of importance to the present work is that, as shown in the graph in Fig. 3, the metal particle
must be highly supersaturated in carbon before graphitic islands can be formed. This occurs between
0.7 and 3.5 ns in the simulation. Also, as the size of the polygons and islands increase the dissolved
carbon concentration decreases, i.e., between 1.8 and 3.5 ns. The number of dissolved carbon atoms
remains constant (at a supersaturated level) once the cap is formed.
The maximum in the dissolved carbon concentration (between 0.7 and 3.5 ns in Fig. 3) shows
that a highly supersaturated carbon concentration is necessary for nucleating small carbon structures
(strings, polygons and small islands). This is consistent with the thermodynamics analysis described
above (cf. Fig. 2). The MD simulations also validate the assertion that smaller carbon islands are less
stable than larger ones. For example, at 3.5 ns in Fig. 3 two graphitic islands (A and B in panel e)
have formed on the surface, but with increasing time the larger island, A, grows bigger (and forms a
SWNT) while the smaller island B redissolves into the metal particle (panels e to h).
It should also be noted that the dissolved carbon is at the lower supersaturated concentration
when the smaller islands redissolve. This is important since, if the carbon concentration does not
decrease during graphitic island nucleation then the smaller islands would not redissolve and would
continue to grow in size. Depending on the growth conditions (e.g., temperature and pressure) this
would lead to more than one SWNT being formed from a single metal particle, or to the metal
particle being encapsulated in a graphene sheet (i.e., if the island does not lift off the particle surface).
Once the particle is encapsulated in a graphene sheet it is poisoned15 and SWNTs do not continue to
grow. Hence, the decrease in dissolved carbon concentration, shown in Figs. 2 and 3, is critical for
the sustained growth of single SWNTs from catalytic metal particles.
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In conclusion, a thermodynamic analysis of the initial stages in SWNT nucleation (i.e.,
precipitation and condensation of carbon atoms on the catalytic particle surface) shows that the metal
particle needs to be highly supersaturated in carbon before small carbon islands can nucleate. In
addition, the dissolved carbon concentration decreases as the islands grow in size.
These
observations are validated by molecular dynamics simulations. The decrease in dissolved carbon
concentration with increasing island size prevents the formation of new islands (which requires high
supersaturation) and also results in smaller, less stable islands redissolving into the metal particle.
This prevents more than one SWNT growing from each metal particle as well as poisoning of the
particle. Larger particles (where there may be areas of highly supersaturated carbon) or high carbon
feedstock pressures may lead to the growth of more than one SWNT per particle or poisoning of the
particle.
The authors are grateful to Arne Rosén for valuable discussions, as well as for time allocated on
the Swedish National Supercomputing facilities and for financial support from the Swedish
Foundation for Strategic Research and the Swedish Research Council.
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Assuming that, the graphite has a circle shape, we have L(N C ) ∝ N C and dL( N C ) ∝
dN C
1
NC
→ 0 when
NC → ∞ .
10
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Figure Captions:
Figure1. Illustration of the three kinds of carbon (C) atoms used in the thermodynamic model of
SWNT nucleation on the catalyst particle surface: Carbon atoms that have condensed (coalesced) to
form the graphitic island, CC; carbon atoms that have precipitated on the particle surface but do not
form part of the island, CP; and carbon atoms that are dissolved in the catalyst particle, CD. The large
spheres are metal atoms and the carbon atoms are shown as ball-and-stick.
Figure 2. Dependence of the dissolved and condensed C atom chemical potentials, µ D and µ C , on
the graphitic cap size.
Figure 3. Typical SWNT nucleation and growth as revealed by MD simulations. The iron cluster
contained 50 atoms (diameter 1 nm) and the temperature is 1000 K. The graph shows the change in
the number of dissolved carbon atoms (CD) during nucleation and growth. The big spheres are metal
atoms and the carbon atoms are shown as ball-and-stick. The horizontal line is shown to guide the
eye.
10