C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6

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Diameter-dependent kinetics of activation and deactivationin carbon nanotube population growth

Mostafa Bedewy, Eric R. Meshot 1, A. John Hart *

Mechanosynthesis Group, Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109, USA

A R T I C L E I N F O

Article history:

Received 8 May 2012

Accepted 27 June 2012

Available online 4 July 2012

0008-6223/$ - see front matter � 2012 Elsevihttp://dx.doi.org/10.1016/j.carbon.2012.06.051

* Corresponding author: Fax: +1 734 615 6146E-mail address: ajohnh@umich.edu (A.J. H

1 Current address: IMEC, Kapeldreef 75, 30

A B S T R A C T

We reveal that the collective growth of vertically aligned carbon nanotube (CNT) forests by

chemical vapor deposition (CVD) is governed by the size-dependent catalytic behavior of

metal nanoparticles, which can be quantitatively related to the activation and deactivation

kinetics of subpopulations of CNTs within the forest. We establish this understanding by

uniquely combining real-time forest height kinetics with ex situ synchrotron X-ray scatter-

ing and mass-attenuation measurements. The growing CNT population is divided into

subpopulations, each having a narrow diameter range, enabling the quantification of the

diameter-dependent population dynamics. We find that the mass kinetics of different

subpopulations are self-similar and are represented by the S-shaped Gompertz model of

population growth, which reveals that smaller diameter CNTs activate more slowly but

have longer catalytic lifetimes. While competition between growth activation and deactiva-

tion kinetics is diameter-dependent, CNTs are held in contact by van der Waals forces, thus

preventing relative slip and resulting in a single collective growth rate of the forest. There-

fore, we hypothesize that mechanical coupling gives rise to the inherent tortuosity of CNTs

within forests and possibly causes structural defects which limit the properties of current

CNT forests in comparison to pristine individual CNTs.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The size-dependent catalytic behavior of metal nanoparticles

[1,2] influences a broad spectrum of technologically impor-

tant processes, such as the reforming of hydrocarbons [3]

and the synthesis of nanotubes and nanowires having con-

trolled dimensions and properties [4,5]. Knowledge of how

reaction rates scale with particle size is required to advance

the understanding of underlying reaction steps and to iden-

tify the limiting aspects for process control. For particles

smaller than �100 nm, the increase in area-to-volume ratio

(Fig. S1), gives rise to classical size effects such as reduction

of the melting point (Gibbs–Thomson effect). For particles

smaller than �5 nm, even more dramatic scaling of surface

energy results from the decrease of the coordination number

er Ltd. All rights reserved

.art).

01 Heverlee, Belgium.

of surface atoms [1]. As a result of these effects [5–7], silicon

nanowires (SiNWs) grown by chemical vapor deposition

(CVD) exhibit diameter-dependent growth rates, which are

explained in terms of the diameter dependence of both the

chemical potential and supersaturation of the catalyst parti-

cle. Because of size effects on catalytic reactions, solubility

and diffusion, individual carbon nanotubes (CNTs) grown by

CVD may exhibit diameter-dependent growth behavior as

well.

However, the potential influence of diameter-dependent

catalytic behavior on CNT growth is still largely not under-

stood, especially when large populations of CNTs (typically

billions of CNTs per cm2) are grown by CVD from substrate-

bound catalysts. Due to the inherent polydispersity of catalyst

nanoparticle sizes [8–10], the population of CNTs growing

.

C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6 5107

from these particles exhibit an intrinsic polydispersity of

diameters [11–14], and possibly of growth rates as well. Cou-

pled chemical and mechanical phenomena potentially arise

as the size-dependent growth behavior of individual CNTs is

combined with interactions among neighboring CNTs due to

intermolecular forces. Hence, elucidating these effects is

important not only for controlling growth of each CNT indi-

vidually, but also for understanding how to tailor functional

hierarchical structures of CNTs, from bundles to micro- and

macroscopic ensembles.

In particular, improved understanding of size-dependent

CNT growth behavior would advance engineering of individ-

ual-CNT devices such as resonators [15], flow cells [16], or

even CNT radios [17,18]. Further, we suggest that understand-

ing the implications of size effects on collective growth is

important to enable precise engineering of CNT growth for

applications that are based on collective properties of large

numbers of CNTs, such as electric interconnects [19,20], ther-

mal interfaces [21], filters [22], sensors [23–26], and structural

composites [27]. Coupling between the hierarchical structure

and the mechanics of CNTs is also important for post-synthe-

sis processing techniques such as the drawing and spinning

of CNTyarns and sheets from vertically-aligned as-grown for-

ests [28–30]. Thus, revealing this size dependence in light of

mechanical competition is crucial for enabling the capability

of truly tailoring the morphological, physical and mechanical

properties of hierarchal CNT structures, such as forests,

yarns, and sheets.

Although previous studies have shown evidence of diame-

ter dependence of CNT growth kinetics [31–33], the size depen-

dence of population growth behavior and the competition

between activation and deactivation kinetics among CNTs

growing in a coordinated structure have not been systemati-

cally studied before. This is likely due to the challenges in (1)

experimentally quantifying the spatiotemporal evolution of

the CNT diameter distribution during growth, (2) in situ moni-

toring of growth behavior (i.e., height and mass kinetics), and

(3) establishing accurate one-to-one correlation between the

CNT diameter and the measured growth behavior.

In fact, there is currently no characterization methodology

capable of monitoring the size evolution of nanostructures

with accurate spatial and temporal correlation between the

nanostructure size and the growth process. Measurement of

CNT diameters and number of walls for both single-wall

(SWCNT) or multi-wall (MWCNT) nanotubes is typically done

using transmission electron microscopy (TEM) imaging

[11,12,14]. However, high resolution spatial mapping of the

CNT diameter distribution within a forest by TEM has proven

difficult due to challenges of sample preparation. In addition,

due to the low throughput of TEM measurements, it is practi-

cally impossible to get accurate descriptive statistics and

probability density functions of size distributions within

CNT populations, which are necessary to determine the

evolution of CNT polydispersity during growth.

On the other hand, in situ monitoring of the growth pro-

cess by measuring forest height kinetics has been achieved

by many methods such as optical interference [34], single-slit

laser diffractography [35], time-resolved reflectivity [36], opti-

cal photography and videography [37,38], and laser triangula-

tion [32,39]. Although real-time TEM imaging has been used to

measure CNT growth kinetics in situ [40], these measure-

ments are for isolated individual-CNT growth in a low-pres-

sure growth environment that is significantly different from

the typical atmospheric pressure growth of CNT forests.

Hence, using known methods, it is not possible to track the

growth kinetics of an individual CNT within a growing popu-

lation such as a film or forest. Raman spectroscopy has also

been used to quantify CNT growth kinetics by integrating

the area under the G-band in situ [41–44]. However, this tech-

nique is only applicable to the study of single CNTs or thin

films and can only give diameter and chirality information

for small SWCNTs (typically smaller than 2.5 nm in diameter)

that fall within the measurement range for the frequency

shift of radial breathing mode (RBM) peaks (about 100–

500 cm�1) [45]. For such small SWCNTs, in situ Raman spec-

troscopy has also been recently used to identify chirality-

dependent growth rates [33,46]; however, this technique fails

to give size-dependent growth kinetics of any CNT that does

not exhibit RBM peaks in the aforementioned range (includ-

ing larger-diameter CNTs like MWCNTs).

In this paper, we present the first technique for metrology

of diameter-dependent CNT population growth behavior, and

we apply this technique to quantify the diameter-dependent

activation and deactivation kinetics of CNT population

growth. Our technique combines real-time measurement of

forest height during growth, with spatially resolved small-an-

gle X-ray scattering (SAXS) and X-ray attenuation measure-

ments of a CNT forest after growth. This builds on the use

of SAXS to quantify CNT diameter distribution and alignment

[13,32,47], as well as our recent utilization of X-ray attenua-

tion to measure forest mass-kinetics of growth [13]. We now

discretize the forest into CNT subpopulations each having dif-

ferent diameters and track the growth kinetics of each sub-

population within the total population. The time evolution

of number density and total mass per unit area are accurately

mapped as a function of CNT diameter in the growing CNT

forest. Growth of smaller diameter CNTs activates more

slowly and exhibits a longer lifetime than larger diameter

CNTs. We suggest that attraction forces from van der Waals

(VDW) interactions between CNTs are sufficiently strong to

cause CNTs to adopt a collective average growth rate by pre-

venting slip between neighboring CNTs, in spite of diame-

ter-dependent growth rates that are predicted by analytical

models of individual CNT growth. We conclude that the scal-

ing of these mechanical constraints, along with the diameter-

dependent growth behavior, is responsible for the tortuosity

of CNTs within forests.

2. Experimental section

CNT forests were grown by CVD from substrate-bound Fe cat-

alyst nanoparticles, which were made by dewetting a multi-

layer thin film (1 nm Fe/10 nm Al2O3/300 nm SiO2/675 lm Si)

by thermal annealing in a reducing atmosphere. For process-

ing, a cold-wall reactor (modified SabreTube, Absolute Nano)

was operated according to the recipe shown in Fig. S2 [37].

The input feedstock was a mixture of hydrogen (310 sccm

H2) and helium (180 sccm He) for dewetting, and ethylene

(120 sccm C2H4) was added during the CNT growth step.

5108 C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6

Fig. S2 shows the schematics and time-temperature-flow

sequences. Source gases of He (99.999%, PurityPlus), H2

(99.999%, PurityPlus), and C2H4 (99.999%, PurityPlus) were

used. Real-time height kinetics were measured by a non-con-

tact laser probe (LK-G152, Keyence), as described previously

[32,39].

Synchrotron X-ray characterization was performed ex situ

in the G1 beamline at Cornell High Energy Synchrotron Source

(CHESS), with a monochromatic beam (energy = 10 ± 0.1 keV

and wavelength �0.13 nm), having a height of �10 lm fo-

cused using a single-bounce monocapillary). The sample

was moved vertically using a motorized stage so that forest

height was scanned (from top to bottom) at an increment of

10 lm. Simultaneously, we collected scattering patterns on a

2D area detector (Flicam CCD) as well as the intensity values

upstream and downstream of the forest (Fig. 1A).

Fig. 1 – (A) Schematic of the setup for collecting

simultaneous X-ray scattering and attenuation data. (B)

Schematic of a typical forest and placement of a discrete

slice parallel to the substrate. (C) SEM of CNT forest

morphology (Inset: photograph of a CNT forest). (D) Forest

growth kinetics showing the time evolution of both the

collective forest height and total mass per unit area. The

Gompertz model of population growth is used to fit the

mass-based growth kinetics.

Scattering patterns were utilized to calculate the inte-

grated intensity of line scans between ±10� of the maximum

intensity direction of the inverse space parameter q, and the

mathematical model adopted for fitting these line scans con-

siders the scattering intensity of hollow cylinders. As in our

previous work, we use the probability density function (PDF)

of a log-normal distribution of diameters [13,32] to accurately

represent the polydispersity in CNT diameters, such that:

PDFðDÞ ¼ 1

Drffiffiffiffiffiffi2pp exp

�ðln D� lÞ2

2r2

" #; ð1Þ

where the parameters l and r are related to the mean (D) and

variance (rD) of the CNT diameter distribution as

l ¼ ln D� 12

ln 1þ r2D

D2

� �; and ð2Þ

r2 ¼ ln 1þ r2D

D2

� �: ð3Þ

3. Methodology and results

In order to determine the population dynamics of forest

growth, we use synchrotron X-rays (Fig. 1A) and discretize a

representative CNT forest into 10-lm-thick horizontal slices

parallel to the substrate, as shown in Fig. 1B. Because of the

accuracy of sample alignment procedure and the small size

of the X-ray beam [13], we were able to apply our character-

ization methodology to tens of different CNT forest samples,

some of them as short as 100 lm, and sample dimensions

ranged from 1 to 10 mm. The typical morphology of tortuous

CNTs within a forest is shown in Fig. 1C. Probing the forest

with X-rays and fitting mathematical models to SAXS data

allow us to quantify (within each slice) the diameter distribu-

tion [13,47] and the average orientation, which is expressed by

the orientation parameter (based on the mean square cosine

of the angle with the vertical axis) [13,47–49]. For each slice,

the mass density is calculated based on the Beer-Lambert–

Bouguer law of mass attenuation of X-rays [13]. Combining

these results with real-time forest height measurements un-

iquely lets us calculate the time evolution of the total mass

of the growing CNT population (Fig. 1D), which is a more

accurate representation of the reaction progress than the

height kinetics because the alignment and number density

of the growing CNTs change with time [13,48]. Importantly,

the S-shaped population kinetics (i.e., the time evolution of

population size), which follows the Gompertz model of popu-

lation growth (Fig. 1D), represents a polydisperse population

of CNT sizes within each slice. As in our previous work, the

CNT outer diameter (D), the ratio of inner to outer diameter

(c), and the probability density function (PDF) of diameter dis-

tribution (Fig. 2A) are obtained from fitting SAXS patterns

using a model for log-normally distributed core–shell cylin-

ders [13,32,48].

We then analyze the size-dependent growth behavior of

subpopulations of CNTs via spatiotemporal mapping of the

CNT number density and diameter distribution throughout

the forest. For each slice, we divide the CNT population into

a series of discrete ‘‘bins’’ with 0.5 nm range, and then deter-

mine the time evolution of mass and number density for each

Fig. 2 – Measured diameter-dependent CNT population growth kinetics based on analysis of X-ray scattering, absorption, and

height kinetics data. (A) PDFs for the lognormal distribution of outer CNT diameters, shown for the top and bottom slices,

illustrating the division into subpopulations within each forest slice. (B) Schematic of the CNT population in a slice, which is

broken into a summation of subpopulations with discrete diameter values. (C) Time evolution of the cumulative areal mass

density of each CNT diameter bin within the forest. (D) Time evolution of the normalized cumulative areal mass density of

each CNT diameter (Inset: 3D-plot of the same data).

C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6 5109

subpopulation. Based on the measured diameter PDFs for a

forest grown according to the recipe shown in Fig. S2, approx-

imately 90% of the population (by number) comprises CNTs

having diameters within the 5–20 nm range (calculated by

numerical integration of the PDFs and plotted in Fig. S3).

Hence, for our analysis, we consider that a diameter range

from 1 to 40 nm completely represents the population.

Each quantity, such as mass density or number density, is

designated two subscript indices i and j. The first index, i, is

the slice number, where i = 1 corresponds to the top-most for-

est slice and i = 156 corresponds to the bottom-most slice (for-

est height = 156 · 10 lm = 1.56 mm). The second index, j, is the

bin number, where j = 1 is the first bin having CNT diameters in

the 1–1.5 nm range and j = 78 is the last bin having CNT diam-

eters in the 39.5–40 nm range.

The mass density of each slice is calculated using the Beer–

Lambert–Bouguer law of mass attenuation (Eq. (4)), which re-

lates the mass density of the slice (qi) to the X-ray intensity up-

stream of the CNT forest (I0), the X-ray intensity downstream of

the CNT forest (Ii), and the CNT forest thickness in the direction

of the X-ray beam (t) (as shown in Fig. 1A). This equation uses

the total mass attenuation coefficient of atomic carbon (l/q),

taking both scattering and absorption into account [50]. By inte-

gration with respect to slice thickness in the vertical direction

(z-direction), the mass per unit substrate area, or areal density

(mi) for each slice is calculated (Eq. (5)). Then, the cumulative

areal density of the forest (Mi) at a certain position across forest

height is calculated by summation over all slices above that po-

sition (Eq. (6)),

qi ¼lnðIo=IiÞtðl=qÞ ; ð4Þ

mi ¼ qidz; ð5Þ

Mi ¼Xi

i¼1

qidz: ð6Þ

Now, the ratio between the number of CNTs that belong to

each subpopulation (Nij) and the total number of CNTs in a

slice (Ni) is calculated (Eq. (7)), based on the PDF shown in

Fig. 2A,

Nij

Ni

� �¼Z

slice i

PDFidD: ð7Þ

Combining the calculated areal density (mass per unit sub-

strate area) and the corrected average CNT length (Dli) of each

slice [13,49] with the CNT size distribution, we calculate the

total number of CNTs (Ni) within each slice (Eq. (8)). This total

number can be used to calculate the number of CNTs (Nij)

belonging to each diameter bin (Eq. (9)), which is in turn used

to calculate the effective areal density (mij) represented by the

5110 C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6

CNT subpopulation within each diameter bin (Eq. (10)), as well

as the cumulative areal density (Mj) of each bin (Eq. (11)):

NiðzÞ ¼miðzÞPj

j¼1

Nij

Nj

� �qg

p4 ðD

2j � ðDjcðzÞÞ2ÞDliðzÞ

; ð8Þ

Nij ¼ NiNij

Ni

� �; ð9Þ

mij ¼ Nijqg

p4ðD2

j � ðDjcÞ2ÞDli; ð10Þ

MjðzÞ ¼Xi

i¼1

mij: ð11Þ

The spatial maps of these quantities across the forest height

are then converted into the time domain via the real-time

height kinetics measurements ðdzdt Þi. Hence, the time evolution

of the areal and number density kinetics of CNTs is calculated

for the total population, as well as the subpopulations in each

diameter bin, according to Eqs. (12)–(15):

MðtÞ ¼Z t¼ti

t¼0

dmi

dzi

dzdt

� �i

dt; ð12Þ

Fig. 3 – Measured mass kinetics for selected CNT diameters, and

scale (A), and logarithmic scale (B). Box markers indicate the loca

Gompertz curves illustrating the effect of each parameter on th

dNðtiÞdt

¼ dNðziÞdz

� �dzdt

� �; ð13Þ

MjðtÞ ¼Z t¼ti

t¼0

dmij

dzi

dzdt

� �i

dt; ð14Þ

dNijðtÞdt

¼ dNijðzÞdzi

� �dzdt

� �i

: ð15Þ

By discretizing the CNT population within each slice, we

then calculate the diameter-dependent time-resolved kinetics

of cumulative areal density, as shown in Fig. 2. More results

for the evolution of both the number density and the cumula-

tive areal mass density versus forest height (Fig. S4) and ver-

sus growth time (Fig. S5) are shown in the Supporting

Information. Like the cumulative mass kinetics for the entire

CNT population (Fig. 1D) [13], we find that the mass kinetics of

each diameter also follow an S-shaped curve, which can be

represented by the Gompertz model.

The diameter-dependent kinetics of cumulative areal

mass density, shown in Fig. 2C, are normalized as shown in

Fig. 2D. All curves have the characteristic S shape; however,

the slope at the end of forest growth (the time of collective

forest growth termination [48]) is generally larger for smaller

corresponding fits of the Gompertz model, plotted on linear

tion of the inflection point on each curve in (A). (C) Calculated

e curve shape. The equation is maðtÞ ¼ a:eð�e�k:ðt�sÞÞ .

Fig. 4 – Diameter dependence of the kinetic parameters

extracted from the Gompertz model using the curve fits in

Fig. 3: (A) Change of the asymptote and the maximum

growth rate (at inflection). (B) Change of the time offset (s)

and the rate parameter (k). (C) Change of the growth time

needed for achieving 10% and 90% of the final mass of each

subpopulation (see Fig. S6), as well as the change of the

catalyst lifetime, defined as the difference between the

aforementioned two growth times.

Fig. 5 – Analysis of van der Waals attractive forces between a

pair of CNTs within a forest. (A) Schematic showing two bent

CNTs in contact, modeled as parallel cylinders having

diameters D1 and D2 and length L, separated by distance d (the

graphene layer spacing 0.335 nm is chosen). (B) Scaling of the

calculated VDW force per unit contact length (L) as a function

of CNT diameter (D2) for different CNT diameters (D1).

C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6 5111

CNTs. This indicates that larger diameter CNTs terminate first

and that smaller CNTs may still be growing when the forest

terminates, though these subpopulations do not have suffi-

cient number density to maintain the self-supporting struc-

ture of the forest.

We find that, like the whole CNT population (Fig. 1D),

the population kinetics of each CNT diameter can be repre-

sented by the Gompertz model [51] of population growth

(Eq. (16)),

maðtÞ ¼ aeð�e�kðt�sÞÞ: ð16Þ

In this equation, a is the asymptotic final cumulative mass

of growth, k is a population rate parameter, and s is the

time offset of the inflection point. The inflection point is

the time at which the rate of change of slope of the curve

stops increasing and starts decreasing, and the location of

this point relative to the final value is an inherent property

of the Gompertz curve (37% of its final value). Fig. 3A and B

shows that good fits for our experimental results are ob-

tained using Gompertz curves, indicating that the popula-

tion growth dynamics for all subpopulations follow an

asymmetric sigmoid function.

5112 C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6

4. Discussion

Although the size evolution of catalyst nanoparticles during

growth such as by diffusion through the catalyst support lay-

ers [8] could explain the shift in diameter distribution towards

a smaller diameter (Fig. 2A), the significant change in CNT

number density (first increasing rapidly, then decreasing grad-

ually) shown in Figs. 4S and 5S indicates that catalyst activa-

tion and deactivation are inherently diameter-dependent.

Our findings also highlight the asymmetry of CNT activation

and deactivation kinetics, and their importance to overall

growth of the population. It is important here to define activa-

tion as the stage of growth during which the number of actively

growing CNTs within each diameter bin (0.5 nm range)

increases with time. Conversely, the deactivation stage is when

the opposite happens, i.e., the number of growing CNTs

decreases with time due to termination of individual CNTs

within the subpopulation (see Fig. S5A). In other words, activa-

tion is when the rate of change of the slope for cumulative

mass kinetics is positive (acceleration), and deactivation is

when the second derivative of cumulative mass kinetics is

negative (deceleration). The inflection of the mass versus time

occurs when the maximum density has been reached, and

this is an early indication of the onset of growth termination.

The effect of changing each of the three equation param-

eters (a, k and s) on the shape of the curve is shown in

Fig. 3C. The value of the asymptote (a) is directly proportional

to the final mass of a given population. A low value of the

population rate parameter (k) means that the slope of the

population kinetics at inflection is small, i.e. a lower value

of k indicates that the population growth is more spread on

the time axis. A higher time parameter (s) means that it takes

longer for the population to reach its maximum density, i.e.,

to reach the inflection point. Because the Gompertz model

has only three parameters, the degree of asymmetry is fixed.

As a result, the location of the inflection point is always at

37% of the population asymptote (a). Nevertheless, the good-

ness of the Gompertz fits shown in Fig. 3 indicates that this

attribute is reflected in our data as well.

We plot the parameters of the Gompertz model fits versus

CNT diameter (Fig. 4). Both the asymptote (a) and the maximum

population growth rate (kae ) at inflection increase with diameter,

reach a maximum (representing the mode of the complete CNT

population), and decrease again (Fig. 4A). This non-monotonic

trend is determined by the relative sizes of the CNT diameter

subpopulations, as described by the lognormal diameter

distribution of the whole CNT population (Fig. 2A). Even though

the asymptote and the maximum growth rate each have a

maximum value, the monotonic variation of the parameters s

and k (Fig. 4B) indicates diameter-dependent scaling of CNT

growth behavior. The time of inflection (s) is inversely related

to CNT diameter, so it takes longer for the subpopulations that

have small diameters to reach 37% of their total mass at

termination. Conversely, the population rate constant k

increases with diameter, demonstrating that populations of

larger CNTs grow at a faster rate.

As a guide, both sets of data in Fig. 4B were fit with allome-

tric curves (y = axb), which is a flexible and versatile scaling

formula and has been used to describe size-dependent behav-

ior of chemical systems such as catalytic activity of nanopar-

ticles [52,53] and hydriding and dehydriding kinetics of

nanoparticles [54]. Fitting results give exponent (b) values of

�0.15 and 0.18 for the s and the k curves, respectively. These

scaling results demonstrate that the balance between the

activation–deactivation competition is dependent on diame-

ter, and that the average catalyst lifetimes vary monotonically

with diameter as well.

It is also important to note that the diameter-dependent

growth parameters plotted in Fig. 4 describe population behav-

ior (i.e., the mass kinetics of a subpopulation of CNTs) rather

than the height (or length) kinetics of individual CNTs. For

example, the rate population constant (k) is considered a nor-

malized population growth rate (with respect to a that

represents the quantity of CNTs), so that it can be used to com-

pare the growth dynamics of different subpopulations accu-

mulating different final quantities of CNTs by the end of

growth. The rate constant is not an indication of the growth

kinetics of individual CNTs, because it represents the collective

dynamics of activation and deactivation of many CNTs belong-

ing to a given subpopulation. In fact, our methodology assumes

that both the apparent forest height kinetics and the tortuos-

ity-corrected CNT length kinetics do not depend on CNT diam-

eter. Rather, we suggest that the lengthening kinetics are the

same for all individual CNTs, which is explained later in con-

text of mechanical coupling of CNTs during collective growth.

Because a direct correlation between the size of the catalyst

nanoparticle and CNT diameter has been widely established

[9,40,55–58], we can use our findings to further understand

the diameter-dependence of catalytic activity of nanoparticles

for CNT growth. We define the catalyst lifetime as the time it

takes for the cumulative areal mass density to increase from

10% to 90% of its final value (Fig. 4C). This is considered to be

the region for steady-state growth, where the slope (rate of

mass increase) does not significantly change, thus excluding

both the rapid increase of slope during the early activation

period and the rapid decrease of slope during the final deacti-

vation period. Fig. 4C thus demonstrates that populations of

smaller CNTs have a longer catalyst lifetime, while larger

CNTs deactivate (stop growing) first.

Because the catalyst thin film dewets rapidly into a popu-

lation of particles with a well-defined size distribution [9], we

suggest that the slower activation kinetics of smaller CNTs

are not likely to arise from formation dynamics that are diam-

eter-dependent. Further, it has been discussed that the decel-

eration and final termination of growing CNTs are caused by

catalyst poisoning [59,60], overcoating with carbon [61], diffu-

sion of catalyst through underlying support layer (e.g., Al2O3)

[8,48], particle coarsening by Ostwald ripening [10], and build-

up of stresses at the catalyst-CNT interface due to mechanical

coupling within the forest [62]. To our knowledge, our mea-

surement of diameter-dependent catalytic lifetimes is the

first experimental proof that the CNT deactivation mecha-

nism is size-dependent.

Moreover, while all the CNTs in the population are as-

sumed to grow at a collective (average) rate that represents

the increase of forest height, the growth rate (lengthening

kinetics) of individual isolated CNTs has been shown to de-

pend on the CNT diameter, based on theory and experiments.

For instance, it was reported that smaller CNTs grow faster

than larger CNTs due to enhanced diffusion in smaller catalyst

C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6 5113

particles, resulting from the presence of a viscous state

around the melting point [31]. Nevertheless, this work focused

on small Fe clusters having diameters smaller than 3 nm,

where straight CNTs were grown at low densities on quartz

to favor a horizontal orientation (parallel to the substrate). In

contrast, it has been predicted theoretically that smaller CNTs

should have slower growth rates as a result of having smaller

surface-area-to-circumference ratio [14,36]. The diameter

dependence of CNT growth rate predicted by analytical mod-

els depends on assumed values of constants such as reaction

rates and activation energies. These predictions may be incon-

clusive because the constant values themselves could be

diameter-dependent, as has been proposed recently [14].

And, importantly, mechanical interactions among neighbor-

ing CNTs that arise during the typical coordinated growth of

ensembles have not been taken into consideration.

In principle, the discrepancy between the expected indi-

vidual diameter-dependent CNT growth rates and the single

average growth rate of the population can be explained in

two ways. First, individual CNTs could slide axially with re-

spect to one another to compensate for the possible variation

in rates. However, if slip occurred, the forest-top surface

would not be uniform and flat, and there would be a signifi-

cant gradient in average diameter throughout the forest. Lar-

ger diameter CNTs would dominate the upper portion of the

forest, and the number density of CNTs would increase from

top to bottom, which is opposite to the observed trend. The

second case is where the resultant growth rates of individual

CNTs are not significantly diameter-dependent, i.e., all CNTs

grow at almost the same lengthening rate. As a result, the

CNT forest grows upward as a unified solid body, and individ-

ual CNTs that terminate are pulled from the catalyst inter-

face. We suggest that this is more plausible, due to the

strong mechanical coupling between neighboring CNTs.

Therefore, we explain the difference between size-depen-

dent growth behavior of individual CNTs and the collective

behavior of a forest by considering the strength of mechanical

coupling between CNTs within the forest. Indeed, mechanical

coupling between neighboring CNTs is responsible for the

self-supporting aligned morphology of forests. This occurs

once the CNT number density reaches a critical value for

self-organization, after which the CNTs grow collectively at

the same average rate [48].

We support our claims by calculating the attractive VDW

force between two CNTs. We consider the CNTs to be parallel

cylinders having diameters of D1 an D2, length L, and separa-

tion d (Fig. 5A), according to Eq. (17) [63,64]:

FVDW ¼ �pCq1q2

ð8ffiffiffi2pÞd5=2

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD1D2

D1 þ D2

� �sð17Þ

where

A ¼ pCq1q2; ð18Þ

is the Hamaker constant (�10�19 J for condensed phases).

In Fig. 5B, we plot the diameter-dependent values of the

VDW attractive force per unit contact length L between two

CNTs in contact (d = 0.335 nm) having diameters of D1 = 5, 10

and 15 nm and D2 = 2–40 nm. The results of the model are

considered an upper bound because the CNT is modeled as

a solid cylinder, as opposed to the hollow CNT geometry,

and also because the model considers the case of perfectly

parallel CNTs. Also, our VDW force estimations are calculated

based on a simple two-body model, which might underesti-

mate the actual collective multi-body interactions present

within a forest [65]. These values show that VDW interactions

could apply lateral forces in the range of 10s of nN on CNTs,

for any appreciable contact length (>100 nm). The interfacial

shear strength between CNTs due to VDW interactions was

previously estimated to be in the range of 0.05–0.25 GPa [66],

and the static coefficient of friction between two CNTs was

experimentally measured to be 0.2, which is two orders of

magnitude higher than the kinetic coefficient of friction.[67]

Hence, we conclude that VDW forces are large enough to

prevent slip between CNTs in contact.

As a result of these lateral interactions, axial forces that

are one order of magnitude less (�nN) could arise due to static

friction. This is comparable to the force output of 0.16 nN per

CNT that was approximated by measurements of how much

weight could be lifted by a growing CNT forest [68]. And, in

this previous study, the applied pressure resulted in signifi-

cantly more defective CNTs [68]. We believe that the inherent

collective effects dictate the hierarchical forest morphology

comprising of tortuous CNT bundles, typically observed by

electron microscopy [69].

Although, the alignment might slightly depend on diame-

ter, this has never been demonstrated and is difficult to exper-

imentally ascertain. Even if tortuosity is diameter-dependent,

this dependence is likely to be minor (if any) because of the

complex organization of CNTs and mechanics of bundling.

Hence, the inherent assumption in our methodology that

the alignment (tortuosity) of CNTs within the forest is not

diameter-dependent is justified; as a result, we assume that

all CNTs have the same alignment-based correction function

[13,49] for inferring lengthening kinetics. This assumption is

also supported by morphological studies for CNT forests,

using electron microscopy [13,48,49] and ultra-small angle

X-ray scattering [70], which showed that the bundling behav-

ior is dominant and that the same bundle, which is held

together by VDW interactions, could include CNTs of any

diameters, as long as they are in proximity.

The longest CNTs reported have been grown individually

under alignment by gas flows, without effects of mechanical

coupling, and interestingly these CNTs are of high structural

quality [71]. Hence, the overall role of mechanical competition

in collective CNT growth, and the effect of forces on CNT for-

mation, may be a limiting aspect of CNT forest growth. Forces

may also have an important influence on the properties of

CNT forests, such as electrical and thermal conductivity.

Notably, the properties of CNT forests are typically far below

predictions based on the scaling of properties of individual

CNTs [19,72–74]. Hence, further study is required to elucidate

the dynamic forces applied on CNTs during growth and

understanding their effects on structural defects and resul-

tant properties.

5. Conclusions

Quantification of the size-dependence of CNT growth kinetics

is essential to build a mechanistic understanding of the

5114 C A R B O N 5 0 ( 2 0 1 2 ) 5 1 0 6 – 5 1 1 6

growth process. We have presented the first experimental

technique that enables the study of diameter-dependent acti-

vation and deactivation kinetics of CNT populations. The

technique is non-destructive and can be implemented in situ

during growth, as it is based on combining SAXS measure-

ments, with mass attenuation and real-time forest height

measurements. We find that CNT subpopulations with smal-

ler diameters have slow activation rates and long catalyst

lifetimes compared to subpopulations with larger CNTs and

that the S-shaped Gompertz model can accurately represent

all stages of population growth. Further, interactions due to

attractive VDW forces pin neighboring CNTs in contact, pre-

venting slip among CNTs that grow at different rates. The

combination of diameter-dependent growth behavior and

mechanical coupling causes the typical tortuous CNT mor-

phology in dense forests, and may cause structural defects

in the CNTs. This complex interplay must be taken into ac-

count in CNT growth models and in developing growth recipes

for CNT forests with improved organization and properties.

Acknowledgments

We thank Arthur R. Woll, Erik Polsen and Michael Reinker for

help with X-ray scattering data collection, and Namiko

Yamamoto of MIT for preparation of catalyst substrates.

Financial support was provided by the Department of Energy

Office of Basic Sciences (DE-SC0004927). X-ray scattering was

performed in the G1 beamline at the Cornell High-Energy

Synchrotron Source (CHESS), which is supported by the NSF

and NIH-NIGMS under NSF Grant DMR-0936384. Electron

microscopy was performed at the University of Michigan Elec-

tron Microbeam Analysis Laboratory (EMAL).

Appendix A. Supplementary data

Supplementary data associated with this article can be found,

in the online version, at http://dx.doi.org/10.1016/j.car-

bon.2012.06. 051.

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