Article
pubs.acs.org/JPCC
Model for Self-Rolling of an Aluminosilicate Sheet into a SingleWalled Imogolite Nanotube
Rafael I. González,†,‡ Ricardo Ramírez,§,‡ José Rogan,†,‡ Juan Alejandro Valdivia,†,‡ Francisco Munoz,†,‡
Felipe Valencia,†,‡ Max Ramírez,†,‡ and Miguel Kiwi*,†,‡
†
Departamento de Física, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile 7800024
Facultad de Física, Universidad Católica de Chile, Casilla 306, Santiago, Chile 7820436
‡
Centro para el Desarrollo de la Nanociencia y la Nanotecnología, CEDENNA, Avda. Ecuador 3493, Santiago, Chile 9170124
§
ABSTRACT: Imogolite is an attractive inorganic nanotube, formed from weathered
volcanic ashes, that also can be synthesized in nearly monodisperse diameters. It has found
a variety of uses, among them as an effective arsenic retention agent, as a catalyst support
and as a constituent of nanowires. However, long after its successful synthesis, the details of
the way it is achieved are not fully understood. Here we develop a model of the synthesis,
which starts with a planar aluminosilicate sheet that is allowed to evolve freely, by means of
classical molecular dynamics, until it achieves its minimum energy configuration. The
minimal structures that the system thus adopts are tubular, scrolled, and more complex
conformations as well, depending mainly on temperature. The minimal nanotubular
configurations that we obtain are monodispersed in diameter and quite similar in diameter
both to those of weathered natural volcanic ashes and to the ones that are synthesized in
the laboratory. A tendency toward nanotube agglomeration is also observed, in agreement
with experiment.
■
INTRODUCTION
Since the discovery by Iijima1 of carbon nanotubes (C-NTs),
and their inorganic counterpart (inorganic concentric polyhedral and cylindrical structures of tungsten disulfide) by
Tenne et al.,2 cylindrical structures have been intensely
investigated by experimentalists and theorists. These hollow
cylinders are very attractive because of their fascinating
properties and their practical uses. C-NTs, and several
inorganic ones, are fabricated by electric arc discharge, laser
ablation, or chemical vapor deposition processes, while
inorganic oxide nanotubes, like the imogolite aluminosilicate
that is our present focus of attention, are created mainly by lowtemperature liquid phase chemical processes.3
Among the various NTs, imogolite is especially attractive for
several reasons. In the C-NT synthesis, the graphitic sheet
strain energy required to bend it into a NT decreases
monotonically, as the nanotube diameter increases, thus
precluding the possibility of tuning the NT diameter. With
imogolite, on the contrary, highly monodisperse NT diameters
are created. These NTs have a large arsenic retention capacity4
and, based on the experience acquired with 50 nm diameter
halloysite clay nanotubes,5 imogolite may eventually be used as
a vehicle for drug delivery. Imogolite also shows potential as
additive of transparent polymers, such as molecular sieves,6−8 as
support for insulating polymers,9 as a constituent of nanowires,10,11 as catalyst support,12−14 and as a base for organic−
inorganic nanohybrides.15−21 Moreover, recently Kang et
al.22,23 demonstrated their use as membranes and gas
adsorbents.
© 2014 American Chemical Society
Single-walled imogolite is a clay NT with chemical
composition Al2SiO7H4. It is found in weathered volcanic
ashes,24 whose conformation is illustrated in Figure 1. This
structure, of basic relevance to the study of imogolite, was put
forward in 1972 by Cradwick et al.25 It consists of a curved
gibbsite cylinder with ortho-silicic acid, coordinated via oxygen
with three aluminum atoms. At this point it is also worth
Figure 1. (a) Imogolite unit cell with N = 12 repetitions of the 28
atom circular sector, of angle 2π/N, marked by the black continuous
line. (b) Lateral view of the 28 atom structure that is angularly
repeated to form imogolite. Periodic repetitions of the unit cell are
imposed in the axial direction. Lz is the axial length of the unit cell; H:
light gray, O: red, Si: yellow, and Al: pink.
Received: August 26, 2014
Revised: November 11, 2014
Published: November 13, 2014
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Figure 2. Illustration of the imogolite formation process that we simulate. (1) Planar aluminosilicate sheet; (2) Initial scrolling stages; (3) The NT is
completed; (4) Axial and lateral view of the NT after relaxation; it is noticed that the defects at the NT seam, illustrated in (3), have disappeared. Lz
is the axial length of the unit cell.
mentioning that Tamura and Kawamura,26 already in 2002,
used molecular dynamics to model the structure of tubular
gibbsite, finding that the total energy of the tubular gibbsite
molecules, relative to that of flat gibbsite, tends to zero as the
radius of the tube increases. In other words, the gibbsite energy
does not have a minumum as a function of radius, which
precludes diameter tuning. Shortly after the work of Cradwick
et al.25 became known, Farmer et al.27 published a protocol for
the synthesis of imogolite. Independently of the diverse
synthesis procedures that have been reported afterward, the
imogolite diameters turned out to be highly monodisperse.27−31
Consequently, it is of interest to explain this behavior on the
basis of a microscopic model calculation,32 as the one we report
here.
The chemical formula for the Cradwick unit cell, ordered
from the outside of the NT inward, is [(OH)3Al2O3SiOH]2N,
where N is the number of repetitions of the circular sector, of
angle 2π/N, that are required to build the NT cylinder
illustrated in Figure 1b. Moreover, the values N ≈ 10 for natural
and N ≈ 12 for synthetic imogolite have been reported.27,30,33,34 For the latter, the measured external diameter
is 2.3 nm, with an average length of ≈100 nm. The imogolite
diameter can be modified by the replacement of Si by
Ge30,32,34−40 and, in this way, fine-tune the NT properties
and capabilities, as reported by Mukherjee et al.,30 who
obtained 3.3 nm diameter Ge NTs, of lengths of the order of 15
nm.
Yucelen et al.41−44 contributed to elucidate the role of
molecular precursors and nanoscale intermediaries, between 1−
3 nm in size, in the formation of single-walled aluminosilicate
NTs. They also determined experimentally the relation
between precursor structure and shape with the final imogolite
NT curvature. Moreover, they, as well as Maillet et al.,45
formulated a generalized kinetic model for the formation and
growth of single-walled metal oxide nanotubes. On the other
hand, Levard et al.46,47 made significant contributions to the
understanding of the role of the precursor shape and
composition in the formation of imogolite, and Bac et al.48
synthesized single-walled hollow aluminogermanate nanospheres.
In spite of the fact that the work by Cradwick et al.25 and
Farmer et al.27 are 40 years old, and more than 30 years have
passed since Wada et al.35 published the protocol for the
synthesis of Ge-imogolite, the interest in the subject has not
faded.39,41,42,44−46,49−54 Moreover, Maillet et al.45,55 reported in
2010 the synthesis of double-walled Ge-imogolite, and later on,
Thill et al.37 showed that the formation of double walled Siimogolite NTs, as well as that of Ge-imogolite thicker than
double-walled NTs, is quite unlikely. In 2012, Thill et al.38
reported that, in the synthesis process of imogolite-like NTs, a
very small proportion of nanoscrolls formed when the Si
content in the mixture of Si and Ge is 10%. They may be
envisioned as an intermediate stage between double- and triplewall NTs.
All these results suggest that the self-rolling mechanism, as a
path toward the formation of imogolite NTs, is a subject worth
exploring from a theoretical point of view. What we report
below are the results of classical molecular dynamics
simulations. We start with an ideal planar aluminosilicate
sheet and study the dynamics of the self-rolling that gives rise to
imogolite NTs, and related curved structures, at various
temperatures. We also compare these structures with the
relaxed ones that are obtained, starting with tubular initial
conditions (TIC) after their energy is minimized.
As mentioned above, the synthesis of imogolite is mainly
carried out by low-temperature liquid phase chemical
processes.3 However, we choose to treat these problems
separately since the simulation of the liquid sysnthesis implies a
detailed study of the action of the solvent. In addition, it is
significantly more intricate, as it implies systems an order of
magnitude larger than the ones we report here, and we leave
these issues for future work.
This paper is organized as follows: after this Introduction we
describe the method we used in Method. Next, the results
obtained are given in Results, and the paper is closed with
Summary and Conclusions.
■
METHOD
Our main objectives are to describe the dynamics of the rolling
up process that starts with a flat aluminosilicate sheet and ends
up as an imogolite NT or a related structure, and to find the
minimum energy configurations, these structures do adopt,
over a wide temperature range (10 K ≤ T ≤ 368 K). For the
time being, we ignore the role played by precursors.41,42,53 Our
present objective is illustrated in Figure 2, where the following
are illustrated: (1) flat aluminosilicate sheet we denominate as
planar initial conditions (PIC), (2) an intermediate stage of the
bending process, (3) the closing of the structure into an N = 12
imogolite NT, which is allowed to relax in order to reach the
perfect tubular conformation shown in (4), where both an axial
and a side view are presented. These boundary conditions are
adopted in order that the formation of a seamless tubular
conformation is possible, in spite of the fact that the aluminum
atoms at the edges are not fully coordinated.
To achieve the above outlined objectives, the principal tool
we use is molecular dynamics (MD) simulations. These MD
simulations, as well as the structural relaxations, were carried
out using the large-scale atomic/molecular massively parallel
simulator (LAMMPS) code.56 For the atomic interactions, the
CLAYFF potential57 is used, since it has been proven to be
adequate to model aluminosilicate (imogolite) nanotubes.32,36,58−60
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The CLAYFF potential, developed by Cygan et al.,57
incorporates the charges of every single atom, the van der
Waals interaction, and a harmonic potential for the O−H group
stretching. Analytically it is given by
E=
e2
4π ϵ0
∑
i≠j
In order to check our results, we also used tubular initial
conditions (TIC). We chose to use the Fast Inertial Relaxation
Engine (FIRE),63 in combination with conjugate gradient
minimizations, to optimize the NT conformation and the
length of the simulation cell, with encouraging results. The
LAMMPS implementation of FIRE does not allow to vary the
length of the simulation box; however, it does implement
conjugate gradient. On the other hand, FIRE yields lower
energy structures. Because of these reasons, we combined both
methods, cell length and structural optimization, successively
and repeatedly to achieve both better conformations and cell
lengths. For the aluminosilicate structure, we adopted the
model by Cradwick,25 with 9 ≤ N ≤ 24 values. Each one of
these structures was relaxed using the FIRE algorithm,63−65 as
implemented in the LAMMPS code, in order to have a
representative set of structures to compare the rolling
simulations with already reported results, varying the simulation
parameters and carrying out convergence tests. We started our
simulations with 30 slightly different tubular imogolite NTs;
that is, they differed just by random displacements (of less than
0.1 Å) of the atoms from their “bulk” equilibrium positions.
This process was carried out for each N value, the number of
repetitions of the circular sector structure illustrated in Figure
1b.
⎡⎛
qiqj
+
rij
12
⎛ R ⎞6 ⎤
R 0, ij ⎞
⎟⎟ − 2⎜⎜ 0, ij ⎟⎟ ⎥
⎥
r
⎝ rij ⎠ ⎦
⎣⎝ ij ⎠
∑ εij⎢⎢⎜⎜
i≠j
+ kij(rij − r0)2
(1)
where the qi are the partial charges, obtained by means of
quantum mechanics calculations, e is the electron charge, and ε0
is the dielectric vacuum permittivity. The second summation is
the van der Waals contribution, where R0,ij and εij are empirical
parameters derived by fitting to bulk structural and physical
properties, and rij are the distances between atoms i and j. The
last term in eq 1 describes the O−H bond stretch energy by
means of a simple harmonic interaction, where the kij are the
stretching constants, and r0 is the equilibrium O−H distance,
given in Table 1. The work by Cygan et al.57 does not
incorporate charge transfer and redistribution, and here we use
the same approach in order to avoid unphysical results.
Table 1. CLAYFF Parameters;57 “Bond Parameters”
Parameterize the Bond Stretching
■
RESULTS
In Figure 3a we show the imogolite energy that we obtain for
TIC as a function of N, and compare it with the ones that result
keeping fixed the simulation cell length Lz = 8.40 Å, as done by
Konduri et al.36,58 It is seen that the minimum corresponds to
N = 10, in agreement with first-principles calculations,66 and
with the configuration of natural imogolite. It is also apparent
nonbond parameters
species
hydroxyl H
hydroxyl O
bridging O
octahedral Al
tetrahedral Si
charge(e)
ε (eV)
0.425
0
−0.950
6.7
−1.050
6.7
1.575
5.7
2.100
8.0
bond parameters
×
×
×
×
10−3
10−3
10−8
10−8
R0(Å)
0
3.5532
3.5532
4.7943
3.7064
species i
species j
k (eV/Å2)
r0 (Å)
hydroxyl O
hydroxyl H
24.03
1.0
The Coulomb forces are long-ranged and, thus, converge
slowly as a function of the size of the simulation box, requiring
special techniques to handle them and to evaluate their
contribution since the size of the simulation box is a crucial
issue. Usually they are treated by means of an Ewald sum,61 but
that requires computer times that are proportional to the
square of the number of atoms. Here we handle them by means
of the Particle−Particle−Particle Mesh (P3 M) method,62
which is a Fourier-based Ewald summation procedure,
significantly more efficient, and that allows for parallelization.
To ensure accuracy we took six repetitions of the unit cell along
the symmetry axis to properly account for these long-range
Coulomb interactions. Moreover, the use of periodic boundary
conditions in the axis direction contributes to mantain the
atomic coordination.
The flat aluminosilicate is allowed to evolve freely in 2D, at a
given temperature, to seek its minimum energy conformation.
Once this putative minimal energy is reached, a 3D Gaussian
velocity distribution, consistent with the given temperature, is
applied to the system, which is allowed to evolve for 200 ps,
and we use a time step of 1 fs in all our calculations. This
procedure is carried out for a large number of randomly
different seeds. In all these cases the aluminosilicate sheet rolled
up, adopting a variety of shapes, but mainly forming nanotubes
and nanoscrolls.
Figure 3. Energy E as a function of N. The blue × correspond to the
case where the length Lz of the unit cell was kept fixed at 8.40 Å. The
open black circles are the values of E/N obtained for the relaxed values
of Lz. Only minor differences are observed. (b) Unit cell length Lz per
angular repetition N, as a function of N. The completely relaxed
structure corresponds to T = 0. The nonzero temperature values were
obtained as a NPT average at 1 bar.
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that the difference in energetics between keeping the cell size
fixed and freeing it is not very significant at T = 0.
There is general agreement in the literature on the structure
and composition of the imogolite unit cell. However, the value
of N varies between N = 10 (natural) and N = 11−15
(synthetic).42 In addition, Yucelen et al.41−43 showed that the
NT diameter can be controlled by the replacement of the
anions involved in the synthesis,42 and Lee et al.67 confirmed
recently that the variation of the temperature also plays a major
role. But, to the best of our knowledge, the length of the unit
cell as a function of temperature and NT diameter has received
little attention. The first experimental measurements25,35
reported a length Lz of 8.40 Å. However, in 2005 Mukherjee
et al.30 reported a measured value of 8.51 Å, while ab initio
calculations by Demichelis et al.66 used a length of 8.45 Å, and
Li et al.68 adopted an even larger value of 8.70 Å.
Since imogolite is stable at room temperature and only loses
its tubular structure69 around 770 K, we investigated its thermal
behavior as a function of N and of temperature T, up to 368 K,
which is the temperature at which the synthesis is carried out
experimentally. In Figure 3b the unit cell length Lz is plotted as
a function of N for T = 0 (green), and for several final
temperatures, Tf, where Tf = 10, 150, 300, and 368 K, and the
pressure was kept constant at 1 atm by means of the Nosé−
Hoover algorithm. Next we focus our attention on the N = 12
case, which is the one more frequently obtained in experiments.
After structural relaxation, we obtain for N = 12 NT lengths of
8.39, 8.41, 8.44, and 8.50 Å at 10, 150, 300, and 368 K,
respectively. In addition, it is worth remarking that the values
we obtain coincide with those measured experimentally.30
Consequently, the different experimental results on Lz can be
assigned to the temperature variation.
Planar and Tubular Initial Conditions. Our main
concern is the way in which an aluminosilicate sheet could
roll up to form imogolite NTs. Therefore, we start with the
planar (2D) counterpart of the unit cell illustrated in Figure 1
outstretched in the x and in the z-(axial) directions, while in the
z direction periodic boundary conditions are adopted. In order
to shed light on the rolling process we start with planar initial
conditions (PIC) and allow the system to evolve. This is carried
out in two steps: first, with periodic boundary conditions for
the 2D unit cell along the z-axis, we apply a temperature ramp
from 1 K up to a given final temperature Tf (Tf = 10, 150, 300,
and 368 K), at a rate of 7.4 K/ps, restricting the system to
remain planar. Once the system reaches Tf, and keeping the
temperature fixed during 200 ps, random initial velocities are
given to the atoms, and allowing them to move in 3D in contact
with a thermostat. Shortly after the planar constraint is
removed, that is within the next 50 ps, rolling up was observed.
The system was then allowed to evolve for 200 ps. The average
total energy of the configurations was calculated for 50 different
initial velocity realization on the planar sheet, and the lowest
energy configuration was kept. This process was briefly
described above and illustrated in Figure 2.
In Figure 4 we plot the total energy per circular sector
structure (illustrated in Figure 1b) of the minimal energy
conformation we obtain, for temperatures of 10, 150, 300, and
368 K for both planar (PIC) and tubular initial conditions
(TIC). The latter correspond to the perfect imogolite NT of
Figure 1a. The 10 K structure was investigated to eliminate
thermal effects, and in this way we also avoid defects along the
NT seam. In contrast, 368 K is the temperature at which the
NT are grown in the laboratory.36,39 At T = 10 K the energies
Figure 4. Thermodynamic average of E/N for temperatures of 10, 150,
300, and 368 K. The black x correspond to tubular (TIC), and the red
circles to planar (PIC), initial conditions. Snapshots of the scrolled
conformations, obtained for the minimum energy, are provided for
representative N values. The dashed horizontal line corresponds to the
total energy of two interacting N = 11 NTs at the same temperature.
for the N = 12−18 NTs obtained with PIC and TIC show little
difference, as can be seen in Figure 4a, since the NT generated
with PIC become cylindrical and practically defect free. For N =
8−11 the seam of the PIC NTs are not defect free, which
reflects in an energy increase. The defects consist in H atoms
misplaced along the outer wall, and OH groups pointing
inward. It is also relevant to point out that a small finite
temperature is sufficient to shift the minimum energy from N =
10 to N = 11, as can be observed in Figure 3 for the TIC case.
For N = 19, we observe an energy peak and thereafter, for N >
19, a transition to a double tube (“binocular”) structure, thereby
reducing considerably its energy in relation to larger diameter
single nanotubes. Experimentally, a NT agglomeration during
growth has been observed.30 In this context, we calculated the
energy of two parallel N = 11 NTs that are close to each other,
which provides a lower energy bound. At higher temperatures
(see Figure 4b−d), a tendency to the formation of nanoscrolls
rather than nanotubes is apparent, suggesting that partial
charges and their Coulomb interaction are key ingredients.
These nanoscrolls have been reported38 for Ge-imogolite (with
0.1 Si/[Si + Ge]), but not for pure Si imogolite. Nevertheless,
the energy of two coupled N = 11 imogolite NTs, indicated by
the green dashed lines in Figure 4, is favorable when compared
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to all conformations that form spontaneously. Hence, the NT
packing is an issue that does deserve attention.
At this point it is convenient to indicate that to calculate the
energy of the two coupled N = 11 imogolite NTs we did as
follows: (i) the optimized N = 11 NT was copied identically
and laid 2.5 nm appart and parallel to the first NT, with
periodic boundary conditions along the tube axis; (ii) next, the
two tube system was optimized using the conjugate gradient
method, followed by 200 ps of MD at the corresponding
temperature. The value of N = 11 was chosen because it is
intermediate between N = 10 (the imogolite found in volcanic
ashes) and N = 12, the one synthesized experimentally.
Moreover, the formation of Si and Ge double wall NTs was
recently addressed by Guimaraes et al.70 using the selfconsistent charge density functional tight-binding method.
They concluded that double wall NTs are energetically
unfavorable and that single-wall NTs prefer to adopt zigzag
chirality. We obtained similar results using classical molecular
dynamics.
In Figure 4b the T = 150 K results are presented, and we
observe that for 8 ≤ N ≤ 14 almost perfect NTs form, which
moreover, are very similar in energy per atom and very close to
the N = 11 minimum, independently of the choice of PIC or
TIC. In contrast with the 10 K results, kinetic effects help with
the healing of the nanotube seam. A nanoscroll first forms for N
= 15, followed by “binocular” configurations for larger N.
However, for 23 ≤ N ≤ 27, a double tube structure is more
favorable. For N = 22, the nanoscroll structure yields the
minimal energy over the whole range of 8 ≤ N ≤ 30 values.
However, the energy of two coupled NTs is slightly more
favorable.
The dynamics at 300 and 368 K are illustrated in Figure 4c,d.
At 300 K, the local minimum, N = 9 for PIC, is present, and
when its structure is inspected (see inset), a rather coarse match
at the seams is clearly visible. However, the energies for N = 10
and N = 11 are very similar to this local minimum. For 300 K
and N ≥ 12, only nanoscrolls form, which yields a decrease of
the energy as a function of N up to N = 22, the global minimum
for our calculations at T = 300 K, due to the fact that more OH
groups interact. Much the same occurs at T = 368 K, but the
minimum at N = 10 is much sharper, several local minima are
present, and both scrolled and double NTs are generated, as
displayed in the insets of Figure 4d.
In the above-discussed simulations it is apparent that as the
temperature increases the maximum value of E/N decreases.
These maxima coincide with the N values above which the NTs
fail to close as a single cylindrical tube. This behavior is
illustrated in Figure 5, where we plot the N value for the
maximum energy as a function of T for PIC (N > 10) and the
global minimum for TIC. For PIC, a linear decrease of the
largest E/N versus T is observed. The implication is that kinetic
effects limit the NT diameter (maximum N). This is in
agreement with the results reported by Lee et al.67 who
observed that by varying the temperature the diameter of the
NTs can be tuned. On the other hand, the minimum obtained
using TIC sets a lower bound for the NT diameters, generating
the rather monodisperse tubes observed experimentally.
To stress the role played by temperature in Figure 6a,b, we
sketch the structure of representative low lying energy NTs and
indicate their respective energies at T = 150 and 300 K. At 150
K there is a predominance of closed NTs for N < 15. The same
trend is observed at 300 K, but limited to N < 13. For T = 150,
a global minimum is present for N = 22, which corresponds to a
Figure 5. Number of repetitions N of the local energy maxima for PIC,
and energy minima for TIC as a function of temperature.
Figure 6. Families of conformations and of their energies, obtained
using PIC; (a) at 150 K; (b) at 300 K. The red dashed line indicates
the lowest energies for the various structures.
nanoscroll in a predominantly double tube scenario. This is in
contrast to what occurs at 300 K, where the majority of the
conformations for N > 13 are nanoscrolls and only a few double
tubes are present, which underscores the importance of kinetic
effects.
■
SUMMARY AND CONCLUSIONS
The main objective of this contribution is to shed light on the
dynamics of the rolling process of an aluminosilicate sheet into
imogolite-like nanostructures of various diameters and to
determine the relevance temperature has on this procedure.
To achieve this goal, we start with a planar aluminosilicate sheet
and allow it to evolve spontaneously by means of classical
molecular dynamics. We also construct tubular NTs to use as
reference and control of our simulations.
The first item we addressed is the relevance of keeping fixed
the length of the unit cell that is repeated using periodic
boundary conditions, and compare with results obtained when
this length is allowed to relax in the axial direction. We found
that the length of the unit cell along the imogolite symmetry
axis does not have a significant relevance for the energy of the
imogolite NTs, however, it explains the difference between
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several experimental results. In fact, the values we obtained for
these lengths are in good agreement with experiment.
When the simulations are started with planar initial
conditions, we obtain several possible rolled conformations as
the end result: nanotubes, nanoscrolls, and double nanotubes
and nanoscrolls (“binocular” like configurations), depending on
diameter and temperature. The minimal energy tubular
structure at 150 K is obtained for N = 11, while the natural
one corresponds to N ≈ 10 and the synthetic one, created at T
≈ 368 K, is of N ≈ 12. All in all, a rich assortment of
configurations is obtained as the temperature and radius (N
values) are varied. It is also worth remarking that kinetic factors
and agglomeration are issues that play a significant role in the
synthesis of imogolite. All these features do emerge naturally in
our simulations.
An additional feature that our simulations do yield is the
diameter monodipersion that is observed in both natural and
synthetic imogolites. This feature, which is only slightly affected
by temperature variations in the range that experiments are
carried out,67 can be traced to kinetic effects that limit imogolite
NT diameters.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
This work was supported by the Fondo Nacional de
́
Investigaciones Cientificas
y Tecnológicas (FONDECYT,
Chile) under Grants #11110510 (F.M.), #3140526 (R.I.G.),
#1110630 (R.R.), #1110135 (J.A.V.), #1120399 and 1130272
(M.K. and J.R.), and Financiamiento Basal para Centros
́
Cientificos
y Tecnológicos de Excelencia (R.R., J.R., J.A.V.,
F.M., F.V., M.R., and M.K.). F.V. was supported by CONICYT
Doctoral Fellowship Grant #21140948.
■
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