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Kinetic Monte Carlo modelling of the initial stages ofzeolite synthesisZhang, X.
DOI:10.6100/IR724487
Published: 01/01/2012
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Citation for published version (APA):Zhang, X. (2012). Kinetic Monte Carlo modelling of the initial stages of zeolite synthesis Eindhoven: TechnischeUniversiteit Eindhoven DOI: 10.6100/IR724487
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Download date: 25. Jun. 2018
Kinetic Monte Carlo Modelling of The Initial Stages
of Zeolite Synthesis
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen
op woensdag 25 januari 2012 om 16.00 uur
door
Xueqing Zhang
geboren te Jinan, China
Dit proefschrift is goedgekeurd door de promotor:
prof.dr. R.A. van Santen
Copromotor:
dr. A.P.J. Jansen
This work is supported by the Netherlands Organisation for Scientific Research (NWO).
Copyright c⃝2012 by Xueqing Zhang
Printed by Ipskamp Drukkers
Cover design: L. Wang and X. Zhang
A catalogue record is available from the Eindhoven University of Technology Library.
ISBN: 978-90-386-3081-6
To my wife Liping
Contents
1 Introduction 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Simulation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Scope of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Theory of the Off-lattice Kinetic Monte Carlo 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 The Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Process-type Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Unimolecular Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Bimolecular Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.5 Composite Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.6 Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 KMC Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Asynchronous Updating of Particle Positions . . . . . . . . . . . . . . 24
2.3.2 Determining Reaction Times . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Determining the Position of the Reaction . . . . . . . . . . . . . . . . 28
2.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Mechanism of Silicate Oligomerization 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Model and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Theory of Continuum kMC . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Model of the Oligomerization . . . . . . . . . . . . . . . . . . . . . . 44
CONTENTS
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 COSMO and Explicit-Water Model at Neutral pH . . . . . . . . . . . 54
3.3.2 Explicit-Water and Mean-Field Model at Neutral pH . . . . . . . . . 55
3.3.3 Explicit-Water Model at Different pH and temperature conditions . . 61
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Effects of Counterions 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Silicate Oligomerization and Gelation 97
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Model and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Summary 111
List of Publications 115
Curriculum Vitae 117
Acknowledgements 119
vi
Chapter 1
Introduction
1.1 General Introduction
Since their discovery zeolites have attracted very wide scientific interest because of their
many applications to important processes such as gas separation, softening of water, catal-
ysis in oil refining, petrochemistry and fine chemistry.1,2
Zeolites were discovered in 1756 by the Swedish mineralogist Cronstedt, who named
them from the Greek words zein and lithos, meaning boiling stone.1,2 From 1862 early
efforts were made to synthesis zeolite by mimicking geological conditions (T > 200 ◦C,
p > 100 bar). Large scale hydrothermal synthesis of zeolite started in the 1940’s but
received considerable attention only in the last past decades, and have today resulted in
many commercial materials.3
Considerable efforts were made to understand the zeolite synthesis process to control their
pore structure and morphology.4–10 However, the process is complicated, involving surfac-
tant self-assembly in solution, mesophases, and silicate condensation reactions. Therefore,
the synthesis of zeolite is a very challenging task. A variety of experimental and theoret-
ical techniques have been developed to study the silica-based condensation processes,6,8–18
however, the formation mechanisms are still poorly understood.19,20 The early stages of so-
lution oligomerization play a decisive role in determining the final structure.21 A detailed
investigation of this process is, therefore, very valuable. The essential difficulty of study-
Introduction
ing the prenucleation process arises from the fact that the silicate oligomers are typically
of the size of several Si(OH)4 molecules, which is hardly accessible to most of the current
experimental methods. Even if they are detected by microscopic techniques, the structural
and reactive properties may not be distinguished due to their small size. Furthermore, the
species freely move throughout the available volume of solution, reducing the change of their
appearance in the volume being examined.21 Thus, it is extremely challenging to experi-
mentally probe the crucial early stages of the synthesis at the molecular level. The current
theoretical models also have many drawbacks. Electronic structure calculations15,22,23 and
molecular dynamics (MD)17 have been used to probe the formation of zeolites and meso-
porous materials. Information about energetics of chemical reactions can be obtained by
using DFT calculations, but it is very difficult to predict the kinetics just from the reaction
profiles. Stable structures of silicate oligomers can be obtained from DFT calculations, but
the most stable oligomers might not be the most preferable products. Moreover, DFT and
MD methods are computationally very expensive and restricted to very small system and
short simulation time (on the order of pico- or nanoseconds), thus relevant statistical infor-
mation cannot be extracted. The time scale for initial stage of zeolite formation is on the
order of hours or even longer, which is not accessible to MD or DFT simulations. Monte
Carlo models can access larger length and time scale.9,10,24,25 Hybrid Monte Carlo,9 lattice
based coarse-grained Monte Carlo,10 and reactive Monte Carlo25 models have been used to
probe the silicate oligomerization reactions. However, the current Monte Carlo methods are
still not adequate to represent reactions in solutions. Potentials or force fields have been
used in the Monte Carlo studies mentioned above to describe the particle interactions. The
application of potentials allows for the simulation of large silicate clusters, which, however,
falls short of detailed information of small oligomers. The fitted energy parameters require
further calibration. Another drawback is that there is no real time in these methods. In the
methods above, how the effects of pH and template molecules can be adequately represented
remains somewhat problematic. The current difficulties in modelling reactions in solutions
motivated us to develop a new method. In this thesis, we will present a new form of off-
lattice kinetic Monte Carlo method, which is particularly suitable for study of reactions in
solutions.
2
1.2 Simulation Methodology
1.2 Simulation Methodology
The effective modelling of the initial stages of zeolite formation requires a method that can
simulate events at microscopic length and macroscopic time scale. Kinetic Monte Carlo
has the potential to shed light on the mechanisms of silicate oligomerization and gelation
processes.
In this thesis we develop an off-lattice kinetic Monte Carlo (kMC) theory,26 which we
call continuum kMC, to model silicate oligomerization reactions in water solution. In this
theory, we take the general approach and apply it to reactions in solutions. We show that
we can simplify the kMC simulations in such a way that the reactions can be determined
independently from the simulations, just as for lattice-gas kMC. We treat the diffusion of
molecules in the solution analytically. Because we then only need to simulate the reactions
explicitly, the time that a simulation takes is drastically reduced. The details of this method
are presented in Chapter 2.
We use a lattice kinetic Monte Carlo model27 to investigate a latter stage of zeolite
synthesis, the aggregation of silicate oligomers and gelation process. In the lattice kinetic
Monte Carlo simulations the system is treated as a grid. A grid is a collection of sites. Each
site has a label, which characterizes its properties like vacant, occupied, what adsorbate,
etc.27,28 In this thesis, a configuration of a silicate oligomer is then given by a particular
distribution of labels. The change of the labels, meaning the evolution of a system, is
given by diffusions and reactions of the silicate oligomers. This change can mathematically
be formulated by the chemical Master Equation (Eq. 1.1) and can be derived from first
principles. The chemical Master Equation describes the configurational change of a system
as function of time.
dPα
dt=∑β
[WαβPβ −WβαPα] . (1.1)
Pα(β) is the probability of the system being in a configuration α(β) and Wαβ(βα) is the rate
constant of the change of configuration β into α (α into β). The rate constants determine
the probability of a particular configuration and the speed of its creation and destruction.
Values for the rate constants can come from density functional theory calculations.
3
Introduction
1.3 Scope of This Thesis
The formation of zeolites consists of several stages: first an oligomerization process which
eventually leads to the formation of sub-colloidal particles, second the nucleation process,
and finally crystal growth. This thesis deals with the theoretical investigations of the initial
stages of zeolite synthesis, which is the silicate oligomerization and gelation. During the first
hours of oligomerization, various silicate oligomers are formed in solution. The dominant
species depends sensitively on the reaction conditions. It has long been known that the
synthesis condition directly influence the resulting crystal lattice. The important factors
that control the zeolite synthesis are pH and the temperature of the solution, and the
presence of counter ions and template molecules. In general, high pH values increase the
crystal growth rates and shorten the nucleation period. The temperature can alter the zeolite
structure as well as the induction period and crystal growth kinetics. Structure direction
occurs when inorganic or organic molecules are used to direct the crystallization towards a
specific zeolite structure. Structure-directing agents are generally inorganic cations such as
Na+, K+, Li+, Ca2+ and organic molecules like TPA+ TMA+ or TEA+.29
Although our knowledge of the general aspects of zeolite synthesis has improved signif-
icantly since the discovery of these materials, the mechanism by which silicate oligomers
interact to form clusters in the early stages of the process is still not known precisely. Sev-
eral questions will be answered in this thesis. What is the dominant silicate species in the
early oligomerization process? How do the pH and temperature of the solution influence
the early oligomerization? Do counterions play an important role in determining the sil-
icate structures? What is the growth mechanism of silicate species? How do the silicate
oligomers aggregate? How does the silicate gelation proceed? The difficulty in answering
those questions is mostly due to the high complexity of the synthesis process. The main
goal of this thesis is therefore to gain insight into the mechanism of the initial stages of
zeolite synthesis.
Bibliography
[1] Barrer, R. M. Hydrothermal Chemistry of Zeolites ; Academic Press: London, 1982.
[2] Murugavel, R.; Walawalkar, M. G.; Dan, M.; Roesky, H. W.; Rao, C. N. R. Acc. Chem.
4
BIBLIOGRAPHY
Res. 2004, 37, 763.
[3] Thomas, J. M.; Thomas, W. J. Principles and Practice of Heterogeneous Catalysis ;
VCH: New York, 1997.
[4] Kirschhock, C. E. A.; Ravishankar, R.; Verspeurt, F.; Grobet, P. J.; Jacobs, P. A.;
Martens, J. A. J. Phys. Chem. B 1999, 103, 4965.
[5] Depla, A.; Lesthaeghe, D.; van Erp, T. S.; Aerts, A.; Houthoofd, K.; Fan, F.; Li, C.;
Speybroeck, V. V.; Waroquier, M.; Kirschhock, C. E. A.; Martens, J. A. J. Phys. Chem.
C 2011, 115, 3562.
[6] de Moor, P. P. E. A.; Beelen, T. P. M.; van Santen, R. A.; Beck, L. W.; Davis, M. E.
J. Phys. Chem. B 2000, 104, 7600.
[7] Catlow, C. R. A.; Bromley, S. T.; Hamad, S.; Mora-Fonz, M.; Sokol, A. A.; Wood-
ley, S. M. Phys. Chem. Chem. Phys. 2010, 12, 786.
[8] Knight, C. T. G. J. Chem. Soc., Dalton Trans. 1988, 1457.
[9] Wu, M. G.; Deem, M. W. J. Chem. Phys. 2002, 116, 2125.
[10] Jorge, M.; Auerbach, S. M.; Monson, P. A. J. Am. Chem. Soc. 2005, 127, 14388.
[11] Kirschhock, C. E. A.; Ravishankar, R.; Looveren, L. V.; Jacobs, P. A.; Martens, J. A.
J. Phys. Chem. B 1999, 103, 4972.
[12] Kirschhock, C. E. A.; Ravishankar, R.; Jacobs, P. A.; Martens, J. A. J. Phys. Chem.
B 1999, 103, 11021.
[13] Depla, A.; Verheyen, E.; Veyfeyken, A.; Houteghem, M. V.; Houthoofd, K.; Spey-
broeck, V. V.; Waroquier, M.; Kirschhock, C. E. A.; Martens, J. A. J. Phys. Chem. C
2011, 115, 11077.
[14] Mora-Fonz, M. J.; Catlow, C. R. A.; Lewis, D. W. Angew. Chem. Int. Ed. 2005, 44,
3082.
[15] Trinh, T. T.; Jansen, A. P. J.; van Santen, R. A.; Meijer, E. J. Phys. Chem. Chem.
Phys. 2009, 11, 5092.
5
Introduction
[16] de Moor, P. P. E. A.; Beelen, T. P. M.; van Santen, R. A. J. Phys. Chem. B 1999,
103, 1639.
[17] Rao, N. Z.; Gelb, L. D. J. Phys. Chem. B 2004, 108, 12418.
[18] Verstraelen, T.; Szyja, B. M.; Lesthaeghe, D.; Declerck, R.; Speybroeck, V. V.; Waro-
quier, M.; Jansen, A. P. J.; Aerts, A.; Follens, L. R. A.; Martens, J. A.; Kirschhock, C.
E. A.; van Santen, R. A. Topics in Catalysis 2009, 52, 1261.
[19] Auerbach, S. M.; Ford, M. H.; Monson, P. Curr. Opin. Colloid Interface Sci. 2005, 10,
220.
[20] van Santen, R. A. Nature 2006, 444, 46.
[21] Erdemir, D.; Lee, A. Y.; Myerson, A. S. Acc. Chem. Res. 2009, 42, 621.
[22] Pereira, J. C. G.; Catlow, C. R. A.; Price, G. D. Chem. Commun. 1998, 1387.
[23] Trinh, T. T.; Jansen, A. P. J.; van Santen, R. A. J. Phys. Chem. B 2006, 110, 23099.
[24] Schumacher, C.; Seaton, N. A. Adsorption 2005, 11, 643.
[25] Malani, A.; Auerbach, S. M.; Monson, P. A. J. Phys. Chem. Lett. 2010, 1, 3219.
[26] Zhang, X.-Q.; Jansen, A. P. J. Phys. Rev. E 2010, 82, 046704.
[27] Lukkien, J. J.; Segers, J. P. L.; Hilbers, P. A. J.; Gelten, R. J.; Jansen, A. P. J. Phys.
Rev. E 1998, 58, 2598.
[28] Zhang, X. Q.; Offermans, W. K.; van Santen, R. A.; Jansen, A. P. J.; Lins, A. S. U.;
Imbihl, R. Phys. Rev. B 2010, 82, 113401.
[29] Houssin, C. J.-M. Y. Nanoparticles in Zeolite Synthesis ; 2003.
6
Chapter 2
Theory of the Off-lattice Kinetic
Monte Carlo
ABSTRACT
In this chapter, we present an off-lattice kinetic Monte Carlo method, which
is useful to simulate reactions in solutions. We derive the method from first-
principles. We assume that diffusion leads to a Gaussian distribution for the
position of the particles. This allows us to deal with the diffusion analytically,
and we only need to simulate the reactive processes. The rate constants of these
reactions can be computed before a simulation is started, and need not be com-
puted on-the-fly as in other off-lattice kinetic Monte Carlo methods. We show
how solvent molecules can be removed from the simulations, which minimizes
the number of particles that have to be simulated explicitly. We present the re-
lation with the customary macroscopic rate equations, and compare the results
of these equations and our method on a variation of the Lotka model.
This chapter is based on: X. Q. Zhang and A. P. J. Jansen, Physical Review E 2010, 82, 046704.
Theory of the Off-lattice Kinetic Monte Carlo
2.1 Introduction
Kinetic Monte Carlo (kMC) simulations are increasingly being used to study the kinetics
of catalytic processes. They do not have the drawbacks of the older macroscopic equations
that are based on a mean-field approximation, which assumes a homogeneous distribution
of the reactants and an absence of fluctuations. In fact, kMC can be derived from first
principles and give for a given model practically exact results; the few assumptions in
the derivations are usually correct.1,2 Of course, computational costs of kMC are higher
than those for macroscopic equations, but they are only modest compared to for example
electronic structure calculations.
Many kMC simulations use a lattice-gas model. The translational symmetry allows for a
drastic simplification of the models that one uses; the number of processes becomes limited
because of the symmetry, and their kinetic parameters can be determined independently
from the simulation itself. This means that kMC simulations require generally only very
modest computer resources, and they can be applied to quite large systems.3
The remarks above certainly apply to many KMC simulations used for studying reactions
on surfaces. A more general approach in which no a priori assumption on the processes and
the symmetry of the system are made (in particular no lattice-gas model) has been used
as well.4–6 The drawback of that approach is that the determination of the processes and
their rate constants become part of the simulations. This slows down the simulations by
many orders of magnitude. If electronic structure calculations are used for this, then it is
no longer possible to study kinetics properly, although it can reveal unusual mechanisms for
some processes.4 Kinetics can still be studied with a force field, but this has only been done
with kMC for few systems.7–9
In this chapter, we take the general approach and apply it to reactions in solutions. We
will show that we can simplify the kMC simulations in such a way that the reactions can
be determined independently from the simulations, just as for the lattice-gas kMC. We
treat the diffusion of molecules in the solution analytically. Because we then only need to
simulate the reactions explicitly, the time that a simulation takes is drastically reduced. We
call the resulting kMC method continuum kMC. It has the same advantage with respect to
macroscopic rate equations as lattice-gas kMC.
8
2.2 Theory
This chapter is structured as follows. Section 2.2 presents the theory of continuum kMC.
Section 2.2.1 gives the derivation of the master equation for the most general case. The
equation forms the basis for kMC. In section 2.2.2 we present process-type reduction, which
is a coarse-graining method that eliminates the explicit handling of diffusion in continuum
kMC. Sections 2.2.3 and 2.2.4 derive expressions for the kMC rate constants of uni- and
bimolecular reactions, respectively. Section 2.2.5 generalizes the derivation from point-like
particles to real molecules. Section 2.2.6 relates continuum kMC to rate equations, and
shows how solvent molecules that may appear as reactants in reactions can be eliminated
from the formalism. Section 2.3 describes our algorithm for continuum kMC. It consists of
the algorithm itself (section 2.3.1), the method to determine the time when (section 2.3.2),
and the place where a reaction takes place (section 2.3.3). Section 2.4 presents simulations
of an adaptation of the Lotka model. It shows that continuum kMC gives clearly different
results from rate equations for the model, and discusses the reasons for that. Finally,
section 2.5 gives a brief summary.
2.2 Theory
2.2.1 The Master Equation
The derivation of the master equation is usually based on the observation that there is
a separation between the time scale on which reactions take place and the time scale of
much faster motions like vibrations.10,11 The longer time scale of reactions defines states,
in which the system is localized in configuration space, and the transitions between them
can be described by a master equation. The rates of the individual transitions can each
be computed separately by one of the methods of chemical kinetics; e.g., Transition State
Theory (TST).10–12 We present here a different derivation that incorporates all process at
the same time. It has only been outlined in the literature yet, so we give it here in some
detail.2,13,14 It is a generalization of the derivation of Variational TST (VTST); i.e., we
partition phase space in more than two regions,15–17 and it is an alternative to the derivation
using projection operators.18,19
In line with the idea of different time scales mentioned above, we start with identifying
the regions in configuration space where the fast motions take place. We will comment
9
Theory of the Off-lattice Kinetic Monte Carlo
however at the end of this section on how to generalize the approach. Figure 2.1 shows a
sketch of a potential-energy surface (PES) of an arbitrary system. We assume that only
the electronic ground state is relevant, so that the PES is a single-valued function of the
positions of all the atoms in the system. The points in the figure indicate the minima of
the PES. Each minimum of the PES has a catchment region. This is the set of all points
that lead to the minimum if one follows the gradient of the PES downhill.20
Figure 2.1: A sketch of a potential-energy surface of an arbitrary system and its corresponding
graph. The points are minima. The edges in the graph connect minima that have
catchment regions that border on each other. They correspond to reactions or other
activated processes. The thin lines on the left depict the borders of the catchment
regions.
We now partition phase space into these catchment regions and then extend each catch-
ment region with the conjugate momenta. Let’s call C the configuration space of a system
and P its phase space.21,22 The minima of the PES are points in configuration space. We
define Cα to be the catchment region of minimum α. This catchment region is a subset of
configuration space C, and all catchment regions form a partitioning of the configuration
space.
C =∪α
Cα. (2.1)
10
2.2 Theory
(There is a small difficulty with those points of configuration space that do not lead to
minima, but to saddle points, and with maxima. These points are irrelevant because the
number of such points are vanishing small with respect to the other points. They are found
where two or more catchment regions meet, and we can arbitrarily assign them to one of
these catchment regions.) With q the set of all coordinates and p the set of all conjugate
momenta we can extend the catchment region Cα to a corresponding region in phase space
Rα as follows.
Rα = {(q,p) ∈ P|q ∈ Cα}. (2.2)
We then have for phase space
P =∪α
Rα. (2.3)
If we use the regions Rα, we can derive the master equation exactly as for the lattice-gas
model.
The probability to find the system in region Rα is given by
Pα(t) =
∫Rα
dq dp
hDρ(q,p, t), (2.4)
where h is Planck’s constant, D is the number of degrees of freedom, and ρ is the phase space
density. The denominator hD is not needed for a purely classical description of the kinetics.
However, it makes the transition from a classical to a quantum mechanical description
easier.21
The master equation tells us how these probabilities Pα change in time. Differentiating
Eq. (2.4) yieldsdPα
dt=
∫Rα
dq dp
hD
∂ρ
∂t(q,p, t). (2.5)
This can be transformed using the Liouville-equation into22
dPα
dt=
∫Rα
dq dp
hD
D∑i=1
[∂ρ
∂pi
∂H
∂qi− ∂ρ
∂qi
∂H
∂pi
], (2.6)
where H is the system’s Hamiltonian, which we assume to have the form
H =D∑i=1
p2i2mi
+ V (q), (2.7)
with V the PES. The integrals over the conjugate momenta can be done for the terms with
derivatives of the Hamiltonian with respect to the coordinates. This shows that these terms
11
Theory of the Off-lattice Kinetic Monte Carlo
become zero, because ρ goes to zero for any of its variables going to ±∞. (Otherwise it
would not be integrable). The integrals over the coordinates of the terms with derivates of
the Hamiltonian with respect to the momenta can be converted to a surface integral using
the divergence theorem.23 This yields
dPα
dt= −
∫Sα
dS
∫ ∞
−∞
dp
hD
D∑i=1
nipimi
ρ, (2.8)
where the first integration is a surface integral over the surface of Rα, and ni are the
components of the outward pointing normal of that surface. As pi/mi = qi, we see that
the summation in the last expression is the flux through Sα in the direction of the outward
pointing normal (see Figure 2.2).
Rα
Sγα
βαS
Rβ
Rγ
Figure 2.2: Schematic drawing of the partitioning of phase space into regions R, each of which
corresponds to some catchment region of the potential-energy surface. The process
that changes α into β corresponds to a flow from Rα to Rβ. The transition probability
Wβα for this process equals the flux through the surface Sβα, separating Rα from
Rβ, divided by the probability to find the system in Rα.
The final step is now to decompose this flux in two ways. First, we split the surface Sα
12
2.2 Theory
into sections Sα = ∪βSβα, where Sβα is the surface separating Rα from Rβ. Second, we
distinguish between an outward flux,∑
i nipi/mi > 0, and an inward flux,∑
i nipi/mi < 0.
This gives then the master equation
dPα
dt=∑β
[WαβPβ −WβαPα] , (2.9)
with transition probabilities for the process α → β defined by
Wβα =
∫SβαdS∫∞−∞
dphD
[∑Di=1 ni
pimi
]Θ[∑D
i=1 nipimi
]ρ∫
Rαdq∫∞−∞
dphD ρ.
The function Θ is the Heaviside step function.24
Although we will use Eq. (2.10) in what follows, we want to show here also a more familiar
form in which the transition probabilities can be written. We assume that ρ can locally be
approximated by a Boltzmann-distribution
ρ ∝ exp
[− H
kBT
], (2.10)
where T is the temperature and kB is the Boltzmann-constant. We also assume that we
can define Sβα and the coordinates in such a way that ni = 0, except for one coordinate i,
called the reaction coordinate, for which ni = 1. These assumptions make the derivation
easier, but are not essential. The integral of the momentum corresponding to the reaction
coordinate can then be done and the result is
Wβα =kBT
h
Q‡
Q, (2.11)
with
Q‡ ≡∫Sβα
dS
∫ ∞
−∞
dp1 . . . dpi−1dpi+1 . . . dpDhD−1
× exp
[− H
kBT
], (2.12)
Q ≡∫Rα
dq
∫ ∞
−∞
dp
hDexp
[− H
kBT
]. (2.13)
We see that this is an expression that is formally identical to the TST expression for rate
constants.25 There are differences in the definition of the partition functions Q and Q‡,
but they can generally be neglected. For example, it is quite common that the PES has a
13
Theory of the Off-lattice Kinetic Monte Carlo
well-defined minimum in Rα and on Sβα, and that it can be replaced by a quadratic form in
the integrals above. The borders of the integrals can then be extended to infinity and the
normal partition functions for vibrations are obtained. This is sometimes called harmonic
TST.26
TheW ’s indicate how fast the system moves from (the catchment region of) one minimum
to another. We will often call them therefore rate constants. The system can only move
from minimum α to minimum β if the catchment region of these minima border on each
other. Only in such a case we have Wβα = 0. The right-hand-side of Figure 2.1 shows
the minima of the PES as points. Two minima are connected if their catchment regions
border on each other, and the system can move from one to the other without having to
go through a third catchment region. The result is the graph in Figure 2.1. The vertices
of the graph are the minima of the PES and the edges indicate how the system can move
from one minimum to another.
Although we have presented the partitioning of phase space based on the catchment
regions of the PES, this is formally not required. In fact, we have not used this particular
partitioning in the derivation up to Eq. (2.10) anywhere. One can in principle partition phase
space in any way one likes and derive a master equation. It is the partitioning that then
defines the processes that the master equation describes. Of course, most partitionings lead
to processes that are hard to interpret physically, but there are variations in the partitioning
above that are useful. For example, by taking the union of catchment regions separated only
by low barriers, as in Section 2.2.2, Eqs. (2.16) and (2.18) follow immediately.
The surface Sβα was split to distinguish fluxes in opposite directions. If there is a tra-
jectory of the system that crosses the surface and then recrosses it, then no reaction has
occurred, but both crossings contribute to the rate constants of α → β and β → α. The idea
of VTST is to move Sβα to remove recrossings and to minimize the rate constants.12,15–17 It
can be shown that when we have a canonical ensemble, this is equivalent to locating Sβα at
a maximum of the Gibbs energy along the reaction coordinate.27,28 In this paper we assume
that the effect of such variations can be neglected. As our derivation is a generalization
of VTST, it has the same limitations and possible ways to deal with them. We refer to
Chapter 4 of12 for a fuller discussion.
14
2.2 Theory
2.2.2 Process-type Reduction
The changes that correspond to edges of the graph in Figure 2.1 correspond to a number
of different types of processes. The most important changes for us are chemical reactions,
but different minima can also arise from diffusion of atoms or molecules, reorientations
of molecules, and conformational changes of a molecule. In fact most changes will not
be reactions, because they normally have high activation barriers (small rate constants W )
relatively to other possible changes. Consequently, most computer time in a kMC simulation
will be spent on other changes than reactions. This is undesirable. In this subsection we
introduce an idea of a kMC method that does only reactions.
To separate the chemical reactions from the low-barrier processes we partition the minima
of the PES (see Figure 2.3). All minima within one group are connected by low-barrier
process, and to get from one group to another at least one chemical reaction has to take
place. We adapt our notation by replacing α by (a, r) with a indicating the various groups,
and r the various minima within a group. The master equation in terms of these new labels
becomesdP(a,r)
dt=∑(b,s)
[W(a,r)(b,s)P(b,s) −W(b,s)(a,r)P(a,r)
]. (2.14)
This master equation still has low-barrier processes, which are characterized by rate con-
stants W(a,r)(b,s) = 0 with a = b. There are also reactions; i.e., a = b. To identify the
high-barrier processes with chemical reactions is correct if all molecules are “point-like” and
the density is not too large, but it may not be correct if we are dealing with large molecules
or high densities. In that case the partition should not only be based on chemical reactions,
but on other high-barrier processes as well.
Instead of P(a,r) we introduce
πa ≡∑r
P(a,r). (2.15)
It is possible to write down a master equation for πa.
dπa
dt=
∑r
dP(a,r)
dt
=∑r
∑(b,s)
[W(a,r)(b,s)P(b,s) −W(b,s)(a,r)P(a,r)
]=
∑b
[ωabπb − ωbaπa] (2.16)
15
Theory of the Off-lattice Kinetic Monte Carlo
A
B
C
D
A
C
D
B
Figure 2.3: The fat lines in these plots partition the minima of the potential-energy surface
into groups that are separated by reactions. Minima within a group are connected
by diffusion, reorientations, or conformational changes. The trajectory on the left
shows two reactions, AB and CD, with low-barrier processes in between. In process-
type reduction these intermediate processes are replaced by one step (dashed arrows)
that is done analytically.
with
ωab =∑r,s
W(a,r)(b,s)
P(b,s)
πb
. (2.17)
The ratio P(b,s)/πb is a conditional probability that the system is at minimum (b, s) if we
know that the system is in one minimum belonging to group b. The rate constant ωab is
then the sum of the rate constants of all reactions from group b to group a weighted with
this conditional probability.
The rate constants in the master equation (5.1) are given by Eq. (2.10) with the ap-
propriate change in notation. Because the denominator for W(a,r)(b,s) is equal to P(b,s) we
get
ωab =
∫Sab
dS∫∞−∞
dphD
[∑Di=1 ni
pimi
]Θ[∑D
i=1 nipimi
]ρ∫
Rbdq∫∞−∞
dphD ρ.
(2.18)
Rb =∑
s R(b,s), Sab is the surface bordering on Ra and Rb, and ni is component i of the
16
2.2 Theory
outward-pointing surface normal of Rb. (Actually, we could have gotten this expression and
master equation (2.16) directly by partitioning phase space in regions Ra.)
Because the approach here reduces the number of types of processes that we have to
handle explicitly in the kMC simulations, we call it process-type reduction. It is useful if we
can easily compute the new rate constants ωab. This is possible if we can determine the rate
constants W(a,r)(b,s) with a = b before a kMC simulation and if it is easy to compute P(b,s)/πb
during the simulation. We can determine the rate constants W(a,r)(b,s) if, for example, the
reaction rates W(a,r)(b,s) do not depend, at least approximately, on r and s. If the reaction
takes place in the gas phase or in a solvent, then its rate constant may depend only little on
the precise position of all the atoms and molecules. For the ratio P(b,s)/πb it may be possible
to derive analytical expressions based on simple models of diffusion, reorientation, and
conformational changes. Alternatively, we might be able to work with Eq. (2.18) directly.
2.2.3 Unimolecular Reactions
To make progress we need to specify our system in more detail. We assume that our system
consists of atoms and molecules that can react with each other and a larger number of inert
atoms and molecules. We also assume that all these atoms and molecules can be regarded
as “point-like” particles. An extension of our method to particles with an internal structure
is possible, but at first we want to deal with this simpler case. We are only interested in
the reacting atoms and molecules, but the other atoms and molecules are important for
the diffusion of the reacting particles. Such a system as we are describing here could be
a solution or a gas mixture. In such a system the groups are defined by the number of
particles.
The simplest reaction to deal with is a unimolecular reaction of the type A → B. Such
a reaction allows us to give a simple expression for ωab starting from Eq. (2.17). We first
note that when a reaction takes place at minimum (b, s) there is only one minimum (a, r)
that the system can go to. If there would be another minimum (a, r′) with r = r′ that
the system might be able to go to, then this would be not a simple reaction but a reaction
combined with some diffusion. We are excluding this. As a consequence the summation
over r in Eq. (2.17) has just one term for which W(a,r)(b,s) = 0.
The second observation is that such a reaction can take place everywhere, at any time,
17
Theory of the Off-lattice Kinetic Monte Carlo
and always with the same rate constant. This means that W(a,r)(b,s) = 0 has always the
same non-zero value. If we call that value Wuni, then we have
ωab =∑r,s
W(a,r)(b,s)
P(b,s)
πb
= Wuni
∑s
P(b,s)
πb
= Wuni. (2.19)
2.2.4 Bimolecular Reactions
For bimolecular reactions A + B → C the reasoning for unimolecular reactions does not
work. A bimolecular reaction is not possible in most of the minima in a group, because they
correspond to situations where the particles are too far apart to be able to react. These
minima are the ones that have been removed in Figure 2.3 when going from the left to the
right part. The particles have to get together first before they can react. This means that
we have to work with Eq. (2.18).
The first step is to get an expression for ρ(q,p, t). We can split off the momenta as
follows.21,22
ρ = ρ(q,p, t) ∝ ρ(q, t)∏i
exp
[− p2i2mikBT
]. (2.20)
As long as the particles stay apart so that they don’t interact, they only diffuse. We can
then take
ρ(q, t) ∝∏i
exp
[−(qi − q
(0)i )2
2σ2i
](2.21)
with
σ2i = 2Dit (2.22)
and Di the diffusion constant of the particle of coordinate qi. This expression holds for
diffusing particle that have coordinates q(0)i at time t = 0. For particles to react they have
to get close to each other. This will increase the energy, because there will be an activation
barrier. The configuration space density will have to be modified in the region where the
particles are close together. We assume that when the particles approach each other the
change in ρ(q, t) can be given by a Boltzmann factor so that
ρ(q, t) ∝ exp
[−V (q)
kBT
]∏i
exp
[−(qi − q
(0)i )2
2σ2i
]. (2.23)
The normalization constants that are missing in these expression can be ignored, because
they cancel in Eq. (2.18).
18
2.2 Theory
To evaluate ωab we can assume that we have only one particle A and one particle B. If
there are more particles, then we get a rate constant for the reaction of each A-B pair that
is the same. By looking at just two particles the mathematics is simplified substantially.
With two particle the surface Sab depends only on the distance between the particles. In
fact, it is defined as the set of points in phase space for which the distance between the
particles is the distance in the transition state of the reaction.
The integrals are easiest to evaluate if we transform to center-of-mass and relative coordi-
nates. The integrals over the momenta of the center-of-mass in numerator and denominator
of Eq. (2.18) cancel. For relative coordinates it is best to transform to spherical coordinates.
The difference in the integrals over the conjugate momenta is that in the denominator the
integral over the conjugate momentum pr of the distance between the particles is from −∞to ∞. In the numerator however it is from −∞ to 0. This is because the particle have to
get closer together to react. After integration of the momenta we are left with
ωab =
(kBT
2πµ
)1/2∫Sab
dX dr dθ dφ r2 sin θ ρ(X, r, θ, φ, t)∫CbdX dr dθ dφ r2 sin θ ρ(X, r, θ, φ, t)
. (2.24)
with X the center-of-mass coordinates, and (r, θ, φ) the relative coordinates in spherical
form. As explained above, for the relative coodinates Sab is a sphere with a radius that is
equal to the distance between the particles in the transition state. Cb is the interior of that
sphere.
The integrals over the center-of-mass coordinates in Eq. (2.24) also cancel. For the
remaining integrals we need Eq. (2.23). For the denominator we assume that the PES is
constant over the integration area. This is not the case for the area where the particles get
close to each other, but the contribution of that region to the integral can be ignored. For
the numerator the PES has the value corresponding to the energy of the transition state of
the reaction. The final result is then
ωab =
(kBT
2πµ
)1/2
exp
[−Ebar
kBT
](2.25)
× 4R2TS√
16π(DA +DB)3t3exp
(− |x(0)
B − x(0)A |2
4(DA +DB)t
)
when we assume that RTS is small compared to |x(0)B −x
(0)A |. Ebar is the height of the barrier
for the reaction, and RTS the distance between the particles at the transition state.
19
Theory of the Off-lattice Kinetic Monte Carlo
2.2.5 Composite Particles
Suppose A and B are again point-like, but that the diffusion has led to an equilibrium
situation. This changes the configuration space density ρ(q, t) because the exponent of the
diffusion in Eq. (2.23) becomes a constant. In the derivation above we still get Eq. (2.24),
because nothing changes for the conjugate momenta. The integrals over the coordinates do
change. In fact they become much simpler. We get
ω(eq)ab =
(kBT
2πµ
)1/24πR2
TS
L3exp
[−Ebar
kBT
](2.26)
when we assume that the particles move in a cubic box with sides of length L. If we compare
this to the expression in Eq. (2.25) we see that the effect of the diffusion is to change the
rate constant by a factor
fdiff =L3
[4π(DA +DB)t]3/2exp
(− |x(0)
B − x(0)A |2
4(DA +DB)t
). (2.27)
The important point now is that we assume that the rate constant when the system is
not yet at equilibrium is
ωab = krxfdiff (2.28)
with krx the rate constant at equilibrium for any type of particle. Indeed, if we use the
normal TST expression
k =kBT
h
Q‡
Qexp
[−Ebar
kBT
](2.29)
and evaluate it for a reaction A+B → C of “point-like” particles, then we get back Eq. (2.24).
2.2.6 Rate Equations
If we look at ω(eq)ab , Eq. (2.26), it seems that as the box in which the particles move becomes
bigger, the particles react more slowly. This is indeed the case, and as it should be. If we
change the size of the box then the number of reactions that actually occurs per unit time
scales with the box’s volume L3. The number of pairs of A and B that can react, however,
scales with the square of the volume of the box L6. As a consequence the rate constant has
to scale with L−3, as it does. This may not be immediately apparent if we look at Eq. (2.28).
20
2.2 Theory
However, the rate constant krx contains partition functions for translations which does lead
to a L−3 dependence of ωab.
The necessity of such dependence can also be shown by deriving the rate equations
in terms of concentrations. These should have rate constants that do not have this L
dependence. To see that this is indeed so we first write the rate equations in terms of
numbers of particles.dNA
dt= −ωNANB, (2.30)
with ω the rate equation that we have derived in the previous sections. To get concentrations
we have to divide by L3.
d[A]
dt=
1
L3
dNA
dt
= −(L3ω)NA
L3
NB
L3. = −(L3ω)[A][B]. (2.31)
We see that the L dependence of ω that we have in the rate equation in terms of numbers
of particles cancels against a L3 factor.
The factor fdiff shows a L3 dependence in Eq. (2.27), but for long times this factor has
to go to 1, because the results for the rate constant should go to the equilibrium expression
krx. It is clear that the exponent in the expression for fdiff goes to 1. This does not hold
for the ratio. The problem here is that we have assumed a Gaussian dependence of the
configuration space density ρ(q, t). This is correct as long as the diffusion length is small
compared to the size of the box. The expressions above should therefore only by used when
L ≫√
4π(DA +DB)t.
If the solvent is only a spectator in the reactions, then we can practically ignore it. The
presence of a solvent may change the rate constants, but that effect can be taken into account
by simply modifying the intrinsic rate constant krx in Eq. (2.28). If the solvent actually takes
place in the reaction, but we do not want to include it explicitly in our simulations, then
the expressions for the rate constants above need to be modified. There are two cases.
To focus our minds let’s deal with water that participates in acid-base reactions. We
assume that there is a particle Zp that can donate a proton to H2O (or OH−). The particle
Zp is then transformed into Z. There is also a reverse reactions where Z gets back a proton
from H2O (or H3O+). I assume that there is an equilibrium
Zp + S Z + Sp (2.32)
21
Theory of the Off-lattice Kinetic Monte Carlo
with S either H2O (Sp is then H3O+) or OH− (Sp is then H2O). The different possibilities
for S mean that there are really two different equilibria.
We can write a macroscopic rate equations for this equilibrium.
d[Zp]
dt= −k1[Zp][S] + k−1[Z][Sp]. (2.33)
Here k1 and k−1 are macroscopic rate constants. They are related to the rate constants for
a kMC simulation via kn = ωnV with ωn the kMC rate constant and V the volume of a
simulation box (see Eq. (2.31)). In a kMC simulation we work with discrete particles Z and
Zp. For the number of particles we have
dNZp
dt= −k1NZp[S] + k−1NZ[Sp]. (2.34)
We do not want to include the particles S and Sp explicitly in the simulation. The expression
above shows that we can accomplish this by replacing the (two) equilibria above by
Zp Z (2.35)
with rate constants k1[S] = ω1NS and k−1[Sp] = ω−1NSp with NS (NSp) the number of S
(Sp) in the simulation box if we would include them in the simulation explicitly. (Note
that we still have two equilibria or four reactions because of different possibilities for S.)
Multiplying ω±1 by the number of solvent particles NS or NSp means that the rate constant
no longer depends on the size of the simulation box. The number of particles combines with
the volume of the simulation box (L3 in the rate constant krx) to give a density. This is
appropriate, because the reaction effectively has become a unimolecular reaction.
Solvent molecules can also be formed by a reaction (e.g., if we have a condensation
reaction), or be reacted away by the reverse of such a reaction. We can write this as
X Y+ S (2.36)
with S the solvent molecule. We can again write a macroscopic rate equations for this
equilibrium.d[X]
dt= −k1[X] + k−1[Y][S]. (2.37)
For the number of particles we have
dNX
dt= −k1NX + k−1NY[S]. (2.38)
22
2.3 KMC Algorithms
Again we do not want to include the particles S explicitly in the simulation. The expression
above shows that we can accomplish this by replacing the equilibrium above by
X Y (2.39)
with rate constants k1 = ω1 and k−1[S] = ω−1NS. The rate constant of the first reaction
does not change, because it is a unimolecular reaction (i.e., of the type A → . . .).
Note that because we have derived the expressions here using concentrations, we have
implicitly assumed that our system is homogeneous; i.e., both our particles of interest and
the solvent molecules are randomly distributed. This means we have taken fdiff = 1. An
alternative derivation that only assumes that the solvent molecules are homogeneously dis-
tributed can also be given. Suppose we have a reaction X+S, with S a solvent molecule.
The rate constant for a particular molecule X with a particular molecule S is given by
ω = krxfdiff . We can get the rate constant for a particular molecule X with all molecules S
by summing ω over all molecules S. This summation does not affect the rate constant krx,
but summing fdiff gives us
L3
[4π(DX +DS)t]3/2
∑n
exp
[− |rX − rS,n|2
4(DX +DS)t
](2.40)
with rS,n the position of solvent molecule n just after a reaction. We now assume that
the solvent molecules are randomly distributed. We can then replace the summation by an
integral. The result is
L3
[4π(DX +DS)t]3/2
∫drS ρS exp
[− |rX − rS|2
4(DX +DS)t
](2.41)
= L3ρS = NS. (2.42)
Here ρS is the density of the solvent molecules. We see that we get the same result as above;
we have to multiply the rate constant by the number of solvent molecules and set fdiff = 1.
2.3 KMC Algorithms
Because the rate constants depend on time, we need to use the first-reaction method
(FRM),13,29,30 but the method needs to be adapted from the lattice-gas version. There
are two possibilities. The first is to make a list of all reactions, determine which reaction
23
Theory of the Off-lattice Kinetic Monte Carlo
occurs first, do that reaction, update the position of all particles that did not react, and
then repeat this procedure. Because of the updating of the particle positions, we have to
deal with the diffusion of the particles explicitly. We do not want to do this. It is also not
necessary. If we store when and where each particle is created, then updating the position
of the particles after each reaction can be avoided as explained in the next section.
2.3.1 Asynchronous Updating of Particle Positions
Suppose that we only know the positions of the particles at the time they were created; i.e.,
at the beginning of a simulation and when they were formed. The FRM algorithm then
looks as follows.
1. Initialize the simulation.
2. Determine the next reaction to occur.
3. Update the system, and repeat at 2, unless the end of the simulation is reached (e.g.,
no more reactions, or no more time).
Initializing the simulation consists of the following steps.
1.1. Generate initial positions ri,0 of the particles.
1.2. Set the time t to some initial value t0.
1.3. Choose conditions when to stop the simulation.
1.4. Make a list Lpart of particle positions and times when the particles were at the corre-
sponding positions.
1.5. Make a list Lrx containing all reactions and times when the reaction will take places
(see the end of this Section and Section 2.3.2).
Determining the next reaction to occur involves looking in Lrx for the reaction that occurs
first. We define tn as the time of the nth reaction to take place. We have ti > tj if and only
if i > j.
24
2.3 KMC Algorithms
Updating the system when a reaction (say number n) takes place involves the next series
of steps.
3.1. Determine the position where the reaction takes place (see the end of this Section and
Section 2.3.3).
3.2. Remove the reacting particles from Lpart and their reactions from Lrx.
3.3. Add the particles that are formed to Lpart and their reactions to Lrx. (This involves
mainly determining when the new reactions take place. This is explained at the end
of this Section and Section 2.3.2.)
3.4. Change time to t = tn.
In this algorithm it is only known where the particles are at the beginning of the simula-
tion, when they are created, and when they react. The list Lpart consists of pairs (ri, τi) that
indicate the particle i was at position ri at time τi. This list and also Lrx is not computed
anew after each reaction, but both are updated. This should make this algorithm faster.
Lrx is implemented as a binary tree of reactions and the times when they will occur.31 It is
ordered based on these times. When a new particle is created all its reactions are determined
and added to Lrx. As is standard in FRM,30 a reaction is only removed when it is about to
occur and a check reveals that a particle in the reaction no longer exists.
The determination of the place where a bimolecular reaction takes place changes a bit.
Let’s call the reacting particles again 1 and 2. If they are at position r1 at time τ1 and
position r2 at time τ2, respectively, then diffusion will bring them to position r at a time t
with t > τi with probability
1
[4πDi(t− τi)]3/2exp
[− |r− ri|2
4Di(t− τi)
]. (2.43)
So they will both be at position r with a probability proportional to
exp
[− |r− r1|2
4D1(t− τ1)
]. exp
[− |r− r2|2
4D2(t− τ2)
]. (2.44)
This is a probability distribution centered at
D2(t− τ2)r1,n−1 +D1(t− τ1)r2,n−1
D1(t− τ1) +D2(t− τ2)(2.45)
25
Theory of the Off-lattice Kinetic Monte Carlo
with a width √2D1(t− τ1)D2(t− τ2)
D1(t− τ1) +D2(t− τ2)(2.46)
in all directions.
For the rate constant of a bimolecular reaction we get instead of Eqs. (2.27) and (2.28)
ω = krxL3
[4π(D1(t− τ1) +D2(t− τ2))]3/2(2.47)
× exp
[− |r1 − r2|2
4(D1(t− τ1) +D2(t− τ2))
]for t > τ1, τ2. The reaction time is given by∫ trx
τ2
dt ω = − ln r (2.48)
assuming τ2 > τ1 and with r a random number from the interval [0, 1⟩. Substitutions of theexpression for ω gives us for the integral∫ trx
τ2
dt ω =krxL
3
4π(D1 +D2)|r2 − r1|(2.49)
×
[erf
(|r2 − r1|
2√
D1(τ2 − τ1)
)
−erf
(|r2 − r1|
2√
(D1 +D2)trx −D1τ1 −D2τ2
)]with erf the error function.32 This integral is limited to
0 ≤∫ trx
τ2
dt ω (2.50)
≤ krxL3
4π(D1 +D2)|r2 − r1|erf
(|r2 − r1|
2√
D1(τ2 − τ1)
).
We see that there is a probability that the reaction will never take place.
2.3.2 Determining Reaction Times
The reaction time of a unimolecular reaction is given by the usual expression30
∆t = − 1
krxln r (2.51)
26
2.3 KMC Algorithms
with ∆t the time between the beginning of the simulation or the time that the particle
was formed, r a random number from the interval [0, 1⟩, and krx the rate constant of the
reaction.
To get the reaction times for bimolecular reactions we have to solve Eq. (2.48). We can
write this equation as
erfc
(α√∆t
)= β (2.52)
with
β = − 4π(D1 +D2)|r2 − r1| ln rkrxL3
+ erfc
(|r2 − r1|
2√D1(τ2 − τ1)
)(2.53)
and
∆t = trx −D1τ1 +D2τ2D1 +D2
. (2.54)
The function erfc is the error function complement.32 It equals one minus the error function.
We have α, β > 0 so that 0 ≤ erfc(α/√∆t) ≤ 1 for ∆t > 0. This means that Eq. (2.52) has
no solution if β > 1. This is something that we have already seen.
We don’t know an analytical solution of Eq. (2.52), so we will look at numerical methods.
The function erfc(α/√∆t) is a monotonically increasing function of non-negative ∆t from
0 to 1. We would prefer to use Newton-Raphson to get the solution, because it converges
very rapidly to the solution.33 Unfortunately, that is not always possible. The derivative
is a monotonically increasing function for 0 ≤ ∆t ≤ 2α2/3. This means that on that
interval Newton-Raphson can be used. If we are on this interval, but above the solution,
then Newton-Raphson will approach the solution from above. If we are on this interval, but
below the solution, then the first step Newton-Raphson will bring us above the solution. This
may bring us outside the interval, which should be checked. If we are outside the interval,
then we should bracket and use bisection.33 The whole procedure then looks as follows. We
start by bracketing the solution, then do Newton-Raphson if we have 0 ≤ ∆t ≤ 2α2/3, or a
bisection if not.
27
Theory of the Off-lattice Kinetic Monte Carlo
2.3.3 Determining the Position of the Reaction
For a unimolecular reaction this is quite easy. The particle simply diffuses and then reacts
at whatever place it will be. So if a particle is at position r′ at time τ , then diffusion will
bring it to position r at a time t with t > τ with probability
1
[4πD(t− τ)]3/2exp
[− |r− r′|2
4D(t− τ)
], (2.55)
where D is its diffusion constant.
For a bimolecular reaction it becomes a bit more difficult. Let’s call the reacting particles
again 1 and 2. If they are at position r1 at time τ1 and position r2 at time τ2, respectively,
then diffusion will bring them to position r at a time t with t > τi with probability
1
[4πDi(t− τi)]3/2exp
[− |r− ri|2
4Di(t− τi)
]. (2.56)
So they will both be at position r with a probability proportional to
exp
[− |r− r1|2
4D1(t− τ1)
]. exp
[− |r− r2|2
4D2(t− τ2)
]. (2.57)
This is a probability distribution centered at
D2(t− τ2)r1,n−1 +D1(t− τ1)r2,n−1
D1(t− τ1) +D2(t− τ2)(2.58)
with a width √2D1(t− τ1)D2(t− τ2)
D1(t− τ1) +D2(t− τ2)(2.59)
in all directions. (We have seen this already in section 2.3.1.)
2.4 Illustrative Example
We present here results for a variation of the Lotka model to illustrate the method described
in the previous sections.34 In the original two-dimensional lattice-gas version there are two
types of particles; A and B. The A’s adsorb with rate constant ξ, the B’s desorb with rate
constant 1− ξ, and when there is an A next to a B then it is immediately transformed into
a B as well. The rate equations for this model show a steady state with probably that a
28
2.4 Illustrative Example
site is occupied by an A equal to 0 and by a B equal to ξ. (The probability for A goes to
0 as the rate constant for the transformation of A to B goes to infinity.) If the parameter ξ
is small however, a kMC simulation shows very well defined oscillation.
We have here no lattice with sites, so we have adapted the model. We have three reactions
S → A, (2.60)
B → S, (2.61)
A + B → 2B. (2.62)
Here the particle S has the same function as a vacant site. We have given it therefore also a
very small diffusion constant of 1.25·10−11 A2/s. The A’s and B’s have been given a diffusion
constant of 1.25 · 10−2 A2/s. This is still quite a small diffusion constant. The idea is to
try to mimic the behavior of the the two-dimensional lattice-gas model. That model has
avalanches in which clusters of A’s are converted into B’s at the same time. A fast diffusion
mixes up the system and destroys such clusters before they have been able to convert. The
rate constants for the reactions are 0.005 s−1, 0.995 s−1, and 158 s−1, respectively. The last
one is not infinite, but still much faster than the others.
Figure 2.4 shows how the concentrations change as a function of time. Initially there are
only B’s; 262144 B particles in a cubic box of size 160× 160× 160 A3. The rate equations
predict a steady state with concentrations [A] = 1.54 · 10−9/A3and [B] = 3.2 · 10−4/A
3. We
see that the concentration of B is close to this value, but that the concentration of A is much
larger. There are also quite large fluctuations. In fact, if we Fourier-transform the steady
state part of Figure 2.4 we get a peak around 0.07 s−1 (see Figure 2.5), which indicates that
we may have oscillations. They are however much less well-defined as in the two-dimensional
lattice-gas model, but comparable to the three-dimensional lattice-gas model.34
The oscillations have the same origin as oscillations in the two-dimensional lattice-gas
model. In the lattice-gas model most A+B → 2B reactions occur in the form of avalanches;
i.e., an A next to a B is converted into a B, another A next to that first A is then also
transformed into a B, etc. We see such avalanches here as well. The number of A’s that are
transformed in one go can become quite large,34 but here they remain much smaller than
for the lattice-gas model. We define the size of an avalanche as the number of A + B →2B reactions between consecutive formations of an A. Figure 2.6 shows the probability
distribution of the size of these avalanches. The reason why they are so much smaller than
29
Theory of the Off-lattice Kinetic Monte Carlo
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100
B
A
time (s)
3o
num
ber
of p
artic
les
(1/A
)
Figure 2.4: Concentration of A and B for the Lotka model as a function of time. The initial
concentration of B’s is 6.4× 10−2 particles/A3.
those in the original lattice-gas model is that it takes some time for each A + B → 2B
reaction to occur in the continuum kMC simulations, mainly because the particles have to
diffuse to each other to react, whereas in the lattice-gas model they start as neighbors and
react immediately. The avalanches form clusters of B’s and a clear segregation of A’s and
B’s as can be seen in Figure 2.7.
The difference between the lattice-gas and the continuum kMC simulations can be made
smaller by using a larger grid with lattice points closer together. The drawback is that the
computer time increases. The reason is that the number of diffusional hops that need to
be simulated in the lattice-gas model is inverse proportional to the square of the distance
between the lattice points.1,2
It may be that the oscillations in the continuum kMC simulations become better defined
when the system size is increased. When we decrease the size of the system, all properties
remain the same, except that the power spectrum becomes noisier. Unfortunately, we
30
2.5 Summary
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
frequency (1/s)
Fou
rier−
tran
sfor
med
con
cent
ratio
n (a
.u.)
Figure 2.5: Power spectrum (in arbitrary units) of the steady state concentration of A.
could not increase the size substantially as the simulation times would become too large.
Decreasing the parameter ξ, which gave better defined oscillations in the two-dimensional
lattice-gas model, shifted the peak in the power spectrum to smaller values, but did not
make it less noisy. This is similar to the three-dimensional lattice-gas model.
2.5 Summary
We have presented here a form of kMC simulations, which we call continuum kMC, that
should be useful to simulate reactions in solution. As for lattice-gas kMC, the rate constants
of the reactions can be determined prior to the simulation, so that the simulation itself takes
little computer time, or can be done on large systems.
We have derived the method from the master equation that described the evolution of
the system as hops from one minimum of the potential-energy surface to a neighboring
31
Theory of the Off-lattice Kinetic Monte Carlo
0.0001
0.001
0.01
0.1
1
1 10 100
0.00001
0.000001
number of particles
prob
abili
ty
Figure 2.6: Probability distribution P (s) as a function of the size s of the A+B → 2B avalanches.
one. This master equation is coarse-grained by using an analytical approach to the diffusion
of the particles. This leads to a new master equation that describes only the chemical
reactions, and no other processes. The diffusion is incorporated in the expression for the
rate constants. Solvent molecules need not be included explicitly in the simulations. Their
effect can be incorporated in the rate constants as well.
The algorithm that we have used is an adaptation of the first-reaction method. The
positions of the particles are not updated. At most two positions of each particle are
generated during a simulation; the position where the particle is formed, and the position
where it reacts and ceases to exist. For both positions there is a corresponding time.
We have illustrated the method using a Lotka model. This model shows kinetics that is
clearly different from that obtained from the rate equations. The reason for that is that the
system is not homogeneous. There are clusters of particles, and all particles in one cluster
react at about the same time.
32
BIBLIOGRAPHY
Figure 2.7: Snapshot of a simulation of the Lotka model showing the clusters of A’s (light spheres)
and B’s (dark spheres) that are formed.
We think that continuum kMC will be useful for many other systems. In Chapter 3
and Chapter 4, we present the application of this model to more complicate systems, the
formation of small silicate oligomers from Si(OH)4. This is the initial stage of the formation
of zeolites. An important aspect is the effect of template molecules, other cations, pH, and
temperature. All this can easily be included in our method.
Bibliography
[1] The kinetic Monte Carlo website, http://www.catalysis.nl/˜chembond/kMC/.
[2] A. P. J. Jansen, An introduction to Monte Carlo simulations of surface reactions, Los
Alamos Preprint Server; http://arXiv.org/, paperno. cond-mat/0303028 (2003).
33
Theory of the Off-lattice Kinetic Monte Carlo
[3] Salazar, R.; Jansen, A. P. J.; Kuzovkov, V. N. Phys. Rev. E 2004, 69, 031604.
[4] Henkelman, G.; Jonsson, H. J. Chem. Phys. 2001, 115, 9657.
[5] Xu, L.; Henkelman, G. J. Chem. Phys. 2008, 129, 114104.
[6] Xu, L.; Mei, D.; Henkelman, G. J. Chem. Phys. 2009, 131, 244520.
[7] Much, F.; Ahr, M.; Biehl, M.; Kinzel, W. Comp. Phys. Comm. 2002, 147, 226.
[8] Middleton, T. F.; Wales, D. J. J. Chem. Phys. 2004, 120, 8134.
[9] Pedersen, A.; Henkelman, G.; Schiotz, J. S.; Jonsson, H. New J. Phys. 2009, 11,
073034.
[10] Berry, R. S.; Breitengraser-Kunz, R. Phys. Rev. Lett. 1995, 74, 3951.
[11] Breitengraser-Kunz, R.; Berry, R. S. J. Chem. Phys. 1995, 103, 1904.
[12] Laidler, K. J., Chemical Kinetics (Harper and Row, New York, 1987).
[13] Jansen, A. P. J. Comput. Phys. Comm. 1995, 86, 1.
[14] Gelten, R. J.; van Santen, R. A.; Jansen, A. P. J., in Molecular Dynamics: From
Classical To Quantum Methods, edited by Balbuena, P. B.; Seminario, J. M. (Elsevier,
Amsterdam, 1999), pp. 737–784.
[15] Keck, J. C. J. Chem. Phys. 1960, 32, 1035.
[16] Pechukas, P., in Dynamics Of Molecular Collisions, Part B, edited by W. Miller
(Plenum Press, New York, 1976), pp. 269–322.
[17] Truhlar, D. G.; Isaacson, A. D.; Garrett, B. C., in Theory Of Chemical Reaction Dy-
namics, Part IV, edited by M. Baer (CRC Press, Boca Raton, 1985), pp. 65–138.
[18] Prigogine, I., Introduction To Thermodynamics Of Irreversible Processes (Interscience
Publishers, New York, 1968).
[19] van Vliet, C., Equilibrium And Non-Equilibrium Statistical Mechanics (World Scientific
Publishing Co., Singapore, 2008).
[20] Mezey, P. G., Potential Energy Hypersurfaces (Elsevier, Amsterdam, 1987).
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[21] McQuarrie, D. A., Statistical Mechanics (Harper, New York, 1976).
[22] Becker, R., Theorie der Warme (Springer, Berlin, 1985).
[23] Kreyszig, E., Advanced Engineering Mathematics (Wiley, New York, 1993).
[24] Zemanian, A. H., Distribution Theory and Transform Analysis (Dover, New York,
1987).
[25] van Santen, R. A.; Niemantsverdriet, J. W., Chemical Kinetics and Catalysis (Plenum
Press, New York, 1995).
[26] Henkelman, G.; Johannesson, G.; Jonsson, H., in Progress In Theoretical Chemistry
and Physics, edited by S. D. Schwarts (Kluwer, London, 2000).
[27] Garrett,B. C.; Truhlar, D. G. J. Am. Chem. Soc. 1979, 101, 5207.
[28] Garrett, B. C.; Truhlar, D. G. J. Am. Chem. Soc. 1980, 102, 2559.
[29] Gillespie, D. T. J. Comp. Phys. 1976, 22, 403.
[30] Lukkien, J. J.; Segers, J. P. L.; Hilbers, P. A. J.; Gelten, R. J.; Jansen, A. P. J. Phys.
Rev. E 1998, 58, 2598.
[31] Knuth, D. E., The Art Of Computer Programming, Volume III: Sorting and Searching
(Addison-Wesley, Reading, 1973).
[32] Abramowitz, M.; Stegun, I. A., Handbook Of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables (Dover, New York, 1965).
[33] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes.
The Art Of Scientific Computing (Cambridge University Press, Cambridge, 1989).
[34] Hovi, J.-P.; Jansen, A. P. J.; Nieminen, R. M. Phys. Rev. E 1997, 55, 4170.
35
Theory of the Off-lattice Kinetic Monte Carlo
36
Chapter 3
Mechanism of Silicate
Oligomerization
ABSTRACT
The mechanism of the initial stage of silicate oligomerization from solution is
still not well understood. Here we use an off-lattice kinetic Monte Carlo (kMC)
approach called continuum kMC to model silicate oligomerization in water so-
lution. The parameters required for kMC are obtained from density functional
theory (DFT) calculations. The evolution of silicate oligomers and their role in
the oligomerization process are investigated. Results reveal that near-neutral
pH favors linear growth, while a higher pH facilitates ring closure. The silicate
oligomerization rate is the fastest at pH 8. The temperature is found to increase
the growth rate and alter the pathway of oligomerization. The proposed pH and
temperature-dependent mechanism should lead to strategies for the synthesis of
silicate-based materials.
This chapter is based on: X. Q. Zhang, T. T. Trinh, R. A. van Santen and A. P. J. Jansen, Journal of
the American Chemical Society, 2011, 133, 6613.
Mechanism of Silicate Oligomerization
3.1 Introduction
Understanding how zeolites form is of fundamental scientific and technological importance.1–3
Although numerous experimental and theoretical studies have been devoted to investigat-
ing the prenucleation process of siliceous zeolite formation,4,5 the mechanisms governing the
transformation of small silicate molecules into oligomers are still poorly understood.2,6 The
very early stages of solution oligomerization play a decisive role in determining the final
structure.7 Thus, higher levels of control over nucleation cannot be achieved without un-
derstanding the fundamentals of the elementary steps of silicate oligomerization. However,
a detailed investigation of this process is still missing. The essential difficulty of studying
the prenucleation process arises from the fact that the silicate oligomers are typically of the
size of several Si(OH)4 molecules, which is hardly accessible to most of the current exper-
imental methods. Even if they are detected by microscopic techniques, the structural and
reactive properties may not be distinguished due to their small size. Furthermore, some of
the species exist for extremely short times and freely move throughout the available volume
of solution, reducing the change of their appearance in the volume being examined.7
A variety of spectroscopies and scattering techniques have been used to study the silica-
based condensation reactions,8–10 but the information they give is generally incomplete
and indirect.11 Knight et al. used 29Si nuclear magnetic resonance (NMR) spectroscopy to
study condensed silicate species present in aqueous solutions.10 Using mass spectrometry,
Pelster et al. investigated the temporal evolution of silicate species during hydrolysis and
condensation of silicates.12 Depla et al. and Fan et al. presented UV-Raman investigations
of the initial oligomerization reactions of silica the sol-gel process.13,14 However, because
of the multitude of simultaneous reactions in solution, it is difficult to extract information
about individual events using only experimental data.15
Many models have been developed for modelling the early stages of solid formation. Wu
and Deem introduced a Monte Carlo model for silicate solutions for investigation of the
nucleation process during zeolite synthesis in the absence of a structure directing agent.16
A force field was used to simulate the formation of covalent bonds. Chemical potentials
for Si and O are implicitly related to the pH of the system. The solvent effects were mod-
eled by applying a distance-dependent dielectric constant. Schumacher et al. presented a
Monte Carlo method for simulation of hydrothermal synthesis of periodic mesoporous sil-
38
3.1 Introduction
ica (PMS).17,18 Using simplified potentials this model enables the simulation, at an atomic
level, of the entire process of the synthesis of templated PMS. The pH effects was taken into
account implicitly in the reaction probabilities during the simulation. They also simulated
the adsorption properties of the PMS models using Grand Canonical Monte Carlo simula-
tion. More recently, Malani et al. presented a reactive Monte Carlo model, which is useful
for modelling silicate oligomerization.19 They have obtained agreement for the evolution of
the Qn distribution upon comparing the simulation results to experimental observations.
Lattice-gas kinetic Monte Carlo models were also used to model the crystal nucleation. The
method developed by Frenkel et al. has been used to give reliable results of the crystal
nucleation and growth.20–22 Jorge et al. presented a lattice-gas kinetic Monte Carlo model
describing the formation of silica nanoparticles.23 They showed qualitative agreement with
published experimental observations.
Here we compare our method with the models published earlier. Potentials or force
fields have been used in all the modelling studies mentioned above to describe the particle
interactions. The application of potentials allows for the simulation of large silicate clusters,
which, however, falls short of detailed information of small oligomers. In this work DFT is
used to predict the interaction and reaction details, which are the input of the subsequent
kMC simulation. This allows us to track more detailed information, especially for the
unstable species (such as the reactant complex and intermediate species). On the other hand,
the calculation of a reaction barrier using DFT is easier than creating an efficient potential
or force field for a certain type of material. This widens the range of applications of our
model. The influence of water molecules is modeled explicitly in the DFT calculations and
is incorporated in the rate constants in the kMC simulations. However, we stress that the
particles are coarse-grained in the kMC simulations in order to make them computationally
tractable. Thus, we cannot simulate the latter crystal-like zeolite frameworks. The off-lattice
MC methods of Schumacher et al. and Wu and Deem mentioned above are equilibrium
algorithms that are interpreted by rare event theory.16–18 The drawback is that there is no
real time in these methods. The same holds for the work of Malani et al.19 An advantage of
our kMC method is that the diffusion of molecules in the solution can be treated analytically.
This allows the simulation itself to take little computer time or to be done on large systems.
In the works of Frenkel et al. and Jorge et al., using a lattice gas, this is not the case
and the simulations are much more time-consuming.20–23 With the free energy barrier and a
recrossing coefficient calculated by the Frenkel method, these results would give a rate. With
39
Mechanism of Silicate Oligomerization
a recrossing coefficient estimated from transition state theory, they would give an estimated
rate. Another advantage of our model is that some important factors that influence the
reactions in solution (pH and structure directing agents) can more easily be included (effects
of structure directing agents will be shown in another work). Furthermore, the pH of the
solution is modeled more explicitly than in the earlier works, as described in the Model
and Methods section. Lattice-gas models also fall short of structural information of silicate
oligomers, such as five-coordinated silicate, 3-ring, and 4-ring, which are important in the
early stages of zeolite formation.
Many other theoretical methods, including electronic structure calculations15,24,25 and
molecular dynamics (MD),26 have been used to probe the formation of zeolites and meso-
porous materials. Information about energetics of chemical reactions can be obtained by
using DFT calculations, but it is very difficult to predict the kinetics just from the reaction
profiles. Stable structures of silicate oligomers can be obtained from DFT calculations, but
the most stable oligomers might not be the most preferable products. Moreover, DFT and
MD methods are computationally very expensive and restricted to very small systems and
short simulation times (on the order of pico- or nanoseconds); thus, relevant statistical in-
formation cannot be extracted. The time scale for the initial stages of zeolite formation is
on the order of hours or even longer, which is not accessible to MD or DFT simulations.
Failure of the current techniques in investigating oligomerization from solution motivated
us to develop a new approach. The effective modelling of silicate oligomerization in solution
requires a method that can simulate events at microscopic length and macroscopic time
scale. Given the experimental and theoretical difficulties, the off-lattice kinetic Monte Carlo
method provides an alternative way to gain key insights into the prenucleation process. Here
we use a kinetic Monte Carlo (kMC) theory,27 which we call continuum kMC, to model
silicate oligomerization reactions in water solution. In this theory, we take the general
approach and apply it to reactions in solutions. We show that we can simplify the kMC
simulations in such a way that the reactions can be determined independently from the
simulations, just as for lattice-gas kMC. We treat the diffusion of molecules in the solution
analytically. Because we then only need to simulate the reactions explicitly, the time that a
simulation takes is drastically reduced. The model overcomes the limitations of the models
mentioned above. In comparison to MD and DFT simulations continuum kMC can access
longer time scales, are computationally inexpensive, and are more flexible than lattice Monte
40
3.1 Introduction
Carlo models. It is also more realistic than the methods of rate equations. In an early
study,27 we compared the results of rate equations and our method on a variation of the
Lotka model. This model shows kinetics that is clearly different from that obtained from
the rate equations. The reason for that is that the system is not homogeneous. For the case
of silicate oligomerization reactions, the concentrations of some types of species are very low
and fluctuate strongly, and thus the rate equations do not work as properly as off-lattice
kMC. In this study, we compare the method with simulations of mean field approximations
and show the differences.
The formation of zeolites consists of several stages: first, an oligomerization process which
eventually leads to the formation of subcolloidal particles, second, the nucleation process,
and finally crystal growth.28 In this work we focus on the early stages of silicate oligomer-
ization. Further development of a clearer picture of prenucleation may help determine the
optimum conditions necessary for the effective organization within the silicate clusters.7 In
addition, a greater understanding of these processes may lead to an increase in the nucle-
ation rate and avoiding the formation of undesired structures. The basic aim of this work
is therefore to understand the mechanisms by which the silicate oligomers are formed in
solution. The evolution of cluster-size distribution and the effects of pH and temperature
on the oligomerization process are investigated.
This chapter is organized as follows. Model and Methods describes the model we used for
the silicate solution system and the simulation techniques employed. Results and Discussion
presents the results obtained from the simulations. The results section begins with an
analysis of the formation of silicate species implicated in the formation of zeolites. The
effects of pH and temperature on the oligomerization process are discussed, and the preferred
conditions for key silicate species are obtained. In Conclusions, we present our conclusions
and a brief outlook about future goals.
41
Mechanism of Silicate Oligomerization
3.2 Model and Methods
3.2.1 Theory of Continuum kMC
The kMC simulations were carried out using our newly developed continuum kMC.27 We
derived the method from first principles. We assume that diffusion leads to a Gaussian
distribution for the position of the particles. This allows us to deal with the diffusion
analytically, and we only need to simulate the reactive processes, so that the simulation
itself takes little computer time or can be done on large systems. Of course, computational
costs of kMC are higher than those for macroscopic equations, but they are only modest
compared to, for example, electronic structure calculations.
We have derived the method from the master equation that described the evolution of
the system as hops from one minimum of the potential-energy surface to a neighboring one.
This master equation is coarse-grained by using an analytical approach to the diffusion of the
particles. This leads to a new master equation that describes only the chemical reactions,
and no other processes. The diffusion is incorporated in the expression for the rate constants.
The rate constants then depend on the distance between reacting particles at times before
the reaction occurs. Solvent molecules need not be included explicitly in the simulations.
Their effect can be incorporated in the rate constants as well. The reaction rate constants
can be computed before a simulation is started and need not be computed on-the-fly as in
other off-lattice kinetic Monte Carlo methods. The short-range interactions are included in
the DFT calculations. So their effect is incorporated in the values of the rate constants for
the reactions. Long-range interactions have been neglected. All oligomers are regarded as
pointlike particles in our simulations. Therefore, there is no excluded-volume effect that is
present in lattice-gas kMC. We have compared our continuum kMC with lattice-gas kMC
for the formation of dimers only and found that the effect of overlap between particles
is negligible, provided the concentrations are not too high. The separation between the
oligomers must be clearly larger than the size of the oligomers. More detailed information
about the continuum kMC method can be found in chapter 2.
An important advantage of continuum kMC is that solvent molecules can be removed
from the simulations, which minimizes the number of particles that have to be simulated
explicitly. If the solvent is only a spectator in the reactions, then we can practically ignore
42
3.2 Model and Methods
it. We need to know the relation between the kMC and the macroscopic rate constants. For
a reaction A + B → C we have
d[A]
dt= −k[A][B] (3.1)
with k being the macroscopic rate constant. In a kMC simulation we work with discrete
particles. We have to multiply by L3 with L being the side length of the simulation box.
Then we havedNA
dt= − k
L3NANB = −ωNANB (3.2)
where NA and NB are the numbers of A and B respectively, and ω is the kMC rate constant.
It is related to the macroscopic rate constant via k = ωL3. (note that the kMC rate constant
can become dependent on the size of the simulation box).
For the case of silicate solution, water molecules participate in acid-base reactions. Sili-
cate species can donate a proton to OH− (or H2O). For example, Si(OH)4 is then transformed
into Si(OH)3O−. There is also a reverse reaction where Si(OH)3O
− gets back a proton from
H2O (or H3O+). These processes can be given by the following two example equations
Si(OH)4 + (OH)− Si(OH)3O− + H2O (3.3)
with macroscopic rate constants k1 and k−1 (reverse process), and,
Si(OH)4 + H2O Si(OH)3O− + H3O
+ (3.4)
with macroscopic rate constants k2 and k−2 (reverse process). We can write macroscopic
rate equations for these equilibria. Let us take the first case as an example:
d[Si(OH)4]
dt= −k1[Si(OH)4][OH−] + k−1[Si(OH)3O
−][H2O] (3.5)
The solvent actually takes part in the reaction, but if we do not want to include it
explicitly in our simulations, then the rate constants above need to be modified. The
expression above shows that we can accomplish this by replacing the equilibria above with
Si(OH)4 Si(OH)3O− (high pH) (3.6)
43
Mechanism of Silicate Oligomerization
with rate constants k1[(OH)−] = ω1N(OH)− and k−1[H2O] = ω−1NH2O, with kn = ωnL3 [see
Eq. 3.2], and ωn the kMC rate constants. The N(OH)− and NH2O are the numbers of (OH)−
and H2O in the simulation box, which we assume to be constant. We also have
Si(OH)4 Si(OH)3O− (low pH) (3.7)
with rate constants k2[H2O] = ω2NH2O and k−2[H+] = ω−2NH+ . [(OH)−] and [H+] are
determined by the pH of solution, such as [(OH)−]=10−7 mol/L at pH = 7.
3.2.2 Model of the Oligomerization
kMC is very efficient, allowing for the simulations of large systems and long simulation
times with modest computational work. In our simulations, the silicate-solution system
contains up to 6000 silicate monomers, which is large enough to give good statistics, and
the total simulation time is up to thousands of seconds, which is long enough for a realistic
description of the initial stage of zeolite formation. The simulation box is 215× 215× 215
A3, and the initial monomer concentration is 1 mol/L, which is usually used in experiments.
The reaction rate constants of all possible condensation and reverse reactions were obtained
from DFT calculations, which were published earlier.24,25 Two models were used to calculate
the reaction barriers, one in which the solvent effect was treated by using the continuum
solvation COSMO method implemented in the GAUSSIAN03 package,24,29 which we call the
COSMO model, and the other one in which Car-Parrinello molecular dynamics simulations
were applied with explicit modelling of water molecules,25,30 which we call the explicit-water
model.
In an early study we reported, in agreement with other previously published works,31 the
reaction mechanism of oligomers containing up to four Si atoms, including the calculation
of reaction activation energies.24 To avoid unnecessary complexity, we have not considered
the doubly ionized species, such as [Si(OH)2O2]-2 and [Si2(OH)4O3]
-2. Even though doubly
ionized species exist at very high pH, they are relatively unreactive in oligomerization.32
It has been shown that there are two different mechanisms: one in which the growing
oligomer is negatively charged, as shown in Figure 3.1, and a neutral one in which all
oligomers are neutral, as shown in Figure 3.2.24 In a high-pH environment, the system is
dominated by anionic species. Thermodynamic calculations show that in solution the [OH]−
44
3.2 Model and Methods
-60
-40
-20
0
20
40
60
80
+
Dimer + H 2O
Dimer...H 2O
TS2
Intermediate
Reactants TS1
Reactant complex
E(k
J/m
ol)
TS1
TS2
+
+
Figure 3.1: Schematic of the anionic mechanism of the dimerization reaction.
ion will deprotonate the monomeric species to form the monocharged anion [Si(OH)3O]−.
The condensation reactions proceeded through a two-step mechanism with formation of a
pentacoordinated intermediates, as shown in Figure 3.1. The first step is the formation of
the SiO-Si bond between two monomers, and the second step is the removal of water to form
the dimer species. In the first step, the anion [Si(OH)3O]− will approach the monomer to a
minimum distance to form a structure stabilized by three strong hydrogen bonds, which we
call reactant complex. The transition state corresponds to formation of the SiO-Si bond.24
In this step, a reaction intermediate, which we call intermediate species, is formed with
a pentacoordinated silicon. This was also reported by other researchers.31 Hydrogen is
transferred at the same time that a hydroxyl group starts to leave. As a result, the water
molecule will be the leaving group and the product is again an anion that can either form
a neutral dimer or initiate another condensation reaction to form a trimer.
The dimerization reaction can also occur via neutral reactant species, as shown in Figure
3.2. Two molecules approach through formation of hydrogen bonds at a minimum distance.
This complex rearranges via a transition state with an intermolecular hydrogen transfer. The
activation energy of this step is very high due to strong interference of the hydroxyl proton.
After hydrogen transfer, the water fragment leaves the molecule to form the dimer. The
5-fold silicon complex is not observed in this neutral route with lateral attack. The reaction
for two anionic monomers (both reactants are charged), which we call a double-anionic
45
Mechanism of Silicate Oligomerization
-40
-20
0
20
40
60
80
100
120
140
160
180
Dimer + H 2O
Dimer...H 2OTS
Reactant complex
E(k
J/m
ol)
+ +
Reactants
TS
Figure 3.2: Schematic of the neutral mechanism of the dimerization reaction.
mechanism, is rather unfavorable.5,24 The pathway of trimerization and tetramerization is
the same as that of dimerization.
Formation of the three-membered ring (3-ring) has been suggested before to occur via
an intramolecular condensation reaction. Intramolecular hydrogen bridges between the hy-
droxyl groups of the molecules have to be broken to create a geometry so that internal
ring closure can actually happen. This causes the unfavorable energies of intermediates
(a pre-transition-state configuration) and the intermediate with five-coordinated Si.24 The
ring closure reaction may also take place via a hydrogen transfer mechanism between neu-
tral species. The neutral linear trimer changes conformation. The two ends of the chain
approach each other. For the transition state, which is very similar to the case of the dimer,
a hydrogen transfers to a hydroxyl group. After that, a water molecule will leave the cluster
and a 3-ring is formed. The case of four-membered ring (4-ring) formation is similar to that
of the 3-ring mechanism.
The formation of silicate oligomers, with reaction rate constants, can be described by the
following equations. The following symbols are used; · · · for the reactant complex, –O– for
the intermediate species, · · ·H2O for the leaving water, △ for the 3-ring, ⊥ for the branched
46
3.2 Model and Methods
tetramer, and, ♢ for the 4-ring.
Si(OH)4 + (OH)−1.4844× 104
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.3525× 10−2Si(OH)3O
− + H2O (3.8)
Si(OH)4 + H2O2.5169× 10−16
GGGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGGG
1.4172× 104Si(OH)3O
− + H3O+ (3.9)
Si(OH)4 + Si(OH)3O− 1.8161× 107GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
4.8332× 108[Si(OH)4· · · Si(OH)3O]− (3.10)
[Si(OH)4· · · Si(OH)3O]−2.2716× 104
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.4034× 106[Si(OH)4–O–Si(OH)3]
− (3.11)
[Si(OH)4–O–Si(OH)3]− 1.0308× 103GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
7.5716× 101Si2O7H5
−· · ·H2O (3.12)
Si2O7H5−· · ·H2O
3.3094× 1011GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.7537× 1012Si2O7H5
− + H2O (3.13)
Si(OH)4 + Si(OH)49.110× 106
GGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGG
7.2929× 1012Si(OH)4· · · Si(OH)4 (3.14)
Si(OH)4· · · Si(OH)48.1221× 10−7
GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
2.6139× 10−8Si2O7H6· · ·H2O (3.15)
Si2O7H6· · ·H2O2.1176× 1010
GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.7515× 1012Si2O7H6 + H2O (3.16)
Si2O7H6 + (OH)−1.4335× 104
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.3030× 10−2Si2O7H5
− + H2O (3.17)
Si2O7H6 + H2O2.4262× 10−16
GGGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGGG
1.3629× 104Si2O7H5
− + H3O+ (3.18)
Si2O7H5− + Si(OH)4
1.5971× 107GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.7239× 108[Si2O7H5· · · Si(OH)4]
− (3.19)
Si2O7H6 + Si(OH)3O− 1.6009× 107GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
6.3686× 104[Si2O7H5· · · Si(OH)4]
− (3.20)
47
Mechanism of Silicate Oligomerization
[Si2O7H5· · · Si(OH)4]− 3.2031× 104GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.4034× 106[Si2O6H5–O–Si(OH)4]
− (3.21)
[Si2O6H5–O–Si(OH)4]− 1.4535× 103GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
2.9932× 100Si3O10H7
−· · ·H2O (3.22)
Si3O10H7−· · ·H2O
6.5801× 1011GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.6393× 1012Si3O10H7
− + H2O (3.23)
Si3O10H8 + (OH)−1.4136× 104
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.2838× 10−2Si3O10H7
− + H2O (3.24)
Si3O10H8 + H2O2.3907× 10−16
GGGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGGG
1.3418× 104Si3O10H7
− + H3O+ (3.25)
Si2O7H6 + Si(OH)41.5955× 107
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
2.3470× 1011Si2O7H6· · · Si(OH)4 (3.26)
Si2O7H6· · · Si(OH)45.7601× 10−7
GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
1.8537× 10−8Si3O10H8· · ·H2O (3.27)
Si3O10H8· · ·H2O1.1804× 1011
GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.6383× 1012Si3O10H8 + H2O (3.28)
[Si3O10H7]− 2.2716× 104GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
4.2104× 1010[Si3O9H7–O–]−(△) (3.29)
[Si3O9H7–O–]−(△)2.2716× 104
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
2.3527× 101Si3O9H5
−(△)· · ·H2O (3.30)
Si3O9H5−(△)· · ·H2O
8.3715× 1010GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.6590× 1012Si3O9H5
−(△) + H2O (3.31)
Si3O10H8
5.1971× 10−8
GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.3282× 10−8Si3O9H6(△)· · ·H2O (3.32)
Si3O9H6(△)· · ·H2O4.6665× 1011
GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.6578× 1012Si3O9H6(△) + H2O (3.33)
Si3O10H8 + Si(OH)3O− 1.5109× 107GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
5.0058× 105[Si3O10H8· · · Si(OH)3O]− (3.34)
48
3.2 Model and Methods
Si3O10H7− + Si(OH)4
1.5117× 107GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
3.7989× 109[Si3O10H8· · · Si(OH)3O]− (3.35)
[Si3O10H8· · · Si(OH)3O]−1.7239× 108
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.5554× 107[Si3O10H8–O–Si(OH)3]
− (3.36)
[Si3O10H8–O–Si(OH)3]− 2.8900× 103GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
7.5716× 10−1Si4O13H9
−· · ·H2O (3.37)
[Si4O13H9· · ·H2O]−4.6665× 1011
GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.5785× 1012Si4O13H9
− + H2O (3.38)
Si4O13H10 + (OH)−1.4030× 104
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.2838× 10−2Si4O13H9
− + H2O (3.39)
Si4O13H10 + H2O2.3907× 10−16
GGGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGGG
1.3305× 104Si4O13H9
− + H3O+ (3.40)
Si3O10H8 + Si(OH)3O− 1.5109× 107GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
5.5480× 106[Si3O10H8· · · Si(OH)3O]−(⊥) (3.41)
Si3O10H7− + Si(OH)4
1.5117× 107GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
2.4308× 108[Si3O10H8· · · Si(OH)3O]−(⊥) (3.42)
[Si3O10H8· · · Si(OH)3O]−(⊥)2.4308× 108
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
2.5177× 105[Si3O10H8–O–Si(OH)3]
−(⊥) (3.43)
[Si3O10H8–O–Si(OH)3]−(⊥)
2.0496× 103GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.7281× 10−2Si4O13H9
−(⊥)· · ·H2O (3.44)
Si4O13H9−(⊥)· · ·H2O
2.4308× 108GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
7.5785× 1012Si4O13H9
−(⊥) + H2O (3.45)
Si4O13H9− 1.1425× 104GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.5018× 1010[Si4O12H9–O–]−(♢) (3.46)
[Si4O12H9–O–]−(♢)6.3686× 104
GGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGG
1.9153× 10−1Si4O12H7
−(♢)· · ·H2O (3.47)
49
Mechanism of Silicate Oligomerization
Si4O12H7−(♢)· · ·H2O
8.3715× 1010GGGGGGGGGGGGGGGGGGBFGGGGGGGGGGGGGGGGGG
7.5898× 1012Si4O12H7
−(♢) + H2O (3.48)
The corresponding activation energies and rate constants of oligomerization and reverse
reactions are listed in Table 3.1.
Table 3.1: Calculated activation energies [kJ mol−1] and kMC rate constants of reactions given
by Eq. 3.8 to Eq. 3.48. The unit of the rate constants is s−1. The first column
is the reactions numbered corresponding to the Equations 3.8 to 3.48 listed above.
Eact(1) and Rate(1) are the activation energy and rate constant of forward reaction,
respectively. Eact(-1) and Rate(-1) are the activation energy and rate constant of
reverse reaction, respectively. The temperature is 350 K and pH = 7 (pH-neutral
environment). The activation energies were obtained from COSMO model.
Reactions Eact(1) Rate(1) Eact(-1) Rate(-1)
1 0 1.484×104 99 1.353×10−2
2 191 2.517×10−16 0 1.417×104
3 0 1.816×107 28 4.83×108
4 57 2.271×104 45 1.403×106
5 66 1.031×103 87 7.572×10−1
6 9 3.309×1011 0 7.754×1012
7 2 9.110×106 0 7.293×1012
8 127 8.122×10−7 137 2.614×10−8
9 17 2.118×1010 0 7.752×1012
10 0 1.434×104 99 1.303×102
11 191 2.426×10−16 0 1.363×104
12 0 1.597×107 31 1.724×108
13 0 1.601×107 54 6.369×104
14 56 3.203×104 45 1.403×106
15 65 1.454×103 83 2.993×100
16 7 6.580×1011 0 7.639×1012
17 0 1.414×104 99 1.283×10−2
18 191 2.391×10−16 0 1.342×104
19 0 1.596×107 10 2.347×1011
20 128 5.760×10−7 138 1.854×10−8
21 12 1.180×1011 0 7.638×1012
50
3.2 Model and Methods
22 57 2.272×104 15 4.210×1010
23 57 2.272×104 77 2.353×101
24 13 8.372×1010 0 7.659×1012
25 135 5.197×10−8 134 7.328×10−8
26 8 4.667×1011 0 7.658×1012
27 0 1.512×107 48 5.006×105
28 0 1.512×107 22 3.799×109
29 31 1.724×108 38 1.555×107
30 63 2.890×103 87 7.572×10−1
31 8 4.667×1011 0 7.579×1012
32 0 1.403×104 99 1.284×10−2
33 191 2.391×10−16 0 1.331×104
34 0 1.511×107 41 5.548×106
35 0 1.512×107 30 2.431×108
36 30 2.431×108 50 2.518×105
37 64 2.050×103 98 1.728×10−2
38 30 2.431×108 0 7.579×1012
39 59 1.143×104 18 1.502×1010
40 54 6.369×104 91 1.915×10−1
41 13 8.372×1010 0 7.590×1012
The rate constants were calculated by equation,
k = νexp
[−Eact
kBT
]. (3.49)
with Eact the activation energy and ν the prefactor. The prefactors are calculated for the
following three cases. The first case is a unimolecular reaction; ν1 = kBTh
, where kB is
Boltzmann constant and h is Planck’s constant. The second case is a bimolecular reaction;
ν2 =
√(kBT/2πµ)4πR2
V, where µ = m1m2
(m1+m2), m1 and m2 are the masses of the reactants,
R is the distance between the reactants in the transition state, and V is the volume of
the simulation box. And the third case is a bimolecular reaction with solvent molecule as a
reactant; ν3 = ν2Nsolvent. Nsolvent is the number of solvent molecules. Here the reactions with
solvent molecules were modeled implicitly. The number of H3O+ and OH− molecules are
determined by pH. Such as the number of OH− molecules, N(OH)− = NAV 10(pH−14). More
51
Mechanism of Silicate Oligomerization
details can be found in ref.27 The activation energies listed in Table 3.1 were obtained from
Density Functional Theory (DFT), with the B3LYP hybrid exchange-correlation functional.
The B3LYP method has been reported to provide excellent descriptions of various reaction
profiles and particularly of geometries, heats of reaction, activation energies, and vibrational
properties of various molecules. The basis set used to expand the molecular orbital was
all electron type 6-31+G(d,p). The solvation effect was included by using the continuum
solvation COSMO method.24
We also did simulations with the Car-Parrinello method as implemented in the CPMD
package.? In agreement with the COSMO model and other published results, silicate
oligomerization proceeds through an anionic intermediate species. Therefore, we used this
preferred anionic pathway again to investigate the formation of silicate oligomers. We con-
sidered a system consisting of one silicic acid Si(OH)4 and its deprotonated form Si(OH)3O−
with 64 water molecules. The simulation cell was a periodically replicated cubic box with
a size corresponding to a density of solution around 1 g cm−3 at ambient conditions. The
temperature was set at T = 350 K imposed with a Nose-Hoover thermostat. The electronic
structure was calculated using the Kohn-Sham formulation of density functional theory
(DFT) with the BLYP functional. BLYP has proven to give an accurate description of
the structure and dynamics of water and silicate-water interaction. Electronic states are
expanded in plane waves with a wavenumber of up to 70 Ry. The mass associated with
the fictitious electronic degree-of-freedom is 700 a.u. The time-step in the numerically in-
tegrated equations-of-motion is 0.145 fs.25 The parameters from explicit-water model are
listed in Table 4.1.
Table 3.2: Calculated activation energies [kJ mol−1] and prefactors of reactions from Eq. 3.8 to
Eq. 3.48. The first column is the reactions numbered corresponding to the Equations
3.8 to 3.48 listed above. Eact(1) and Pref(1) are the activation energy and prefactor
of forward reaction respectively. Eact(-1) and Pref(-1) are the activation energy and
prefactor of reverse reaction respectively. The temperature is 350 K and pH = 7
(pH-neutral environment). The activation energies were obtained from explicit-water
model.
Reactions Eact(1) Pref(1) Rate(1) Eact(-1) Pref(-1) Rate(-1)
1 0 1.484×104 1.484×104 99 8.050×1012 1.353×102
2 191 8.040×1012 2.517×10−16 0 1.417×104 1.417×104
3 0 1.820×107 1.816×107 28 7.293×1012 4.833×108
52
3.2 Model and Methods
4 44 7.293×1012 1.979×106 25 7.293×1012 1.355×109
5 42 7.293×1012 3.935×106 77 7.293×1012 2.353×101
6 9 7.293×1012 3.309×1011 0 7.750×1012 7.754×1012
7 2 1.810×107 9.110×106 0 7.293×1012 7.923×1012
8 127 7.293×1012 8.122×10−7 137 7.293×1012 2.614×10−8
9 17 2.118×1010 2.118×1010 0 7.752×1012 7.751×1012
10 0 1.434×104 1.434×104 99 7.750×1012 1.303×10−2
11 191 7.750×1012 2.426×10−16 0 1.360×104 1.363×104
12 0 1.600×107 1.597×107 31 7.293×1012 1.724×108
13 0 1.600×107 1.601×107 54 7.293×1012 6.369×104
14 43 7.293×1012 2.790×106 23 7.293×1012 2.694×109
15 33 7.293×1012 8.671×107 69 7.293×1012 3.677×102
16 7 7.293×1012 6.580×1011 0 7.640×1012 7.639×1012
17 0 1.410×104 1.414×104 99 7.640×1012 1.284×10−2
18 191 7.640×1012 2.391×10−16 0 1.340×104 1.342×104
19 0 1.600×107 1.596×107 10 7.293×1012 2.347×1011
20 128 7.293×1012 5.760×10−7 138 7.293×1012 1.854×10−8
21 12 7.293×1012 1.180×1011 0 7.640×1012 7.638×1012
22 35 7.293×1012 4.362×107 11 7.293×1012 1.665×1011
23 48 7.293×1012 5.006×105 72 7.293×1012 1.312×102
24 13 7.293×1012 8.372×1010 0 7.660×1012 7.659×1012
25 135 7.293×1012 5.197×10−8 134 7.293×1012 7.328×10−8
26 8 7.293×1012 4.667×1011 0 7.660×1012 7.658×1012
27 0 1.510×107 1.511×107 48 7.293×1012 5.006×105
28 0 1.510×107 1.512×107 22 7.293×1012 3.799×109
29 31 7.293×1012 1.724×108 38 7.293×1012 1.555×107
30 63 7.293×1012 2.890×103 87 7.293×1012 7.572×10−1
31 8 7.293×1012 4.667×1011 0 7.580×1012 7.579×1012
32 0 1.400×104 1.403×104 99 7.580×1012 1.284×10−2
33 191 7.580×1012 2.391×10−16 0 1.330×104 1.331×104
34 0 1.510×107 1.511×107 41 7.293×1012 5.548×106
35 0 1.510×107 1.512×107 30 7.293×1012 2.431×108
36 69 7.293×1012 3.677×102 11 7.293×1012 1.665×1011
53
Mechanism of Silicate Oligomerization
37 43 7.293×1012 2.790×106 121 7.293×1012 6.384×10−6
38 30 7.293×1012 2.431×108 0 7.293×1012 7.579×1012
39 64 7.293×1012 2.050×103 17 7.293×1012 2.118×1010
40 48 7.293×1012 5.006×105 111 7.293×1012 1.984×10−4
41 13 7.293×1012 8.372×1010 0 7.590×1012 7.590×1012
The calculation of the prefactors is the same as described above. In this work, we also
study the silicate oligomerization at other pHs and temperatures. The change of pH conse-
quently changes the number of H3O+ and OH− ions. The pH only changes the rate constants
of bimolecular reactions that with solvent molecule as a reactant. The changed rate con-
stants can be easily calculated by changing Nsolvent. Nsolvent can be water molecules, H3O+
or OH− ions. The others remain the same. Thus the rate constants for other pHs are not
shown. The rate constant is related to temperature by the Eq. 3.49. The rate constants at
other temperatures can be calculated from the available prefactors, and are thus not shown
here. The diffusion constants are 2.00×10−17 m2/s, 1.10×10−17 m2/s, 7.65×10−18 m2/s and
5.65×10−18 m2/s for monomers, dimers, trimers and tetramers respectively.
3.3 Results and Discussion
3.3.1 COSMO and Explicit-Water Model at Neutral pH
In our kMC simulation, the initial concentration of silicate monomers is 1 mol/L. The forma-
tion of zeolite particles is initiated when the solution is heated to a temperature of 350 K.28
The temperature in this simulation is therefore set at 350 K. The rate constants are shown in
Table 1 in Table 3.1. Figure 3.3 shows the concentration of branched tetramers as a function
of time. It is clear that the system almost fully transformed into branched tetramers (about
87 percent after 100 s) for the COSMO model, while, the explicit-water model shows the
near absence of branched tetramer. Experimentally, there are plenty of other species, such
as 3-ring and 4-ring species. This reveals that the COSMO model is inadequate in modelling
silicate oligomerization in water, which was also predicted by other researchers.11 A number
of calculations of silicate oligomerization from water solutions were recently done by using
the COSMO model.15,24,33 How the environment, especially the solvent, can be adequately
54
3.3 Results and Discussion
represented remains somewhat problematic. In the modelling of anionic silicate species by
Catlow et al. it did not prove possible to provide a sufficiently detailed representation of
the solvent to obtain results that were comparable with experiment.11 To model the solvent
more accurately, Catlow et al. then treated a few water molecules explicitly, which create
the first solvation layer around the anion. The remainder of the solvent was modeled by
the COSMO approach. Excellent results for the deprotonation of the silicate monomer were
then obtained.11 To study the influence of solvation on silicate oligomerization reactions,
Schaffer et al. used a hybrid implicit/explicit hydration model that explicitly accounted
for water in the calculations. Their results on the silicate dimer cluster revealed a marked
change in both the mechanism and energetics of the reactions.31 More recently, we reported
the role of water in silicate oligomerization reactions, in which all the water molecules were
modeled explicitly.25 The results for the kinetics based on that approach are discussed in the
next section. The inclusion of explicit water molecules changes the kinetics of the reactions.
Formation of some of the species becomes relatively unfavorable. The most distinct case
is the branched tetramer. There is a near absence of branched tetramer when the water
molecules are treated explicitly, due to the fact that the formation of branched tetramers
are rather unfavorable in this case. Consequently, formation of other species, 3-ring and
4-ring, becomes more favorable, as shown below.
3.3.2 Explicit-Water and Mean-Field Model at Neutral pH
The following kMC simulations were done based on the parameters shown in Table 4.1,
which were obtained from the explicit-water model. The kMC simulation starts with 6000
silicate monomers, and the simulation box is 215×215×215 A3. The pH of the solution
determines the number of OH− and H3O+ ions, and consequently determines the conversion
rate of neutral and anionic species. pH 7 and temperature T 350 K are used for the first
simulation, and we will take this simulation as reference for simulations with different pH
and temperature.
Figure 3.4 shows the concentration of monomeric (neutral and anionic) and linear species
(including dimer, trimer, and tetramer) as a function of time, in comparison with results
from mean field approximation. Monomeric species are initially abundant; thus, dimeriza-
tion is the dominant process. From the curve we can see the fast consumption of monomeric
55
Mechanism of Silicate Oligomerization
0 20 40 60 80 100
0.00
0.05
0.10
0.15
0.20
0.25
Con
cent
ratio
n (m
ol/L
)
Time (s)
COSMO
Explicit-water
Figure 3.3: Concentration of branched tetramers as a function of time at pH 7 and temperature
350 K. The rate constants were obtained from the COSMO model. The height of
the peaks in the insert is 1.6×10−4 mol/L and corresponds to the formation of one
molecule.
species. The system almost runs out of free monomeric species after 0.005 s, containing in-
stead many small silicate oligomers as described below. Although monomeric species are
produced by hydrolysis reactions, only a few of them can be observed, because the hydrol-
ysis process is slow, while the produced monomeric species are consumed immediately by
oligomerization reactions. This suggests that the monomeric species are more reactive than
highly condensed oligomers.
The dimer is the first stable product of oligomerization. The process from monomer
to dimer through two transition states is very fast. It is finished in 0.005 s. From 0.005
to 0.05 , the dimer dominates the species population. After dimerization three-membered
silicon species (linear trimer and 3-ring) emerge. However, the concentrations are low and
the species do not exist long. The linear trimer participates in two types of reactions. It
can either further oligomerize to form the linear tetramer (or branched tetramer) or form
a 3-ring by ring closure. The linear trimers are therefore consumed quickly. The linear
tetramer is the largest linear species in this model; the only route available for consumption
56
3.3 Results and Discussion
0.000 0.001 0.002 0.003 0.004 0.005
0.0
0.2
0.4
0.6
0.8
1.0
Con
cent
ratio
n (m
ol/L
)
Time (s)
Monomers(a)
Continuum kMC
Mean field approximation
0.0 0.1 0.2 0.3 0.4 0.5-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.05
0.10
0.15
0.20
0.25
Time (s)
Linear trimerDimer
Linear tetramer
Con
cent
ratio
n (m
ol/L
)
Time (s)
Dimer
Linear tetramer
Linear trimer
Continuum kMCMean field approximation
(b) (c)
Figure 3.4: Concentration of monomers (a) and linear species (b) as a function of time at pH
= 7 and temperature 350 K (explicit-water model) in comparison with results from
mean field approximation (a and c).
of linear tetramers is ring closure. Thus, it can only form 4-rings rather than convert into
a linear pentamer (larger species will be considered in our future work). After 0.1 s, the
linear tetramer becomes dominant. Linear trimers can easily be converted, and the linear
growth is favored. This suggests a clear tendency to form linear tetramers. The change in
concentration becomes small after about 0.5 s. This is the first period of interest, which we
call phase A.
We also did simulations with the mean field approximation for comparison to continuum
kMC. The rate constants used in the mean field simulations are the same with those of
kMC, except for the volume dependence of kMC rate constants for bimolecular reactions
(see Model and Methods). Figure 3.4 shows clear differences between the two models. The
57
Mechanism of Silicate Oligomerization
most distinct difference is that the consumption of monomers (see Figure 3.4a) and formation
of linear tetramers (see Figure 3.4c) are faster for the mean field approximation. The reason
for this is that the species are assumed to distribute homogenously and the diffusion is
assumed to be infinitely fast in the mean field simulations. While in the continuum kMC
simulations the particles have to diffuse to get close enough before they are able to react.
Thus the linear growth, the bimolecular reactions, are slower for continuum kMC than for
mean field.
0.0 0.2 0.4 0.6 0.8 1.0
0.000
0.005
0.010
0.015
Mean field approximation
0 2 4 6 8 10
0.000
0.005
0.010
0.015
0.020
0.025
Time (s)
Linear trimer
3-ring
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimer
3-ring
Continuum kMC(a) (b)
0 50 100 150 200 250 300
0.00
0.05
0.10
0.15
0.20
0.25
Mean field approximation Continuum kMC
0 200 400 600 800 1000
0.00
0.05
0.10
0.15
0.20
0.25
Time (s)
Linear tetramer
4-ring
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear tetramer
4-ring
(c) (d)
Figure 3.5: Formation of 3-rings (a) and 4-rings (c) at pH 7 and temperature 350 K (explicit-
water model) with comparison to the results from a mean field approximation (b and
d).
Ring structured species are very important to zeolite formation. The 3-ring is the smallest
closed structure that can be formed in a silicate structure. The 4-ring is commonly found
in most of the zeolitic structures; 61 zeolites have the 4-ring as part of their structure.34
From Figure 3.5 we can see that the concentration of the 3-ring follows that of the linear
58
3.3 Results and Discussion
trimer with only a small delay. This suggests that the 3-ring formation is easy; meanwhile,
it is also easy for a 3-ring to reopen again to form a linear trimer. On the other hand,
further linear growth leading to the linear tetramer is more favorable. Therefore, only a
few of three-silicon-membered species are left after 0.5 s. This is consistent with published
experimental results. NMR studies of reaction gels find solution-phase 3-rings.35 However,
the presence of such rings in zeolitic structures is rare. Our calculated results reveal that
3-rings are easy to reopen to support formation of 4-rings. The 3-rings do not directly
participate in the growth of zeolite frameworks but serve as a source for the growth of larger
species. Maybe this is why there are very few 3-rings present in the zeolite frameworks.
Small species are consumed rapidly. This is again consistent with published experimental
studies. Icopini and co-workers36 reported that [SiO2]n≤3, where the subscripted n equals
the number of silica tetrahedra in the polymeric molecule, decreases rapidly and approaches
constant values soon after the beginning of the experiment.
4-rings emerge at the same time as linear tetramers (see Figure 3.5). The concentration
of linear tetramers keeps increasing in the first 0.5 s, indicating that linear growth is faster
than 4-ring formation. Unlike the situation of 3-rings, the concentration of 4-rings does not
follow that of the linear tetramer. This indicates that both the formation and ring reopening
for a 4-ring are more difficult than for a 3-ring, and the 4-ring is more stable than the 3-ring.
The stable structure makes the 4-ring a popular structure in the zeolite frameworks.
The average particle size increases rapidly during the first 0.5 s. After that, the oligomer-
ization process is dominated by 4-ring formation. This is in agreement with UV-Raman
studies. Depla et al. have found that the 4-rings are the dominant species in the initial
oligomerization of the silica sol-gel process.13 The 4-rings are formed rapidly at early times,
during which the system has abundant supporting species. After 300 s about 95 percent of
the silicate species are transformed into 4-rings, and the system achieves equilibrium. This
is the second period of interest, which we call phase B. At the end of the simulation the
rates of the forward and reverse reactions are equal and the system is at steady state. It is
also possible to form larger species like pentamer, hexamer, etc. In that case some amount
of 4-rings might be consumed, but in this model four-membered oligomers are the largest
molecules considered.
The presence and formation mechanism of branched tetrameric species in the early stage
of silicate oligomerization is still problematic. Pereira et al. showed that it is much eas-
59
Mechanism of Silicate Oligomerization
ier to form linear than branched tetramers.37 We have also found that the formation of
linear tetramers and 4-rings is favored over that of branched tetramers.25 This demon-
strates that the formation of branched tetramer is rather unfavorable. The formation
of branched tetramer24,25,37–39 and branched cyclic tetramer (a 3-ring condensed with a
monomer)12,31,35,37,40–42 has been reported by a number of studies. Using density func-
tional theory, Pereira et al. reported structural and energetic properties of both branched
and branched cyclic tetramers.37 They found that the branched tetramer is more stable
than the branched cyclic tetramer. Schaffer et al. reported in more detail the formation
of branched cyclic tetramers by three pathways.31 They found that the pathway from a
branched tetramer is the most favorable. In this work we found that the formation of
branched tetramers is very unfavorable. Therefore, the formation pathway of branched
cyclic tetramers from branched tetramers is not operable (because the concentration of
reactants is very low), although the activation energy of this pathway is low.
Figure 3.5 shows a clear difference with the mean field approximation. Apart from
the fluctuations, the values of concentration and the time scales are also different. The
concentration of linear trimers is about two times that of 3-rings. The transformation from
linear trimer/tetramer to 3-ring/4-ring is slower than that of continuum kMC. The reason
is explained below.
0.00 0.02 0.04 0.06 0.08 0.10
0.000
0.005
0.010
0.015
0.020
0.025
Con
cent
ratio
n (m
ol/L
)
Time (s)
R(tetramer)
R(dimer)
R(trimer)
(a)
0 1 2 3 4 5
0.0
5.0x10-5
1.0x10-4
1.5x10-4
2.0x10-4
2.5x10-4
3.0x10-4
0
2.8x10-8mol/L
Continuum kMC
Con
cent
ratio
n (m
ol/L
)
Time (s)
Mean field
Intermediate species(b)
Figure 3.6: Concentration of reactant complexes (a) and intermediate species (b) as a function
of time at pH = 7 and temperature 350 K (explicit-water model).
Figure 3.6 shows the concentration of the unstable species. The reactant complex of
60
3.3 Results and Discussion
the dimer, trimer and tetramer are symbolized by R(dimer), R(trimer), and R(tetramer)
respectively. Apparently, reactant complexes are not stable. More R(dimer) is formed in the
first 0.002 s, due to the fact that the supporting species (monomers) is more abundant. The
sharp increase in concentration of R(dimer) corresponds to the fast decrease of monomers.
Figure 3.6 shows only the intermediate that leads to the dimer. The concentration changes
of the other intermediate species (not shown here) are similar to the curve shown in Figure
3.6. It is apparent that the intermediate species are even less stable than the reactant
complexes. The concentration shows only fluctuations. They exist for extremely short
times, and they will not be detectable experimentally. The peak values of the concentration
of the two models are almost the same: 1.6×10−4mol/L, as shown in Figure 3.6. These
are in good agreement. However, for the result of mean field simulation, the concentration
keeps constant value (very low) after the first peak. This is the reason the ring closure
is slow for the mean field simulations. The ring closure reactions that occur through a
intermediate are always unimolecular reactions. The particle simply diffuse and then reacts
at whatever place it will be; it does not depend on the rate of diffusion. Therefore, the
3-ring and 4-ring formations from kMC are faster than those from mean field simulations.
The particles are modeled explicitly in the kMC simulations. For the case of unstable
species, such as intermediate species, the concentration is either zero or nonzero in the kMC
simulations. However, in the mean field approach, the concentration is always nonzero
(very low concentration). Mean field assumes a homogeneous distribution of reactants and
an absence of fluctuations. However, this is not the real case. In this work we show that
the heterogeneous distribution and the fluctuations are important. The continuum kinetic
Monte Carlo simulations are a more realistic representation of the experimental situation.
3.3.3 Explicit-Water Model at Different pH and temperature con-
ditions
High pH
For the case of high pH, the evolution of monomers, reactant complexes, and intermediate
species are similar to those at neutral pH and are thus not shown in the following subsections.
Figure 3.7 shows how the concentration of key silicate species changes with time at pH
61
Mechanism of Silicate Oligomerization
0 2 4 6 8 100.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Con
cent
ratio
n (m
ol/L
)
Time (s)
Dimer
Linear tetramer
Linear trimer
(a)
0 2 4 6 8 10
0.00
0.01
0.02
0.03
0.04
0.05
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimer
3-ring
(b)
0 20 40 60 80 100 120 140 160 180 200
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear tetramer
4-ring
(c)
Figure 3.7: Evolution in the concentration of key species: linear species (a), formation of 3-ring
(b), and formation of 4-ring (c), at pH 8 and temperature 350 K.
8. The formation of dimers is similar to the case of neutral pH. The consumption needs
a longer time than at pH 7, as indicated by the curves. The dimers are mainly consumed
by further linear growth, while the linear growth is slower at pH 8 than at pH 7. Linear
tetramers need 2.5 s to become dominating, which is longer than at pH 7. Phase A lasts
longer. This is due to the fact that more species are ionized at pH = 8, which consequently
results in the insufficient supply of neutral monomers, and thus the linear growth (a neutral
species reacts with an anionic species) becomes slower. The formation of 3-rings is different
from that at neutral pH. High concentrations of linear trimer do not lead to pronounced
formation of 3-rings in the first 2 s, as can be seen from Figure 3.7. This demonstrates that
the linear trimer prefers further linear growth and a consequent 4-ring formation rather than
3-ring formation at high pH, and thus the formation of 3-ring becomes unfavorable. There
are fewer linear tetramers formed at pH 8. Interestingly, the phase B is shorter at high pH
62
3.3 Results and Discussion
although the earlier phase A is longer. It is noteworthy that the process of transformation
from monomers to 4-rings at high pH is different. At near neutral pH, the silicate species
are first almost fully transformed into linear tetramers, and followed by 4-rings formation.
At high pH, these two processes, linear growth and 4-ring formation, occur simultaneously.
The reason for this is that the linear growth is dominated by the anionic mechanism (neutral
species react with anionic species). Neutral pH yields a favorable ratio of neutral/anionic
species for linear growth. However, the ring closure occurs mainly through a single anionic
linear species. At high pH, there are more anionic species, which consequently increases the
rate of ring closure. Therefore, the formation of 4-rings is faster at high pH (pH 8), and
this reduces the temporary concentration of linear tetramer.
We then increase the pH to 9 (results are shown in Figure 3.8). There are fewer linear
species formed with respect to the results of pH 8. Linear growth is slow and unfavorable
at pH 9. This is again because there are fewer neutral species at high pH. Phase A is
much longer. A number of linear trimers are formed, which, however, prefers further linear
growth rather than ring closure. Thus a low concentration of 3-rings is found. The ring
closure prefers high pH; thus, the 4-ring formation should be faster at higher pH. However
phase B becomes longer at pH 9. This is due to linear growth being very slow (again
due to insufficient supply of neutral monomers), and the 4-ring formation is limited by the
formation of linear tetramers. The formation of both linear and ringed species is slower
when the pH is increased to 10 (data not shown here); the population is dominated by the
small species. Because of this, most of the species are anionic, and thus the linear growth
is rather unfavorable at pH 10. The rate-determining step is the linear growth at high pH.
From the results presented above, we can conclude that the size of the silicate oligomers
decreases with increasing pH (from 8 to 10). The same trend was found for larger sil-
icate clusters in experimental studies. The cluster size was reported to decrease with
pH.23,43,44 Using in situ small-angle X-ray scattering (SAXS) and small-angle neutron scat-
tering (SANS) Fedeyko et al. studied the formation of silicate nanoparticles in basic solutions
of tetraalkylammonium cations (TAA).43 They found that the particles have a core-shell
structure with silica at the core and the TAA cations at the shell. The particle core size
is nearly independent of the size of the TAA cation but decreases with pH, suggesting the
electrostatic forces are a key element controlling their size and stability. Using a lattice
model Jorge et al. studied the formation of silicate nanoparticles.23 They found that more
63
Mechanism of Silicate Oligomerization
0 10 20 30 40 50
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Con
cent
ratio
n (m
ol/L
)
Time (s)
Dimer
Linear tetramer
Linear trimer
(a)
0 5 10 15 20 25 30
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimer
3-ring
(b)
0 20 40 60 80 100 120 140 160 180 200
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear tetramer
4-ring
(c)
Figure 3.8: Evolution in the concentration of key species: linear species (a), formation of 3-ring
(b), and formation of 4-ring (c), at pH = 9 and temperature 350 K (explicit-water
model).
of the neutral silica monomers become ionized at high pH. A consequence of this is a signifi-
cant increase in the particle charge on the surface and subsequent coverage by TPA cations.
This layer is stabilized by electrostatic attractions between these cations and the negatively
charged silica surface. Therefore, higher pH means that a protective TPA layer forms, and
hence inhibiting growth.23 The silicate clusters of the published works above are larger and
on the scale of several or tens of nanometers. However, no details of this have been reported
before for small silicate oligomers. In this work, we found the same trend, but for a different
reason. At high pH, the growing oligomers are anionic, the small species are also anionic,
and thus the oligomerization can only occur through the double-anionic mechanism. The
activation barrier for this is very high.5,24 The high activation barrier of the double-anionic
mechanism prevents the oligomers from further growing at very high pH. Therefore, the
64
3.3 Results and Discussion
decrease in cluster size with pH is due to the high activation barrier of the double-anionic
mechanism. This may also play a role in the growing mechanism of lager clusters and may
account for the phenomenon that cluster size decreases with increasing pH to some extent.
Comparing the transformations from linear tetramer to 4-ring at different pHs, we can
see a shift in the peak of linear tetramer to lower concentration as the pH increases. This
means that the formation of linear tetramer is more favorable at low pH. Different from
the situation of neutral pH, the increase in concentration of 4-rings is accompanied by a
decrease in concentration of linear species, including dimer, linear trimer, and tetramer.
The distribution of species is wider. This means that the formation of 4-ring is limited by
the linear growth. This trend increases with increasing pH.
The changes in concentration collected at high pH showed significant differences with
respect to those collected at neutral pH. The maximum values of concentration for the 3-
rings are considerately higher at neutral pH. This means that the high pH does not favor
formation of 3-rings but does favor 4-ring formation. An interesting finding at pH 8 is that
the formation of 4-rings is most rapid, indicating a faster particle growth at pH 8 than at
pH 7 and 9. At pH 7, the ring closure is slow, and at pH 9, the linear growth is slow, while
at a pH value of 8, the molar ratio between neutral and anionic species makes the anionic
mechanism of linear growth and ring closure the most favorable choice. This is consistent
with results reported by Tleugabulova et al.45 Using fluorescence anisotropy decay analysis
Tleugabulova and co-workers evaluated formation and growth mechanisms of silicate. They
found a faster particle growth at pH 8.2 than at pH 9.2 and more rapid condensation of
silicate as the pH approaches neutrality. Icopini et al. also found that the oligomerization
rate is more rapid at near neutral pH.36 Overall, this pH dependent behavior is consistent
with the silicate particle growth mechanism at larger scales (several or tens of nanometers)
reported experimentally.45,46
Low pH
We also investigated silicate oligomerization under acidic conditions. The variations in
concentration of key silicate species at pH 6 are shown in Figure 3.9. The processes of
formation and consumption of dimers and trimers are faster than those under the other
conditions studied above. The system finishes phase A after only 0.02 s. The linear tetramers
65
Mechanism of Silicate Oligomerization
are formed quickly, while their consumption is much slower. This indicates that lower pH
favors the formation of linear tetramers. Perhaps further linear growth is also favorable at
low pH. However, in this work the linear tetramer is the largest linear species. Although a
great deal of linear trimers are formed early, they prefer further linear growth. Formation of
3-rings is then unfavorable under these conditions. The transformation from linear tetramer
to 4-ring is very slow. The system needs more than 6000 s to finish phase B, which is
much slower compared to the cases of higher pH. The rate-determining step is thus 4-ring
formation.
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Con
cent
ratio
n (m
ol/L
)
Time (s)
Dimer
Linear tetramer
Linear trimer
(a)
0.00 0.05 0.10 0.15 0.20
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimer
3-ring
(b)
0 1000 2000 3000 4000 5000 6000
0.00
0.05
0.10
0.15
0.20
0.25
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear tetramer
4-ring
(C)
Figure 3.9: Evolution in the concentration of key species: linear species (a), formation of 3-ring
(b), and formation of 4-ring (c), at pH = 6 and temperature 350 K (explicit-water
model).
We decreased the pH further to 5 (data not shown). Only a few 4-rings are observed
after thousands of seconds. The system is dominated by linear species. The effect of pH on
the rate of linear growth can be seen by comparing the maximum values of concentration of
66
3.3 Results and Discussion
linear tetramers. The results in this subsection reveal that low pH favors linear growth, while
ring closure becomes unfavorable. As the pH increases, more of the neutral silicate species
become ionized. A consequence of this is a significant increase in the possibility of both
linear growth through the anionic mechanism and ring closure. As the pH increases further,
most of the silicate species are ionized, which however increases the activation barrier of
linear growth again.
5 6 7 8 9 10
O
ligom
eriz
atio
n ra
te
pH
Figure 3.10: Effects of pH value on the silicate oligomerization rate.
The pH-dependent silicate oligomerization found in this work is in excellent agreement
with previous experimental works.46 Lin et al. found that the hydrolysis and condensation
rate of the silica species are pH dependent. They plotted the effects of pH value on the
silicate condensation rate. At pH > 2, the condensation rate increases with pH until pH 8
and then decreases again. Under acidic conditions, silica species are the less condensed linear
oligomers, while in alkaline solution the silica species are the more cross-linked clusters.46
In this work, although the length scales are different, we found the same trend. Figure 5.2
shows the effects of pH value on the silicate oligomerization rate. The silicate oligomerization
rate is the fastest at pH 8, and the rate decreases with an increase or decrease of pH. We
also found that the linear oligomers are favored in acidic conditions, while ring species are
favored in alkaline conditions. These phenomena are due to the fact that the pH controls the
distribution of neutral and anionic species and consequently determines the oligomerization
rate and species population. At low pH, most of the species are neutral, and thus ring closure
is unfavorable. This is the reason that silica species are the less condensed linear oligomers
under acidic conditions. The silicate species become ionized under alkaline conditions, which
67
Mechanism of Silicate Oligomerization
favors ring closure. The total growth is the combination of linear growth and ring closure.
Results show that the distribution at pH 8 is optimum for the total growth rate of silicate
oligomers.
Temperature Effect
Usually, formation of zeolite crystals occurs upon heating of the solution, making temper-
ature a key variable to be studied. The temperature effects are introduced into this model
via the transition state theory (TST). Zhdanov47 reported the first measurements on crys-
tal linear growth rates and showed directly for the first time the effect of temperature in
increasing growth rate. Experimental studies showed that the average particle sizes increase
with temperature.43 Theoretical works also found that increasing the temperature allows
for further silica particle growth.23 In this work, in addition to an increase in the growth
rate, the temperature is found to alter the pathway of oligomerization. However, we cannot
compare with the previous studies upon the cluster sizes.
Figure 3.11 and Figure 3.12 present the change in concentration with time for linear
species and 3-rings at pH = 7 and temperatures 400 and 450 K respectively. In comparison
to the case of 350 K, both the linear growth and the ring closure are faster at high temper-
ature. The concentration of 3-rings does not follow that of linear trimer. This means that
temperature 400 K does not favor 3-ring formation. 3-rings are unstable at high tempera-
tures. Meanwhile, 400 K favors formation of 4-rings, as shown in Figure 3.13. This indicates
that the silicate condensation rate is increased by increasing the temperature to 400 K.
450 K is also in the range of typical temperatures of zeolite synthesis.11 Interestingly, the
maximum value of concentration of linear tetramers is greatly reduced at 450 K. The con-
centration curves for linear tetramers indicate a substantial difference in the growth pathway
at different temperatures. At low temperature, the small silicate species first transformed
into linear tetramers, and the second step is 4-ring formation. At higher temperature, the
linear growth and ring closure occur simultaneously. For the formation of 4-rings, the over-
all concentration trend is similar to that at lower temperature. The difference is that the
ring closure occurs more rapidly at high temperature. Phase B only takes about 1 s. Here
we conclusively demonstrate that the overall silicate oligomerization rate increases with
temperature. When the temperature is increased to 450 K, we see a few more branched
68
3.3 Results and Discussion
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
C
once
ntra
tion
(mol
/L)
Time (s)
Dimer
Linear tetramer
Linear trimer
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.02
0.04
0.06
0.08
0.10
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimer
3-ring
(b)
Figure 3.11: Change in concentration with time for linear species (a) and 3-rings (b) at pH = 7
and temperature 400 K.
tetramers formed.
Preferred Conditions
The pH was reported to control the stability of silica nanoparticles and, hence, determines
their size distribution on the scale of nanoparticles.23 Here we report the pH-controlled
distributions of small silicate oligomers. In solution, monomeric silicate molecules undergo
condensation reactions that lead to the formation of silicate oligomers, which depends on
the conditions (mainly pH and temperature) of the solutions. Figure 3.14 gives a schematic
of how each oligomer could be built up from the monomers and summarizes the preferred
conditions of key silicate species. The reactant complexes and intermediate species are
omitted for clarity. The preferred conditions are the most favorable conditions at which
a certain type of silicate oligomer is formed. The silicate oligomerization under various
conditions (pH ranges from 5 to 10) have been studied. The preferred condition for the
dimer is a low pH value of 6. It is noteworthy that the linear species prefer low pH, while
ringed species prefer higher pH. This compares well with previous theoretical works. Wu
and Deem reported that the pH value affects the critical cluster size and the nucleation
barrier through the oxygen chemical potential.16 They found that a decrease of pH leads to
favorable dimerization. Malani et al. found that a high concentration of OH groups favors
ring formation.19 The oligomerization is a combined action of linear growth and ring closure.
69
Mechanism of Silicate Oligomerization
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimer
Dimer
Linear tetramer
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimer
3-ring
(b)
Figure 3.12: Change in concentration with time for linear species (a) and 3-rings (b) at pH = 7
and temperature 450 K.
Therefore, the pH at which the rate of silicate oligomerization is most rapid is neither very
high nor very low. Form the discussions above, preferred pH lies between 6 and 8 for different
species. This is in agreement with experimental results again. Experimentally, the silicate
oligomerization rate is most rapid at near pH 8.45,46 The formation of branched tetramers
can be favored at a temperature of 450 K, as described in the next subsection. The preferred
conditions for the key species of the early stage of zeolite formation are obtained, which can
accelerate this stage.
3.4 Conclusions
Most importantly, we have developed a continuum (off-lattice) kinetic Monte Carlo model
to study the oligomerization reactions of large scale silicate-solution systems, which opens
the way to study many other important problems occurring in solutions on the atomic
length and macroscopic time scale. The present study demonstrates that continuum kMC
theory is able to provide detailed information regarding the early stage of zeolite formation.
Comparing continuum kMC and mean field approximations on the silica-solution system, we
conclude that the mean field approximation is rate-limited by intermediate species. Results
reveal that the COSMO model is not adequate in modelling silicate oligomerization from
water solution, and thus water molecules have to be considered explicitly. We demonstrate
70
3.4 Conclusions
0 2 4 6 8 10
0.00
0.05
0.10
0.15
0.20
0.25
C
once
ntra
tion
(mol
/L)
Time (s)
Linear tetramer
4-ring
(a)
0.0 0.5 1.0 1.5 2.0
0.00
0.05
0.10
0.15
0.20
0.25
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear tetramer
4-ring
(b)
Figure 3.13: Formation of 4-rings at pH = 7 (explicit-water model) and temperatures 400 K (a)
and 450 K (b).
that pH and temperature greatly influence the oligomerization rate and pathway. Therefore,
silicate oligomerization can be controlled by varying the pH and temperature of the solution.
A significant finding is that near neutral pH favors linear growth, because the linear growth
is mainly driven by an anionic mechanism in which there is one neutral and one anionic
reactant, while a higher pH makes the silicate species anionic, which facilitates ring closure.
In the case of pH 7, the species oligomerize first to linear tetramers and then close to form
4-rings, while at high pH the linear growth and ring closure occur simultaneously. The
total growth rate is a interplay between linear growth and ring closure. pH 8 is found to
be the optimum value that takes care of both linear growth and ring closure, and hence
the silicate oligomerization is the fastest at pH = 8. The decrease of cluster size with
pH is due to the fact that the double-anionic mechanism operable is very slow. The rate-
determining steps are ring closure, at very low pH, and linear growth, at very high pH.
Preferred conditions necessary for effective oligomerization that can accelerate the initial
stage of silicate oligomerization and as a result avoid the formation of undesired species
have been obtained.
71
Mechanism of Silicate Oligomerization
Figure 3.14: Summary of the preferred conditions for the formation of key species. The pH
range considered is 5 to 10.
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75
Mechanism of Silicate Oligomerization
76
Chapter 4
Effects of Counterions
ABSTRACT
In this chapter, we present an investigation of silicate oligomerization in water
solution in the presence of counterions (Li+ and NH4+) using continuum kinetic
Monte Carlo (kMC). The parameters required for kMC are obtained from den-
sity functional theory (DFT) calculations. Formation of cyclic trimers (3-rings)
and branched tetramers have been reported by using several models, including
DFT in the gas phase, DFT with COSMO treating the solvent, Car-Parrinello
simulations with explicit water molecules, and also experimental results. How-
ever, the results are substantially different. Moreover, in many other studies,
no branched tetramer was observed. In this chapter, the silicate oligomerization
from monomer to 3-ring and branched tetramer, and the roles of counterions in
these processes, are discussed in detail. The presence of counterions is found to
influence the oligomerization rate and selectivity, and a structure-directing role
of counterions is found. Results reveal that Li+ favors formation of branched
tetramers over 3-rings, whereas NH4+ directs the formation of 3-rings.
This chapter is based on: X. Q. Zhang, T. T. Trinh, R. A. van Santen and A. P. J. Jansen, The Journal
of Physical Chemistry C, 2011, 115, 9561.
Effects of Counterions
4.1 Introduction
A molecular-scale understanding of silicate growth is critical to the development of micro-
porous materials, such as zeolites.1,2 The very early stages of solution oligomerization play
a decisive role in determining the final structure.3 Thus, higher levels of control over nucle-
ation cannot be achieved without understanding the fundamentals of the elementary steps
of silicate oligomerization. Although numerous experimental and theoretical studies have
been devoted to investigating the prenucleation process of siliceous zeolite formation,1–5 the
mechanisms governing the transformation of small silicate molecules into oligomers are still
not well understood.
Monitoring the early stages of zeolite synthesis still remains a challenge. It has long been
known that cationic species directly influence the resulting crystal lattice. A variety of spec-
troscopies and scattering techniques have been used to study the silica-based condensation
reactions,6–8 but the information they give is generally incomplete and indirect.9 Because
of the multitude of simultaneous reactions in solution, it is difficult to extract information
about individual events using only experimental data.10 An excellent review of the achieve-
ments of the zeolite formation have been presented by Cundy and Cox.11 Wu and Deem
introduced a Monte Carlo model for silicate solutions for investigation of the nucleation
process during zeolite synthesis in the absence of a structure-directing agent.12 A force field
was used to simulate the formation of covalent bonds. Chemical potentials for Si and O are
implicitly related to the pH of the system. The solvent effects were modeled by applying a
distance-dependent dielectric constant. Schumacher et al. presented a similar method for
simulation of hydrothermal synthesis of periodic mesoporous silica (PMS).13 Using simpli-
fied potentials, this model enables the simulation, at an atomic level, of the entire process
of the synthesis of templated PMS. More recently, Malani et al. presented a reactive Monte
Carlo model, which is useful for modelling silicate oligomerization.14 They have obtained
agreement for the evolution of the Qn distribution upon comparing the simulation results to
experimental observations. Jorge et al. presented a lattice-gas kinetic Monte Carlo model
describing the formation of silica nanoparticles.15 They showed qualitative agreement with
published experimental observations. Here we compare our method with the models pub-
lished earlier. Potentials or force fields have been used in all the modelling studies mentioned
above to describe the particle interactions. The application of potentials allows for the sim-
ulation of large silicate clusters, which, however, falls short of detailed information of small
78
4.1 Introduction
oligomers. In this work, DFT is used to predict the interaction and reaction details, which
are the input of the subsequent kMC simulation. This allows us to track more detailed infor-
mation, especially for the unstable species (such as the reactant complex and intermediate
species). There is no real time in the off-lattice MC methods mentioned above. An advan-
tage of our kMC method is that the diffusion of molecules in the solution can be treated
analytically. This allows the simulation itself to take little computer time or to be done on
large systems. For the lattice-gas models, the simulations are much more time-consuming.
Lattice-gas models also fall short of structural information of silicate oligomers, such as
five-coordinated silicate, 3-ring and 4-ring, which are important to the early stage of zeolite
formation. Another advantage of our model is that the important factors that influence
the reactions in solutions (including pH, temperature, and structural-directing agents) can
easily be included.
Electronic structure calculations,10,16–18 and molecular dynamics (MD),19 have also been
used to probe the formation of zeolitic materials. Using ab-initio techniques, Pereira et al.
investigated the mechanisms and energetics of condensation of silica-based clusters. Their
calculated activation energies accord well with those measured for silica condensation in sol-
gel systems.10,16 Rao and Gelb have performed large-scale molecular dynamics simulations
of the polymerization of silicic acid in aqueous solution. The structural details of silicate
clusters formed are in qualitative agreement with experimental NMR data.19 Typically, it is
difficult for MD methods to model chemical reactions. Moreover, DFT and MD methods are
computationally very expensive and restricted to very small systems and short simulation
times: thus, relevant statistical information cannot be extracted.
Here, we use our newly developed kinetic Monte Carlo (kMC) theory,20 which we call
continuum kMC, to study the effects of counterions on silicate oligomerization reactions in
water solution. In this theory, we start with the general approach and apply it to reactions
in solutions. We show that we can then simplify the kMC simulations in such a way that
the rate constants of reactions can be determined independently from the simulations, just
as for the lattice-gas kMC. We treat the diffusion of molecules in the solution analytically.
Because we then only need to simulate the reactions explicitly, the time that a simulation
takes is drastically reduced.
The formation of zeolites consists of several stages: first, an oligomerization process
that eventually leads to the formation of subcolloidal particles, second, the nucleation pro-
79
Effects of Counterions
cess, and finally, crystal growth. During the first hours of oligomerization, various silicate
oligomers are formed in solution. The dominant species depends sensitively on the reac-
tion conditions, the pH of the solution, the temperature, and the structure-directing agents
(SDAs). Structure direction occurs when inorganic or organic molecules are used to direct
the crystallization toward a specific zeolite structure. Structure-directing agents are gen-
erally inorganic cations, such as Na+, K+, Li+, and Ca2+, and organic molecules, such as
TPA+ TMA+ or TEA+.21 Using different ammonium salts as templating agents leads to
different products. Burkett et al. identified the cubic octamer, prismatic hexamer, and
cyclic-trimer as the predominate products when using TMA+, TEA+, and TPA+, respec-
tively.22,23 It is known that solutions of Na+, TMA+, and TBA+ hydroxides do not lead to
silicalite-1 formation, whereas those of TPA+ do selectively generate the MFI framework
type of silicalite-1.24,25 TMA+ leads to the synthesis of layered silica phases,26 while the
TBA+ derivative gives rise to several different zeolites.24 Zwijnenburg et al. demonstrated
how substituting part of the silicon cations in silica for pairs of aluminium and (alkali metal)
cations can change the relative energetic ordering of competing structure-types.27 The for-
mation of small silicate oligomers in solution in the presence of structure-directing agents
is an important current issue in zeolite synthesis. However, detailed information regarding
the structure-directing role of counterions (Li+ and NH4+) is still missing. The basic aim
of this chapter is, therefore, to understand the mechanisms by which the silicate oligomers
are formed in the presence of counterions in water.
4.2 Computational Methods
The kMC simulations were carried out using our newly developed continuum kMC.20 We
derived the method from first principles. We have derived the method from the master
equation that described the evolution of the system as hops from one minimum of the
potential-energy surface to a neighboring one. This master equation is coarse grained by
using an analytical approach to the diffusion of the particles. This leads to a new master
equation that describes only the chemical reactions, and no other processes. We assumed
that diffusion leads to a Gaussian distribution for the position of the particles. This allows us
to deal with the diffusion analytically, and we only need to simulate the reactive processes,
so that the simulation itself takes little computer time, or can be done on large systems.
80
4.2 Computational Methods
The rate constants of the reactions can be computed before a simulation is started and
need not be computed on-the-fly as in other off-lattice kinetic Monte Carlo methods. The
short-range interactions are included in the DFT calculations, so their effect is incorporated
in the values of the rate constants for the reactions. Long-range interactions have been
neglected. More detailed information about the continuum kMC method can be found in
chapter 2.
Water actually plays an important role in the structure direction encountered in zeolite
synthesis. Interactions of water molecules with cations are part of the template effect and,
therefore, are of crucial importance. Therefore, the water molecules are considered explicitly
in the calculation of the rate constants.25 We considered a system consisting of various
neutral and deprotonated silicate oligomers with 64 water molecules. The counterions are
Li+ and NH4+. The counterions are assumed to be in the neighborhood of the reacting silica
molecules. The simulation cell is a periodically replicated cubic box with a size corresponding
to a density of solution around 1 g cm−3 at ambient conditions. pH 7 and temperature T
350 K are used for all the kMC simulations.
The rate constants of silicate oligomerization reactions in the presence of counterions
(Li+ and NH4+) were determined from density functional theory based molecular dynamics
simulations.25 In the case of dimerization, we considered a system consisting of one silicic
acid Si(OH)4 and its deprotonated form Si(OH)3O− with 64 water molecules. The simulation
cell was a periodically replicated cubic box with a size corresponding to a density of solution
around 1 g cm−3 at ambient conditions. The temperature was set at T = 350 K imposed
with a Nose-Hoover thermostat. The parameters obtained with presence of counterions Li+
and NH4+ are listed in Table 4.1 and Table 4.2 respectively.
Table 4.1: Calculated activation energies [kJ mol−1], with presence of Li+ counterions , and
prefactors of reactions from Eq. 3.8 to Eq. 3.48. The first column is the reactions
numbered corresponding to the Equations 3.8 to 3.48 listed above. Eact(1) and Pref(1)
are the activation energy and prefactor of forward reaction respectively. Eact(-1) and
Pref(-1) are the activation energy and prefactor of reverse reaction respectively. The
temperature is 350 K and pH = 7.
Reactions Eact(1) Pref(1) Eact(-1) Pref(-1)
1 0 1.484×104 99 8.050×1012
2 191 8.040×1012 0 1.417×104
81
Effects of Counterions
3 0 1.820×107 28 7.293×1012
4 70 7.293×1012 61 7.293×1012
5 89 7.293×1012 121 7.293×1012
6 9 7.293×1012 0 7.750×1012
7 0 1.434×104 99 7.750×1012
8 191 7.750×1012 0 1.360×104
9 0 1.600×107 31 7.293×1012
10 0 1.600×107 54 7.293×1012
11 78 7.293×1012 66 7.293×1012
12 96 7.293×1012 112 7.293×1012
13 7 7.293×1012 0 7.640×1012
14 0 1.410×104 99 7.640×1012
15 191 7.640×1012 0 1.340×104
16 83 7.293×1012 66 7.293×1012
17 94 7.293×1012 91 7.293×1012
18 13 7.293×1012 0 7.660×1012
19 0 1.510×107 41 7.293×1012
20 0 1.510×107 30 7.293×1012
21 60 7.293×1012 65 7.293×1012
22 85 7.293×1012 98 7.293×1012
23 30 7.293×1012 0 7.293×1012
Some of the energies in the tables are corrected, because the silicate condensation reac-
tions are endothermic in the simulations with water molecules treated explicitly.18,25 This
is questionable, because it would mean that the oligomers will not form. We attribute this
to the contribution of entropy reduction form the water shell around the silicate molecules.
Leung et al.34 have examined the source of the entropy reduction which occurs when a
gas-phase molecule is moved into solution and have concluded that the main contribution
to the entropy change comes from cavity formation in the solvent. The change in entropy
mainly comes from the rearrangement of the water shell, that is dominated by hydrogen
bonds. The larger the molecule, the larger the water shell around it, and hence the change
in water shell is larger for a larger molecule. This indicates that the addition of counterions
contribute to the entropy change, and the contribution from NH4+ is larger than that from
82
4.2 Computational Methods
Li+, due to the larger volume of NH4+. To make the reactions exothermic, we reduce the
entropy contribution from the water shell and the counterions . The entropy contribution of
these three sets of parameters (without counter ion, with Li+ and with NH4+) are reduced
somewhat arbitrarily by 50 kJ/mol, 60 kJ/mol and 70 kJ/mol respectively. The counterions
have contribution to the intermediate states as well. We then reduce the energies of the
intermediate states of the two sets of parameters (with Li+ and with NH4+) by 50 kJ/mol
and 60 kJ/mol respectively. Although it is somewhat arbitrary, it is reasonable. If the
energy contributions are reduced less, the reactions will not take place. If they are reduced
more, dimers are too stable, as a result no further oligomerization reactions occur.
In this work, we only study the silicate oligomerization at neutral pH and temperature of
350 K. The change of pH consequently changes the number of H3O+ and OH− ions. The pH
only changes the rate constants of bimolecular reactions with solvent molecule as a reactant.
The changed rate constants can be easily calculated by changing Nsolvent. Nsolvent can be
the number of water molecules, H3O+ or OH− ions as described in the next section. The
others remain the same. The rate constant is related to temperature by the Eq. 3.49. The
rate constants at other temperatures can be calculated from the available prefactors. Thus
we give the activation energies and prefactors, the readers can calculate the rate constants
for other pHs and temperatures themselves.
Table 4.2: Calculated activation energies [kJ mol−1], with presence of NH4+ counterions , and
prefactors of reactions from Eq. 3.8 to Eq. 3.48. The first column is the reactions
numbered corresponding to the Equations 3.8 to 3.48 listed above. Eact(1) and Pref(1)
are the activation energy and prefactor of forward reaction respectively. Eact(-1) and
Pref(-1) are the activation energy and prefactor of reverse reaction respectively. The
temperature is 350 K and pH = 7.
Reactions Eact(1) Pref(1) Eact(-1) Pref(-1)
1 0 1.484×104 99 8.050×1012
2 191 8.040×1012 0 1.417×104
3 0 1.820×107 28 7.293×1012
4 106 7.293×1012 81 7.293×1012
5 95 7.293×1012 115 7.293×1012
6 9 7.293×1012 0 7.750×1012
7 0 1.434×104 99 7.750×1012
83
Effects of Counterions
8 191 7.750×1012 0 1.360×104
9 0 1.600×107 31 7.293×1012
10 0 1.600×107 54 7.293×1012
11 63 7.293×1012 72 7.293×1012
12 82 7.293×1012 119 7.293×1012
13 7 7.293×1012 0 7.640×1012
14 0 1.410×104 99 7.640×1012
15 191 7.640×1012 0 1.340×104
16 54 7.293×1012 66 7.293×1012
17 94 7.293×1012 105 7.293×1012
18 13 7.293×1012 0 7.660×1012
19 0 1.510×107 41 7.293×1012
20 0 1.510×107 30 7.293×1012
21 62 7.293×1012 76 7.293×1012
22 102 7.293×1012 108 7.293×1012
23 30 7.293×1012 0 7.293×1012
4.3 Results
0 2 4 6 8 10
0.0
5.0x10-3
1.0x10-2
1.5x10-2
2.0x10-2
2.5x10-2
3.0x10-2
3.5x10-2
Con
cent
ratio
n (m
ol/L
)
Time (s)
Dimer
Linear trimer
0 2 4 6 8 10
0.00
0.05
0.10
0.15
0.20
Con
cent
ratio
n (m
ol/L
)
Time (s)
3-ring
Figure 4.1: Concentration of linear species and 3-rings as a function of time without counter ion.
Monomeric species are initially abundant; thus, dimerization is the dominant process.
84
4.3 Results
The system almost runs out of free monomeric species after 0.005 s, containing instead
many small silicate oligomers as described below. This suggests that the monomeric species
are more reactive than highly condensed oligomers. The dimer is the first stable product of
oligomerization. The reactant complexes and intermediate species are very unstable; their
concentrations just show fluctuations (not shown). The process from monomer to dimer
through two transition states is very fast, as indicated by the sharp increase in concentration
of dimers (see Figure 4.1). The dimers participate in the further monomer addition (further
oligomerization); the fast decrease period (0.005-3.0 s) results in a low concentration as
shown in Figure 4.1. Followed by the dimerization, three-membered silicon species (linear
trimer and 3-ring) emerge. The linear trimer participates in two types of reactions. It can
further oligomerize to either form the branched tetramer (or linear tetramer) or form a 3-
ring by ring closure. In this work, we did not include the linear tetramer (and larger linear
species), because no rate constants for their formation are available. Including larger species
is useful, but in this work, we focus on the effects of counterions , and thus larger species
are not critical. The linear trimer is the largest linear species in this model; the routes
available for consumption of the linear trimer are ring closure and reacting with a monomer
to form a branched tetramer. Comparison of the concentration changes of the dimer and
linear trimer shows that dimers are metastable, and the dimers are converted into linear
trimers by monomer addition. Linear trimers are relatively more stable. In an early study,
we included the linear tetramer.28 We showed that there are fewer linear trimers formed,
due to further linear growth (formation of the linear tetramer) being favorable. The trimers
are easily converted into linear tetramers. In this work, there are fewer routes available,
which result in the slow consumption of linear trimers.
The cyclic trimer, which we call 3-ring, is the smallest closed structure that can be formed
in a silicate structure. Figure 4.1 shows the change in concentration of 3-rings with time.
The concentration of 3-rings increases rapidly and reaches a value of about 0.2 mol/L (60
percent of the all Si) after 3 s. This suggests a clear tendency to form 3-rings over other
species. Recently, we reported that the formation of 4-rings is more favorable over 3-rings.28
In this work, we show that formation of 3-rings is more favorable than branched tetramers.
Therefore, in the early stage of silicate oligomerization and without counterions, the 4-ring
is the most favorable product, the 3-ring is less favorable, and formation of the branched
tetramer is rather unfavorable.
85
Effects of Counterions
0 2 4 6 8 10
0.0
5.0x10-3
1.0x10-2
1.5x10-2
2.0x10-2
Con
cent
ratio
n (m
ol/L
)
Time (s)
Dimer
Linear trimer
0 2 4 6 8 10
0.0
2.0x10-5
4.0x10-5
6.0x10-5
8.0x10-5
1.0x10-4
1.2x10-4
1.4x10-4
1.6x10-4
1.8x10-4
Con
cent
ratio
n (m
ol/L
)
Time (s)
3-ring
0 2 4 6 8 10
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
C
once
ntra
tion
(mol
/L)
Time (s)
Branched tetramer
Figure 4.2: Evolution in concentration of key species in the presence of Li+ counterions.
Figure 4.2 shows the changes in concentrations of key species as a function of time in
the presence of Li+ counterions. Unlike the case without counter ion, the concentration
of dimers and linear trimers increase slowly at early times. As has been predicted by ab-
initio molecular dynamics simulations, the presence of Li+ increases the activation barrier of
linear growth.25 Thus, the formation of both dimers and linear trimers is slower (see Figure
4.1) compared with that without counterion. This indicates that the formation of linear
species is relatively unfavorable. Interestingly, the dimers become more stable (compared
with linear trimers) in the presence of Li+, as can be seen from the figure that there are more
dimers left after the system has reached equilibrium. The case of linear trimers is different.
There are fewer linear trimers compared with the situation in the absence of counterions.
86
4.3 Results
This means that the presence of Li+ slows down the formation rate of linear species.
The situation of 3-rings is completely different form that without counter ion as shown
in Figure 4.2. Only a few of the 3-rings can be observed. This means that the formation
of 3-rings is rather unfavorable in the presence of Li+. This is consistent with our earlier
work.25 From DFT predictions of the reaction barriers, we found that Li+ favors branched
tetramers over 3-rings. However, the reactions depend also on the concentration of reactant
species and the condition of the solution. Thus, the extent to which a certain type of species
is favored cannot be known from only reaction barrier predictions. Here, we show that there
is almost no 3-rings formed. Instead, formation of branched tetramers is favorable, as shown
in Figure 4.2. The concentration of branched tetramers increases rapidly in the first 3 s,
and then remains nearly constant. This conclusively demonstrates that the presence of Li+
greatly favors formation of branched tetramers over 3-rings.
0 2 4 6 8 10
0.0
2.0x10-4
4.0x10-4
6.0x10-4
8.0x10-4
1.0x10-3
1.2x10-3
1.4x10-3
Con
cent
ratio
n (m
ol/L
)
Time (s)
Linear trimerDimer
Figure 4.3: Concentration of linear species as a function of time in the presence of NH4+ coun-
terions .
From Figure 4.3, we can see that the change in concentration of linear species in the
presence of NH4+ is very different from the two previous cases. Only a few dimers and
linear trimers can be observed. The newly formed linear species immediately participate
in further oligomerization reactions or the reverse reactions. This indicates that the NH4+
counterions destabilize the linear species.
Fast formation of 3-rings is followed by a period with very slow changes of the concentra-
87
Effects of Counterions
0 50 100 150 200
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Con
cent
ratio
n (m
ol/L
)
Time (s)
3-ring
Branched tetramer
Continuum kMC
0 50 100 150 200
0.00
0.05
0.10
0.15
0.20
0.25
Con
cent
ratio
n (m
ol/L
)
Time (s)
3-ring
Branched tetramer
Mean field approximation
Figure 4.4: Evolution in concentration of 3-rings and branched tetramers in the presence of NH4+
counterions and compared with results from mean field approximation.
tions (see Figure 4.4). The concentration of 3-rings reaches 0.3 mol/L after 200 s. This is
higher than in the other cases studied. However, the time the system needs to reach equilib-
rium is much longer. This suggests that the presence of NH4+ does favor the formation of
3-rings, but there might be a competing process that slows down the total oligomerization
rate. The formation of branched tetramers is very fast, as indicated by the sharp increase in
concentration in the first few seconds. The concentration then decreases fast and is followed
by a gradual decrease. This indicates that the presence of NH4+ does favor the formation
of branched tetramers, but they are unstable at this condition. This shows that NH4+
ions stabilize the 3-rings but destabilize the branched tetramers. Although formation of
branched tetramers is fast, they are metastable, and they serve as a source for 3-rings. The
competition results in the slow formation of 3-rings.
4.4 Discussion
The presence and formation mechanism of the branched tetramer in the early stage of sili-
cate oligomerization is still problematic. In an early study, we have found that the branched
tetramer is very stable when the water molecules are treated using the COSMO model.17
More recently, we found that the formation of the branched tetramer is unfavorable when
the water molecules are modeled explicitly.18,28 Although many predictions based on DFT
88
4.4 Discussion
calculations have been reported, an agreement is still not reached. Stable structures of
silicate oligomers can be obtained from DFT calculations, but the most stable oligomers
might not be the most prevalent products. Further confirmation using kMC is, therefore,
necessary. We believe that the combination of DFT method used, together with continuum
kinetic Monte Carlo, allows the silicate oligomerization from solution to be modeled reliably.
No branched tetramer was observed in this simulation without a counterion. In agreement
with experimental evidence, Pereira et al. showed that it is much easier to form the lin-
ear rather than the branched tetramer.16 We have also found that the formation of linear
tetramers and 4-rings is more favorable over branched tetramers without a counterion.28
This demonstrates again that the formation of the branched tetramer is rather unfavorable.
The formation of branched tetramer16–18,25,29,30 and the branched cyclic tetramer (a 3-ring
connects with a monomer)16,31–33,35,36 has been reported by a number of studies. Using
density functional theory, Pereira et al. reported structural and energetic properties of both
branched and branched cyclic tetramers.16 They found that the branched tetramer is more
stable than branched cyclic tetramer. Charles et al. reported in more detail the formation
of the branched cyclic tetramer by three pathways. They found that the pathway from a
branched tetramer is the most favorable.31 However, the real reactions depend not only on
reaction barriers but also on the concentration of reactant species and condition of the solu-
tion. Our kMC results showed that the formation of branched tetramers is very unfavorable
(without structure-directing agents).28 Therefore, the formation pathway of the branched
cyclic tetramer from the branched tetramer is not operable (because the concentration of
reactants is very low), although the activation energy of this pathway is low. This suggests
that the branched four-silicon species is likely present in form of branched cyclic tetramer
(without the presence of structure-directing agents). The discussions above suggest that
the branched cyclic tetramer may be formed from a 3-ring (that reacts with a monomer),
although the activation energy is relatively high.
The presence of counterions does not favor the formation of linear species. This is con-
sistent with DFT predictions. With Li+, the calculated activation barrier for the first step
of dimerization is 70 kJ/mol, whereas the intermediate state is found to have a free energy
of 59 kJ/mol relative to the reactant state. For comparison, the calculated barrier and
intermediate free energy for the system without the cation is significantly lower: 44 and 18
kJ/mol, respectively.18 NH4+ counterions also destabilize the intermediate and transition
state relative to the reactant state.25 The barrier of the SiO-Si bond formation (106 kJ/mol)
89
Effects of Counterions
and intermediate free energy (85 kJ/mol) in the presence of the NH4+ cation is substantially
higher than in the system without the cation (44 and 18 kJ/mol, respectively). Both are
also significantly higher (30-40 kJ/mol) than for the first step of the dimerization reaction
in the presence of the Li+ cation. Apparently the effect of stabilization of the reactant state
and/or the weakening of the SiO-Si bond is significantly stronger with NH4+.
Results from DFT calculations have predicted that, with NH4+ in solution, the free
energy profiles for the 3-ring and branched tetramer are very similar. Therefore, the 3-ring
formation is competing with the formation of higher branched silica oligomers.25 However,
we could not provide the information about the final dominant product using only DFT.
Here, we see the competing process, and show that the final dominant product is the 3-rings.
Comparing the concentration of 3-rings in Figure 4.1, Figure 4.2 and Figure 4.4 we see that
there are more 3-rings formed with the presence of NH4+. Increasing the number of 3-rings
is important for the synthesis of low framework density (FD) zeolites. The range of the
observed FD values depends on the type and relative number of the smallest rings in the
tetrahedral networks.37 The 3-ring is, of course, the smallest ring that can be formed in the
zeolites and molecular sieves. The minimum framework density increases with the size of
the smallest rings in the network. Without structure-directing agent, the formation of the
3-ring is unfavorable. Recently, Han et al. and Jiang et al. synthesized low FD zeolites
with 3-ring and double 3-ring building units, respectively.38,39 Because Si does not favor the
formation of 3-rings because of the high Si-O-Si stress, other elements, such as Be, Zn, and
Ge, have been introduced into zeolite frameworks to facilitate the formation of 3-rings.38 In
the case of germanium silicate zeolites, the presence of germanium is a handicap from an
economical and the structure-stability point of view. In recent years, it has been shown that,
by choosing a more optimized structure-directing agent, it is possible to decrease or even
completely remove the germanium in the synthesis process.39 In this work, we show that
the presence of NH4+ counterions in solution does favor the formation of silicate 3-rings.
This may lead to the strategy for the synthesis of Germanium-free and low FD zeolites.
We take the special case, the competing process, as an example to compare with the result
from mean field approximation and to show why kMC is necessary. The rate constants
used in the mean field simulations are the same as those of kMC. Figure 4.4 shows two
distinct differences between the two models. The first is the time scale. The formation
of 3-rings (also the consumption of branched tetramers) is much slower for the mean field
90
4.4 Discussion
approximation. The second difference is that there are more branched tetramers formed
in the mean field approximation. The reason for these is that the species are assumed
to distribute homogenously and the diffusion is assumed infinitely fast in the mean field
simulations. In the continuum kMC simulations the particles are modeled explicitly, they
have to diffuse to get close enough before they are able to react. The ring closure reactions
that occur through a intermediate are always unimolecular reactions. The particle simply
diffuse and then reacts at whatever place it will be; it does not depend on the rate of
diffusion. For the intermediate species, the kMC shows fluctuations, while mean field gives
a very low concentration. The formation of 3-rings is rate limited by intermediate species,
as discussed in another study.28 Therefore, the 3-ring formation from kMC is faster than
from mean field simulations. The formation of branched tetramers (bimolecular reactions)
is diffusion limited in the continuum kMC simulations, and are, therefore, slower. Thus, the
heterogeneous distribution and the fluctuations are important.
Figure 4.5: Schematic of the structure-directing role of counterions . The reactant complexes
and intermediate species are omitted for clarity.
From the results presented above, we discuss the structure-directing role of the coun-
91
Effects of Counterions
terions . The two cations have different effects on the activation barriers; thus, different
dominant species are expected. The yellow line guides the reaction pathway without counter
ion (see Figure 4.5). As has been reported previously, formation of branched tetramers is
unfavorable without counter ion. The formation of 3-rings is much more favorable than
branched tetramers at this condition. The presence of Li+ counterions completely changes
the situation, as indicated by the green line. There are more branched tetramers formed.
Li+ ions stabilize the branched tetramers, and then no 3-ring is observed. Interestingly,
NH4+ counterions play a role somewhere between the situations with Li+ counterions and
without counter ion. As guided by the purple line, the presence of NH4+ favors the forma-
tion of branched tetramers; however, they are metastable at this condition. 3-rings become
relatively more stable with NH4+. The branched tetramers break down to support the
formation of 3-rings.
4.5 Conclusions
Comparing the results obtained from the models without and with counterions (Li+ and
NH4+), we found the following key points: (1) The dominant species depends sensitively on
the counterions . (2) Li+ counterions slow down the linear growth, while the dimers become
relatively stable. (3) Li+ favors formation of branched tetramers over 3-rings. (4) NH4+
counterions destabilize the linear species and favor the further oligomerization. (5) NH4+
favors the formation branched tetramers; however, they are metastable. The presence of
NH4+ stabilizes the 3-rings with a competing process, as predicted by DFT calculations,
which slows down the total oligomerization rate. The 3-rings dominate at long times.
Overall, a clear picture of structure direction role of counterions (Li+ and NH4+) in the
early stage of silicate oligomerization is obtained. These findings may lead to strategies for
the synthesis of low FD zeolites.
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95
Effects of Counterions
96
Chapter 5
Silicate Oligomerization and Gelation
ABSTRACT
We present a lattice-gas kinetic Monte Carlo model to investigate the formation
of silicate oligomers, their aggregation and the subsequent gelation process. In
the early oligomerization stage, the 3-rings are metastable, 5-rings and 6-rings
are formed in very small quantities, 4-rings are abundant species, linear and
branched species are transformed into more compact structures. Results reveal
that the gelation proceed from 4-ring containing species. A significant amount
of 6-rings, sharing Si with 4-rings, form in the aging stage. This reveals the
formation mechanism of silicate rings and clusters during zeolite synthesis.
Silicate Oligomerization and Gelation
5.1 Introduction
Understanding silicate oligomerization and gelation is of fundamental scientific and tech-
nological importance.1–3 Condensation of silicate monomers leads to oligomeric units that
condense further in different reaction steps to larger species and clusters. The mechanisms
governing these processes are still not well understood.2,4 The essential difficulty of studying
the pre-nucleation process arises from the fact that the silicate oligomers are typically of
the size of several Si(OH)4 molecules, which is hard to access by most of the current exper-
imental methods. Even if they are detected by microscopic techniques, the structural and
reactive properties may not be distinguished due to their small size. Furthermore, some of
the species exist for extremely short times and freely move throughout the available vol-
ume of solution, reducing the change of their appearance in the volume being examined.5
The very early stages of solid-state formation from solution can be crucial in determining
the properties of the resulting solids. Thus, higher levels of control over nucleation can-
not be achieved without understanding the fundamentals of the elementary steps of silicate
oligomerization.
A variety of spectroscopies and scattering techniques have been used to study the silica-
based condensation reactions,6–10 however the observation of these processes is still prob-
lematic due to the lack of adequate techniques to directly monitor the formation of silicate
oligomers. Knight et al. used 29Si nuclear magnetic resonance (NMR) spectroscopy to study
condensed silicate species present in aqueous solutions.8 Using mass spectrometry, Pelster et
al. investigated the temporal evolution of silicate species during hydrolysis and condensation
of silicates.10,11 However, because of the multitude of simultaneous reactions in solution, it
is difficult to extract information about individual events using only experimental data.12
Many theoretical methods have been used to model the early stages of solid forma-
tion.12–17 Typically it is difficult for MD methods to model chemical reactions. Information
about energetics of chemical reactions can be obtained by using DFT calculations, but
kinetics cannot be predicted. Stable structures of silicate oligomers can be obtained from
DFT calculations, but the most stable oligomers might not be the most preferable products.
Moreover, DFT and MD methods are computationally very expensive and restricted to very
small system and short simulation time (on the order of pico- or nanoseconds), thus rele-
vant statistical information cannot be extracted. The time scale for initial stage of zeolite
98
5.2 Model and Methods
formation is on the order of hours or even longer, which is not accessible to MD or DFT sim-
ulations. Monte Carlo models have emerged and have been proved to be good methods for
modelling such systems.13–15,18–21 Wu and Deem introduced a Monte Carlo model for silicate
solutions for investigation of the nucleation process during zeolite synthesis in the absence
of a structure directing agent.14 Malani et al. presented a reactive Monte Carlo model,
which is useful for modelling silicate oligomerization.15 They have obtained agreement for
the evolution of the Qn distribution upon comparing the simulation results to experimental
observations. Lattice-gas Monte Carlo models were also used to model the formation of
silica nanoparticles.13,18 More recently, White et al. studied the initial stages of gel/cluster
formation by using a coarse-grained Monte Carlo simulation, which represents all oligomers
as point particles. They used quantum chemical-based interaction (dimerization) energies,
which were determined through density functional theory computations.22
The effective modelling of the silicate oligomerization in solution requires a method that
can simulate events at microscopic length and macroscopic time scale. Kinetic Monte Carlo
method has the potential to shed light on the mechanism of silicate oligomerization and
gelation process. At the early stage of silicate oligomerization, various of silicate oligomers
can be formed, including linear, branched and ringed species. We have studied the silicate
oligomerization from solution up to 4 silicate species.16,19,20 Insight into the formation mech-
anism of larger silicate clusters is valuable. However, in the previous studies, DFT could not
access larger clusters, continuum kMC could model larger systems, but larger clusters could
not be predicted due to the coarse-graining effect. Here we use a lattice-gas kinetic Monte
Carlo model to give insight into the mechanism of aggregation of silicate oligomers and
the subsequent gelation. We use more accurate energy parameters than a previous Monte
Carlo study.22 We demonstrate the successful application of multiscale simulation method
to the silicate oligomerization and gelation processes, and reveal mechanistic information
regarding the formation of silicate clusters.
5.2 Model and Methods
We use a lattice kinetic Monte Carlo model23 to investigate the formation and aggregation
of silicate oligomers and the gelation process. In the lattice kinetic Monte Carlo simulations
the system is treated as a grid. A grid is a collection of sites. Each site has a label, that
99
Silicate Oligomerization and Gelation
characterizes its properties; i.e., vacant, or occupied by some chemical species. A configu-
ration of a silicate oligomer is then given by a particular distribution of labels. The change
of the labels, meaning the evolution of a system, is given by diffusion and reactions of the
silicate oligomers. This change can mathematically be formulated by the chemical Master
Equation (5.1) and can be derived from first principles. The chemical Master Equation
describes the configurational change of a system as function of time.
dPα
dt=∑β
[WαβPβ −WβαPα] . (5.1)
Pα(β) is the probability of the system being in a configuration α(β) and Wαβ(βα) is the rate
constant of the change of configuration β into α (α into β). The rate constants determine the
probability of a particular configuration and the speed of its creation and destruction. Values
for the rate constants can come from electronic structure calculations or from experiments.
We have used the body-centered cubic (BCC) lattice with a distance between the cubic
lattice points of 3.0 A (to represent the Si-O-Si bond longth, the O atoms are omitted for
simplicity of the model) in the kMC simulations. This lattice allows the formation of 3-, 4-,
5-, and 6-rings. Earlier studies have used this lattice as well,18 but have also used a simple
cubic lattice.13,22 The simulation box is 50× 50× 50 lattice points, which is 150× 150× 150
A3 in size, and the initial monomer concentration is about 1 mol/L (there are 2000 lattice
sites occupied by monomers), which is usually used in experiments. The reaction barriers
were obtained from DFT calculations, which were published earlier.16,19 Each lattice site
is occupied by a silicate monomer or vacant. The OH groups are course-grained into the
silicon atoms.
The silicate oligomerization consists of a series of reactions, including linear growth re-
actions and ring closure reactions, each of which may have its own activation energy. The
activation energies are crucial parameters of the input of kMC. In the study of White et al.22
the quantum chemical-based dimerization energy for all reactions was used. In our early
studies, the activation barriers for linear growth, 3-ring formation and 4-ring formation were
found to be quite different.16,24 It is therefore important to use different activation barriers
for linear growth, 3-ring formation and larger ring formation.
The initial monomers simply diffuse before they are able to react (become next nearest
neighbors). When two silicon atoms become nearest neighbors, they are chemically bonded.
100
5.3 Results and Discussion
The linear growth occurs via monomer addition or aggregation of dimers and trimers. For-
mation of the silicate rings (3-ring, 4-ring, 5-ring and 6-ring) occur via an intramolecular
condensation reaction as has been suggested before.16,24 The formed linear, ringed, and
branched species can also diffuse, and aggregate to form larger ones. The change in confor-
mation of aggregated silicate species is allowed. This is modeled by, for example, rotating
of newly aggregated 4-ring to form more bonds (keeping the formed bonds). The activation
energies (and also for the reverse process) for linear growth, 3-ring formation, and larger
ring formation, are 61 (77, reverse), 72 (72, reverse), and 95 (111, reverse) in kJ/mol, re-
spectively. These parameters were obtained from the literature.16,19 The diffusion of clusters
with more than ten Si atoms was not modeled, because that would add much complexity
to the model. The pH of the solution is modeled implicitly, assuming that two monomers
can always react via the anionic mechanism,24 in which there is on neutral and one anionic
reactant.
5.3 Results and Discussion
Figure 5.1 shows the the evolution of the system. The system starts with monomers
only. They are distributed randomly in a cubic box with periodic boundary conditions.
Monomeric species are initially abundant, thus dimerization that leads to the first stable
product, the dimer, is fast. The formed dimers can initiate another reaction, the formation
of trimers. The linear growth can lead to dimer, trimer, tetramer, pentamer and hexamer.
Formation of the silicate rings (3-ring, 4-ring, 5-ring and 6-ring) occur via an intramolecu-
lar condensation reaction. Monomers are abundant at this stage, the monomer addition is
therefore the dominant process. Monomers are consumed rapidly, resulting in many small
silicate oligomers. Pentamers and hexamers are rare. The dominant linear species are dimer,
trimer and tetramer. The reverse processes are also possible. There are also monomers pro-
duced by hydrolysis reactions. At t = 0.02 s, there are lots of 3-rings formed as shown in
the figure. At the same time, 4-rings emerge. With the increase of time, the number of
4-rings increases rapidly, the 3-rings are consumed at the same time. A multitude of silicate
structures is formed, these can be linear, branched, cyclic, branched cyclic, and double rings.
We can see from the figure that the 4-ring containing species starts to grow larger at t =
0.1 s. They continue to grow to form the larger clusters, the liner species and the 3-rings
101
Silicate Oligomerization and Gelation
are almost completely consumed.
Figure 5.1: Schematic of the evolution of the system studied. The different types of silicon
atoms are color coded as follows: green, monomers and linear species; pink, 3-rings;
yellow, 4-rings and aggregated 4-ring containing silicate clusters. Oxygen atoms are
omitted for clarity.
Figure 5.2 presents the change in rates of linear growth (and the reverse processes) with
time. The forward process is initially faster. The rate of the reverse follows that of the for-
ward process, and finally both have the same rates. This indicates fast monomer exchanges
between silicate oligomers. The rate of monomer exchanges for dimer is much faster than for
trimer and tetramer (ratedimer > ratetrimer > rateteramer). Pelster et al. have used electrospray
ionization mass spectrometry (ESI-MS) in connection with isotopically labeled silicates to
study the interconversion process between oligomers.25 They have focused on two cagelike
species: the cubic octamer in the presence of tetramethylammonium (TMA+) and the pris-
matic hexamer in the presence of tetraethylammonium (TEA+). For both species a second,
identical solution, but made from 29Si-enriched silica was prepared in parallel. After the
two solutions were combined, exchange of the 29Si atoms between the silicate oligomers was
observed. They have found that the interconversion of even very stable oligomers in solution
is rapid, and the interconversion processes between silicates in solution are fast in compar-
102
5.3 Results and Discussion
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.00
0.05
0.10
0.15
0.20
rate
(s-1)
t (s)
dimerization reverse
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.000
0.005
0.010
rate
(s-1)
t (s)
trimerization reverse
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.000
0.002
0.004
rate
(s-1)
t (s)
tetremerization reverse
Figure 5.2: Change in rate with time of silicate dimer, linear trimer and linear tetramer.
ison with the timescale of zeolite formation.25 They also found that the exchange for the
hexamer proceeds by about two orders of magnitude faster than for the octamer. Here, the
fast interconversions (monomer exchanges) between silicate oligomers (dimer, linear- trimer
and tetramer) are presented. The rates of monomer exchanges are found to be faster for
smaller silicate oligomers. Thus, our calculated results are consistent with those obtained
form mass spectrometric analysis.
At early times, formation of 5-ring and 6-ring is rare. 4-rings dominate the ring popula-
tion. The 6-ring is also a common structure of many zeolites. 4-rings can be formed during
the early oligomerization stage, but 6-rings form later. The aggregated silicate species re-
arrange their conformations. There are lots of 6-rings formed during this process as shown
in Figure 5.3 where we highlighted the 6-rings. From the enlarged graph (the insert of
Figure 5.3) we can see that the 6-rings are formed via sharing silicon with 4-rings. It is very
common to find the 6-rings sharing silicon with 4-ring in the zeolite structures. In both
103
Silicate Oligomerization and Gelation
theoretical and experimental studies,26,27 there are lots of 5-rings formed in the presence of
TPA template molecules. The formation of 5-ring is rare in this simulation. The presence
of TPA template molecules may stabilize the 5-ring or lower the activation energy of 5-ring
formation, this will be studied in another work. This may tentatively lead to the conclusion
that the presence of TPA template favors formation of 5-rings.
Figure 5.3: 6-rings (highlighted) in aggregates.
Figure 5.4 shows the change in fraction of silicate oligomers as a function of time. The
system departs from 100 percent of monomers. The monomer addition to form linear species
is fast. The linear species are meatastable as can be seen from Figure 5.4. The linear species
are transformed into rings to be more condensed. At early times the formation of 3-ring is
faster than the 4-rings. However 3-rings are not stable, they reopen to form linear trimer,
and serve as source for the growth of larger species. This is in good agreement with our
earlier work. In another study, we demonstrated by using an off-lattice kinetic Monte
Carlo model the silicate oligomerization in water solution.19 The 3-ring was found to be
104
5.3 Results and Discussion
metastable, and 4-rings were the dominating species. However the larger species were not
included in that work. In the present work, we focus on the aggregation and formation of
larger silicate species. Figure 5.4 demonstrates that the formation 4-rings is very fast in
the first 0.5 s, when supporting species are abundant. The fraction of 6-rings follows the
increase of 4-rings. The total percentage is high than 1, which is due to the fact that 6-rings
share Si with 4-rings. Formation of 6-rings mainly occur through aging of silicate clusters.
As already reported, the use of different ammonium salts as the organic template leads
to different products. Using mass spectrometry, Pelster et al. reported that the presence
of TMA+ favored the formation of double 4-ring, whereas in TEA+ containing samples, the
equilibrium is pushed to the side of double 3-ring.10 Formation of double rings is therefore
very sensitive to the template molecules.11 Mass spectrometry studies showed lots of double
3-ring and double 4-rings with present of template molecules.10 In this study, although lots
of 3-rings are formed at early stage, the formation of double 3-ring is unfavorable. This may
tentatively lead to the conclusion that the formation of double 3-rings is rather unfavorable
in the absence of template molecules.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Frac
tion
Time(s)
Monomers and linear species
3-rings also in aggregates
4-rings also in aggregates
6-rings also in aggregates
Figure 5.4: Change in the fraction of silicate oligomers as a function of time. Note: 4-rings and
6-rings share Si.
105
Silicate Oligomerization and Gelation
0 10 20 30 40 500.0
0.1
0.2
0.3
0.4
0.5
0.6
Frac
tion
Size of the clusters
0.01 s
0.02 s
0.05 s
0.1 s
0.5 s
1.0 s
2.0 s
3.0 s
6.0 s
Figure 5.5: Cluster size distribution at different simulation times. There is a shift to higher
values, by 0.05 in step, for clarity.
Figure 5.5 gives the cluster size distribution at different simulation times. At t = 0.01
s, the major species are monomers and dimers, and the largest cluster is 9 Si in size. The
fraction of monomer and dimer decreases rapidly with time, which results in a fast growth
of the clusters. After 2 s, the species population is dominated by the clusters ranging from
9 to 23 Si in size. The largest clusters are 8, 11, 14, 21 37, at time 0.001, 0.002, 0.005,
0.1, 0.5 s, respectively. The largest cluster size increases rapidly during the first 0.5 s, and
remains 39 for the rest of the simulation time. This reveals that, in the first 0.05 s, the
dominant processes are dimerization and trimerization. After that, larger species form, and
the aggregation proceeds, which results in the fast increase in the cluster size, as shown in
Figure 5.5. The fraction of the silicate clusters smaller than 8 in size are very small after
2 s. This is in agreement with the analysis of the change in average cluster size with time,
as presented in Figure 5.6. The average cluster size increases to 13.5 Si at t = 2 s, and is
106
5.3 Results and Discussion
followed by a slow increase. Although the largest cluster stops growing after 1 s, there are
still some amount of small species which can aggregate to increase the average cluster size.
0 1 2 3 4 5 60
2
4
6
8
10
12
14
16
A
vera
ge c
lust
er s
ize
Time (s)
Figure 5.6: The change in average cluster size with time.
Finally, we give a schematic representation of the growth pathway (Figure 5.7). In
solutions, monomeric silicate molecules undergo condensation reactions that lead to the
formation of silicate oligomers. These oligomers can be linear and ringed species. The
linear species cannot be longer than 6 silicon atoms, the largest ring can be a 6-ring. The
initial species population is dominated by 4-ring containing species, as shown in Figure
5.7. These variety of oligomers then aggregate to form larger ones. The firstly aggregated
structure is quite open. Many silicate clusters have branches. The aging occurs after the
aggregates formed. As the aging proceeds, the clusters tend to be more condensed. Both
4-rings and 6-rings were found to be crucial to the zeolites formation. 6-rings are formed
during aging, and share Si with 4-rings. The cooperative behavior between 4-rings and
6-rings is important to the formation of many kinds of zeolite.
Depla et al. presented UV-Raman investigation of the initial silicate oligomerization
reactions. In their work, chain growth, cyclodimerization, and branching were identified.
107
Silicate Oligomerization and Gelation
Chains did not grow longer than pentamer, and ring sizes were limited to 6-rings. 4-rings
were abundant species. Gelation proceeded from branched 4-rings and branched chains. In
the present work, the calculated results are in good agreement with what is found experi-
mentally. We also found that the 4-rings dominant the ring population. The formation of
clusters proceeds form 4-ring containing species.
Figure 5.7: The reaction pathway leading to silicate clusters.
5.4 Conclusions
The silicate oligomerization and gelation have been studied using lattice kinetic Monte
Carlo. Good agreement with experimental studies have been obtained. The following key
points were found. (1) Results reveal that linear species tend to close to form rings. (2)
3-rings are metastable, the formed 3-rings reopen to support formation of larger species. (3)
4-rings dominant the ring population. (4) 5-rings and 6-rings are rare in the early stage.
108
BIBLIOGRAPHY
(5) The aggregation of silicate oligomers is followed by aging that leads to more condensed
silicate clusters. (6) The gelation proceeds from 4-ring containing structures. (7) 6-rings are
mainly formed during aging of of the silicate clusters (8) Fast monomer exchanges between
silicate oligomers are found, which is consistent with experimental findings.
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110
Summary
Kinetic Monte Carlo Modelling of The Initial Stages of Zeolite Synthesis
The very early stages of solid-state formation from solution can be crucial in determining
the properties of the resulting solids. Thus, higher levels of control over nucleation cannot
be achieved without understanding the fundamentals of the elementary steps of zeolite
synthesis. The mechanisms governing the transformation of small silicate molecules into
clusters are still far from being understood. In this thesis the silicate oligomerization is
studied using our newly developed method, their aggregation and the subsequent gelation
are also discussed.
A form of kMC simulations, which we call continuum kMC, that is useful to simulate
reactions in solution is presented (Chapter 2). In this method, the rate constants of the
reactions can be determined prior to the simulation, so that the simulation itself takes little
computer time, or can be done on large systems. We have derived the method from the
master equation that described the evolution of the system as hops from one minimum of
the potential-energy surface to a neighboring one. This master equation is coarse-grained
by using an analytical approach to the diffusion of the particles. This leads to a new master
equation that describes only the chemical reactions, and no other processes. The diffusion
is incorporated in the expression for the rate constants. Solvent molecules need not be
included explicitly in the simulations. Their effect can be incorporated in the rate constants
as well. An important aspect of zeolite synthesis is the effect of template molecules, other
cations, pH, and temperature. All this can easily be included in our method. The new
method is proved to be successful in Chapter 3 and Chapter 4. We think that continuum
kMC will be useful for many other systems. This may open the way to study many other
important problems occurring in solutions on the atomic length and macroscopic time scale.
Summary
In Chapter 3, we report an investigation of oligomerization reactions of large scale silicate-
solution systems, using the method developed in Chapter 2. The calculated results demon-
strate that the continuum kMC theory is able to provide detailed information regarding the
early stage of zeolite formation. Comparing continuum kMC and mean field approximations
on the silica-solution system, we conclude that the mean field approximation is rate-limited
by intermediate species. We demonstrate that pH and temperature greatly influence the
oligomerization rate and pathway. Therefore, silicate oligomerization can be controlled by
varying the pH and temperature of the solution. A significant finding is that near neutral pH
favors linear growth, because the linear growth is mainly driven by an anionic mechanism
in which there is one neutral and one anionic reactant, while a higher pH makes the silicate
species anionic, which facilitates ring closure. In the case of pH 7, the species oligomerize
first to linear tetramers and then close to form 4-rings, while at high pH the linear growth
and ring closure occurs simultaneously. The total growth rate is a interplay between linear
growth and ring closure. pH 8 is found to be the optimum value that takes care of both
linear growth and ring closure, and hence the silicate oligomerization is the fastest at pH 8.
The decrease of cluster size with pH is due to the fact that the double-anionic mechanism
operable is very slow. The rate-determining steps are ring closure, at very low pH, and
linear growth, at very high pH. Preferred conditions necessary for effective oligomerization
that can accelerate the initial stage of silicate oligomerization and as a result avoid the
formation of undesired species have been obtained.
The silicate oligomerization with presence of counterions is investigate in Chapter 4. A
comparison of the results in the presence of counter ions (Li+ and NH4+) with those obtained
in Chapter 3 is presented. The dominant species depends sensitively on the counterions.
Li+ counterions slow down the linear growth, while the dimers become relatively stable.
Li+ favors formation of branched tetramers over 3-rings. NH4+ counterions destabilize the
linear species and favor the further oligomerization. NH4+ favors the formation branched
tetramers; however, they are metastable. The presence of NH4+ stabilizes the 3-rings with a
competing process, as predicted by DFT calculations, which slows down the total oligomer-
ization rate. The 3-rings dominate at long times. In this chapter, a clear picture of structure
direction role of counter ions (Li+ and NH4+) in the early stage of silicate oligomerization
is presented.
Extending our research to a larger scale would be valuable. However the continuum
112
kinetic Monte Carlo method could not probe the formation mechanism of larger species,
due to that the silicate oligomers are regarded as pointlike particles. Chapter 5 presents a
lattice kinetic Monte Carlo study of the silicate oligomerization and gelation, in which the
silicate clusters are much larger than those of the previous chapters. The lattice kinetic
Monte Carlo results reveal that the linear species tend to close to form rings. 3-rings
are metastable, the formed 3-rings reopen to support formation of larger species. 4-rings
dominant the ring population. The aggregation of silicate oligomers is followed by aging
that leads to more condensed silicate clusters. The gelation proceeds from 4-ring containing
structures. 6-rings are mainly formed during aging of the silicate clusters. These findings
are in good agreement with experimental results.
We believe that our research have provided valuable insight into the mechanism of the
initial stages of zeolite synthesis.
113
Summary
114
List of Publications
1. X. Q. Zhang, T. T. Trinh, R. A. van Santen and A. P. J. Jansen, Structure-Directing Role
of Counterions in the Initial Stage of Zeolite Synthesis,The Journal of Physical Chemistry
C, 2011, 115, 9561.
2. X. Q. Zhang, T. T. Trinh, R. A. van Santen and A. P. J. Jansen, Mechanism of the
Initial Stage of Silicate Oligomerization, Journal of the American Chemical Society, 2011,
133, 6613.
3. X. Q. Zhang and A. P. J. Jansen, Kinetic Monte Carlo Method for Simulating Reactions
in Solutions, Physical Review E 2010, 82, 046704.
4. X. Q. Zhang, W. K. Offermans, R. A. van Santen, A. P. J. Jansen, A. Scheibe, U. Lins and
R. Imbihl, Frozen Thermal Fluctuations in Adsorbate-induced Step Restructuring, Physical
Review B, 2010, 82, 113401.
5. X. Q. Zhang, T. T. Trinh, R. A. van Santen and A. P. J. Jansen, Reply to “Comment
on ‘Structure Directing Role of Counterions in the Initial Stage of Zeolite Synthesis” The
Journal of Physical Chemistry C, accepted for publication.
6. X. Q. Zhang, R. A. van Santen and A. P. J. Jansen, Kinetic Monte Carlo Modelling of
Silicate Oligomerization and Early Gelation, in preparation.
Others:
1. X. Q. Zhang, H. Li and K. M. Liew, The Structures and Electrical Transport Properties
of Germanium Nanowires Encapsulated in Carbon Nanotubes, Journal of Applied Physics,
2007, 102, 073709.
2. X. Q. Zhang, H. Li, K. M. Liew, Y. F. Li and F. W Sun, Helical Shell Structures of
List of Publications
Ni–Al Alloy Nanowires and Their Electronic Transport Properties, Chinese Physics Letters,
2007, 24, 1701.
3. H. Li, X. Q. Zhang, F. W. Sun, Y. F. Li, K. M. Liew and X. Q. He, Theoretical Study of
the Electrical Transport of Nickel Nanowires and a Single Atomic Chain, Journal of Applied
Physics, 2007, 102, 013702.
4. H. Li, X. Q. Zhang and K. M. Liew, Structures and Electronic Transport of Water
Molecular Nanotubes Embedded in Carbon Nanotubes, Journal of Chemical Physics, 2008,
128, 034707.
5. H. Li and X. Q. Zhang, The Dependence of Electronic Transport on Compressive Defor-
mation of C60 molecule, Physics Letters A, 2008, 372, 4294.
6. H. Li, X.Q. Zhang, K.M. Liew and X.F. Liu, Structures of Water Molecular Nanotube
Induced by Axial Tensile Strains, Physics Letters A, 2008, 372, 6288.
116
Curriculum Vitae
Xueqing Zhang was born on the 29th of October 1982 in Jinan, China. After he got his
bachelor’s dgree at the Yantai Normal University in 2005, he continued his master study at
the Ocean University of China. His research project was modeling structures and electronic
transport properties of low dimensional nanomaterials, supervised by Prof. Li Hui. In
January 2008 he defended his thesis and got his master’s degree. On 4th of February 2008,
he started his Ph.D. research under supervision of Prof. Rutger van Santen and Dr. Tonek
Jansen, at the Eindhoven University of Technology. His research project dealt with the
theoretical understanding of the initial stages of zeolite synthesis. The most important
results of this work are described in this thesis.
Curriculum Vitae
118
Acknowledgements
First and foremost I would like to express my deepest gratitude to Prof. Rutger van Santen
for giving me the opportunity to work in his research group. Rutger, thank you very much
for the many stimulating discussions and for your continuous support and help during my
Ph.D. I am very grateful to Dr. Tonek Jansen for his careful guidance and for providing me
with his extensive knowledge in the field of kinetic Monte Carlo simulations, which is the
main method used in this thesis. Dear Rutger and Tonek, without your contribution and
the stimulating discussions we had, a lot of the results described in this thesis would not
have been obtained.
I am very grateful to the members of the defense committee (Prof. Hans Niemantsver-
driet, Prof. Johan Lukkien, Prof. Thijs Michels, Prof. Evert Jan Meijer, and Prof.
Veronique Van Speybroeck) for the time they spent in reading the thesis and for their
helpful suggestions.
Prof. Emiel Hensen is specially acknowledged for giving me the opportunity the continue
my research as a postdoc. Thanks to Emma for her patience in administrative questions
and kind help.
I thank the current and former members of the group for their welcoming attitude and the
nice atmosphere they create in the group. A number of them I want to mention specifically.
Shuxia, thanks for many scientific discussions, for being a nice officemate, and for sharing
ideas. Thuat, thanks a lot for your nice DFT results and useful discussions.
Peter Vassilev and Bouke, as the system administrators, your responds are always helpful
and in time, thanks a lot. I also thank Cristina, Weiyu, Tianwei, Guanna, Sharan, Bartek,
Yang Gang, Ionel, Olus, Evgeny, Ivo, Minhaj, Ojwang, for lots of useful scientific discussions.
Acknowledgements
I also thank other group members, Gao Lu (also Liu Rong), Zhang Yi, Liu Peng, Li
Meiqin, Zhang Zhijun, Li Xiaobo, Guo Meiling, Quek Xianyang, Guan Yejun, Yang Jie,
Wu Leilei (and Zhu Min), Yue Chaochao, Zhang Yanmei, Alessandro, Arjan, Pieter, Tang
Haodong, Xie Rongjun, Wang Peiyuan, for their cooperation, helpful discussion and leisure
activities. I also extend my gratitude to all people who were not mentioned but supported
me during my time in SMK.
Special acknowledgements to Lou Xianwen, Tian Mingwen, Xue Lijing, Song Liguo and
Lv Kangbo for your kind help during my Ph.D study, and to Donglin for your contribution
to this thesis.
I also thank many other friends, Huan Rubin, Zhang Lianzhi, Ma Piming, Cai Xiaoxia,
Xu Wei, Chen Jiaqi, Guan Qingling, Li Weizhen, Hu Xin, Zhou Qian, Tsoi Shufen, Jeroen,
Guo Mingyu, Lin Jianbin, Wang Feng, Chen Yulan, Wu Jing, Li Yun, Han Yang, Jiu
Tonggang, Zhang Xiaoying, Jia Changwen, Xie Rui, Wang Qi, Gao Chuanbo, Li Yingyuan,
Tan Lianghui, Chen Delei, Kong Chuipeng, Sun Chunxia, Dong Weifu for sharing ideas and
the nice party time.
I would like to express my appreciation to all friends in the Netherlands for their friendship
and support, Wu Yan, Liu Ying, Wu Jingyuan, Wang Fei, Gao Yan, Sun Fengwei, Shen Ju,
Yang Lei, Han Jinping, Gu Bing, Chai Yajing, Milos, Feng Tao, Zhang Lili, Li Ping, Liu
Zhen, Wang Cuiping, Jin Yi, Li Qian, Han Jungong, Wang Chunmei. My special thanks
to the card playing committee, Lou Xianwen, Tian Mingwen, Xue Lijing, Lu Gang, Ma
Piming, and thanks to Mingwen and Lijing for the place and food support. Thanks to the
badminton people, tennis people, and the taekwondo club. I really enjoyed playing with
you. Merijn, thanks for your training. Thanks to the family of Han and Tine, who make
my stay in Eindhoven colorful. I am also grateful to many people that I can not mention
all here.
My special thanks to Prof. Li Hui at the Ocean University of China (now at the Shandong
University), who brought me into the world of theoretical modelling.
I am deeply indebted to my parents who have always provided me with continuous support
and encouragement. To my grandparents and other relatives, thank you all. Dear grandma,
you are gone but not forgotten. To my parents in-law, sister in-law, and other in-laws, thank
you very much for everything you have done for me.
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Last but certainly not least, I would like to express my appreciation to my wife Liping. I
could not have finished this thesis without your love and support! Dear Liping, I love you!
And all the best wishes to our baby who is swimming in mommy’s womb. Baby, ··{Ï
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