Kinetic Monte Carlo modelling of the initial stages of … Monte Carlo Modelling of The Initial...

128
Kinetic Monte Carlo modelling of the initial stages of zeolite synthesis Zhang, X. DOI: 10.6100/IR724487 Published: 01/01/2012 Document Version Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Zhang, X. (2012). Kinetic Monte Carlo modelling of the initial stages of zeolite synthesis Eindhoven: Technische Universiteit Eindhoven DOI: 10.6100/IR724487 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 25. Jun. 2018

Transcript of Kinetic Monte Carlo modelling of the initial stages of … Monte Carlo Modelling of The Initial...

Page 1: Kinetic Monte Carlo modelling of the initial stages of … Monte Carlo Modelling of The Initial Stages of Zeolite Synthesis PROEFSCHRIFT ter verkrijging van de graad van doctor aan

Kinetic Monte Carlo modelling of the initial stages ofzeolite synthesisZhang, X.

DOI:10.6100/IR724487

Published: 01/01/2012

Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

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

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 25. Jun. 2018

Page 2: Kinetic Monte Carlo modelling of the initial stages of … Monte Carlo Modelling of The Initial Stages of Zeolite Synthesis PROEFSCHRIFT ter verkrijging van de graad van doctor aan

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

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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

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To my wife Liping

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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

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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

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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-

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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

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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

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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

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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

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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

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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.

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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

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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

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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)

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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

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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).

Sγα

βαS

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

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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

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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.

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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)

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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

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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,

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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).

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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.

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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).

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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)

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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)

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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

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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.

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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)

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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)

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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.

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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

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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

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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

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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

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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

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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

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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).

34

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BIBLIOGRAPHY

[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

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Theory of the Off-lattice Kinetic Monte Carlo

36

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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.

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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

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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

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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

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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.

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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

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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)

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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]−

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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

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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

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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)

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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)

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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)

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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-

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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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BIBLIOGRAPHY

[40] Cho, H.; Felmy, A. R.; Craciun, R.; Keenum, J. P.; Shah, N.; Dixon, D. A. J. Am.

Chem. Soc. 2006, 128, 2324.

[41] Gomes, J. R. B.; Cordeiro, M. N. D. S.; Jorge, M. Geochim. Cosmochim. Acta 2008,

72, 4421.

[42] Knight, C. T. Zeolites 1990, 10, 140.

[43] Fedeyko, J. M.; Rimer, J. D.; Lobo, R. F.; Vlachos, D. G. J. Phys. Chem. B 2004,

108, 12271.

[44] Yang, S.; Navrotsky, A.; Wesolowski, D. J.; Pople, J. A. Chem. Mater. 2004, 16, 210.

[45] Tleugabulova, D.; Duft, A. M.; Zhang, Z.; Chen, Y.; Brook, M. A.; Brennan, J. D.

Langmuir 2004, 20, 5924.

[46] Lin, H.-P.; Mou, C.-Y. Acc. Chem. Res. 2002, 35, 927.

[47] Zhdanov, S. P. ACS Adv. Chem. Ser. 1971, 101, 20.

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Mechanism of Silicate Oligomerization

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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.

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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

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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-

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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.

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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

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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

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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

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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.

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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.

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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.

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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-

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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

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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)

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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

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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-

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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.

Bibliography

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[5] Mora-Fonz, M. J.; Catlow, C. R. A.; Lewis, D. W. Angew. Chem. Int. Ed. 2005, 44,

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[6] de Moor, P. P. E. A.; Beelen, T. P. M.; van Santen, R. A. J. Phys. Chem. B 1999,

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ley, S. M. Phys. Chem. Chem. Phys. 2010, 12, 786.

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[11] Cundy, C S.; Cox, P. A. Micropor. Mesopor. Mater. 2005, 82, 1.

[12] Wu, M. G.; Deem, M. W. J. Chem. Phys. 2002, 116, 2125.

[13] Schumacher, C.; Gonzalez, J.; Wright, P. A.; Seaton, N. A. J. Phys. Chem. B 2006,

110, 319.

[14] Malani, A.; Auerbach, S. M.; Monson, P. A. J. Phys. Chem. Lett. 2010, 1, 3219.

[15] Jorge, M.; Auerbach, S. M.; Monson, P. A. J. Am. Chem. Soc. 2005, 127, 14388.

[16] Pereira, J. C. G.; Catlow, C. R. A.; Price, G. D. J. Phys. Chem. A 1999, 103, 3268.

[17] Trinh, T. T.; Jansen, A. P. J.; van Santen, R. A. J. Phys. Chem. B 2006, 110, 23099.

[18] Trinh, T. T.; Jansen, A. P. J.; van Santen, R. A.; Meijer, E. J. Phys. Chem. Chem.

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Effects of Counterions

[19] Rao, N. Z.; Gelb, L. D. J. Phys. Chem. B 2004, 108, 12418.

[20] Zhang, X.-Q.; Jansen, A. P. J. Phys. Rev. E 2010, 82, 046704.

[21] Houssin, C. J.-M. Y. Nanoparticles in Zeolite Synthesis ; 2003.

[22] Burkett, S. L.; Davis, M. E. Chem. Mater. 1995, 7, 920.

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nardi, S. Microporous Mesoporous Mater. 2004, 74, 59.

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Chem. Soc. 2006, 128, 2324.

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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.

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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

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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

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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.

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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

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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-

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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

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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

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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.

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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

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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.

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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.

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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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2007, 129, 15414.

[2] Auerbach, S. M.; Ford, M. H.; Monson, P. Curr. Opin. Colloid Interface Sci. 2005, 10,

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[3] Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979.

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[11] Pelster, S. A.; Schuth, F.; Schrader, W. Anal. Chem. 2007, 79, 6005.

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[13] Jorge, M.; Auerbach, S. M.; Monson, P. A. J. Am. Chem. Soc. 2005, 127, 14388.

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[14] Wu, M. G.; Deem, M. W. J. Chem. Phys. 2002, 116, 2125.

[15] Malani, A.; Auerbach, S. M.; Monson, P. A. J. Phys. Chem. Lett. 2010, 1, 3219.

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Phys. 2009, 11, 5092.

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[19] Zhang, X.-Q.; Trinh, T. T.; van Santen, R. A.; Jansen, A. P. J. J. Am. Chem. Soc.

2011, 133, 6613.

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2011, 115, 9561.

[21] Zhang, X.-Q.; Jansen, A. P. J. Phys. Rev. E 2010, 82, 046704.

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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.

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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

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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.

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Summary

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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

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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.

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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.

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Curriculum Vitae

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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.

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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

���

121