E cient Approaches to Modelling and Characterizing the ... · Self-Organization of Nanomaterials ... around the world. Nowadays, many disciplines are involved in this process: engi-neering,

Efficient Approaches to Modelling

and Characterizing the

Self-Organization of Nanomaterials

Effiziente Ansatze zur Modellierung und Charakterisierung der

Selbstorganisation von Nanomaterialien

Der Naturwissenschaftlichen Fakultat

der Friedrich-Alexander-Universitat Erlangen-Nurnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Patrick Duchstein

aus Frankfurt am Main

Als Dissertation genehmigt

von der Naturwissenschaftlichen Fakultat

der Friedrich-Alexander-Universitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 14. April 2016

Vorsitzender des Promotionsorgans: Prof. Dr. Jorn Wilms

Gutachter: Prof. Dr. Dirk Zahn

Prof. Dr. Tim Clark

Acknowledgments

Writing this PhD thesis has been a long and challenging journey, which would not

have been possible without the help of others.

First and foremost I would like to thank my supervisor, Prof. Dirk Zahn. For

the continuous and longlasting support throughout the years. For the always

inspiring abundance of creativity, constantly supplying me with new ideas and

perspectives. For the positive attitude, always having a few supportive and mo-

tivating words ready. For giving me the opportunity to jointly attend several

international conferences. For the enjoyable coffee talks, and the very amicable

atmosphere. Thanks a lot!

Furthermore, I thank Prof. Rudiger Kniep for the support during the first part

of my PhD thesis in Dresden, for providing me with funding, and for inspiring my

work dealing with biomineralization.

More thanks go to Prof. Tim Clark and Prof. Wolfgang Peukert for scientific

collaboration.

I would also like to thank my co-workers and friends, emphasizing Theodor

Milek, Philipp Ectors, and Alexander Urban for long scientific (and non-scientific)

discussions, along with plenty of constructive criticism, always leading to new

perspectives and insights. Moreover, I thank Hanno Dietrich, Markus Walther,

Tina Kollmann, Konstantin Weber, Nico van Eikema Hommes, and Christina

Ebensperger, for the very enjoyable times at work.

From the former Dresden group, I thank Agnieszka Kawska and Oliver Hochrein

for introducing me to molecular dynamics simulations.

I thank my family, and all my friends from Erlangen and Dresden, for continu-

ously supporting me throughout the last years.

iii

List of publications

[1] Agnieszka Kawska, Patrick Duchstein, Oliver Hochrein, andDirk Zahn. Atomistic Mechanisms of ZnO Aggregation from EthanolicSolution: Ion Association, Proton Transfer, and Self-Organization. NanoLetters, 8(8):2336–2340, August 2008.∗†

[2] Patrick Duchstein, Oliver Hochrein, and Dirk Zahn. AutomatedMotif Identification in Solids. Zeitschrift fur anorganische und allgemeineChemie, 634(11):2035, 2008.

[3] Agnieszka Kawska, Patrick Duchstein, Oliver Hochrein, andDirk Zahn. From Ion Aggregation to Nanocrystal (Self-)Organization: aTransferable Simulation Platform. Zeitschrift fur anorganische und allge-meine Chemie, 634(11):2017, 2008.

[4] Patrick Duchstein, Oliver Hochrein, and Dirk Zahn. Motif Iden-tification in Materials Simulations. Zeitschrift fur anorganische und allge-meine Chemie, 635(4-5):649–652, 2009.∗

[5] Agnieszka Kawska, Patrick Duchstein, Oliver Hochrein, andDirk Zahn. Atomistic modelling of ion aggregation from solution and theself-organization of nanocrystals and nanocomposite biomaterials. Chem-istry Central Journal, 3:P33, June 2009.

[6] Theodor Milek, Patrick Duchstein, Gotthard Seifert, and DirkZahn. Motif Reconstruction in Clusters and Layers: Benchmarks for theKawska-Zahn Approach to Model Crystal Formation. ChemPhysChem, 11(4):847–852, 2010.∗†

v


[7] Jurgen Brickmann, Raffaella Paparcone, Simon Kokolakis,Dirk Zahn, Patrick Duchstein, Wilder Carrillo-Cabrera, PaulSimon, and Rudiger Kniep. Fluorapatite-Gelatine Nanocomposite Su-perstructures: New Insights into a Biomimetic System of High Complexity.ChemPhysChem, 11(9):1851–1853, 2010.

[8] Patrick Duchstein and Dirk Zahn. Atomistic modeling of apatite-collagen composites from molecular dynamics simulations extended to hy-perspace. Journal of Molecular Modeling, 17(1):73–79, January 2011.∗

[9] David M. Benoit, Philipp Ectors, Patrick Duchstein, JosefBreu, and Dirk Zahn. A new polymorph (IV) of benzamide: struc-tural characterization and mechanism of the I-IV phase transition. ChemicalPhysics Letters, 514:274–277, August 2011.

[10] Annu Thomas, Elena Rosseeva, Oliver Hochrein, WilderCarrillo-Cabrera, Paul Simon, Patrick Duchstein, Dirk Zahn,and Rudiger Kniep. Mimicking the Growth of a Pathologic Biomineral:Shape Development and Structures of Calcium Oxalate Dihydrate in thePresence of Polyacrylic Acid. Chemistry – A European Journal, 18(13):4000–4009, March 2012.

[11] Patrick Duchstein, Christian Neiss, Andreas Gorling, and DirkZahn. Molecular mechanics modeling of azobenzene-based photoswitches.Journal of Molecular Modeling, 18(6):2479–2482, June 2012.

[12] Theodor Milek, Patrick Duchstein, and Dirk Zahn. Mit Simula-tionen Nanokristallen und -kompositen auf der Spur. Nachrichten aus derChemie, 60(9):868–871, September 2012.

[13] Patrick Duchstein, Rudiger Kniep, and Dirk Zahn. On the func-tion of saccharides during the nucleation of calcium carbonate-protein bio-composites. Crystal Growth & Design, 13(11):4885–4889, September 2013.∗

[14] Philipp Ectors, Patrick Duchstein, and Dirk Zahn. NucleationMechanisms of a Polymorphic Molecular Crystal: Solvent-Dependent Struc-tural Evolution of Benzamide Aggregates. Crystal Growth & Design, 14(6):2972–2976, April 2014.∗†

vi


[15] Philipp Ectors, Patrick Duchstein, and Dirk Zahn. From oligomerstowards a racemic crystal: molecular simulation of DL-norleucine crystalnucleation from solution. CrystEngComm, 17:6884–6889, 2015.∗†

[16] Patrick Duchstein, Theodor Milek, and Dirk Zahn. MolecularMechanisms of ZnO Nanoparticle Dispersion in Solution: Modeling of Sur-factant Association, Electrostatic Shielding and Counter Ion Dynamics.PLoS ONE, 10(5):e0125872+, May 2015.∗

[17] Patrick Duchstein, Tim Clark, and Dirk Zahn. Atomistic Modelingof a KRT35/KRT85 Keratin Dimer: folding in aqueous solution and un-folding under tensile load. Physical Chemistry Chemical Physics, 17:21880–21884, June 2015.

[18] Dirk Zahn, and Patrick Duchstein. Multi-scale modelling of deforma-tion and fracture in a biomimetic apatite-protein composite: Molecular-scaleprocesses lead to resilience at the µm-scale. Accepted.

∗Publication is part of this thesis†Investigator for the motif recognition

vii

Contents

1 Introduction 1

1.1 Motif recognition in atomistic simulations . . . . . . . . . . . . . 2

1.2 Biomineralization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Apatite-collagen composites . . . . . . . . . . . . . . . . . 6

1.2.3 Otoconia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Preliminary studies . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Zinc oxide nanoparticles . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Electrostatic interactions . . . . . . . . . . . . . . . . . . . 12

1.3.2 Van der Waals attraction . . . . . . . . . . . . . . . . . . . 12

2 Motif-recognition in atomistic simulations 19

2.1 Motif Identification in Materials Simulations . . . . . . . . . . . . 21

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . 25

2.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Motif Reconstruction in Clusters and Layers . . . . . . . . . . . . 31

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.3 Simulation Details . . . . . . . . . . . . . . . . . . . . . . 36

2.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Atomistic Mechanisms of ZnO Aggregation from Ethanolic Solution 49

2.4 Nucleation mechanisms of a polymorphic molecular crystal . . . . 65

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4.2 Simulation Methods . . . . . . . . . . . . . . . . . . . . . 66

2.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

ix

Contents

2.5 Molecular simulation of DL-norleucine crystal nucleation from so-

lution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.5.2 Models and Methods . . . . . . . . . . . . . . . . . . . . . 83

2.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3 Biomineralization 95

3.1 Modeling of Apatite-Collagen Composites from Hyperspace Simu-

lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.1.3 Simulation details . . . . . . . . . . . . . . . . . . . . . . . 103

3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.2 Function of Saccharides in Calcium Carbonate-Protein Biocomposites113

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 114


3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4 Zinc Oxide Nanoparticles 127

4.1 Molecular Mechanisms of ZnO Nanoparticle Dispersion in Solution 129

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 130


4.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5 Summary and conclusion 141

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.3 Deutsche Zusammenfassung . . . . . . . . . . . . . . . . . . . . . 145

5.4 Schlusswort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

x

1 Introduction

The design of innovative materials poses an ongoing challenge to researchers

around the world. Nowadays, many disciplines are involved in this process: engi-

neering, physics, chemistry, biology, and computer science, amongst others.

Looking back to the beginning of materials research, in the early eighteenth

century, chemists started to classify materials according to their origin. They

introduced three different kingdoms: vegetables/plants, animals, and minerals.[1]

More criteria were applied to further subdivide these classes into a full taxonomy,

such as the mode of extraction and preparation, and what it could be applied for.

Apart from the manifold of forms and structures in the inanimate world, in living

nature, evolution has created a rich pool of specialized materials. These include

functional tissue, serving specific purposes: spider silk is strong but preserves a

high elasticity whilst being adhesive to insects; the cornea refracts light whilst

being able to bend to focus on objects at varying distances; horn features specific

shapes and surfaces, being stiff enough to endure fights in the animal kingdom;

bone and teeth are even harder than horn, but nevertheless not too brittle. Besides

a very high availability of the resources required for their synthesis, many of these

materials feature built-in repair mechanisms.

The appealing properties of biomaterials serve as inspiration for man-made

materials. The use of bio-inspired processes and systems for engineering purposes

is called biomimetics, or bionics. Naturally, the biological systems have to be

well-understood to serve as template for the industry.

The present work aims to establish a new level of understanding of solid materi-

als, their principles of self-organization, along with their intrinsic physicochemical

properties. Within chapter 1, a short introduction to the topics and systems under

investigation will be given, with references to works published in the respective

field by the author. Chapter 2 deals with a novel method of quantitatively and

qualitatively analyzing atomistic structures, and applications for rationalizing the

1

1 Introduction

evolution of forming crystal nuclei. In chapter 3, we shift to composite materials,

and provide insights into two different model systems of high complexity. Chap-

ter 4 describes the modelling of a functionalized ZnO nanoparticle, and methods

of analyzing its stabilization in solution. Finally, a summary of the results and

gained insights, along with an outlook on future perspectives, is given in chapter 5.

1.1 Motif recognition in atomistic simulations

The quickly increasing computing power enables researchers to access processes

and reactions taking place on increasing scales. Simulation times up to the ms

range, and system sizes of up to 109 atoms are now becoming possible. The in-

creasing simulation scale poses a challenge to the person carrying out the analysis,

requiring sophisticated, automated methods for analysis.

Using 5 years on a dedicated supercomputer, Matsumoto et al. [2] simulated the

nucleation of ice crystals from liquid water, by running a long-term brute force

simulation. More efficient simulation techniques, including those discussed in the

present work, yield significantly sped-up molecular simulations. With increasing

model complexity now accessible, characterization protocolls become more and

more important. The formation of crystal nuclei was investigated by monitor-

ing the lifetime of hydrogen bonds and coordination numbers of water molecules

within the simulation box. Whilst this simple methodology is suited for the sce-

nario it was applied to, there is a clear need for more generalized approaches for

analysis of complex systems.

Several criteria have to be satisfied by such an automated analysis method:

scalability, robustness, and wide applicability. Scalability denotes the applicability

to a range from small-scale to large-scale trajectories, related to either trajectory

length or number of particles. Robustness refers to the function a priori, i. e.,

without any previous knowledge or other parameterization, as well as the fault

tolerance of the underlying algorithm, whilst a wide applicability is important to

cope with all possible simulation scenarios—organics, inorganics, and composites

including both.

Common approches for the analysis of crystallization processes and phase tran-

sitions include

2

1.1 Motif recognition in atomistic simulations

• Common Neighbor Analysis (CNA) [3]

• Centrosymmetry Parameter Analysis (CSP) [4]

• Common Neighborhood Parameter (CNP) [5]

• Bond Angle Analysis (BAA) [6]

CNA is a method which assigns a prevalent crystal system (typically face centered

cubic (FCC), body centered cubic (BCC) or hexagonal close packed (HCP))

to the structure. Second-order atomic “neighborhoods” are evaluated for

each atom, counting the number of direct neighbors (i. e., atoms within a

predefined cutoff σ) for each atom. For each pair of atoms three parameters

are evaluated: whether the atoms are direct neighbors (i: true or false), the

number of mutual neighbors j the currently evaluated pair shares, as well as

the number of bonds k (neighborhood relationships) between these mutual

neighbors.

For each specific triple (i, j, k), the total number of neighborhood relation-

ships is counted, and the ratio of normalized abundance of numerically equal

triples gives a clear indication for the underlying crystal structure. Whilst

this method features a clear distinguishment between simple packing motifs,

it is not applicable for more complex systems such as molecular crystals, or

composites.

CSP quantifies the deviation of an atom from its ideal lattice position. It must

only be applied to centrosymmetric crystal structures such as fcc, bcc, and

hcp. Atoms positioned opposite each other with respect to a central atom

are identified, and their pairwise vectors to the central atom are evaluated.

For an fcc lattice, the centrosymmetry parameter is defined as follows:

P =6∑i=1

|Ri + Ri+6|2

where Ri and Ri+6 are the vectors from two oppositely placed atoms to the

central atom. It is obvious that these vectors sum up to the zero vector in

a perfect lattice, whereas the higher the CSP is for a given atom, the more

it deviates from the ideal lattice position, resulting from either a stacking

fault, or its position at the surface.

3

1 Introduction

CNP is a combination of both aforementioned methods, CNA and CSP. It is

defined in the following way:

Qi =1

ni

ni∑j=1

∣∣∣∣∣nij∑k=1

(Rik + Rjk)

∣∣∣∣∣2

where j iterates over the nearest neighbors of atom i, k iterates over the nij

common nearest neighbors between atoms i and j. Rik is the vector between

the atoms i and k.

BAA assigns a local crystal structure (bcc, fcc, or hcp) type to an atomic neigh-

borhood. Angular distributions between nearest neighbors with respect to

a central atom are evaluated, and binned with a bin-width ε. Counting the

occurrences of each angle in a given structure allows implications on the

underlying crystal lattice.

While these methods are applicable without any parameterization, a common

shortcoming is lack of universal applicability. CNA and CNP are well-suited for

analysis of the underlying crystal structure (fcc, bcc, or hcp), whilst CSP and CNP

highlight the deviation from a perfect crystallographic lattice. However, none of

these methods is suited for analyzing more sophisticated crystal structures, or

even molecular (organic) crystals.

Molecular crystals are of relevance in most pharmaceuticals, as over 90% are ad-

ministered in microcrystalline powder form, usually compressed to tablets. With

a higher number of molecules in the unit cell, which can be found in solvates, or

hydrates, the complexity of motif recognition increases drastically.

These shortcomings, together with an increasing complexity of simulation sce-

narios with respect to time and length scales, indicates a demand for new methods

of analysis. Ideally, such a method would be able to automatically find and ex-

tract arbitrary motifs within a simulation trajectory. In this work, we present

an approach which is able to overcome the above-mentioned shortcomings. A

detailed description of the method is given in section 2.1, along with proof-of-

concept studies highlighting NaCl motif formation during crystallization from the

melt, and an analysis of grains and grain boundaries in polycrystalline aluminum.

Moreover, we present several applications of our approach: the analysis of Cu

4

1.2 Biomineralization

cluster nucleation in the gas phase (section 2.2), ZnO aggregation from ethanolic

solution (section 2.3), benzamide aggregation (section 2.4), and the evolution of

dl-norleucine crystal nuclei from solution (section 2.5).

Our approach allows to gain a new level of understanding of mechanisms of

nucleation and self-organization on the atomistic scale. In the next section, we

present a larger and far more complex system than the aforementioned ones: the

nucleation of calcium salts on polypeptides. In this scenario, the interplay of inor-

ganic salts with an organic matrix forms an interface with increasing complexity.

Nucleation takes place on a longer timescale (miliseconds to seconds vs picosec-

onds), the area of the atomic interface between inorganics and organics is large,

and moreover exerts a dominating influence on crystal nucleation.


1.2.1 Background

Biominerals are hard, mineralized tissues which can be found in all kinds of or-

ganisms. Most prominent examples are mollusc shells, bone, and teeth. They

consist of a polymeric organic phase, such as collagen or keratin, supporting an

inorganic crystalline component, such as silica, iron oxides, metal sulfides, calcium

phosphate, or calcium carbonate. This combination of two or more constituent

materials, leading to properties different from those of an isolated component,

classifies them as composite materials. Many biominerals feature a hierarchical

structure. As a consequence, they exhibit different, and often specialized charac-

teristics, such as a specific shape and structure, leading to mechanic properties

outperforming their purely inorganic counterparts.

These characteristics, along with a high availability of their constituent mate-

rials open up pathways for a cost-efficient synthesis. Therefore, biominerals serve

as an important inspiration for modern materials. To make use of the constitut-

ing components, to mimic the structure, and to tune materials’ properties even

further, it is indispensable to gain an in-depth understanding of the composition,

structure, and organization of these biomaterials.

Minerals containing calcium carbonate form the most important class of solid

biomaterials. They are most abundant in marine lifeforms, such as sea urchins,

5

1 Introduction

mollusks, sponges, corals, and crustaceans.[7] Calcium carbonate exists in three

different polymorphs: aragonite, vaterite, and calcite. Calcite is the most stable

and abundant form, whereas vaterite is rather rare in biominerals. Besides these

crystalline phases, a hydrated amorphous phase exists, and is stabilized by crayfish

as a reservoir for calcium ions,[8] or used as a precursor for calcite or aragonite.[9].

Historical investigations on biomineralization

Although first investigations, especially on the formation of bone and cartilage,

were carried out as early as the 1920s, biomineralization as a field of modern re-

search was mainly established in the second half of the twentieth century. Evolv-

ing diffraction methods such as X-ray and neutron diffraction made it possible

to analyze the atomic structure at a very high level of detail. In 1920, De Jong

[10] discovered that bone and apatite feature the same X-ray diffraction pattern.

It took 40 more years until the crystallographic structure of hydroxyapatite was

resolved by means of X-ray diffraction.[11]

The close cohesion between an organic matrix and inorganic materials in bone

was first shown by transmission electron microscopy, leading to the insight that

organic fibrils are filled up with inorganic matter.[12, 13] Shortly after, similar

investigations were performed on dentin.[14]

Moreover, investigations on biomineralization compounds mainly containing

calcium carbonate as inorganic matter were carried out by Clarke and Wheeler

[15] and Bøggild [16], investigating the composition of marine sponges and mollusk

shells.

For a more comprehensive overview on biomineralization, such as the history,

underlying biological mechanisms, processes and properties, as well as biomimetic

approaches to innovatively modelling new materials, we refer to the printed lite-

rature.[17–20]

Within this thesis, two classes of biomaterials are being investigated further:

apatite-collagen composites, and calcium carbonate-collagen composites.

1.2.2 Apatite-collagen composites

Apatite-containing biominerals are abundant, among them are bone, cartilage,

dentin, and cementum. They all are composite materials, consisting of varying

6


amounts of collagen plus small amounts of non-fibrous proteins as the organic

phase, and calcium hydrogen phosphate as the inorganic phase. Enamel, the hard

and brittle covering tissue of human teeth, consists of amelogenins and enamelins

as the organic phase, as well as hydroxyapatite.

1.2.3 Otoconia

Otoconia are biominerals contained in the inner ear of vertebrates, in the saccule

and in the utricle. Their function is related to the sensation of accelleration,

allowing for the perception of movements in either horizontal (saccule) or vertical

(utricle) direction. They consist of calcium carbonate in different polymorphs:

aragonite or sometimes vaterite in fish, and calcite in mammals. Otoconia are

connected to the sensory system via hair cells, causing them to excert a shearing

force on the underlying cells when the organism accellerates. Nerves sense these

forces and transmit the information to the brain.

1.2.4 Preliminary studies

Experimental works

Kniep and co-workers have successfully been investigating the nature of biomin-

eralization for a long time. First studies were dealing with the morphology of

biogenous apatite-collagen composites.[21, 22] Double-diffusion experiments with

calcium and phosphate ions into a gelatine matrix demonstrated a high degree of

self-organization during composite growth, as shown in figure 1.1.[21] The mor-

phogenesis starts with elongated hexagonal-prismatic seeds with an aspect ratio

of 5:1 (length:diameter) and a length of up to 30µm, developing further towards

dumbbell-shaped aggregates, and develops into a notched sphere as final structure.

Further experiments of Kniep and co-workers with biogenic (carbonated) ap-

atite yielded additional particles with a completely different morphology.[24] These

pure calcite-gelatine composites exhibit different stages of growth compared to the

apatite-collagen ones, while still displaying a high degree of self-organization. Six

rhombohedral branches evolve towards a cone-like structure, with a belly in the

1Picture taken with permission from [23]. Copyright c© 2006 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim

7

1 Introduction

Figure 1.1: Superpositioned growth stages of a fluorapatite-gelatine nanocompos-ite. Aggregation starts as a hexagonal prism, continues to the shapeof a dumbbell, and finally becomes a notched sphere.1

center, and three planar faces situated on each end, rotated by 60 degrees toward

each other, as shown in figure 1.2. The particle morphology shows very close

resemblance to mammalian Otoconia.

Figure 1.2: Calcite-gelatine growth stages. Starting from six rhombohedralbranches, calcite gelatine crystals develop a cone-like structure withthree planar faces on each end.2


8


Simulations

Kawska et al. carried out simulations of calcium phosphate aggregation on a

collagen triple helix.[25, 26] Docking of single ions on the protein showed that

calcium penetrates the triple helical structure and stabilizes in the inner core

by means of salt bridges, substituting the collagenous network of intermolecular

hydrogen bonds, and thus stiffening the helix by creating stronger bonds in the

center. Conversely, phosphate ions laterally attach to the collagen triple helix

without penetrating its inner structure, causing the helix to slightly bend around

the attached ion.[25] See figure 1.3 for an illustration.

Figure 1.3: Single calcium- and phosphate-ions docked on a collagen triple helix.3

As an extension of the previous investigations, Kawska et al. simulated fluorap-

atite nucleation along a collagen triple helix, applying an iterative growth scheme:

the Kawska-Zahn simulation methodology.[27] Recurring motifs of three calcium

ions arranged in a triangle and coordinated by a fluor ion in the center could be

found in the center of the collagen triple helix. These triangular motifs were ar-

ranged perpendicularly to the c-axis of the apatite crystal structure. Even though

the aggregate was still largely amorphous, these results lead to the conclusion that

collagen supports the formation of the apatite crystal structure, and thus acts as

a nucleation seed.

The calcium carbonate particles from Kniep’s experiments, along with the simu-

lations from Kawska et al. provided inspiration to transfer the previous simulation

methodology to a new combination of materials, namely calcium carbonate and

collagen. Details of these simulations along with the results are described in sec-

tion 3.2.


9

1 Introduction

Furthermore, having learned about the interplay of calcium phosphate and col-

lagen from the previous simulations,[26] we set up a scale-up model mimicking a

bulk composite system suited for more sophisticated simulations. Details of this

system can be found in section 3.1. The resulting system has been used success-

fully to study shearing and subsequent relaxation processes of apatite-collagen

composites.[28]

1.3 Zinc oxide nanoparticles

The preceding section, along with the publications, demonstrated how crystals

nucleate, and how inorganic matter interplays with organic biomolecules. Fur-

thermore, we provided a method to analyze simulation results of crystal growth.

In this section, we are shifting from biominerals to man-made composites. This im-

plies that the molecular component is typically not a nucleating agent, but rather

represent surfactants that inhibit the growth of nanoparticles, and also provides

cross-links between them. By the prominent example of ZnO-surfactant systems,

we elaborate on a method to analyze functionalized nanoparticles in solution, thus

characterizing the solute-solvent interfaces.

Zinc oxide exhibits a couple of interesting physical, chemical and biological

properties and applications. In old times, the romans exploited it as a source

for zinc, for the production of brass—which is an alloy consisting of copper and

zinc. Nowadays, most of industrially produced zinc oxide is used by the rubber

industry. They apply zinc oxide as an addon to reduce vulcanisation time and

increase elasticity, since it catalyses the formation of sulfur cross-links between

the polymer chains. Furthermore, zinc oxide nanoparticles are commonly used in

sunscreen, where they absorb ultraviolet light.

Further, in the case of dentistry, zinc oxide is used. There, zinc oxide powder

is mixed with oil, acting as a plasticizer, and eugenol acting antiseptically, and

for the alleviation of pain. This composition yields a stable cement which can be

applied as a temporary dental filling material.

Nanoparticles can be functionalized to increase their spectrum of applications.

This implies covering the surface with surfactants, exhibiting different properties

than the uncoated particle. See figure 1.4 for a sketched representation. These sur-

10


face modifications enable particles to interact with their environment in a different

way, changing their solubility, self-diffusion, mechanical properties, and promoting

or hindering particle agglomeration.

Figure 1.4: Nanoparticle with surfactants

Dispersed functionalized nanoparticles and their intrinsic properties have been

studied extensively for the case of biological surfactants, in both experiment and

simulation. For example, DNA molecules have been used to direct nanoparticle

self-assembly into different crystallographic arrangements.[29–31] Complementary

single-stranded DNA molecules were attached to the particles, providing for mu-

tual self-recognition of the DNA-strands, and leading to superstructures in various

crystal lattices.

Simulations of functionalized gold nanoparticles coated with DNA molecules

have been carried out by Lee and Schatz [32]. Particles exhibiting a diameter of

1.8 nm were coated with four single stranded DNA molecules, running molecular

dynamics simulations for 8 ns. It was shown that DNA molecules attach perpen-

dicular to the particle surface. Whilst the radius of the particle including the

attached molecules was about 49 A, they tried to deduce the effective radius reff

of the particle solely from the radius of gyration of the attached DNA strands,

leading to a reduced radius of only 29 A. Investigating the sodium concentration

within a 3 nm radius around the particle, they found an increase of 20% compared

to the bulk concentration.

11

1 Introduction

Sknepnek et al. [33] modelled a nanoparticle functionalized with triblock-copolymers,

using a coarse-grained approach. After solvating these particles in a periodic

simulation box, they observed different types of ordered superstructures (square

columnar, hexagonal columnar, layered hexagonal, cubic, or gyroid), depending

on particle concentration and affinity.

1.3.1 Electrostatic interactions

Coating the surface with ‘sticky’ surfactants may hinder particles from coagu-

lating, or even coalescing, and thus change material properties. For example,

choosing a surfactant able to recognize particular crystallographic surfaces leads

to elongated growth along the uncoated axes, thus allowing growth control. An

even more effective way of completely preventing agglomeration is the functional-

ization of particles with charged surfactants. This leads to electrostatic shielding,

interplaying with sterical hinderance, and hence stabilizing a dispersed colloidal

suspension.

1.3.2 Van der Waals attraction

The potential energy between two separated particles is the sum of attraction and

repulsion:

U = Uattr + Urep

These interactions can be related to the distance r between the particles:

U(r) =A

rm− B

rn

where A/rm is the repulsive part, B/rn is the attractive part. Due to the fact that

attractive forces propagate over a larger area than repulsive forces, m > n is

commonly assumed. Setting m to 12 and n to 6 results in the Lennard-Jones type

of interaction potential.

These van der Waals interactions between atoms, molecules, or particles are

weak forces, in contrast to electrostatic and covalent interactions. Whilst the

repulsive part stems from the overlap of solvent shells or individual atoms (also

referred to as Pauli repulsion), the attractive part is caused by fluctuating dipole

12


moments, here scaled with the inverse sixth power of the distance between two par-

ticles. The dipole-dipole interactions can be subdivided in three types: permanent-

permanent (Keesom forces), permanent-induced (Debye forces), and induced-

induced (London forces). The third type, also referred to as London dispersion

force (LDF), is the one which in most cases exhibits the highest contribution to

the intermolecular interaction energy, compared to the other types.

London [34] suggested computing the interaction energy V between two identical

molecules as a function of the ionization potential I, the polarizability α, and the

intermolecular distance r:

V = −3

4

α2I

r6

This approach raises the question of an efficient way of computing the polar-

izability α. For small molecules, polarizabilities can be computed by means of

quantum mechanical methods. For larger molecules, as well as particles on the

nanometer scale, these computational efforts are infeasible. Therefore we propose

a method which is able to estimate polarizabilities from the fluctuations of the

dipole moment of the colloid. Whilst the Zinc oxide particle we investigated[35]

already exhibits an intrinsic permanent dipole moment, the surrounding halo of

counter ions accounts for these fluctuations.

As an extension to the LDF for colloidal systems, DLVO theory, named after

Derjaguin, Landau, Verwey and Overbeek, provides an estimation of the inter-

action energy of colloids by combining van der Waals attractive forces with the

repulsive double layer force. To calculate the former, the Hamaker constant of the

compound is required – which can in turn be computed by the specific van der

Waals interaction energy of the particle.

Section 4.1 gives a detailed description of the simulations we carried out on Zinc

oxide particles functionalized with poly(ethylene oxide)-block-poly(methacrylic

acid). We demonstrate a novel method suited for evaluating the effective charge

of the colloid in solution, as well as its hydrodynamic radius. Moreover, we outline

the assessment of net dipole moments and particle polarizability, without the need

for the application of an external electric field.

13

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17

1 Introduction

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18

2 Motif-recognition in atomistic

simulations

19

2.1 Motif Identification in Materials Simulations


Patrick Duchstein, Oliver Hochrein, and Dirk Zahn

This section has previously been published:Title Motif Identification in Materials Simulations

Author Patrick Duchstein, Oliver Hochrein, and Dirk ZahnJournal Zeitschrift fur allgemeine und anorganische Chemie

Publisher John Wiley and SonsVolume 635

Issue 4–5Pages 649–652Date April 2009DOI 10.1002/zaac.200900013

Copyright c© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Reprintedwith permission. Figures, tables, sections, and references have been renumbered.

Abstract We present an algorithm for the identification of structural motifs on

large scale atomistic simulation systems of solid materials. Given an arbitrary mo-

tif (model), the geometric hashing paradigm is used to find corresponding matches

in a large structure (target). To account for real materials, the algorithm tolerates

motif deformation and different orientations. Application to solid state science is

illustrated by the examples of crystal growth simulations and grain analysis in

polycrystalline structures.

21

2 Motif-recognition in atomistic simulations

2.1.1 Introduction

In the past decades, atomistic simulation studies evolved to a powerful tool for

the investigation of an increasing manifold of aspects of solid state chemistry

and materials science. While initially confined to unit cells or small supercells,

modern simulation protocols nowadays allow the molecular dynamics investigation

of systems comprising of millions of atoms. On this basis, we can now address

phenomena such as grain- and phase boundary analyses.

To tackle the complexity of large simulation systems, it is insufficient to rely

on the increasing performance of computers. Much more than computational

resources, human resources, i. e. smart algorithms for evaluating atomic interac-

tions [1], identifying structure candidates [2, 3], bridging the time-scales of forma-

tion/transformation processes [4] and for analyzing the results are needed. The

latter issue is closely connected to the visualization of simulations systems. While

configurations of hundreds to thousands of atoms may be illustrated by simple

ball-and-stick models, partially ordered systems of millions of atoms require more

sophisticated approaches to structural analysis and visualization.

Herein, we present a new method for the topological analysis of atomistic con-

figurations and trajectories from molecular dynamics simulations, which aims at

an automated identification of motifs. For the characterization of unit cells, the

classification of specific polyhedra proved a versatile concept for understanding

crystal structures. To identify such motifs, common software relies on testing all

possibilities by performing nested loops over all atoms.

2.1.2 Theory

Geometric hashing [5] was originally invented for image recognition tasks, pro-

viding a fast way to match a set of motifs onto a target picture. Unlike 2-

dimensional images, molecular dynamics simulations provide trajectories, i. e.3-

dimensional data sets of atomic coordinates as a function of time. The extension

to 3-dimensional molecular recognition tasks, was first suggested by Nussinov and

Wolfson [6], and later applied to protein backbones [7].

A further feature relevant to motif recognition processes in solid state simula-

tion systems is the consideration of periodic boundary conditions, which play an

integral role in most computational setups. We account for periodic boundary

22


conditions by duplicating the outmost atoms of the simulation box next to the

faces within a distance of half of the maximum distance of two atoms of the model

Algorithmic Procedure

For object recognition to work, the motif to be found, the model, has to be con-

verted into a representation which can be quickly compared with substructures

of the target object. Several different approaches for the model representation

have been presented in the past [6]. For our purpose, the representation has to

be rotationally and translationally invariant. Therefore, we chose a representa-

tion similar to the one suggested by Bachar et al. [7]. The conversion into our

representation is the first step of the algorithm, the preprocessing step.

Step 1: Preprocessing For each pair (ai, aj) of atoms of the model within a

distance dij < r1 called a reference pair, we record the distances from ai and aj

to all other atoms ak within the distances dik, djk < r2 (with r2 < r1). The

atomic indices i, j, k are stored into a hash table, using the distances triplet

D = (dij, dik, djk) along with the atomic species as hash keys. The introduc-

tion of a distance tolerance τ within the hash function is important to account

for noise, i. e.slight displacements of the atoms in the target. (Technically, this is

accomplished by considering not only the buckets which correspond to the hash

key under investigation, but also buckets which correspond to adjacent hash keys).

Step 2: Initial Matching In this step, atomic triplets of the target are matched

against the model hash. Each atom triplet of the target, taking into account

the radii r1 and r2 as defined above, is matched against the model hash. For

each matching atom triplet of the model that corresponds to the target triplet, we

compute the geometric transformation (rotation and translation) that is necessary

to project the model triplet onto the target triplet. From this, displacement vectors

and rotation matrices for all geometric transformations of atomic triplets, along

with correspondences of atoms of the model within the target, are obtained. These

geometric transformations are inserted into two three-dimensional arrays with a

predefined number of bins. The bin width ideally corresponds to τ .

23


Step 3: Clustering the Transformations All geometric transformations that

were grouped in one bin are compared to each other, and, moreover, with the

transformations in the neighboring bins. All transformations that are similar to

each other, i. e. the application to all model atoms always projects them onto their

corresponding target atoms within a specific tolerance τ , are clustered together.

Clusters exhibiting a large number of equivalent transformations define a match

of the model structure within the target structure.

Step 4: Cleaning up Matches A major problem arising in step 3 are doublets:

the same model atom might correspond to several different target atoms, and

vice versa. Therefore it is vital to count the number of correspondences of each

model/target atom pair within the transformation cluster. The highest number

of correspondences sets the final correspondence for a match. In many cases, a

successfully identified match overlaps with a neighboring motif. This fact has

to be considered carefully, to allow finding a consistent set of matches in a well-

defined target structure. Undervaluing the overlap count of a motif in a designated

crystal structure leads to gaps and other artifacts in the set of matches, whilst

overvaluing may produce inconsistencies. An extreme case of overlapping is a

complete copy of the match, along with different transformations. In this case,

it can be assumed that the motif exhibits rotational symmetry, and one of the

matches can be deleted. The number of identical atoms in two overlapping motifs

may be adjusted from zero to all minus one.

Distortion Tolerance and Screening Radii

A crucial feature which is required for matching motifs of solid crystals is the

availability of an adjustable tolerance value, which signifies how close the motif in

the simulated structure is related to the motif in the “perfect” crystal structure.

Large deviations in the atomic positions may easily lead to falsely identified motifs

if the tolerance value is set too high. If, on the other hand, the tolerance value is

too low, a correct motif with only slight deviations in the atomic positions might

not be found. Since we deal with a priori unknown structures and trajectories, no

general assumption of the error distribution can be done.

24


We define τ as the maximal difference that the distance of a pair of atoms in the

target structure may deviate from the corresponding distance in the model. During

step 2, the target distances (dtij, dtik, d

tjk) are hence matched to distance triplets in

the model structure +/- the tolerance, i.e. (dmij ± τ, dmik ± τ, dmjk ± τ). Moreover, in

step 3 the same criterion is used to decide whether two transformations are similar

or not. The optimal values for the tolerance and the screening radii depend on

the system under investigation. For both simulation systems described in the

following, 0.5 A, 5 Aand 20 A, were found as suitable parameters for τ , r1 and r2,

respectively.

Few other noise models have been proclaimed so far, including gaussian noise

distribution as described by Wolfson and Rigoutsos [8]. For the sake of transfer-

ability, here we choose the simple tolerance criterion described above. The more

intuitive root mean square deviation must be computed for complete matches, i.e.

as a postprocessing step. The same applies to the minimum- and maximum single

atomic displacements of each motif match.

2.1.3 Results and Discussion

To demonstrate the general use of our motif identification algorithm, a variety of

different simulation scenarios was explored. The test cases discussed in the fol-

lowing include the analysis of a polycrystalline structure and the time-dependent

study of crystal self-organization both from the melt and from solution.

Polycrystalline Structures

The analysis of grains plays an important role in computational materials science.

As a text case we used an artificially prepared configuration reflecting an alu-

minium polycrystal of particularly fine domain fragmentation. This structure was

prepared by a molecular dynamics run of the supercooled melt at high pressure

and is not intended to mimick a real structure. Instead, artificially small grain

sizes are taken as a critical performance test of the motif identification algorithm

for discriminating ordered domains.

Apart from identifying the octahedral motifs of the fcc structure, our algorithm

also allows to illustrate different orientations. This feature is particularly useful

to illustrate different grains (figure 2.1, left). Here, different colorization has been

25


used to discriminate different angles of the motif with respect to the axes of the

crystal in the final configuration. Rotational symmetry and mirror symmetry were

considered to color the motifs consistently.

Figure 2.1: Simulation system of 17500 Al atoms in a polycrystalline structure.Left: octahedral motifs colorized according to their orientation to thecoordinate system of the simulation box. Right: Highlighting of atomswhich are shared by several motifs of different orientation. Dependingon the chosen error tolerance, point defects and tilt grain boundariesare identified, whilst stacking faults (green circles) remained elusive.

Moreover, grain boundaries can be identified by first searching for ordered do-

mains, and afterwards highlighting all atoms which are shared by several grains.

This requires counting the number of motifs each atom takes part in. Deviations

from the value in the single crystal indicate point defects, dislocation, stacking

faults and domain boundaries. This analysis is sensitive to the choice of the error

tolerance τ . Figure 2.1, right, illustrates the assessment of differently oriented

grains and point defects, whereas stacking faults are invisible in this representa-

tion (but may be identified from the visualization of the motifs as in Figure 2.1,

left).

26


Time-Dependent Studies of Motif Formation during Crystal Growth

Based on a recent molecular dynamics simulation study of NaCl crystallization

from the melt [9] we furthermore applied the motif identification algorithm to

demonstrate time-dependent structure analysis. Figure 2.2 illustrates the forma-

tion of rocksalt motifs of different orientation during the nucleation of a NaCl

single crystal. This is particularly beneficial for the stages of crystal formation

which involves pre-critical aggregates of different orientation. The latter com-

pete until a sufficiently large set of identically oriented motifs form a post-critical

aggregate.

Apart from pure visualization, more complicated structures may also require

the quantitative analysis of the number of motifs as a function of aggregate size.

An example for such studies is given by our recent work on ZnO aggregation from

solution, in which the rate of chair- and boat-type motif formation was analyzed

separately [10, 11]. From this, the evolution towards a 3:1 ratio while building up

the wurtzite structure was explored.

2.1.4 Conclusions

We described an algorithm for the identification of structural motifs in large simu-

lation systems. The approach was demonstrated for two rather different simulation

models covering crucial aspects of solid state and materials research. To provide

general applicability, our algorithm comprises of only a very limited number of

parameters, which for most applications can easily be estimated on the basis of

the radial distribution functions.

The model motif, which is to be recognized in a large structure, may in principle

be chosen freely. Ho wever, for most applications the search for small building

blocks seems to be most adequate. Larger structures which may be devised in

several small motifs may be identified by an analysis of orientation correlation of

adjacent motifs. Our motif identification method is available free of charge; in

case of interest please contact [email protected].

27


Figure 2.2: Formation and orientation of NaCl cubes in a simulation system of270 ion pairs. During the early stage of crystal aggregation from themelt, motifs of mismatching orientation compete until a larger set ofcorrelated motifs constitutes a post-critical nucleus (Na: yellow; Cl:purple).

28

References

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29


proteins. Protein engineering, 6(3):279–288, April 1993.

[8] H. J. Wolfson and I. Rigoutsos. Geometric hashing: an overview. IEEE

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[9] Dirk Zahn. Atomistic Mechanisms of Phase Separation and Formation

of Solid Solutions: Model Studies of NaCl, NaCl–NaF, and Na(Cl1-xBrx)

Crystallization from the Melt. J. Phys. Chem. B, 111(19):5249–5253, April

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(2):024513+, January 2006.

30

2.2 Motif Reconstruction in Clusters and Layers

2.2 Motif Reconstruction in Clusters and Layers:Benchmarks for the Kawska-Zahn Approach toModel Crystal Formation

Theodor Milek, Patrick Duchstein, Gotthard Seifert,and Dirk Zahn

This section has previously been published:Title Motif Reconstruction in Clusters and Layers: Benchmarks for

the Kawska-Zahn Approach to Model Crystal FormationAuthor Theodor Milek, Patrick Duchstein, Gotthard Seifert, and Dirk

ZahnJournal ChemPhysChem

Publisher John Wiley and SonsVolume 11

Issue 4Pages 847–852Date Mar 15, 2010DOI 10.1002/cphc.200900907

Copyright c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Reprintedwith permission. Figures, tables, sections, and references have been renumbered.

Abstract A recently developed atomistic simulation scheme for investigating

ion aggregation from solution is transferred to the morphogenesis of metal clus-

ters grown from the vapor and layers deposited on a substrate surface. Both

systems are chosen as benchmark models for intense motif reorganization during

aggregate/layer growth. The applied simulation method does not necessarily in-

volve global energy minimization after each growth event, but instead describes

crystal growth as a series of structurally related configurations which may also in-

clude local energy minima. Apart from the particularly favorable high-symmetry

configurations known from experiments and global energy minimization, we also

demonstrate the investigation of transient structures. In the spirit of Ostwald’s

step rule, a continuous evolution of the aggregate/layer structure during crystal

growth is observed.

31


2.2.1 Introduction

Crystal nucleation poses an ongoing challenge to both experiment and theory.

While experimental studies offer a wide variety of insights at the macro- and

mesoscopic scale, investigations of the initial stage of crystal formation, that is,

the atomistic level of detail, often remain elusive.[1] Here molecular dynamics

simulation approaches come into play as they offer the resolution in both space

and time desired for detailed mechanistic analyses. Ideally this should encompass

ion/atom association, aggregate formation, the nucleation of structural motifs

and postcritical crystal growth. However, the computational effort is typically

immense and often special techniques are needed to make simulations possible.

To reduce the complexity of crystal nucleation processes to a minimum, the first

simulation studies of aggregate growth were focused on cluster formation from the

vapor.[2] By now, metal clusters of up several tens of particles have become test

cases for global energy minimization strategies. About ten years ago, Frenkel and

coworkers pioneered the study of crystallization from the melt by means of free

energy approaches.[3–5] Therein the system is driven along a predefined reaction

coordinate to enhance the kinetics of the processes. Further, reaction-coordinate-

free methods like the transition path sampling molecular dynamics scheme were

applied to crystal nucleation from the melt.[6, 7] While both approaches proved

successful for nucleation in pure or only marginally diluted systems,[7] crystal

aggregation from solutions of low solute concentrations imply additional challenges

to computer simulation.[8] Apart from the energy barriers related to association

of already nearby particles, crystal nucleation from solution also implies solute

diffusion to the forming aggregate. To cope with nucleation processes in dilute

solutions, we recently developed a simulation approach that mimics such diffusion

processes in an approximate manner.[9] While the uptake of solutes is modeled by

a simple docking-type approach, incorporation into the aggregate is explored by

detailed atomistic simulation.

In particular during the very early stage of aggregate formation, considerable

structural changes are observed. We recently developed a molecular dynamics

simulation approach that allows the investigation of motif formation, ripening re-

actions and motif reorganization, the nucleation of ordered domains and growth

of nanocrystals of up to hundreds of ions.[9–11] Basically, such size-induced evo-

32


lution of structures was anticipated in terms of the Ostwald’s step rule already

one century ago. Yet we are still at the beginning of a detailed atomistic under-

standing.

As kinetic hindering may prevent full relaxation to the global energy minimum,

realistic approaches to aggregate growth must also include local energy minima.

Control of this diversity is far from trivial and in principle requires a full account

of the conditions of the corresponding nucleation experiment – which would in-

crease the computational demand by many orders of magnitude. In what follows,

we transfer our simulation approach to the formation of copper clusters from the

vapor and to the growth of Au layers on Cu substrates. Both systems exhibit

particularly drastic size-induced structural transitions and have been intensively

studied from both experiments and global energy minimization searches. These

systems hence reflect reliable benchmarks to crystal growth simulations based on

series of structurally related (not necessarily global) minimum energy configura-

tions and help to shed light into the interplay of particularly stable structures of

high-symmetry and the related transient configurations.

2.2.2 Theory

While the most realistic account of crystal formation processes would call for

straight-forward molecular dynamics simulations, for most systems the required

computational resources are out of reach by many orders of magnitudes. To tackle

the time-length scale we recently introduced a simulation protocol that is divided

in several steps each reducing different aspects of system complexity. Within an

iterative procedure, aggregate growth is explored particle-by-particle, typically

starting from the very beginning, that is, the association of a pair of atoms or

ions.

Each iteration corresponds to the (putative) uptake of a new particle which

diffusion to the forming aggregate is treated in an implicit manner to avoid large-

scale atomistic simulations of atmospheres or ionic solutions. The migration of

particles to the aggregate or a substrate layer is mimicked by placing atoms/ions

near the forming crystal and the analysis of putative adsorption sites from simple

steepest descend energy minimization. In this step, molecules of the solvent/gas

atmosphere are omitted and the aggregate is kept fixed to avoid artificial defor-

33


mation from the impact of the incoming particle. Depending on the symmetry

of the system under investigation, the incoming atom/ion is placed on a random

position on a sphere, cylinder or layer to mimic diffusion to an aggregate, growth

controlling molecule or a substrate layer, respectively (figure 2.3).

Figure 2.3: Upper panel: docking procedure for modeling particle diffusion to theforming aggregate (left), a growth controlling molecule (middle) and toa substrate layer (right). The association of new ions/atoms is followedby detailed relaxation (lower panel) of the whole system, includingsubstrate and solvent (if required) degrees of freedom. Starting fromthe association of a pair, crystal growth is modeled particle-by-particleby repeating the docking/relaxation procedures iteratively.

After the uptake of a new particle, the aggregate is subjected to relaxation. In

this step all degrees of freedom are considered, including the atoms of substrates

(figure 2.3, right) or, if applicable, solvent molecules and further solutes (refer to

refs. [9–12] for detailed discussions on solvent reintroduction and consideration

of ripening reactions). The relaxation procedure is inspired from simulated an-

nealing but does not impose rigorous confirmation of finding the global energy

34


minimum. Instead we perform a finite temperature molecular dynamics run, pick

the configuration of minimal energy from this trajectory and quench it to a local

energy minimum.

The duration of these runs and the underlying temperature represent (linked)

parameters which are not trivial to assess. Clearly, sufficiently long relaxation runs

would allow the system to adopt the global energy minimum configuration. Dating

back to Ostwald’s seminal work 100 years ago, many crystal growth processes are

however known to proceed via intermediate structures which are local energy

minima. To rationalize this phenomenon it is helpful to imagine a cluster of

n particles to which a new atom is associated. In case the aggregate of n + 1

particles exhibits local energy minima that are similar in energy yet different in

structure from the global minimum, it is likely that aggregate growth occurs via

the structurally most related configurations. The latter can be reached via no or

only small energy barriers, whilst large structural rearrangements are hindered by

high barriers. For each growth step, this implies memory effects to the previous

aggregate arrangement or to a given substrate layer.

In principle, a full account of the interplay of structurally related local minima

and the global energy minimum configuration requires relaxation runs directly

corresponding to the respective nucleation experiment. As in many cases this is

computationally not feasible we use an approximate approach based on elevated

temperature and reasonably short relaxation runs. To control structural relax-

ation during aggregate growth, here we suggest the analysis of potential energy

per particle as a function of aggregate size. For aggregate formation from the va-

por, it is reasonable to expect the energy to essentially smoothly decrease towards

convergence to the energy of formation of the bulk crystal. As exceptions to this

rule, polyhedra of high symmetry are well known to represent particularly stable

structures. From both, experiments dedicated to cluster formation from the vapor

and from rigorous global energy minimization studies, a series of “magic numbers”

have been identified as the size of particularly stable aggregates.2 Hence, in ad-

dition to providing a continuous series of aggregate structures, realistic growth

simulations must also account for particularly stable configurations of high sym-

metry.

35


2.2.3 Simulation Details

The atomic interactions were modeled by the embedded atom approach of Sut-

ton and Chen[13] which consists of a repulsive pair potential (first term) and an

attractive multi-body potential which reflects the atomic coordination through

spherical averaging [equation (2.1)]:

Epot = ε

[∑i<j

(a

rij

)n− c

∑i

√ρi

](2.1)

The local density ρ for atom i is taken as equation (2.2):

ρi =∑j 6=i

(a

rij

)m(2.2)

The parameters for gold and copper are denoted in table 2.1, including the

interpolated Au–Cu parameters which were adopted from ref. [14].

ε [eV] a A c m N

Au–Au 1.2794 4.079 34.428 6 9Au–Cu 1.2586 3.848 36.996 7 9.5Cu–Cu 1.2382 3.631 39.755 8 10

Table 2.1: Parameters of the empirical potential energy terms used.

For both model systems a time step of 10 fs was found to be appropriate for

the molecular dynamics simulations. The clusters were investigated by constant-

temperature simulations in the vapor phase. The Cu substrate layers were mim-

icked by 16× 16× 3 unit cells to which periodic boundary conditions are applied

within the xy-plane.

2.2.4 Results

Copper Clusters

Starting from the association of a pair of copper atoms, we performed 25 inde-

pendent series of particle-by-particle growth runs. As the computational demand

increases with the total number of atoms to the power of two we decided to

36


stop aggregate growth after reaching 200 atoms. This proved sufficient to explore

the interplay of competing motifs originating from Mackay icosahedra,[15] Marks

decahedra[16] and the material’s tendency to arrange on an fcc lattice.

Several temperature values and time intervals were tested for the relaxation

runs in-between the aggregate growth steps. At 400 K and relaxation periods of

25 ns the aggregate structure was found to evolve via the particularly favorable

high-symmetry configurations,[2] whilst exhibiting structurally similar transient

states.1 This feature may be monitored by plotting the potential energy per

particle as a function of time (figure 2.4).

Figure 2.4: Potential energy of the aggregate taken per atom and as a functionof the aggregate size. The curve reflects averages over all growth runsfor which the energy values were calculated after relaxation of thecorresponding aggregate structure. The experimental values for thebulk and the surface energy terms were taken from refs. [17, 18].

1While these characteristics remain unchanged for longer relaxation runs (or by increasingthe temperature to 500 K), the choice of too short relaxation intervals give rise to artificialhigh-energy configurations. During further growth such structures change into more favorablearrangements in an uncontrolled manner, that is, by producing steep cascades in the energyprofile. As the latter differ for independent growth runs, such characteristics represent a clearsignature of insufficient relaxation. All structures discussed in the following were obtained from25 ns relaxation runs which ensured convergence of the energy profiles.

37


On the basis of classical nucleation theory (CNT), the aggregate energy may

be associated to a bulk and a surface energy term. In most applications of CNT,

the nuclei are taken to be spherical and the two energy terms are written as a

function of the radius. As the clusters obtained in our growth runs change from

spherical to cylindric and cubic shapes, we decided to use a shape-independent

formulation of CNT. For this purpose, the surface and the bulk energy terms are

considered as a function of the number of particles in the aggregate N . For all

of the observed shapes, N scales linearly with the volume whilst the surface area

scales with N 2/3. This allows a shape independent fit of the energy per particle

based on equation (2.3):

E(N)

N=csurface ·N 2/3 − µbulk ·N

N(2.3)

The corresponding fit was performed for all growth runs and the energy profile

based on the average surface and bulk terms is shown in figure 2.4. To compare our

results with experimental studies based on conventional CNT relying on spherical

aggregate shape, we use equation (2.4):

N = ρ · 4

3πr3 and 4πr2 · γ = csurface ·N 2/3 (2.4)

to rewrite equation (2.1) as equation (2.5):

E(N)

N=

3√

36π · ρ−2/3 · γ ·N−1/3 − µbulk (2.5)

with ρ and γ being the particle density and the surface energy density, respec-

tively. From our simulations both, surface energy density and the bulk energy were

found in reasonable agreement with the experiment.[17, 18] Clearly, the usage of

(semi-)empirical force-fields poses general limitations to accuracy. To reduce arti-

facts to a minimum, small clusters of fewer than 10 atoms are not considered for

the CNT fit. For such small clusters, the potential energy is particularly sensitive

to the electronic shell structure and ab initio approaches are more appropriate.

It is noteworthy that only the configurations which are identical or very close to

the symmetric icosaeder structures exhibit significant deviations from the simple

CNT fit of E(N). The kinks in the energy profile reflect particularly stable Marks

decahedra (Cu75) and Mackay icosahedra of which all of the well-known “magic

38


numbers” of 13, 55, 147, (309,. . . ) particles may be clearly observed from our

aggregate growth simulations reaching up to 200 atoms.[15] Despite the appar-

ently simple evolution of the energy profile during aggregate growth, the clusters

experience rather drastic changes. Both the Mackay icosahedra and the Marks

decahedra are characterized by five-ring motifs in the aggregate center. This is in

controversy to the fcc-lattice of the bulk material and considerable rearrangements

of the whole cluster must occur during aggregate growth.

The morphogenesis may be quantitatively demonstrated by plotting the ratio of

icosahedral, decahedral and cuboctahedral (fcc structure) motifs as a function of

size (figure 2.5) using a recently developed method for the automatic identification

of motifs.[19] Small clusters of 55 or less copper atoms are dominated by five-

ring motifs originating to the Cu13 Mackay icosahedra (figure 2.5 and figure 2.6).

At later stages of aggregate growth a reorganization of the five-ring motifs is

observed leading to the symmetric Cu75 decahedron. While the evolution from

Cu55 to Cu75 reflects a rotation of the five-ring motifs, in some simulation runs

a complete decay of the five-fold symmetry is observed during the formation of

a Cu118 cuboctrahedron (figure 2.6b). This may be considered as a signature of

the increasing importance of fcc-type motifs with increasing aggregate size. Apart

from the dominance of icosahedra near the “magic numbers” of 13, 55 and 147

particles, the observation of (temporary) decahedral and cuboctahedral motifs

agrees nicely with the findings of Reinhard et al. whose experiments indicate that

such structures may exist even in clusters of only 1 nm diameter.[20]

Please note that the related energy values are in line with the profile of the

CNT fit and hence reflect configurations of “regular” stability. During further

aggregate growth the morphogenesis is therefore redirected in favor of five-ring

motifs to reach the next particularly stable Mackay icosahedra (here Cu147). In

other terms, the clusters tendency to arrange as fcc-type structures is poisoned by

the particularly stable Mackay icosahedra (and the Marks decahedra), and thus

fails to fully dominate nuclei of up to 200 copper atoms. This is in full agreement

with the experimentally established knowledge of magic numbers up to aggregates

counting a few thousands of atoms.[2]

39


Figure 2.5: Average occurrence of icosahedral (red curve), decahedral (green) andcuboctahedral (blue) type structures.

Gold Deposition on Copper Layers

To explore the suitability of our simulation approach for the investigation of epi-

taxy we performed two series of Au aggregate growth runs in the presence of Cu

substrate layers. Two types of substrate layers were investigated. In full agree-

ment to the experimental evidence, our simulations demonstrated the association

of identically oriented (111) gold layers to (111) copper substrates. However,

a more peculiar morphogenesis is observed for gold deposition on (100) copper

layers. Diffraction experiments indicate the formation of identically oriented Au

layers only for the first monolayer, whilst further gold deposition induces the for-

mation of (111) multilayers.[21–25]

The latter phenomenon may be attributed to the competition of the stress

induced by atoms placed on a single lattice and the energy differences related

to unfavorable atomic coordination at dislocations and interfaces. Indeed our

growth simulations exhibit frequent rearrangements of (100)- and (111)-adlayer-

type motifs which begin after the association of just a few Au atoms and terminate

with the completion of a (111) mono-layer. Exemplary snapshots are shown in

figure 2.7. While small “islands” of Au atoms tend to follow the [100] orientation

40


Figure 2.6: a) Evolution of the Cu55 Mackay icosahedron to a Cu75 Marks dec-ahedron. b) Evolution of the Cu75 Marks decahedron to the Cu147

Mackay icosahedron. Within the range of 119 to 136 particles, theclusters adopt fcc type structures.

41


of the Cu substrate layer (figure 2.7a), after the agglomeration of more than 7 Au

atoms we observed the shifting of columns along the [011] or the [011] direction

of the substrate layer. Depending on the island size, this leads to a dynamic

interplay of (100)- and (111)-adlayer-type motifs. With increasing deposition of

gold atoms the adlayer gradually evolves in favor of the (111)-adlayer-type motifs

in which traces of (100)-adlayer-type motifs remain in terms of stacking faults.

While our simulations nicely agree with the observed rearrangement of (100)-

and (111)-adlayer-type motifs, it should be stated that experimental evidences

also indicate the formation of a Cu3Au monolayer motifs as an intermediate

structure.[21–25] This alloying is known to occur via the exchange of Cu and

Au atoms which is almost completely reversed in the course of further gold as-

sociation.[21, 23] In contrast to these findings, our simulations only describe the

reorganization of a (100) Au monolayer during its growth to several multilayers,

that is, the formation of (111) multilayers. While a few Cu–Au exchange pro-

cesses were observed during the relaxation runs as temporary phenomena, the

formation of a Cu3Au monolayer surely requires more accurate modeling of the

Au–Cu interactions.

So far, the available empirical Au–Cu models do not provide a proper account

of the alloying potentials[14, 26] whereas quantum calculations indeed confirm

the experimentally observed exchange of Au and Cu atoms.[27] Unfortunately,

the combination of accurate DFT calculations and our molecular dynamics sim-

ulations of many nanoseconds implies rather unfeasible computational demands.

Nevertheless, within the limited accuracy of the empirical force fields used, we con-

sider the observed evolution of gold adlayer motifs to be in reasonable agreement

with the experiment. This particularly holds for the key focus of the manuscript,

that is, the competition of structural motifs and its interplay with surface and in-

terface energies. For this, we suggest that statistical quality, that is, the reasonable

sampling of structures, is much more important than highly accurate procedures

for the evaluation of explicit interactions.

42


Figure 2.7: a) Snapshot of an early stage of Au deposition on a (100) Cu substratelayer. Here the small island of gold atoms fully adopts the structuralarrangement of the substrate. b) Snapshot of a later stage of Audeposition on a (100) Cu substrate layer. Stress-induced shifting ofAu columns along the [011] or the [011] direction of the substratelattice caused the reorganization of the gold atoms in favor of a (111)adlayer.

43

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[3] Stefan Auer and Daan Frenkel. Prediction of absolute crystal-

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[4] Pieter Rein ten Wolde and Daan Frenkel. Homogeneous nucleation

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47

2.3 Atomistic Mechanisms of ZnO Aggregation from Ethanolic Solution

2.3 Atomistic Mechanisms of ZnO Aggregationfrom Ethanolic Solution: Ion Association,Proton Transfer, and Self-Organization

Agnieszka Kawska, Patrick Duchstein, Oliver Hochrein,and Dirk Zahn

This section has previously been published:Title Atomistic Mechanisms of ZnO Aggregation from Ethanolic

Solution: Ion Association, Proton Transfer, andSelf-Organization

Author Agnieszka Kawska, Patrick Duchstein, Oliver Hochrein, and DirkZahn

Journal Nano LettersPublisher American Chemical Society

Volume 8Issue 8

Pages 2336–2340Date June 24, 2008DOI 10.1021/nl801169x

Copyright c© 2008 American Chemical Society. Reprinted (adapted) with permis-sion from [1]. Figures, tables, sections, and references have been renumbered.

Abstract We report on atomistic simulations related to the nucleation of zinc

oxide nanocrystals from ethanolic solution. The underlying mechanisms are ex-

plored from the very initial stage of Zn2+ and OH− ion association to the formation

of nanometer-sized aggregates counting up to 250 ions. The embryonic aggregates

consist of zinc and hydroxide ions, only. At later stages of aggregate growth, pro-

ton transfer reactions at the aggregate–solvent interface account for the formation

of O2−2 ions and induce the precipitation of zinc oxide. After the association of

around 150 ions, ZnO domains were found to nucleate in the central region of

the [Znx(OH)yOz]2x−y−2z aggregates. In the course of further ion association and

condensation reactions, progressive self-organization leads to an extended core in

which the ions are arranged according to the wurtzite structure.

49


Apart from its wide use for UV protection, nanocrystals of zinc oxide exhibit

a large variety of interesting potential applications including room temperature

UV lasers,[2] LEDs,[3] solar cells,[4] and sensors.[5] Moreover, they also represent

a protype model for the exploration of oxide nanocrystal precipitation from so-

lution. Most of the related sol-gel processes are based on alcoholic solutions of

Zn2+ and OH− ions plus counterions originating to the reactant materials).[6, 7]

Nevertheless, stable Zn(OH)2 crystals are typically not observed.[8] Instead, the

most common precipitation reactions involve proton transfer leading to the for-

mation of oxide ions and the crystallization of ZnO. The key to the understanding

of ionic self-organization from solution and its interplay with ion-solvent interac-

tions originates to the atomistic scale. For such investigations,in particular when

related to the very early stage of ion association and crystal nucleation from so-

lution, advanced molecular simulation techniques proved to be a promising tool

and considerably extended our mechanistic understanding.[9–11]

One of these approaches is represented by our recently developed simulation

scheme for aggregate growth from solution[11] which is divided in subsequent

steps as illustrated in figure 2.8. In each iteration, single ions are added to an ag-

gregate which is then subjected to relaxation processes, including proton transfer

reactions. Choosing a single ion as the initial aggregate allows the investigation

of ion association and self-organization from its very beginning.

For each aggregate growth attempt, the first step (i) is given by a docking-

like approach for searching putative sites for ion association. The ionic species

is chosen randomly as Zn2+ or OH− ions (imposing a 1:2 ratio). To mimic the

effect of ion migration from solution, the incoming ion is placed on the surface

of a sphere which envelops the aggregate. In keeping the ionic positions of the

central aggregate fixed, the solvent molecules are removed, and a putative site for

ion association is identified from steepest descend minimization of the interactions

of the aggregate and the incoming ion. In the next steps (ii and iii), the solvent

is reintroduced. Subsequent relaxation by simulated annealing is first applied to

the solvent molecules only (ii) and then to the whole system (iii).[11]

Steps i–iii are modeled by means of empirical force-fields and do not require

quantum calculations. However, as an extension to the original aggregate growth

scheme[11] the resulting configuration of step iii is scanned for hydrogen bonds

which might induce proton transfer reactions. In the new iteration step (iv) the

50


Figure 2.8: Simulation Scheme for Aggregate Growth

51


H· · ·O distances of the hydrogen bonds interconnecting the OH− ions are calcu-

lated, and the shortest hydrogen bond is taken as a candidate for proton transfer.

The putative reaction A–OH· · ·OH → A–O2 + H2O (A = aggregate) is then ex-

plored from combined quantum/classical molecular mechanics calculations. For

the sake of computational feasibility (the possibility of proton transfer is explored

after every ion association event), we only compute the energy levels of the re-

actant and the product state. This covers the essential aspect of the putative

reaction, that is, whether proton transfer is exothermic or not. Both possible

states are relaxed from simulated annealing before the energetically more favor-

able constellation is taken as the starting point for the next aggregate growth

event.

∆EH+-transfer =Eqmvacuum(O2− + H2O)− Eqm

vaccuum(OH− + OH−)

− {Eclvacuum(O2− + H2O)− Ecl

vacuum(OH− + OH−)}

+ Eclsolvent([Znx(OH)y−2Oz+1]2x−y−2z + H2O)

− Eclsolvent([Znx(OHyOz]

2x−y−2z

(2.6)

Hamiltonian of the Quantum/Classical (qm/cl) Calculations. The first terms

originate to the quantum calculation of the reacting subsystem in vacuum. From

this, the corresponding energy levels of the classical force-field description is sub-

tracted (second line) to avoid double counting in the classical modeling of the

ion-ion interactions. The last two terms reflect the ion-ion and ion-solvent in-

teractions of the aggregate before and after proton transfer, respectively. During

relaxation of the putative product state (involving the formation of O2− and H2O),

the water molecule was always found to migrate to the ethanolic solution. In the

vacuum calculations related to the product state, the distances of the O2− ion and

the water molecule were therefore taken as infinite.

The quantum calculations of the electronic ground states were performed using

the Gaussian package at the 6–311** basis set and second order Moller-Plesset per-

turbation theory for considering electron correlation. For the classical molecular

mechanics simulations, empirical force fields[12–14] were taken from the literature

as described in detail in Supporting Information. To mimic a bulk ethanolic so-

lution, the aggregate is placed in a cubic box (≈ 4 × 4 × 4nm3) of 750 ethanol

52


molecules. Periodic boundary conditions are applied, and temperature and pres-

sure are constrained to 300 K and 1 atm, respectively. For the molecular dynamics

simulations, a time-step of 1 fs was found to be appropriate. For each aggregate

growth iteration, the relaxation process following the association of the newly

added ion amounts to a simulation time of 100 ps (see ref [11] for more details).

The following analysis is based on three independent growth runs. While the illus-

trations show representative configurations of a single growth run, the presented

statistics correspond to the data from all simulations runs.

The very initial stage of the aggregation process corresponds to the association

of Zn2+ and OH− ions from the ethanolic solution. Strikingly, self-organization

may already be observed in aggregates counting less than 20 ions (Figure 2.9).

In this premature ordering, the zinc ions are coordinated by OH− ions arranged

as (incomplete) octahedra similar to the rocksalt structure. In this constellation,

the hydrogen atoms of the OH− groups points to the solvent and Hδ+ · · ·Zn2+

contacts are avoided. Consequently, the formation of extended domains of such

motifs is clearly hindered. Indeed, the preordering represents a very temporary

feature which changes dramatically in the course of local O2− ion formation as

discussed in the following.

At the aggregate surface, the hydroxide ions predominantly form hydrogen

bonds with the ethanolic solution. However, in the course of further OH− ion

association temporary OH− · · ·OH− interconnections leading to proton transfer

reactions are observed. Figure 2.10 shows the first proton transfer event observed

from one of our aggregate growth simulations. A newly associated hydroxide ion

experiences electrostatic bonding to two Zn2+ ions and hydrogen bonding to the

hydrogen atom of a neighboring OH− ion. Along this hydrogen bond, proton

transfer was found to be exothermic. The resulting H2O molecule is only loosely

affiliated to the aggregate and eventually diffuses into the ethanolic solution. On

the other hand, the O2− ion remains in the aggregate and gives rise to dramatic

structural rearrangements.

A first insight into the effect of O2− ion formation on the relaxation of the

aggregate may be obtained from the [Zn24(OH)40O2]4+ agglomerate illustrated in

figure 2.9 (bottom). Already after the formation of only two O2− ions, the ear-

lier observed preordering (octahedral coordination) of the aggregate is rearranged

in favor of the wurtzite structure (tetrahedral coordination). Further aggregate

53


Figure 2.9: Snapshots of the early stage of aggregate formation (solvent moleculesare not shown). Aggregates counting only a few tens of ions compriseof Zn2+ and OH− ions, only. Self-organization of these ions leads tomotifs exhibiting similarities to the rocksalt structure. At later stagesof the aggregate growth O2− ions result from proton transfer reac-tions (see also figure 2.10). This process is accompanied by dramaticstructural rearrangements.

54


Figure 2.10: Onset of ZnO formation by condensation reactions. At the aggregatesurface, a newly associated hydroxide ion (top right) experiences hy-drogen bonding to the hydrogen atom of an adjacent OH− ion. Thishydrogen bond is used for exothermic proton transfer. The result-ing water molecule migrates into the solvent, whereas the O2− ionremains within the aggregate.

55


growth leads to a systematically increasing dominance of oxide ions. Though

the hydroxide ions represent the major anionic species at the aggregate surface,

agglomerates counting more than around 150 ions exhibit a core which is fully

sparse of OH−. The ionic arrangement in this core region may be described by

six-membered rings, formed as either chair- or boat-type constellations. The au-

tomated identification of these motifs is based on a geometric hashing algorithm

as described in Supporting Information.

Figure 2.11a illustrates the beginning of ZnO ordering in the aggregate center.

At this stage of aggregate growth, most of the six-ring motifs are randomly ori-

ented (see figure 2.12 for the total number of chair and boat motifs as a function

of aggregate size). However, the two highlighted chair-type motifs exhibit a stag-

gered arrangement and represent the onset of orientation correlation between the

six rings. Indeed, the staggered arrangement does not undergo substantial changes

during further crystal growth and is hence suggested as a center of stability. This

assumption is supported by the observation of progressive ordering in the same

manner starting from this nucleus. Figure 2.11b shows the aggregate after the

uptake of about 50 more ions. Adjacent to the staggered six-ring motifs high-

lighted in figure 2.11a, further chair motifs of matching orientation were formed.

The ionic ordering of the ZnO nanocrystal hence nucleates in the center of the

agglomerates and, in the course of further aggregate growth, propagates within

the increasing bulk domain.

In the hexagonal wurtzite structure, the ABAB stacking of layers of Zn–O six

ring along the c axis gives rise to boat motifs connecting adjacent (001) layers

(figure 2.12, top right). This feature helps to discriminate the ionic arrangement

from the cubic sphalerite structure. In sphalerite, the individual (001) layers are

identical to those of the wurtzite structure but stacked in terms of ABCABC. As

a consequence, neighboring (001) layers are connected by chair rather than boat

motifs. We performed an analysis of the number of chair and boat motifs as a

function of the aggregate size (figure 2.12). Despite the small numbers of ions

available, self-organization already causes chair and boat motif formation roughly

at the 1:3 ratio present in the ideal wurtzite structure. To illustrate the resulting

stacking of layers, a close up of the ordered core of the [Zn79(OH)72O43] aggregate

is also shown at different orientation in the top left of figure 2.11b.

56


Figure 2.11: Nucleation and growth of domains of the wurtzite structure. Thecharacteristic six rings are highlighted in blue and discussed morequantitatively in figure 2.12. (a) Self-organization of the ZnO crystalstructure is initiated in the aggregate center which contains no OH−

ions. The two staggered six ring form a center of stability and giverise to further ordering in favor of the wurtzite structure. The uppersix ring is highlighted for better visibility. (b) At later stages of ag-gregate growth, progressive self-organization results in the formationof an extended core comprising of Zn2+ and O2− ions ordered in chairand boat motifs corresponding to the wurtzite structure. For bettervisibility, only chair-type six-ring motifs are highlighted.

57


Figure 2.12: Number of chair (blue) and boat (green) motifs as a function of theaggregate size. The slopes of the dotted lines indicate chair and boatmotif formation at a ratio of 1:3 as corresponding to the ideal wurtzitestructure. The later is illustrated by the inset in the top left whichdepicts the 3 chair and 9 boat motifs coordinating each ion (yellow).

58


A geometric analysis of the Zn–O six rings may be given by the interionic

distances and angles. In the aggregate core, the nearest neighbor Zn2+ · · ·O2−

distance was found to vary from 1.8 to 2.1 A, whereas the Zn2+ · · ·OH− distances

range from 2.0 to 2.2 A. From crystal structure refinements of bulk ZnO, the

corresponding distances were found as 1.9746(1) A (basal) and 1.9857(1) A (api-

cal), respectively.[15] Moreover, the Zn–O–Zn/O–Zn–O angles of neighboring ion

triples in the aggregate center were found to range from 100 to 120◦ which also is

in reasonable agreement with experiments related to the bulk material (108.11(2)

and 110.79(2)◦, respectively). Detailed statistics are provided in Supporting In-

formation.

At the aggregate-solvent interface, ionic ordering is much less pronounced com-

pared with the nanocrystal core. However, a typical feature of the aggregate-

solvent interface is given by the vanishing occurrence of O2− species and the

dominance of OH− ions which hydrogen atom points toward the ethanolic sol-

vent. Figure 2.13 depicts radial distribution functions calculated for the solvent-

ion distances of the [Zn79(OH)72O43] aggregate shown in figure 2.11. The upper

plot refers to hydrogen bonding of the OH group of ethanol to the anions of the

nanocrystal. Ethanol–oxide ion hydrogen bonds are only rarely observed. The

O2− ions are thus separated from the solvent and occur only in the inner part

of the nanocrystal. Scanning the whole series of aggregates obtained from our

calculations, we found at most one single oxide ion at the interface to the sol-

vent. The OH− ions are hence the dominating anionic species at the aggregate

surface. In average, the hydroxide ion–solvent interactions involve one to two

H(ethanol)· · ·O(OH−) and at most one O(ethanol)· · ·H(OH−) hydrogen bond.

On the other hand, the Zn2+ ions of the aggregate surface form 1-2 electrostatic

bonds to the oxygen atoms of the ethanol molecules.

As part of our modeling approximations, the ethanolic solution embedding the

aggregates is taken to be free of further ions. While the dissociation of water and

Zn2+/OH− ions from the aggregate are considered during the relaxation runs, a

quantitative assessment of the interplay of ion uptake/dissociation exceeds the

scope of the present work. The different chemical potentials of the ionic species

depend on the respective concentration in the solution and so does the size of

the critical nucleus. However, from our work, two important aspects of aggregate

stabilization can be identified: (i) ripening reactions leading to the formation of

59


Figure 2.13: Radial distribution functions of the solvent–ion distances as obtainedfor the aggregate depicted in figure 2.11b. Anion–solvent interactionsare only observed for OH− ions; the O2− ions are located in the innercore of the aggregate.

O2− ions and (ii) nucleation and growth of a ZnO domain of wurtzite crystal

structure type.

In conclusion, we elucidated the atomistic mechanisms of the aggregation of

(wurtzite-type) zinc oxide nanocrystals from ethanolic solution. The very initial

stage of this process is given by Zn2+ and OH− ions association and the formation

of [ZnxOHy]2x−y agglomerates exhibiting octahedral coordination constellations.

These aggregates of less than 50 ions correspond to precursors which undergo A–

OH− · · ·OH− → A–O2− + H2O (A = aggregate) proton transfer reactions. While

the resulting water molecule migrates into the solution, the formation of O2− ions

implies dramatic changes within the aggregate. Premature ordering is largely

reversed, and after the association of around 150 ions, wurtzite-type (tetrahedral

coordination constellations) ZnO domains were found to nucleate in the central

region of the nanocrystal. While the aggregate surface mainly consists of Zn2+ and

OH− ions, the core comprises Zn2+ and O2− ions, only. The latter are ordered in

Zn–O six rings according to chair and boat motifs which undergo self-organization

in favor of the final crystal structure. Aggregates counting more than 200 ions

60


already exhibit a nanometer-sized core of the wurtzite structure which grows in

the course of further ion association and condensation reactions.

61

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63

2.4 Nucleation mechanisms of a polymorphic molecular crystal

2.4 Nucleation mechanisms of a polymorphicmolecular crystal: solvent-dependent structuralevolution of benzamide aggregates

Philipp Ectors, Patrick Duchstein, and Dirk Zahn

This section has previously been published:Title Nucleation mechanisms of a polymorphic molecular crystal:

solvent-dependent structural evolution of benzamide aggregatesAuthor Philipp Ectors, Patrick Duchstein, and Dirk ZahnJournal Crystal Growth & Design

Publisher American Chemical SocietyVolume 14

Issue 6Pages 2972–2976Date April 16, 2014DOI 10.1021/cg500247c

Copyright c© 2014 American Chemical Society. Reprinted (adapted) with permis-sion from [1]. Figures, tables, sections, and references have been renumbered.

Abstract We report on molecular dynamics simulations dedicated to the ag-

gregation and the early stage of nucleation of benzamide molecular crystals. As

suggested by an earlier study of the molecular interactions in different benzamide

polymorphs[2], we find the solvent to take a prominent role for polymorph con-

trol. The underlying driving forces are now rationalized by in-depth investigation

of the formation of dimers, subsequent molecule association, and self-organization

into aggregates of up to 100 benzamide molecules. Within the forming nuclei we

identify the development of ordered molecular motifs as characteristic for either

the P1 or P3 polymorph structure. From the different structural evolution related

to the two nucleation scenarios, the preference of the stable P1 form is observed

in aqueous solution, whilst precipitation from the vapour exhibits a tendency for

the P3 polymorph of benzamide. While benzamide aggregation from the vapour

is mainly driven by hydrogen bonding between benzamide molecules, aggregation

in aqueous solution is dominated by segregation of the hydrophobic moieties and

π − π interactions of the benzene rings.

65


2.4.1 Introduction

The standard approach to tackling molecular crystal polymorphism is still trial

and error. For industrial purposes, the testing of different solvents and ther-

modynamic conditions was even automated such that robots systematically probe

suitable synthesis routes. Meanwhile, in-depth studies of the underlying molecular

interactions, often based on computer simulations, are developing into an increas-

ingly rational approach to control molecular crystal polymorphism. This applies

to (i) the theoretical assessment of the manifold of different polymorphs[3–5] and,

more recently, (ii) molecular simulation as tools aiming at the rationalization of

the nucleation mechanisms leading to the various crystal structures.[6–12]

By the example of an experimentally well-established benchmark system, benza-

mide, we recently demonstrated the analysis of molecular interactions in different

polymorphic forms and its possible implications for polymorph control.[2] From

this, the favoring of either P1 or P3 forms of benzamide was attributed to different

layer–layer interactions within periodic models of the bulk crystals. Guided by this

postulated mechanism of P1 versus P3 polymorph selectivity, in the present study,

we aim at the in-depth understanding of the whole morphogenesis of a forming

aggregate, i. e., molecule-by-molecule association and subsequent self-organization

in favor of crystalline motifs. To account for the role of the solvent, two different

nucleation routes are investigated discriminating the amphiphilic nature of benza-

mide molecules during precipitation from polar solvent or nonpolar environment.

2.4.2 Simulation Methods

Force Fields The molecular interactions are mimicked by empirical potential

energy functions that allow efficient sampling of aggregate structures at reasonable

accuracy. The force field used for benzamide is based on fully flexible molecular

models using the general AMBER force field (GAFF).[13] We, however, refitted

both the atomic charges and the amide–benzene torsion with respect to quantum

calculations (see figure 2.14 and table 2.2). For the partial charge calculation, the

benzamide dimer was used as reference, as all experimentally confirmed crystal

structures[14–17] show this very distinct motif. The dimer structure was optimized

at the MP2 6-311+G(d,p) level and then used to calculate a restricted ESP-fit[18]

(table 2.2). Using the same quantum approach, force-field parameters for the

66


amide–benzene torsion were derived. The corresponding interactions are described

by the below torsion potential plus the nonbonding interactions as denoted in

table 2.2, using the usual 0.83 and 0.5 scaling factors for the Coulomb and the

Lennard-Jones potentials, respectively.

Vtorsion = 10kJ/mol · [1 + cos(2ϕ− π)] + 1.4kJ/mol · [1 + cos(6ϕ)]

Figure 2.14: Atomic Representation of Benzamide as also used in table 2.22.

atom GAFF type charge ε [kJ/mol] σ [A]

N(1) N -0.716853 0.71128 3.25000H(2) HN 0.331928 0.06569 1.06908C(3) C 0.588880 0.35982 3.39968O(4) O -0.542819 0.87864 2.95993C(5) CA -0.038452 0.35982 3.39968C(6) CA -0.089859 0.35982 3.39968C(7) CA -0.168663 0.35982 3.39968C(8) CA -0.118405 0.35982 3.39968H(9) HA 0.122741 0.06275 2.59965H(10) HA 0.145213 0.06275 2.59965H(11) HA 0.144929 0.06275 2.59965

Table 2.2: Atom Types and Force-Field Parameters of Benzamide

Using this optimized force field, we were able to reproduce all crystal structures

of benzamide polymorphs with an accuracy in potential energy of 1 kJ/mol per

molecule (as compared to quantum calculations of the P1 and P3 structures).[2, 19]

2The torsion around the C(3)–C(5) axis has been reparameterized.

67


Note that the P1 and P3 polymorphs are energetically equivalent within these error

margins. Moreover, the force field has been successfully used to predict metastable

defects in form 1 of benzamide, in good agreement with NMR experiments.[19]

As the standard GAFF parameters were adopted for the nonbonded interaction

models of the force field, mixing intermolecular interaction parameters – as em-

ployed for the benzamide-water interactions – is possible. In what follows, water

is described by the standard TIP3P model.[20]

Molecular Dynamics Simulation The molecular dynamics simulations were car-

ried out using a time step of 1 fs. Explicit Coulomb summation was used for the

aggregates modeled in the gas phase, whereas a (shifted-force) cutoff of 1 nm was

found appropriate for the aqueous solutions. Molecule-by-molecule association

was investigated by means of a recently developed simulations scheme.[12, 21]

The Kawska-Zahn approach reflects an iterative procedure that allows investi-

gating aggregate evolution molecule-by-molecule starting with the formation of

a dimer. Each growth step is devised into two processes, (i) the diffusion of a

solute molecule to reach the aggregate and (ii) solute association and aggregate

reorganization. The first step (i) reflects a random walk of initially dispersed

solute molecules until close proximity to the forming aggregate. To avoid com-

putationally demanding molecular dynamics simulation of this process, we mimic

the outcome of solute diffusion to the aggregate by a simple docking procedure.

Starting with the formation of a benzamide dimer, the Kawska-Zahn approach

models each association step by placing a solute on a random position in prox-

imity of the forming aggregate to account for unbiased solute diffusion leading

to aggregate-solute contacts. Next, possible association or solute rejection is ex-

plored from simple potential energy minimization, which is implemented as rigid

aggregate-solute docking in the absence of solvent molecules. Finally (ii), aggre-

gate relaxation is explored from unconstrained molecular dynamics simulations

allowing full flexibility of the aggregate and considering the solvent. Thus, an

effective Monte Carlo-type procedure is used for docking new solutes to a forming

aggregate, while, after each solute association, the relaxation of the aggregate is

explored in detail from molecular dynamics simulations.[12, 21] When exploring

aggregation from solution, the relaxation step (ii) is implemented by placing the

aggregate in a bulk model of explicit water molecules treated as a cubic box of

68


≈ 5 nm cell edge length. After 50 ps of constant pressure, constant temperature

simulations at ambient conditions, aggregate relaxation is explored from 250 ps

simulated-annealing runs at constant volume.

In particular, during the relaxation runs for aggregates grown from the vapor,

it is useful to not impose ambient conditions, but to start simulated annealing

from a lower simulation temperature (200 K) to avoid excessive dissociation rates

experienced for the newly added solute. This allows for more careful relaxation

after the rather simplistic docking procedure and thus reduces failed attempts

during the Monte Carlo-type steps mimicking putative aggregate growth steps.

Several degrees of such “precooling” were tested for the annealing runs (260, 220,

and 200 K) along with an increase in the sampling times up to 10 ns.

We also draw the reader’s attention to the fact that a rigorous grand-canonical

Monte Carlo procedure would imply accepting/rejecting growth attempts on the

basis of the free energy of solute association to the forming aggregate. While, in

principle, accessible to molecular simulation, here, we choose a (computationally

much cheaper) acceptance criterion, namely, the direct inspection if the solute gets

incorporated into the aggregate or is dissociated upon relaxation in solution. This

intuitive criterion was found to perform excellently for a wide range of solutions[12,

21] with the exception of very high concentrations such as that of overconcentrated

NaCl in water.[21]

Motif Recognition We used a motif-recognition algorithm to identify crystalline

features in the growing aggregates.[22] The identification of crystalline motifs in

only partially ordered nuclei is a nontrivial challenge, and particular care is re-

quired to avoid biasing the analysis by a too intuitive choice of the underlying

order parameters. Indeed, the search for a specific molecular arrangement (such

as the hydrogen-bonded dimers in the present work) is always subject to the chosen

tolerance (root-mean-square deviation). Nonetheless, a much less biased analysis

can be performed on a relative scale when comparing equivalent motifs of different

polymorphs using identical tolerance. While the high-energy form P2 is of no rel-

evance for the present study, the two most stable polymorphs of benzamide are il-

lustrated in figure 2.15. Both structures are composed of hydrogen-bonded dimers

that form hydrogen-bonded sheets within the (001) plane. P1 and P3, however,

differ in terms of the contact angle and distance of the neighboring molecules of

69


adjacent sheets. Thus, trimers comprising a hydrogen-bonded dimer (highlighted

in red in figure 2.15) and an adjacent molecule “attached” via a benzene-benzene

contact represent the smallest molecular motif that allows discrimination between

P1- and P3-type ordering. Comparing a putative dimer/trimer motif in the ag-

gregate to the corresponding dimer/trimer motifs in the bulk crystal structures

as reference, a root-mean-square deviation of less than 1.3 Awas chosen as the

threshold for crystalline motif recognition. This treshold was found to provide

best discrimination between P1- and P3-type motifs.

Figure 2.15: Illustration of the P1 and P3 structures of benzamide.[14, 15] Whilethe units cells are shown as gray boxes, hydrogen-bonded dimers con-stituting the key building blocks are highlighted in red. To discrimi-nate P1 and P3 motifs, the orientation of such dimers with adjacentbenzamide molecules is analyzed (dashed green line). The tilt anglesare 0A◦ and 74A◦ in P1 and P3, respectively.

2.4.3 Results

Guided by the suggested preference of P1- and P3-type polymorph growth in polar

and nonpolar solutions, respectively, we explored two setups of benzamide precip-

itation. Molecule-by-molecule association up to the aggregation of 125 benzamide

molecules was explored (a) in the vapor as a proxy to nonpolar solution (several

nonpolar solvents were tested for the relaxation of a small series of aggregates

demonstrating vacuum as an excellent proxy for such solvation. An important

exception is benzene, which does π–π stacking with the solute and thus interacts

more selectively) and (b) in aqueous solution. For each setup, three indepen-

70


dent growth runs were performed to provide at least qualitative insights into the

statistical manifold of possible nucleation and growth routes. The mechanisms

discussed in the following were found in all of these simulation runs.

Benzamide aggregation from the vapor is dominated by hydrogen bonding,

which is particularly evident for aggregates of less than about 10 molecules. A

representative growth run is illustrated in figure 2.16a. Upon further aggregate

growth, benzene-benzene contacts are also observed; however, π–π stacking plays

a minor role even in the largest aggregates explored comprising 125 molecules

(figure 2.16b). As a consequence, the aggregates mainly reflect an agglomerate

of hydrogen-bonded dimers, which tend to arrange in favor of hydrogen-bonded

sheets in the course of aggregate growth. However, this ordering is observed only

partially (i. e., locally for bulk-like domains), while the outer boundaries of the

aggregate are characterized by benzene groups at the surface to avoid undercoor-

dinated hydrogen bond acceptors and donors. Indeed, up to 125 molecules, the

overall aggregate shape is close to spherical and thus mainly controlled by surface

tension.

On the other hand, benzamide aggregation from aqueous solution is found to

be driven by two roughly equally strong interactions. Figure 2.17a illustrates the

morphogenesis of a representative growth run. In particular, for the dimer, but

also the binding of trimers, tetramers, etc., we clearly observe an interplay of

hydrophobic segregation of the benzene moieties and hydrogen bonding between

the amide groups. As a consequence, hydrogen-bonded benzamide dimer motifs

are observed at much later stages of aggregate growth. The largest aggregate

explored, comprising 125 benzamide molecules, is illustrated in more detail in

figure 2.17b. Whereas the onset of ordering is observed in the inner part of the

aggregate, the outer boundaries are poorly ordered and exhibit amide-amide and

amide-water hydrogen bonds at roughly equal rates. In contrast to the rigorous

benzamide-vacuum interface, this leads to a weakly ordered and more extended

benzamide-water interface region.

Despite the still small number of ordered motifs observed during these early

stages of aggregation, it is already possible to discriminate the different rates of

hydrogen-bonded benzamide dimer formation as identified for aggregation from

the vapor and in aqueous solution. Figure 2.18 shows the absolute number of such

motifs as a function of aggregate size as sampled from three independent growth

71


Figure 2.16: (a) Snapshots of aggregate growth from the vapor. Starting from thehydrogen-bonded dimer compact, almost spherical hydrogen-bondedclusters are formed. Colors: H (white), C (gray), N (blue), andO (red). The hydrogen atoms of the benzene ring are omitted forclarity. (b) Aggregate of 125 benzamide molecules as precipitatedfrom the vapor at different representations, but identical orientation.Left to right: all molecules, highlighting of dimer motifs, highlightingof parallel dimers (sheets) highlighting P1 (red) versus P3 (green)type motifs.

72


Figure 2.17: (a) Snapshots of benzamide aggregate growth from aqueous solu-tion. Solute-solute interactions are characterized by the interplayof hydrophobic segregation and hydrogen-bonded dimer formation.Colors: H (white), C (gray), N (blue), and O (red). The solventmolecules and the hydrogen atoms of the benzene ring are omittedfor clarity. (b) Aggregate of 125 benzene molecules as precipitatedfrom aqueous solution at different representations, but identical ori-entation. Left to right: all molecules, highlighting of dimer motifs,highlighting of parallel dimers (sheets) highlighting P1 (red) versusP3 (green) type motifs.

73


runs for precipitation both from the vapor and in water. Note that this ordering

increases gradually and not by spontaneous disorder-order transitions. While the

dimers reflect the smallest and most simple motif of the molecular crystal, more

subtle discrimination between P1- and P3-type polymorph nucleation requires the

analysis of trimer motifs, as outlined in figure 2.15. Even for aggregates comprising

more than 100 molecules, the overall number of such motifs is rather small (see

figure 2.16b and figure 2.17b for illustration). Nonetheless, for aggregate growth

from the vapor, we can identify the biasing in favor of the metastable polymorph

P3. The overall sampling from all aggregate runs up to 125 molecules shows 80%

of all trimer motifs are of P3-type. On the other hand, the analogous analysis

of aggregation from aqueous solution exhibits only a weak bias in favor of 60%

P1-type motifs.

Thus, even for the very early stages of aggregation explored in this study, we

can relate the favoring of different polymorphic structures to the preferential for-

mation of P1- versus P3-type trimer motifs, in agreement to experimental and

other theoretical evidence, both collected for macroscale crystallites.[2, 19] With

growing aggregate size, the importance of surface/interface effects diminishes and

the quantitative P1/P3 motif ratio surely changes (to more and more pronounced

favoring). Our findings should, therefore, be considered from a more qualitative

point of view, i. e., the selectivity of P1- versus P3-type trimers as a consequence of

different driving forces to solute association in polar and nonpolar solutions. This

constitutes a bias in forming aggregates, which ordering was found to increase

continuously. Even if aggregate growth at later stages should involve an abrupt

disorder-order transformation, we still argue that the observed motifs reflect nu-

cleation seeds giving rise to similar P1 versus P3 selectivity.

Employing the Kawska-Zahn approach, we investigated benzamide molecular

crystal nucleation as a series of molecule-by-molecule association events. Our

simulation procedure starts with the very early stage of dimer association, fol-

lowed by aggregation of up to 125 molecules. Within the growing nuclei, we

clearly observe the self-organization in favor of either P1- or P3-type polymorphs,

depending on the solvent environment. In agreement with the available experi-

mental experience, our study thus shows the suitability of our simulation approach

to rationalize the mechanisms of molecular crystal aggregation and the nucleation

of competing polymorphic structures.

74


Figure 2.18: (a) Motif recognition of hydrogen-bonded dimers in aggregates grownfrom the vapor (red) and in water (blue) as a function of aggregatesize. In aqueous solution, benzamide-benzamide hydrogen bondingcompetes with hydrophobic segregation. Aggregate ordering in favorof dimer motifs thus occurs at later stages, once a desolvated ag-gregate core is developing. See also figure 2.15 and figure 2.16. (b)Motif recognition of sheets comprising two parallel hydrogen-bondeddimers, shown for aggregates grown from the vapor (red) and in wa-ter (blue) as a function of aggregate size. See also figure 2.15 andfigure 2.16.

75


While the investigated benzamide aggregates undergo rapid relaxation, followed

by vibrations and rotation of the whole aggregate, we point out that the case of

prenucleation clusters is also accessible to our method. This was recently elabo-

rated by de Yoreo and co-workers for calcium carbonate aggregation in water,[23]

who identified liquid-crystalline-type aggregate structures that were sampled from

parallel-replica molecular dynamics simulations.

The authors declare no competing financial interest.

76

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[5] Sarah S. Price. Computed Crystal Energy Landscapes for Understanding

and Predicting Organic Crystal Structures and Polymorphism. Acc. Chem.

Res., 42(1):117–126, October 2008.




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[7] A. Gavezzotti, G. Filippini, J. Kroon, B. P. van Eijck,

and P. Klewinghaus. The Crystal Polymorphism of Tetrolic Acid

(CH3Cı£34CCOOH): A Molecular Dynamics Study of Precursors in Solu-

77


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[8] Peter G. Vekilov. Dense Liquid Precursor for the Nucleation of Ordered

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[9] Stefano Piana, Manijeh Reyhani, and Julian D. Gale. Simulat-

ing micrometre-scale crystal growth from solution. Nature, 438(7064):70–73,

November 2005.

[10] Wolfgang Lechner, Christoph Dellago, and Peter G. Bolhuis.

Role of the Prestructured Surface Cloud in Crystal Nucleation. Physical Re-

view Letters, 106(8):085701+, February 2011.

[11] Matteo Salvalaglio, Thomas Vetter, Federico Giberti, Marco

Mazzotti, and Michele Parrinello. Uncovering Molecular Details of

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(41):17221–17233, October 2012.

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80

2.5 Molecular simulation of DL-norleucine crystal nucleation from solution

2.5 From oligomers towards a racemic crystal:molecular simulation of DL-norleucine crystalnucleation from solution

Philipp Ectors, Patrick Duchstein, and Dirk Zahn

This section has previously been published:Title From oligomers towards a racemic crystal: molecular simulation

of dl-norleucine crystal nucleation from solutionAuthor Philipp Ectors, Patrick Duchstein, and Dirk ZahnJournal CrystEngComm

Publisher Royal Society of ChemistryDate November 20, 2014DOI 10.1039/C4CE02078B

Reproduced from Ref. [1] with permission from The Royal Society of Chemistry.Figures, tables, sections, and references have been renumbered.

Abstract We elucidate the evolution of forming nuclei leading towards a racemic

molecular crystal by means of computer simulation. Our simulations start with the

association of d- and l-norleucine molecules and subsequent formation of aggre-

gates and thus provide molecular scale insights into the early stages of nucleation

from (octanol) solution. During aggregate evolution from micelles to bilayers and

eventually nanoparticles comprising staggered bilayers we identified a step-wise

increase in molecular ordering. While the key structural motifs in the early aggre-

gates are given by hydrogen bonded dimers, after association of ≈150 molecules

a solid-solid phase transition leads to ordering of the hydrogen-bonded network

in favour of the final crystal structure. This secondary nucleation event seems to

be of key importance for changing the initially random arrangement of d and l

enantiomers in the small (<150 molecules) aggregates to an ordered arrangement,

that is alternating -d-d- and -l-l-alignments within the staggered bilayers of the

molecular crystal.

81


2.5.1 Introduction

The profound (atomic level of detail) investigation of nucleation mechanisms from

computer simulation could break new grounds to understand and possibly control

crystal formation processes. Along this line, molecular crystals are of peculiar

interest because of their immense importance to the pharmaceutical industry. On

the other hand, the polymorphism frequently encountered in molecular crystals

also poses a major challenge to molecular simulation.

Pioneered by Gavezzotti, Jansen and others, decades of simulation work on

exploring the energy landscapes that constitute possible crystal structures re-

vealed the abundance of local energy minima, each corresponding to a possible

polymorph.[2–4] For ionic compounds the different packing into crystal structures

often imply considerable energy differences and many structural candidates can be

eliminated because of unfavourable electrostatics. Molecular crystals instead tend

to comparably small energy differences upon different structural arrangements and

thus exhibit even more pronounced polymorphism.[2–4]

Thus, apart from searching possible structures of the finally obtained crystal,

in particular for molecular compounds, we suggest simulation approaches which

consider the whole formation process [5–11]. On such basis, the choice for one of

the abundant structural candidates encountered for molecular crystals is given by

both relaxation to (possibly local) energy minima and structural similarity along

the self-organization process. To obtain such heuristic information, molecular

simulations must start from the association of individual molecules to dimers,

trimers etc. and eventually describe the evolution of molecular clusters from pre-

to post-critical nuclei, whilst providing an explicit account of putative structural

transitions along the nucleation process [11].

Using a recently developed simulation method [11, 12], here we demonstrate

the analysis of molecular crystal nucleation from solution by the example of d-

and l-norleucine as examples. This compound is not of explicit pharmaceutical

activity, but serves as a particularly reliable benchmark model for the evolution

of a nucleating molecular crystal. Based on a newly developed molecular interac-

tion model of high accuracy, norleucine was successfully subjected to a series of

simulation studies of both bulk and nanocrystalline polymorphs [13–15].

82


2.5.2 Models and Methods

All molecular interaction models were chosen as non-reactive force-fields taken

from the literature. For d- and l-norleucine, Anwar and coworkers developed a

model of particularly high accuracy, being able to tackle subtle differences (<1 kJ

mol−1) in the energy landscape of molecular crystal polymorphs [13]. As solvents,

we explored n-octane and n-octanol in which force field parameters were adopted

from the GAFF list [16]. Aggregate relaxation in solution was explored in periodic

models of ≈ 8 × 8 × 8 nm3 mimicking the bulk octane and octanol solutions at

300 K and 1 atm.

For the molecular dynamics simulations a time step of 1 fs was used in com-

bination with a Berendsen thermostat with a coupling constant of 5 ps. The

simulations in the gas phase and in weakly polar solvents call for careful evalua-

tion of Coulomb interactions which long-range nature motivated the choice of a 2

nm cut-off, whilst 1.2 nm were found as still appropriate for preliminary relaxation

of the scale-up models that will be discussed later.

The evolution of forming aggregates is studied by means of molecule by molecule

association and relaxation based on the Kawska-Zahn approach [11, 12]. Thus,

within an iterative procedure, d- and l-type molecules (at equal occurrence) are

docked to a forming aggregate (starting from the dimer) from randomly chosen

incoming vectors. After initial quenching to the nearest energy minimum, 5 ns

simulated annealing runs are performed to allow aggregate relaxation. While

in particular for the small oligomers this step may to lead to the global energy

minimum, we reveal that the annealing runs were only chosen to provide energy

convergence to an energy minimum close, but not necessarily identical to the

global one. This tolerance does not only boost simulation times drastically, but

also appears more realistic for the complexity inherent to crystal nucleation from

solution. Structural transitions requiring the crossing of large energy barriers

are disfavoured and we instead observe a heuristic evolution of the forming nuclei

which also includes local energy minima (see ref. [11, 17] for an extended discussion

of this issue). To provide at least a qualitative account of the immense manifold

of aggregate configurations, three independent simulation runs were performed for

each of the scenarios discussed below.

83


2.5.3 Results

Early stages of nucleation: molecular-by-molecule association

Employing the Kawska-Zahn approach, the evolution of forming nuclei was first

explored as molecule by molecule association, i. e.at the maximum level of detail.

This was accomplished in the vapour, octane and in octanol solution. For all

scenarios, the nucleation mechanisms were found as very similar. This motivated

the development of computationally efficient approximate routes to mimic the role

of the solvents as a post-processing step. The aggregates discussed in the following

were depicted from a simulation run of nucleation from the vapour, embedded in

octanol solution and characterized after 5 ns relaxation runs.

A representative simulation run showing the evolution of a forming aggregate

from 2 to 222 molecules is illustrated in figure 2.19.

In both vapour and non-polar solution, the driving force to norleucine associa-

tion is hydrogen bonding which was found to lead to a continuous evolution from

micelles to disc-shaped bilayers structures. Upon aggregation of ≈ 150 (160 in

the simulation run shown in figure figure 2.19(b)) we observed the onset of the

formation of a second layer. The underlying mechanism was identified as molecule

association to local kinks, initially reflecting a defect of increasing size within a

single bilayer. However, after the accumulation of ≈ 50 molecules a new layer is

formed and the defect heals in favour of two staggered bilayers.

It is worthy to note that both the disc-like single bilayer and structures of

staggered bilayers show significant boundary effects. To ensure a maximum num-

ber if amide-amide hydrogen bonding (and thus avoid dangling hydrogen bond

acceptors/donors), single bilayer filaments show ‘round’ boundaries. Moreover,

staggered bilayers may receive stabilization from (partially) arranging them as

‘loops’ (see also figure 2.20 and figure 2.21). Both of these arrangements resem-

ble the micelle type structures observed for small oligomers and emphasize the

importance of hydrogen bonding as the primary driving force to norleucine aggre-

gation from non-polar solvent. While this essentially reflects the overall number

of hydrogen bonds, ordering of the hydrogen bonded network as discussed in the

following is suggested as a secondary driving force that, in combination with even

weaker van der Waals contacts of the alkyl chains, finally leads to the molecular

crystal structure.

84


Figure 2.19: (a) d/l-norleucine oligomers as observed from molecule-by-moleculeassociation in non-polar solvent or from the vapor. With increas-ing size, the nucleation mechanism involves gradual evolution fromhydrogen-bonded micelles to a bilayer. (b) Later stages of D/L-norleucine nucleation as observed from molecule-by-molecule asso-ciation in non-polar solvent or from the vapor. After incorporationof ≈ 150 molecules within a single hydrogen-bonded bilayer, a secondbilayer nucleates from defects in the pristine bilayer. With increasingstabilization of the ad-grown bilayer, the initially substantial defectsin the first bilayer gradually recover.

85


Figure 2.20: (a) Self-organization of the hydrogen bonded network within the bi-layers. Top: side view as in figure 2.19. Bottom: viewing directionnormal to the bilayer. While initially arranging as discs, later stagesof aggregate growth show the nucleation of d/l-norleucine dimerdomains (highlighted in red and by dotted boundaries). The inter-faces of these triangular domains favour molecular arrangements as inhydrogen-bonded trimers (highlighted molecules) and were identifiedas nucleation sites for the formation of the final crystal structure. (b)Top: domains of dimer and trimer structures as identified in the form-ing nuclei shown in figure 2.20(a). Bottom: corresponding hydrogenbonding patterns (side view). The crystal structure reflects alter-nating stacks of enantiopure sheets (d and l indicated by blue andgreen arrows, respectively) comprising trimer motifs. Note that thedimers favour triangular domain boundaries, while the trimers favourrectangular domains. The former leads to the pentagon-shaped bi-layers shown in figure 2.20(a) (50 − 150 molecules) and the latter(> 150 molecules) represents a bonding pattern found in the alphadl-norleucine crystal structure (the underlying unit cell is highlightedin blue).

86


Figure 2.21: (a) Structural evolution of a d/l-norleucine aggregate using the up-scaling method. Based on building blocks comprising 10 moleculesthe mechanistic features of the molecule-by-molecule nucleation studyare nicely reproduced. With increasing size the aggregate shape de-velops from sheets to structures with roughly equal dimensions inall directions. (b) Illustration of d/l-norleucine nucleation from oc-tanol. The forming nucleus comprises 500 molecules and reflects theevolution from micelles, increasingly ordered bilayers to nestled loopsof hydrogen bonded bilayers.

87


This concept is supported by the observation that the forming bilayer gradu-

ally evolves from round discs (figure 2.20(a), left) to structures with increasingly

well-defined edges and domain boundaries. Along this line, the ordering of the

hydrogen bonded network increases drastically. While the bilayer comprising 50

molecules exhibits little ordering, the bilayer of 150 molecules is partitioned into

triangular domains (figure 2.20(a), middle).

These domains do not match the crystal structure of any known polymorph,

but are best described as arrays of dimer motifs in which pairs are connected

by two hydrogen bonds, whilst interactions of neighbouring dimers involve only

single hydrogen bonds. This however reflects an intermediate type of ordering

en route to the eventual crystal structure. With increasing aggregate size we

find the grain boundaries of dimer domains to preferentially arrange as chains of

hydrogen bonded trimers. Thus, the bulk (α polymorph) crystal structure, which

comprises alternating rows of pure d-d-d and l-l-l trimer motifs, nucleates at

the boundaries of well-ordered, yet transient domains. Growth and decay of the

domains is subject to fluctuations; however with increasing aggregate size we find

the trimer domains to grow at the cost of dimer domains. The underlying solid-

solid phase transition was identified as 60◦ molecular rotations within half of the

involved dimers to reconnect the hydrogen bonded network without altering the

total number of bonds (figure 2.20(a), see also ESI for a more detailed description

of dimer and trimer motif identification).

Later stages of nucleation: growth steps using building blocks

Based on the detailed analysis of aggregate evolution described above, we now

elucidate the possibility to use speed-up models that would allow for the study

of later stages of nuclei growth. An intuitive approach is to use small aggregates

as obtained from molecule-by-molecule association as building blocks. Clearly,

depending on the size of the used building blocks this approach introduces in-

creasing bias with respect to single molecule attachment. Comparing building

blocks of 10, 20 and 40 molecules, we identified the association of decamers as

suitable up-scaling, whilst the nucleation pathway increasingly deviates from the

route observed by molecule-by-molecule association when using larger building

blocks.

88


While the early stages of nucleation are surely based on the association of in-

dividual molecules, it should be revealed that stages of crystal formation likely

involve the incorporation of small aggregates or even the coalescence of several

nuclei. The suitable account of the latter types of process requires consideration

of appropriate statistics of building blocks of different sizes and is beyond the

scope of the present study. In what follows, the association of decamers should

instead be interpreted as a speed-up technique mimicking molecule-by-molecule

growth.

A characteristic simulation run based on docking decamers is illustrated in fig-

ure 2.21(a). Boosting simulation speed by a factor of ten we find all characteristics

of the early stages of micelle and single bilayer formation, as well as nucleation of

further bilayers, are reproduced. Moreover, this approach allows studying the for-

mation of aggregates counting 1000 molecules and thus provides a more extended

account of bilayer staggering as nestled loops leading to a compact habit (rather

than sheets).

While the formation of the hydrogen bonded network in accord with the molecu-

lar crystal structure was already observed in aggregates comprising≈ 150 molecules,

the analysis of ordering d- and l-enatiomers requires exploring later stages of ag-

gregate growth. In the bulk racemic crystal (for all known polymorphs), favourable

packing of the alkyl chains was identified as the driving force to the formation of

enantio-pure rows along [010], whilst d- and l-type rows alternate along the [100]

direction [18]. On the other hand, an entirely random arrangement of d- and l-

enatiomers within the same hydrogen bonded network is entropically favourable.

It is intuitive to expect the energetic favouring of the proper packing of alkyl

chains to develop only gradually with increasing size of trimer domains, whilst

the initial stages of nucleation still are dominated by the association of molecules

from random directions and should thus tend towards random d/l ordering.

This is indeed the case. In terms of the trimer motifs discussed above, random

D/L sequences imply only 2/8 = 25% enatio-pure trimers, whilst the final crystal

comprises 100% pure D-D-D and L-L-L trimers. Inspecting snapshots of aggre-

gates up to 1000 molecules we find only subtle preferences of enatio-pure trimers.

Indeed, a statistics over all trimer motifs leads to 26% and could be argued as not

significant to show a trend to enatio-pure rows. However, exploring the stability

of the different trimer motifs we found a much more pronounced picture. Trimer

89


motifs remaining intact upon association of 1,2 and 3 further molecules to the

aggregate were observed to be 33, 39 and 51% enantio-pure, respectively. For the

scale-up model, aggregate reorganization after the association of the 10-molecules

sized building blocks is even larger and here we find trimer stability exclusively

for the enantio-pure species (up to aggregate growth by 30 molecules).

2.5.4 Conclusion

Tackling crystal nucleation from molecular simulations remains an ongoing chal-

lenge. However, present simulation techniques allow elucidation of the evolution

of forming nuclei at the molecular level of detail, whilst going from dimers to

nanoparticles of more than 5 nm dimensions. While the obtained aggregates are

still too small to unravel the final ordering in terms of one of the polymorphs of

d/l norleucine, which show only subtle differences in ordering the alkyl chains,

the observed metamorphosis from micelles to bilayers to nested loops of staggered

bilayers demonstrates the profound mechanistic understanding accessible for the

early stages of crystal formation. Interestingly, initial ordering involves structural

motifs different from those of the final crystal. A solid-solid phase transition is

identified as a secondary nucleation step, and in the forming aggregates the two

competing domains are subject to considerable fluctuations. Our simulations show

the underlying propagation of the phase front is slightly biased in favour of order-

ing d- and l-type enantiomers in trimer domains, whilst the domains comprising

dimers show no enantio-selectivity. In other terms, we argue that the secondary

nucleation step is of crucial importance for understanding how the forming aggre-

gates evolves from a random D/L agglomerate to an ordered racemic crystal.

90

References

[1] Philipp Ectors, Patrick Duchstein, and Dirk Zahn. From oligomers

towards a racemic crystal: molecular simulation of dl-norleucine crystal nu-

cleation from solution. CrystEngComm, 2015.

[2] Angelo Gavezzotti. Are Crystal Structures Predictable? Acc. Chem.

Res., 27(10):309–314, October 1994.

[3] J. Christian Schon and Martin Jansen. Auf dem Weg zur Synthese-

planung in der Festkorperchemie: Vorhersage existenzfahiger Strukturkandi-

daten mit Verfahren zur globalen Strukturoptimierung. Angewandte Chemie,

108(12):1358–1377, June 1996.

[4] Sarah S. Price. Computed Crystal Energy Landscapes for Understanding

and Predicting Organic Crystal Structures and Polymorphism. Acc. Chem.

Res., 42(1):117–126, October 2008.




Physics, 104(24):9932–9947, June 1996.

[6] A. Gavezzotti, G. Filippini, J. Kroon, B. P. van Eijck,

and P. Klewinghaus. The Crystal Polymorphism of Tetrolic Acid

(CH3Cı£34CCOOH): A Molecular Dynamics Study of Precursors in Solu-

tion, and a Crystal Structure Generation. Chem. Eur. J., 3(6):893–899, June

1997.

[7] Peter G. Vekilov. Dense Liquid Precursor for the Nucleation of Ordered

Solid Phases from Solution. Crystal Growth & Design, 4(4):671–685, July

91


2004.

[8] Stefano Piana, Manijeh Reyhani, and Julian D. Gale. Simulat-

ing micrometre-scale crystal growth from solution. Nature, 438(7064):70–73,

November 2005.

[9] Wolfgang Lechner, Christoph Dellago, and Peter G. Bolhuis.

Role of the Prestructured Surface Cloud in Crystal Nucleation. Physical Re-

view Letters, 106(8):085701+, February 2011.

[10] Matteo Salvalaglio, Thomas Vetter, Federico Giberti, Marco

Mazzotti, and Michele Parrinello. Uncovering Molecular Details of

Urea Crystal Growth in the Presence of Additives. J. Am. Chem. Soc., 134

(41):17221–17233, October 2012.



Ed., 50(9):1996–2013, February 2011.




(2):024513+, January 2006.

[13] Sigrid C. Tuble, Jamshed Anwar, and Julian D. Gale. An Approach

to Developing a Force Field for Molecular Simulation of Martensitic Phase

Transitions between Phases with Subtle Differences in Energy and Structure.

J. Am. Chem. Soc., 126(1):396–405, January 2004.

[14] Dirk Zahn and Jamshed Anwar. Collective displacements in a molecular

crystal polymorphic transformation. RSC Adv., 3(31):12810–12815, 2013.

[15] Dirk Zahn and Jamshed Anwar. Size-Dependent Phase Stability of a

Molecular Nanocrystal: a Proxy for Investigating the Early Stages of Crys-

tallization. Chem. Eur. J., 17(40):11186–11192, September 2011.

92


[16] Junmei Wang, Romain M. Wolf, James W. Caldwell, Peter A.

Kollman, and David A. Case. Development and testing of a general amber

force field. J. Comput. Chem., 25(9):1157–1174, July 2004.

[17] Theodor Milek, Patrick Duchstein, Gotthard Seifert, and Dirk

Zahn. Motif Reconstruction in Clusters and Layers: Benchmarks for the

Kawska-Zahn Approach to Model Crystal Formation. ChemPhysChem, 11

(4):847–852, 2010.

[18] Carl H. Gorbitz, Kristian Vestli, and Roberto Orlando. A solu-

tion to the observed Z’ = 2 preference in the crystal structures of hydrophobic

amino acids. Acta Cryst. B, 65(3):393–400, June 2009.

93

3 Biomineralization

95

3.1 Modeling of Apatite-Collagen Composites from Hyperspace Simulations

3.1 Atomistic Modeling of Apatite-CollagenComposites from Molecular DynamicsSimulations Extended to Hyperspace

Patrick Duchstein, and Dirk Zahn

This section has previously been published:Title Atomistic modeling of apatite-collagen composites from

molecular dynamics simulations extended to hyperspaceAuthor Patrick Duchstein, and Dirk ZahnJournal Journal of Molecular Modeling

Publisher SpringerVolume 17

Issue 1Pages 73–79Date April 06, 2009DOI 10.1007/s00894-010-0707-7

With kind permission from Springer Science+Business Media: [1]. Copyright c©Springer-Verlag 2010. Reprinted with permission. Figures, tables, sections, andreferences have been renumbered.

Abstract The preparation of atomistic models of apatite- collagen composite

mimicking enamel at length scales in the range of 1 to 10 nanometers is out-

lined. This biocomposite is characterized by a peculiar interplay of the collagen

triplehelix and the apatite crystal structure. Structural coherence is however only

obtained after drastic rearrangements, namely the depletion of protein-protein hy-

drogen bonds and the incorporation of calcium triangles which are stabilized by

salt-bridges with the collagen molecule. Starting from an isolated collagen triple

helix and a single-crystalline apatite structure, a composite model is obtained by

gradually merging the two components viaan additional (hyperspace) coordinate.

This approach allows smooth structural relaxation of both components whilst

avoiding singularities in potential energy due to atomic overlap.

97

3 Biomineralization

3.1.1 Introduction

Crystal nucleation and growth, its structure and habitus as well as the resulting

materials properties are of fundamental interest in physics, chemistry and ma-

terials science, but also in a specific discipline of biology - the investigation of

biominerals. While nucleation processes and materials properties are well char-

acterized at the macroscopic and mesoscopic scale by a wealth of experimental

evidence, this does not apply to the microscopic scale, which is much more diffi-

cult to access from the experiment. For understanding mechanisms at the atomic

level of detail computer simulations have proven to be a very powerful tool. Molec-

ular dynamics simulations may easily achieve the atomistic resolution and hence

appear particularly suited for detailed mechanistic investigations.

For complex materials, there are immense obstacles to these else wise very

appealing perspectives of atomistic simulations. On the one hand, information of

the atomic structure is typically not accessible at full detail. Setting up simulation

models from intuition is a tricky task – and may severely bias the reliability of the

results. On the other hand, detailed structures may be obtained by exploring the

nucleation process itself, but this requires tackling the time-length scale problem

inherent to molecular dynamics simulations of crystal nucleation [2].

Here, we focus on collagen-hydroxyapatite composites of about 2 weight-% col-

lagen content mimicking the predominant part of human enamel. The importance

of this material leads to a large number of experimental and theoretical studies

[3–5]. In particular, we investigated the mechanisms of aggregate formation and

growth-control by collagen fibers at the atomistic scale [6, 7]. From this we iden-

tified triangles of calcium ions coordinating a fluoride ion in/near its center as a

peculiar motif of the apatite crystal structure which formation is promoted by ion

association to the biomolecule. Moreover, the collagen triple-helices also induce

orientation control to these motifs giving rise to the alignment of the triplehelices

along the c-axis of the apatite crystal [3, 7] (cf. figure 3.1).

The aim of the present work is to outline the design of scale-up models of

bio(mimetic) composites, based on building-rules that are derived from earlier

investigations of the nucleation of small aggregates and its interplay with the

organic component. While studies of ion-by-ion association to collagen are limited

to the very early stage of composite growth, i. e., aggregates of a few hundred ions,

98


Figure 3.1: Illustration of biomimetic apatite-collagen composites of about 2weight-% collagen. The collagen triplehelices are arranged parallelto the c-axis of the apatite crystal. To restraint system complexity, inthe present study a single-crystalline apatite model is used. Based onexperimental evidence, a more sophisticated model should consider anano-mosaic structure of apatite crystallites [4]

scale-up modeling can pave the way to much larger atomistic models. The latter

shall mimic the bulk composite at length scales up to tens of nanometers without

losing the atomistic level of detail.

3.1.2 Theory

The key difficulty of preparing an atomistic model of a collagen-apatite composite

is related to the structural relaxation of both components upon collagen incorpo-

ration into an apatite crystal. While the atomic structures of both, isolated col-

lagen triple-helices and the apatite crystal are well-known, our information about

the structure of collagen incorporated into apatite is much less detailed. Recent

molecular simulation studies revealed that collagen rear- ranges in order to best

incorporate calcium ions [8]. During the nucleation of apatite-collagen compos-

ites protein- protein hydrogen bonds are broken and replaced by even stronger

electrostatic interactions, namely salt-bridges stabilizing Ca3X (X = F−, OH−)

motifs [7]. Thus, apatite-collagen biominerals represent a prominent example for

99

3 Biomineralization

composites which atomistic structure shows important differences from that of the

isolated components.

This situation is somewhat similar to the often observed difference in protein

structure when comparing its configurations in vacuum and in aqueous solution.

Yet, the common procedure for preparing an atomistic model of a protein (or other

solute) in solution is based on the isolated systems, i. e. the protein structure after

relaxation in the vapor state and an equilibrated solvent box. When introducing

the solute, overlapping water molecules are simply cut. The resulting system

then needs structural relaxation to adopt to the solvent-solute interactions. In

a solution, this may easily be accomplished by constant-temperature, constant-

pressure molecular dynamics simulations.

Transferring this to molecule incorporation into a solid (here: collagen into ap-

atite) is in principle possible, yet the structural relaxation now becomes a critical

issue. Unlike in a liquid solvent, in the embedding solid ion/molecule diffusion is

drastically slower and typically elusive to conventional molecular dynamics simu-

lations. Moreover, the removal of charged moieties should be balanced in order to

maintain charge neutrality.

Here, we tackle these problems by using a combination of the approach out-

lined above and a method for gradually introducing solute-solvent interactions

by model extension to hyper-space coordinates. The latter procedure was origi-

nally invented by Roux and coworkers for performing free energy calculations via

thermodynamic integration in 4-dimensional space [9]. Therein, the extra dimen-

sion is used to first ‘separate’ solute and solvent, and then gradually merge both

systems by reducing the ‘displacement’ along the 4th dimension. Hence, there

are two physically meaningful states, i. e.displacement along the 4th dimension

η equal to zero (the solvated solute) and η being larger than the cutoff-distance

rcut of evaluating intermolecular forces. The latter state corresponds to two sepa-

rate simulation systems (solute and solvent fully separated). Within the interval

0 < rcut the solute-solvent interactions are artificially flattened as the underlying

atomic distances rij are calculated from:

rij =√

(xi − xj)2 + (yi − yj)2 + (zi − zj)2 + (ηi − ηj)2

100


whilst

ηij =

0 for solvent atoms

η for solute atoms

and

Fij =qiqj4πε0

(1

rij

)2

+ 12Aij

(1

rij

)13

− 6Bij

(1

rij

)7

Here qi, qj, Aij and Bij denote the partial charges and the van-der-Waals pa-

rameters, respectively. The forces between atoms i and j are functions of 1/rij and

would become infinite for atoms which overlap in 3-dimensional space. The ex-

tension to 4-dimensional space is a mathematical trick to avoid such singularities

and to flatten the interaction potentials in a controlled manner.

For solute incorporation into a liquid we only need to smoothly change η from

rcut to zero within a constant-temperature, constant-pressure molecular dynamics

run. For the introduction of a molecule into a crystal this procedure only allows

partial relaxation. Without the removal of ions from the crystal the atomic overlap

becomes critical as soon as η gets lower than the corresponding van-der-Waals

radius. Further reduction of η would lead to strong repulsive forces and eventually

the destruction of the crystal model. To avoid this, we suggest the following

simulation procedure (see figure 3.2):

1. Starting from η = rcut we perform constant-temperature, constant-pressure

molecular dynamics run in which η is gradually reduced. The average po-

tential energy is computed as a function of η and the simulation protocol is

switched to step 2 as soon as Epot(η) starts to increase.

2. Ions overlapping within a distance delimiter d with the incorporated molecule

are removed. Cations and anions are treated differently to maintain charge

neutrality of the total system. We use dcation = danion + ∆d with |∆d| always

being the smallest value that allows charge neutrality for a given danion. The

latter is a free parameter which is screened for optimal choices as described

in the following.

3. The additional coordinate η is then reduced to zero and the potential energy

Epot(η = 0) is calculated for a given parameter danion.

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3 Biomineralization

Figure 3.2: Workflow description of the model preparation. Two separate sys-tems, the apatite crystal and a collagen triplehelix, are linked via ahyperspace coordinate η. By gradually reducing η from infinity (orbetter the cut-off distance used for evaluating the atomic interactions)collagen-apatite interactions are introduced smoothly. While the ex-tension to hyperspace avoids singularities in potential energy, atomicoverlaps are increasingly penalized with decreasing η. To provide sys-tem stability and exothermic incorporation energy, unfavorable ionsare removed from the apatite model (see also figure 3.3).

102


4. The above procedure is performed for a series of distance delimiters danion

typically within 0 < danion < rvdw. For the resulting series of model systems

the incorporation energy Eincorp = Epot(η = 0)−Epot(η = rcut) is evaluated.

Here, Epot(η = rcut) refers to the truncated composite from which the col-

lagen molecules were cut (based on the optimized structure for η = 0). The

potential energy is taken from a single-point calculation without structural

relaxation. Finally, the configuration of maximum incorporation energy is

suggested as the most suitable atomistic model of the composite.

3.1.3 Simulation details

The atomic interactions were mimicked by empirical forcefields as described in

[7]. Ewald summation is used for the runs related to the investigation of the

final model only. During model preparation, shifted-force potentials with a cut-off

radius of 12A were used. The molecular dynamics simulations were performed in

the constant-temperature, constant-pressure ensemble applying room temperature

and ambient pressure. A time-step of 1 fs was used.

As a starting point, a 10 × 10 × 12 super cell of mono- clinic hydroxyapatite

was prepared using periodic boundary conditions. In analogy to our previous

study of apatite aggregation on collagen, the triplehelical backbone is mimicked

by a (Gly-Pro-Hyp)12 model. Radmer and Klein [10] and Persikov et al. [11]

established this simplified approximant as the most stable sequence occurring in

the collagen helical domain that captures all the significant features of its unique

characteristics.

The hyperspace coordinate was initially chosen as 12 A (the cut-off distance)

and then reduced at a rate of 0.1 A per picosecond. From comparing different

rates, this value was found to be sufficiently low to allow structural relaxation of

both sub-systems in case that the collagen is oriented approximately parallel to the

apatite c-axis prior to starting the molecular dynamics simulations. The distance

delimiter danion was scanned in steps of 0.01 A within the interval 1.0 A− 5.0 A.

For the evaluation of the bulk moduli of the single crystalline hydroxyapatite

and the composite model, we first performed structural relaxation runs at ambient

pressure and zero Kelvin. The volume was then varied from 0.99 to 1.01 times the

103

3 Biomineralization

equilibrium value, and the corresponding potential energy was interpolated by a

harmonic fit in order to assess the bulk modulus as the second derivative.

3.1.4 Results

A series of apatite-collagen composites were prepared using different distance de-

limiters danion for cutting ions which overlap with the collagen triple-helix. The

potential energy as a function of the hyperspace coordinate is shown for differ-

ent values of danion (figure 3.3). The solid line corresponds to danion = 0, i. e.,

to merging the collagen molecule and the apatite crystal without cutting ions.

The strong repulsive forces imply very unfavorable potential energy levels and

eventually cause the destruction of the crystalline order. The optimal composite

model is obtained for danion = 1.8 A (∆d = +0.5 A). Much lower, yet exother-

mic incorporation energies are obtained for larger distance delimiters for cutting

overlapping ions. As a limiting case this includes configurations in which the col-

lagen triple-helix is laterally attached to the surface of a large cavity in the apatite

crystal.

While collagen adsorption to an apatite surface implies only minor structural

changes and comparably low incorporation (or better association) energy, the

composite model obtained from our simulation procedure using optimal cutting

delimiters (danion = 1.8 A, dcation = 2.3 A) exhibits strong structural changes

which reflect an optimized interplay of both components forming the composite

as described in the following.

In the optimized composite model, the collagen triple-helix exhibits a parallel

orientation with respect to the c-axis of apatite (figure 3.4). Within the ab-

plane, the biomolecule is located next to a row of Ca3OH motifs which tend to

be enwrapped by the triple-helix. This arrangement shows remarkably small dis-

tortion of the apatite crystal. Indeed, the latter largely maintains its crystalline

order and structural changes are only observed in close proximity of the collagen

molecule. Such stable (both in terms of structural coherence and incorporation

energy) configurations are however only obtained for model preparation runs in

which the biomolecule is directed along the c-axis prior to the incorporation pro-

cess. Simulation runs for which collagen incorporation into apatite is attempted at

orientation which strongly differ from this alignment, failed to provide reasonable

104


Figure 3.3: Potential energy as a function of the hyperspace coordinate. The solidline corresponds to merging the collagen molecule and a single crys-talline apatite model, which implies atomic overlap and destruction ofthe crystal. By removing apatite ions (whilst maintaining charge neu-trality) collagen incorporation becomes favorable, and induces onlylocal structural changes of the apatite crystal. The dashed line re-flects the maximum incorporation energy which is obtained by settingthe distance criteria for avoiding atomic overlap to danion = 1.8 A anddcation = 2.3 A. As indicated by the dotted curve, for larger cut-off dis-tances the incorporation energy reduces considerably and eventuallyevolves to the energy of adsorption obtained for placing the collagentriple helix on the surface of an apatite crystal.

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3 Biomineralization

composite models (destruction of the apatite crystal). Whilst small misfit angles

were found to be corrected, excessive tilting would probably require extremely low

incorporation rates η to allow structural relaxation.

Focusing on the optimized apatite-collagen composite model as illustrated in

figure 3.4, an additional molecular dynamics run applying ambient pressure and a

cycle of gradual heating/cooling was performed. Starting from room temperature,

100 K increments were implemented every 25 ps. Upon heating up to 1000 K and

subsequent cooling to 0 K at the same rate, the composite model underwent

only minor structural changes hinting at the stability of the interplay of both

components.

A striking signature of this interplay is the dissociation of intramolecular hydro-

gen bonds of the biomolecule and their replacement by salt bridges which stabilize

Ca3OH motifs. Qualitatively, this was anticipated from our earlier investigations

of Ca3F motif incorporation into collagen based on (fluorapatite) aggregate growth

simulations [7]. Here, we can provide sufficient statistics to perform a quantita-

tive analysis. Figure 3.5 shows the radial distribution functions (rdf) of the H–O

distances within the collagen triplehelix. For the isolated triple helix, integration

of the rdf of the hydrogen atoms – carbonyl oxygen atoms indicate a coordina-

tion number of 0.8, i. e., in average about one hydrogen bond per carbonyl group.

These are almost entirely depleted in the course of apatite-collagen composite

formation. On the other hand, Ca2+–O bridges are formed, leading to a coordi-

nation of the incorporated calcium ions by about 3 oxygen atoms. Integration of

the Ca2+–O radial distribution functions yields 1.1 and 1.7 contacts to carbonyl

oxygen atoms and to the hydroxyl group of the Hyp residues, respectively. The

Hyp residues hence change from hydrogen bond donors to acceptors of Ca2+–O

bridges. This is implemented by local rearrangement of the Hyp side-chain, and

does not compromise the overall triplehelical structure of the collagen molecule.

For the Ca3OH motifs embedded in the collagen triple helix an occurrence

distribution of the tilt angle of the normal vectors of the calcium triangles with

respect to the c-axis of the apatite crystal was performed (see figure 3.6). From

this a tendency in favor of parallel alignment may be deduced which again is in

reasonable agreement to our previous simulation study related to the growth of

aggregates counting a few hundred ions [7].

106


(a)

Figure 3.4: Relaxed model of the apatite-collagen composite. The collagen triple-helix embeds a large number of triangles of calcium ions in/near whichcenter a hydroxyl ion is located. Hence, this motif of the apatite crystalis largely preserved. Defects and distortions of the apatite crystal areonly observed in close proximity of the biomolecule. This hints at thecoherent structural interplay of collagen and apatite (after relaxationof both components). In accordance to previous findings, we observedthe replacement of protein-protein hydrogen bonds and the incorpo-ration of calcium triangles which are stabilized by salt-bridges withthe collagen molecule. (a) viewing direction along [010], (b) viewingdirection along [001].

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3 Biomineralization

Figure 3.5: Upper panel: radial distribution functions of the H–O distances relatedto hydrogen bonds within the collagen triplehelix. For the isolatedtriple helix, integration of the rdf of the hydrogen atoms - carbonyloxygen atoms indicate a coordination number of 0.8. These hydrogenbonds are almost entirely depleted in the course of apatite-collagencomposite formation. Lower panel: the hydrogen bonds are replacedby Ca2+–O bridges leading to a coordination of the incorporated cal-cium ions by about 3 oxygen atoms (1.1 contacts to carbonyl oxygenatoms and 1.7 to the hydroxyl group of the Hyp residues) of the col-lagen molecule.

108


Figure 3.6: Occurrence distribution of the tilt angle (with respect to the c-axis ofapatite) of the normal vectors of Ca3OH triangles embedded in thecollagen triple helix. The tilt angles in the single crystalline matrixarising from thermal fluctuations take a maximum of a few degrees,only.

Collagen incorporation into apatite is attributed to a series of unique materials

properties, including hierarchical growth mechanisms and reduced brittleness [2].

The latter aspect may be illustrated from calculating the bulk moduli of pure

apatite and the composite model, respectively (table 3.1). Comparison with the

experiment shows qualitative agreement only. Surely, our single-crystalline simu-

lation models cannot provide quantitative agreement with the much more complex

hydroxyapatite polycrystals and enamel biominerals, respectively. From a quali-

tative point of view, the mechanism of hardness reduction of apatite by collagen

incorporation is nevertheless demonstrated.

3.1.5 Conclusions

While our simulation model surely cannot account for the whole complexity of

enamel, we nevertheless consider the presented simulation procedure as a reason-

able approach to the preparation of nanometer scale apatite-collagen composite

models. Indeed, our atomistic models mimic important aspects of enamel at

length scales in the range of 1 to 10 nanometers. On length scales of 1 nm and

below, we demonstrated the modeling of the peculiar interplay of the collagen

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3 Biomineralization

B0 / GPa (simulation model) B0 / GPa (experiment) [12]

HAP 101.2 (single crystal) 88.0 (synthetic polycrystal)HAP-collagencomposite

86.6 63.1 (bovine enamel)

relative change -14% -28%

Table 3.1: Comparison of the bulk moduli calculated from our simulation modelswith the available experimental data [12]. While the single-crystallinesimulation models cannot provide quantitative agreement with themuch more complex hydroxyapatite polycrystals and enamel samples,qualitatively the mechanism of hardness reduction of apatite by colla-gen incorporation is nevertheless demonstrated.

triplehelix and the apatite crystal structure. This includes orientation alignment

of both components and details of the atomistic structure such as the depletion of

protein-protein hydrogen bonds and the incorporation of calcium triangles which

are stabilized by salt-bridges to the collagen molecule. On the other hand, the

presented composite model accounts for dimensions of about 5 × 5 × 8 nm3 and

may hence serve as a starting point to exploring materials properties of the bulk

composite by means of atomistic simulations.

110

References

[1] Patrick Duchstein and Dirk Zahn. Atomistic modeling of apatite-

collagen composites from molecular dynamics simulations extended to hyper-

space. Journal of Molecular Modeling, April 2010.

[2] Dirk Zahn, Oliver Hochrein, Agnieszka Kawska, Gotthard

Seifert, Yuri Grin, Rudiger Kniep, and Stefano Leoni. Extending

the scope of ’in silico experiments’: Theoretical approaches for the investiga-

tion of reaction mechanisms, nucleation events and phase transitions. Science

and Technology of Advanced Materials, pages 434+, July 2007.

[3] J. C. Elliott. Structure and chemistry of the apatites and other calcium

orthophosphates. Elsevier, 1994. ISBN 9780444815828.

[4] Rudiger Kniep and Paul Simon. Fluorapatite-Gelatine-Nanocomposites:

Self-Organized Morphogenesis, Real Structure and Relations to Natural Hard

Materials. In Kensuke Naka, editor, Biomineralization I, volume 270 of

Topics in Current Chemistry, pages 73–125. Springer Berlin Heidelberg, 2007.

[5] Dirk Zahn, Oliver Hochrein, Agnieszka Kawska, Jurgen Brick-

mann, and Rudiger Kniep. Towards an atomistic understanding of apatite-

collagen biomaterials: linking molecular simulation studies of complex-,

crystal- and composite-formation to experimental findings. Journal of Ma-

terials Science, 42(21):8966–8973, 2007.





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45(12):1911–1915, March 2006.









2006.

[9] Regis Pomes, Elan Eisenmesser, Carol B. Post, and Benoıt Roux.

Calculating excess chemical potentials using dynamic simulations in the fourth

dimension. The Journal of Chemical Physics, 111(8):3387–3395, 1999.

[10] Randall J. Radmer and Teri E. Klein. Triple Helical Structure and

Stabilization of Collagen-like Molecules with 4(R)-Hydroxyproline in the Xaa

Position. Biophysical Journal, 90(2):578–588, January 2006.

[11] Anton V. Persikov, John A. M. Ramshaw, Alan Kirkpatrick, and

Barbara Brodsky. Amino Acid Propensities for the Collagen Triple-

Helix†. Biochemistry, 39(48):14960–14967, November 2000.

[12] Dale E. Grenoble, J. Lawrence Katz, Karl L. Dunn, Robert S.

Gilmore, and K. Linga Murty. The elastic properties of hard tissues and

apatites. J. Biomed. Mater. Res., 6(3):221–233, May 1972.

112

3.2 Function of Saccharides in Calcium Carbonate-Protein Biocomposites

3.2 On the Function of Saccharides during theNucleation of Calcium Carbonate-ProteinBiocomposites

Patrick Duchstein, Rudiger Kniep, and Dirk Zahn

This section has previously been published:Title On the Function of Saccharides during the Nucleation of Calcium

Carbonate–Protein BiocompositesAuthor Patrick Duchstein, Rudiger Kniep, and Dirk ZahnJournal Crystal Growth & Design

Publisher American Chemical SocietyVolume 13

Issue 11Pages 4885–4889Date September 13, 2013DOI 10.1021/cg401070h

Copyright c© 2013 American Chemical Society. Reprinted (adapted) with per-

mission from [1]. Figures, tables, sections, and references have been renumbered.

Abstract The molecular mechanism of calcium carbonate nucleation in the

presence of various types of collageneous proteins is unravelled from computer

simulation of ion-by-ion association steps. Single calcium ions are incorporated

in the triplehelix by formation of salt bridges to carbonyl and hydroxyl groups

of collagen, while single carbonate ions tend to bind laterally to the biomolecule.

However, upon multiple ion association, the self-organization of the forming ag-

gregate strongly depends on the triple-helical collagenous strand. In absence of

glycosylated lysine residues, we observed that carbonate ions bind to calcium ions

that are already incorporated into the triple helix and eventually cause the un-

folding of the protein. On the other hand, otolin-1, a specific, collagen-like protein

found in biogenic calcite-based composites such as otoconia, comprises a partic-

ularly high degree of glycosylated amino acids which avoid such “destructive”

calcium–carbonate contacts by providing alternative association sites more lateral

to the backbone. This leads to the formation of a saccharide–calcium carbonate

agglomerate that does not compromise the protein’s triple helix and constitutes

the organic–inorganic interface of the nucleating biocomposite.

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3 Biomineralization

3.2.1 Introduction

Bone, teeth, and otoconia (functional biominerals of the human body) are compos-

ite materials, comprising an inorganic phase which closely interacts with organic

tissue, in many cases collagen [2–6]. These fiber proteins are constituted by a

triple-helical backbone which, as a simple approximant, can be mimicked by a

(Hyp-Pro-Gly)n polypeptide. However, biogenic collagen obtained from biomin-

erals is known to comprise of more complex (X-Y-Gly) sequences giving rise to

an analogous triple helix but offering a variety of side groups along the protein

backbone. The most prominent feature of this more lateral part of collagen is

given by saccharide groups. From NMR experiments, saccharides have recently

been identified as the dominant part of the organic–inorganic interface in bone

and teeth [7]. While the explicit function of these residues remained unknown so

far, it is intuitive to assume that millions of years of evolution did not place the

saccharide groups coincidentally.

This idea is supported by the identification of different saccharide contents in

different types of biominerals. Glycosylation of collagen fibers takes place in the

endoplasmatic reticulum, before triple helices are formed. Lysine becomes hydrox-

ylated by lysyl hydroxylase, and then glycosylated by β-(1-0)-galactosyl- and α-

(1-2)-glucosyltransferase, resulting in α-d-glucopyranosyl-β-d-galactopyranosyl-

hydroxylysine (GGH) [8]. The degree of glycosylation of collagen fibers varies

with its type and location of synthesis. For example, in collagen type X, 65% of

the lysine residues become hydroxylated, and of those, 97% become glycosylated

[9]. While structurally very similar, otolin-1 contains an even larger content of

lysine residues and may thus be expected to exhibit a particularly high degree of

glycosylation. Otolin-1 is a collagen-like protein specific to calcite-based biomin-

erals such as otoconia (a biomineral being part of the acceleration sensors in the

inner ear).

While there is a large body of experimental evidence related to the importance

of saccharides for inducing ion aggregation, we lack mechanistic knowledge of how

this process and its interplay with the nucleation and growth of a hierarchical

composite actually works. To look into the molecular scale mechanisms, computer

simulations have proven a powerful tool of investigation. In what follows, we

build on simulation models and algorithms developed earlier and provide a direct

114


comparison of ion association to a small series of collagen models. Such molecular

models cannot account for the full complexity of the organic matrix, yet they allow

a focus on specific aspects of the biomolecules and are, thus, particularly suited

to elaborate detailed mechanistic insights.


To explore the role of saccharide side groups in collagen-based biominerals, we

transfer a recently presented molecular simulation protocol mimicking ion-by-ion

association and self-organization during apatite–collagen nucleation [10] to the

aggregation of calcium carbonate–collagen composites. In full analogy to Kawska

et al. [10], empirical potentials are used to describe the atomic interactions of

calcium carbonate [11], water [12], and the biomolecules [13, 14]. The molecu-

lar dynamics simulations are based on a recently developed atomistic simulation

scheme for investigating crystal growth from solution[15]. The Kawska-Zahn ap-

proach reflects an iterative procedure for tackling nucleation from very dilute

solutions mimicking ion diffusion to the aggregate by a docking procedure and in-

vestigating aggregate growth and relaxation from simulated annealing molecular

dynamics simulation runs after each ion association step [16]. On the basis of three

independent growth runs, we investigate the association of up to 200 ions to three

different collagen models of different complexity. Similar to previous simplifica-

tions of collagen models [10], the triple-helical backbone of human Otolin-1 and

collagen-X were mimicked by a subsequence, residues 176–208 of uniprot accession

no. A6NHN0 and 389–421 residues of P23206, respectively. The collagenous do-

main was assumed to be a parallel homo trimer and all lysine residues in position

Y of GLY|CYS–X–Y were replaced by GGH.

3.2.3 Results

To allow direct comparability to a previous study of apatite–collagen composite

nucleation, we initially adopted the simple (Gly-Pro-Hyp)n collagen model from

Kawska et al. [10]. Indeed, the association of a single calcium or carbonate ion

to this nonglycosylated collagen triple helix was found in full analogy to calcium

and phosphate ion association: the Ca2+ ions are incorporated into the helical

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3 Biomineralization

center, form salt-bridges with carbonyl and hydroxyl groups and thus stiffen the

triple-helical structure, as shown in figure 3.7. On the other hand, the larger PO3−4

and CO2−3 ions attach laterally to the helix.

A striking discrepancy of apatite-(Gly-Pro-Hyp)n and calcium carbonate-(Gly-

Pro-Hyp)n composite nucleation is observed upon further ion association. The

simulations related to apatite showed that the phosphate ions remain at lateral

positions, while the incorporation of [Ca3F] motifs into the collagen backbone

keeps the overall structure of the triple helix intact. In contrast to this successful

interplay of apatite motifs and biomolecular structure, the association of calcium

and carbonate ions leads to increasing unfolding of the triple helix. Calcium ions

that are incorporated in the (Gly-Pro-Hyp)n triple helix tend to drag carbonate

ions from the otherwise lateral association site. As a consequence, Cax(CO3)2x−2yy

aggregates are formed within the collagen backbone, thus pushing the peptide

strands apart. Figure 3.7b illustrates this process as observed for the CaCO3

ion pair (left), and increasingly dramatic, during the association of further ions

(middle) until the helical structure is finally disrupted (right).

Arguably, this finding could be related to the strong simplification of our colla-

gen model, relying on a (Gly-Pro-Hyp)n polypeptide approximant, rather than on

a collagen species actually present in the vestibular system [17]. We thus repeated

our aggregate growth simulations in order to investigate the interplay of otolin-1

with calcium and carbonate ion association. Indeed, for glycosylated otolin-1, the

picture is substantially different: figure 3.8 shows that the disaccharide chains and

the protonated amine groups of lysine offer particularly favorable association sites

for carbonate ions. Favorable hydrogen bonding provided lateral to the triple he-

lix acts as a selective shield for CO2−3 ion association. While single Ca2+ ions may

still be incorporated inside the triple helix, no ion pairs, triples, etc. are formed

within the “interior” of otolin-1, thus keeping the triple-helical structure intact.

The nucleation of Cax(CO3)2x−2yy aggregates instead occurs at the contact regions

of disaccharide side chains and the aqueous solution. Upon formation of larger

ion agglomerates, this leads to the intergrowth of (largely disordered) calcium

carbonate with the glyclosylated side chains of otolin. Figure 3.9 illustrates this

phenomenon for a later stage of aggregate growth (comprising 184 ions), clearly

indicating the intact triple-helical structure of the protein.

116


Figure 3.7: (a) Association of a single calcium (yellow, left) and a single carbonate(gray/green, right) ion to nonglycosylated collagen in aqueous solution(15.000 water molecules, not shown). The purple ribbons indicate thebackbone of the triple-helical peptide. (b) The association of severalcalcium (yellow) and carbonate (gray/green) ions to the nonglycosy-lated collagen leads to distortion and finally to the unfolding of thecollagen triple helix. For clarity, solvent molecules (15.000 H2O) arenot shown.

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3 Biomineralization

Figure 3.8: Calcium and carbonate ion association to the glycosylated otolinmodel in aqueous solution (solvent not shown). The inset at the rightshows the coordination of a carbonate ion by hydrogen bonding to anamino group (left) and to several saccharide residues (top, bottom, andright). This induces the formation of calcium carbonate clusters later-ally bound to otolin without compromising its triple-helical structure(see also figure 3.7 and figure 3.9).

118


Figure 3.9: Calcium carbonate aggregation promoted by glycosylated otolin. Thesame configuration is shown at two representations demonstrating theintact triple helix (left) and the intergrowth of ionic clusters and thesaccharide residues (right, highlighted in blue).

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3 Biomineralization

To support this mechanistic concept, we performed further aggregate growth

simulations using a nonglycosylated model of the otolin-1 sequence described

above. For this “control experiment”, it is indeed observed that calcium car-

bonate ion clusters form in between the collagen strands and finally disrupt the

triple helix in full analogy to the (Gly-Pro-Hyp)n polypeptide approximant, as

discussed earlier. Even stronger evidence was collected from comparing two in-

dependent simulation runs dedicated to calcium carbonate and hydroxy-apatite

aggregation to glycosylated collagen-X. The corresponding triple helix exhibits a

smaller degree of glycosylation (≈ 3% of all amino acids) than otolin (≈ 12% of

all amino acids). Strikingly, glycosylated collagen-X was found to induce calcium

phosphate association and the nucleation of apatite motifs in full analogy to the

(Gly-Pro-Hyp)n polypeptide approximant, while calcium carbonate aggregation

still leads to disintegration of the protein at late stages of aggregate growth (as

compared to nonglycosylated collagen). Accordingly, collagen glycosylation plays

a minor role for calcium phosphate-based composite formation but is of such criti-

cal relevance to carbonate association that a specialized collagen type, the heavily

glycosylated otolin, is needed for structural integer carbonate-based composites.

The comparison of the different mechanisms of apatite association to collagen

and calcium carbonate association to otolin furthermore hints at a different struc-

tural interplay. The hierarchical nature of apatite–collagen composites could be

rationalized by apatite motif orientation induced by collagen, namely, by correla-

tion of the crystallographic c axis of apatite and the long axis of the protein fiber

[10] and by the generation of intrinsic electric fields taking over control of further

hierarchical developments [18]. In contrast to this, the disaccharide side chains

of otolin provide much less structural ordering to the forming calcium carbonate

aggregates. The analysis of the distances of Ca2+–O (from CO2−3 ) contacts and co-

ordination numbers averaged over the full aggregate growth trajectory (figure 3.10)

reveals only weak ordering. Thus, the very first nucleation step can be described

as the formation of areas of largely disordered (“amorphous”) calcium carbonate,

a metastable state which is often discussed as a precursor phase for the formation

of calcium carbonate-based biominerals [19–21]. The actual formation of calcite,

which is expected at much later stages of aggregate growth, is not observed on the

basis our simulations, which are limited to the infancy of nucleation. However,

120


this does not rule out a putative role of the biomolecule during metastable–stable

phase transformations of more mature calcium carbonate states.

Figure 3.10: Occurence profile calculated for Ca–O distances within the (largelydisordered) ion clusters associated to otolin (solid curve) (cf. fig-ure 3.9). The dashed lines indicate the corresponding distances ascharacteristic for the calcite and aragonite crystal structures. Takinginto account Ca2+ contacts to oxygen atoms of CO2−

3 groups, saccha-ride hydroxy groups, and water molecules, the averaged coordinationnumber of the organic–inorganic interface is 6.97, while the coordina-tion numbers of calcium by oxygen in the crystal structures of calciteand aragonite are 6 and 7, respectively.

3.2.4 Conclusions

For the nucleation of calcium carbonate-based biominerals, the glycosylation of

collagen seems to be an important biological benefit. The simulation results clearly

show that saccharide groups attached to collagen triple helices not only strongly

interact with calcium and carbonate ions but also are necessary to guarantee the

structural integrity of the fibrous collagen macromolecules. In case the triple-

helices are not or only insufficiently glycosylated, the scenario of collagen disrup-

tion is the preferred reaction mechanism. With this, the first and basic insight is

given for an open problem which was provided already in 1996 [22], and which is

still pending [19], the function of polysaccharides in biomineralisation.

More recent investigations in biogenic and biomimetic otoconia (calcite-based

biominerals in the inner ear of vertebrates) revealed the presence of parallel ar-

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3 Biomineralization

rangements of undistorted (linear) fibrils of the organic component within the

composite system [23, 24]. The biomimetic otoconia have preferably been grown

by double-diffusion in gelatin-gel matrices, which, at first glance, seems to be con-

tradictory to our present calculations, showing the destructive force of calcium

carbonate during attachment at/in nonglycosylated collagen. However, commer-

cial gelatin still contains 0.5–1.0 wt.% of covalently bound saccharides [25], which

are not detached during gelatin processing from collagen and which help for cal-

cium carbonate nucleation without disruption of the triple-helices structure.

Despite the enormous complexity intrinsic to biogenic systems, molecular sim-

ulations dedicated to selected aspects of biomineral formation may provide mech-

anistic insights at a unique level of detail [16]. On this basis, at least a qualitative

understanding of composite nucleation, the development of hierarchical structures,

and the resulting materials properties are within reach [10, 26]. Here, we show

that disaccharide groups attached to collagen represent a crucial modification, or

better to say, functionalization of the biomolecule. In absence, or at insufficient

different degree of glycosylated lysine residues, collagen was observed to unfold

during calcium carbonate based composite formation, while otolin-1 comprises a

sufficient amount of “shielding” disaccharide groups to ensure structural integrity

of the triple-helix. Moreover, our simulations show that the organic–inorganic in-

terface of the forming composite is constituted by the intergrowth of glycosylated

collagen and largely disordered calcium carbonate.

122

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126

4 Zinc Oxide Nanoparticles

127

4.1 Molecular Mechanisms of ZnO Nanoparticle Dispersion in Solution

4.1 Molecular Mechanisms of ZnO NanoparticleDispersion in Solution: Modeling of SurfactantAssociation, Electrostatic Shielding and CounterIon Dynamics

Patrick Duchstein, Theodor Milek, and Dirk Zahn

This section has previously been published:Title Molecular Mechanisms of ZnO Nanoparticle Dispersion in Solution:

Modeling of Surfactant Association, Electrostatic Shielding andCounter Ion Dynamics

Author Patrick Duchstein, Theodor Milek, and Dirk ZahnJournal PLoS ONE

Issue 5Pages e0125872+Date May 11, 2015DOI 10.1371/journal.pone.0125872

Reprinted with permission. Images, tables, sections, and references have beenrenumbered.

Abstract Molecular models of 5 nm-sized ZnO/Zn(OH)2 core-shell nanopar-

ticles were derived as scale-up models (based on modelling of ion-by-ion aggre-

gation and self-organization, cf. Kawska et al. [1]) and subjected to mechanis-

tic analyses of surface stabilization by block-copolymers. The latter comprise

a poly-methacrylate chain accounting for strong surfactant association to the

nanoparticle by hydrogen bonding and salt-bridges. While dangling poly-ethylene

oxide chains provide only a limited degree of sterical hindering to nanoparticle

agglomeration, the key mechanism of surface stabilization is electrostatic shield-

ing arising from the acrylates and a halo of Na+ counter ions associated to the

nanoparticle. Molecular dynamics simulations reveal different solvent shells and

distance-dependent mobility of ions and solvent molecules. From this, we provide

a molecular rationale of effective particle size, net charge and polarizability of the

nanoparticles in solution.

129


4.1.1 Introduction

Nanoparticle stabilization and functionalization constitute key steps to nanoma-

terial production. To account for the hindering of nanoparticle agglomeration,

the role of stabilizing molecules has been characterized by two intuitive concepts:

the uptake of solutes may be avoided by i) sterical hinderance and ii) electrostatic

shielding. The same mechanistic concepts are commonly discussed to account

for stabilizing dispersions of nanoparticles in solution. Both mechanistic pictures

are well-established from experiments, e. g. by comparing particle size and hydro-

dynamic radius and by assessing the net charge of colloids from zeta-potentials

[2, 3].

In-depth understanding of the underlying processes on the molecular scale, that

is the mechanisms of surfactant association to the particle, the structuring of sol-

vent shells and the distribution of counter ions near a nanoparticle, is rather

sparse. While hard to access experimentally, molecular simulations in principle

appear very suited to tackle this issue. However, even when employing computa-

tionally efficient molecular models (rather than quantum chemistry approaches)

the limited time- and length scales still inherent to molecular simulations call for

smart simulation techniques. Such an approach was recently presented by Vlugt

and coworkers using the example of metal nanoparticles stabilzed by alkylthiols

[4, 5]. While this provided profound insights into nanoparticle dispersion on the

basis of purely sterical hindering, in the present work we report on molecular

simulations related to nanoparticle stabilization from the interplay of steric and

electrostatic shielding. For this purpose, we focus on zinc oxide nanoparticles in

ethanolic solution stabilized by partially charged block-copolymers, thus referring

to a widely used nanosystem [6].


Molecular simulations related to ZnO nanoparticles synthesized from the vapor

may be started from simply cutting hexagonal rods off the single crystal struc-

ture [7]. However, ZnO nucleation from solution typically leads to ZnO/Zn(OH)2

core/shell nanoparticles with more complex structures. To provide realistic models

we build on a recent simulation study related to zinc and hydroxide ion associa-

tion in ethanolic solution [1], ripening reactions and the nucleation of aggregates

130


comprising ≈ 100 ions. From this, a scale-up model of a ZnO/Zn(OH)2 core/shell

nanoparticle was prepared as described in detail as supplementary information.

The surfactant molecules were chosen as small polymers, namely poly(ethylene

oxide-block -methacrylate), (EO8-b-MAA8)8−. While the polyethylene oxide chain

is uncharged, each of the acrylate units exhibits a –1 charge, thus resulting in

a net charge of –8 per surfactant molecule. The MAA8 block is chosen for its

good binding affinity to ZnO, whilst the EO8 block shows poor affinity to the

nanoparticle (and is thus expected to be dangling into the solution).

Molecular dynamics simulations were performed on the basis of empirical in-

teraction potentials adopted from earlier studies of ZnO nucleation from solution

[1] and bismuth oxide association to polyacrylate [8] (cf. supplementary informa-

tion). The simulation temperature and pressure were fixed to 300 K and 1 atm,

respectively. A time-step of 1 fs was applied throughout the molecular dynamics

simulations. In dependent simulation runs, three colloid models were explored

in ethanolic solution, mimicked by a cubic cell of (≈ 10 nm)3 comprising 8,442

ethanol molecules and subjected to periodic boundary conditions.

The association of surfactant molecules to our nanocrystal models was treated

in full analogy to the uptake of ions to a forming aggregate as reported earlier

[1, 9, 10]. Transferring the Kawska-Zahn approach to surface functionalization we

thus mimic surfactant diffusion to the nanoparticle implicitly by applying random

incoming vectors and random surfactant molecule orientation. Placing surfactant

molecules in 1.5 nm distance from the nanoparticle, we then explore molecular

docking from detailed molecular simulation. This is first performed in absence of

solvent molecules (keeping the nanoparticle fixed) to obtain a putative association

structure from simple energy minimization. The nanoparticle-surfactant system

is immersed into ethanolic solution and propagated by simulated annealing runs

to allow the overall system (including solvent and Na+ counter ions) to relax. The

sodium ions are initially placed randomly. Subsequently, their positions are prop-

agated by performing nanosecond-scale molecular dynamics simulations to ensure

convergence of the ionic distribution in the simulation cell. Three independent

runs of this kind were performed to provide at least qualitative account for the

abundance of possible nanoparticle-surfactant structures possible even for a single

ZnO/Zn(OH)2 nanorod model.

131


4.1.3 Results

The resulting colloid models differ in terms of surfactant association sites and the

overall number of (EO8-b-MAA8)8− polymers that stably bind to the ZnO/Zn(OH)2

nanoparticle. Surfactant association is dominated by the acrylate units which

form a COO−–Zn2+ salt bridge and up to 1–2 hydrogen bonds per methylacrylate

monomer. On the other hand, the PEO chains were found to not bind to the

ZnO surface, but fluctuate considerably and are best described as tangling into

the embedding solution (figure 4.1). While association to the faces normal to the

polar axis of the ZnO crystallite is only slightly preferred to binding laterally,

we however observe a strong favoring of the top, i. e.(0001) over the bottom face

indicative of selective binding of the negatively charged surfactants to (partially)

compensate the dipole moment of the ZnO nanoparticle [11]. Statistics for binding

to top, bottom and lateral faces are given for all model systems in table 4.1.

Model Ratio of surface EO8-b-MAA8 bound to Normalized ratio

areas {0001}{1010} (0001) (0001) {1010} n{0001}

0.705·n{1010}

I 0.705 3 0 7 1.2II 0.705 5 1 4 4.3III 0.705 4 2 9 1.9

Table 4.1: Ratio of surface areas normal to the polar axis of ZnO with respectto lateral faces and absolute numbers of surfactants bound to the top,bottom and lateral faces. For comparison of association to the top andto lateral faces, the numbers of surfactants were put in relation to thecorresponding surface areas.

The counter ions (Na+) were placed in the solution and allowed to diffuse freely.

While ion distribution was found to reach a dynamic equilibrium after a few tens

of nanoseconds, additional 100 ns runs were used for the analyses described in

the following. While the number of sodium ions was chosen to fully compen-

sate the colloid charges, only a fraction form stable COO−–Na+ salt bridges and

thus ‘behave’ akin to the surfactant molecules. The remaining sodium ions were

found to be loosely associated and are best described as a dynamic equilibrium of

Na+ association to surfactant molecules, migration within a nm-sized halo around

the nanoparticle and dispersion into the bulk solution (figure 4.1). To quantify

132


Figure 4.1: Illustration of the colloid model II highlighting the pathways of Na+

counterions (yellow) within 0.5 ns. While the solvent is not shown forclarity, the effective colloid dimensions including the halo is illustratedas a surrounding surface. Atom colors: Zn (cyan), O (red), H (white),Na (yellow). Polymer colors: EO (pink), MA (green).

133


this phenomenon we calculated the Na+ diffusion coefficient as a function of the

ion-colloid distance. Figure 4.2(a) shows the continuous increase of ionic mobil-

ity within a distance of ≈ 1.8 nm and a plateau region at farer distance from

the colloid. While the latter corresponds to sodium ions not associated to the

nanoparticle, Na+ that bind to the nanoparticle or its surfactants molecules ex-

hibit a diffusion constant of zero and may thus be considered as surfactants (hence

reducing the net charge of the colloid drastically). In-between, that is from 0

to 1.8 nm distance, Na+ ions are loosely associated and constitute a positively

charged halo which also reduces the effective charge of the colloid.

(a) (b)

Figure 4.2: (a) Diffusion coefficient (blue) of the Na+ ions as a function of thedistance to the colloid and fitted switching function (black) used forquantification of the degree of ion association to the colloid (modelII). (b) same as left, but additionally illustrating the number of Na+

ions (red curve). The vertical blue line denotes the surface distancedelimiter discriminating halo and bulk counter ions. The horizontalred lines point out the number of counter ions directly associated withthe particle (58.7), and the total number of Na+ ions including thehalo (74.9). All data was averaged over 100 ns. See supplementary foranalogous plots of models I and III.

We suggest to use the distance dependence of ionic mobility for quantification

of the degree of Na+ association to the nanoparticle. For this purpose, we fitted

the diffusion constants to a switching function fswitch(r):

D(r) = Dbulk solution · fswitch(r) = Dbulk solution · tanh(a · r) (4.1)

134


where the use of a tangents hyperbolicus as the functional form of the switching

function allows accurate fitting using a single parameter a. The same switching

function is applied to attribute a weighted charge to the effective charge of the

colloid:

qNa+ effectivehalo = +1 · {1− fswitch(r)} (4.2)

Qeffectivecolloid = Qsurfactants +

∑i

qNa+ effectivehalo (ri) (4.3)

This allows a reasonable estimate of the effective colloid charge including contri-

butions from its halo of loosely associated counter ions (figure 4.2(b)). Moreover,

we suggest the fitted switching functions for assessing the effective radius of the

colloids in solution. As the switching function reaches 1 only for infinite distance,

we propose 0.95 as delimiter for discriminating the halo from the bulk solution.

This value is somewhat arbitrary, and error margins of ±0.1 nm result from con-

sidering alternative switching function delimiters of 0.9 or 0.975, respectively. For

all three models investigated, the effective radii and net charges are reported in

table 4.2.

Model Polymers Na+ 〈Na+〉 d(halo) Reff Qeffcolloid 〈p〉 α

bound bound in halo in A in A in e in eA in eA2V−1

I 10 50 17.5 19.7 50.5 -25.9 240.1 3.67 · 105

II 11 59 16.2 18.4 49.1 -25.7 233.6 4.70 · 105

III 15 90 15.0 17.8 48.4 -14.0 275.8 5.09 · 105

Table 4.2: Number of block-copolymers and sodium ions bound as surfactants tothe nanoparticles, number of charge carriers and thickness of the haloof counterions, effective radius and colloid charge, net dipole momentand polarizability as assessed for all model systems investigated.

The above considerations refer to the predominant colloid-colloid interactions,

i. e.Coulomb repulsion because of the negative net charges. However, the mobil-

ity of the sodium ions within the ≈ 1.8 nm sized halo around the nanoparticle

also allows for a molecular scale account of the van-der-Waals interactions. Us-

ing the effective charge model denoted in equation (4.3), we calculated the dipole

moments of the colloids in solution as functions of time. The corresponding occur-

rence profiles show (i) a net dipole moment stemming from ZnO, and only partially

135


compensated by hydroxylation and inhomogeneous distributions of the surfactants

and counterions, and (ii) considerable fluctuations of the overall dipole moment

which is attributed to fluctuations within the halo of Na+ countercharges (fig-

ure 4.3). Finding (ii) reflects the variability of the dipole moment ∆p = p(t)−〈p〉that arises from the distribution of mobile counterions in the solvation halo and

thus reflects a general phenomenon of any charged colloid systems. Without ac-

tually applying an external electric field, our simulations allow the calculation of

colloid polarizability α by relating the energy of the dipole field ∆p stemming

from polarizing the halo of counterions to Boltzmann statistics employed for the

occurrence profile h(∆p) as illustrated in figure 4.3. This leads to:

h(∆p) = h0 · e−(p−〈p〉)2

2σ = h0 · e−

12α−1(∆p)2

kBT (4.4)

(a) (b)

Figure 4.3: (a) Occurrence profile of the dipole moment as sampled from the sol-vated colloid model II. The width of the Gaussian fit (black curve)is used to estimate the polarizability of the halo of counterions. Seesupplementary for analogous plots of models I and III. (b) Dipole mo-ment of the ZnO/Zn(OH)2 core-shell nanoparticle (red curve) and thecolloid including surfactants and the halo of counterions (blue curve)as functions of time. In agreement with ab initio calculations relatedto bulk ZnO crystal we observe that surface passivation with hydrox-ides already reduces the dipole moment by a factor of almost 10 [10].Surfactants and counterions lead to further reduction by about 50%.

For all of the three colloid solutions investigated, we find a nice Gaussian type

occurrence profile of the dipole moment and were thus able to assess the polar-

136


izability α at good accuracy (figure 4.3(a)). Strikingly, colloid polarizability is

constant within very small margins, whilst the values of the effective charge and

the dipole moments differ considerably within our set of models (table 4.2). This

may be rationalized by the roughly equal thickness and charge density of the halo

of counterions, each comprising 15–18 Na+ ions. As a simple estimate for trans-

ferring our findings to different sizes of ZnO nanoparticles, it is thus intuitive to

assume that the halo thickness remains constant for different nanoparticle sizes,

while the number of mobile ions in the halo and thus polarizability should scale

linearly with the particle surface area.

4.1.4 Conclusion

In conclusion we elucidated the atomic mechanisms of ZnO nanoparticle stabi-

lization by charged surfactants and its interplay with Na+ counterions from the

embedding ethanolic solution. A large fraction of the sodium ions tends to bind

to the nanoparticle-surfactant molecules system, and may hence be considered

as surfactants as well. However, within a 1.8 nm sized halo around the colloid,

we identified loosely associated Na+ which account for a polarizable ‘cloud’ of

charges. Assessment of the underlying polarizability from molecular simulation –

and thus molecular scale insights into van-der-Waals interactions of the colloids

– is outlined from inexpensive sampling of the spontaneous fluctuations of the

dipole moment.

While our simulations were based on a limited set of colloid models and one type

of solvent (ethanol) only, from a qualitative viewpoint it is nevertheless intuitive

to suggest the observed mechanisms to be of general relevance for rationalizing

electrostatic nanoparticle shielding. We would be enthusiastic to offer our ZnO-

polymer data for comparison with erperiments – or to provide similar modelling

work for other systems.

137

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139

5 Summary and conclusion

5.1 Summary

Within this work, we have developed a series of methods for simulation and anal-

ysis of materials, along with specific applications. Ranging from simulations of

crystal nucleation studies to bulk composite materials to functionalized colloidal

particles, we also present a set of innovative methods for examining atomistic

structures, as well as materials properties.

In section 2.1 we provided a novel method suited to the investigation of trajecto-

ries and structures resulting from complex molecular simulations. This approach

was based on the geometric hashing paradigm, and able to identify custom motifs

in atomistic structures and trajectories. Since the target scenarios in most cases

do not exhibit perfect lattices, a distortion tolerance criterion ε was introduced.

The method greatly simplifies the analysis procedure of structures and trajectories

amounting to gigabytes of molecular simulation results, by providing qualitative,

as well as quantitative insights into structural relationships. Many of the reported

insights would not be accessible by means of other quantitative analysis methods,

let alone visual inspection.

As a proof of concept, we demonstrated that this approach can successfully be

applied to identify grain boundaries in polycrystalline aluminum, and the evolu-

tion of rock salt motifs of varying orientations during the crystallization of NaCl

from the melt. Furthermore, we applied our approach to the nucleation of copper

clusters (see section 2.2), the formation of zinc oxide nuclei (see section 2.3), as

well as benzamide (see section 2.4), and dl-Norleucine (see section 2.5).

While the method was perfectly suited for the systems under investigation, and

was an important part of the computer-aided analysis delivering quantitative in-

sights, it is still based on human intuition of which motifs to search for. However,

141


more sophisticated methods, allowing for the automatic identify recurring struc-

tural relationships, would be of use. Multidimensional analysis methods could

have a high potential of overcoming this shortcoming in the future.

After investigating “smaller” systems, we proceeded to more complex materi-

als. Chapter 3 deals with biomineralization, and the formation of biogenous com-

posites. In section 3.1, we described the setup of realistic bulk apatite–collagen

nanocomposites. This composite model is highly suited for further simulation

studies, and has already been applied by Dirk Zahn and Eric Bitzek to investigate

elastic and plastic deformation under tensile load.

To achieve a plausible model, we invented a new method, involving an addi-

tional (virtual) “hyperspace” coordinate for the computation of pairwise atomic

distances. This approach allows atoms to temporarily overlap in real (three-

dimensional) space, without exceeding interatomic potentials, or even causing

singularities. It therefore makes it possible to interweave two compounds, and

thus obtain a plausible composite model. The method exhibits high potential to

be further developed, since it allows “tunneling” energy barriers on a very short

timescale, and therefore, acts as an efficient sampling strategy. Possible fields of

future applications are protein-ligand docking, as well as protein folding.

While further exploring the field of biominerals, we shifted from calcium phosphate-

collagen biocomposites to calcium carbonate-collagen composites (see section 3.2).

By applying the Kawska-Zahn simulation method, we shed light onto a set of

posttranslational modifications of collagen, including the glycosylation of collagen

triple helices. We showed how the structural integrity of collagen fibers can be

destroyed by carbonate ion incorporation, and how disaccharide side-chains “pro-

tect” the structure. For the first time, we suggested a possible purpose of collagen

glycosylation in vivo, which had continuously been requested in the literature.

While at first it may sound reasonable to take our results for granted, one cannot

do so, because biological systems are by far more complex than our abstracted

models. More substances could be involved in the nucleation process, and molecu-

lar simulation in particular cannot cover them all. Since literature on the specific

compound is sparse, more experimental expertise is needed as a solid basis for

further understanding of the system, and to provide evidence for the results.

Going from biomimetic to man-made nanocomposites, we investigated physic-

ochemical properties of colloidal zinc oxide nanoparticles (see section 4.1). We

142

5.2 Conclusion

provide molecular scale insights into nanoparticle stabilization by electrostatic

shielding, using surfactants and counter ions. Moreover, a methodology was de-

veloped which gives access to the effective charge and radius, along with dipole

moments, and the polarizability of a functionalized particle by means of molecu-

lar dynamics simulations. The insights gained exhibit high potential for further

theoretical studies. Future investigations could cover the scenario of two (or even

more) particles, and how they interact with each other, mutually influencing their

dipole moments and the electrical double layer before particle-particle contacts

are established.

5.2 Conclusion

Atomistic simulations form an integral component of contemporary materials re-

search. Experimental investigations on the atomistic scale are often a challenging

task, and are typically not well suited for time resolved analyses. Simulations may

provide insights on the composition, arrangement, and formation of materials con-

stituents, and may even supply detailed information on underlying mechanisms—

ranging from the synthesis up to mechanic and electronic properties. Furthermore,

theoretical models derived from simulation can help predict dedicated materials’

properties, opening up new pathways for synthesis planning.

Covering a very broad field of different topics, materials, and their intrinsic

properties, we made a scientific contribution to the field of materials simulations,

ranging from the mechanisms of crystal nucleation and composite formation to

the invention of novel analysis methods. Considerable developments in both algo-

rithms and hardware have made computer simulations valuable tools to comple-

ment experiments, and to shed light on molecular scale mechanisms, helping in

the rational design of materials. The variety of methods invented within the scope

of this thesis, ranging from preparation to analysis of materials, shows the broad

applicability of atomistic simulations, as well as fields of application beyond pure

dynamics simulations.

While much further progress is needed to make molecular simulations a rigor-

ous route to tailor-made materials, the presented algorithms for model creation,

simulation and analysis are suggested as useful steps in this direction.

143

5.3 Deutsche Zusammenfassung

5.3 Deutsche Zusammenfassung

Die vorliegende Arbeit beschaftigt sich mit Methoden zur Simulation und Analy-

se von Materialien. Schwerpunkte liegen bei der Nukleation von Kristallen, dem

atomaren Aufbau von Kompositfestkorpern, sowie funktionalisierten kolloidalen

Nanopartikeln. Weiterhin stellen wir innovative Ansatze zur Analyse von ato-

mistischen Strukturen aus Simulationen, sowie Methoden zur Bestimmung von

Materialeigenschaften, vor.

Die Auswertung von Strukturen auf atomarer Skala aus Simulationen sowie Tra-

jektorien stellt mitunter eine große Herausforderung dar. Multivariate Analysever-

fahren ermoglichen hierbei, automatisiert relevante Informationen zu extrahieren,

welche mit bloßem Auge nicht sofort sichtbar sind. In Abschnitt 2.1 wird eine neue

Methode zur Analyse von Simulationsdaten vorgestellt. Diese ist in der Lage, unter

Berucksichtigung eines Toleranzfaktors beliebige Motive aus atomistischen Struk-

turen zu extrahieren. Insbesondere bei der Simulation von Nukleationsprozessen

wird so ermoglicht, die Entstehung von Kristallisationskeimen qualitativ sowie

quantitativ zu belegen.

Die von uns entwickelte Methode fand in mehreren Szenarien erfolgreich Anwen-

dung: um Korngrenzen in polykristallinem Aluminium zu finden (Abschnitt 2.1),

zur Analyse einer Kristallisation von Kochsalz aus der Schmelze (Abschnitt 2.1),

bei der Nukleation von Kupferclustern aus der Gasphase (Abschnitt 2.2), Zink-

oxid aus ethanolischer Losung (Abschnitt 2.3), sowie um die Entstehung von Mo-

lekulkristallen von Benzamid (Abschnitt 2.4) und dl-Norleucin (Abschnitt 2.5)

zu untersuchen.

Eine Einschrankung der Methode besteht jedoch hinsichtlich der gesuchten Mo-

tive: die sinnvolle Auswahl derselben bleibt der Erfahrung und Intuition des jewei-

ligen Anwenders uberlassen. Um die Analyse zu erleichtern ware es wunschenswert,

eine automatisierte Identifikation von”interessanten“ Motiven zu implementieren.

Multivariate Verfahren konnen hierbei eine Rolle spielen.

Darauf aufbauend wurden weitere Studien mit deutlich komplexeren Materi-

alverbindungen durchgefuhrt. In Kapitel 3 wird auf die Biomineralisation, die

Entstehung von biologisch relevanten Kompositmaterialien, und das Zusammen-

wirken derselben, eingegangen. In Abschnitt 3.1 wird die Erstellung eines realisti-

schen Apatit-Kollagen-Nanokomposit-Modells vorgestellt. Dieses modellierte Bio-

145


komposit fand bereits in einer weiteren Studie von Zahn und Bitzek erfolgreich

Anwendung, welche sich mit den elastischen und plastischen Verformungseigen-

schaften unter mechanischer Belastung auseinandersetzt.

Fur die Modellierung wurde eine neuartige Simulationsmethode implementiert,

welche eine zusatzliche (virtuelle)”Hyperraum“-Koordinate zur Berechnung der

paarweisen Atomabstande verwendet. Dies ermoglicht ein zeitweises”Uberlappen“

von Atomstrukturen im dreidimensionalen Raum, ohne die zugehorigen Paarpo-

tentiale zu verletzen. Die Methode ermoglicht, zwei Materialien miteinander zu

verflechten, und somit ein realistisches Modell eines Verbundwerkstoffs zu erhal-

ten. Die Methode besitzt ein hohes Potential fur weitere Anwendungsgebiete, da

sie gestattet, hohe Energiebarrieren in sehr kurzer Simulationszeit zu uberwinden,

und somit ein effektives Sampling moglich macht. Zukunftige Anwendungsfelder

sind das Protein-Ligand-Docking, sowie die Proteinfaltung.

In weitergehenden Simulationen zur Biomineralisation widmeten wir uns der

Nukleation von Calciumcarbonat-Kollagen-Kompositen (Abschnitt 3.2). Unter Zu-

hilfenahme eines von Kawska und Zahn entwickelten Aggregationsalgorithmus

wurden posttranslationale Modifikationen entlang eines Kollagenstrangs unter-

sucht, insbesondere die Glykosylierung von Lysin-Aminosauren. Es zeigte sich,

wie die tripelhelikale Struktur vom Kollagen durch Carbonat-Ionen zerstort wird,

und wie angehangte Disaccharid-Gruppen mittels Abschirmung zum Erhalt der

Struktur beitragen. Erstmals wurde ein moglicher biologischer Nutzen der Glyko-

sylierung von Kollagenfasern vorgeschlagen – in der Literatur fanden sich bislang

keinerlei Hinweise auf einen moglichen Zweck derselben. Biologische Systeme sind

jedoch in der Regel deutlich komplexer als die hier betrachteten abstrahierten Mo-

delle, und beinhalten viele weitere unbekannte Einflussfaktoren sowie Substanzen;

in der Simulation lasst sich diese Vielschichtigkeit kaum abbilden, vereinfachte Mo-

delle mussen fur einen Erkenntnisgewinn herhalten. Insbesondere aufgrund der

geringen Literaturdichte zum behandelten Thema waren weitere experimentelle

Untersuchungen wunschenswert, um die Ergebnisse zu validieren und auszubau-

en.

Das letzte Kapitel fuhrte die Simulationen von biomimetischen hin zu kunstlich

hergestellten Nanokompositen (Abschnitt 4.1). Hier wurden physikochemische Ei-

genschaften von kolloidalen Zinkoxid-Nanopartikeln untersucht. Neue Erkennt-

nisse uber die Stabilisierung von Nanopartikeln mittels elektrostatischer Abschir-

146

5.4 Schlusswort

mung von oberflachenaktiven Substanzen und Gegenionen wurden gewonnen. Wei-

terhin entwickelten wir eine Methode zur Ermittlung von effektiver Ladung und

effektivem Radius von Kolloiden, sowie der Polarisierbarkeit eines funktionalisier-

ten Partikels. Die so gewonnenen Einsichten konnen als Ausgangspunkt fur wei-

tere theoretische Untersuchungen verwendet werden. Eine mogliche Erweiterung

besteht in der Modellierung von zwei oder mehr Partikeln in einem System, deren

Dipolmomente sowie elektrochemische Doppelschicht sich gegenseitig beeinflussen,

bevor es zur Ausbildung von direkten Partikel-Partikel-Kontakten kommt.

5.4 Schlusswort

Detaillierte atomistische Simulationen sind aus der modernen Materialforschung

nicht mehr wegzudenken. Sie bieten Einblicke in die Zusammensetzung und in-

nere Ordnung und konnen Erklarungen zugrundeliegender Mechanismen liefern

– angefangen von der Synthese bis hin zu mechanischen oder elektronischen Ei-

genschaften. Solcherlei Erkenntnisse lassen sich, wenn uberhaupt, allein durch ex-

perimentelle Untersuchungen oft nur mit hohem Aufwand erzielen. Weitergehend

lassen sich mit Hilfe von aus der Simulation gewonnenen theoretischen Modellen

Vorhersagen uber dedizierte Materialeigenschaften treffen. Dadurch eroffnen sich

neue Perspektiven fur die Syntheseplanung.

In der vorliegenden Arbeit wurde auf ein breites Spektrum von Materialien

und Materialeigenschaften eingegangen, vielschichtige Erkenntnisse wurden ge-

wonnen. Viele verschiedene Modellsysteme wurden unter die Lupe genommen,

und ein aktiver wissenschaftlicher Beitrag zur modernen theoretischen Material-

forschung geleistet. Weiterentwicklungen von Algorithmen und Hardware bieten

einen Erkenntnisgewinn durch Simulationen, welcher es ermoglicht, experimentelle

Ergebnisse auf molekularer Ebene zu validieren und zu erganzen, und das geziel-

te Materialdesign nachhaltig zu beeinflussen. Die Vielfalt der im Rahmen dieser

Arbeit entwickelten Methoden zur Erstellung und Analyse von Werkstoffmodellen

zeigt, wozu atomistische Simulationen in der Lage sind und wo Anwendungsfelder

jenseits der klassischen Dynamiksimulationen liegen.

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Sicherlich sind noch viele weitere zukunftige Entwicklungsschritte notwendig bis

die Simulation einen unverzichtbaren Teil zur Gestaltung maßgeschneiderter Ma-

terialien darstellen wird – die hier vorgestellten Algorithmen zur Modellerstellung,

-simulation und -analyse konnen hierfur eine gute Grundlage liefern.

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