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A Thesis

entitled

Molecular Dynamics Simulations of Dodecanethiol Coated Gold Nanoparticles on

Organic Liquid Toluene

Nitun Nirjhar Poddar

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the

Master of Science Degree in Engineering

______Dr. Jacques Amar, Committee Chair

______Dr. Mohammed Niamat, Committee Member

______Dr. Gursel Serpen, Committee Member

______Dr. Patricia R. Komuniecki, Dean College of Graduate Studies

The University of Toledo

December 2013

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of Molecular Dynamics Simulations of Dodecanethiol Coated Gold Nanoparticles on

Organic Liquid Toluene

Nitun Nirjhar Poddar

Submitted to the Graduate Faculty as partial fulfillment of the requirements for Master of Science Degree in Engineering

The University of Toledo December 2013

Colloidal gold nanoparticles may be used in a variety of applications ranging from solar cells to sensors to catalysis and drug delivery. In particular, ordered gold nanoparticle thin-films have been proposed as efficient coupling layers which may be used to maximize the efficiency of photovoltaic solar cells. In addition, recently a drop- drying method to self-assemble a well-ordered monolayer of Au nanoparticles has been developed. In this method, a monodisperse solution of gold nanoparticles whose size has been selected via chemical synthesis, and which are coated with organic ligands to prevent aggregation and precipitation is used. Using this method a monolayer of Au nanoparticles can be made that covers the full area of a 2” silicon wafer. However, the interaction between the gold nanoparticles and the interface, which plays a key role in determining the film-quality, is not well understood.

One of the main objectives of the work presented in this thesis is to obtain a better fundamental understanding of the structure and interactions of ligand-coated Au nanoparticles at a solvent-vapor interface. In particular, motivated by experimental and simulation results, we have carried out extensive molecular dynamics simulations of dodecanethiol (DDT)-coated Au nanoparticles in both bulk toluene and at the toluene- iii vapor interface. Our simulations were carried out using the public domain software

LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) along with the

OPLS (Optimized Potential for Liquid Simulation) force field which is a force-field which has been specifically developed for simulating organic liquids.

Our simulations indicate that, in contrast to previous simulations of decanethiol and octadecanethiol coated Au nanoparticles in water, the dodecanethiol-coated nanoparticle (NP) sits relatively low in the toluene, such that the Au core never penetrates above the interface, while the thiols only partially stick out above the interface.

Somewhat surprisingly, we also found that the nanoparticle is not spherical at the interface, but instead the nanoparticle width is larger than the nanoparticle height. Both of these results may be explained by the fact that the nanoparticle ligands are attracted to toluene. Due to this attraction, the ligands above the NP core, which stick out above the interface, are more compressed than those on the side or bottom which extend into liquid toluene. These results also indicate that the nanoparticle rolling mechanism, which has been observed in other experiments and which may greatly enhance the diffusivity of Au nanoparticles during the island self-assembly process, does not apply in this case.

In addition to determining the nanoparticle position and shape at the interface, we have also determined the interfacial adsorption energy for a nanoparticle at the interface.

The resulting relatively large value for the binding energy was found to be consistent with experimental results for nanoparticle island-growth with high excess thiol concentration, for which a diffusion length (corresponding to the distance a nanoparticle can diffuse before leaving the interface and returning to solution due to thermal fluctuations) much larger than the inter-island spacing was observed. By comparing the

iv radial toluene density near the nanoparticle at the interface with that near the nanoparticle in bulk, and also taking into account the particle position with respect to the interface, the van der Waals corrections to the binding energy were also calculated. Since diffusion plays an important role in in drop-drying, we have also calculated the diffusion coefficient for DDT-coated Au NP at the toluene-vapor interface. Somewhat surprisingly, we find that - perhaps due to the fact that the nanoparticle is almost fully submerged - the two-dimensional diffusion coefficient at the toluene-vapor interface is in good agreement with the Stokes-Einstein prediction for 3D bulk diffusion.

To my dearest dad for all his dreams.

Acknowledgements

I am very grateful to Prof. Jacques G. Amar for his patient guidance and generous help as my advisor during my graduate studies here at the University of Toledo. He has been a great mentor and has always guided me throughout my work. I could not have finished this project without his help.

I express my sincere thanks to our group members Dr. Yunsic Shim and Bradley

Hubartt, who helped me solve specific problems during the simulations and results analysis.

I deeply appreciate my committee members Dr. Gursel Serpen, Dr. Mohammed

Niamat, and Dr. Jacques G. Amar for attending my defense.

I am really grateful to my family who has always been there to encourage and support me during my studies and work here at the University of Toledo.

Table of Contents

Abstract ...... iii

Acknowledgements ...... vi

Table of Contents ...... vii

List of Figures...... x

1 Introduction ...... 1

1.1 Scope of thesis ...... 5

2 Molecular Dynamics ...... 7

2.1 Basic idea of molecular dynamics ...... 7

2.2 Constant temperature (NVT) molecular dynamics...... 8

2.2.1 Velocity-Verlet Algorithm...... 9

2.2.2 Molecular dynamics ensembles...... 11

2.2.2.1 Microcanonical ensemble (NVE): ...... 11

2.2.2.2 Extended Langrangian Formalism/Adding Stochastic forces and

velocities (NVT) ...... 12

2.2.2.3 Isothermal–isobaric ensemble...... 13

2.3 LAMMPS molecular dynamics simulator ...... 14

2.3.1 Input script ...... 14

2.3.2 Settings...... 15

vii

2.4 VMD (Visual Molecular Dynamics) ...... 19

2.5 Periodic Boundary Conditions ...... 20

2.6 Minimum Image Convention ...... 22

2.7 Molecular Dynamics Potentials ...... 23

2.7.1 Lennard-Jones Potential ...... 23

2.7.2 OPLS force field potential ...... 26

2.7.2.1 OPLS Functional form: ...... 27

2.7.2.2 Derivation of the parameters ...... 28

2.7.2.2.1 Bonds and angles ...... 28

2.7.2.2.2 Dihedrals ...... 29

2.7.2.2.3 Charges ...... 31

2.7.2.2.4 Van der Waals parameters ...... 31

3 Methods ...... 33

3.1 Previous work on nanoparticle-solvent interactions ...... 33

3.2 Determining the position of dodecanethiol-coated Au nanoparticles at toluene-

air interface ...... 34

3.2.1 Creation of dodecanethiol-coated nanoparticle (NP) ...... 35

3.2.2 Creation of toluene liquid ...... 38

3.2.3 Positioning the nanoparticle above liquid toluene ...... 45

3.2.4 Nanoparticle lowered into Toluene ...... 46

3.3 Oscillation of nanoparticle at the interface ...... 50

3.3.1 How Z was calculated ...... 51

3.4 Calculating adsorption energy ...... 60

viii

3.5 Calculating surface tension ...... 61

3.6 .XYZ and .mol2 File format ...... 64

3.7 Radial toluene density distribution r) ...... 67

4 Results...... 70

4.1 Position of nanoparticle at interface ...... 70

4.2 Adsorption energy of Nanoparticle at interface...... 80

4.3.1 Results for Adsorption energy ...... 80

4.3.2 Surface tension for bulk toluene and bulk dodecanethiol ...... 83

4.3.3 Radial Toluene density and van der Waals correction...... 85

4.3.4 Nanoparticle diffusion at interface ...... 90

5 Conclusion and Future Works ...... 92

References...... 97

List of Figures

1-1: Au nanoparticles in toluene solutions ...... 2

1-2: Schematic showing drop-drying process in which islands nucleate and self-assemble at the toluene-air interface as the droplet dries...... 3

1-3: Scanning-electron-microscope picture of self-assembled Au nanoparticle film made via drop-drying of 6 nm Au nanoparticles coated with dodecanethiol and suspended in toluene...... 4

2-1: VMD simulation and setup ...... 20

2-2: Periodic images of a simulation box in 2 d...... 21

2-3: Interaction between closest pairs...... 23

2-4: Lennard-Jones potential curve ...... 25

2-5: Schematic showing bond and angle interactions between atoms i,j and i,j,k respectively along with corresponding force constants...... 29

2-6: Diagram showing dihedral and improper interactions between atoms i,j,k,l including corresponding force constants ...... 30

3-1: Equilibrated polymer-coated alkanethiol NPs of diameter (from top to bottom) 2.0,

4.0, and 8.0 nm at the water-vapor interface for S-(CH2)9-CH3, S-(CH2)9-COOH, S-

(CH2)17-CH3, and S-(CH2)17-COOH (from left to right). (From Ref. 8) ...... 34

3-2 : Buckminster fullerenes: (a) C28. (b) C32. (c) C50. (d) C60. (e) C70 ...... 35

3-3: Picture of buckyball C540 configuration...... 36

3-4: Single dodecanethiol ligand ...... 37

3-5 : Perspective view of nanoparticle with dodecanethiol-coated ligands...... 38

3-6: Toluene molecule as created in our simulation ...... 39

3-7: Toluene lattice as created in our simulation...... 41

3-8: Picture of 5.6nm cubic toluene liquid box...... 42

3-9 : Toluene liquid created after replication process...... 43

3-10: Liquid-vapor interface for liquid toluene...... 44

3-11: Dodecanethiol-coated Au nanoparticles submerged in toluene solution ...... 49

3-12: Plot of Z vs time ...... 51

3-13: Density profile of toluene with the nanoparticle included ...... 54

3-14: Density profile of toluene with the nanoparticle excluded ...... 55

3-15: Similar triangle technique used to calculate the interface ...... 59

3-16 : Simulation system to calculate surface tension of toluene ...... 62

3-17 : Simulation system to calculate surface tension of dedecanethiol ...... 63

4-1: (a) Picture of NP equilibrated at toluene-vapor interface (b) Same NP with toluenes removed for clarity...... 71

4-2: Center of mass of the nanoparticle as a function of time ...... 72

4-3: Distance of center of mass of nanoparticle from interface vs time ...... 73

4-4 : Distance of highest sulfur of nanoparticle from interface vs time ...... 74

4-5 (a): Nanoparticle at its lowest point in toluene (b) Nanoparticle at its highest point in toluene after equilibration...... 75

4-6: Oscillation of the nanoparticle at toluene-vapor interface after 6.5 ns of equilibration

...... 76

4-7 : Nanoparticle submerged in bulk toluene...... 76

4-8 (a): Dodecanethiol ligand stretched out the most in our simulation...... 80

4-8 (b): Dodecanethiol ligand compressed the most in our simulation ...... 80

4-9 : Potential energy plot of nanoparticle submerged in bulk liquid toluene ...... 81

4-10 : Potential energy as function of time for a nanoparticle at the toluene-vapor interface...... 83

4-11: Calculated radial toluene densities (r) relative to bulk toluene density 0 obtained from MD simulations of dodecanethiol-coated 3 nm core radius Au NP in bulk toluene

(solid curve) and adsorbed at toluene-vapor interface (dashed curve). Distance of center of adsorbed NP from interface is z = 3.5 nm...... 87

4-12 : Mean-square displacement of NP as function of time. Slope of fit is 3.18 x 10-5

A2/fs...... 90

xii

Chapter 1

Introduction

Colloidal gold nanoparticles have been utilized for centuries by artists due to the vibrant colors produced by their interaction with visible light. More recently, due to their optical and electronic properties - which may be tuned by adjusting their size, shape, and or surface properties – colloidal gold nanoparticles have also been used in a variety of other applications including photovoltaic solar cells, sensors, catalysis, and drug delivery.

Of particular interest are their optical properties which are due to the existence of surface plasmons and which may be adjusted to maximize the optical absorption over a given range of frequencies and wavelengths. For example, due to the strong coupling between gold nanoparticles and light, gold nanoparticle thin-films have been proposed as efficient coupling layers which may be used to maximize the efficiency of photovoltaic solar cells.

Recently, a drop-drying method to self-assemble a well-ordered monolayer of Au nanoparticles has been developed1. In this method, a monodisperse solution of gold nanoparticles(such as shown in Fig. 1-1 below), whose size has been selected via chemical synthesis, and which are coated with organic ligands such as dodecanethiol to prevent aggregation and precipitation, is used.

During this process, no sophisticated instruments are required except a substrate, a micropipette, and a gold nanoparticle solution. In addition, using this method a monolayer of Au nanoparticles can be made that covers the full area of a 2” silicon wafer.

As part of the interfacial self-assembly mechanism, solvents evaporate rapidly and the gold nanoparticles adsorb at the interface due to the surfactant-like behavior of the

Figure 1-1: Au nanoparticles in toluene solutions nanoparticles which reduces the surface tension of the interface. After the droplet has completely dried and a well-ordered monolayer has been deposited, the ligands may then be removed chemically, leaving a well-ordered film of Au nanoparticles with unique optical and electronic properties.

Fig. 1-2 shows a schematic of the self-assembly mechanism process. As can be seen, as the droplet dries Au nanoparticles are swept up and adsorbed at the toluene-air interface, and then diffuse and nucleate islands of toluene (see middle panel on right) which then continue to diffuse and grow (as nanoparticles attach to their sides) to form a complete monolayer as the droplet continues to evaporate.

Self-Assembly Mechanism time

Evaporation 2D nucleation and growth

Toluene

Substrate

As the solvent evaporates, nanoparticles arrive at the liquid-air interface. Nanoparticles stick to the interface due to a surfactant-like effect. 2D assembly follows.

4 Bigioni et al. Nature Materials 2006, 5, 265–270

Figure 1-2: Schematic showing drop-drying process in which islands nucleate and self- assemble at the toluene-air interface as the droplet dries.

The resulting compact film typically exhibits long-range order, is void free and exhibits very few defects. As already noted, this process can be carried out with different size, and even composition of nanoparticles and with different ligands attached to the nanoparticles, although in the case considered here dodecanethiol ligands attached to a

Au core are considered. In general, the core of the nanoparticle determines the physical properties of the nanoparticle while the shell determines the chemical properties. Fig. 1-3 below shows an example of the extremely well-ordered films which may be produced using this method. The sharp hexagonally-arranged peaks in the Fast Fourier Transform

(FFT) of the nanoparticle positions (see lower inset) also strongly demonstrate the existence of long-range order in these films.

Fig. 1-3: Scanning-electron-microscope picture of self-assembled Au nanoparticle film made via drop-drying of 6 nm Au nanoparticles coated with dodecanethiol and suspended in toluene.

As a result of the strong plasmonic properties of the resulting film after the dodecanethiol has been chemically removed, self-assembled Au nanoparticle films have been proposed as an “impedance-matching” layer to enhance the coupling of the incident electromagnetic field to the semiconductor film in photovoltaic devices. In addition, interfacial colloidal self-assembly promises to enable fast, inexpensive and facile patterning of nanoscale objects far exceeding the limits of conventional lithography. This will have significant nanotechnological impact in diverse fields such as ultra-thin film coatings, catalysis, optoelectronics, sensors, and ultra-high density magnetic storage.

Furthermore, the advancement of a fundamental theory of nanoparticle interactions (both with the interface and with each other) will create opportunities to develop new strategies for assembling novel nanocomposite thin films and structures from a wide range of technologically important materials. 4

Accordingly, it is important to improve our understanding of the entire process of self-assembly during drop-drying including the processes of single nanoparticle

(monomer) adsorption and island adsorption, island diffusion and growth, and monolayer self-assembly since a more complete understanding of these underlying processes may allow the development of improved methods for the formation of well-ordered and technologically useful nanoparticle thin films. As a first contribution to a more detailed understanding of these processes, in this thesis we present the results of molecular dynamics simulations of dodecanethiol-coated Au nanoparticles at the toluene-vapor interface, which have been carried out in order to improve our understanding of the interactions between single nanoparticles (monomers) and the toluene-air interface, as well as of the structure and position of the adsorbed nanoparticles.

1.1 Scope of thesis

An outline of the organization and topics discussed in this thesis are as follows:

In Chapter 2, we present an introduction to the methods of molecular dynamics simulation, along with concepts of periodic boundary conditions and minimum image convention. Details of the velocity-Verlet integration algorithm, and different ensemble types such as NVT and NVE are described. In addition, we discuss the LAMMPs simulation software, as well as the Lennard-Jones and OPLS potentials used in our simulation.

In Chapter 3, we provide details of our molecular dynamics setup using

LAMMPS in our simulation. In particular, we discuss how liquid toluene and dodecanethiol-coated Au nanoparticle were simulated, along with the methods used to 5 find the position of the nanoparticle with respect to the toluene-vapor interface, as well as to calculate the binding energy, diffusion coefficient, and surface tension.

In Chapter 4, we present our simulation results. In particular, results for the position and dynamics of the nanoparticle at the toluene-vapor interface, along with the shape of the nanoparticle at the toluene-vapor interface and in bulk toluene are presented.

Our results for the nanoparticle adsorption energy, surface tension and diffusion coefficient are also presented.

In Chapter 5, we summarize our conclusions, and discuss possible future work.

Chapter 2

Molecular Dynamics

2.1 Basic idea of molecular dynamics

The last decade has seen a rapid advancement in the speed of computers. Thanks to Moore’s law which states that the number of transistors on an integrated circuit doubles every two years, the power of computers has surged in the last decade. This improvement in computer technology has been taken advantage of by scientists. In physics, computer simulations are being carried out in order to test theories, and experiments. Simulation is thus the middle ground between theory and experiment.

Computer simulations thus work by taking an abstract model of a particular system, and simulating it in order to estimate the performance of systems too complex for analytical solutions.

Molecular dynamics is a computer simulation technique in which the evolution of a set of interacting atoms can be followed by integrating the equations of motion. Using this method, the laws of classical mechanics, such as Newton’s equations, are followed for each atom i in a system of N atoms as shown below:

Eq. 2-1

Here, m is the mass of the atom, a is the acceleration given by d2r/dt2, and F is the force acting on it due to its interaction with the other atoms. The forces that act on the atoms are generally obtained either from classical interatomic potentials or from quantum mechanical calculations. These two approaches are called classical and quantum molecular dynamics.

Molecular dynamics allows the microscopic properties of a system to be directly studied through simulations. Thus, by integrating the equations of motion, molecular dynamics can generate information at the molecular or microscopic level including atom positions and velocities in a material. The conversion of this microscopic information to macroscopic data such as pressure, temperature or energy requires statistical mechanics.

The design of a molecular dynamics simulation should account for the available computational power. Simulation size such as the number of particles, timestep, and the total time duration must be selected so that the calculation can finish within a reasonable time. However, the simulation must be long enough to be relevant to the time scales of the natural process being studied in order to make statistically valid conclusions from the simulation.

2.2 Constant temperature (NVT) molecular dynamics

While molecular dynamics simulations are often carried out at constant particle number

N, system volume V, and energy E (micro-canonical ensemble) in many cases it is desirable to carry out molecular dynamics simulations at constant temperature T rather

8 than at constant energy. For instance, many typical laboratory experiments are carried out at constant temperature and constant pressure. The temperature T can be related to the average of the kinetic energy as follows:

Eq.2-2

-5 Here pi (mi) is the momentum (mass) of atom i, while k = 8.617 x 10 eV/K is

Boltzmann’s constant. In conventional constant-energy molecular dynamics, the temperature can only be found by carrying out the simulations and calculating the average kinetic energy. To resolve this situation, constant temperature and constant volume (NVT) simulation methods have been developed.

2.2.1 Velocity-Verlet Algorithm

The Verlet algorithm is a numerical method used to integrate Newton’s equations of motion, and calculate trajectories of particles in molecular dynamics simulations. The algorithm has been rediscovered many times, with the most recent discovery by Verlet in

1960 for molecular dynamics. Using the algorithm, the evolution of the position and velocities of the atoms in a system may be calculated as follows:

 Step 1 – Evaluate current force on particle i.

 Step 2 – Compute r at new time using the equation below :

Eq. 2-3(a)

 Step 3 – Calculate the new accelerations ai (t + Δt) from the interaction

potential based on the particle’s new position xi(t + Δt) 9

 Step 4 – Using old and new accelerations, calculate new velocities as

shown in equation below :

Eq. 2-3(b)

 Step 5 – Update acceleration, velocity, and positions to its new value from what

was computed in this step.

Each of the above steps is performed sequentially for each atom in the system during each iteration. After finishing the last step, the simulation time t is incremented by the timestep Δt. However, since a typical vibration period is 10-12 seconds or 1 picosecond, and for numerical accuracy the time Δt between calculations of a set of atomic coordinates/velocities must be significantly shorter than this, in MD simulations of solid materials the typical timestep is the order of 10-15 seconds or 1 femtosecond.

During a molecular dynamics (MD) simulation, the force acting on each of the particles due to its neighbors must be calculated at each timestep. Thus, an MD simulation uses a large portion of its computational time finding the atomic neighbors in order to calculate the forces on each atom. As a result, this technique is extremely expensive computationally and serial (single-processor) MD simulations are usually applied to small systems of up to a few tens-of-thousands of atoms and for short periods of time up to a few nanoseconds. Despite these drawbacks, molecular dynamics simulations continue to be very useful and popular since –

a) The method is very easy to implement.

b) If the potential is correct, the simulation results will be very correct.

c) It can study very short length- and time-scales which cannot be easily studied

experimentally.

2.2.2 Molecular dynamics ensembles

An ensemble consists of a large number of virtual copies of a system considered all at once, and each represents a possible state the real system might be in. There exist many different types of ensembles :

2.2.2.1 Microcanonical ensemble (NVE):

This is a thermodynamic state where there are a fixed number of atoms, N, a fixed volume, V, and a fixed energy, E. All copies or distinct microstates of the system have the same number of atoms, the same volume and fixed energy. The benefit of the ensemble is that it allows one to calculate average values for thermodynamic properties.

Thermodynamically, a microcanonical ensemble corresponds to an isolated system with constant energy. As a consequence, the total energy in the system does not fluctuate, and the system can only access those molecular states that correspond to that value of E. The total number of microscopic states corresponding to a certain value of the system energy is called the degeneracy of the system and is denoted by

Eq. 2-4

A real system is not isolated, and is at a temperature determined by its surroundings. As discussed, in the NVE ensemble, the temperature of a system may be calculated using equation 2-4 above.

Each value of the energy E in the microcanonical ensemble corresponds to a different temperature and so to calculate the dynamics of the system at a given temperature, the energy needs to be set correctly. To set the temperature, we need to consider the interaction of a system with the outside world at temperature T.

2.2.2.2 Extended Langrangian Formalism/Adding Stochastic forces and velocities

(NVT)

Simulations at constant temperature are very useful as it allows one to study behavior of systems at different temperatures. There are various methods to perform a constant temperature MD simulation. As the temperature of a system is related to the kinetic energy of the particles, one can control the temperature by controlling the particle velocities. In particular, in one method the velocities are rescaled each timestep according to v‘ = v. One such thermostat is the Berendsen thermostat. Here, the velocity scaling parameter  is given by

Eq. 2-5 where Δt is the time step, T is the current temperature, T0 is the desired temperature and

is a time constant. Another method is to constrain the velocities by a Gaussian constraint method. In a third method the temperature is held constant by a heat bath. In this method, the velocity of a randomly selected particle is replaced by one chosen from the Maxwell-Boltzmann distribution. This is the same as collision with a particle in an imaginary heat bath similar to the Langevin thermostat. In this case, the interaction between the simulated system and the heat bath is modeled by an interchange of energy between them.

2.2.2.3 Isothermal–isobaric ensemble

The isothermal-isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature T and constant pressure P. It is also called the NPT ensemble, where the number of particles N is also kept constant. This ensemble plays an important role in chemistry and chemical reactions that are performed under constant pressure conditions. The partition function can be written as the weighted sum of the partition function of canonical ensemble, Z(N,V,T) .

Eq. 2-6 where β = 1 / kBT (kB is the Boltzmann constant), and V is the volume of the system.

There are several candidates for the normalization factor C, e.g., C = N/V, or C = βP.

These choices make the partition function a nondimensional quantity. The differences vanish in the thermodynamic limit, i.e in the limit of infinite number of particles. The characteristic state function of this ensemble is the Gibbs free energy,

Eq. 2-7

This themodynamic potential is related to the Helmholtz free energy (logarithm of the canonical partition function), F, in the following way:

G = F + PV

Eq. 2-8

2.3 LAMMPS molecular dynamics simulator

The simulations carried out as part of this thesis were run using the LAMMPS molecular dynamics simulator. LAMMPS stand for Large-scale Atomic Molecular

Massively Parallel Simulator, and is an open-source software written in C++ and developed at Sandia National Laboratory. LAMMPs works by accepting an input file of commands, and then using these commands to carry out the simulation. Often, data files are required by LAMMPS in order to read in the atom configuration set up by the user as restart files output by LAMMPS. Files output by LAMMPs can be analyzed through object oriented languages such as Python and C scripts in order to study physical properties of interest. LAMMPs also outputs files such as .xyz files which can be visualized through VMD or Jmol. Researchers widely use LAMMPS in order to study atomic, polyatomic, biological, metallic, or granular microscopic systems.

2.3.1 Input script

A sample LAMMPS input script of the type used in our simulations is shown below: units real atom_style full boundary p p p processors 4 4 3 pair_style lj/cut/coul/long 10.0 10.0 bond_style harmonic angle_style harmonic #class2 dihedral_style opls print "Between dihedrals and read data" read_restart restart.5100000 kspace_style pppm 1e-6 special_bonds lj/coul 0.0 0.0 0.5

14 group toluene type 1 2 3 4 group rest type 1 2 3 4 6 7 8 velocity gold set 0.0 0.0 -0.00005 sum no units box velocity all create 300.0 4928459 mom yes rot yes dist gaussian balance z 0.2567 0.4456 fix 1 rest nvt temp 300.0 300.0 100.0 fix 2 gold rigid single neighbor 2.0 bin neigh_modify delay 0 every 1 check yes timestep 1 thermo_style custom step temp press vol pe ke etotal thermo 50 dump 1 all xyz 3 movie.test.*.xyz dump 9 all custom 4000 center.custom4400-4500k_nve.unwrapped id type xu yu zu dump_modify 9 sort 1 restart 100000 restart.* compute cofm np com compute toldipole toluene property/atom q z variable toldipoleProd atom c_toldipole[1]*c_toldipole[2] compute toldipoleZSum all reduce sum v_toldipoleProd variable tolueneDipoleZValue equal c_toldipoleZSum/26973 thermo 50 run 1000

2.3.2 Settings

The LAMMPS input script starts with the “style” of atoms to be used in the simulation. This determines what attributes are associated with the atoms. This command is to be used before the simulation setup through the “read_data” or “create_box” commands. Once a style is assigned, it cannot be changed. The choice of style also 15 affects what quantities are stored by each atom, and what quantities are communicated between processors. The atom style used in our simulation was “atomic” which is used for coarse-grained liquids, or solids, or metals. This style defines point particles, while the mass of the particle needs to be assigned on a per particle basis.

Next, the force fields to be used are defined in the input script. For instance, the pair_style command sets the formula LAMMPS uses to compute pairwise interactions. In

LAMMPS, pair potentials are defined between pairs of atoms that are within a cutoff distance, and the set of active interactions. In our simulation, we used “lj/cut/coul/long” as the pairwise potential. The potential computes the standard 12/6 Lennard-Jones potential using the equation below:

Eq. 2-9 where r < rc is the distance between a pair of atoms and rc is the cutoff.

This style also includes a Coulombic pairwise interaction (due to the existence of a net charge on each atom) given by the equation below:

Eq. 2-10

In the above equation, r is the distance between the pair of atoms, C is an energy- conversion constant, qi and qj are the charges on the two atoms, and  is the dielectric constant which can be set by the dielectric command. Since the Coulomb interaction is a long-range interaction, an additional damping factor is applied to the Coulombic term so it can be used in conjunction with the “kspace”_style command and its “pppm” option

16 which are used to efficiently calculate the Coulomb interactions between charges in different periodic images of the system. The Coulombic cutoff specified for this style means that pairwise interactions within this distance are computed directly, while interactions outside that distance are computed in reciprocal space.

The bond_style command determines the formula LAMMPS uses to compute bond interactions between pairs of atoms. In LAMMPS, a bond differs from a pairwise interaction in that bonds are defined between specified pairs of atoms and remain in force for the duration of the simulation. The list of bonded atoms is read in by a read_data or read_restart command from a data or restart file. The coefficients associated with a bond style can be specified in a data or restart file via the “bond_coeff” command. In our simulation, we used the “harmonic” bond style. The harmonic bond style uses the following potential :

Eq. 2-11 where r0 is the equilibrium bond distance.

The angle_style command sets the formula LAMMPS uses to compute interactions between triplets of atoms , which remain in force for the entire duration of the simulation. The list of angle triplets are read in via a “read_data” or “read_restart” command from a user-created data file, or binary restart file. The angle style used in this simulation was harmonic. The following equation is used to calculate the potential in this case:

Eq. 2-12

17 where 0 is the equilibrium value of the angle, and K is a prefactor.

The dihedral_style command determines the formula LAMMPS uses to compute dihedral interactions between quadruplets of atoms, which remain in force for the duration of the simulation. Similar to angle styles, the list of dihedral quadruplets are read in by a “read_data” or “read_restart” command from a data or restart file. The coefficients associated with a dihedral style can be specified in a data or restart file or via the dihedral coeff command. In our simulations, we used the dihedral style “opls”. The equation below is used to compute the potential in this case:

Eq. 2-13

In the above equation, the angle  is the angle between the two sets of planes ABC and

BCD defined by each atom quadruplet ABCD, while the four constants K1, K2, K3, K4 were entered from the restart file used by LAMMPS.

We also used kspace_style which defines a long-range solver for LAMMPS to use in order to compute long-range Coulombic interactions. The pppm style we used creates a particle-particle particle-mesh solver which maps the atom charge to a 3d mesh, and uses

3d Fast Fourier Transforms (FFTs) to solve Poisson’s equation on the mesh, and then interpolates electric fields on the mesh points back to the atoms.

In LAMMPS, we also create atoms using the “read_data” or “create_atoms” or

“lattice” commands. LAMMPS can create atoms sitting on a lattice, via the lattice and create_atoms command. For our purposes, we first created either the atoms in the toluene solvent or the dodecanethiol atoms surrounding the nanoparticle by first writing and running several programs which would create an initial text file which specified each 18 atom type, coordinate, molecular type etc. and then reading the atom coordinates in via the read_data or read_restart files command. We also utilized the replicate command to create a bigger box size from an already equilibrated system. The replicate command takes the atoms in a simulation box, and then creates copies of the system in the x, y, and z directions as needed.

In order to update atom positions or velocities due to time integration, control temperature, apply a force to atoms, or enforce boundary conditions, LAMMPS uses the

“fix” command. There are more than a dozen fixes defined in LAMMPS, and these are used to do the basic computations that drive a simulation. In our simulation, we used fixes such as NVT, NVE, and NPT. The corresponding molecular dynamics ensembles of the fixes are described in the previous sections.

There is also a command called “run” which runs the dynamics for a specified number of time steps. The output of the program is a dump or restart file. The output files give a log file of thermodynamic info, output of atom coordinates, velocities, and other per-atom quantities such as energy, stress parameters, etc. It is also possible to obtain a snapshot in native xyz format which can be viewed using VMD, or Jmol. The post- processing of these output files are done using program written in high-level languages like C/C++ or Java, or Python.

2.4 VMD (Visual Molecular Dynamics)

VMD is a molecular modeling and visualization program that was extensively used throughout this thesis to view and analyze the results of our molecular dynamics simulations. It also includes tools for working with volumetric data, sequence data, and

19 arbitrary graphics objects. In this project, VMD was used to see clips of the simulation as it took place. This was useful in figuring out problems with the simulation, and also to get an idea of what was taking place.

VMD is specifically designed for visualizing biological and molecular systems. It was developed by the Theoretical Biophysics group at the University of Illinois at Urbana

Champaign. VMD provides a way to animate a collection of data. In this project,

LAMMPS provide a collection of coordinates of atoms in an .xyz file format over several time steps or frames. VMD takes those coordinates and create a clip of the simulation over several frames.

Figure 2-1: VMD simulation and setup

2.5 Periodic Boundary Conditions

Computer simulation programs are run to predict and study the properties of a system in bulk. However, we are not interested in surface effects usually. Our simulation tracks only a small number of particles in order to not slow down the computation. As a

20 result, most molecules are near the edge of the sample, and thus at the surface. Therefore, in order to avoid surface effects in our computations, our system size would have to be extremely large to ensure that the surface has only a small influence on the bulk properties. However such a system would be too large to simulate.

We can eliminate undesired surface or finite-size effects from our computation by using periodic boundary conditions. In periodic boundary conditions, the cubical simulation box is replicated throughout space to form an infinite lattice. During simulation, when a molecule moves in the central box, its periodic image in every one of the other boxes moves with exactly the same orientation in the same way. Thus, as a molecule leaves one face of the central box, its image will enter the opposite face. The system has no surface as there are no walls at the boundary of the central box. The central box simply forms a convenient coordinate system to measure locations of the molecules.

An example of a two-dimensional periodic system is shown in Fig. 2-2 below. As the particle moves through a boundary, its corresponding image moves across their corresponding boundaries. The number of particles in the central box is conserved.

Figure 2-2: Periodic images of a simulation box in 2 d.

It is not necessary to store the coordinates of all images in a simulation, but only those that are in the central box. When a molecule leaves the box by crossing a boundary, we note the identical molecule entering from the opposite side.

2.6 Minimum Image Convention

Periodic boundary conditions allow us to carry out molecular dynamics simulations using a relatively small system. However, this may lead to another problem.

During molecular dynamics simulations, we need to calculate the pairwise force or energy between all interacting pairs of atoms. If the simulation system has N molecules, then we need to calculate N(N-1) interactions between each pair of atoms. This problem grows if we have an infinite number of boxes in every direction.

To solve this problem, we utilize the minimum image convention. Let us suppose that we are using a potential with a finite range, i.e. when separated by a distance equal or larger than a cutoff distance Rc, two particles do not interact with each other. Also, let us assume that we are using a box whose size is larger than 2Rc in each x, or y, or z direction. When these conditions are satisfied, we can conclude that at most one among all pairs formed by a particle i in the box, and the set of all periodic images of another particle j will be within the cutoff Rc and thus interact. To demonstrate this, let us assume that “i” interact with two images j1 and j2 of j. Two images must be separated by the translation vector bringing one box into another, whose length is at least 2Rc by hypothesis. In order to interact with both j1, and j2, i should be within a distance of Rc from each of them which is impossible since they are separated by more than 2Rc.

When these conditions are satisfied, we use the minimum image criterion. Thus, among all possible images of particle j to interact with, select the closest, and throw away all the others. Only the closest is a candidate to interact. Other images do not interact.

This condition simplifies a molecular dynamics program. However, we need to make sure that the box size is at least 2Rc in all directions when Periodic Boundary Conditions are in effect.

Figure 2-3: Interaction between closest pairs.

2.7 Molecular Dynamics Potentials

2.7.1 Lennard-Jones Potential

The Lennard_Jones potential is a common potential which is used to represent the forces between molecules in a fluid. The potential approximates the true intermolecular potential curve. Such a potential is attractive at long distances and very repulsive if the molecules get too close. As an example, consider two non-bonded atoms at infinite distance, and not interacting. The atoms are brought closer to each other with minimal energy input to a certain distance, r. At this distance, the atoms have an attractive force

23 between them. The attractive force between the two atoms brings them even further together until they reach an equilibrium distance at which their minimum bonding potential is reached. To further decrease the distance between both atoms, additional energy is required because the atoms overlap, and the atoms’ electrons are forced to occupy each other’s orbitals, and as a result the repulsive forces act and push them further apart. At these distances, the force of repulsion is greater than the force of attraction.

The most common form for a Lennard-Jones potential may be written as:

Eq. 2-14 where V is the intermolecular potential between the two atoms or molecules, r is the distance between the two particles, ϵ is the well depth and a measure of how strongly the two particles attract each other, and σ is a measure of the range of interaction. In the equation above, the 1/r12 term is repulsive, and describes the Pauli repulsion at short ranges due to overlapping of electron orbitals, whereas the 1/r6 term is the attractive long- range term, and describes the attraction at long ranges due to the Van der Waals force or dispersion force.

The stability of an arrangement of atoms is a function of the Lennard-Jones separation distance. As the separation distance decreases below equilibrium, the potential energy becomes increasingly positive, while the force is repulsive. Such a large potential energy is energetically unfavorable as this indicates an overlapping of atomic orbitals. However, at long separation distances, the potential energy is negative and approaches zero as the separation distance increases to infinity and force is attractive. As the separation between the two particles reaches a distance slightly greater than σ, the potential energy reaches a 24 minimum value, and the force at this point is zero. The pair of particles is most stable at this distance, and will remain in that orientation until an external force is exerted upon them.

Fig 2-4: Lennard-Jones potential curve

One practical problem with the Lennard-Jones potential is that it has infinite range. Thus it is impossible for two particles to go far enough apart to escape their attractive force. The force gets infinitely weak at large distance. From a computational point of view, this means the force must be calculated for every pair of particles in the system. This will increase the computational time for large systems. Thus, in MD simulations, the LJ potential is typically cut off at a finite distance of around 3 σ. In our simulation, we used the Lennard-Jones potential in order to calculate the pairwise potential between our non-bonded atom pairs. The coefficients and cutoffs were entered through a data file into LAMMPS.

2.7.2 OPLS force field potential

The force field refers to the parameters of the potential used to calculate the forces in Molecular Dynamics simulations. It is used to create a certain number of atom types which can be used to describe any bonds, angles, impropers, dihedrals and long-range interactions. More atom-types than elements are used since the chemical surroundings greatly influence the parameters. The angle between carbons in alkyne side chains will differ from that of carbons in a benzene ring, and thus two atom types will need to be defined for the same atoms. OPLS (optimized potential for liquid simulations) [3, 4, 5] is a force field parameter used to define organic, and biomolecular systems. The OPLS force field parameters were optimized by testing against properties of pure organic liquids, especially heats of vaporization, densities, and free energies of hydration.

Most force fields in widespread use for macromolecular systems have a similar form including harmonic bond stretching and angle bending, Fourier terms for torsional energetics, and Coulomb plus Lennard-Jones terms for intermolecular and intramolecular non-bonded interactions. Anharmonic and cross-terms may also be added.

Also, one would like to add instantaneous polarization effects. However these are not widely adopted due to increased computational demands and due to the absence of fully polarizable force fields. The main differences between non-polarizable force fields are mainly choices on numbers of interaction sites and origin and extent of testing of the parameters in energy expression.

In OPLS, both nonbonded and torsional energy parameters were derived to reproduce gas-phase structures and conformational energetics from ab initio RHF/6-

31G* calculations and observed thermodynamic properties of organic liquids. Multiple

26 compounds of the same type were considered in the fitting process to avoid biasing the torsional parameters for particular molecules.

2.7.2.1 OPLS Functional form:

Based on the AMBER potential, Jorgensen et al. [3] created the OPLS force field using the following functional form :

Eq. 2-15(a)

Eq. 2-15(b)

Eq. 2-15(c)

Eq. 2-15(d)

Eq. 2-15(e)

½ ½ with the combining rules Aij = (Aii Ajj ) and Cij = (Cii Cjj )

In the above equations, bonds and angles are described by harmonic potentials, since they are very strong and fluctuate slightly around their equilibrium values at room temperature. The dihedral potential is described via a cosine expansion and may take any value within 360o depending on the height of the barrier between the low energy conformations, which make the precision of the dihedral potential barrier crucial for many polymer properties. Dihedral potentials always have symmetry around 180o. The long range interactions are only counted for atoms three or more bonds apart. They consist of Coulomb and Lennard-Jones two-body interaction terms. The Lennard Jones potential is a combination of attractive Van der Waals forces due to dipole-dipole interactions and empirical repulsive forces due to Pauli repulsion. The intramolecular nonbonded interactions Enonbonded are counted only for atoms three or more bonds apart. 1,

4 interactions are scaled down by the fudge factor fij = 0.5, otherwise fij = 1.0. All the interaction sites are centered on the atoms, and there are no pairs.

2.7.2.2 Derivation of the parameters

2.7.2.2.1 Bonds and angles

The equilibrium values for bonds as well as angles are simply taken from x-ray data. The values for the force constants are derived by fitting to experimental vibration frequency data. Although it is encouraged that the geometries of simple molecules match the experimental data as well as possible after energy minimization, it is believed that in most cases the difference is negligible.

Figure 2-5: Schematic showing bond and angle interactions between atoms i,j and i,j,k respectively along with corresponding force constants.

2.7.2.2.2 Dihedrals

There are two different approaches as to how to calculate dihedral potentials. One approach is to optimize the dihedral potential for the simplest possible molecule, and then apply it to larger ones containing the same dihedral. The other approach is to optimize the dihedral parameters to best describe a large number of different molecules. The second method, although seems more accurate, can lead to dependence on the set of chosen molecules. Both these approaches compute dihedral parameters from ab initio methods as follows :

i. Ab initio calculations

- Scan dihedral or improper of interest

- Optimize geometry at each step

- Calculate change in potential energy

For this purpose, either perturbation, or restricted Hartree-Fock or hybrid methods between Hartree-Fock and density functional theory are used. The basis sets chosen for the geometry optimization are at least 6-31g and can go up to 6-311g, depending on the size of the molecule in question. 29

Figure 2-6: Diagram showing dihedral and improper interactions between atoms i,j,k,l including corresponding force constants

ii. Potential Energy according to MD

- Set dihedral parameters to zero

- Compute potential energy of each optimized configuration

Note that this requires the knowledge of all other force field parameters in the molecule.

iii. Fitting of the parameters

- Subtract MD results from ab initio results

- Obtain parameters from fitting a dihedral function to the resulting curve

The difference between ab initio and MD results corresponds exactly to the influence of the dihedral, since all other interactions are already included in the force field. The most common mathematical representation of dihedrals are:

a) Proper dihedrals

Eq. 2-16

b) Ryckaert-Belleman dihedrals

Eq. 2-17 where 

iv. Checking the results

- Rerun MD simulations using new parameters

- Compare MD and ab initio results

2.7.2.2.3 Charges

In order to obtain partial charges of a molecule whose values will not be altered during the MD runs, it is important to ensure that the charge calculations are done in equilibrium conformation. This may be achieved by optimizing the geometry using any ab initio method with a basis set better than or equal to 6-31G. The charges can then be obtained by fitting to electrostatic potential. Adjusting the partial charge at the centers of the nuclei should be done in such a fashion that the electrostatic potential given by the wave function is best reproduced.

2.7.2.2.4 Van der Waals parameters

This is the area where the difference between OPLS/Amber and other force fields comes into play. Jorgensen et al. [3] pioneered the development of force field parameters for organic molecules focusing on systems explicitly taking the solvent into account. As a matter of fact, the OPLS/AMBER force field for peptides and proteins takes most bond angle and dihedral parameters from the force fields developed by Weiner et al [19]. This is achieved by carrying out Monte Carlo simulations on organic liquids, e.g CH4 or C3H6, and then empirically adjusting the Lennard Jones parameters to match the experimental densities and enthalpies of vaporization. A difficult issue is the factor to scale down the

Lennard Jones 1-4 interactions. This has to be done because the 1/r12 term would lead to

31 unphysically high repulsions. The value chosen is somewhat arbitrary and force field dependent. The same holds true for the combination rules for the Lennard Jones parameters, i.e the choice of Lennard Jones parameters for the interaction between different atom types. The two main competing methods are:

- Lorentz-Bertelot

Eq. 2-18(a)

Eq. 2-18(b)

- Geometrical Average

Eq. 2-18(c)

Eq. 2-18(d)

The method should be chosen in accordance with the force field to be used.

Chapter 3

Methods

3.1 Previous work on nanoparticle-solvent interactions

Before discussing the methods we have used to carry out our simulations of dodecanethiol-coated Au nanoparticles, we first discuss previous work. We note that

Lane and Grest [8] have previously carried out extensive simulations using the OPLS potential of a variety of coated spherical nanoparticles of different sizes both in solution and at the liquid-vapor interface in both decane and water solvents, with several different types of ligands including decanethiol, octadecanethiol. As can be seen in Fig. 3-1 below, depending on the particle size and coating, the particle may either stay well above the interface or penetrate deep into the solvent. However, in this work the specific solvent (toluene) and coating (dodecanethiol) and nanoparticle size (6.0 nm) considered here were not studied. In addition, estimates of the binding energy and of the effects of the van der Waals interaction were not included, nor were quantitative measures of the nanoparticle shape and location with respect to the interface presented. In addition, the diffusion constant for adsorbed NPs at the interface was not calculated. We now discuss in detail some of the methods we have used to carry out our simulations o

dodecanethiol-coated Au nanoparticles at the liquid-vapor interface and in bulk, along with the methods used to create our nanoparticle and toluene systems.

Fig. 3-1: Equilibrated polymer-coated alkanethiol NPs of diameter (from top to bottom) 2.0, 4.0, and 8.0 nm at the water-vapor interface for S-(CH2)9-CH3, S-(CH2)9-COOH, S- (CH2)17-CH3, and S-(CH2)17-COOH (from left to right). (From Ref. 8)

3.2 Determining the position of dodecanethiol-coated Au

nanoparticles at toluene-air interface

In order to carry out our simulations of Au nanoparticles at the toluene-air interface, we first equilibrated a sample of 15,000 toluene molecules at atmospheric pressure and room temperature in a cubic box with periodic boundary conditions (PBCs) using constant pressure and temperature (NPT) dynamics, as discussed in more detail in

Section 3.2.2. We then created two interfaces by doubling the vertical box size. A dodecanethiol-coated 3 nm radius nanoparticle was then introduced at the interface and was allowed to equilibrate in order to determine its equilibrium position and fluctuations,

34 as well as the binding energy at the interface. Here we outline in more detail some of the steps involved in carrying out these simulations.

3.2.1 Creation of dodecanethiol-coated nanoparticle (NP)

As in previous work carried out by Lane and Grest [8], we assumed that the packing of the dodecanethiol chains on the Au NP is similar to that for buckminsterfullerene (see Fig. 3-2 below). In addition, since the NP is coated with dodecanethiol, as also assumed by Lane and Grest, we do not include the Au core in our simulations. We note that previously Lane and Grest studied nanoparticles with diameter d = 4 nm coated with 72 alkanes in H2O using the OPLS potential. Scaling this to the size of our nanoparticles (d = 6 nm) and assuming the same packing density leads to

2 approximately (6/4) x 372 ≈ 540 alkanes which corresponds to a C540 fullerene.

Accordingly we represented the binding sites for the S atoms in the thiol chains to the Au core by a C540 fullerene structure. Fig. 3-3 shows a picture of the C540 structure used in our simulations, which was based on a structure obtained from an online fullerene library.

Fig. 3-2 : Buckminster fullerenes: (a) C28. (b) C32. (c) C50. (d) C60. (e) C70

Since the radius of the C540 buckyball is 1 nm while the radius of the Au core used in experiments is 3 nm, in our simulations we scaled the coordinates by a factor of 3 in order to increase the radius of the Au core to 3 nm. This increased the separation between

35 the dodecanethiols, and gave us the nanoparticle size we wanted. Then, we attached a dodecanethiol next to each carbon of the buckyball.

Fig. 3-3: Picture of buckyball C540 configuration.

Avogadro is a public domain software used as a molecule editor and visualizer. The software is used for Computational Chemistry, Molecular Modeling, Materials Science and related areas. Using Avogadro we were able to create a model of the dodecanethiol molecule from its chemical structure with the correct coordinates. Fig. 3-4 shows a picture of a dodecanethiol (C12H23S) molecule created with Avogadro.

For each of the 540 dodecanethiol molecules attached to the C540 shell representing the edges of the Au core of the NP, we removed the hydrogen atom bonded to the sulfur of dodecanethiol, and attached the sulfur to the carbon, with a bond length of

1.81 nm. We however first had to rotate the dodecanethiol around its axis to make sure

Figure 3-4: Single dodecanethiol ligand that the dodecanethiol is along the axis corresponding to the carbon to which it is being attached. In order to do so, we first determined the unit vector ĝ = (gx,gy,gz) corresponding to the orientation of our basis dodecanethiol. For each C atom in the C540, the corresponding radial unit vector ŵ = (wx,wy,wz) = ri/|ri| was also determined. The coordinates of all of the dodecanethiol prototype atoms were then shifted so that the S atom was at the desired position, while the angle  for rotation was determined using the expression Cos  = ŵ•ĝ. The corresponding rotation axis u = (ux,uy,uz) was then determined from the cross-product u = g x w, e.g. ux = gywz-gzwy, uy = gzwx-gxwz, uz = gxwy-gywx. The coordinates ri = (xi,yi,zi) of each atom of the dodecanethiol molecule were then rotated by an angle  around the rotation axis u, using the transformation r’ =

R r, where the rotation matrix is given by:

This transformation was carried out for each dodecanethiol molecule using a program written in C which rotated the basic coordinates of our prototype thiol as described above. Fig. 3-5 shows a picture of the coated NP.

Figure 3-5 : Perspective view of nanoparticle with dodecanethiol-coated ligands

3.2.2 Creation of toluene liquid

As mentioned above, we first created a sample of bulk toluene at room temperature and atmospheric pressure by carrying out NPT simulations of an initial structure consisting of 15,000 toluene molecules on a cubic lattice. Each toluene molecule, with chemical formula CH3C6H5, was created based on the known structure and bond lengths. Fig. 3-6 below shows a picture of a single toluene molecule. As can be seen, toluene has a ring-like structure with a methyl group at one end.

The equilibrium C-C bond length in the benzene ring of toluene is 0.14 nm. The equilibrium C-H bond length in toluene is 0.108nm. The equilibrium bond length between C-C where one carbon is the carbon of benzene and the other is carbon of CH3 is

0.151nm. Using C code, a program was written to create a LAMMPS input file which

38 contained the coordinates of each of the atoms in each molecule, as well as the corresponding bonding structure for each molecule.

Figure 3-6: Toluene molecule as created in our simulation

In the picture in Fig. 3-6 above, each black sphere represents a carbon atom, and each white sphere represents a hydrogen atom. The atoms 9, 8, 3, 4, 5, 10 are aromatic carbons of the benzene ring, and are henceforth referred to as CA. Atoms labeled 7, 2, 1,

6, 11 are Hydrogen bonded to carbons of the benzene and are referred to as HA. The atom labeled 12 is the carbon of the CH3 methyl group and is referred to as CT. The hydrogen atoms bonded to CT are labeled 13, 14, and 15. They will be referred to as HT.

The equilibrium angle between HT-CT-CA is the typical tetrahedral bonding angle of

109.5 degrees.

The equilibrium angle between HT-CT-HT is 107.8 degrees. The equilibrium angle between CA-CA-CA is 120 degrees. The equilibrium angle between HA-CA-CA is 120 degrees as well. The equilibrium angles between CA-CA-CT are 120 degrees also.

Based on the bond length and angles between atoms, the coordinates of a single toluene molecule were created. In addition to the equilibrium bond angles and bond

39 distances, dihedral torsion was also taken into account in our simulations. This was done by including stiffness constants from the OPLS potential corresponding to the angle  between two planes of atoms, where

Edihedral = V1(1 + cos )/2 + V2(1 - cos 2)/2 + V3(1 + cos 3)/2 + V4(1 - cos 4)/2

Eq. 3-1 and the values of V1 – V4 are given by the potential. For example, torsion corresponding to a dihedral angle corresponding to out-of-plane motion of the aromatic H’s was included. The hydrogens (HT) bonded to a carbon (CT) also have a dihedral angle along planes of 120 degrees. Dihedral angle is defined as the angle between two planes. In creating a single toluene molecule, each of the hydrogen of CH3 was separated by 120 degrees with one at zero degrees from the benzene-ring plane or just above the benzene ring. The other two are on either side of the plane looking down. From a side-view each of the CH3 C-H bonds should again be 120 degrees.

While many previous research studies have modeled the benzene ring of toluene as rigid for simplicity, in our work we allowed all of the toluene atoms to stretch - where bond length would change, bend – where bond angles between atoms would change due to bending, and rotate – due to change in dihedral or torsional angle between planes.

Once a single toluene was created, an array of toluene was then created by placing them on a lattice. A distance of 12 A (Angstrom) was placed between each toluene atom in each direction. At first, the length l (maximum of x coordinate – minimum of x coordinate for a single toluene), thickness t (maximum of y coordinate – minimum of y coordinate), and width w (maximum of z coordinate – minimum of z coordinate) was

40 calculated. Then a distance of 12 Angstrom, was added to each dimension. This represented the lattice constants in the three directions. More specifically,

a = l + 12; b = t + 12 c = w + 12 where a, b, and c correspond to the lattice constants in the x, y, and z directions respectively. Initially, a 10 x 10 x 10 lattice of molecules was created, and thus 1000 toluene molecules were placed on the lattice. Fig. 3-7 below shows the toluene molecules placed on the lattice with a separation of 12 Angstroms between them. The dimensions of the box was made cubic by taking the largest dimension, which happened to be its length along the x axis, and setting the other two box dimensions equal to that length.

Figure 3-7: Toluene lattice as created in our simulation.

After the toluene atoms were placed on the lattice, we carried out NPT molecular dynamics simulations at 1 atmosphere pressure and 300 Kelvin (or room temperature) with periodic boundary conditions in which the number of atoms, pressure and temperature were kept constant. The size of the NPT simulation box shrank significantly as the system was equilibrated. A picture of toluene equilibrated at room temperature for

500 picoseconds is shown below in Fig. 3-8. As can be seen from the picture, the box shrank while the molecules became disordered and formed a liquid-like structure. After the system was equilibrated, it was then used to create a larger system so that we could study the adsorption of a dodecanethiol-coated Au nanoparticle with a 3 nm core radius.

The amount of toluene in the Fig. 3-8 below is 1000 molecules, or 15000 toluene atoms.

A larger system of toluene was next created where there were 27000 molecules of toluene or 405,000 toluene atoms.

This was done using the LAMMPS replicate command which takes a simulation box, and creates copies of it in each dimension as specified. A picture of the larger

Figure 3-8: Picture of 5.6nm cubic toluene liquid box. system toluene is shown in Fig. 3-9.

After liquid toluene was created, it was equilibrated under NPT where the pressure was 1 atmosphere, and the temperature was 300K or room temperature.

Equilibration was done for 0.5 ns to allow the liquid to shrink or expand, and attain a pressure of 1 atm. The box size obtained at the end of the equilibration was 17.0 nm x

17.0 nm x 17.0 nm.

Next, we created a liquid vapor interface by increasing the size of the box in the z direction and doing image bit shifting to take care of the periodic boundary conditions in the z direction. We increased the size of the box in the restart file where we changed the size of the box along the z direction. Bit shifting was done because some molecules are on the periodic boundary of the simulation box. That is, there are liquid toluene molecules which have part of their atoms on the top of the box and others at the bottom.

This is because we have periodic boundary conditions in x, y, and z directions.

Periodic boundary conditions are a set of boundary conditions that are used to simulate a large system. A unit cell or simulation box of a geometry in 3 dimensions are defined where an object passes through one face of the unit cell, and reappears on the opposite face with the same velocity.

Figure 3-9 : Toluene liquid created after replication process.

The simulation is an infinite tiling of the system. As discussed in the Introduction

(Chapter 1), the minimum-image convention is a form of periodic boundary condition bookkeeping where each individual particle in the simulation interacts only with the closest image of remaining particles in the system. Because of periodic boundary conditions, atoms which straddle the boundaries see their images on the other size of the 43 box. Just increasing the size of the box will prevent the atoms from seeing their image on the other side of the box. Thus, the image bits have to be changed as well. Each atom has three image bits in x, y, and z direction. We go over each atom and check if their z image bit is the same. If they aren’t, we set each atom in the molecule to the lowest image bit, and add the length of the box to the z coordinate. This essentially positions the atom to the top of the box, or above the liquid-vapor interface, while giving the atom a lower z image bit number as should be the case. Thus, a part of the molecule which is at the bottom of the box sees its image on the other size of the box at the top, and interacts with it. Otherwise, bonds get broken if the image bit and positions are not changed.

Figure 3-10: Liquid-vapor interface for liquid toluene..

Once the liquid-vapor interface was created, it was equilibrated for 2 to 3 ns using

NVT dynamics (constant box size) at 300K or room temperature. Fig. 3-10 shows the resulting liquid vapor interface created for toluene.

3.2.3 Positioning the nanoparticle above liquid toluene

After the liquid-vapor interface was created, the nanoparticle was inserted into the simulation box. To do this, all toluene molecules within a certain radius of the nanoparticle were deleted. This was done by removing all references to the molecules within that zone. That is, all atom coordinates, bonds, angles and dihedral values specified for the molecules were deleted. Next, the nanoparticle was positioned a few nanometers above the liquid-vapor interface of toluene. This was done essentially by combining the restart files of toluene and the nanoparticle. All the atom ids of the dodecanethiols in the restart files of the nanoparticle were shifted by the highest atomic id number of toluene molecules in the restart file for toluene. This prevented one atom of toluene and dodecanethiol from having the same atomic id number, as this will create errors in compilation of the program. As the atom ids of the dodecanethiol was shifted by a certain number – the highest atom id of toluene, all references to dodecanethiol in other sections of the restart file also needed to be shifted by a certain number. Other sections that needed the atom id numbers are the “Bonds” section of the restart file, “Angles” section of the restart file, and the “Dihedrals” section of the restart file. These sections for toluene and dihedrals in their respective restart files were also combined. This was done by copying all bond, or angle, or dihedral data from the restart files, but shifting the bond ids or angle ids or dihedral ids for dodecanethiols atoms, and shifting the atoms id of dodecanethiols they reference to by the highest atom id of toluene atoms.

The coordinates of each of the atoms of the nanoparticle also had to be shifted according to the following formula :

Nanoparticle_atom_coordinate - Old_center_of_mass_of_nanoparticle + new_center_of_mass_of_nanoparticle 45

In the above formula, the coordinate of the old center of mass of the nanoparticle was subtracted from coordinates of individual nanoparticle atoms, and the new center of the nanoparticle was added to it. This essentially shifted the nanoparticle from its old center to its new center above the liquid toluene.

The center of the nanoparticle was calculated by averaging the position of all the sulfur atoms which were bonded to the positions of non-existent carbon. Fig. 3-10 above shows the nanoparticle situated above the liquid vapor interface.

3.2.4 Nanoparticle lowered into Toluene

After the nanoparticle was created, NVT molecular dynamics simulations were carried out on the entire system. However, as the nanoparticle was positioned high above the liquid toluene, it did not feel the force from liquid toluene to be drawn into it. To allow the nanoparticle to be attracted to the liquid, the nanoparticle had to be lowered into the simulation box. This was done by giving the atoms in the nanoparticle a small net velocity downwards of 0.00005 Angstrom/femtosecond until the nanoparticle was close enough to liquid toluene. In LAMMPs, this was implemented using the following command :

velocity gold set 0.0 0.0 -0.00005 sum no units box

In the above line, gold refers to the sulfur atoms of the nanoparticle. The sulfur atoms are rigid, meaning all the sulfurs in the nanoparticle moves as a group. Thus, giving them a net velocity moves all the sulfurs along with the rest of the nanoparticle.

The velocity created was 0.0 in the x direction, 0.0 in the y direction, and -0.00005 in the z direction. “sum no” means that the velocity specified is not to be added to the existing velocity of the atom but is to be replaced by the velocity specified. “units box” just

46 specify that we are dealing with a box rather than a lattice where the units are

Angstrom/femtosecond rather than lattice units of spacing per femtosecond. The command above reset the velocity of the nanoparticle in each step of the simulation, and lowered it by 0.00005 Angstrom each step.

After the nanoparticle was brought a few angstroms away from the liquid vapor interface of toluene, it was allowed to be attracted to the liquid. However, the nanoparticle had a net momentum from when it was being lowered into the liquid. Thus, the momentum of the system needed to be reset. This was accomplished via the following command:

velocity all create 300.0 4928459 mom yes rot yes dist Gaussian

The velocity create command creates an ensemble of velocities using a random number generator with the specified seed as the specified temperature. This essentially resets the velocities to make the temperature which is a function of kinetic energy 300

Kelvin. The momentum of the system is set to zero by “mom yes”. Here, all atoms of the systems had their momentum set to zero. The rotational momentum of the system was also set to zero via “rot yes” where “rot” stands for rotation.

However, before this command was ran, the restart file had to be altered to make all the atom ids of the system consecutive. That is, all atom id numbers should be from a range x to z with no gap. That is, if the first id number is 1 and the last is 2000, all atoms in the system should have a unique id starting from 1 and ending with 2000 with no gap in between. For example, the id numbers “1, 2, 3, 4, 5, 6” is consecutive while “1, 3, 5, 6” isn’t as in the second instance, 2 and 4 are missing numbers for the progression.

To do so, a C program was written to read the restart file containing coordinates and bond and angle information of toluene and the nanoparticle, and the atom ids were changed. This is done by making the atom ids in the “Atoms” section of the restart file consecutive. Here, old atom ids were stored for each atom, and a new atom id for it was generated based on its position in the file. If atom with atom id 3434 was the 300th atom in the atom section of the restart file, its atom id was changed from 3434 to 300. As the old and new ids were saved by the program, this can be used later on. In the “Bonds” section of the restart file, the bonds refer to the old atom ids. Here, the old atom ids were replaced by its corresponding new atom id. The “Angles” and “Dihedrals” sections were similarly changed by updating the old atom id with the new consecutive atom id.

After the momentum of the system was set to zero, and the nanoparticle was brought close to liquid toluene, NVT molecular dynamics simulations were carried out at

T = 300K (room temperature). The simulation was allowed to run for 8 nanoseconds, which took a few weeks to complete. We ran the simulation in steps of 0.5 nanoseconds, and saved configurations of the nanoparticle in restart file for use in later runs. A description of the restart files and how they are used is provided in the introduction of the thesis.

The nanoparticle was slowly attracted into the liquid and eventually entered the liquid. A program was written to analyze the movement of the nanoparticle as it was attracted into Toluene. The program read in coordinates of the nanoparticle from dump file created by LAMMPs, and output the center of mass of the nanoparticle as it was drawn into the liquid.by averaging the coordinates of the atoms of the nanoparticle.

The graph in Fig. 3-12 gives a plot of the center of mass of the nanoparticle as a function of time. As can be seen from the graph, the nanoparticle was strongly attracted to the liquid toluene for the first 4 ns, before it started to settle down in the liquid. After the first 4 ns, the nanoparticle merely bobbed up and down in the liquid. From pictures of the simulation, we observed that the nanoparticle was absorbed deep into liquid toluene, but was not completely submerged. The gold core of the nanoparticle was mostly below the liquid vapor interface of toluene, and oscillated up and down in the liquid, with only a

Figure 3-11: Dodecanethiol-coated Au nanoparticles submerged in toluene solution fraction of the gold core coming above the interface of liquid toluene at times during the oscillation. A picture of the nanoparticle submerged into liquid toluene is shown in Fig.

3-11. As can be seen from the picture, the nanoparticle is mostly inside toluene, with a fraction of the thiol ligands sticking out.

3.3 Oscillation of nanoparticle at the interface

To calculate how much the nanoparticle was sticking out above the interface, we calculated the center of mass (CM) of the nanoparticle as well as the position of the interface of liquid toluene. Using these results we were able to calculated the relative position Z of the interface relative to the CM of the NP over the length of the simulation, where

Z = z coordinate of the interface of liquid toluene – z coordinate of the center of mass of the nanoparticle.

Fig 3-12 shows our results for the time-dependence of Z(t) during the course of our simulation. As can be seen, the liquid-vapor interface is initially far below the center- of-mass of the nanoparticle and so z is negative. For the first nanosecond, the nanoparticle’s center was above the liquid. Then it started to go down into the liquid and the value of Z became positive. It eventually plateaued between 30 to 32 Angstroms, which means the distance of the nanoparticle center from the interface above it hovered around 30 - 32 Angstroms. Incidentally, the average distance between the sulfur of the nanoparticle and the center of the nanoparticle is 32 Angstrom. Thus, the nanoparticle oscillated with its gold core just inside the liquid toluene.

Figure 3-12: Plot of Z vs time

3.3.1 How Z was calculated

Z was calculated in our simulation by going over xyz movie files dumped by

LAMMPS. “.xyz” is a file format of LAMMPs where the type of the atom and its coordinates are dumped during an iteration of Lammps. In the LAMMPs input script, it was specified how frequently the coordinates of the atoms of toluene and the nanoparticle needed to be dumped. The format of an xyz file is as follows :

Number of atoms Atoms Type1 xcoor ycoor zcoor Type2 xcoor ycoor zcoor Type2 xcoor ycoor zcoor Type3 xcoor ycoor zcoor

In a movie file, coordinates of the file were dumped after a certain number of timesteps, and thus the file format above repeats itself until the run is over. Each run

51 produced a movie file with a profile of coordinates of toluene and the nanoparticle. In our simulation, the nanoparticles was type 5 or higher, while the toluene atoms were 4 and lower. Thus we were able to distinguish between toluene atoms and dodecanethiol of the nanoparticle. In our simulation, we dumped the coordinates of the atoms in our simulation every 20000 md steps or femtoseconds. Thus, each xyz movie file created had coordinates dumped every 20000 femtoseconds. Below is an example of a moviefile in our simulation:

425115 Atoms 3 250.792 215.618 233.955 3 249.432 215.711 231.98 1 249.549 216.593 232.6 1 250.518 216.615 233.642 ...... 425115 Atoms 3 247.4 220.168 235.507 3 249.847 220.124 234.884 1 249.365 219.54 235.657 1 247.97 219.275 235.381

Each frame starts with the number of atoms in the frame that is 425,115 atoms following by the string “Atoms”. Next, the types of the atom (first column) as well as their coordinates (column 2, 3, and 4) are specified. When a new frame is to be started, the number of atoms and the string “Atoms” is written into the file, followed by coordinates for each atom. This goes on until the simulation ends.

From each frame, we calculated the center of mass of the nanoparticle. This was done by averaging the positions of the sulfurs. Each sulfur in the nanoparticle was of type

5. Thus, a C program was written which went over a frame of the xyz movie file, and 52 obtained the positions of the sulfur atoms. All positions or coordinates of the sulfur atoms were added, and when a frame ended, the sum was divided by 540. Thus, the center of mass of the nanoparticle was calculated. We knew when a frame ends because each frame has a certain number of lines. That is, in our simulation there are 425,115 atoms.

Thus, each frame has 425,115 lines plus two lines at the beginning of the frame. Each frame, as a result had 425,117 lines. The program reads the number of lines that were written, and once a count of 435,117 was reached, the program realizes that it has reached the end of the frame, and completes its calculation for the frame. We calculated the centers of mass from the positions of the sulfurs as the sulfurs are modeled as rigid bodies. This means that during each timestep the total force and torque on each rigid body was calculated as the sum of the forces and torques on its constituent particles. The coordinates, velocities, and the orientations of the atoms in each body are updated so that the body moves and rotates as a single entity. Thus, we obtained the center of mass of the nanoparticle by averaging the positions of the sulfurs.

The calculation of the liquid-vapor interface of toluene was more complicated, and we needed to develop an algorithm to find the interface from coordinates of toluene atoms in each frame. All the toluene atoms were of type 4 and below. Thus, we read through a frame and looked for atoms of type 4 and below to calculate the interface. At first, density profile for a frame is created. The algorithm to create density profile for a frame is as follows:

Num[1000]; If (atom i == toluene atom){ J = (int) (z coordinate of atom i/ 1.0) Num[j]++; } For (i = 0; i<1000; i++){ 53

Print ( i+0.5, Num[i]); }

In the above code, the program reads through each toluene atom and finds its z coordinate. It converts the z coordinate which is a double to an integer and bins it. That is, it counts the number of times the z coordinate, a double converted to an integer, occurs. Thus, the code above takes a slab that is 1 Angstrom in length along the z axis, and counts the number of toluene atoms that occurs in that slab. The program then outputs this value. A density profile calculated using the algorithm above is shown below in Fig. 3-13. :

Figure 3-13: Density profile of toluene with the nanoparticle included

The density profile above gives a count of the number of atoms as a function of the z coordinates. The number of atoms in a slab reaches equilibrium between z values of

150 Angstrom and 200 Angstrom. This is the height of the toluene liquid from the bottom interface up till where the nanoparticle sits. The number of toluene atom is almost evenly

54 distributed between 150 Angstrom and 200 Angstrom. In fact, the number of atoms on each slab of 1 Angstrom gives the density of liquid toluene, which should be the case.

The Fig. 3-13 above does not descend directly from its equilibrium value to zero.

This is because there is a nanoparticle that skews the density profile. The nanoparticle sits inside toluene, and displaces the toluene atoms. Thus, we see the dip in the figure above.

To remove the dip, we only calculated the density profile for atoms which were 62

Angstrom away from the nanoparticle only. Thus, we only counted the toluene atoms on the outer edge of the liquid. This prevents the nanoparticle from altering the density profile. The Fig. 3-14 below gives the modified density profile.

Figure 3-14: Density profile of toluene with the nanoparticle excluded

As can be seen from the picture, the density profile does not have the dip, and it falls sharply to zero. The count of atoms falls to zero when we reach the interface, and there are no toluene molecules present. We define the position of the interface as the point where the count of atoms is midway between the values at the two extremes

(corresponding to half of the toluene density or 1300 atoms per slab in this case). The algorithm to calculate the modified density profile is shown below:

Num[1000]; If (atom i == toluene atom){ dx = absolute(x_i – x_cm); if (dx> Lx/2) dx=Lx-dx; (minimum image convention for box of size Lx) dy = absolute(y[i] - ycm); if (dy> Ly/2) dy=Ly-dy; (minimum image convention for box of size Lx) dr=square root(dx*dx+dy*dy); if (dr < 62)continue; else j = (int) (z coordinate of atom i/ 1.0) Num[j]++; } For (i = 0; i<1000; i++){ Print ( i+0.5, Num[i]); }

In the above algorithm, an array called Num of size 1000 is created. This is the bin where the count of toluene atoms will be stored for each slab along the z axis. If an atom is a toluene atom, it will be of type 4 or less in the xyz movie file. Such atoms are read in by the C program. Next, its distance from the center of the nanoparticle is calculated. This is done by first calculating dx, which is the distance of the atom from the center of the nanoparticle along the x axis. Thus, dx = x_i – x_cm, where x_i is the x coordinate of the atom, and x_cm is the x coordinate of the center of mass of the nanoparticle. We take the absolute value of dx. Next we apply the minimum image convention. That is, if dx is more than half the size of liquid toluene along x axis, there is a much closer image from the center of the nanoparticle for the same atom on the other size of the box. This is called minimum image convention as is described in the introduction. We take the closer distance than the farther away one. Thus if dx is greater

56 than Lx/2 (where Lx is the size of the box in the x direction) dx = Lx-dx, and we take the mirror image of the atom on the other side of the box. Similar calculations are done for dy, which is the distance between the center of mass of the nanoparticle and the toluene atom along the y axis. Next, we calculate dr, where dr is the distance between the center of mass of the nanoparticle and the toluene atom along the xy plane. Thus, dr is defined as dr = (dx*dx + dy*dy)1/2. If dr is less than 62 A (which is 12 A larger than the radius of the nanoparticle plus ligands) we ignore the atom from our calculation as it’s close to or in the region of the nanoparticle. However, if dr is greater than 62 A, we bin the atom to calculate the density profile as described before.

From the density profile in Fig. 3-14, we can see that the number of toluene atoms in a vertical slab of 1 Angstrom reaches an equilibrium value of approximately 1300 atoms in a 1 Angstrom height. This corresponds to the density of toluene as discussed before. The density value of 1300 atoms per slab is reached when we move from the interface and enter liquid toluene. Interface is the point when the number of toluene per slab starts to move from 0 to 1300. Usually, the interface is the midpoint of when this jump is made. We assume the midpoint was around 674 toluene atoms. Thus, we took the corresponding z coordinate that gave this value, which is where the liquid-vapor interface of toluene starts. Also note that there are two jumps in Fig. 3-14. This is because we have two interfaces – one below liquid toluene, and one above liquid toluene. We are interested in the top interface.

To calculate the interface of liquid toluene in each frame in our xyz movie file, we use the following algorithm: numatom[10000]; zcoor[10000]; 57

for(i=10000;i--;i>=0){ if(z1>674){ n2=numatom[i]; z2=zcor[i]; break; } else { n1=numatom[i]; z1=zcor[i]; } zint=z1-1.0*(674-n1)/(n2-n1);

In the above algorithm, we had array numatom and zcoor, each of size 1000 and they store the count of atoms in each slab of size 1 Angstrom as you moved up the axis as well as the z coordinate that gave the value. That is, numatom stores the y axis value in Fig. 3-

14, while zcoor stores the corresponding x axis value. Once these values are stored, we try to find the point where the number of atoms value of 674 was crossed. To do that, we create a loop and start from the last element stored in the two arrays. We store the point where we receive an atom count greater than 674, as well as the point that came before.

Thus, n1 and z1 will give us the point before when the number of atoms crossed 674.

Here, z1 stores the z coordinate that corresponds to atom count stored in n1 of the immediate value less than or equal to 674. And z2 stores the z coordinate that corresponds to atom count stored in n2 of the first atom count greater than 674 in a slab of height 1 Angstrom. We then solve similar triangles to find the point where 674 was crossed.

The Fig. 3-15 below shows how the similar triangle was used to calculate the z coordinate that gave a value of 674 atoms per slab of 1 Angstrom.

Figure 3-15: Similar triangle technique used to calculate the interface

In Fig. 3-15 above, we want to find z_int, which is the z coordinate of the interface position. We solve similar triangles as follows:

(z_int – z1) / (674 – n1) = (z2 – z1) / (n2 – n1) Or, (z_int – z1) / (674 – n1) = -1.0 / (n2 – n1), as z2-z1 = -1.0 for they are neighboring slabs each of heigh 1 Angstrom, and z2 < z1 Or, z_int – z1 = -1.0 * (674 – n1) / (n2 – n1) Or, z_int = z1 -1.0 * (674 – n1) / (n2 – n1)

Thus, we find z_int, which is the exact location of z coordinate where the count of atoms of 674 was reached. This is the interface position after which the toluene density reached an equilibrium in liquid toluene.

We calculated z_int for each frame in an xyz file. Each xyz file had several frames, each taken after a period of 20,000 femtosecond. Each z_int was calculated using the algorithm discussed before, where we first populated numatoms and zcoor arrays. Thus, our code automatically calculated the z_int or z coordinate of the interface by going over coordinates of atoms in an xyz file.

3.4 Calculating adsorption energy

In order to calculate the adsorption energy with which the nanoparticle was bound to the interface, we needed two systems, one in which the nanoparticle is at the center of toluene, and another where the nanoparticle is at the interface. In order to minimize the amount of computation, we started with our simulation box which was 17 x 17 x 34 nm in size and then reduced this size to 11 x 11 x 34 nm. In order to do this, we removed any toluene molecule which was 15 nm below the upper liquid-vapor interface. We also removed all toluene molecules which were farther than 5.5 nanometers from the center of the nanoparticle along the xy plane. This shrunk our box to 11 x 11 x 34 nm in size with

15 nm toluene in the vertical direction, thus leading to a smaller system, but with the nanoparticle still at the toluene liquid-vapor interface. We next ran Nose-Hoover NVT molecular dynamics on this simulation box, in order to determine the average potential energy for this system.

In order to determine the interfacial adsorption energy for a nanoparticle, we then carried out a similar simulation on a system with exactly the same number of toluene and

DDT molecules but with the nanoparticle at the center of the box. To create such a system we first started with the 11 x 11 x 34 nm system described above. We then removed toluene molecules from the bottom of the liquid, and placed them several angstroms above the toluene interface, while the number of toluene molecules moved was selected so as to put the nanoparticle at the center. We did the repositioning in the usual way in our Laamps restart file. We went through the restart file, and increased the z coordinate of toluene molecules which we chose to move to the top. Once we had such a configuration with a smaller toluene liquid block above a larger toluene block with a

60 nanoparticle in it, we pushed the liquid block on top with a small velocity until it collided with the larger block. Next, we ran NVT molecular dynamics and started taking data after the system had equilibrated for 2-3 nanoseconds. We note that the number of molecules in both the simulation boxes are identical which is necessary in order to compare the binding energy. The detailed results of our calculations for the binding energy are given in the next Chapter (Chapter 4 – Results).

3.5 Calculating surface tension

Since we expect that the binding energy is related to the difference in surface tension between the toluene-vapor interface and the DDT-coated NP interface, in order to check the accuracy of our OPLS potential, we calculated the surface tension of pure liquid toluene as well as that of pure liquid dodecanethiol. In each case a bulk sample with a liquid-vapor interface was created and then equilibrated at room temperature using NVT molecular dynamics, while the pressure tensor components Pxx, Pyy, and Pzz were calculated and used to determine the surface tension. The results of our surface tension calculations are given in Chapter 4 – Results.

In order to calculate the surface tension of liquid toluene, we used a previously equilibrated simulation box of liquid toluene of size 17 nm x 17 nm x 17 nm. We took that box, and created a liquid vapor interface, doubling the size of the box in the z direction. Fig. 3-16 (a) below shows our simulation box for toluene with the liquid-vapor interface.

Next we carried out NVT molecular dynamics in order to equilibrate the system at room temperature and then started taking measurements of the pressure tensors after the first 1 or 2 ns. Through LAMMPS, we output the pressure tensor values TO a dump file,

61 and used the resulting average to give us an estimate of the surface tension for our simulated toluene.

To calculate the surface tension of dodecanethiol, a similar procedure was followed. In particular, we first extracted the configuration of a single dodecanethiol from the restart file containing the nanoparticle. We also obtained the bonds, dihedrals and angles from the restart file for the particular molecule as well and created a new restart file.

Figure 3-16 : Simulation system to calculate surface tension of toluene

We next put the single dodecanethiol molecule on a lattice, and replicated it to create a 10 x 10 x 10 lattice of molecules. We equilibrated these dodecanethiol molecules for a while using NPT molecular dynamics at room temperature and 1 atmosphere. Next we

62 increased the size of the box by a factor of 12 ( 2 x 2 x 3) by replicating the dodecanethiol molecules. We noted that the box is not cubic when we do that given that dodecanethiol molecule is assymetric. So we found the center of dodecanethiol liquid box, and removed everything that is 14.4 nms away from the center of the box along x, y and z axes. We ran

NPT molecular dynamics on this dodecanethiol simulation box, allowing the box to expand or contract while keeping the box cubic in shape. To measure the surface tension, we next created a liquid vapor interface. While creating the liquid vapor interface, we needed to unwrap the dodecanethiol molecule to make sure the image bits reflect the position of the atoms. We explained unwrapping on section 3.2.2 of this thesis. Fig. 3-17 below shows the liquid vapor interface of liquid dodecanethiol.

Figure 3-17 : Simulation system to calculate surface tension of dedecanethiol

Once the liquid vapor interface was created, we ran NVT molecular dynamics on the system. In a way similar to toluene, we initially saw a spike in the potential energy, and total energy of liquid dodecanethiol. We let the system equilibrate for a few nanoseconds, 63 and then started taking data on surface tension of the system, namely Pxx, Pyy, Pzz values.

Note that the surface tension value we got was for dodecanethiol, and not the nanoparticle. The nanoparticle has dodecanethiol molecules on a sphere of gold, and we don’t have such a system when calculating surface tension of dodecanethiol. The

“Results” section of this thesis gives a more detailed description of our final surface tension results for dodecanethiol.

3.6 .XYZ and .mol2 File format

In order to create representations of the molecular simulation taking place during

MD, we used several public-domain software packages including Jmol, and VMD

(Visual Molecular Dynamics). VMD used the xyz file format, while Jmol allowed many different types of file formats. For our project, we used .xyz and .mol2 file formats with

Jmol. In this section, we give the file format of .xyz and mol2 files.

The xyz file format has the following structure –

No of atoms Atoms Atom-type xcoor ycoor zcoor Atom-type xcoor ycoor zcoor Atom-type xcoor ycoor zcoor ……

That is in the first line, we specify the number of atoms we have in our simulations. In the next line, the put in the word “Atoms” or a blank. Next, we start specifying the coordinates, and the type of the atoms in the format shown above. These files can be output through LAMMPS, and viewed via VMD or Jmol.

One drawback of the .xyz file format is that it does not specify anything other than the coordinates of the atoms. Thus, bonds between atoms are drawn erratically based on distance between atoms rather than actual existence of bonds. To overcome this problem during visualization of atoms, we use the .mol2 file format.

The .mol2 file can have a lot of information in it such as bond, center of mass information, molecule information, etc in several sections. For our purposes, we had only three sections titled molecule, atom and bond. Each section has its own format. A sample of the .mol2 file used in this project is given below :

@MOLECULE benzene 12 12 1 0 0 SMALL NO_CHARGES @ATOM 1 C1 1.207 2.091 0.000 C.ar 2 C2 2.414 1.394 0.000 C.ar 3 C3 2.414 0.000 0.000 C.ar ...... @BOND 1 1 2 ar 2 1 6 ar 3 2 3 ar .. … In the above sample .mol2 file, we have three sections which begin with

@. The three sections are titled @MOLECULE,

@ATOM, and @BOND. The format of

@MOLECULE is given below: mol_name num_atoms mol_type 65 charge_type [status_bits [mol_comment]

In the above format, there are six data lines. The first data line is the name of the molecule. The second data line is the number of atoms in the simulation for the molecule.

The third line is the name of the molecule type. The types of molecules are “SMALL,

BIOPOLYMER, PROTEIN, NUCLEIC_ACID, SACCHARIDE”. For our purposes, we used the type “SMALL”. The fourth data line is the type of charges associated with the molecule. The types are – “NO_CHARGES, DEL_RE, GASTEIGER, GAST_HUCK,

HUCKEL, PULLMAN, GAUSS80_CHARGES, AMPAC_CHARGES,

MULLIKEN_CHARGES, DICT_ CHARGES, MMFF94_CHARGES,

USER_CHARGES”. In our mol2 files, we used “NO_CHARGES” as the type of charge.

Status bit are the internal status bits associated with the molecule, and should not be set by the user. Mol_comment is the comment associated with the molecule. For our purposes, we needed to specify only the first 4 lines in our file.

The second section in our .mol2 file specifies the atoms and their coordinates.

The format is as follows : atom_id atom_name x y z atom_type [subst_id [subst_name [charge [status_bit]]]]

Atom_id is the id associated with the atom, and is referenced by other sections such as the bonds section where bonds for that specific atom is specified. Atom_name is the name of the atom. This can be any value and is not used later on in the file. The x, y, and z values are the coordinates of the atom. Atom_type is the type of the atom. The various types of atoms recognized by the software are given in the reference below2. The other values for the atom section are not mandatory, and was not used by us.

The third and final section in our .mol2 file is the bond section. The format for the section is as follows: bond_id origin_atom_id target_atom_id bond_type [status_bits]

In the above format, bond_id is the id of the bond at the time it was created. Any number or string can be specified for this. For our purposes, the bond_id was the count of the bond. The origin_atom_id and target_atom_id are the ids for the atoms and need to reference the atom_id in the atom section in order to create bonds for those atoms. The other field is the bond type, which can be - “1”, meaning single bond, “2” meaning double bond, “3” meaning triple bond, “am” meaning amide, “ar” meaning aromatic,

“du” meaning dummy, “un”, meaning unknown, and “nc” meaning not connected. For our purposes, we used the type “1” or single and “ar” for aromatic bonds in the benzene ring.

3.7 Radial toluene density distribution r)

To find out how much the toluene penetrates into the nanoparticle core, and calculate the effects of the Van der Waals interaction energy on the interfacial binding energy, the radial density of toluene was also calculated, for both a nanoparticle in the bulk as well as a nanoparticle at the interface. Radial toluene density gives the number of toluene atoms in a shell of width 1 Angstrom radially from the center of the nanoparticle.

Thus, we obtain a count of the number of toluene atoms at various radii from the center of the nanoparticle. The algorithm to obtain the radial distribution of the toluene is given below:

Loop over all toluene atoms{ Store x, y, and z coordinates of toluene atom i.

Calculate R = |r_i – r_com| , i.e the distance of atom i from the center of the nanoparticle N = (int) R / 1.0 Count[N]++ If atom is Hydrogen, then CountH[N]++ If atom is Carbon, then CountC[N]++ }

Output: Loop over all N values{ Plot Count[N] against R Plot CountH[N] against R Plot CountC[N] against R

XX = (4/3)*pi* ( [(N+1)*Delta_r]^3 - [N*Delta_r]^3) Plot C(N) / XX against R }

In the above algorithm, we loop over all toluene atoms in a certain configuration or timestep. For our purposes, we use an xyz file which contains the coordinates of toluene and dodecanethiol atoms. Next, we obtain the distance of the toluene atom from the center of the nanoparticle. We divide the distance obtained by the shell radius, which is

1.0 in this case. This gives the bin that the toluene atom belongs to. We maintain an array called “Count” which stores the number of toluene atoms belonging to a certain bin. We also maintain a count for the number of hydrogen atoms, and carbon atoms belonging to a certain shell.

In the second part of the program, we print out the number of toluene atoms, or carbon and hydrogen atoms belonging to a certain shell. We also calculate the volume of the shell using the formula for the volume of a sphere which is (4/3)  r3. Thus we subtract the volume of the outer shell from the innter one to give the volume of the region the toluene atoms belong to. We normalize the count of number of toluene atoms belonging to a shell by dividing by the volume of the shell. We plot a graph of the 68 normalized toluene atom count against the radius from the center of the nanoparticle. Our results for the radial density of toluene surrounding a nanoparticle in bulk toluene and at the interface are presented in the Results section of the thesis.

Chapter 4

Results

4.1 Position of nanoparticle at interface

In order to determine the equilibrium position of a dodecanethiol (DDT) coated

Au nanoparticle (NP) adsorbed at the toluene-vapor interface, we first created a toluene sample with two interfaces (one above and one below) in a 17 nm x 17 nm x 34 nm box with periodic conditions, as described in more detail in Chapter 3. We then placed our

DDT-coated NP above the top interface, and allowed it to approach the interface and equilibrate via molecular dynamics (MD) at room temperature for 12.4 ns. As can be seen in Fig. 4-2, the nanoparticle was attracted deep into the liquid toluene, until the core was entirely submerged inside the liquid, and then it oscillated up and down. Also shown in

Fig. 4-1(b), is a picture with the toluene removed, in order to better view the shape of the

NP at the interface. As can be seen, it looks basically spherical, although a more quantitative analysis (see below) indicates a definite asymmetry, due to the interface, between the vertical and horizontal directions. As can be seen from Fig. 4-1(a), the thiol- coated gold nanoparticle is embedded deep into the liquid toluene, although a

(a) (b) Figure 4-1: (a) Picture of NP equilibrated at toluene-vapor interface (b) Same NP with toluenes removed for clarity.

significant fraction of the DDT ligands are sticking out above the interface. Fig. 4-2 shows the z-coordinate of the center-of-mass of the NP as a function of time as it approaches the interface.

In order to carry out a quantitative analysis of the nanoparticle position and oscillations with respect to the toluene-vapor interface, we have also measured the relative distance z = zNP(t) – zint(t) between the center of mass of the nanoparticle and the interfacial position as a function of time as shown in Fig. 4-3. As previously

71 discussed in Chapter 3, the interfacial position zint(t) was determined from the dependence of the toluene density on height.

Figure 4-2: Center of mass of the nanoparticle as a function of time

As can seen in Fig. 4-3, the nanoparticle starts above the interface, and then is quickly attracted to the liquid toluene in the first 2 nanoseconds. For the next 4 ns, it continues to settle down at the toluene-vapor interface. For the remaining 6 ns, it oscillates due to thermal fluctuations and appears to be almost completely equilibrated.

As can be seen in Fig. 4-3, since z is greater than 3 nm, the core is fully submerged in the toluene. However, since the average value of the displacement z of the center-of-mass below the interface (35 ± 3 A) is less than 40 Angstroms, while the average radius of the nanoparticle (ligands) is around 45.7 A, and the maximum ligand radius is approximately 49 A, some of the thiols are always sticking out of the liquid toluene.

The Fig. 4-4 below shows the distance of the highest positioned sulfur to the interface of liquid Toluene. The Fig. 4-3 has the same shape as Fig. 4-4 above showing the distance of the center of the nanoparticle to the interface.

Figure 4-3: Distance of center of mass of nanoparticle from interface vs time

Figure 4-3 above confirms that the nanoparticle position equilibrates after the first 3 or 4 ns, and then merely bobs up and down. According to the plot below, the z-coordinate of the highest S atom on the nanoparticle relative to the interface plateaus around 0

Angstrom. This means that the gold core is oscillating up and down with only a fraction of the gold core below the interface.

Fig. 4-5(a) below shows the nanoparticle when it is at its deepest position in liquid toluene. The distance of the center of the nanoparticle to the toluene interface is

38.3 A as it oscillates in liquid toluene, and this is the highest value reached. Fig. 4-5(b) shows the nanoparticle at its highest point when it is exposed the most in air after it has equilibrated. The distance of the center of nanoparticle to the interface in this case is 28.2

A. The nanoparticle oscillates in liquid toluene after it settles down. Fig. 4-3 below shows the oscillation of the nanoparticle along the z axis after it has settled down.

Figure 4-4 : Distance of highest sulfur of nanoparticle from interface vs time

The motion of the nanoparticle at the interface was also analyzed in order to test if the nanoparticle was rolling in liquid toluene, since in a previous study9 of uncoated nanoparticles in a thin-layer of film, an unusual rolling behavior leading to anomalous diffusion had been observed. In order to carry out the analysis, we used xyz files containing coordinates of the sulfur atoms over the duration of the run. It was found that the sulfur merely moves along the liquid in a fashion similar to diffusion rather than of rolling. This makes sense as the nanoparticle is deep inside liquid toluene. Rolling would be possible if the nanoparticle was sitting well above the interface or, as in Ref. 9 if the nanoparticle were not coated with dodecanethiols.

(a) (b)

Figure 4-5 (a): Nanoparticle at its lowest point in toluene (b) Nanoparticle at its highest point in toluene after equilibration.

4.2 Shape of nanoparticle at interface and in bulk

A further analysis of our molecular dynamics results was also carried out in order to find out the shape of nanoparticle both at the toluene-vapor interface and in bulk toluene. A picture of the nanoparticle at the interface is shown in Fig. 4-5(a) while Fig. 4-7 below shows the nanoparticle in bulk. As discussed in Chapter 3 (Methods), the nanoparticle was moved from the interface to the center of the liquid toluene (using the methods described in Chapter 3) in order to study the shape of the nanoparticle in bulk.

Figure 4-6: Oscillation of the nanoparticle at toluene-vapor interface after 6.5 ns of equilibration

Figure 4-7 : Nanoparticle submerged in bulk toluene.

As part of our analysis, the average radius of nanoparticle (including the thiol ligands) was then calculated when the nanoparticle was at the interface, and also when it was submerged in liquid toluene. The average radius was calculated as follows –

Eq. 4-1

tip In this equation the quantity ri gives the coordinates(xi, yi, zi) of the farthest atom (from the center of the NP) in each dodecanethiol chain i, while is the coordinate(xc, yc, zc) of the center of mass of the nanoparticle, which is calculated by averaging the coordinates of the sulfur atoms of the dodecanethiols. Once these coordinates are available, the Pythagorean distance of the farthest thiol from the center of the nanoparticle is calculated for each thiol atom. This distance is added up and divided by 540 to give the average radius. Later on, we will show that there are thiols which have a much higher radius than the average radius, as well as thiols with a much lower radius than the average radius.

We extracted the coordinates of the thiols from the xyz files of a certain configuration when the nanoparticle was at the interface, and when the nanoparticle was at the center of liquid toluene. The average radius based on the definition above when the nanoparticle is at the interface is 45.9 A, while it is only slightly less (45.75 A) when the nanoparticle is at the center of liquid toluene. Thus, the average radii for the two situations are almost similar. 77

We also calculated the mean-square radius according to the formula below :

Eq. 4-2

That is, we calculate the radius for each thiol given by the distance from the center of mass of the thiol to its outermost atom, and then square it for that thiol. We do that for each thiol, taking the square of the radius, and sum it for each thiol. Next we divide the square of the radius by 540 to get Rsqav. We did this for the nanoparticle at both the interface and when it was at the center of liquid toluene. When the nanoparticle was

2 at the interface, the Rsqav value is 2111.199219 A while the corresponding value was

2096.793213 A2 for the nanoparticle at the center of liquid toluene.

To compare RAvg and Rsqav, we calculated the mean-square fluctuation of the NP radius <R2>. The formula to calculate it is as follows :

2 <(R) > = RsquareAng – (RAvg * RAvg)

Eq. 4-3

The mean-square fluctuation of the radius <(R)2> was found to be equal to 2.63 A at the interface, while in bulk toluene a somewhat larger value (3.13 A) was found.

Calculations were also carried out to determine the asymmetry of the particle shape at the interface. In particular, both the width of the nanoparticle, as well as the height were calculated. The width was calculated as follows:

W/2 = MAX [(x2 + y2)1/2] over all the thiol atoms

Eq. 4-4

In the above equation, x is equal to difference between x coordinate of a thiol atom, and the center of mass of the nanoparticle. Similarly, y is equal to the difference 78 between the y coordinate of that thiol atom, and the center of mass of the nanoparticle.

We also calculated the height of the nanoparticle to determine how spherical the nanoparticle is. The height was calculated using the formula –

H/2 = MAX ( | - | ) over all the thiol atoms

Eq. 4-5

In the above equation, zi is the z coordinate of a thiol atom, while is the z coordinate of the center of mass of the nanoparticle. Thus, we take the distance of a thiol atom to the center of the nanoparticle along the z axis. We store the highest distance, and call it the height or H/2 of the nanoparticle.

At the interface, we found that the width of the nanoparticle (W = 48.2 A) is higher than the height (H = 46.7 A) which indicates a somewhat asymmetric shape. This is most likely due to the fact that the dodecanethiol molecules are attracted to liquid toluene, and as a result the thiols at the top of the NP do not stick out as much from the core, thus making the height smaller than the width. In contrast, there was a much smaller difference (most likely due to fluctuations) measured between the width and height of the nanoparticle (NP) in bulk toluene away from the interface.

In our simulation, the distance of the sulfur to the center of the nanoparticle is around 32 A. The maximum distance of the sulfur to the nanoparticle is 32.74 A, and the smallest distance is 31.75 A. This difference is due to the asymmetry of the C540 buckeyball configuration we have used to attach the thiol molecules to the Au core. Over the course of the simulation, the shape of the dodecanethiols of the nanoparticle also varies. The maximum length of the thiol (distance of the farthest thiol atom to the sulfur)

79 is 16.43 A while the minimum length of the thiol is 7.27 A. Fig. 4-8(a) and Fig. 4-8(b) below show the thiol molecules with maximum length and minimum length.

Figure 4-8 (a): Dodecanethiol ligand stretched out the most in our simulation

Figure 4-8 (b): Dodecanethiol ligand compressed the most in our simulation

4.2 Adsorption energy of Nanoparticle at interface

4.3.1 Results for Adsorption energy

Besides the shape and position of the NP at the interface, one important quantity which we measured in our simulations was the adsorption energy of the nanoparticle.

Adsorption is the adhesion of atoms, ions, or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the adsorbate on the surface of the

80 adsorbent. This is different from absorption in which a fluid permeates or is dissolved by a liquid or solid. Similar to surface tension, adsorption is a consequence of surface energy.

In our simulation, we measure the adsorption energy by calculating the potential energy of the nanoparticle when it is at the interface and comparing with the potential energy when it is submerged at the center of liquid toluene. To do that, we take our configuration of the nanoparticle when it is at the interface, and average its potential energy value over several nanoseconds. In this case, we obtained the potential energy averaged over 7 nanoseconds when it was at the interface. In a similar way, we obtained the average potential energy (averaged over 2.5 nanoseconds) when the nanoparticle was at the center of liquid toluene. The graphs in Fig. 4-9 and 4-10 below show the potential energy as a function of time in each case. The details of setting up the simulation to calculate the adsorption energy were already discussed in Section 3.4.

Figure 4-9 : Potential energy plot of nanoparticle submerged in bulk liquid toluene 81

Averaging over the entire run of 2.5 ns, we found that the average potential energy for the system when the nanoparticle is in bulk toluene is 867709.36 Kcal/mol with a standard deviation of 347.8 Kcal/mol. However, since the average time for the energy fluctuations corresp corresponds to the typical time for the potential energy to oscillate around its average value), then the error in the potential energy for this case is approximately 347.8

Kcal/mol divided by (N )1/2 (where N ≈ 302 is the number of correlation times during which the energy oscillated in the course of the run) or 20.0 Kcal/mol. In contrast, the average potential energy for the system when the nanoparticle is at the interface of liquid toluene is 867645.15 Kcal/mol with a standard deviation of 351.7 Kcal/mol, while in this case, since N ≈ 351, the error in the potential energy is estimated to be approximately

18.77 Kcal/mol. Combining the error in the average potential energy at the interface

(18.77 Kcal/mol) with the error in the average potential energy in the bulk (20.0

Kcal/mol) leads to a combined error for the sum or difference of these two energies of

(18.772 + 20.02)1/2 Kcal/mol or 27.4 Kcal/mol.

The difference between the potential energy of the two systems is 64.21 Kcal/mol which implies (using the conversion factor 1 Kcal/mol = 0.04336 eV) a calculated interfacial adsorption energy for a 6 nm DDT-coated Au NP in toluene of approximately

2.8 ± 1.2 eV. This is the adsorption energy or binding energy of the system. We note that this value is significantly higher than a previous experimental estimate of 0.3 eV which was obtained for the case of very low excess thiol density. However, it is consistent with experiments using a significantly higher excess thiol density for which a high density of thiols on the NP is expected as in our simulations, in which it was found

82 that the diffusion length (corresponding to the distance a NP can diffuse at the interface before desorbing) is of the order of microns or larger.

Figure 4-10 : Potential energy as function of time for a nanoparticle at the toluene-vapor interface

Thus, this relatively high adsorption energy is due to the combination of a large thiol density on the NP (due to the large excess thiol concentration) as well as the relatively strong attraction between dodecanethiol molecules on the NP and toluene. We note however, that the inclusion of long-range van der Waals interactions (which were not included in our simulations) is expected to reduce this estimate by approximately 0.4 eV.

We give a discussion of the Van der Waals interactions, and its impact on adsorption energy in section 4.4.

4.3.2 Surface tension for bulk toluene and bulk dodecanethiol

While the adsorption energy of a dodecanethiol-coated Au NP at the toluene-vapor interface may be related to the large thiol density on the NP as well as the strong attraction between dodecanethiol molecules and toluene, it is also indirectly related to the

83 surface tension of liquid toluene as well as that of dodecanethiol. Accordingly, we have also tested the OPLS potential used in our simulations by calculating the surface tension values for liquid toluene and liquid dodecanethiol and comparing with experiment. The surface tension of liquid toluene was obtained by creating a block of toluene, and then creating a toluene air interface by doubling the size of the box in the z direction. We next measured the pressure tensor values Pxx, Pyy, Pzz in order to determine the surface tension of the liquid. The formula to calculate surface tension from pressure tensor is as follows:

Surface tension = (Lz / 2) (Pzz – (Pxx + Pyy) / 2)

Eq. 4-6

Thus, in our simulation, we calculated the average pressure tensor values, and then plugged these averages into the Eq. 4-6 above. For toluene, we obtained Pxx = -

45.671 atm, Pyy = -46.089 atm, Pzz = -35.151 atm. The size of our box in the z direction was 340.70 Angstrom. Thus, the surface tension we obtained was 1831.57 Atm-A or

18.56 dynes/cm. The actual surface tension of liquid Toluene was 28.52 dynes/cm.1 This gives a difference of 34.9% between actual and simulation values for the surface tension.

This difference is due to the fact that, while the OPLS potential is typically quite accurate when used to model most quantities involving organic liquids, for the case of surface tension all empirical potentials give values which are too low. The reason for this is that most of these potentials, including the OPLS potential, do not include the effects of polarizability which plays an important role in determining the surface tension. We note however, that by slightly modifying our simulation to include improper dihedral angle contributions to the OPLS potential, a slightly higher value (approximately 20.0

84 dynes/cm) in good agreement with results found in Ref. 2 (Caleman, Mareen et al) was obtained. Nevertheless, this value is still significantly lower than the experimental value of 28.52 dynes/cm.

In a similar way, the surface tension of pure bulk liquid dodecanethiol was also calculated. The value of Pxx was -59.62 atm, Pyy = -60.34 atm, and Pzz = -45.82 atm. The height of the simulation box containing dodecanethiol was 262.40 A. This gave a surface tension value of 1857.59 atm A or 18.82 dynes/cm. The actual surface tension of dodecanethiol (measured by Prof. Bigioni’s group) was approximately 28.0 dynes/cm.3

This gives a difference of 37%. As already noted, much of the difference between the real and simulated surface tension can be attributed to the limitations of the OPLS potential.

4.3.3 Radial Toluene density and van der Waals correction

In this thesis, as well as in the previous studies by Lane and Grest [8], of the shape of ligand-coated NPs at the water-air and decane-air interface, the interaction between the nanoparticle ligands and the surrounding solvent was simulated using OPLS potentials, which include Lennard-Jones interactions, Coulomb interactions, bond- bending and bond-stretching interactions, and proper and improper dihedral interactions.

However, in both our MD simulations and those of Lane and Grest for Au NPs in water, the interaction between the Au core and the solvent was not included. The neglect of this interaction was based on the assumption that it is relatively weak due in part to the

“large” distance between the outer thiols and the inner core as well as the fast 1/r6 decay of the van der Waals interaction. However, due to the relatively large Hamaker constant for Au, as well as the relatively large size of our Au NPs compared to the dodecanethiol

85 radius, such an approximation may not be correct. Accordingly, it is of interest to try to estimate to what extent the effects of the long-range van der Waals interaction between the Au core and the “outer” solvent may affect the value of the NP binding energy obtained in our MD simulations.

In order to carry out an estimate of the effect of van der Waals forces between the nanoparticle and the solvent, we have compared the radial density i(r) of toluene surrounding the Au core of the NP at the interface with the corresponding radial toluene density b(r) for a nanoparticle in bulk toluene. To determine the radial toluene density

(r) in each case, we calculated the number of toluene molecules within a shell of radius r to r + dr, where dr = 1 Angstrom. We next normalized the density by dividing by the volume of the shell. This gave us the concentration of toluene molecules at a certain radius r from the nanoparticle’s center. We then divided this value by the bulk density 0 of toluene to obtain the relative radial toluene density.

Fix. 4-11 shows a comparison between our results for the relative radial density

i(r)/0 for a NP at the interface and those for the relative radial density b(r)/0 for a NP in bulk toluene. As can be seen, in both cases the radial toluene density saturates at a distance of approximately 4.9 nm which indicates that the outer radius of the ligands is approximately 4.9 nm. However, for the NP adsorbed at the toluene-vapor interface the saturation value of the density is significantly lower than the bulk density, due to the presence of the interface.

Fig. 4-11: Calculated radial toluene densities (r) relative to bulk toluene density 0 obtained from MD simulations of dodecanethiol-coated 3 nm core radius Au NP in bulk toluene (solid curve) and adsorbed at toluene-vapor interface (dashed curve). Distance of center of adsorbed NP from interface is z = 3.5 nm.

We note that in general the total reduction E in the interfacial binding energy due to the

van der Waals interaction between the Au core and the solvent may be written:22

 E  dR [' (R)  ' (R)]4R2V (R)  b i R R 1 c

Eq. 4-7  where R1 is the radius of the Au core and Rc > R1 is some inner cutoff radius, and VR1(R)

corresponds to the van der Waals potential at a distance R from a spherical core of radius

7,22 R1 which may be written,

4A R3 V (R)   12 1 R 3(R2  R2)3 1 1

Eq. 4-8 

1/2 where A12 = (A11 A22) = 0.98 eV is the Hamaker constant for the interaction between

Au and toluene, where A11 = 2.8 eV is the Hamaker constant for Au and A22 = 0.34 eV is

the Hamaker constant for toluene.

We note that for R greater than the outer radius of the thiols, ’b(R) = 1.

Accordingly, for ease of evaluation the van der Waals interaction energy may be

rewritten in the form,

E  E  E in out

Eq. 4-9 

where

R 2 E  dR [' (R)  ' (R)]4R2V (R) in  b i R R 1 c

Eq. 4-10 

corresponds to the contribution near the NP core, which may be calculated using the

radial toluene densities obtained in our simulations, and

 E  dR cos1(z /R)4R2V (R) out  R R 1 2

Eq. 4-11 

corresponds to the contribution far away from the NP where the toluene density below

the interface is equal to the bulk toluene density and the toluene density above the

interface is equal to zero (inset of Fig. 4-11). As a result, in this region the relative radial

-1 solvent density at a distance R is given by ’i(R) = 1 - cos (z/R)/. Using these

expressions with Rc = 4.1 nm and R2 = 5.5 nm, along with the results shown in Fig. 4-11

for ’i(R) and ’b(R), leads to the estimate Ein ≈ - 0.23 eV. Similarly, again taking R2 =

5.5 nm and assuming z ≈ 3.5 nm we find Eout ≈ - 0.17 eV. Combining these results

leads to an estimate for the overall reduction of the NP binding energy due to the van der

Waals interaction E ≈ 0.40 eV. While this is significantly smaller than our estimate Eb

= 2.8 ± 1.2 eV for the adsorption or binding energy obtained from our MD simulations, it

leads to a revised estimate for the binding energy given by Eb = 2.4 ± 1.2 eV.

We note that for the case of drop-drying experiments with significantly smaller

excess thiol concentrations - for which the ligand density on the NP cores is expected to

be significantly smaller than in our simulations - this correction to the binding energy

will be much more significant since in this case the binding energy is significantly

smaller. As an example, for the case of drop-drying experiments with a relatively low

excess thiol concentration (corresponding to a significantly lower thiol-density on the NP

core than assumed in our simulations) experimental estimates for the total binding energy

(which automatically includes any van der Waals corrections) ranged from 0.1 ev to 0.3 eV. Thus, in this low thiol concentration case we expect that the van der Waals corrections will play a more significant role in reducing the binding energy.

4.3.4 Nanoparticle diffusion at interface

Since the rate of nanoparticle (monomer) diffusion at the toluene-vapor interface plays a crucial role in determining the island density and size in drop-drying experiments it is of significant interest to determine the diffusion constant. In order to do so, we calculated the 2D diffusion coefficient of the nanoparticle based on our molecular simulation results, over the period of time after which the NP was fully equilibrated at the interface.

Fig. 4-12 : Mean-square displacement of NP as function of time. Slope of fit is 3.18 x 10-5 A2/fs.

The expression used to calculate the diffusion coefficient D in two-dimensions may be written as follows:

2 2 2 r (t) = (x - xo) + (y – y0) = 4 t

Eq. 4-12 where r2 (t) is the mean-square displacement of the particle from its initial position after a time t, and x(t) and y(t) are the positions of the NP at time t, and x0 and y0 are the initial

NP coordinates. Thus, the diffusion coefficient may be obtained by performing a linear fit to the dependence of the mean-square displacement. As can be seen in Fig. 4-12, a linear fit give for the slope S, S = 4 D = 3.18 x 10-5 A2/fs. Dividing by 4 and converting to μ2/sec we obtain for the diffusion of a DDT-coated 6 nm Au nanoparticle at the toluene-air interface,

D = 79.5 μ2/sec

We note that, somewhat surprisingly, this value is very close to the expected value for 3D diffusion of a DDT-coated 6 nm Au nanoparticle in bulk toluene predicted by the Stokes-

Einstein relation D = kT/6r, where  is the toluene bulk viscosity and r = 5 nm is the hydrodynamic radius which includes the thiols. The closeness of our measured value to that predicted for bulk diffusion is most likely due to the fact that only a small portion of the NP sticks out above the interface, and that in addition, no rolling motion has been observed. The closeness of this value to the Stokes-Einstein estimate for bulk diffusion, also indicates accuracy of the OPLS potential in duplicating the bulk viscosity of toluene.

In addition, we note that since our result confirms the Stokes-Einstein estimate, this value also justifies the value used for the NP diffusion rate in kinetic Monte Carlo simulations

Au NP island nucleation and growth which are being carried out at the University of

Toledo.

Chapter 5

Conclusion and Future work

In this project, we studied the behavior of a single gold nanoparticle coated with dodecanethiol ligands in liquid toluene. We sought to understand how the nanoparticle interacts with the interface, how deep it sits as well as other physical properties such as the binding energy, nanoparticle shape, and diffusion constant. This is the first step in trying to understand how a film of nanoparticles self-assembles when a liquid toluene solution evaporates over a surface.

Through our molecular dynamics simulations, we were able to find out where the nanoparticle sits on liquid toluene, and how it interacts with the interface. In particular, we found that the dodecanethiol-coated nanoparticle sits relatively low in the toluene, with a distance between the NP center-of mass and the toluene-vapor interface of approximately 34.7 ± 1.5 A, while the thiols stick out above the interface a maximum distance of approximately 17 A. In addition, although the position of the NP with respect to the interface fluctuates somewhat, the Au core never penetrates above the interface.

We note that this result may be contrasted with previous results obtained by Lane and

Grest for the case of decanethiol-coated Au nanoparticles in water (see Fig. 3-1) in which the nanoparticles are high above the interface, with the ligands only partially

92 submerged. The reason for this difference is that in our simulations the dodecanethiol ligands are strongly attracted to toluene, and as a result the nanoparticle remains almost completely submerged at the interface. Our results also indicate that the nanoparticle rolling mechanism, which has been observed in other experiments and which may greatly enhance the diffusivity of Au nanoparticles during the island self-assembly process, does not apply in this case.

As part of our molecular dynamics simulations, we also analyzed the shape of the nanoparticle at the interface. Somewhat surprisingly, we found that the nanoparticle is not spherical at the interface, but instead the nanoparticle width is larger than the nanoparticle height when it is at the interface. This may again be explained by the fact that the nanoparticle ligands are attracted to toluene. Due to this attraction, the ligands above the NP core, which stick out above the interface, are more compressed than those on the side or bottom which extend into liquid toluene.

In addition to determining the nanoparticle position and shape at the interface, we have also determined the binding energy or interfacial adsorption energy for a nanoparticle at the interface. We found a value of 2.8 ± 1.2 eV which is significantly larger than the value of kT at room temperature (0.025 eV), and thus essentially means that the nanoparticle is strongly bound to the interface and as a result will never leave the interface. We note that this result is consistent with experimental results for high excess thiol concentration in solution, which also correspond to a large thiol concentration on the

Au NP core as was assumed in our simulations, and for which a diffusion length

(corresponding to the distance a NP can diffuse before leaving the interface and returning to solution due to thermal fluctuations) much larger than the interisland spacing was

93 observed. In addition, the radial toluene density surrounding the NP both at the interface and in bulk has been calculated.

By combining these results with the known Hamaker constants for Au and for toluene, we have also estimated the corrections to the binding energy due to the long- range van der Waals interaction. Our results indicate that while the correction is significant (approximately 0.4 eV) the corresponding reduction of the binding energy of the NP to the toluene-vapor interface does not significantly reduce the diffusion length compared to typical experimental time- and length-scales. However, it is likely that the van der Waals interaction will somewhat affect the position of the NP at the interface. In addition, it is likely that for other systems (such as decanethiol-coated Au nanoparticles in H2O which have recently been studied by Lane and Grest [8], the van der Waals correction will be significantly larger (approximately 1 eV or even more) and thus significantly influence both the binding energy and position of the NP at the interface.

Thus, our study of the effects of the van der Waals interaction on the binding energy and particle-position at the interface, for dodecanethiol-coated Au nanoparticles in toluene, is likely to have important ramifications for a variety of other related systems.

Finally, since it plays an important role in determining the island-density and average island-size in drop-drying experiments, we have also used our molecular dynamics simulation results to estimate the two-dimensional diffusion coefficient for a

DDT-coated Au NP at the toluene-vapor interface. Surprisingly, our results indicate that the two-dimensional diffusion coefficient is approximately equal to 80 μ2/sec which is in good agreement with the Stokes-Einstein prediction of 73 μ2/sec for 3D bulk diffusion of dodecanethiol-coated gold nanoparticles in liquid toluene as calculated by Bigioni et al1.

These results are of particular interest since they may be used in future kinetic Monte

Carlo simulations of drop-drying experiments in order to model the NP island self- assembly process over extended time- and length-scales. In addition, our results show clearly that the NP rolling mechanism, which had been previously observed to occur in experiments on non-coated NPs in thin liquid films, does not apply in this case.

While our simulations have provided significant insight into the shape, position, binding energy, and diffusion of a nanoparticle at the toluene-vapor interface, in the future it would be of interest to pursue a number of different extensions of this work. For example, in order to better understand the process of island formation, it would be of interest to study the interaction between two or more nanoparticles at the interface, in order to determine an effective interaction between nanoparticles as well as the binding energies for nanoparticles in a cluster. Such a calculation would be of particular interest for the simple case of a dimer since it may be compared with theoretical predictions made by Khan et al, and thus allow us to determine the accuracy of these predictions. We note that in order to do carry out such a calculation, the van der Waals interaction between nanoparticle Au cores would also have to be taken into account. Since diffusion and coalescence of nanoparticle islands also play an important role in the self-assembly process, it would also be interesting to extend our results for the diffusion of a single nanoparticle at the interface, in order to study the diffusion of nanoparticle islands, and thus determine the dependence of the diffusion coefficient on cluster size. Finally, it would also be of interest to directly include the van der Waals interaction between the Au core and the surrounding toluene in future simulations, in order to test our estimates for the magnitude of the van der Waals corrections to the binding energy as well as to

95 determine to what extent the van der Waals interaction affects the position of the particle at the interface.

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