Mayur Suri - University of Minnesotadtraian/thesis-suri.pdf · a thesis submitted to the faculty of...

UNIVERSITY OF MINNESOTA

This is to certify that I have examined this copy of a master’s thesis by

Mayur Suri

and have found that it is complete and satisfactory in all respects,and that any and all revisions required by the final

examining committee have been made.

Traian Dumitrica

Name of Faculty Adviser(s)

Signature of Faculty Adviser(s)

Date

GRADUATE SCHOOL

Phase-Transition Assisted Deposition ofPassivated Nanospheres

A THESISSUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL

OF THE UNIVERSITY OF MINNESOTABY

Mayur Suri

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OFMASTER OF SCIENCE

Traian Dumitrica, Adviser

c© Mayur Suri 2008

i

Abstract

Large-scale atomistic simulations considering a 5 nm in radius H-passivated

Si nanosphere that impacts with relatively low energies onto a H-passivated

Si substrate reveal a transition between two fundamental collision modes. At

impacting speeds of less than ∼ 1000 m/s, particle-reflection dominates. At

this lower-impact speed, no significant phase transition within the particle is

observed. Adhesion is primarily motivated by bond formation between particle

and substrate. Therefore, passivated nanoparticles could not be captured. At

increased speeds the partial onset in the nanosphere of a β-tin phase on the

approach followed by amorphous-Si phase on the recoil is an efficient dissipa-

tive route that promotes particle-capture. In spite of significant deformation,

the integrity of the deposited nanosphere is retained. This project explains

the efficient fabrication of nanoparticulate films by deposition under hyper-

sonic speeds. Preliminary impact simulations are also performed for C and SiC

nanoparticles.

CONTENTS ii

Contents

List of Figures v

1 Introduction to Molecular Dynamics 1

1.1 The Molecular Dynamics Method . . . . . . . . . . . . . . . . . . . . 2

1.2 The velocity-Verlet Time Evolution Algorithm . . . . . . . . . . . . . 3

1.3 Periodic Boundary Conditions and Linked-List Method . . . . . . . . 6

1.4 Temperature Control Algorithm . . . . . . . . . . . . . . . . . . . . . 8

1.5 The Interatomic Potential . . . . . . . . . . . . . . . . . . . . . . . . 11

2 The Tersoff Potential 13

2.1 The Functional Form of the Tersoff Potential . . . . . . . . . . . . . . 14

3 Experimental Deposition of Nanoparticles 21

4 Modeling the Nanoparticle – Substrate System 27

4.1 The Nanocluster and Substrate . . . . . . . . . . . . . . . . . . . . . 27

4.2 Surface Passivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Simulation Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Molecular Dynamics Results - Impact of Hydrogenated Si Nanopar-

ticles on Bare and Hydrogenated Si Substrate 41

6 More Molecular Dynamics Results - Impact of Si31,075N413, C90,326 and

Si12,080C12,080 Nanoparticles on Bare Si Substrate 54

6.1 Si31,075N413 Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.2 C90,326 and Si12,080C12,080 Nanoparticles . . . . . . . . . . . . . . . . . 58

CONTENTS iii

7 Future Work 62

References 64

LIST OF FIGURES iv

List of Figures

1 Flow chart for the molecular dynamics method . . . . . . . . . . . . . 4

2 2-D representation of Periodic Boundary Conditions . . . . . . . . . . 7

3 Dynamics of linked-list assignment . . . . . . . . . . . . . . . . . . . 9

4 Comparison of cohesive energies of various bulk-Si structures . . . . . 14

5 Structure of the highly compressed BC-8 state of Si . . . . . . . . . . 17

6 Bonding in the Murty-Atwater Classical Potential . . . . . . . . . . . 18

7 Tersoff Paramteres for Si, H, and N used in this work . . . . . . . . . 19

8 Test Molecules for Tersoff Potential . . . . . . . . . . . . . . . . . . . 20

9 Schematics for the hypersonic plasma particle deposition process . . . 22

10 Schematics for the ocused deposition process . . . . . . . . . . . . . . 23

11 Gear filled with SiC nanoparticles . . . . . . . . . . . . . . . . . . . . 23

12 Ball-and stick representation of a Si nanoparticle of 1 nm in radius . . 28

13 Perspective on the simulated nanoparticle − surface system . . . . . . 29

14 Center of mass (z) with respect to height of the substrate (d) . . . . . 33

15 Example of a Phase Space Plot . . . . . . . . . . . . . . . . . . . . . 33

16 Plot of instantaneous particle temperature at 2000 m/s impact . . . . 34

17 Shape of the plane of contact . . . . . . . . . . . . . . . . . . . . . . 35

18 Evolution of contact radius for 900 m/s impact . . . . . . . . . . . . 36

19 Region of contact stress in the particle (shown in red) . . . . . . . . . 37

20 Evolution of contact stress for 2000 m/s impact . . . . . . . . . . . . 37

21 Atoms in strained nanocluster as a function of cohesive energy − grey

atoms correspond to CE of 4.6 eV while light blue atoms have a CE of

∼ 4.3 eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

LIST OF FIGURES v

22 Atoms with coordination number 6 versus time . . . . . . . . . . . . 39

23 Radial distribution function . . . . . . . . . . . . . . . . . . . . . . . 40

24 Phase space trajectories for three impacting speeds: (a) 900 m/s, (b)

1, 300 m/s, and 2, 000 m/s. The nanosphere approaches the substrate

with positive V , touches down at δ = 0, reaches V = 0 at maximum

penetration, and recoils with negative V . . . . . . . . . . . . . . . . 43

25 MD simulations of H-passivated Si nanosphere impacting onto a H-

passivated Si substrate show two collision modes: (a) reflection and

(b) capture. The middle frames show the maximum penetration in-

stant. Only a cross-section is shown and H atoms are not represented.

The color code carries the local PE, with blue (gray) and pink (light

gray) representing atoms with PE larger and smaller than 4.4 eV/atom,

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

26 Time evolution of (a) contact radius and (b) contact-stress. Variation

of nanosphere’s (c) temperature, and (d) potential energy under two

impacting speeds, V0 = 900 m/s (lower two curves) and V0 = 2, 000 m/s

(upper two curves), and two surface types, H-passivated nanosphere

and substrate (black) and H-passivated nanosphere and bare substrate

(gray). Down arrows mark the particle release instants . . . . . . . . 47

LIST OF FIGURES vi

27 Pair-correlation function of the H-passivated Si nanosphere at maxi-

mum penetration for (a) V0 = 900 m/s and (b) V0 = 2, 000 m/s. Ver-

tical bars mark the neighbor position in cubic Si. (c) Pair-correlation

and (d) bond-angle distributions in the most deformed conical region.

Vertical bars mark neighbor positions and bond angles in bulk β-tin

Si. (e) Unit cells for cubic-diamond (left) and β-tin Si (right) . . . . . 49

28 RADF at the end of 30 ps for impact at 2000 m/s with hydrogen on

both surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

29 BADF at the end of 30 ps for impact at 2000 m/s with hydrogen on

both surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

30 Time evolution of contact energy . . . . . . . . . . . . . . . . . . . . 55

31 Time evolution of contact area for different surface chemistries . . . . 56

32 Time evolution of coordination number plot for SiN . . . . . . . . . . 57

33 Phase space trajectories at 2,000 m/s . . . . . . . . . . . . . . . . . . 57

34 Phase space trajectories for C90,326 nanoparticle . . . . . . . . . . . . 59

35 Restitution for impact of Si nanoparticle with hydrogen on both surfaces 60

36 Phase space trajectories for Si12,080C12,080 nanoparticle . . . . . . . . . 61

1 Introduction to Molecular Dynamics

For nearly seven decades now, computers have been used as a mathematical labo-

ratory for simulating real-world experiments. A plethora of microscopic simulation

techniques have evolved over time, examples of which are [39]:

− Monte Carlo (MC) simulations

− Classical molecular dynamics (MD)

− Molecular dynamics combined with density function theory (DFT)

− Discrete methods such as Cellular automata, Lattice-Boltzmann method.

Broadly speaking these methods can be largely classified into two categories

Stochastic and Deterministic. Monte Carlo is a prime example of a stochastic method.

MC simulations probe the system by small trial changes to the particle positions. The

energy difference between one configuration and the next is measured, and it is this

difference in energy between successive configurations that is used as a benchmark

for accepting or rejecting the trial move [43]. This algorithm is called the Metropolis

algorithm. MD on the other hand is a deterministic method. It is a technique for

computing the equilibrium and transport properties for many body systems [2]. In

classical MD, the motion of individual atoms within the system is determined by

solving classical equations of motion. The idea that Newtonian mechanics − given

a known potential constraint and initial state of a system − can effectively predict

molecular motion is, in fact, an eighteenth century concept. Laplace [39] foresaw this

when he wrote:

“Given for one instance an intelligence which could comprehend all

the forces by which nature is animated and respective situations of the

1

1.1 The Molecular Dynamics Method

beings who compose it an intelligence sufficiently vast to submit these

data for analysis it would embrace in the same formula the movements

of the greatest bodies of the universe and those of the lightest atoms”

1.1 The Molecular Dynamics Method

Solving the classical equations of motion, step-by-step, is the basis of the Molecular

dynamics method. The equations are:

mi∂2ri

∂2t= fi (1.1)

fi = −∂U (RN)

∂ri

(1.2)

where mi stands for the mass of the atom, ri is the coordinate of the atom i. The

force acting on the atom is the negative derivative of the potential energy function

U (RN). RN = (r1, r2....rN) encompasses the whole set of N atoms.

Using a Molecular Dynamics simulation entails making some choices. The most

important choices are listed below.

1. Choice of Interatomic potential − U(RN).

2. Choice of time integrator to solve equation (1.1).

3. Choice of Periodic Boundary Conditions (PBC).

4. Statistical constraints, like constant energy or temperature, imposed on the

system.

Each of theses choices will be explained in some detail in later sections. In par-

ticular, parameters that were used in this project will be emphasized.

2

1.2 The velocity-Verlet Time Evolution Algorithm

A flow chart to sequentially map out steps in a MD simulation is good way of

understanding the process.

Figure 1: Flow chart for the molecular dynamics method


Since the central equation relating force on atoms to acceleration is a second or-

der differential equation, namely (1.1), a suitable time integrator is crucial to the

3


simulation’s success. Choice of the integrator dictates, among other things, energy

conservation (short term or long term), accuracy in trajectory, length of permitted

step size, storage requirement etc. Ever since it was it was first introduced in 1960,

the Verlet algorithm has been a popular choice. For simplicity the algorithm is dis-

cussed for a single particle of coordinate x and velocity v. The following description

of the algorithm closely follows references [15] and [20].

The Verlet equations are:

x(t +4t) = x(t) + v(t)4t +a(t)4t2

2+

b(t)4t3

6+ o(4t4) (1.3)

x(t−4t) = x(t)− v(t)4t +a(t)4t2

2− b(t)4t3

6+ o(4t4) (1.4)

Adding equations (1.3) and (1.4) gives:

x(t +4t) = 2x(t)− x(t−4t) + a(t)4t2 + o(4t4) (1.5)

To calculate the velocity from the equations for position, the following equation is

used:

v(t) =(x(t +4t) + x(t−4t)) + o(4t3)

24t(1.6)

Note that the above equation does not estimate the instantaneous velocity for a

given position and also requires the storage of two timesteps to estimate the velocity.

Further, it is one order less accurate than the equations for position.

The TROCADERO program [40], used for all simulations here, uses a slightly

modified version called the velocity-Verlet algorithm. It is a related scheme for time

progression which stores information from only one time step to estimate the velocity.

4

1.3 Periodic Boundary Conditions and Linked-List Method

Reference [15], details the derivation of the velocity-Verlet equations. For brevity, only

the equations are listed below :

x(t +4t) = x(t) + v(t)4t +a(t)4t2

2(1.7)

v(t +4t) = v(t) +a(t) + a(t +4t)

24t (1.8)


Periodic Boundary Conditions are a crucial concept necessary to maintain the rele-

vance of simulation results to empirical observations. The following summary is based

largely on reference [2].

Consider a simulation of about 1000 atoms, arranged in a 10×10×10 cubic matrix.

As can be readily noted this arrangement leaves nearly half of all atoms in the system

on the surface of the cube. Surface atoms carry the baggage of more reactivity, due to

the presence of unsaturated bonds. Such fundamental physical differences will alter

the properties of the system, compared to a experimentally observed nanocrystal of

1000 atoms, which is likely found within a larger arrangement of atoms.

To tackle this problem, the cube is embedded in a matrix of its own replicas

and each atom sees its closest neighbors. If an atom leaves the basic simulation box,

another incoming image atom substitutes for it. This artificially imposed periodicity is

called periodic boundary conditions (PBC). The figure below gives a two dimensional

representation of PBC. The central cell is the actual simulation box, enmeshed by its

replicas. Note the circle in the final frame that indicates the newly - paired atoms.

Most classical potentials have a finite spatial range, beyond which they do not

5


Figure 2: 2-D representation of Periodic Boundary Conditions

affect the interatomic forces. Mathematically this interatomic potential, V (rij), can

be written as:

V (rij) = 0 for rij > rcutoff

rcutoff corresponds to the maximum distance for which the potential function is

effective.

During the course of the simulation, a large number of pair-wise distance calcu-

lations would be necessary, just to determine whether a pair of atoms do or do not

interact. Just to give an idea about the scale of these iterations, if the system has N

elements, 0.5N(N − 1) distinct pairs of atoms arise. This presents a huge computa-

tional stumbling block to faster simulations. In reference [50], Verlet first suggested

the Linked-List algorithm designed to deal with this issue. The main steps are:

1. Creating a skin surrounding each atom, such that the radius of the cut-off sphere

rlist is a slightly greater than rcutoff .

2. During subsequent MD steps, only rij < rlist are considered.

6

1.4 Temperature Control Algorithm

3. These atoms are linked together in a list. A head atom is associated with each

cell.

4. Subsequent iterations consider only the atoms within a linked list.

5. The cells are redefined every few time steps keeping in mind the motion of the

atoms within the system.

Figure 3 represents this process. The red circles represent atoms outside of the

link-list zone. The outer circle indicates the extent of spatial range of the linked-list.

The inner circle indicates the potential’s cut off limit. All atoms within the outer

circle are part of the linked list, but only the blue atoms are allowed to interact with

the central atom. It is important to note that linked lists are a valid concept only as

long as the red atoms do not enter the inner circle. When this happens, the list has

to be remade to account for new entrants.

Once atoms move out to encroach into the rcutoff zone, as the second frame

shows, the list has to be redefined to link only relevant atoms. This is schematically

represented in the third frame. The shape of the cell is shown as a circle only for

convenience of representation. This shape can vary; in fact, TROCADERO uses

cubical cells.


Two kinds of statistical ensembles used in our simulations are NVE and NVT. NVE

stands for constant number of atoms, constant volume, and constant energy. NVT

stands for constant number of atoms, constant volume, and constant temperature.

Reference [43] lists multiple methods of temperature control such as:

1. Differential Thermostat

7


(a) Initial cell assignment. (b) Migration of atoms into cell.

(c) Redefining original cell to form newlinked list.

Figure 3: Dynamics of linked-list assignment

8


2. Proportional Thermostat

3. Stochastic Thermostat

4. Integral Thermostat.

The Langevin thermostat used in this project is an example of stochastic thermo-

stat. In this method, select degrees of freedom in the system are subject to collisions

with virtual particles which act as “dampers”. This is theoretically similar to sim-

ulating Brownian motion in a field of viscous force, and has parallels in the field

of aerosol particulate simulation [8]. The reason for using Langevin Thermostat is

because it has been determined to be well suited for both low speed and high speed

collision simulations [24]. The following description of the Langevin thermostat is

taken mostly from references [6] and [30].

The Langevin stochastic equation is:

∂vi

∂t= −∂U

∂ri

− γvi + G (1.9)

Here, as before, vi stands for a component of velocity, ri is the coordinate of the

atom i and U is the potential energy. γ is the friction coeffecient (similar to viscos-

ity), and G (random Gaussian noise) is determined by second fluctuation-dissipation

theorm, as:

〈Gi(ti)Gj(tj)〉 ≥ 2γKbTδijδ(ti − tj) (1.10)

Here Kb is Boltzmann constant, T represents the instant temperature. As γ

increases, so do thermal fluctuations, and loss of particle memory. Reference [43]

elucidates how this method is applied to MD :

9

1.5 The Interatomic Potential

1. Particles collide against imaginary damping particles. The velocities of these

imaginary particles are determined by a Maxwell-Boltzmann velocity distribution at

To, which is the reference temperature for the system.

2. The memory of trajectory they have is lost (motion is gradually randomized).

3. If collision frequency is too high a stronger loss of particle memory results, leading

to a decay of dynamic system functions.


Potential energy functions define the interaction between atoms in a system. If a

potential function is designed to estimate the interaction of N atoms, it is called

a N-body potential function. The simplest form it can assume is, of course, a 2-

body potential. The very first MD simulations for atoms by A. Rahman in 1964

[33] used the Lennard-Jones potential to describe the interaction of atoms in Noble

gases. The classic Lennard-Jones potential defines interaction only as a function of

distance. Atoms very close to each other have strong repulsion and when farther apart

dipole-dipole interaction causes attraction between atoms. Mathematically this can

be represented as:

V (r) = 4ε[(σ

r)12 − (

σ

r)6

](1.11)

Here σ is the interatomic distance and ε is the depth of the potential well.

While this form of potential can be used effectively for ionic bonds or metal-

lic bonds, which typically have stable FCC structures, more directional bonds (like

covalent bonds) cannot be suitably described. In this case, the potential function

10


will have to include an angle dependent component, and account for bond order.

Examples of potentials suitable for covalent systems are the Tersoff Potential [44],

Environment Dependent Interatomic Potential [4] and Murty-Atwater Potential [36].

Note that the Tersoff potential chosen for our impact simulations of Si, can not only

describe Si in the cubic-diamond form, but also provides a robust description of Si’s

different pressure induced phases and melting. The choice of potential and its form

are described in the following chapter.

11

2 The Tersoff Potential

As mentioned in the previous chapter, the choice of a suitable interatomic potential

is crucial to the credibility of the molecular dynamics simulation. S.J.Cook et al. [10]

and Balamane et al. [3], did a thorough comparison of the commonly used potentials

for Si. The potentials compared were:

(a) 2 and 3 body Tersoff potential (T2, T3) [44]

(b) Biswas and Hamann (BH) [7]

(c) Stillinger and Webber (SW) [42]

(d) Dodson (DOD) [13]

(e) Pearson, Takai, Halicioglu and Tiller (HTPT) [32]

All potentials reasonably described the cohesive energies and elastic mechanical

properties of diamond cubic Si. However when pressure induced changes in structure

were simulated, stark differences were brought out between the potentials. The first

change of structure when Si is subjected to high pressure ( ≥15 GPa), is to β-tin

phase, where each Si has a coordination number 6. The Tersoff potential correctly

predicted the change of phase and was the only potential to correctly estimate the

bulk modulus of the β-tin phase [3]. The figure below indicates the cohesive energies

of consecutive Si structures formed by pressure induced phase transformation. The

vertical axis corresponds to energies derived from using each of the 5 five potentials

listed above, whereas the horizontal axis corresponds to energies derived from using

density function theory. It can be seen that Tersoff potential correctly estimates the

energies of β-tin structure and the BC-8 structure that follows - at 4.3 eV and 4.4 eV

respectively.

Since this project deals with particle-substrate collisions, a capability to describe

12

2.1 The Functional Form of the Tersoff Potential

Figure 4: Comparison of cohesive energies of various bulk-Si structures

also the amorphization of Si is important. References [3] and [27] both use the Tersoff

potential to describe amorphous Si at 300K. It can be found that amorphous Si

density (2.2901 g/cm3) and average amorphous coordination (4.16) are consistent

with experimental results [3].


Introduced in 1988 by Tersoff [44], the Tersoff potential is one of the most widely used

potentials to describe Si structures. The functional form of the structure is described

below. Details of parameters used in our simulations and our trial runs with simple

molecules to test them follow. The concept behind this potential is based on Linus-

Paulings bond order theory. As explained in reference [44], the strength of a bond

depends directly on the local environment, with more bonds inversely affecting the

13


strength.

The basic form of the potential is:

U =1

2

∑i6=j

Vij (2.12)

Here Vij represents the interaction energy of the bond between atom i and another

atom j located within its neighbor shell.

Vij = fc(rij) [fR(rij) + bijfA(rij)] (2.13)

Here fc, fR and fA are smoothing function, repulsive function, and attractive

function respectively.

fR(r) = A× e−λ1rfA(r) = −B−λ2r

Given the condition, (RD) ≤ r ≤ (R + D),

fC(r) =1

2− 1

2× sin

[π

2× r −R

D

](2.14)

fC(r) becomes unity or zero on either side of this limit. R and D are chosen to include

1st neighbor shells only for Si.

bij =1

(1 + βnεnij)

( 12n

)(2.14)

bij is the special term here, standing for the attractive component of the potential

function. The term εij defines the coordination number of atom i.

14


εij =∑k 6=ij

fC(rij)g(θijk)eλ33(rij−rik)3 (2.14)

In the above function θijk represents the angle formed between the atoms i, j and k,

where k is a 3rd atom found within the neighbor shell of i. In other words the energy

of the bond between i and j is altered by the presence of a 3rd atom k at a distance

of rik from i. The function g(θijk) determines how sharply θijk influences the strength

of Vij.

Figure 5: Structure of the highly compressed BC-8 state of Si

References [27] and [41] both use the Tersoff potential to simulate phase trans-

formations caused by the nano-indentation of Si substrates. Their results, which

closely follows experimentally observed phase transformations with indented Si sub-

15


strates confirm that the Tersoff potential provides a robust description for pressure

induced phase transformations in Si structure. Simulations in this project will not

be restricted to pure Si. To study the effect of surface reactivity on the adhesion of

impacting nanoparticles with the substrate, Si atoms on the particle surface are satu-

rated with hydrogen atoms. To get a better insight into the role of surface chemistry,

impact simulations are also done with nitrogen-saturated Si Nanoparticles. Both of

the above simulations would, consequently, require potentials which are capable of

describing Si-H and Si-N bonds.

The Murty-Atwater 3-body potential [36] which is the most commonly used for Si-

H interaction fundamentally modifies the form of the Tersoff potential. Also separate

parameters are needed for each of the 9 kinds of 3-body interactions that can occur

between Si and H atoms, shown in the figure below. It is computationally demanding

to analyze each 3-body interaction between Si and H atoms to determine which of

the nine categories they fall into (with each category having its own set of interaction

parameters). Also, this potential is not compatible with Si-N bond descriptions.

Figure 6: Bonding in the Murty-Atwater Classical Potential

The potential described by De Brito Mota and co-workers [11] overcomes these

16


shortcomings by retaining the form of the Tersoff potential and modifying its param-

eters to describe different bonds. For example, consider the attraction component

of the Tersoff potential, bij. In the de Brito Mota scheme, bij varies according to

the interacting species. When Si interacts with H, bij is reduced by a factor of 0.78,

whereas when Ai interacts with N, bij is reduced by a factor of 0.65. When like

species interact bij is left unmodified. Also, to our knowledge it is the only potential

to describe Si-H-N interactions successfully within the same functional form. The

parameters used for each element are given below. Small molecules of Si with H and

N were simulated with de Brito Motas parameters, and it was found that their bond

lengths and bond angles compare favorably with experimentally found values. For

example, Si-N bond length was found to be 1.1 A compared to the theoretical value

of 0.98 A and the angle for both simulated and theoretical values centered around

120o. A sampling of these results are also given below.

Figure 7: Tersoff Paramteres for Si, H, and N used in this work

17


(a) Si2H6

(b) SiH4

Figure 8: Test Molecules for Tersoff Potential

18

3 Experimental Deposition of Nanoparticles

Nanoparticle deposition by Hypersonic Plasma Particle Deposition (HPPD) is a

method for deposition of Si particles that was developed in the late 1990’s [37]. A

schematic representation of the deposition system is shown in Figure 9.

A brief explanation of the process is as follows: The thermal plasma is injected

with reactants. The plasmas elemental constituents are forced through the high tem-

perature nozzle where they nucleate into nanoparticles - onto a region of considerably

lower pressure. (Typically the nozzle sees a pressure drop from ∼ 50 KPa to 300 Pa

and a 2,000 K temperature difference.) Such huge pressure and temperature gradients

create a hypersonic nanoparticle flow. Indeed, particle velocities range between 1,000

to 2,000 m/s, before they impact with the substrate. HPPD aims to capitalize on

these high impact velocities to make films with high deposition efficiency and density.

Materials successfully depositied by HPPD include combinations of Si, Ti, C and N

[17].

Methods to implement a more focused deposition of nanoparticles onto selected re-

gions of substrates have been developed by McMurry and coworkers [28], [29]. Aerosol

flows are directed through a series of small orifices in thin film plates. Particles with

high Stokes numbers are filtered out because they are unable to follow the rapid

contraction/expansion needed to pass through a series of such aerodynamic lenses.

Ultimately a highly collimated, focused beam exits the final orifice. Aerodynamic

lenses were used to fabricate microstructures by focused nanoparticle-beam deposi-

tion. This process is represented schematically in Figure 10 [12].

An example of a micromachined Si-gear filled with SiC nanoparticles deposited

by particle-beam [19] is show in Figure 11.

19

Figure 9: Schematics for the hypersonic plasma particle deposition process

Figure 10: Schematics for the ocused deposition process

20

Figure 11: Gear filled with SiC nanoparticles

The fundamental understanding of nanoparticle-surface collision modes in the

low-energy regime (of less than ∼ 1 eV/atom) is of considerable importance because

achieving efficient sticking to surfaces with preservation of a grainy structure are key

issues for creating new materials. Chemical passivation in the gas-phase is essential

for preventing coalescence and large particle growth but makes the deposition step

challenging. Deposition strategies have been developed, some necessitating the cre-

ation of defects in the substrate in order to pin the impinging nanoparticles [31], which

otherwise would be reflected. Remarkably, impact in the newly available regime of

hypersonic speeds leads to efficient capturing of nanoparticles as small as 2 nm in

radii [17].

The achieved deposition of the well-studied surface-coated Si nanospheres [16],

as demonstrated by the produced dense particulate films [17], has been especially

puzzling for a number of reasons: (i) In macrosphere-surface impact, inelastic behavior

is usually due to the nucleation of dislocations [26], which serves as a contact-stress

release mechanism. At the nano-scale this mechanism is unlikely to operate due to

both the small time and size scales. Indeed, the impact duration (τ) can be estimated

with the classical Hertz theory [26] applied to a nanosphere-plane central collision,

as:

21

τ = (5π

4)2/5(

ρ

E)2/5RV −1/5

In this equation, substituting the density of Si of 2, 330 kg/m3, a Young’s modulus

value of 127 GPa, R of 1 nm , and V of 1 m/s, we obtain a contact time duration of

∼ 3 ps (1 ps=10−12 s).

The obtain ps durations for τ means that the nanosphere experiences very high

strain-rates. As indicated by MD investigations, under such extreme conditions nano-

materials are exhibiting new behaviors. For example, a high-rate compression of

metallic nanowires creates amorphization [25] rather than dislocations. Furthermore,

it is also accepted that even under slower applied strain-rates, dislocations cannot

be accommodated in nanoparticles with dimensions below a critical size. Supporting

this point, recent MD [49] obtained that bare Si nanospheres experience a first-order

phase-transition under severe compression. (ii) The Si nanospheres have been me-

chanically compressed with a nanoindentor tip and were found superhard [16], i.e.,

very large pressures, up to four times larger than in bulk were needed to generate

yield. Thus, any plastic mechanism requires high contact pressures and it is not

known whether such pressures are generated during the still low-energetic hypersonic

impact. Finally (iii), the surface chemistry plays an important role as strong adhesion

promotes sticking. In the ideal case, previous MD obtained that bare Si nanospheres

are always captured by the bare Si substrate due to the strong covalent bonding [48].

However, the formation of new Si-Si bonds is hindered when chemically passivated

contacts are involved. Supporting this point, MD indicated that the coalescence of

Si nanoparticles, as seen in references [21] and [22], was significantly slowed down

when surfaces were H-passivated. Thus, a short τ inhibits adhesion and promotes

22

particle reflection.

Hence, pertinent questions to raise here are: How does high velocity impact affect

the structure of the nanoparticle? How does structural transition within the nanopar-

ticle affect adhesion with the substrate? How does particle surface passivity influence

adhesion during hypersonic impact with an equally passive surface? These are the

fundamental questions that this project tries to answer through MD simulations.

23

4 Modeling the Nanoparticle – Substrate System

4.1 The Nanocluster and Substrate

The standard Si nanoparticle used in all simulations has a radius of 5 nm. Si nan-

oclusters, and in general most nanoclusters formed by covalent atomic bonds like C

and Ge have complex structures. The shapes of these nanoclusters are determined

by the number of atoms that form them. Small clusters have high surface to volume

ratio and a number of free bonds for atoms on the surface. These dangling bonds

can cause radical surface re-alignment. To illustrate, below is a figure of an energeti-

cally favorable small Si nanocluster, 1 nm in diameter and comprised of 147 atoms -

equilibrated at 300 K. The shape of the nanocluster is clearly not spherical

Figure 12: Ball-and stick representation of a Si nanoparticle of 1 nm in radius

24

4.2 Surface Passivation

However at sizes of 5 nm and above, experimental and theoretical studies have

shown that the shape of the Si nanoparticle is spherical, with a diamond cubic struc-

ture [16]. To make spherical Si nanospheres, first a cubic block of Si atoms in diamond

structure was created. Spherical particles of the required size were cut off from this

cubic block. These particles were then relaxed, using a conjugate gradient relaxation

scheme, followed by a high temperature (800 K) equilibration over 2000 0.8 fs time

steps, using the Tersoff potential and a velocity verlet time progression scheme. The

cohesive energy of the particle thus obtained was ∼ 4.5 eV/atom.

A (001) type substrate is created with Si atoms arranged in the cubic diamond

configuration. The Tersoff potential has been found to give a good description of this

particular surface orientation [3], with the surface reconstructed with appropriate

dimerization. Substrate size was at 16 × 16 × 5.4 nm. This structure encompassed

71,800 atoms of Si. The bottom 2 layers of the substrate were kept static (i.e. the

3,600 atoms in these layers had their individual velocities nullified). This was done

to withstand the force of particle impact. The layer immediately above, comprising

18,000 atoms had their velocities follow the Langevin dynamics thus enabling them

to act as a thermostat and an energy drain for the system.


One of the central aims of this project is to study the effect that surface passivation

has on the adhesion of nanoparticles to the substrate. This is done by saturating

all the dangling bonds of Si atoms on surface of the particle and substrate with

H atoms. To do this, unsaturated Si bonds on the surface were identified - the

overwhelming majority of surface atoms had just one unsaturated bond, but there

25


Figure 13: Perspective on the simulated nanoparticle − surface system

were a minority of atoms (∼ 90) which had 2 unsaturated bonds. Then, H atoms

were placed at a proximal location of 1.4 A from the Si atoms. This structure

was put through successive cycles of conjugate gradient optimization, before a 500 K

molecular dynamics run, with a 0.8 fs time step. This time step was found to yield

an energetically stable structure, with the H atoms firmly adhered to the surface Si

atoms. A similar procedure was followed for saturating the substrate. This structure

was then optimized for a stable hydrogen terminated (001) surface. The overall

cohesive energy of this substrate was ∼ 12, 192.4 eV.

Now we can discuss the relevance of choosing 30 ps as the overall MD simulation

time. Upon impact, a fraction of the kinetic energy of the particle is converted to

thermal energy in the substrate. The characteristic time for draining this thermal

energy is a good indicator to determine the suitability of the MD simulation time.

26


Reference [47] carries a more detailed discussion of this process, but in brief - consider

the 1-D heat conduction equation:

∂T

∂t= α

∂2T

∂z2(4.14)

T stands for the substrate temperature. If we insert standard values of Si con-

ductivity (α) at 1000 K, which is 0.24 cm2

sand a substrate depth of 5.5 nm, the

characteristic heat conduction time would be about 1.5 ps. Thus we choose a net

simulation time which is 20 times this value.

The choice of a 0.8 fs time scale has more to do with the mechanics of the MD

simulation used. Considering that the mass of Si is ∼27.8 times that of a H atom,

using a large time scale at high temperature simulations may mean that the Si atom

has traversed a much smaller distance than the H atom. Energetically unstable

conditions may arise, especially if the distance between the atoms narrows down

enough to trigger very high repulsion. A smaller time scale was found to be more

feasible under such conditions. For simulations involving pure Si, a time steps of 2 fs

could be used without any instability. Reference [21] also gives a similar explanation

for the choice of time step in simulations involving H atoms.

Before each simulation, both the particle and the whole system are equilibrated at

500 K. This temperature is chosen to reflect actual experimental conditions of HPPD

[19]. Finally, the actual impact is initiated by imparting a translational velocity in

the z-direction to the nanoparticle. As a result the particle then drifts down towards

collision with the substrate. Impact happens after 2-3 ps depending on the initial

velocity. For the whole 30 ps a number of indicators are continuously monitored,

27

4.3 Simulation Indicators

to help understand the process of adhesion or reflection. The next section describes

these indicators.


The quantities which are monitored during the course of the simulation are:

− Phase-Space plots

− Particle temperature

− Particle-substrate contact radius

− Mean distributed contact stress

− Cohesive energy within the particle

− Particle-Surface adhesion energy

− Coordination number within the particle

− Radial distribution function within the particle

− Bond angle distribution function

Phase space plots are an easy graphical way to illustrate the dynamics of particle

impact. Figure 15 is a typical phase space plot for the impact of hydrogentated Si

with a bare substrate at 900 m/s. On the vertical axis is the instantaneous particle

velocity in the z-direction. The horizontal represents the distance between the centre

of the particle and the top layer of the substrate. This quantity is indicated in the

Figure 14. As can be seen in Figure 15, at the beginning of the simulation the particle

hurtles down at velocity of 900 m/s, at a distance of 10 nm above the substrate. When

the radius of the particle is subtracted from the center of mass (CM) values, the plots

give a clear idea of the instantaneous position of the particle with respect to the top

layer of the substrate. In this phase-space plot, the particle ends up bouncing from

28


the substrate, at a reflected speed of 300 m/s.

Figure 14: Center of mass (z) with respect to height of the substrate (d)

The instantaneous particle temperature:

Instantaneous particle temperature is crucial to understanding the transformation

of translational energy upon impact. This is measured by the following equation:

Tp(t) =m

3NpkB

Np∑i=1

v2i (t) (4.14)

Here, Vi is the instantaneous velocity of the ith atom in the particle. To do

away with the effect of translational velocity, Vi is measured in a coordinate system

that originates at the particle. Figure 16 shows temperature fluctuations within the

particle for 2000 m/s impact.

The particle-substrate contact radius:

When the particle impacts the surface, the contact evolves from a single point

of contact to a circular disk of contact between the particle and the surface. The

adhesion beween the particle and the substrate is largely limited to this contact area,

29


Figure 15: Example of a Phase Space Plot

Figure 16: Plot of instantaneous particle temperature at 2000 m/s impact

30


and thus quantifying this area of contact is important. Figure 17 is a picture of the

contact area for 2000 m/s collision between a bare Si particle and a bare substrate.

The figure on the left, is at maximum penetration and the one on the right is at the

final frame of the simulation after 40 ps. Close inspection reveals the contact area

has evolved to a smaller value.

Figure 17: Shape of the plane of contact

Since the contact area is roughly spherical, we can define the contact radius as

the mean distance of the positions of atoms in the plane of contact with respect to

their centre of mass. The equation for this is given by :

a2 =2

N

Na∑i=1

(xi −Xcm)2 + (yi − Ycm)2 (4.14)

Figure 18 shows contact area radius’ fluctuations within the particle for 900m/s im-

pact.

The mean distributed contact stress:

Quantifying stress due to impact is necessary because, high stress levels are needed

to induce a phase transition within the particle and phase transition is a crucial

channel for the flow of impact energy. Once the contact radius is determined, stress

is simply the force acting downward on the substrate divided by the area of contact.

31


Figure 18: Evolution of contact radius for 900 m/s impact

In the figure below, the area used to find the contact stress is high lighted in red.

Figure 20 shows contact stress’ fluctuations within the particle for 900 m/s impact.

The particle cohesive energy (CE):

It can be defined as the net binding energy of an atom with its neighbours. An

atom in crystalline Si has a typical cohesive energy of 4.5 eV, whereas a high

pressure form the β-tin form has a lower cohesive energy of ∼ 4.3 eV [49]. Consider,

for example a 2 nm Si nanoparticle subject to uniaxial compression (36 percent) on

either side. High axial stress causes structural changes within the nanoparticle. This

change is directly represented by changes in the individual cohesive energy of atoms.

In the figure below, grey atoms correspond to CE of 4.6 eV while light blue atoms

have a CE of ∼ 4.3 eV.

The particle - surface interaction energy:

Once all atoms of the particle in contact with the substrate are identified, the

32


Figure 19: Region of contact stress in the particle (shown in red)

Figure 20: Evolution of contact stress for 2000 m/s impact

33


Figure 21: Atoms in strained nanocluster as a function of cohesive energy − greyatoms correspond to CE of 4.6 eV while light blue atoms have a CE of ∼ 4.3 eV.

net energy of all their bonds with atoms of the substrate is recorded. This energy is

divided by the instantaneous contact area to the contact energy per unit area.

The coordination number:

Coordination number is the total number of neighbors for a central atom. Cubic

diamond Si atoms have a coordination of 4. Other forms of Si have different coor-

dination numbers. The β-tin phase, for example has a coordination number of 6.

Monitoring changes in this coordination number is a cue to understand the structural

changes that occur during impact.

The radial distribution function (g):

The RDF is a pair correlation function which describe how, in a structure, its

atoms are arranged radially around each other. In other words, beginning with a

random atom in the system as a centre, it is the average number of atoms one might

expect to find at a particular radial distance from it. RDFs help to easily pinpoint

the atomic structure, especially when the structure is quickly evolving. The following

formula follows reference [1]

34


Figure 22: Atoms with coordination number 6 versus time

g(r) =n(r)

ρ4πr2∆r(4.14)

In the above equation g(r) is the RDF, n(r) is the mean number of atoms in a shell

of width ∆r at distance r, ρ is the mean atom density.

Figure 23: Radial distribution function

35


The bond angle distribution function (BADF):

The BADF simply presents the frequency distribution of bond angles between

atoms of the particle. The BADF for cubic crystalline Si is, for example centered

around 109.5o. This, however is bound to change if the structure itself changes.

In fact, the BADF along with RDF, serves to fingerprint a particular structural

configuration.

36

5 Molecular Dynamics Results - Impact of Hydro-

genated Si Nanoparticles on Bare and Hydro-

genated Si Substrate

In a series of classical MD simulations we examined the microscopic details of the

collision process between a projectile Si nanosphere of 5 nm in radius (R) and a Si

substrate exposing its (001) surface. The considered impacting speeds (V0) were of less

than 2, 000 m/s. To account for the practical conditions of reduced surface reactivity

we considered fully H-coated surfaces [35] [34]. The covalent Si-Si bonding, the

surface chemistry, and the dynamical bonding between the nanosphere and substrate

were described with transferable potentials based on the concept of bond order [44].

This family of interatomic potentials describe very well the most stable phases of

Si in the absence and under external pressure, as well as the Si-H bonding, but

are less for describing fracture. However, the fracture of Si nanospheres was not

identified experimentally [16]. Both the nanosphere, containing 31, 075 Si atoms, and

the substrate, containing 72, 000 Si atoms, were initially equilibrated at 500 K, in

order to mimic the elevated temperature experimental conditions of Rao et al. [38].

As before [48], during collision the last two bottom layers were kept fixed in time and

the substrate temperature was controlled with Langevin dynamics applied to the next

ten atomic layers. All other atoms were followed with a velocity Verlet algorithm. A

time step of 0.8 fs was used for all atoms. Periodic boundary conditions are applied

to the horizontal X and Y directions (the Z impact direction is vertical).

Unlike under energetic impact conditions [18], our MD didn’t lead to fragmenting

37

and spreading of the nanosphere, or cratering of the substrate. Only a few H atoms

were ejected under the highest V0. The obtained collision modes can be identified

in the panels of Fig. 24, presenting the phase space trajectories for the center of

mass of the impacting nanosphere under three values of V0 and three surface-coating

combinations. For the convenience of comparison, the instantaneous speed (V ) was

normalized by V0.

On the approach stage the important parameter is V0. The nanosphere is touching

down and it advances with negative acceleration until a maximum penetration point

is reached. The magnitude of V0 is reflected in the nanosphere deformation measured

when R > Z by δ = R − Z, where Z is the vertical positive displacement of the

nanosphere center of mass measured with respect to the top of the substrate. The

adhesive forces appear secondary since as in the Hertz model [26], the nanosphere

does not show accelerated motion in response to the adhesive contact forces. Addi-

tionally, under the same V0, values for δ presented in Fig. 24 are similar for all three

surface combinations considered: H-passivated nanosphere and substrate (black), H-

passivated nanosphere and bare substrate (gray), and bare nanosphere and substrate

(light gray).

On the recoil stage the outcome depends on both V0 and the chemical reactivity of

surfaces. For V0 = 900 m/s the black and gray curves indicate that the impacting H-

passivated nanosphere is reflected by both the bare and H-passivated substrate. The

collision is practically elastic as V0 restitution is as high as 80%. Our MD showed

that the reflection mode, which eluded previous microscopic investigations [48, 9],

dominates for V0 < 1, 000 m/s when weakly interacting surfaces are involved. As

a useful reference, the light gray curve in Fig. 24(a) indicates that under the same

38

Figure 24: Phase space trajectories for three impacting speeds: (a) 900 m/s, (b)1, 300 m/s, and 2, 000 m/s. The nanosphere approaches the substrate with positiveV , touches down at δ = 0, reaches V = 0 at maximum penetration, and recoils withnegative V

V0 but larger adhesion, the bare nanosphere follows the path of a spiral sink and it

is captured by the bare Si substrate. V and δ form a damped cyclic path towards

the final equilibrium point, which is not reached during the shown 40 ps of MD

time. Thus, energy dissipation is very slow. This mode is the previously identified

soft landing [48, 9], where the deposit maintains its crystalline structure and the

dissipation of the incident energy involves the substrate.

To our surprise, for larger V0 a different behavior occurs: The H-passivated

nanosphere impacting on the bare substrate, Fig. 24(b), and even on the H-passivated

substrate, Fig. 24(c), falls into the path of a pure sink and it is captured with no oscil-

lations. More precisely, this behavior was obtained above the critical V0 of 1, 250 m/s

(1, 550 m/s) for the impact on the bare (H-passivated) substrate. Large δ values can

be noted, indicating that the nanosphere experiences significant deformation. The

39

noted small V0 restitution and the lack of vibrational contact demonstrate efficient

dissipation of the incident energy. It can be also seen that the soft landing mode

undergoes the same qualitative transition, which was the main finding of a previous

work [48].

The collision dynamics is further conveyed in Fig. 25. In a typical reflection the

H-passivated nanosphere experiences deformation over a finite region of circular shape

around the point of contact, as macroscopically expected. At maximum penetration,

Fig. 2(a) (middle), there is a significant potential energy (PE) increase only near the

region of contact, as indicated by the change in color to pink(gray level). This change

appears reversible as the spherical shape is regained and the energetic differences

are washed out after detachment (last frame). The capture mode, Fig. 25(b), is

characterized by a severe deformation of the approaching nanosphere, which assumes

at maximum penetration (middle) a dome-like shape with a large contact region.

There is a significant PE change in a large conical volume of height ∼ R and with the

contact region as base. The irreversible character transpires from the PE distribution

in the final configuration (last frame), and from the sphere cut out by a plane shape

of the end deposit, optimal for adhesion.

Adhesion is quantified geometrically in Figure 26(a), which presents the time-

evolution of the contact radius (a) during the two collision modes. During approach

a increases and reaches its maximum at the maximum penetration instant, regardless

of the surface chemistry. The final a value depends on both V0 and the surface

chemistry. Focusing on the capture mode, we see that a decreases during the recoil,

most significantly when both the nanosphere and substrate are H-passivated. Even

in this case the final a is large, comparable with R. Adhesion was next quantified by

40

Figure 25: MD simulations of H-passivated Si nanosphere impacting onto a H-passivated Si substrate show two collision modes: (a) reflection and (b) capture.The middle frames show the maximum penetration instant. Only a cross-section isshown and H atoms are not represented. The color code carries the local PE, withblue (gray) and pink (light gray) representing atoms with PE larger and smaller than4.4 eV/atom, respectively

41

the finite adhesion energy (γa), defined as the bonding energy measured per contact

area, between the atoms of the nanosphere and those of the substrate. We obtained

0.07 eV/A2 and 0.03 eV/A2 for the H-passivated nanosphere impacting on the bare,

and H-passivated substrate, respectively. For comparison, γa = 0.1 eV/A2 when bare

Si surfaces are involved. Thus, it appears that the capture occurs without substantial

support from the adhesive contacts. Another observation of interest is that after

approach, γa is not changing notably because of the large number of H atoms trapped

in the contact region.

Fig. 26(b) shows the mean distributed contact-stress (pm), computed as the net

vertical force acting on the nanosphere divided by the instantaneous contact area πa2.

Our recorded p data confirms that significant pressures are reached under hypersonic

speeds, sufficient to produce yield according to experimentation [16]. Interestingly,

pm vanishes after the collision not only in the reflection but also in the capture mode.

In the latter, it indicates the occurrence of a stress-relieving transformation.

During collision the incident energy is converted into internal degrees of freedom,

producing temperature (∆Tp) and PE (∆Up) changes in the nanosphere. Examination

of the energy flow confirms the large dissipation in the capture mode. In Figure 26c

the large ∆Tp shows that the particle heats up during and after the 2, 000 m/s collision

and thus a large amount of the incident energy is irreversibly transferred into thermal

agitation. Figure 26(d) demonstrates the occurrence of a plastic change since after

collision there is a significant PE excess with respect to the original crystalline state.

To identify the nature of this plastic change causing ∆Tp and ∆Up rises, we moni-

tored the nanosphere structure by computing its pair-correlation function (g). Partic-

ularly revealing were investigations at the maximum penetration instant, presented in

42

Figure 26: Time evolution of (a) contact radius and (b) contact-stress. Variation ofnanosphere’s (c) temperature, and (d) potential energy under two impacting speeds,V0 = 900 m/s (lower two curves) and V0 = 2, 000 m/s (upper two curves), andtwo surface types, H-passivated nanosphere and substrate (black) and H-passivatednanosphere and bare substrate (gray). Down arrows mark the particle release instants

43

Figure 27. On one hand, under V0 = 900 m/s Figure 27(a) indicates that the cubic Si

structure is maintained, as the main peaks are well centered on the neighbor position

in cubic Si, where each atom bonds to four other equivalent atoms in an undistorted

tetrahedral pattern, Figure 27(e) left. On the other hand, under V0 = 2, 000 m/s

Fig. 26(b) shows the presence of a new structural arrangement, other than cubic Si.

Indeed, there is a widening and simultaneous decrease of the main peak height, as

well as the appearance of a new peak at the interatomic distance (r) of ∼ 3 A. For

more insight, we focused on the conical volume with high PE indicated in Fig.25(b)

middle, where we found 5, 500 sixfold coordinated Si atoms with bond lengths that

did not exceed 2.8 A. In addition to g, we computed the bond angle (BA) distribu-

tion. These results prove that the dominant structural changes correspond to a β-tin

phase of Si, shown in Fig. 27(e) right. This new phase is derived from the cubic Si by

flattening the tetrahedral grouping to the extent that two other atoms are brought

into close proximity, which increases the coordination from 4 to 6. Indeed, in Figure

27(c) g has two sharp peaks at 2.5 and 3.1 A, which match very well the first and

second neighbor distances of β-tin Si. All BA are 109◦ in cubic Si. However, in Figure

27(d) they are grouped around the five distinct BA of the β-tin.

On the basis of this key observation, the observed reflection to capture transition is

interpreted as follows. On the approach stage, nanosphere-substrate adhesion occurs

due to the new Si-Si covalent bonds formed in the contact region. For V0 < 1, 000 m/s,

these bonds rupture under the vigorous backwards motion caused by the largely re-

flected incident energy. However, for V0 > 1, 000 m/s, in the absence of dislocation,

the atomic rearrangement caused by a displacive phase change represents the next

available gateway for absorbing irreversibly a significant amount of the incident en-

44

Figure 27: Pair-correlation function of the H-passivated Si nanosphere at maximumpenetration for (a) V0 = 900 m/s and (b) V0 = 2, 000 m/s. Vertical bars mark theneighbor position in cubic Si. (c) Pair-correlation and (d) bond-angle distributionsin the most deformed conical region. Vertical bars mark neighbor positions and bondangles in bulk β-tin Si. (e) Unit cells for cubic-diamond (left) and β-tin Si (right)

45

ergy. Having a smaller crystallographic c/a0 ratio (0.55, with a0 = 4.73 A) than

cubic-diamond (1.42, with a0 = 3.82 A), the β-tin phase has geometric advantage

as it simultaneously relieves the compressional stress and augments the contact area.

Only the PE stored in elastic deformation is returned on the recoil as coherent reverse

motion of the nanosphere atoms. Capture occurs when the increased adhesion is able

to overcome the weakened recoil motion.

Further monitoring of g demonstrated that, as in the case of bare Si nanospheres [48],

almost no β-tin phase exists in the end deposit. This transition is documented in Fig-

ures 28 and 29. Figure 28 shows the RADF for the particle at the end of 40ps. As

can be seen, there is a small bulge between 3-3.5A, which cannot be seen for bulk

crystalline Si. While amorphization is not very obvious from this figure, the Bond

Angle Distribution shown in 29 is more directly indicative of the amorphization -

with bond angles varying widely about the standard diamond cubic Si angle of 109.5◦

Although not themodynamically favorable, the trapped a-Si is favored here since this

non-crystalline arrangement accommodates both with the unaffected upper spherical

portion of the particle with cubic structure and the large contact, see last frame of

Figure 25(b).

In conclusion, our large-scale MD simulations showed that the poor reactivity of

surfaces prevents the sticking of projectile H-passivated Si nanospheres. However,

sticking is possible when hypersonic speeds are attained. It is accompanied by the

irreversible conversion of the incident energy into thermal agitation, and by the oc-

currence of a phase transition, from diamond to β-tin, the latter eventually evolving

to a-Si. These phase changes, also seen in the MD collision simulations with bare

Si surfaces [48], are not unrealistic. Although the considered impacting energies are

46

Figure 28: RADF at the end of 30 ps for impact at 2000 m/s with hydrogen on bothsurfaces

Figure 29: BADF at the end of 30 ps for impact at 2000 m/s with hydrogen on bothsurfaces

47

relatively low and the nanosphere-substrate contact forces are not large, because of

the nano-size extent of the contact zone the resulting pm are high enough to induce

a first-order phase change. Indeed, both experimental and theoretical examinations

of Si under pressure have revealed the existence of diverse crystalline phases, other

than cubic-diamond. The first such phase is the β-tin state, which emerges under a

pressure of 11 GPa [14]. More recently, the cubic to β-tin phase change was observed

in Si nanocrystals during anvil experiments under a pressure of 22 GPa [46], in good

agreement with our data. The subsequent amorphization of the β-tin on the recoil,

is in contrast with the recovery of a metastable BC8 phase in bulk Si [51] but in

agreement with the amorphization of the β-tin in Si nanocrystals after the release of

the hydrostatic pressure [46]. The MD obtained succession of phase changes on a ps

time scale is of interest for the fundamental understanding [23] of the transformation

kinetics of solid phases in Si nanoparticles. More practically, the identified capture

mediated by phase transition plasticity improves our comprehension of the hypersonic

nanoparticle deposition experiments.

48

6 More Molecular Dynamics Results - Impact of

Si31,075N413, C90,326 and Si12,080C12,080 Nanoparticles

on Bare Si Substrate

While the simulations in this project deal primarily with impact of hydrogenated Si

nanoparticles, preliminary simulations were done with nitrogen coated Si nanopar-

ticles, C nanoparticles and SiC nanoparticles. All three cases present three totally

different impact scenarios, and provide important insights into the effect of chemical

adhesion and structural hardness on the outcome of the impact. The Tersoff potential

already used for all simulations is easily extendible to these elements.

6.1 Si31,075N413 Nanoparticles

As mentioned in Chapter 2, the Tersoff potential and parameters are modified to

account for Si-N interaction. Interaction energy for Si-N is actually stronger than Si-

Si interaction energy. This makes nitrogen coated nanoparticles a very special case.

Coating a Si nanoparticle with N is essentially giving it a very adhesive skin. Thus,

the impact basically follows the pattern of bare Si impact which is adhesion under

all circumstances, albeit with a stronger adhesion to the substrate. The process of

coating itself is akin to the one described in Chapter 3. Initally, the nitrogen is placed

close to Si atoms on the surface with dangling bonds. Successive conjugate gradient

minimization and high temperature MD procedures are carried out to retain the N

on the surface. A time step of 0.8 fs is used through out.

At an impact speed of 2,000 m/s, the particle firmly adheres to the surface. The

49


particle surface adhesion energy was the highest among all cases seen thus far. The

picture below plots this contact energy per unit contact area for 2000 m/s impact.

Bare Si has a unit adhesion energy of 0.1 eV/A, whereas Si-N sees a 50% increase to

0.15 eV/A.

Figure 30: Time evolution of contact energy

This effect only slightly seems to affect the area of contact or the structural defor-

mation. The figures below plot the time-evolution of these 2 parameters. Ultimately,

all particles seem to settle into a nearly even contact area after 40 ps. The structural

deformation represented in Figure 32 is nearly equal for bare Si and nitrogen coated

Si.

50


Figure 31: Time evolution of contact area for different surface chemistries

Figure 32: Time evolution of coordination number plot for SiN

51

6.2 C90,326 and Si12,080C12,080 Nanoparticles

Finally looking at the trajectory of phase space plots for impact at 2,000 m/s,

with different surface chemistries, SiN impact-represented as a solid line in Figure

33, follows the other trajectories before diverging to settle down to a more submerged

position within the substrate.

Figure 33: Phase space trajectories at 2,000 m/s


MD simulations were carried out to understand the impact of C nanospheres with

bare Si substrates. The carbon particles were modeled as nano-meter sized diamond

crystals using the Tersoff paramters for C [45]. The case of C is unique because of

its very high inherent strength, which in turn means there is very sparse structural

deformation. This lack of any structural deformation means an important channel for

the dissipation of internal energy is shut off and most of the impact energy is reflected

back - leading to particle bounce. The simulations were not advanced enough to keep

52


a tab on all parameters like in the previous results. However, the phase space plot

shows the particle bouncing at all three velocities of 900 m/s, 1,600 m/s and 2,000

m/s.

Figure 34: Phase space trajectories for C90,326 nanoparticle

As can be seen from Figure 34, the restitution of impact (reflected velocity/impact

velocity) is very high for C nanoparticles. This is in sharp contrast to the case of Si,

shown in the figure above. For Si, after the onset of phase transition at 1,000 m/s,

the rebounce becomes smaller and smaller until it finally becomes zero. The lack

of palpable structural deformation is directly reflected in the C nanoparticle’s high

restitution.

Finally, when we consider SiC, we found the particle adhered to the surface on all

3 simulations that were performed. Again the simulations were not advanced enough

53


Figure 35: Restitution for impact of Si nanoparticle with hydrogen on both surfaces

54


to map out all system parameters, but the behavior of Si closely mimics the bare

Si behavior. We may assume that the chemical adhesion between the particle and

substrate was too high to allow for any rebounce. The phase space trajectories are

plotted below for SiC impacting at 900 m/s, 1,300 m/s and 1,600 m/s.

Figure 36: Phase space trajectories for Si12,080C12,080 nanoparticle

55

7 Future Work

Studies in this project were limited to studying the impact of one nanoparticle. The

maximum size studied was a radius of 5 nm. These limitations were primarily be-

cause of the computational constraints of simulations involving more than 200,000

atoms. Large scale simulations of multi-particle impact have a huge potential to yield

important results. Particularly interesting would be the simulation of the impact of

newly formed nanoparticles with ones already adhered. This would ultimately give

insights into the process of sintering leading to thin film formation, for the case of

hypersonic deposition. However, carrying out the above simulations would require

for the current code to be parallelized.

Preliminary simulations done in this project with C and SiC nanoparticles can

be extended to thoroughly understand their deposition mechanism. Silicon Carbide

deposition is widely studied as a means to fabricate wear resistant coating [5], hence

simulations in this direction would be particularly useful. The modified Tersoff Po-

tentials cannot describe Si-O interaction suitably. Given the widespread use of SiO2

in the semiconductor industry, simulations with oxygen coated nanoparticles would

be more relevant.

Finally the effect of size on particle deposition is a very interesting aspect to con-

sider. Reference [49] studies the uniaxial compression of Si nanoparticles to quantify

phase transition within them. However, preliminary simulation with particles 1 nm or

less in radius did not find a phase transition to β−tin Si, but rather a skew displace-

ment within the lattice. Hence the outlet for impact energy (namely phase transition)

which has been so crucial to particle adhesion in our simulations may not be equally

relevant for smaller sizes. The size threshold at which the impacting nanoparticle can

56

have an internal structural dislocation, is another important direction to pursue.

57

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