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Molecular-scale studies of mechanical phenomena at the interface between two solid surfaces: from high performance friction to superlubricity and flash heating

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Molecular-scale studies of mechanical phenomena at the interface between two solid surfaces: from high performance friction to superlubricity and flash heating

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Content Molecular-Scale Studies of Mechanical Phenomena at the Interface Between Two
Solid Surfaces: From High Performance Friction to Superlubricity and Flash
Heating
Nariman Piroozan
Advisor: Professor Muhammad Sahimi
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMICAL ENGINEERING)
Mork Family Department of Chemical Engineering and Materials Science
University of Southern California
May 2019
Acknowledgment
The rst several lines of Walt Whitman's 'Song of the Open Road' have always been an
inspiration to me. They harken to mans innate desire to explore and to learn. It is in this
spirit that the research in this Thesis was carried out, and the results were written. There are
a number of people I would like to thank who helped make it possible.
First, I would like to thank my advisor, Professor Muhammad Sahimi. His patience and
dedication to his students is without parallel. In aording his students the freedom to learn
and explore on their own terms, they are able to succeed far beyond their own expectations.
He and I delved into the intricacies of tribology for a variety of dierent materials and made
advances in the eld that we hope others will nd signicant. For that I am eternally grateful
to him.
I would also like to thank the members of my qualifying exam committee, Professors Aiichiro
Nakano, Rajiv Kalia, Katherine Shing, and Malancha Gupta. I appreciate their guidance and
constructive comments that allowed me to conduct the best research within the bounds of my
capability.
I would also like to thank the sta within the Mork Family Department of Chemical Engi-
neering. Andy Chen, the late Martin Olekszyk who left us too early, Heather Alexander, and
Karen Woo were all very gracious and helpful along the way.
Above all, I would like to thank my father, Professor Parham Piroozan. The greatest of
gentlemen, he has borne every burden, overcome countless obstacles, achieved the unachievable,
and has remained steadfast and dedicated to his family throughout his life; the greatest father
a son could ever ask for.
Finally, I would like to thank my late and beloved mother. She has always given me the
strength to persevere and overcome.
1
Contents
Acknowledgment 1
List of Figures 4
Abstract 7
1 Introduction 9
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Molecular Dynamics and Hamiltonian Formulation . . . . . . . . . . . . 11
1.3 Friction at High Performance Parameters . . . . . . . . . . . . . . . . . . 13
1.4 Superlubricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Flash Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 The Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Sliding friction between silicon-carbide surfaces 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Molecular Model of SiC and the Force Field . . . . . . . . . . . . . . . . 20
2.3 Models of Contacting SiC Surfaces . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Determination of Dynamic Friction Force and its Velocity Dependence 23
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Frictionandsuperlubricitybetweensurfacesofpreciousmetalsandgraphene
nanoribbons: A molecular dynamics study 36
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2
3.2 Molecular Models and Force Fields . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Calculation of Dynamic Frictional Forces and Shearing Distance . . . . 42
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Flash Heating and Sliding Friction between Quartz Surfaces 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 The Force Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Models of Contacting Quartz Surfaces . . . . . . . . . . . . . . . . . . . . 58
4.4 Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Bibliography 71
3
List of Figures
1.1 Schematic of an MD simulation model. . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Asperity interaction between two surfaces. . . . . . . . . . . . . . . . . . . . . . 15
2.1 Schematic of the system consisting of the two SiC slabs with the top layer (TL),
top thermostat layer (TTL), the interface between the two slabs, the bottom
thermostat layer (BTL) and the bottom layer (BL). F
n
is the normal force ap-
plied, while V
y
is the sliding velocity that pulls the top layer in the y direction
relative to the bottom slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Dynamic evolution of the temperature at the interface between the two SiC slabs. 24
2.3 Dependence of the average temperature at the interface between two amorphous
slabs on the sliding velocity V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Evolution of the excess free volume at the interface between two amorphous SiC
slabs, and its expansion as the sliding velocity increases. . . . . . . . . . . . . . 26
2.5 Same as in Fig. 4, but in the system that consists of two crystalline slabs. . . . . 26
2.6 Average number of atoms N(z) in thin parallelepipeds perpendicular to the
vertical axis z, and its dependence on the sliding velocity V . z = 0 represents
the interface, and the dip around it is due to formation of excess free volume as
a result of stretching and breaking the SiC bonds there. . . . . . . . . . . . . . . 27
2.7 Dependence of the potential energy of the amorphous system on the sliding
velocity V and sliding distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.8 Dependence of the friction force on the sliding distance and velocity V in the
amorphous system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Same as in Fig. 8, but for friction force between two crystalline slabs. . . . . . . 31
4
2.10 Dependence of the kinetic friction forceF
k
on the sliding velocityV . ForV 35
m/s, dependence of F
k
on V is perfectly logarithmic, while after a transition
region the same type of dependence appears to be roughly followed. . . . . . . . 33
3.1 Schematic of the system consisting of the substrate and GNR. . . . . . . . . . . 41
3.2 Dynamic evolution of the interface temperature in the three pairs at a lifted
height of 2 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Friction force proles for the Au-GNR pair at the lifted height of 1 and 2 nm
nm for both forward and reverse directions. . . . . . . . . . . . . . . . . . . . . 44
3.4 Same as in Figure 3, but for the Ag-GNR pair. . . . . . . . . . . . . . . . . . . . 45
3.5 Same as in Figure 3, but for the Pt-GNR pair. . . . . . . . . . . . . . . . . . . . 45
3.6 The separation eparation distances at which the total potential energies of the
three pairs attain their minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7 Potential energy
uctuation per stick-slip event for the Au-GNR pair in (a)
forward and (b) reverse directions at a lifted height of 1 nm. . . . . . . . . . . . 47
3.8 Same as in Figure 7, but for the Ag-GNR pair. . . . . . . . . . . . . . . . . . . . 48
3.9 Same as in Figure 7, but for the Pt-GNR pair. . . . . . . . . . . . . . . . . . . . 49
3.10 Shearing distance in the Au-GNR pair at the lifted height of 3 nm in forward and
reverse directions. Point 1 signies the initial conformation of the GNR; point
2 represents the maximum deformation prior to slip, while point 3 indicates the
conformation after slip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.11 Same as in Figure 10, but for the Au-GNR pair. . . . . . . . . . . . . . . . . . . 51
3.12 Same as in Figure 10, but for the Pt-GNR pair. . . . . . . . . . . . . . . . . . . 51
3.13 Average friction force for the three pairs in the forward direction. . . . . . . . . 52
3.14 Average friction force for the three pairs in the reverse direction. . . . . . . . . . 53
4.1 Explicit detail of Structures A, B and C of quartz used in the simulation. . . . . 58
4.2 Schematic of the system consisting of the two quartz slabs with the top layer
(TL), top thermostat layer (TTL), the interface between the two slabs, the bot-
tom thermostat layer (BTL) and the bottom layer (BL). F
n
is the normal force
applied, whileV
x
is the sliding velocity that pulls the top layer in they direction
relative to the bottom slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5
4.3 Dynamic evolution of the temperature prole of the thermostatted regions in the
three structures at a sliding velocity of 0.3 m/sec. . . . . . . . . . . . . . . . . . 62
4.4 Dynamic evolution of
ash heating event in Structure A at a sliding velocity of
0.3 m/sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 Maximum interfacial temperature during
ash heating events across the entire
spectrum of sliding velocities and the three Structures. . . . . . . . . . . . . . . 63
4.6 Prole of
ash heating events in Structure C at various velocities. . . . . . . . . 64
4.7 Shear stress-shear strain diagrams for the three structures at various sliding
velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.8 Temperature prole in Structure A immediately before and after
ash heating
events at various sliding velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9 Same as Figure 4-9, except for Structure C. . . . . . . . . . . . . . . . . . . . . 66
4.10 Dependence of the kinetic friction forceF
k
on the sliding velocityV . A logarith-
mic relationship between F
k
and the sliding velocity appears to hold. . . . . . . 68
4.11 Dependence of the eective friction coecient
e
on the sliding velocity V . A
logarithmic relationship between
e
and the sliding velocity appears to hold. . . 69
6
Abstract
Friction is an essential part of human experience, as it is a key component in our ability to
gain traction to walk, stand, work and drive, and also save energy. While friction is certainly
a necessary phenomenon, it presents one glaring problem. We need energy to overcome the
resistance to motion that is caused by frictional forces, and too much of it causes excess cost,
increased energy to perform work and, therefore, increased ineciencies as well. It is the
purpose of this Thesis, then, to present an advance towards two interconnected objectives in
this regard. The rst is a better understanding of the mechanical and energetic precursors that
give rise to friction and, then, to present applicable results in this regard. These objectives
are studied in this Thesis, including the tribological properties of silicon carbide, a material
commonly used in high-temperature conditions, and the superlubric properties of graphene, an
excellent soild lubricant, and its behavior coupled with a series of precious metals. The nal
chapter will study friction between sandstone surfaces, represented by quartz, one of the most
common terrestrial minerals, and its frictional properties at subseismic sliding conditions.
Sliding friction between two SiC surfaces is important, due to its relevance to many practical
applications. It is also important to study whether kinetic friction at the nanoscale follows
Coulomb's law. Since SiC exists both as an amorphous material and with a crystalline structure,
the eect of surface roughness on the kinetic friction may also be signicant. We report the
results of extensive molecular dynamics simulation of sliding friction between surfaces of the
two types of SiC over a wide range of sliding velocities. The amorphous SiC was generated
by the reactive force eld ReaxFF, which was also used to represent the interaction potential
for the simulation of sliding friction. As the sliding velocity increases, bond breaking occurs
at the interface between the two surfaces, leading to their roughening and formation of excess
free volume. They reduce the kinetic friction force, hence resulting in decreasing the dierence
between kinetic friction in the amorphous and crystalline surfaces. The average kinetic friction
force depends nonlinearly on the sliding velocity V , implying that Coulomb's law of friction is
not satised by the surfaces that we study at the nanoscale. The average kinetic friction force
F
k
depends on V as, F
k
/ lnV .
The state of extremely low friction, known as superlubricity, has very important applications
to the development of various types of materials, including those that are invaluable to the
7
goal of reducing energy loss in mechanical systems, and those in complex gearing and bearing
systems. One material that can produce very low friction is graphene that oers distinct
properties as a solid-state lubricant, and can potentially be used as a coating material on
surfaces. We have carried out extensive molecular dynamics (MD) simulations in order to study
and compute the friction force between a graphene nanoribbon and an inert metal surface. The
metal surfaces that we study are those of Au, Ag, and Pt, all of which are used in various
instruments, as well as in various materials employed in the industry. Consistent with very
recent experiments, the Au-graphene system exhibits superlubricity, but the Ag-graphene and
Pt-graphene pairs manifest friction forces higher than that of Au-graphene, although they are
still very small. The MD simulations indicate that the average friction force for graphene on
an Au surface is approximately 1.5 pN, with the corresponding values being about 6 pN and
11 pN for, respectively, Ag and Pt surfaces.
The frictional behavior of faults during seismic events, such as earthquakes, is a topic of
profound importance. The experimentally-observed degradation of shear strengths in silicate
rocks has been hypothesized to emanate from the phenomenon of "
ash heating." Using ex-
tensive MD simulations, we study this phenomenon at interstitial asperities for quartz crystals
and present extensive results on the mechanical and thermal evolution of the material under
shearing and various sliding velocities. Furthermore, we also present the evolution as a func-
tion of varying the thickness of the two layers sliding with respect to each other. We discover
that with an increase in the sliding velocity, the frequency and intensity of
ash heating events
increase as well. This in turn destabalizes the crystalline structure at the interface between
opposing quartz layers, forming an amorphous layer. At low slip rates, the heat generated is
able to diuse away appreciably through the material, resulting in a small temperature rise
and, therefore, weak eect on the overall strength of the material. At higher slip rates, how-
ever, there is not enough time for the heat generated at the interface to dissipate. This in
turn causes an increase in the overall interstitial temperature, which in turn causes decreased
material strength, followed by sharply reduced frictional resistance.
8
Chapter 1
Introduction
1.1 Background
Friction is an essential part of human experience, as it is a key component in our ability to
gain traction in order to walk, stand, work and drive. While friction is certainly a necessary
phenomenon, it presents one glaring problem. We need energy to overcome the resistance to
motion that is caused by frictional forces, and too much of it causes excess costs, increased
energy to performa work and, therefore, increased ineciencies as well. Modern problems,
ranging from climate change, to overpopulation and energy shortages have forced socities to
put an increased premium on a more ecient allocation and application of resources. We see
this in the design of more fuel-ecient cars, ships and aircrafts. In each one of these examples,
the goal of increased eciency will inexorably lead to the need to overcome the challenge of
friction. For example, let us detail a signature case study where frictional forces are important.
When compared to the power produced at the crankshaft, modern transmissions in cars account
for roughly a 15-25 percent loss of power through the drivetrain. The transmission is ultimately
an assortment of gears and gear trains in direct contact with one another, converting angular
speed and torque. In other words, it is a device with a very high number of moving parts in
contact with one another and, therefore, a very high level of friction inherent in its design. Yet,
with this and many more examples in industrial designs, our understanding of the fundamental
nature of friction is still very much incomplete.
The rst controlled experiments, relating to the study of friction, dates back to the 16th
9
century and the work done by Leonardo da Vinci. Although this is when intensive studies
regarding the pheonemnon of friction, progress has been very slow due to the lack of proper
instrumentation to measure friction precisely. To combat this, at the dawn of the industrial
age, scientists and engineers approached the problem of friction from an empirical perspective.
Whether it be in the study of
uid mechanics and the Darcy friction factor, or in study of
compressible
uid
ow and the Atkinson friction factor, the understanding of this phenomenon
was mainly supercial and limited in scope. Many of such models are no longer sucient due
to the advent of modern, high performance industrial products, such as spacecraft, hypersonic
aircraft, and supercars. The design constraints inherent in these products have forced engineers
to have a much clearer understanding of the exact nature of friction and how it can be controlled.
Without such understanding, there will be a clear ceiling with respect to the performance in
the designs of the aforementioned products.
An object at rest generally requires a nite force, referred to as the static friction force,
in order to initiate motion. Upon the initiation of motion, a dierent force is needed in order
to maintain steady sliding motion. We refer to this as the kinetic friction force, which will
be the main focus of this thesis. Experimental studies into the nature of friction and its
importance have seen a dramatic increase in the last few decades. The work done has typically
approached lubricity, i.e. reduction of friction, in one of two ways, namely, the reduction of the
friction force through the application of a lubricant, or the selection of novel materials. This
thesis will focus on the latter approach, using a computational method anchored in molecular
dynamics simulations. Characterization of frictional forces across an interfacial surface has
been done experimentally, but it should be remembered that friction at its most basic level
is an interatomic dynamic. To that end, MD simulations can reveal novel facets of friction
previously unknown, and push our understanding further on a vital topic.
In the year 1814, Pierre Simeon de LaPlace wrote:
"Given for one instant an intelligence which could comprehend all the forces by which nature
is animated and the respecitve situation of the beings who compose it - an intelligence
suciently vast to submit these data to analysis - it would embrace in the same formula the
movements of the greatest bodies of the universe and those of the lightest atoms; for it,
nothing would be uncertain and the future, as the past, would be present to its eyes."
10
The computations inherent in the eld of molecular dynamics encompass a modern real-
ization of this most basic of posits, namely, that the behavior of a system can be computed
if we have a set of initial conditions plus the forces of interactions within the system. Molec-
ular dynamics is essentially just the compuation of the position, velocties and orientations of
molecules at the atmoic level. The key idea here can be reduced to just one word - motion.
The motion of atoms as they translate from point A to B, colliding with one another and with
their container over a given period of time. In this section of this thesis, we delve into the heart
of this eld, its foundation and how it relates to the work we will later present, describe and
discuss.
1.2 Molecular Dynamics and Hamiltonian Formulation
The most fundamental nature of MD simulations can be reduced to a simple discussion of
Newtonian and Hamiltonian dynamics. In Newtonian dynamics, the translational motion of a
spherical molecule is caused by a force exerted by a foreign source. The motion and the applied
force are directly related through the classical equation:
F =m r
i
(1.1)
The acceleration, r
i
, is the second-order derivative of position r with respect to time t. Based
on this equation we can extract Newton's three laws of motioon and determine the mechanics
of acting on an individual molecule in motion. The limitation of Newtonian dynamics rests on
an important point. Although molecular forces and positions change with time, the functional
form of Newton's equation for force is time independent. Equation (1) is invariant under time
translations and, as a result, this single dynamic system is insucient to properly categorize
atoms in motion. Since we expect there to exist a function of both positions and velocities,
whose values are constant with respect to time, a reformulation of classical mechanics is neces-
sary. Indeed, this is exactly what Hamiltonian dynamics represents through the Hamiltonian
H:
H(r
N
;p
N
) = constant: (1.2)
In this equation, p represents the momentum of a molecule i. Since all MD simulations are
isolated systems, we can acknowledge that the total energy E is conserved. If we identify the
11
total energy as the Hamiltonian, then,
H(r
N
;p
N
) =
1
2m
X
i
p
2
i
+U(r
N
) =E (1.3)
with
@H
@r
i
=
@U
@r
i
(1.4)
Consider, now, the total time derivative of the general Hamiltonian (1.2), and recall that
H has no explicit time dependence. Simplifying this system properly will nd that for each
molecule we can derive the equations of motion for Hamiltonian dynamics:
@H
@p
i
=
p
i
m
= _ r
i
(1.5)
and
@H
@r
i
= _ p
i
(1.6)
Equations (1.5) and (1.6) are Hamilton's equations of motion. For a system of N spherical
molecules, let us now simplify Eq. (1.5) further by removing the momentum:
@H
@r
i
=m r
i
(1.7)
Combining Eqs. (1.1) and (1.6), and with the knowledge that the Hamiltonian is an expression
of total energy, we can make the nal step in our derivation towards expressing the equations
of motion in such a way as to make them compatible with the MD theory. In other words, the
objective is to quantify the motion of the molecules such that they can be applied in a stepwise
manner over the course of a given period of time:
F
i
=
@H
@r
i
=
@U
@r
i
(1.8)
Utilizing this approach to dynamics, it is now possible to quantify this problem in a stepwise
manner. Hamilton's reformulation of dynamics allows for the expression of motion between
two atoms i and j in terms of the spatial position r, the potential U, and a given point in
time t. This represents the most fundamental expression of molecular dynamics. From these
parameters, it is now possible to apply the velocity-Verlet algorithms to calculate the force
between atoms i and j and at time t. Figure 1.1 represents a graphical depiction of an MD
system with the position and potential, at a given time, for an arbitrary group of atoms within
a nite volume.
The following facet of MD simulation is arguably just as important as what has already
been discussed. In the study of thermodynamics, we are interested in evaluating systems that
12
Figure 1.1: Schematic of an MD simulation model.
do not appear to change macroscopically. These systems can be desribed through the use of
statistical ensembles. Although many such ensembles exist, two in particular are valuable with
respect to this thesis. In the canonical ensemble, or (NVT ), we have a statistical ensemble
wherein the temperatureT of the system is specied, but the total energy is not known exactly.
Nevertheless, the total number of atoms N and the volumeV of the system are kept constant.
Due to the fact that the system is a function of a conserved quantity, namely, energy, the (NVT )
ensembles do not evolve over time. The microcanonical ensemble, or the (NVE), is a statistical
ensemble wherein the total energyE of the system and the number of atoms within that system
are held constant. Here, the system is totally isolated with respect to other environments. In
other words with the microcanonical ensemble, the number of atoms, the volume and the total
energy are kept constant. Due to the nature of the work in this thesis, as will be described
later, microcanonical ensembles are more appropriate to be used for dynamic simulations.
1.3 Friction at High Performance Parameters
As discussed earlier, friction is the force resisting relative movement between two surfaces in
contact. The two principal classes of friction are sliding and rolling. The former is most relavent
to this thesis and will be its focus. In both rolling and sliding friction, a tangential force F
t
13
in the direction of motion is needed to move the upper body over the stationary lower body.
The upper surface, in almost all cases, applies a normal force F
n
on the lower body. The ratio
between the tangential force F
t
and the normal force R
n
is known as the coecient of friction,
:
=
F
t
F
n
(1.9)
In the past, the problem of friction was approached by engineers empirically. This method has
served well in the design of many designs. Whether they be brakes, tires, or nuts and bolts,
our lack of fundamental understanding of friction has not impeded design engineers. However,
this is beginning to change. As performance criteria rise in the automotive and aerospace
industry, a deeper understanding of friction must be attained. As an example of the nature of
the limitations in our understanding of friction, let us consider the three laws of dry friction:
1. The friction force is proportional to the normal load as given by Eq. (1.9).
2. The friction force is not dependent on the apparent area of the contact solids. In other
words, it is independent of the size of the solid bodies.
3. The friction force is independent of the sliding velocity.
The rst two laws are found today to be satisfactory for many metals, but unsatisfactory
when two polymers are in contact. The third law is less well-founded and also less understood
than the rst two laws. The friction force needed to inititate motion between two bodies is
signicantly greater than what is needed to maintain motion. This force is the aforementioned
static friction. The natural question that stems from this is, what happens at higher velocities?
Is friction force turly independent of motion, or is there much more to this problem than what
has previously been thought?
A phenomenon important to this eld is stick-slip. When this occurs, the friction force
oscillates between two extreme values. Surfaces in contact touch only at points of asperity
interaction, and it is at such points where maximum stress is observed. Coupled with the
normal force of the upper surface upon the lower one, the penetration of this asperity can
be likened to a hardness test. This phase of friction can be colloquially referred to as the
"stick" in the "stick-slip" phenomena. The force will continue to build until that asperity has
passed. When this occurs, we can say that the surfaces have "slipped". Figure (1.2) represents
14
Figure 1.2: Asperity interaction between two surfaces.
a graphical depiction of the adhesive wear model before and after the asperity interaction. An
important question, and ultimately one for the purpose of this thesis, is the detailed nature of
these asperity interactions and deformations, and how they account for the adhesive element of
friction under high-performance conditions for both amorphous and crystalline silicon carbide
(SiC), one of the principal materials used for high performance disc brakes in the industry. The
reason for this is due to the fact that SiC is resistant to abrasive and thermal stresses.
1.4 Superlubricity
While much has been said about the importance of measuring friction accurately and under-
standing the mechanisms inherent in this phenomenon, ultimately the goal that engineers have
is to reduce friction in the design process. The most extreme subset of this goal is known as
superlubricity. This is a regime of motion in which friction nearly vasnishes, while remaining
non-zero. At the micro level, the most important prerequisite for this state of friction to occur
is the presence of repulsive van der waal interactions in a vacuum at extremely low tempera-
tures. Unfortunately, adapting the superlubric state into the macroscale environment remains
a challenge for these and other reasons.
In addition, the use of novel materials is also key in order to achieve such level of reduced
friction. Graphene, being a two-dimensional material, oers unique friction and wear prop-
erties not commonly observed in conventional materials. Besides its well-established thermal
15
and mechanical properties, graphene can serve as a solid or liquid lubricant. Its high chemical
intertness, extreme strength and easy shear capability on its densely-packed and atomically
smooth surface are the most important attributes that account for its remarkable shear capa-
bility. One goal of this thesis is to account for graphenes lubricity characteristics when coupled
with various metals including, gold, silver and platinum.
1.5 Flash Heating
The previous two cases mentioned detail the study of sliding friction under opposing condi-
tions. Between high performance and superlubricity, however, there is another condition with
potentially more impactful consequences. This equally is the nature of sliding friction at sub-
seismic slip rates. While it is generally understood that earthquakes occur as a result of a
fault's weakening over time, the thermal and mechanical reasons for this weakening must be
better understood. The focus of the nal section of this thesis, then, is an investigation into the
mechanical and thermodynamic processes at work in this experimentally observed weakening.
Accounting for these thermal processes, colloquially referred to as "
ash heating", may hold
the answer and applying MD simulations in this regard may uncover them.
1.6 The Structure of this Thesis
Thus, the main goal of this thesis is to understand friction and lubricity at atomic scale.
Not only are these phenomena important to the design of many materials with industrial
applications, they are also not understood well. At the same time, friction is also important
at a vastly larger scales, namely, between the surfaces of faults and fractures in rock. As
such surfaces go into motion, the phenomenon of
ash heating occurs, whereby the apparent
friction coecient reduces with the rising temperature, the stick and slip motions comence,
and ultimately may lead to earthquakes. The phenomenon of
ash heating and its eect of
stick-slip motion in rock is highly important, but not well understood.
In the next chapter we describe our work on friction between two SiC surfaces. Chapter 3
contains the results for superlubricity, while chapter 4 contains the results for
ash heating
16
Chapter 2
Sliding friction between silicon-carbide
surfaces
2.1 Introduction
As already described in Chapter 1, Sliding or kinetic friction between two surfaces in contact,
when one of them is moving, has been a fundamental scientic problem since several centuries
ago
1
and, due to its ubiquity in all types of systems,
2
has gained even more importance in the
modern era. The direction of kinetic friction is the opposite of that of the relative motion of
the two surfaces in contact, but it need not oppose the net external force between them.
One type of kinetic friction is between dry surfaces, which is usually described by Coulomb's
law
3
according to which the magnitude of the kinetic friction force F
k
is given by, F
k
= F
n
,
where is the coecient of kinetic friction, andF
n
is the normal force between the two surfaces
in contact. Thus, according to Coulomb's law, the magnitude of the kinetic friction exerted
through the surface is independent of the magnitude of the sliding velocity V of the surfaces
against each other. Apart from Coulomb's law, however, there are many others that link the
static (contact pressure) and kinematic (velocity and hold time) quantities with the frictional
force, or tangential frictional tractions. Examples include the so-called Shaw's law for saturation
of frictional force,
4
and the rate-and-state family of friction laws.
5
Wet or lubricated sliding friction is the second type of the phenomenon. In this case a
fundamental concept is the so-called Stribeck curve,
6;7
which demonstrates that friction in
uid-
17
lubricated contacts is a nonlinear function of the contact load, the lubricant's viscosity and its
entrainment speed. Nikolai Pavlovich Petrov was presumably the rst
2
who studied the eect
of lubricants upon friction of journal bearings. Petrov recognized that understanding of friction
between sliding lubricated surfaces is obtained by studying it as a problem in hydrodynamics
and derived a simple equation, often referred to as the Petrov's law, for the kinetic friction
force on a journal bearing. In fact, Petrov's law represents the linear part of the Stribeck curve
and holds for high values of the lubrication parameterL, i.e. L = viscosity velocity=pressure.
More generally, the Stokes' equation is used to model lubricated contacting surfaces, with
the result representing an average over the thickness and usually referred to as the Reynolds'
equation. According to Reynolds' equation, F
k
/ V=D, where is the viscosity, and D the
thickness of the lubricating lm. But, in general, wet sliding friction is a complex phenomenon,
due to the fact that at low velocities/high pressure lubrication is mixed with intimate contact
that increases friction considerably, and at very low velocities a pure boundary lubrication
holds. But, neither the Coulomb's law nor any law that governs wet sliding friction takes into
account, at least explicitly, the eect of the chemical structure of the two contacting surfaces,
as well as the morphology of the interfacial area.
While lubricated sliding friction is relatively well understood, and the validity of Reynolds'
law has been demonstrated
8;9
over a range of length scales, work on dry friction is still continu-
ing because the phenomenon of stick-slip is widespread and occurs over widely disparate length
scales, ranging from fault motion in rock that gives rise to seismic events and earthquakes,
10;11
serrated yielding in metals,
12
and curve squeal in rail wheels,
13
to various phenomena at atom-
istic scale, such as abrasive wear on aluminum alloy substrates,
14
friction of krypton monolayers
sliding on a gold surface,
15
friction between silicon (Si) tips in a friction-force microscope and
various carbon compounds,
16
as well on a NaCl (100) surface
17
and unzipping of DNA strands.
18
The most important nding of the research on sliding friction in some materials at atomic scale
has been that, at such scales, it does not necessarily follow Coulomb's law. For example, in
the aforementioned experiments on the NaCl (100) surface, it was found that
18
friction force at
low sliding velocities V varies as, F
k
/ lnV , which is a quite weak dependence on the sliding
velocity.
Since Coulomb's law may be violated at nanoscale, understanding the origin of the violation
requires molecular dynamics (MD) simulation. Such simulations make it also possible to include
18
the precise chemical composition and structure of the sliding surfaces and study their eect.
Several signicant eorts have been made in this direction
1932
for studying sliding friction
between various surfaces. The goal of the present chapter is to report on the results of our
study of the phenomenon between two silicon-carbide (SiC) surfaces, using MD simulations.
Due to its many unique properties, such as high thermal conductivity,
33
thermal shock
resistance,
34
biocompatibility,
35
resistance to acidic and alkali environments,
36
chemical in-
ertness, and high mechanical strength,
37;38
SiC is a highly important material. In addition
to its use in the electronic and computer industries, SiC has also been used for fabricating
nanoporous membranes,
39;40
and nanotubes.
4143
But, what has motivated us to study sliding
friction between two SiC surfaces is its applications to high performance disc brakes, cutting
tools, and blades in gas turbines and jet engines, as well as fabrication of nanoporous materials
explained below. Consider, for example, the development of high performance brakes, particu-
larly for aircrafts. In the past such brakes used carbon-carbon discs
44
that have many excellent
properties. They suer, however, from such disadvantages as insucient stability of friction
caused by humidity and temperature and low oxidation resistance. Thus, carbon-SiC discs were
introduced
4547
that represented a massive leap in the brakes' performance, particularly as it
relates to fatigue resistance. The natural evolution of such brakes is to consider SiC-SiC brakes,
or perhaps ceramic composites containing SiC. In addition, the problem is relevant to sliding
of SiC bers in SiC/graphene/SiC composite.
48
Finally, the phenomenon is important to even
fabrication of nanoporous membranes,
39;40
since the nanoporous SiC layer is deposited on top
of a mesoporous SiC layer (see below), and the static friction between the two layers during
the deposition in
uences the structure and quality of the nal product and, in particular, the
interface between the meso- and nanoporous layers.
We have carried out extensive MD simulations to study dry sliding friction between two
SiC surfaces. Two types of surfaces have been considered. In one we study kinetic friction
at the interface between two amorphous SiC slabs, while in the second case the SiC slabs
have crystalline structure. The goals are to understand the eect of the microstructure of the
surface on the sliding friction, and determine the dependence of the kinetic friction force on
the sliding velocity. One expects, of course, higher friction between two amorphous surfaces
than between the crystalline ones, but the magnitude of the dierence, as well as any other
factor contributing to the dierence other than the roughness of the amorphous surface, is also
19
important to understand.
The rest of this chapter is organized as follows. In the next section we describe the molecular
model of the SiC slabs, as well as the force elds that we utilized in the MD simulations. Sections
2.2 and 2.3 describe the development of the molecular model of the SiC-SiC interface, while
Sec. 2.4 explains the procedure for carrying out the MD simulations of sliding friction between
the two SiC slabs. In Sec. 2.5 we present and discuss the results and their implications. We
then summarize our ndings in Sec. 2.6.
2.2 Molecular Model of SiC and the Force Field
The rst step in the study is generating a molecular model of amorphous SiC. We recently
developed
4951
a model of amorphous SiC using the force eld (FF) ReaxFF
52
that has been
developed for reactive environments. The goal in our previous work
4951
was to mimic the
process by which a nanoporous membrane layer
39;40
made of amorphous SiC is produced by
the pyrolysis of a polymer precursor, allyl-hydridopolycarbosilane (AHPCS), which contains
Si. The primary product of the polymer's pyrolysis is amorphous SiC with covalent Si-C
bonds. Thus, we carried out
49
quantum-mechanical calculations on model materials meant to
capture important reaction steps and structures, in order to estimate the parameters of the
ReaxFF. Then, a molecular model of the AHPCS was developed and ReaxFF was utilized to
simulate its thermal degradation and decomposition as the system was heated up to 1200 K
by the MD simulation. Analysis of the pyrolysis process and its products provided estimates
of various quantities that were in good agreement with the experimental data. Finally, the
system was cooled down to room temperature, with the nal result being an amorphous SiC
lm. The computed properties of the resulting SiC lm, such as its radial distribution function,
X-ray diraction pattern, and connectivity of its atoms were in excellent agreement with the
experimental data. In the present work we use the molecular model of amorphous SiC that we
had generated via the pyrolysis process and MD simulations.
The next step is to select the FF that describes the interactions between the two SiC surfaces.
For this purpose we also used ReaxFF. Both the van der Waals and Coulombic interactions
are included in ReaxFF. In addition it should be kept in mind that ReaxFF eschews explicit
bonds in favor of bond order, which is useful for an environment in which bond breaking occurs
20
regularly. The force applied to the system and the sliding friction heats up the region around
the interface between the two SiC slabs, resulting in continuous breaking of the bonds there,
which is why we opted to utilize ReaxFF that we developed
4951
previously.
2.3 Models of Contacting SiC Surfaces
The unit cell for the amorphous SiC slabs was nonperiodic. Each slab was constructed with
a size 2a 2aa that consisted of 2,560 atoms, equally divided between Si and C, where
a 22

A. The two slabs were identical and, thus, the system consisted of 5,120 atoms. We also
constructed two commensurate ()3C-SiC crystalline slabs, with each slab being a 8b10b4b
superlattice of Si-C crystalline with 2,560 atoms, where b 4:35

A. The structure represented
16 planes of Si-C planes stacked together.
Prior to commencement of the main MD simulations of sliding friction, the energy of each
individual slab was minimized. Then, MD simulation was used to minimize the energy between
the surfaces prior to the commencement of the friction simulation, which produced minimal
dierences in the total potential energy of the system, implying correctness of the original
structure of SiC that we had generated in our previous studies,
4951
and indicating that the
slabs had indeed taken on their equilibrium structure. Next, the energy of the SiC-SiC bilayer
was minimized in order to prepare the system for thermalization. The atoms at the interface
between the two slabs, for both amorphous and crystalline SiC slabs, were in \direct contact"
with one another at the onset of the simulation, by which we mean that the separation distance
between the Si and C atoms in the two slabs at their interface was at the minimum required to
form a bond between them. We then thermalized the system in the NVT ensemble using the
Lagevin thermostat and raised the temperature to 300 K, and then equilibriated it in theNVE
ensemble. This was done in a stepwise manner, with the system thermalized in increments of
50 K and equilibriated after each incremental temperature rise. Duration of the thermalization
and equilibration was 1200 ps.
21
2.4 Molecular Dynamics Simulation
Figure 2.1 provides the details of the molecular system and the anchors to the amorphous
bilayer. The MD simulations were carried out in the (NVE) ensemble using the equilibrated
interface. Sliding friction was generated by forcing the top slab to slide over the bottom one at
a constant velocity V , which allowed us to determine the friction force for a given V . In order
to simulate sliding friction at the interface, we divided the slabs into a series of layers,
5254
referring to them as the top layer (TL), top thermostat layer (TTL), bottom layer (BL), and
bottom thermostat layer (BTL). The TTL and BTL were used to ensure that the upper and
lower parts of the system remain at a constant temperature of 300 K throughout the duration
of the MD simulations, so that generation of heat in the region around the interface can be
clearly attributed to the sliding friction. This was done by rescaling the velocities of the atoms
in the two layers at each time step. The atoms in the BL were xed in all directions to ensure
constant loading, while the atoms in the TL underwent two main events. First, in order to
accurately mimic typical sliding experiments, it is necessary to apply normal loading to the
system. This was accomnplished by applying force in thez direction to all the atoms in the
TL. Two values of the normal forces were used that, at steady state, resulted in the system
being exposed to pressures of approximately 155 MPa and 310 MPa. Second, the atoms within
the group TL were held constant at the prescribed velocity. The atoms directly at the interface
may, however, have a dierent velocity at any given time. A constant velocity was applied in
the y direction to all the atoms in the TL, producing a xed sliding velocity throughout the
course of the MD simulations. Note that, strictly speaking, in the presence of friction, the
relative velocity at the interface in sticking regime (see below) should be zero, even though the
relative motion of the TL and BL would be given by prescribed velocity. In practice, however,
there is always motion, albeit very slow, at the interface during the sticking regime, caused by
shearing.
The same approach was also used with the crystalline bilayer. The simulated velocities were
in the range 10 - 500 m/s, and each simulation run was carried out for 600 - 800 ps that, after
some preliminary simulations, had proven to be suciently long for our purpose. No melting
was observed during any simulation at any velocity, as the melting temperature of SiC is 2730

C. The time step in all the cases was 0.25 fs.
22
Figure 2.1: Schematic of the system consisting of the two SiC slabs with the top layer (TL),
top thermostat layer (TTL), the interface between the two slabs, the bottom thermostat layer
(BTL) and the bottom layer (BL).F
n
is the normal force applied, whileV
y
is the sliding velocity
that pulls the top layer in the y direction relative to the bottom slab.
2.5 DeterminationofDynamicFrictionForceanditsVe-
locity Dependence
We dene the sliding distanceY as the instantaneous dierence between they-positions of the
centers of mass of the two slabs. For all the sliding velocities and distances the temperature in
the TTL and BTL
uctuated by only about 5 percent, with the average values being around
303 K, very close to the set temperature of 300 K. The same type of variations of temperature
with the sliding distance and velocity were obtained in the crystalline material.
Friction produces heat at the interface between the two slabs, however, which increases the
temperature at and near the interface. The MD simulation conrmed this. Figure 2.2 presents
the dynamic evolution of the temperature in the layer of the atoms in the top slab that are in
direct contact (in the sense explained earlier) with those in the bottom slab, which is where the
maximum temperature develops. The results shown in Fig. 2.2 are for the lowest and highest
sliding velocitiesV that we simulated. AsV increases, so also does the heat production caused
by friction, hence increasing the temperature there. The temperature between the interface and
the TTL and BTL in which it is held xed also increases. Figure 2.3 presents the dependence
23
Figure 2.2: Dynamic evolution of the temperature at the interface between the two SiC slabs.
24
Figure 2.3: Dependence of the average temperature at the interface between two amorphous
slabs on the sliding velocity V .
of the average interface temperature on the sliding velocity, indicating very large increase in
the temperature at the interface.
Figure 2.4 presents the state of the two amorphous slabs and the interface between them,
viewed in the x direction (see Fig. 2.1 for the coordinates system), as the sliding velocity V
increases. For low V the two surfaces are in close contact. As V increases, however, bond
breaking due to heating, stretching and deformation occurs at the interface, roughening the
two surfaces, and generating a small gap or free volume at the interface whose extent increases
with increasing sliding velocity. As we show below, this results in smaller frictional forces at the
interface. The corresponding results for the crystalline slabs are shown in Fig. 2.5, indicating
qualitative similarity with Fig. 2.4. The formation of the free volume is similar to the same
type of phenomenon in metallic glasses.
5557
It is known,
55;56
for example, that voids nucleate
from the coalescence of excess free volume generated in shear bands during deformation of some
bulk metallic glass, such as Zr
41:2
Ti
13:8
Cu
12:5
Ni
10:0
Be
22:5
. Excess free volume in a shear band
25
Figure 2.4: Evolution of the excess free volume at the interface between two amorphous SiC
slabs, and its expansion as the sliding velocity increases.
Figure 2.5: Same as in Fig. 4, but in the system that consists of two crystalline slabs.
results in excess free energy relative to a relaxed glass with less free volume.
To better quantify and understand what Figs. 2.4 and 2.5 indicate, we computed the
average number of atoms in thin parallelepipeds perpendicular to the z direction during the
entire simulation, for both the amorphous and crystalline systems. The results are presented in
Fig. 2.6. As expected, far from the interface atz = 0 the total number of atoms is constant. At
z = 0 and in the region around it, however, the number of atoms decreases due to bond breaking
that occurs there. The depth of the \well" that represents the decrease in the number of the
atoms depends on the sliding velocityV , and increases with increasingV . It is precisely due to
this phenomenon that we used the force eld ReaxFF in our MD simulation, even though other
non-reactive FFs, such as the Terso potential,
58;59
are available for SiC and MD simulations
with them are faster.
26
Figure 2.6: Average number of atomsN(z) in thin parallelepipeds perpendicular to the vertical
axis z, and its dependence on the sliding velocity V . z = 0 represents the interface, and the
dip around it is due to formation of excess free volume as a result of stretching and breaking
the SiC bonds there.
27
Figure 2.7: Dependence of the potential energy of the amorphous system on the sliding velocity
V and sliding distance.
Next, we examined the potential energy U of the system, the sum of the energies of both
slabs and the interface, which is an important characteristic of the system. The signicance of
U is that its shape, for any given sliding velocityV , highlights important chemical and physical
phenomena in nanoscale sliding that aect kinetic friction, as U is greatly dependent upon the
nature of the atoms at the interface and their spatial distribution. Due to the constant velocity
V , the kinetic energy, aside from small
uctuations, remains essentially constant, hence enabling
us to determine the kinetic friction force. Because the energy of the two slabs is constant as
they slide relative to each other, the rate of change of U along the sliding direction is linked
directly with the instantaneous kinetic friction force, a fact that we exploit to study dynamic
friction (see below). Because the velocity V is xed, the rate of change of sliding distance Y
with the time is also constant. We also point out that in all the cases the potential energy of
the amorphous system is signicantly higher than its crystalline counterpart, re
ecting a more
unstable bilayer.
28
Figure 2.7 presents the dependence of U on the sliding distance Y for the lowest and high-
est sliding velocities in the amorphous system. The atomistically rough surfaces of the two
slabs give rise to a complex relationship between U and Y . At low velocities the motion is
intermittent, withU varying with the sliding distance periodically, which is typical of stick-slip
motion, stemming from the rough nature of the interface that contributes to the variations in
the friction intensity. Note that such a periodicity in the stick-slip motion has been observed
in experiments
32
by using atomic force microscope. The maxima in U correspond to the slip
forces, whereas the minima represent the stick state. At much higher velocities, however, the
motion is fast and the system does not have enough time to relax. Thus, the variations of U
with the sliding distance is stronger, although they still appear to be quasi-periodic. Note that
the potential energy grows roughly quadratically with the distance in the stick regime
We dene the friction force F
y
per contact area in terms of the potential energy through
the relation
F
y
=
1
A
dU(Y )
dY
; (2.1)
where A is the contact area. Alternatively, one may compute the friction force F
y
directly
using MD calculation's output, as it represents the total lateral atomic force on the upper slab,
divided by the contact area at each MD time step. In other words, F
y
is simply the sum of all
the atomic forces along the y direction. But, only the lateral forces at the interface contribute
toF
y
, as the internal forces cancel each other in the sum. This is the approach we took in this
chapter. F
y
is an important physical quantity because it represents the intensity of the friction
force. Equation (1) and Fig. 2.7 do, however, provide a qualitative picture of what one may
expect for F
y
as a function of the sliding distance Y : at low sliding velocity F
y
should vary
periodically with Y , but as V increases, the system does not have enough time to relax, and
the periodicity is to some extent distorted.
Suppose that the static potential energy corresponding to V = 0 is U
0
(Y ), implying that
U
0
(Y ) represents only the potential energy due to interfacial bonding or adhesion. Thus, the
dierence U(Y )U
0
(Y ) is purely due to the sliding velocity. The force needed to overcome
the energy barriers inU
0
(Y ) isdU
0
(Y )=dY , which is due to the change in the interfacial energy
and represents a lower bound on F
y
, the instantaneous kinetic frictional force during sliding.
Figure 8 presents the directly computed F
y
[i.e., not through Eq. (1)] as a function of the
sliding distanceY and the sliding velocityV . It is clear that our expectations, described earlier
29
Figure 2.8: Dependence of the friction force on the sliding distance and velocity V in the
amorphous system.
based on dU(Y )=dY , for the dierences between F
y
at low and high velocities are precisely
manifested by the results. The results shown in Fig. 8 were computed by applying a normal
force F
n
of 2.85 nano Newton to the system. Qualitatively similar results were obtained when
the magnitude of F
n
was higher and, thus, the results are not shown.
At low sliding velocities the variations in F
y
are periodic, representing stick-slip motion.
During the stick state, the forcedU=dY causes the slab to store strain energy through shearing,
which is then released during slip. The minima inF
y
correspond to the slip forces, whereas the
approach to the maxima ofF
y
represents the stick motion, and they both are larger than those
forV = 0, i.e., the aforementioneddU
0
(Y )=dY . Note that the dierence between the minimum
30
Figure 2.9: Same as in Fig. 8, but for friction force between two crystalline slabs.
and maximum values ofF
y
at low velocities is relatively signicant, whereas the same dierence
is small at high velocities. Once again, similar to the potential energy U(Y ), the force F
y
at
high velocities varies with smaller amplitudes and, due to bond breaking at the interface, its
magnitude also decreases.
Figure 2.9 presents the same results as in Fig. 2.8, but for the crystalline slabs. We rst note
that the range of friction intensities for amorphous SiC is signicantly greater than that of the
crystalline slabs. Due to the crystalline structure, the periodic variations of the friction force,
a characteristic of the stick-slip motion, is even more evident than in the amorphous bilayer.
Similar to the amorphous case, the amplitude of the variations diminishes with increasing sliding
velocity, and the periodic structure is not as well-dened. As the sliding velocity increases,
31
however, the distinction between the amorphous and crystalline interfaces is lost due to the
bond breaking and roughening of the interface and, therefore, at such velocities the friction
forces in both materials are close to each other.
Higher friction forces are associated with increased probability of reaching fatigue stresses,
usually dened as the highest stress that a material can withstand for a given number of cycles
without breaking, which destabilize the material, causing loss of structural integrity. At high
velocities F
y
in the crystalline slabs still exhibits peaks and valleys, but the stick-slip behavior
discussed earlier is less clear due to the
uctuations from the thermal phonons in crystalline
materials, and the fact that there is less time for the system to relax. In addition, at high
velocities thermal phonons are also responsible for damping of slab motion, hence slowing down
the increase inF
y
with the velocityV . Indeed, if we write,U(Y ) =U
0
(Y )+U
e
(Y;V )+U
p
(Y;V ),
whereU
e
is the elastic strain energy, andU
p
is the elastic energy due to phonons, at lowV the
rst two terms of the equation are dominant, whereas at higher velocities, sayV > 25 m/s, it is
only the last term that contributes signicantly toU(Y ) andF
y
, hence explaining the decrease
in the magnitude of F
y
.
The decrease in the friction force with increasing sliding velocity has been seen in experi-
mental studies.
53;54
The reason for this is twofold. One, as already pointed out, is that there
is less time for adhesion to take place at the interface at high sliding velocities and, therefore,
shorter times for the stick-slip motion. The second reason is that even in the crystalline system
the interface becomes increasingly roughened and more similar to an amorphous material, and
a low-density region forms near the interface as the sliding velocity increases. This is then very
much similar to the aforementioned concept of excess free volume; see Fig. 2.6.
The extent of shearing of the system is quantied by the position Y
0
of the center of mass
of the outermost layer, as well as that of the interface layer, Y
i
. Thus, let, Y = Y
0
Y
i
,
which is proportional to the shear strain. In the absence of shearing Y will not change. The
regions in Figs. 2.8 and 2.9 in whichF
y
> 0 represent those for which Y > 0, i.e., forward or
positive shearing of the top slab. The sliding distance from a maximum of Y to its minimum
represents the actual slip distance, which is much smaller than the distance through which Y
increases during the stick motion. This is particularly true at high velocities.
We also calculated the time-average of the instantaneous intensity of frictional force,hF
y
i,
32
Figure 2.10: Dependence of the kinetic friction force F
k
on the sliding velocity V . For V 35
m/s, dependence of F
k
on V is perfectly logarithmic, while after a transition region the same
type of dependence appears to be roughly followed.
over long sliding distances. The kinetic frictional force intensity F
k
is dened by
F
k
=hF
y
i; (2.2)
which is valid for any sliding velocity. Of course, if the potential U(Y ) were exactly symmetric
for allY , thenF
k
= 0, because the sticking and slipping forces would be exactly equal but with
opposite signs. But, as the results presented earlier indicated, U(Y ) is not exactly periodic or
symmetric and, therefore, its slope that yields F
y
varies, i.e., we expect a nonzero F
k
.
Figure 2.10 presents the plot of F
k
versus V , and it is clear that the dependence of friction
force on the sliding velocity is nonlinear. In fact, Fig. 2.10 indicates that the kinetic friction
force F
k
roughly follows, F
k
/ lnV , which had been reported for friction on the NaCl (100)
surfaces
18
at low velocities. According to Fig. 2.10, up to about V 35 m/s F
k
depends
on V as lnV . There seems to be a transition for 35 < V < 55 m/s, beyond which the
33
logarithmic dependence is roughly followed again. Figure 10 also demonstrates the dierence
between sliding frictions in the two types of materials that we study, particularly at low sliding
velocities. According to Fig. 2.10, for V = 0 one has, F
s
=hdU
0
(Y )=dYi 1.8 and 1.4 nano
Newton for, respectively, the amorphous and crystalline surfaces. Coulomb's law asserts that
F
k
is independent of the sliding velocityV . The important implication of Fig. 2.10 is, however,
that F
k
does depend on V . That is, at the nanoscale that we have studied, Coulomb's law is
not satised.
For macroscopic surfaces the friction coecient is dened by, = F
k
=F
n
, where F
n
is
the normal force applied on one of the two slabs. The discussions in the literature on the
sliding friction between two nanoscale surfaces may be divided into two groups. In one group
are
29;53
those that describe the phenomenon in terms of the relations between the friction force
F
y
and its average F
k
and the sliding distance and velocity, as we described earlier, in order
to understand whether macroscopic friction laws are still applicable at the nanoscale. In the
second group are those
60
that use the aforementioned denition of at the nanoscale in order
to extract a dynamic friction coecient. We also point out that using the Prandtl-Tomlinson
model of nanotribology
61;62
one can show straightorwardly that the static value of F
k
, i.e., its
value in the limitV = 0, varies linearly withF
n
, so that the static friction coecient
s
can be
estimated as a result of varying F
n
.
If we were to follow the second group, then, we could obtain an estimate of the static friction
coecient
s
between the two SiC surfaces by extrapolating the results to the limit V = 0. In
that case we nd that for the amorphous bilayer,
s
0:6, whereas for the crystalline surfaces,

s
0:52. Experimental values are in the range
63;64
0:45 0:5.
2.6 Conclusions
Due to its numerous applications, it is of fundamental interest to study the dependence of
kinetic friction between two SiC slabs on the sliding velocity. In addition, since SiC exists
both as an amorphous material and in crystalline form, the eect of surface roughness on the
kinetic friction is also of importance. This chapter reported the results of an extensive study
by MD simulation of sliding friction between the two types of SiC surfaces. With increasing
sliding velocity bond breaking occurs at the interface between the two surfaces, leading to
34
their roughening and formation of excess free volume. The roughening and exces free volume
reduce the kinetic friction force, leading to reduction in the dierence between kinetic friction
in the amorphous and crystalline surfaces. The kinetic friction coecient depends strongly
on the sliding velocity, hence indicating that Coulomb's law of friction is not satised at the
nanoscale. Indeed, the kinetic friction forceF
k
appear to depend logarithmically on the sliding
velocity.
35
Chapter 3
Friction and superlubricity between
surfaces of precious metals and
graphene nanoribbons: A molecular
dynamics study
3.1 Introduction
Friction between various types of surfaces that are used in diverse applications has been studied
for over a century. Friction and wear remain the primary modes of dissipation of mechanical
energy in moving assemblies, such as turbines, engines, and brakes. It is estimated, for example,
that roughly one third of the fuel utilized in automobiles is spent merely to overcome friction.
Furthermore, the eect of wear limits signicantly the lifetime of moving materials. Thus,
the ability to reduce by even a modest amount the frictional forces between moving surfaces
signicantly aids energy savings, which in turn will have important impact on the environment
due to reduced consumption of fossil energy, as well as on the economic feasibility of a vast
multitude of engineering designs.
It is in context of reducing friction and wear that study of lubricity and especially its more
extreme variant, namely, superlubricity, continues to attract attention. An important aspect of
the problem is identifying a material that acts as a lubricant and reduces friction. While one
36
normally thinks of various types of liquids when it comes to lubricants, over the past decade
or so some dry materiash have also manifested promising potential for acting as a lubricant.
One such material is molybdenum disulde, MbS
2
, which is used in constant-velocity joints
and in space vehicles. Its low-friction characteristic is attributed to its layered structure at
molecular level with weak bonding between the layers, which can slide relative to each other
with minimal applied force, hence endowing the MbS
2
surface with low friction. Hexagonal
boron nitride, also called white graphite, is also used in space vehicle as a dry lubricant. A
third solid lubricant, the focus of the present chapter, is graphene.
Graphene, a single atomic layer of graphite, consists of tightly bonded carbon atoms orga-
nized into a hexagonal lattice. The most important feature of graphene is its sp
2
hybridization,
due to which and its 0.142 nanometer-long carbon bonds, it is among the strongest materials
ever discovered, with an ultimate tensile strength of 130 GPa. Moreover, due to the saturation
of its carbon atoms, graphene is chemically inert
6567
, implying the absence of any interfacial
bonding between the carbons and an opposing surface, a important requisite for lubricity.
As discussed and documented by Berman et al.
68
, graphene does indeed exhibit excellent
properties as a solid superlubricant. Thus, due to its excellent mechanical properties and
toughness, one may be able to coat a surface with an ultrathin layer of graphene and, therefore,
suppress essentially any energy dissipation in the system as a result of contact. Superlubric
property of graphene is attributed to two key factors
65;66;69;70
. One is its large lateral stiness
that makes it very dicult, if not impossible, to have a commensurable contact with most solid
surfaces. The second factor is the fact that the incommensurability, when combined with weak
interaction with most materials, generates a state of ultra-low friction when graphene slides
over another material. Strong experimental evidence for this was provided very recently by
Kawai et al.
71
who reported extremely low friction between gold and graphene surfaces.
The understanding and interpretation of the superlubric behavior of graphene must begin
at the molecular scale. Nanoscale friction as a result of nanoscale displacement between two
surfaces has been studied experimentally using atomic force microscope (AFM)
7276
, and has
provided a wealth of information about and insight into the mechanisms of energy dissipation.
The subject has also been reviewed extensively
72;7780
. Moreover, the interatomic interactions
at the interface between two surfaces are of the van der Waals type. The lowest level of lubricity
is obtained if the opposing surfaces take on the Miller index of (111) that provides the ideal
37
geometry necessary for that level.
In this chapter we report on the results of extensive molecular dynamics (MD) simulation
of friction between graphene nanoribbon (GNR) and three surfaces, namely, Au(111), Ag(111)
and Pt(111) substrates. Our objective is to understand the nature of the interactions between
the three substrates and GNR, and the role of an incommensurate surface orientation. In the
simulations we manipulate the GNRs aligned along the [-1,0,1] direction of the metal substrates,
which is typically done in AFM studies
7276
. Furthermore, taking advantage of MD simula-
tion's unique ability, we study the origin of the stick-slip behavior when only van der Waals
interactions are present.
The plan of this chapter is as follows. In the next section the molecular models of the three
precious metals and the GNR, and the force elds for presenting them are described. Section
III describes the details of the MD simulations, while in Section IV we present and discuss the
results. The chapter is summarized in the last section.
3.2 Molecular Models and Force Fields
In an eort to mimic the experimental geometry, we simulate armchair GNRs that consist of
alternating pairs of carbon hexagons. Experiments indicate that such a structure slides best on
a surface with Miller indices that are of analogous geometry, implying, for the precious metals
that we consider, the (111) conguration. The details of the molecular model of each metal
are as follows. Au has a unit cell of length 4.08

A, while the corresponding lengths for those
of Ag and Pt are, respectively, 4.08

A and 3.9

A. The size of the supercells for Au, Ag, and
Pt used in the simulations were, respectively, 182:7 61:2 20:4

A
3
, 183 61:3 20:4

A
3
,
and 181:3 62:8 19:6

A
3
. The same GNRs were used for each metal supercell, which had a
length of 16.13 nm and width of 0.52 nm. The total number of atoms in the Au(111)-GNR,
Ag(111)-GNR, and Pt(111)-GNR systems were, respectively, 14,480, 14,480, and 14,350, with
the number of carbon atoms in the GNR in the three cases being 1845. The peripheral carbon
atoms of the graphene layer were saturated with 133 hydrogen atoms.
To model the various metal-metal interactions, we used the embedded atom model (EAM)
8184
. The model provides accurate description of the structural, mechanical and thermal
38
properties of metallic systems (for a comprehensive review see Ref. [85]). Moreover, using
the EAM, the evaluation of the energy and forces is several orders of magnitude faster than
comparable rst-principle calculations, and accounts accurately for the behavior of an atom
placed in a dened electron density. In the EAM the total energy of the system is the sum of two
terms, namely, a pairwise sum of the interactions between the atoms, and a term representing
the embedding energy for each atomic site. Thus, the total energy is given by
E
EAM
=
1
2
N
X
i=1
N
X
j6=i

ij
(r
ij
) +
N
X
i=1
E
e
i
(
i;h
); (3.1)
where
ij
(r
ij
) is a density-independent, pairwise additive, and short-range core-core pair re-
pulsion between atoms i and j, separated by a distance r
ij
,
i;h
is the electron density of the
host material at atom i due to its remaining atoms, and E
e
i
represents the energy to embed i
into the background electron density . One makes a further simplication by approximating
the density
i;h
by

i;h
=
N
X
j6=i

a
j
(r
ij
); (3.2)
where
a
j
is the electron density contributed by atomj. Given the approximation (2), the energy
is a simple function of the atoms' positions. Note the dierence between
j
and
a
j
: whereas
a
j
represents the contribution to the density from atom j,
j
is the total electron density at atom
j. Since the pair interaction is purely repulsive and of Coulombic type, for atoms of types i
and and j one writes
86
,

ij
=
q
i
(r
ij
)q
j
(r
ij
)
r
ij
; (3.3)
where q
i
is the eective (partial) charge of i that decreases monotonically with the separation
distance. Note that the EAM that we used for the three metals was the same, except for the
dierences in the density based on the unit cell structure, the atomic masses, and the lattice
constants. Foiles et al.
86
give the full parameteric details of the EAM for the three metals
that we consider and, thus, we do not repeat them here.
We used the adaptive intermolecular reactive empirical bond order (AIREBO) force eld
87
, an empirical many-body force eld developed principally for hydrocarbons as a further
development of the second generation REBO (REBO2)
88
. AIREBO adds intermolecular in-
teractions to the purely covalent REBO2 through a set of Lennard-Jones (LJ) potentials for
the three interaction types that are important to our simulations, namely, the C-C, C-H, and
39
H-H. In addition, the interactions between the C and H atoms and those of the metals were
also represented by the LJ 6-12 potential. The LJ parameters for the pairs of unlike atoms
were computed using the Lorentz-Berthelot rules. Thus, the total energy F of the metals plus
the GNRs is given by, E = E
EAM
+E
LJ
+E
AIREBO
, where E
LJ
is meant for the interactions
between C and H atoms with the metals. Table 1 summarizes the LJ parameters (;) that we
used for the interactions between C and H, and the three metals.
Table 1. The LJ parameters used in the simulations
86;89
. k
B
is the Boltzmann's constant.
(

A) =k
B
(K)
Au-graphene
C-Au
2.74 29.01
H-Au
2.74 11.6
Ag-graphene
C-Ag
2.77 26.7
H-Ag
2.77 12.76
Pt-graphene
C-Pt
2.95 83.3
H-Pt
2.95 23.2
3.3 Molecular Dynamics Simulations
Prior to commencement of the main MD simulations of sliding friction, the energy of the entire
system was minimized. The simulation yielded results in which there was virtually no change
in the potential energy of the system relative to its energy before minimization. After the
energy minimization, we began annealing each of the systems consisting of a metal and GNR
40
Figure 3.1: Schematic of the system consisting of the substrate and GNR.
in order to prepare them for the MD simulation at the experimental temperature. This was
done using the Langevin thermostat in the NVE ensemble for 5 ns. The temperature of the
system was raised steadily to 550 K in increments of 100 K. After each increment, the system
was equilibrated prior to further thermalization, and nally quenched at 4.8 K, the experimental
condition
71
, and equilibriated again. By annealing the GNR and the metals' substrate layer,
we ensured that the nal conguration represents an absolute minimum energy state. This is
important in order to obtain the highest level of lubricity at the interface. Note that due to
the very low temperature of the system, thermal eects are not important. The nal step prior
to the initiation of the MD simulations was to apply loading to the graphene layer, which was
done by raising steadily the leading edge of the GNR and setting it at heights of z
0
= 1; 2,
and 3 nm, the heights at which the MD simulations were carried out. The leading edge of the
atoms were then attached to a soft spring, so that they could move in a forward or backward
direction, as was also done in the experiments
71
.
The MD simulations of sliding friction were carried out in the NVE ensemble using the
equilibriated interface, beginning at an initial temperature of 4.8 K. All the simulations were
carried out for each of the three lifting heightsz
0
. Sliding friction was generated by forcing the
GNR to slide over the bottom surface at a constant velocity, which allowed us to determine the
friction force. What is unique about the system is that one must determine a very faint friction.
The friction force is determined through the spring constant k
x
. To do so, a viscous damping
term, m
v was added to each atom in the GNR in order to remove the energy added to the
GNR from the driving velocity. Here, m is the mass of the atom, v is its velocity, and
is the
41
viscous damping parameter, set to be 10 ps
1
. The reason for doing so is helping to remove
the statics typically associated with friction-based MD simulations. We also point out that
applying a force to the system is more appropriate for studying static friction between the two
surfaces, but our focus is on kinetic friction and the evolution of the friction force distribution
in the system.
In order to simulate friction at the interface, we divided the slabs into a series of layers,
referring to them as the bottom layer (BL) and the bottom thermostat (BT). The groups of
atoms that constitute each of the layers are shown in Figure 3.1. The thermostatted layer
was used to ensure that the system remains at a constant temperature of 4.8 K throughout
the duration of the dynamic simulation. We used rescaling of the velocities of the atoms at
each time step. The atoms in the BL were xed in all directions to ensure that no potential
deformation of the metals occurs. The velocity was applied to the GNR atoms at the elevated
leading edge with V
0
= 0:25 m/sec. Figure 3.2 details the potential energy changes over the
course of lifting the leading edge to each of the three heights z
0
. The time step in all cases was
1 fs.
3.4 CalculationofDynamicFrictionalForcesandShear-
ing Distance
Although the temperature in the BL and BT was held constant, due to friction the rest of
the metal and the interface have higher temperatures. Figure 3.2 presents the change in the
temperature at the interface over the entire course of the simulation, indicating that it is roughly
twice as high as the set temperature in the BL and BT, but still very low, which is indicative
of the very low friction there. Note also the higher temperatures in the Ag-GNR and Pt-GNR
systems, when compared with Au-GNR.
As described in the experimental work on Au-GNR system
71
, there is stick-slip motion at
each commensurate/incommensurate transition
90
. The importance of this is two-fold. First,
the behavior re
ects adhesion wherein there is a buildup of the force, which is followed by its
rapid release, or slip. In terms of the MD simulations, the phenomenon is important due to the
magnitude of the potential energy and the temperature, as we use the microcanonical NVE
42
Figure 3.2: Dynamic evolution of the interface temperature in the three pairs at a lifted height
of 2 nm.
ensemble in which the temperature
uctuates.
The simulated friction force was computed by using the follwing equation,
F
x
(t) =k
x
[v
x
tx(t)]: (3.4)
Thus, the simulated friction force is represented as an elastic force that the spring exerts on
the GNR. Here, x(t) represents the average x-coordinate of the lifted end of the GNR. One
end of the spring is attached to the lifted edge of the GNR, while the other end is pulled by a
constant velocity of V
0
. v
x
(t) represents the average velocity of the lifted end of the GNR. For
every lifting height z
0
the force plot was determined as a function of time.
Figures 3.3-3.5 present the dependence on the sliding distance of the friction force for the
three systems at two lifted heights, z
0
= 1 and 2 nm, for both forward and reverse directions.
Pulling the spring in the forward direction, as shown in Fig. 1, has a signicant eect on the
stick-slip behavior, as well as the average friction force. Note that at the indicated elevations,
the forces during forward and reverse motion exhibit similar but not identical trends, which
was also observed in the experimental work on the Au-GNR
71
. For the smallest lifting height,
z
0
= 1 nm, the force oscillations in the forward direction are not particularly smooth, at
least when compared with the corresponding trends in the reverse direction. We attribute this
43
Figure 3.3: Friction force proles for the Au-GNR pair at the lifted height of 1 and 2 nm nm
for both forward and reverse directions.
to the lattice mismatch between the GNR and the metals' substrates, as commensurate and
incommensurate mismatches give rise to nonperiodic behavior
90
. Note the increase in both
the average force, as well as its amplitude in the reverse direction. Both facets of the results
indicate enhanced adhesive behavior in the reverse direction. Moreover, as the lifted height
increases, a signicant dierence in the behavior of the force trace develops, namely, adhesion
becomes more prominent and, as a result, the friction force increases.
When compared with Au-GNR pair, the friction force for both Ag-GNR and Pt-GNR pairs
are noticeably dierent with respect to both its average and the adhesive behavior. One way
of understanding this is true the location of the equilibrium position of the GNR sheet with
respect to the metals' surface. Figure 3.6 presents the dependence of the total potential energy
of the GNR plus the metal surfaces. As the separation vetween the GNR and the metals'
surface increases, the potential energy of the pair decreases sharply, reaching its minimum at a
certain location r
m
, beyond which it increases again. The values of r
m
for Au-GNR, Ag-GNR,
and Pt-GNR pairs are, respectively, 3:05

A, 2:94

A and 2:81

A. Thus, the Pt-GNR interface
experiences the largest friction force because the equilibrium position of the GNR sheet is the
closest to the Pt surface.
Additional insight into the dierences between the three pairss is gained by studying the
44
Figure 3.4: Same as in Figure 3, but for the Ag-GNR pair.
Figure 3.5: Same as in Figure 3, but for the Pt-GNR pair.
45
Figure 3.6: The separation eparation distances at which the total potential energies of the three
pairs attain their minimum.
dynamic changes in their potential energies. Figures 3.7-3.9 present the the potential energy
distribution over one stick-slip transition, computed at a very low velocity, 0.25 m/s. The
distributions reveal the change in the conformation of the elevated GNR: As the nanoribon is
translated in both forward and reverse directions, two phenomena occur. First, as the spring
applies force to the elevated edge, the GNR begins to stretch, hence increasing the potential
energy. Once sucient force has been applied to overcome the adhesive interactions between
the substrate and the GNR surface, the structure moves. After the energy is released, one
expects a decrease in the potential energy. Figure 3.7 shows, however, that very little changes
in the potential energy distibution of the Au-GNR pair over the time interval at the height
of 1 nm, hence indicating very little change in the structural conformation of the GNR per
stick-slip event. As the elevation increases, however, the potential energy changes become
more pronounced, hence indicating that more elastic energy is needed to overcome the adhesive
energy at the interface. In agreement with Figures 3.3-3.5, Figures 3.7-3.9 indicate that, in
general, there are higher levels of the potential energy
uctuation in the reverse direction and,
therefore, stronger structural deformation in the GNR. In addition, compared with the Au-
GNR interface, the potential energy of the Ag-GNR and Pt-GNR pairs manifest
uctuations
with larger amplitudes, which are again in agreement with Figures 3.3-3.6.
Although analysis of the potential energy may be used as an indirect approach for deter-
mining the friction force, it is not by itself a sucient metric. In order to properly quantify the
46
Figure 3.7: Potential energy
uctuation per stick-slip event for the Au-GNR pair in (a) forward
and (b) reverse directions at a lifted height of 1 nm.
47
Figure 3.8: Same as in Figure 7, but for the Ag-GNR pair.
48
Figure 3.9: Same as in Figure 7, but for the Pt-GNR pair.
49
Figure 3.10: Shearing distance in the Au-GNR pair at the lifted height of 3 nm in forward and
reverse directions. Point 1 signies the initial conformation of the GNR; point 2 represents the
maximum deformation prior to slip, while point 3 indicates the conformation after slip.
degree of deformation and, therefore, the eect that both the substrate and the edge elevation
have on the GNR and its friction against the metal surfaces, we must compute and analyze the
shearing distance. Figures 3.10-3.12 present the results for the shearing distance of the lifted
edge of the GNR in both forward and reverse directions for an elevation of 3 nm. In each gure
the point denoted by 1 represents the initial conguration of the GNR prior to applying the
force. Point 2 is indicative of the conformation immediately before the slip event, whereas point
3 represents the nal conformation after the slip event. The shearing distance is then dened
as the dierence between the x-positions at points 1 and 2. The results are fascinating in that
they indicate that the slip event occurs after the spring reaches the critical translation length, in
either the forward or reverse direction, in order to overcome the adhesive energy barrier. These
are consistent with what the potential energy distributions indicated in Figs. 3.7-3.9, namely,
that with an increase in the lifting height z
0
the shearing distance increases markedly, which
is also re
ected in the friction force distributions. Note that the shearing distance behavior
in the Ag-GNR and Pt-GNR pairs indicate that the requisite distance of elongation increases
signicantly, as manifested by the enhanced stick-slip behavior.
50
Figure 3.11: Same as in Figure 10, but for the Au-GNR pair.
Figure 3.12: Same as in Figure 10, but for the Pt-GNR pair.
51
Figure 3.13: Average friction force for the three pairs in the forward direction.
Figures 3.13 and 3.14 present the average friction force for all the elevated heights in the three
pairs. The marked increase in the friction force in Au-GNR and those in Ag and Pt represent
much higher adhesive energy between the latter and GNR. The increase in the friction force
is linear, indicating that for each respective increase in the elevation we have a proportional
increase in the adhesive resistance to the forward or reverse motion of the GNR.
In addition, the average static friction forces - the minimum force immediately before the
layers begin their motion - for both forward and reverse directions were computed. In one case
we considered a
at GNR surface (with no lifting). The results for the (Au-GNR, Ag-GNR,
Pt-GNR) pairs are (0.67, 6.53, 13.9) for the forward direction and (0.745, 6.85, 14.3) for the
reverse direction. In the second case, the force was computed at a lifting height of z
0
= 1 nm.
The results for the forward direction are (0.59, 6.47, 13.79), while for the reverse direction they
are (0.68, 6.72, 14.13), all in pN. Thus, the average static friction force appears to be higher for
the
at GNR than the corresponding value for the lifted GNR. This is an interesting in that
it indicates that the increase in contact area (in the case of
at GNR) supersedes the shearing
stress for an elevation of 1 nm. Clearly, at higher lifting heights higher shearing stress would
be needed.
The denition of what constitutes superlubricity is to some extent subjective. But, if the
mechanical behavior of the interface between Au and GNR represents superlubricity, then,
52
Figure 3.14: Average friction force for the three pairs in the reverse direction.
the changes in the dynamic response, coupled with the calculated friction force distributions,
provide evidence for the "end" of superlubricity as one transitions from the Au(111) to Ag(111)
and to Pt(111) surfaces. In other words, although friction between GNR and Ag and Pt surfaces
is still very low, one must be cautious to refer to them as superlublic, at least compared with
the Au-GNR pair.
3.5 Conclusion
Using extensive molecular dynamics simulations, we studied dynamic friction between graphene
nanoribbon surface and three precious metal substrates, namely, Au(111), Ag(111) and Pt(111),
in order to understand the similarities and dierences between friction in the three systems,
and the factors that cause them. For the Au-GNR system we replicated the experimental setup
of by Kawai et al.
71
, and produced several interesting phenomena regarding the stick-slip
prole for the GNR. In general, at low lifted elevations there is relatively smooth sliding, which
becomes rougher at higher heights. Ag and Pt substrates manifest a transition from Au in
terms of the average friction force. Whereas, in agreement with the very recent experiments
71
, the Au-GNR system does exhibit superlubricity, the friction between GNR and Ag and Pt,
while still very low, is signicantly larger than that in the Au-GNR system. This is clearly
53
demonstrated by the behavior of the shearing distance at higher lifting heights, as well as the
dramatic increase in the elastic energy exerted by the spring on the GNR prior to overcoming
the adhesive energy.
54
Chapter 4
Flash Heating and Sliding Friction
between Quartz Surfaces
4.1 Introduction
As already described in the previous chapters, sliding friction between opposing solid surfaces
has been an enduring scientic problem for many years
9
2. While one aspect of this topic's
utility is related to the fabrication of engineering designs capable of operation at high velocities,
another equally important aspect is its opposing kin, namely, the nature of sliding friction at
subseismic slip rates
93;94
. While it is generally understood that earthquakes occur as a result of
a fault's weakening over time,
95;96
the thermal and mechanical reasons for this weakening must
be better understood. This is the focus of the present chapter.
In the previous two chapters we studied kinetic friction between opposing dry surfaces that,
at macroscopic scales, is generally described via Coulomb's law,
9
7 which states that the the
magnitude of dynamic friction force F
n
is equal to the product of the friction coecient and
the applied normal force F
n
between the two surfaces, F
k
=F
n
. Based on this equation one
surmises, however, that the magnitude of dynamic friction force can be determined independent
of the magnitude of the sliding velocity. Our work in the previous chapters, as well as those
of others have shown that the magnitude of dynamic friction force is in fact a function of
the velocity. Furthermore, other theoretical studies have shown that the sliding velocity is an
55
important aspect of dynamic frictional force. An example of these aforementioned studies is
Shaw's law
9
8.
The purpose of this chapter, however, is a deeper study of dry friction. Indeed, work in
this area continues due to the fact that the phenomenon of stick-slip has wide-ranging and
impactful consequences. Whether it be the topic of this chapter, namely, fault motion in rock-
based materials that can result in the development of seismic events and earthquakes,
99;100
or
to the abrasive wear on carbon-ceramic disk brakes used in Formula One race cars.
101
In any
of these, and other applications, a most important nding is that at the fundamental atomic
scale, the precept of Coulomb's law is not valid. Indeed, the work in this thesis described so
far has shown that in certain cases friction force varies as, F
k
/ lnV:
102;103
Experimental studies have shown that in the transition between subseismic and rapid fault
slips, there is a precipitous decrease in the friction coecient as a function of the velocity.
106
At
low velocities, an important governing factor is thermal eects. Indeed, a major area of research
in the eld of geological phenomena is the study of what is known as
ash heating:
105109
. Flash
heating, as its name may posit, is the sudden spike in the temperature at the interface between
opposing solids, caused by a sudden release of energy. If the materials in question possess low
thermal conductivity, then there exists a signicant risk of thermal fatigue over time.
110112
Thus, if
ash heating results in the interfacial temperature exceeding the melting point of the
material, a resultant weakening of the material is to be expected. But, even in the absence of
melting extreme rise in the temperature can obviously weaken te materials.
As one of the most abundant minerals in Earth's continental crust;
113
as well as its high
melting point, quartz is an indeal material for studying this issue. Sandstones, for example,
are mostly - up to 90 percent - made of quartz. Thus, it can be used in our molecular-scale
comptuational study of the meachisms of subseismic weakening and the resultant onset of plate
slippage and energy release in the form of earthquakes. Indeed, an experimental study with
quartz showed that the friction coecient decreased precipitously with a subsequent increase
in sliding velocity:
114
. Understanding this phenomenon at a deeper, more fundamental level
will require employing sophisticated compuational tools. Thus, as in the previous chapters,
we employ molecular dynamics (MD) simulations that allow us to include the precise chem-
ical composition of the structure under inspection, and study the eect of both thermal and
mechanical properties. Several important advancements have been made to this eect, with
56
regards to the study of sliding friction between varying materials.
115119
The goal, then, of
this chapter is to report the results of extensive MD simulations of the sliding friction between
quartz, i.e., SiO
2
, surfaces. We have studied dry sliding friction between two quartz surfaces.
Three dierent structures have been considered, each with the same interfacial contact area,
but with dierent thicknesses. The goals are to understand the eect of both the thickness and
the sliding velocity on the kinetic friction force, as well as the eect that
ash heating may have
in this regard. Furthermore, the dissipation of thermal energy, in the form of the temperature
prole through the material, will also be studied.
The rest of this chapter is organized as follows. In the next section we detail the force elds
used to describe the SiO
2
slabs in the MD simulations. Section III describes the development
of the molecular model of the SiO
2
-SiO
2
interface, while Sec. IV explains the procedure for
carrying out the MD simulations of sliding friction between the two SiO
2
slabs. In Sec. V we
present and discuss the results and their implications. The chapter is then summarized in Sec.
VI.
4.2 The Force Field
In determining the appropriate force eld (FF) that describes the interactions between the two
SiO
2
surfaces, we took into consideration several factors. First, since one focus of our study
is to understand the eect of mechanical properties of quartz, we needed a FF that does not
eschew explicit bonds; that has been proven capable of accurately simulating the structural
properties of SiO
2
, and is computationally ecient in order to allow for the simulations on the
time scale we required, within a reasonable period of time. To this end, after testing several
dierent FFs, we nally settled on using the FF proposed by Terso,
120;121
which explicitly
characterizes the O-O, O-Si, and Si-Si bonds relevant to quartz. The Terso FF allows for less
variability in the potential energy calculations and, as a result, is much more useful for our
work. All of the parameters of the Terso FF are given in the original references,
120;121
and
need not be repeated here.
57
Figure 4.1: Explicit detail of Structures A, B and C of quartz used in the simulation.
4.3 Models of Contacting Quartz Surfaces
The unit cell for the crystalline ()SiO
2
slabs, used in the simulations are nonperiodic. Three
separate structures were developed of varying thicknesses. Each structure contained a pair of
equally-sized slabs. The thinnest slab, referred to as Structure A, consisted of 10,800 atoms,
equally divided between Si and O with dimensions of 86:8 50:1 32:8

A
3
. The second slab,
referred to as Structure B, contained 27,000 atoms with dimensions of 86:8 50:1 82:

A
3
.
The third and the thickest slab, referred to as Structure C, of ()SiO
2
contained a total of
43,200 atoms with dimensions of 86:8 50:1 131:3

A
3
. Each pair of slabs in each of the three
structures were identical and, thus, the systems consisted of 21,600; 54,000; and 86,400 atoms
for the thin, medium, and large structures, respectively. Figure 4.1 compares the three systems.
The rst step in the MD simulation is to ensure that the energy of each individual slab
was minimized. Following this step, the total energy of the two commensurate surfaces in each
58
system were minimized together, resulting in minimial dierences in the total potential energy
of the system, implying structural stability with respect to the original structures of SiO
2
that
had been produced. With the system having taken its equilibrium conguration, thermalization
commensed. It should be noted that in all the cases, the opposing slabs were in direct contact.
By this we mean that the separation distance between the slabs was small enough to allow for
bonding between opposing Si and O atoms at the interface. We then thermalized the systems
in the NVT ensemble and raised the temperature to 300 K, and then equilibriated it in the
NVE ensemble. This process was performed step by step, with the SiO
2
bilayer thermalized
in increments of 50 K and equilibriated after each incremental temperature increase. The total
time of the subsequent thermalization and equilibration was 7200 ps.
4.4 Molecular Dynamics Simulation
The MD simulation of sliding friction was performed in the NVE ensemble, beginning with
the equilibriated interface at 300 K. Sliding friction across the interface was generated through
the application of a constant velocity V of motion, in order to force the top slab to slide over
the bottom one. Figure 4.2 presents the schematic of how the sliding friction is simulated.
This process allows us to determine the friction force for any given V . In order to be able
to slimulate sliding friction at the interface properly, the slabs were divided into a series of
dierent layers.
103;122124
. The layers, for the purposes of this chapter, will be referred to at
the top layer (TL), the top thermostat layer (TTL), the bottom layer (BL), and the bottom
thermostat layer (BTL). The TTL and BTL layers were used in order to ensure that the system
far from the interface remains at a constant temperature of 300 K throughout the course of
the MD simulations. The function of these regions, as "heat sinks," is that the heat generated
at the interface can be attributed solely to the eect of sliding friction. This is achieved by
rescaling the velocities of the atoms in these two layers at every time step. In order to ensure
that there is symmetry throughout the system, the layers TL and BL were subjected to two
equal and opposite sets of events. First, to ensure constant loading, the atoms in TL were
subjected to force in thez direction, while the atoms in BL were subjected to an equal force
in the +z direction. The resultant sum of the forces was then calculated carefully to obtain a
59
net, constant pressure of 1 GPa. For the second event, the atoms in both groups TL and BL
were held constant at a prescribed velocity, applied in the +x direction to all the atoms in the
group TL, with the same velocity applied in thex direction to all the atoms in the group
BL, resulting in a net velocity V for the system as a whole throughout the course of the MD
simulations.
The simulated velocities were in the range of 0.05-2 m/sec. Each simulation was carried out
for 100-600 ns. Due to the wide range of the velocities, and the fact that we wish to simulation
subseismic velocities, this necessitated very long simulations. The melting point of ()SiO
2
is
roughly 1700

C, but with
ash heating the temperature exceeded the melting point, directly
and only at the points of contact between the two slabs, at the interface. The time step in all
the cases was 10
3
ps. Note that the thickness of the material is important in order to ascertain
any stabilizing or destabilizing in
uence that it may have on the interface. AS described in
Chapter 2, in any tribological system the generation of heat is important. We hypothesize that
the rate of heat propogation through the material is in
uenced by its thickness. Understanding
how this manifests itself is one purpose of this study.
4.5 Results and Discussion
The sliding distance of the two slabs, X, is dened as the dierence between the x positions
of the center of mass of the two slabs. For all the sliding velocities under investigation, the
temperature in the TTL and BTL did not
uctuate by any signicant degree. Indeed, the

uctuations did not exceed 10 percent with an average near 300 K. This is shown in Figure 4.3
that presents the
uctuations in the three structures at a sliding velocity of 0.3 m/sec. Similar
behavior was observed for all the velocities under investigation.
As in all cases concerning studies of friction, the interaction between the slabs results in
the production of heat. This is most acute near the interface between the layers. In Chapter
2 we observed the same phenomenon, and the present study is no exception. Indeed, our MD
simulations conrmed this. Figure 4.4 presents the evolution of the structural conguration
of Structure A at a velocity of 0.3 m/sec. From this perspective, just prior to the onset of
interfacial fracture, there is a large buildup of energy that manifests itself in the form of an
60
Figure 4.2: Schematic of the system consisting of the two quartz slabs with the top layer (TL),
top thermostat layer (TTL), the interface between the two slabs, the bottom thermostat layer
(BTL) and the bottom layer (BL).F
n
is the normal force applied, whileV
x
is the sliding velocity
that pulls the top layer in the y direction relative to the bottom slab.
61
Figure 4.3: Dynamic evolution of the temperature prole of the thermostatted regions in the
three structures at a sliding velocity of 0.3 m/sec.
interfacial cavity during the fracture event. When the two layers are in contact, the interfacial
region becomes amorphous (disordered). However, due to the very high pressure conditions
to which the material is exposed, the cavity quickly disappears. This phenomenon is what we
believe is the root cause of
ash heating at the interface. The eect of this and its contribution
to the weakening of the material is quite fascinating.
In Figure 4.5 we present the maximum temperature at the interface in the Structures A, B,
and C over the course of all the velocities simulated. As the sliding velocity increases under the
aforementioned applied pressure, the interfacial temperature sharply rises up to a velocity of
0.3 m/sec, beyond which there is a gradual tempering of the temperature increase. It should be
noted that at no time during the simulations did the sustained interfacial temperature exceed
the melting point of quartz. In other words, Figure 4.5 does not represent sustained interfacial
temperature, but rather the maximum observed during
ash heating events.
The phenomenon of
ash heating is presented in Figure 4.6, which shows the evolution of

ash heating for stucture C as a function of an increase in the velocity from 0.05 to 0.3 m/sec.
A very fascinating aspect of these results is not only an increase in frequency of
ash heating
events over the course of the simulation, but also an increase in the interfacial temperature,
62
Figure 4.4: Dynamic evolution of
ash heating event in Structure A at a sliding velocity of 0.3
m/sec.
Figure 4.5: Maximum interfacial temperature during
ash heating events across the entire
spectrum of sliding velocities and the three Structures.
63
Figure 4.6: Prole of
ash heating events in Structure C at various velocities.
during each successive
ash. What is most curious about these results is, however, a further
extension of this study. In the lowest velocity after the rst
ash event, the temperature at
the interface reverts to its original level prior to the event's occurence. This indicates that the
rate of heat transfer through the material is sucient so as to allow for this to occur. However,
as the velocity is increased, an interesting pheonomenon occurs. At a velocity of 0.1 and 0.3
m/sec, we observe that due to the increase in the frequency of
ash heating occurences, the
steady-state temperature of the interface after each successive event increases.
In order to understand this, we present in Figure 4.7 the shear stress-shear strain diagrams
for all the three structures at sliding velocities of 0.05, 0.3 and 2 m/sec. To calculate the shear
stress, we extracted the sum of all the forces over the TL and divided by the surface area in
the xy plane. Similar to the method used in Chapter 2, we computed the friction force F
x
directly using the MD simulation output. As this represents the total lateral atomic force on
the TL, we computed the friction force at each successive MD time step. Similarly, in order
to determine the shear strain, we determined the center of mass positions of both the TL and
BL at each MD time step, and from this determined the shearing distance. Using the shearing
distance and the thickness of the material, we determine the angle .
Figure 4.7 reveals the change in the behavior of this system as a function of both the
thickness and velocity. Firstly, we note that the slope of the elastic region is important as it
relates to the shearing strength of the material. Experimental studies on quartz have indicated
a shearing strength of roughly 31.14 GPa. Our results indicate an shear modulus of roughly
29.2 GPa, indicating the accuracy of our simulations. At a velocity of 0.05 m/sec, the highest
64
Figure 4.7: Shear stress-shear strain diagrams for the three structures at various sliding veloc-
ities.
shear stress and strain on the material indicates a signicant friction force at the interface.
Similarly, at the end of the elastic region, i.e., the linear region, there is a clear sign of a stick-
slip event occuring in all the three materials of dierent sizes. Furthermore, the plastic region,
while not strictly linear, exhibits a very short period of lifetime in order to reach the steady
state. This is indicative of a brittle material. As we increase the sliding velocity, however, A
change in such behavior develops. While at low velocities there is a clear transition representing
the "stick-slip" phenomenon, at higher velocities the transition is less evident. Indeed, another
transformation seems to be at work. At V = 0:3 m/sec there is a larger dierence in the shear
stress of the three dierent materials, with Structure C exhibiting the lowest shear stress, and
the longest time for the materials to reach the steady shear stress-shear strain diagram.
The reason for this pheonomenon in the thickest slab needs to be discussed. At highest
velocity simulated, there is a signicant decay in the interfacial stick-slip event, even in the
case of Structure A. As the thickness of quartz transitions from A to C, there is an evolution
in the material from brittle to ductile. Indeed, at the highest velocity, the plastic region of
the stress-strain diagram in Structure C is no longer perfectly straight, with an increase in the
stress even at the point of transition from elasticity to plasticity. This is reminiscent of the
transition from brittle to ductile, as we increase the thickness of the material. The reason for
this pheonomenon will be discussed.
Figures 4.8 and 4.9 provide evidence for the transition from brittle to ductile behavior, as
thickness is increased. In Figure 4.8 we present the temperature proles at 4 dierent points
65
Figure 4.8: Temperature prole in Structure A immediately before and after
ash heating
events at various sliding velocities.
Figure 4.9: Same as Figure 4-9, except for Structure C.
66
during
ash heating events at three dierent velocities. The temperature prole represented as
"initial" represents that at the interface at the beginning of the simulation, which we show as a
reference prole. The next prole represents the interfacial temperature immediately befor the
fracture event. A sharp rise in the temperature is then developed, indicating the
ash occurence.
The prole referred to in the gure as \fracture" represents one at
ash heating. The three
proles discussed do not by themselves represent the reason for any transition between brittle
and ductile, but what follows is far more consequential.
The prole referred to in Figure 4.8 as "slip" indicates the temperature at the interface after
the fracture event, which is a measure of the energy that has propogated from the interface.
At the smallest thickness, we observe a signicant decrease in the interfacial temperature,
indicating a high rate of heat transfer away from the interface. Of course, materials that
operate at lower temperatures exhibit increased strength.
In Figure 4.9, we present the behavior of Structure C, indicating that at the "Slip" event
the temperature at the interface does not decrease nearly as much as it did for Stucture A. This
indicates a decreased rate of heat transfer from the interface and a higher sustained temperature
therein. We believe that this behavior in temperature (heat) propogation, and the fact that
a signicantly higher temperature is observed for Structure C at the interface is the primary
contributing factor to the transition from brittle to ductile behavior that was revealed in Figure
4.7.
While we determined the shearing force for the TL, we uses the method previously described
over the entirety of the top slab in order to determine the frictional force of the system. Figure
4.10 represents the plot of the time-averaged kinetic friction forceF
k
for each structure over the
entire range of the velocities simulated in this Chapter. Note the decay in the friction force that
follows roughly, F
k
/ lnV . Furthermore, if we consider an eective friction coecient
e
as
being equal to the ratio of frictional force, F
k
, divided by the applied normal force, F
n
, we can
gauge an approximate range of values with respect to the friction coecient. In Figure 4.11,
we present the relationship between the eective friction coecient as a function of velocity.
Note the logarithmic decay in the coecient that follows.
Indeed, in the experimental work conducted by Goldsby and Tullis, relating to the eect
of
ash heating on sliding resistance at faults, they also observed this relationship at similar
velocities. What we have proposed in this Chapter is the reason behind this experimentally
67
Figure 4.10: Dependence of the kinetic friction forceF
k
on the sliding velocityV . A logarithmic
relationship between F
k
and the sliding velocity appears to hold.
68
Figure 4.11: Dependence of the eective friction coecient
e
on the sliding velocity V . A
logarithmic relationship between
e
and the sliding velocity appears to hold.
69
and computationally observed phenomenon.
4.6 Conclusions
In this chapter we proposed the mechanism that is the root cause of the interfacial friction force
decay observed in experiemntal studies of quartz. Flash heating events increase in frequency as a
function of increasing sliding velocity, contributing to the increase in the interfacial temperature.
For small thickness, we observed a much more ecient rate of heat transfer from the interface,
manifesting itself in a decreased temperature immediately following
ash heating events. The
eciency of the heat transfer is reduced signicantly with increased thickness, resulting in a
weakened material. The weakening via stress-strain diagrams indicate clearly a decay in the
stress, as well as a transition from brittle to ductile behavior, as the sliding velocity increases.
The transition was most clearly observed for the case of Structure C. The weakening of the
interfacial zones can and the subsequent phenomenon of stick-slip are shown to have their roots
in
ash heating events.
70
Bibliography
1
B. Feeny, A. Guran, N. Hinrichs, and K. Popp, A historical review on dry friction and stick-
slip phenomena, Appl. Mech. Rev. 51, 321 (1998).
2
B.N.J. Persson, Sliding Friction: Physical Principles and Applications (Springer, New York,
2000).
3
J. Halling, ed., Principles of Tribology (MacMillan, London, 1978).
4
M.C. Shaw, A. Ber, and P.A. Mamin, Friction characteristics of sliding surfaces undergoing
subsurface plastic
ow, J. Basic Eng. (June 1960, p. 342; DOI: 10.1115/1.3662595
5
J.R. Rice and A.L. Ruina, Stability of steady frictional slipping, J. Appl. Mech 50, 343
(1983).
6
B. Jacobson, The Stribeck memorial lecture, Tribology Int. 36, 781 (2003).
7
D. Dowson, History of Tribology, 2nd ed. (American Society of Mechanical Engineers, New
York, 1999).
8
B. Bhushan, J.N. Israelachvili, and U. Landman, Nanotribology: friction, wear and lubrica-
tion at the atomic scale, Nature 374, 607 (1995).
9
E.D. Smith, M.O. Robbins, and M. Cieplak, Friction on adsorbed monolayers, Phys. Rev. B
54, 8252 (1996).
10
W.F. Brace and J.D. Byerlee, Stick-slip as a mechanism for earthquakes, Science 153, 990
(1966).
11
C.H. Scholz, Earthquakes and friction laws, Nature 391, 37 (1998).
12
S.D. Mesarovic, Dynamic strain aging and plastic instabilities, J. Mech. Phys. Solids 43,
671 (1995).
13
M.A. Heckel and I.D. Abrahams, Curve squeal of train wheels, part 1: mathematical model
for its generation, J. Sound Vib. 229, 669 (2000).
71
14
L.G. Hector, Jr. and B. Schmid, Simulation of asperity plowing in an atomic force micro-
scope. Part II: Plowing of aluminum alloys, Wear 215, 257 (1998).
15
J. Krim, D.H. Solina, and R. Chiarello, Nanotribology of a Kr monolayer: A quartz-crystal
microbalance study of atomic-scale friction, Phys. Rev. Lett. 66, 181 (1991).
16
O. Zw orner, H. H olscher, U.D. Schwarz, and R. Wiesendanger, The velocity dependence of
frictional forces in point-contact friction, Appl. Phys. A 66, S263 (1998).
17
E. Gnecco, R. Bennewitz, T. Gyalog, Ch. Loppacher, M. Bammerlin, E. Meyer, and H.J.
G ontherodt, Velocity dependence of atomic friction, Phys. Rev. Lett. 84, 1172 (2000).
18
U. Bockelmann, B. Essevaz-Roulet, and F. Heslot, Molecular stick-slip motion revealed by
opening DNA with oiconewton forces, Phys. Rev. Lett. 79, 4489 (1997).
19
J.A. Harrison, C.T. White, R.J. Colton, and D.W. Brenner, Molecular-dynamics simulations
of atomic-scale friction of diamond surfaces, Phys. Rev. B 46, 9700 (1992).
20
J.N. Glosli and G.M. McClelland, Molecular dynamics study of sliding friction of ordered
organic monolayers, Phys. Rev. Lett. 70, 1960 (1993).
21
L. Zhang and H. Tanaka, Towards a deeper understanding of wear and friction on the atomic
scale - a molecular dynamics analysis, Wear 211, 44 (1997).
22
J. Shimizu, H. Eda, M. Toritsune, and E. Ohmura, Molecular dynamics simulation of friction
on atomic scale, Nanotechnology 9, 118 (1998).
23
G. He, M.H. M user, and M.O. Robbins, Adsorbed layers and the origin of static friction,
Science 284, 1650 (1999).
24
M.H. M user and M.O. Robbins, Conditions for static friction between
at crystalline sur-
faces, Phys. Rev. B 61, 2335 (2000).
25
B. Li, P.C. Clapp, J.A. Rifkin, and X.M. Zhang, Molecular dynamics simulations of stick-slip,
J. Appl. Phys. 190, 3090 (2001).
26
M.H. M user, Nature of mechanical instabilities and their eect on kinetic friction, Phys.
Rev. Lett. 89, 224301 (2002).
72
27
J. Zhang and J.B. Sokolo, Adiabatic molecular-dynamics-simulation-method studies of ki-
netic friction, Phys. Rev. E 71, 066125 (2005).
28
E.S. Torres, S. Goncalves, C. Scherer, and M. Kiwi, Nanoscale sliding friction versus com-
mensuration ratio: Molecular dynamics simulations, Phys. Rev. B 73, 035434 (2005).
29
Q. Zhang, Y. Qi, L.G. Hector, Jr., T. Cagin, and W.A. Goddard, III, Atomic simulations
of kinetic friction and its velocity dependence at Al/Al and -Al
2
O
3
/-Al
2
O
3
interfaces,
Phys. Rev. B 72, 045406 (2005).
30
W.K. Kim and M.L. Falk, Accelerated molecular dynamics simulation of low-velocity fric-
tional sliding, Model. Simul. Mater. Sci. Eng. 18, 034003 (2010).
31
X.J. Yang, Molecular dynamics simulations of sliding friction of rigid sphere with single
crystal copper surface, Appl. Mech. Mater. 345, 167 (2013).
32
S. Kawai, A. Benassi, E. Gnecco, H. S ode, R. Pawlak, X. Feng, K. M ullen, D. Passerone, C.
A. Pignedoli, P. Rueux, R. Fasel, and E. Meyer, Superlubricity of graphene nanoribbons
on gold surfaces, Science 351, 957 (2016).
33
Y. Takeda, K. Nakamura, K. Maeda, and Y. Matsushita, Eects of elemental additives on
electrical resistivity of silicon carbide ceramics, J. Am. Ceram. Soc. 70, C-266 (1987).
34
K. Schulz and M. Durst, Advantages of an integrated system for hot gas ltration using rigid
ceramic elements, Filtr. Sep. 31, 25 (1994).
35
A.J. Rosenbloom, D.M. Sipe, Y. Shishkin, Y. Ke, R.P. Devaty, and W.J. Choyke, Nanoporous
SiC: A candidate semi-permeable material for biomedical applications, Biomed. Mi-
crodev. 6, 261 (2004).
36
Z. Li, K. Kusakabe, and S. Morooka, Preparation of thermostable amorphous Si-C-O mem-
brane and its application to gas separation at elevated temperature, J. Membr. Sci. 118,
159 (1996).
37
C.A. Zorman, A.J. Fleischman, A.S. Dewa, M. Mehregany, C. Jacob, S. Nishino, and P.
Pirouz, Epitaxial growth of 3C-SiC lms on 4 in. diam (100) silicon wafers by atmospheric
pressure chemical vapor deposition, J. Appl. Phys. 78, 5136 (1995).
73
38
S.H. Kenawy and W.M.N. Nour, Microstructural evaluation of thermally fatigued SiC-
reinforced Al
2
O
3
/ZrO
2
matrix composites, J. Mater. Sci. 40, 3789 (2005).
39
B. Elyassi, M. Sahimi, and T.T. Tsotsis, Silicon carbide membranes for gas separation
applications, J. Membr. Sci. 288, 290 (2007).
40
B. Elyassi, M. Sahimi, and T.T. Tsotsis, A novel sacricial interlayer-based method for the
preparation of silicon carbide membranes, J. Membr. Sci. 316, 73 (2008).
41
N. Keller, C. Pham-Huu, G. Ehret, V. Keller, and M.J. Ledoux, Synthesis and characteri-
sation of medium surface area silicon carbide nanotubes, Carbon 41, 2131 (2003).
42
S.H. Barghi, T.T. Tsotsis, and M. Sahimi, Hydrogen sorption hysteresis and superior storage
capacity of silicon-carbide nanotubes over their carbon counterparts, Int. J. Hydrog.
Energy 39, 21107 (2014).
43
S.H. Barghi, T.T. Tsotsis, and M. Sahimi, Experimental investigation of hydrogen adsorption
in doped silicon-carbide nanotubes, Int. J. Hydrog. Energy 41, 369 (2016).
44
W. Krenkel and T. Henke, Design of high preformance CMC brake discs, Key Eng. Mater.
164-165, 421 (1999).
45
W. Krenkel, B. Heidenreich, and R. Renz, C/C-SiC composites for advanced friction systems,
Adv. Eng. Mater. 4, 427 (2002).
46
S. Fan, L. Zhang, Y. Xu, L. Cheng, J. Lou, J. Zhang, and L. Yu, Microstructure and
properties of 3D needle-punched carbon/silicon carbide brake materials, Composites Sci.
Technol. 67, 2390 (2007).
47
S. Fan, L. Zhang, Y. Xu, L. Cheng, G. Tian, S. Ke, F. Xu, and H. Liu, Microstructure
and tribological properties of advanced carbon/silicon carbide aircraft brake materials,
Composites Sci. Technol. 68, 3002 (2008).
48
J.B. Wallace, D. Chen, and L. Shao, Carbon displacement-induced single carbon atomic
chain formation and its eects on sliding of SiC bers in SiC/graphene/SiC composite,
Mater. Res. Lett. 4, 55 (2016).
74
49
S. Naserifar, L. Liu, W.A. Goddard, III, T.T. Tsotsis, and M. Sahimi, Toward a process-
based molecular model of SiC membranes. 1. Development of a reactive force eld, J.
Phys. Chem. C 117, 3308 (2013).
50
S. Naserifar, W.A. Goddard, III, L. Liu, T.T. Tsotsis, and M. Sahimi, Toward a process-
based molecular model of SiC membranes. 2. Reactive dynamics simulation of the pyrol-
ysis of polymer precursor to form amorphous SiC, J. Phys. Chem. C 117, 3320 (2013).
51
S. Naserifar, W.A. Goddard, III, T.T. Tsotsis, and M. Sahimi, First principles-based mul-
tiparadigm, multiscale strategy for simulating complex materials processes with applica-
tions to amorphous SiC lms, J. Chem. Phys. 142, 174703 (2015).
52
A.C.T. van Duin, S. Dasgupta, F. Lorant, and W.A. Goddard, ReaxFF:. A reactive force
eld for hydrocarbons, J. Phys. Chem. A 105, 9396 (2001).
53
A. Li, Y. Liu, and I. Szlufarska, Eects of interfacial bonding on friction and wear at sil-
ica/silica interfaces, Tribol. Lett. 56, 481 (2014).
54
C. Hu, M. Bai, J. Lv, Z. Kou, and X. Li, Molecular dynamics simulation on the tribology
properties of two hard nanoparticles (diamond and silicon dioxide) conned by two iron
blocks, Tribol. Int. 90, 297 (2015).
55
C. Nagel, K. R atzke, E. Schmidtke, J. Wol, U. Geyer, and F. Faupel, Free-volume changes
in the bulk metallic glass Zr
46:7
Ti
8:3
Cu
7:5
Ni
10
Be
27:5
and the undercooled liquid, Phys.
Rev. B 57, 10224 (1998).
56
W.J. Wright, T.C. Hufnagel, and W.D. Nix, Free volume coalescence and void formation in
shear bands in metallic glass, J. Appl. Phys. 93, 1432 (2003).
57
A.R. Yavari, A. Le Moulec, A. Inoue, N. Nishiyama, N. Lupu, E. Matsubara, W.J. Botta, G.
Vaughan, M. Di Michiel, and A. Kvick, Excess free volume in metallic glasses measured
by X-ray diraction, Acta Mater. 53, 1611 (2005).
58
J. Terso, Empirical interatomic potential for carbon, with applications to amorphous car-
bon, Phys. Rev. Lett. 61, 2879 (1988).
75
59
J. Terso, New empirical approach for the structure and energy of covalent systems, Phys.
Rev. B 37, 6991 (1988).
60
Y. Wang, J. Xu, J. Zhang, Q. Chen, Y. Ootani, Y. Higuchi, N. Ozawa, J.M. Martin, K.
Adachi, and M. Kubo, Tribochemical reactions and graphitization of diamond-like carbon
against alumina give volcano-type temperature dependence of friction coecients: A
tight-binding quantum chemical molecular dynamics simulation, Carbon133, 350 (2018).
61
L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen K orper, Z. Agnew. Math.
Mechanik 8, 85 (1928).
62
G.A. Tomlinson, A molecular theory of friction, Phil. Mag. 7, 905 (1929).
63
X. Zhao, Y. Liu, Q. Wen, and Y. Wang, Frictional performance of silicon carbide under
dierent lubrication conditions, Friction 2, 58 (2014).
64
S. Lafon-Placette, K. Delb e, J. Denape, and M. Ferrato, Tribological characterization of
silicon carbide and carbon materials, J. Europ. Ceramic Soc. 35, 1147 (2015).
65
Verhoeven G S, Dienwiebel M, Frenken M J W M (2004) Model calculations of superlubricity
of graphite. Phys. Rev. B 70:165418.
66
M user M H (2003) Structural lubricity: Role of dimension and symmetry. Europhys. Lett.
66:97.
67
Cai J, Rueux P, Jaafar R, Bieri M, Braun T, Blankenburg S, Muoth M, Seitsonen A P,
Saleh M, Feng X, M ullen K, Fasel R (2010) Atomically precise bottom-up fabrication of
graphene nanoribbons. Nature 466:470.
68
Berman D, Erdemir A, Sumant A V (2014) Graphene: a new emerging lubricant. Mater.
Today 17:31.
69
Matsushita K, Matsukawa H, Sasaki N (2005) Atomic scale friction between clean graphite
surfaces. Solid State Commun. 136:51.
70
Hod O (2012) Interlayer commensurability and superlubricity in rigid layered materials.
Phys. Rev. B 86:075444.
76
71
Kawai S, Benassi A, Gnecco E, S ode H, Pawlak R, Feng X, M ullen K, Passerone D, Pignedoli
C A, Rueux P, Fasel R, Meyer E (2016) Superlubricity of graphene nanoribbons on gold
surfaces. Science 351:957.
72
Manini N, Braun O, Tosatti E, Guerra R, Vanossi A (2016) Friction and nonlinear dynamics.
J. Phys.: Condens. Matter. 28: 293001.
73
Dietzel D, Ritter C, M onningho T, Fuchs H, Schirmeisen A, Schwarz U D (2008) Frictional
duality observed during nanoparticle sliding. Phys. Rev. Lett. 101:125505.
74
Brndiar J, Turansk y R, Dietzel D, Schirmeisen A, Stich I (2011) Understanding frictional
duality and bi-duality: Sb-nanoparticles on HOPG. Nanotechnology 22:085704.
75
Dietzel D, Feldmann M, Schwarz U D, Fuchs H, Schirmeisen N (2013) Scaling laws of struc-
tural lubricity: Phys. Rev. Lett. 111:235502.
76
Mougin K, Gnecco E, Rao A, Cuberes M T, Jayaraman S, McFarland E W, Haidara H, Meyer
E (2008) Manipulation of gold nanoparticles: In
uence of surface chemistry, temperature,
and environment (vacuum versus ambient atmosphere). Langmuir 24:1577.
77
Krim J (2012) Friction and energy dissipation mechanisms in adsorbed molecules and molec-
ularly thin lms. Adv. Phys. 61:155.
78
Vanossi A, Manini N, Urbakh M, Zapperi S, Tosatti E (2013) Modeling friction: From
nanoscale to mesoscale. Rev. Mod. Phys. 85:529.
79
Park J Y, Salmeron M (2014) Fundamental aspects of energy dissipation in friction. Chem.
Rev. 114:677.
80
Manini N, Mistura G, Paolicelli G, Tosatti E, Vanossi A (2017) Current trends in the physics
of nanoscale friction. Adv. Phys.: X 2:569.
81
Daw M S, Baskes M I (1984) Embedded-atom method: Derivation and application to im-
purities, surfaces, and other defects in metal. Phys. Rev. B 29:6443.
82
Daw M S, Foiles S (1993) The embedded-atom method: a review of theory and applications.
Mat. Sci. Engr. Rep. 9:251.
77
83
Norskov J K (1979) Electron structure of single and interacting hydrogen impurities in free-
electron-like metals. Phys. Rev. B 20:446.
84
Norskov J K, Lang N D (1980) Eective-medium theory of chemical binding: Application
to chemisorption. Phys. Rev. B 21:2131.
85
M. Sahimi (2003) Heterogeneous Materials II. Springer, New York, chapter 9.
86
Foiles S M, Baskes M I, Daw M S (1986) Embedded-atom-method functions for the fcc
metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys. Rev. B 13:7983.
87
Stuart S J, Tutein A B, Harrison J A (2000) A reactive potential for hydrocarbons with
intermolecular interactions. J. Chem. Phys. 112:6472.
88
Brenner D W, Shenderova O, Harrison J A, Stuart S J, Ni B, Sinnott S B (2002) A second-
generation reactive empirical bond order (REBO) potential energy expression for hydro-
carbons. J. Phys.: Condens. Mater. 14:783.
89
Chien S.-K., Yang Y.-T., Chen C.-K. (2011) A molecular dynamics study of the mechanical
properties of graphene nanoribbon-embedded gold composites. Nanoscale 3:4307.
90
Woods C R, Britnell L, Eckmann A, Ma R S, Lu J C, Guo H M, Lin X, Yu G L, Cao Y,
Gorbachev R V, Kretinin A V, Park J, Ponomarenko L A, Katsnelson M I, Gornostyrev
Yu N, Watanabe K, Taniguchi T, Casiraghi C, Gao H.-J., Geim A K, Novoselov K S (2014)
Commensurate-incommensurate transition in graphene on hexagonal boron nitride. Nat.
Phys. 10:451.
91
Gigli L, Manini N, Tosatti E, Guerra R, Vanossi A (2018) Lifted graphene nanoribbons on
gold: from smooth sliding to multiple stick-slip regimes. Nanoscale 10:2073.
92
B. Feeny, A. Guran, N. Hinrichs, and K. Popp, Appl. Mech. Rev. 51, 321 (1998).
93
L. Smeraglia, A. Billi, E. Carminati, A. Cavallo, G. Di Toro, E. Spagnuolo, and F. Zorzi,
Ultra-thin clay layers facilitate seismic slip in carbonate faults, Sci. Rep. 7, 664 (2017).
94
C. Collettini, C. Viti, T. Tesei, and S. Mollo, Geology 41, 927 (2013).
95
H. Sone and T. Shimamoto, Nature Geosci. 2, 705 (2009).
78
96
J.R. Rice, J. Geophys. Res. 111, B05311 (2006).
97
J. Halling, ed., Principles of Tribology (MacMillan, London, 1978).
98
M.C. Shaw, A. Ber, and P.A. Mamin, J. Basic Eng. (June 1960), p. 342; DOI: 10.1115/1.3662595
99
W.F. Brace and J.D. Byerlee, Science 153, 990 (1966).
100
C.H. Scholz, Nature 391, 37 (1998).
101
W. Krenkel, and N. Langhof, Ceramic matrix composites for high performance friction
applications, Proceedings of the IV Advanced Ceramics and Applications Conference.
(2017).
102
U. Bockelmann, B. Essevaz-Roulet, and F. Heslot, Phys. Rev. Lett. 79, 4489 (1997).
103
N. Piroozan, S. Naserifar, and M. Sahimi, J. Applied Physics 125, (2019).
104
H. Kitajima, F.M. Chester, and J.S. Chester, J. Geophys. Res. Solid Earth 116, B08309
(2011).
105
G. Di Toro, R. Han, T. Hirose, N. De Paola, S. Nielsen, K. Mizoguchi, F. Ferri, M. Cocco,
and T. Shimamoto, Nature 471, 494 (2011).
106
E. Spagnuolo, O. Plmper, M. Violay, A. Cavallo, and G. Di Toro, Sci. Rep. 5, 16112 (2015).
107
H. Kitajima, J.S. Chester, F.M. Chester, and T. Shimamoto, J. Geophys. Res. 115, 408
(2010).
108
C.W. Mase and L. Smith, J. Geophys. Res. 92, 6249 (1987).
109
A.H. Lachenbruch, J. Geophys. Res. 85, 6097 (1980).
110
A. Molinari, Y. Estrin, and S. Mercier, J. Tribol. 121, 35 (1999).
111
S. Lim, M. Ashby, and J. Brunton, 37, 767 (1989).
112
C.M.M. Ettles, J. Tribol. 108, 98 (1986).
113
R.S. Anderson, and S.P. Anderson, Geomorphology: The mechanics and chemistry of land-
scapes, Cambridge University Press. p. 187 (2010).
79
114
D.L. Goldsby and T.E. Tullis, Geophys. Res. Lett. 29, 1844 (2002).
115
M.H. M user and M.O. Robbins, Phys. Rev. B 61, 2335 (2000).
116
B. Li, P.C. Clapp, J.A. Rifkin, and X.M. Zhang, J. Appl. Phys. 190, 3090 (2001).
117
M.H. M user, Phys. Rev. Lett. 89, 224301 (2002).
118
J. Zhang and J.B. Sokolo, Phys. Rev. E 71, 066125 (2005).
119
E.S. Torres, S. Goncalves, C. Scherer, and M. Kiwi, Phys. Rev. B 73, 035434 (2005).
120
J. Terso, Phys. Rev. Lett. 61, 2879 (1988).
121
J. Terso, Phys. Rev. B 37, 6991 (1988).
122
A.C.T. van Duin, S. Dasgupta, F. Lorant, and W.A. Goddard, J. Phys. Chem. A 105,
9396 (2001).
123
A. Li, Y. Liu, and I. Szlufarska, Tribol. Lett. 56, 481 (2014).
124
C. Hu, M. Bai, J. Lv, Z. Kou, and X. Li, Tribol. Int. 90, 297 (2015).
80

Abstract (if available)

Abstract Friction is an essential part of human experience, as it is a key component in our ability to gain traction to walk, stand, work and drive, and also save energy. While friction is certainly a necessary phenomenon, it presents one glaring problem. We need energy to overcome the resistance to motion that is caused by frictional forces, and too much of it causes excess cost, increased energy to perform work and, therefore, increased inefficiencies as well. It is the purpose of this Thesis, then, to present an advance towards two interconnected objectives in this regard. The first is a better understanding of the mechanical and energetic precursors that give rise to friction and, then, to present applicable results in this regard. These objectives are studied in this Thesis, including the tribological properties of silicon carbide, a material commonly used in high-temperature conditions, and the superlubric properties of graphene, an excellent solid lubricant, and its behavior coupled with a series of precious metals. The final chapter will study friction between sandstone surfaces, represented by quartz, one of the most common terrestrial minerals, and its frictional properties at subseismic sliding conditions. ❧ Sliding friction between two SiC surfaces is important, due to its relevance to many practical applications. It is also important to study whether kinetic friction at the nanoscale follows Coulomb's law. Since SiC exists both as an amorphous material and with a crystalline structure, the effect of surface roughness on the kinetic friction may also be significant. We report the results of extensive molecular dynamics simulation of sliding friction between surfaces of the two types of SiC over a wide range of sliding velocities. The amorphous SiC was generated by the reactive force field ReaxFF, which was also used to represent the interaction potential for the simulation of sliding friction. As the sliding velocity increases, bond breaking occurs at the interface between the two surfaces, leading to their roughening and formation of excess free volume. They reduce the kinetic friction force, hence resulting in decreasing the difference between kinetic friction in the amorphous and crystalline surfaces. The average kinetic friction force depends nonlinearly on the sliding velocity V, implying that Coulomb's law of friction is not satisfied by the surfaces that we study at the nanoscale. The average kinetic friction force Fₖ depends on V as, Fₖ ∝ ln V. ❧ The state of extremely low friction, known as superlubricity, has very important applications to the development of various types of materials, including those that are invaluable to the goal of reducing energy loss in mechanical systems, and those in complex gearing and bearing systems. One material that can produce very low friction is graphene that offers distinct properties as a solid-state lubricant, and can potentially be used as a coating material on surfaces. We have carried out extensive molecular dynamics (MD) simulations in order to study and compute the friction force between a graphene nanoribbon and an inert metal surface. The metal surfaces that we study are those of Au, Ag, and Pt, all of which are used in various instruments, as well as in various materials employed in the industry. Consistent with very recent experiments, the Au-graphene system exhibits superlubricity, but the Ag-graphene and Pt-graphene pairs manifest friction forces higher than that of Au-graphene, although they are still very small. The MD simulations indicate that the average friction force for graphene on an Au surface is approximately 1.5 pN, with the corresponding values being about 6 pN and 11 pN for, respectively, Ag and Pt surfaces. ❧ The frictional behavior of faults during seismic events, such as earthquakes, is a topic of profound importance. The experimentally-observed degradation of shear strengths in silicate rocks has been hypothesized to emanate from the phenomenon of “flash heating.” Using extensive MD simulations, we study this phenomenon at interstitial asperities for quartz crystals and present extensive results on the mechanical and thermal evolution of the material under shearing and various sliding velocities. Furthermore, we also present the evolution as a function of varying the thickness of the two layers sliding with respect to each other. We discover that with an increase in the sliding velocity, the frequency and intensity of flash heating events increase as well. This in turn destabilizes the crystalline structure at the interface between opposing quartz layers, forming an amorphous layer. At low slip rates, the heat generated is able to diffuse away appreciably through the material, resulting in a small temperature rise and, therefore, weak effect on the overall strength of the material. At higher slip rates, however, there is not enough time for the heat generated at the interface to dissipate. This in turn causes an increase in the overall interstitial temperature, which in turn causes decreased material strength, followed by sharply reduced frictional resistance.

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Creator Piroozan, Nariman (author)

Core Title Molecular-scale studies of mechanical phenomena at the interface between two solid surfaces: from high performance friction to superlubricity and flash heating

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publication Date 04/25/2019

Defense Date 03/22/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag computational physics,molecular dynamics,OAI-PMH Harvest,tribology

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sahimi, Muhammad (committee chair), Jha, Birendra (committee member), Nakano, Aiichiro (committee member)

Creator Email piroozan@usc.edu,piroozan2@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-143591

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Legacy Identifier etd-PiroozanNa-7242.pdf

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Document Type Dissertation

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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