Reaction Rate Theory Applied to the Stress and Velocity Dependence of

Atomic-Scale Wear

Zac Milne

The qualifying exam will be held June 3rd from 1-3 p.m. in the MEAM conference room

1

Contents

Section Title Page

Abstract 2

1 Introduction 3

2 The history and derivation of the Eyring equation 5

3 Studies reveal atom-by-atom wear. 8

4 Atomic simulations show that an elastic contact can follow Boltzmann

statistics 16

5 The current theory of atomistic wear as a function of sliding distance

shows no dependence on velocity. 24

6 The interfacial shear stress has a velocity dependence in Briscoe and

Evans’ theory. 25

7 Experimental studies disagree on the velocity dependence of wear. 29

8 Preliminary data at various sliding velocities suggests a velocity

dependence of wear. 32

9 Future work is proposed to probe the velocity and temperature

dependence of wear. 35

10 Conclusions 37

References 39

2

Abstract

Since the demonstration of atom-by-atom wear of atomic force microscope tips,

research has been conducted to understand the physical mechanisms behind it. An

Eyring equation has been proposed and validated for the dependence of wear on

normal stress, but the dependence on velocity and temperature has not been explored

systematically. This paper reviews the history and derivation of the Eyring equation,

and illustrates how it has been applied to the study of atomic-scale wear. The Eyring

equation is derived under specific assumptions, notably that the reactants have a

Boltzmann distribution of energy. A recent article reports molecular dynamics studies

which help to elucidate the validity of this assumption for atom scale solid contacts.

Researchers who have applied the Eyring equation to model atom-by-atom wear have

proposed an activation-barrier-lowering shear stress-velocity relationship which

originates in a study by Briscoe and Evans on the friction of Langmuir Blodgett films.

This work is briefly discussed and the validity of its conclusions as applied to wear are

critiqued. An atomic force microscopy study of wear versus velocity conducted by

Bhushan et al. is discussed and analyzed. This article also reports preliminary data for

the wear dependence of velocity, and it is shown that the existing theory may not

completely explain this aspect of wear. Future work is proposed to probe the

dependencies of wear on velocity and temperature.

3

1. Introduction

The advent of micro/nano electromechanical systems (M/NEMS) has brought about

an increase of research activity in the area of nano and micro mechanics. The field of

nanotribology contributes to this pursuit by studying the physical mechanisms

underlying friction and wear in nanoscale contacts. It has been shown that the

macroscale laws of friction and wear cannot simply be extended to the nanoscale, [1, 3-

5] so there is a large effort to understand this divergence and its implications for the

lifespan of M/NEMS. Wear of surfaces occurs in several distinct regimes, including

abrasive, fretting, fatigue, and atom-by-atom wear (also called “atomistic wear”). The

latter wear regime has been the subject of recent research thanks to the use of tools

such as the atomic force microscope (AFM) and the transmission electron microscope

(TEM) which allow researchers to probe nanoscale contacts with a vast array of

analytical tools [1, 3-6].

Atom-by-atom wear can be described as the attrition of atoms from one surface to

another in a discrete (i.e., not bulk) fashion. This has been demonstrated in published

work [1,3-6], and visually documented and quantified in a transmission electron

microscope (TEM) by Jacobs et al. [3]. Figure 1 below illustrates traces of the profile of

a silicon AFM tip after several sliding increments, as well as a lattice-resolved picture of

the very end of the tip, demonstrating that the tip is indeed being worn.

When atom-by-atom wear is the dominating wear regime, a mean field theory can

be used to describe the atomic process of bonding which leads to wear. The Eyring

equation, derived in the early 20th century, has been successful in describing rates of

chemical reaction. Atom-by-atom wear is assumed to occur as a chemical reaction, and

thus the Eyring equation is a promising candidate to describe it. This article explores

the history and derivation of the Eyring equation and cites how it has been applied to

the study of atom-by-atom wear in a mean field sense.

The Eyring equation involves the use of statistical mechanics. Two surfaces undergo

sliding contact to induce wear, and the atoms in a single asperity contact are assumed

4

to have a Boltzmann distribution of energies. A recent molecular dynamics study,

reviewed in section 4, has indeed found that the Boltzmann assumption is an accurate

one over a wide range of conditions.

Fig 1. Periodic high-resolution TEM images demonstrate gradual surface evolution without sub-surface

damage. a. Four traces of successive profiles of one of the four asperities tested, shown at 200 nm

intervals of sliding distance, overlaid on a TEM image of the asperity just after the final interval, indicated

by the red trace. Inset: detailed view of the traces, demonstrating gradual wear of the surface, often

with less than 1 nm change in the asperity height per sliding interval. b. Representative lattice-resolved

image of the same asperity shows no evidence of dislocations or defects in the sub-surface silicon lattice,

even in highly worn areas [3].

The mean field Eyring model discussed in this manuscript is theorized to be

dependent on interfacial shear stress (frictional), which itself is a function of sliding

velocity. The form of the velocity dependence as derived by Briscoe and Evans [7] is

explored in detail in section 6. Some preliminary data on the velocity dependence of

wear is discussed and it is concluded that additional focused work is required to make a

sound judgment on the validity of the theorized relationship between velocity and wear,

and to understand whether a different model altogether is required.

5

2. The history and derivation of the Eyring equation

Even though the mechanism of atom-by-atom wear occurs with orders-of-magnitude

fewer atoms in comparison with macroscale wear phenomena, the theory assumes that

a sufficient number of atoms are present such that the process can be described as

stochastic and can therefore be modeled by statistical mechanics. As a result,

researchers [1, 3-5] have proposed the use of reaction rate theory (RRT) which has

been successful in describing a number of chemical rate processes. The application of

RRT to model atom-by-atom wear will be the main focus of this paper. This section will

briefly cover the development of RRT and outline a non-rigorous derivation of the

Eyring equation.

Reaction rate theory began as a response to the mounting set of data which

exhibited an exponential dependence of reaction rates on temperature. Arrhenius was

the first to propose the formula on a purely empirical basis, where is the

rate constant with units of s-1, is an experimentally-determined constant, is the

ideal gas constant, is the absolute temperature, and is the activation energy. This

formula was used widely by the second decade of the 20th century [8]. However, the

activation energy and the pre-exponential A had no scientific backing, and thus

several researchers at that time made attempts to derive their physical significance.

In 1931, Henry Eyring and Michael Polanyi published a seminal article that

delineated the origins of the activation energy (also sometimes denoted as or

) [9]. Their major insight was that, in a chemical ensemble undergoing reactions, the

activated species are in equilibrium with the reactants. Marcelin [10] proposed a

potential surface for the ensemble of reactants and products with coordinates

completely described by each species’ position and momentum (2N degrees of freedom

where N is the number of atoms; 2 because there is motion along the reaction

coordinate only), which can be pictorially represented as in Figure 2, similarly to how

Eyring and Polanyi also illustrated it.

6

At any concave down cusp of the energy versus reaction coordinate curve, the

reacting species are in what is called their “activated state,” which Eyring and Polanyi

proposed was in quasi-equilibrium with reactants and products. The term quasi denotes

the fact that the activated complexes do not achieve a Boltzmann distribution of

energies, but their concentration relative to the reactants, which are in a Boltzmann

distribution, is a constant called the “equilibrium constant”. Therefore the constant

can be written as:

Fig 2. A one-dimensional, two step (bonding and wearing) representation of the potential surface for a

chemical ensemble. The illustrations represent: (far left) a bondless contact, (middle left) a bond formed

between the tip and substrate after the atom overcomes ΔEbond, (middle right) that bond broken after

overcoming ΔEbreak and, (far right) the atom from the tip bonded to the substrate. The total difference in

energy between the initial (unworn) state to the final (worn) state is ΔEwear. This is the energy required to

increase the entropy dS and (in the case of wear) increase the enthalpy of the system.

ΔEbond

ΔEbreak

ΔEwear

Reaction coordinate

Energ

y

1.1

7

where is the concentration of the activated species and and are the

concentrations of the reactants respectively. The rate of production of product

( is

the concentration of products and is time) is some constant multiplied by the

concentration of activated complexes:

.

The constant can be taken as the speed at which activated species decay into

products, which is their characteristic vibration frequency because at the cusp, any

perturbation will either send the complexes into products, or back to reactants. Many

studies have assumed that the rate of transformation into products, the “forward rate”,

was much higher than the rate of transformation back into reactants, so that this latter

process can be suitably ignored, effectively setting the proportion of the forward rate to

unity [8].

Using a statistical mechanical approach, Eyring derived an expression for the

equilibrium constant of a binary reaction, which is the fraction of particles with an

energy higher than an activation energy (the minimum energy required to force a

reaction) to the rest of the particles in the system. He arrived at the ratio of activated

species to reactant species:

where is Boltzmann’s constant and is the temperature of the system at equilibrium.

Putting these results together, the final expression for the rate constant becomes:

where is the vibrational frequency of the activated complex. The reaction rate is the

rate at which reactants become products normalized by the concentration of the

reactants.

The assumptions inherent to this derivation are that: 1) the reaction takes place

along the path of least energy, 2) the initial and final states are linked through a

1.2

1.3

1.4

8

smooth (continuously differentiable), continuous path, 3) the probability that the

reaction goes to products is much higher than the reaction decaying back into

reactants, 4) the initial and activated states are in equilibrium, and 5) there is one less

degree of freedom at the activated state which is vibrational freedom.

Currently there is no rigorous framework mathematically expounding in a molecular

kinetics framework how the Eyring equation can apply to wear. The greatest obstacle to

doing so is that the stress landscape of two atomic solids in contact is complex and still

not fully understood. Additionally, atomistic wear removes a relatively small number of

atoms compared to the number of atoms that react in a binary chemical reaction and it

is not completely certain that a Boltzmann distribution can be applied to such a small

sample. Another problem is that the limits of temperature, velocity, stress, and

activation energy within which atomistic wear can occur are not yet known and need to

be further studied. There are several articles which derive the prefactor for different

atomistic reactions (i.e. atomic hopping or dislocation glide [11, 12]). Perhaps the

closest analogy to wear is that of atomic hopping. Vineyard provided the derivation of

the prefactor for atomic hopping into lattice vacancies [13], but no analogous

derivation has been given to elucidate to rate process for wear events at the interface

of two sliding surfaces, and it is not clear that such a derivation is possible. Despite this

shortcoming, as will be shown in the following sections, the theory has been applied

successfully in a mean field sense to model atomistic wear.

3. Studies reveal the existence of atom-by-atom wear.

Wear is a process by which changes in the physical and chemical makeup of solids

are induced by the sliding of one body over another. Although on the macroscale wear

can be abrasive or dominated by plastic deformation and fracture, [14-16] on the

nanoscale and at low loads it is often atomistic in nature. Therefore, its origins are

undoubtedly on a scale that requires further insight than typical continuum approaches

can lend. Even so, continuum-derived theories of wear such as the Archard wear law

have been applied to ever smaller scales since its conception by J. Archard in 1953

[14]. Archard’s law is expressed as:

9

where is the total volume of material removed, is a constant, is the normal load,

is the sliding distance, and is the hardness of the surfaces.

Although on the macroscale, and on the nanoscale for a few studies of fracture, this

equation has applied very well [17-20], it is almost entirely empirical and thus does not

provide a complete picture of wear at all sliding distances and normal loads on the

nanoscale [1-5, 21]. Understanding the nanoscale mechanisms of wear is critical to the

successful deployment of micro and nano electromechanical devices (M/NEMS).

Because of their relatively high surface-area-to-volume ratio, adhesion of nanoscale

devices can play a greater role. Intuitively, interfacial stress is directly related to

adhesion because it is related to surface energy. It is also obvious that wear has a

dependence on interfacial stress: higher stress equates to higher wear. Currently, the

relations between all of these characteristics of nanoscale contacts are the subject of an

emerging body of research. This section will focus on those studies that have elucidated

the deviations of nanoscale wear from Archard’s law and which also found that their

data was in good correlation with reaction rate theory.

Recent studies of nanoscale wear have demonstrated that, in some cases, material

is removed in an atom-by-atom fashion. Gotsmann and Lantz performed sliding tests in

an atomic force microscope (AFM) of silicon tips on polyaryletherketone spun cast on

silicon counter-surface [1]. The use of an AFM to perform nano-wear experiments is

desirable because wear occurs at the single asperity level as depicted in Figure 3.

Fig 3. Illustration of the representation of a macroscale asperity system by a single asperity AFM probe

[22] [ 23].

2

10

The authors performed sliding wear tests for hundreds of meters of sliding distance,

taking pull off force measurements every 62 cm of sliding. Figure 4 depicts a typical pull

of force measurement taken in an AFM. The tip approaches the sample and a snap in

occurs. The tip is then retracted from the sample and adhesion between it and the

substrate pulls the tip down, which is sensed by the AFM photodiode which outputs this

signal as a displacement of the cantilever from its equilibrium position.

The authors assumed that the tip was shaped like a truncated cone, and that the

pull of force they measured is proportional to the radius of the flat portion of the

truncated cone, which was measured at the end of each experiment i.e.,

. The strength of this assumption lies in the fact that as the radius of the tip

increases, more material comes into contact with the substrate, thus strengthening the

adhesion. Although this assumption has not been explicitly tested, it simplified the wear

test because they could perform sliding and pull-off forces all within the AFM, never

needing to take the AFM chip out. Wear versus sliding distance data were taken for 11

tips, data for 5 of which are shown in Figure 5.

Fig 4. Illustration of a measurement. a. The contact and retraction of an AFM probe, which

deflects due to the adhesive force between tip and sample. b. Representation of a pull off force curve

[3] [ 24].

Figure 5 shows that a free exponential fit to the data (blue curves), using a least

squares analysis with Archard’s equation for a conical tip , ( is the radius of

the tips , and are exponents which equal 1/3 for a conical tip), revealed a much

closer correlation than that obtained by applying the Archard wear equation with n and

Substrate

11

m equal to 1/3 to it (red curves), especially in the low load case of 5 nN, where the

Archard equation deviates strongly from the data.

Fig 5. Data from Gotsmann et al. of

tip radius versus sliding distance for

silicon flat-punch AFM tips on a

polyaryletherketone spun cast on

silicon counter surface for various

normal forces. The black curve

represents experimental results, while

the red and blue curves are the free

exponential fit and Archard wear law

fit, respectively [1].

The lack of correlation between Archard’s wear equation and the physical data taken

by Gotsmann et al. and others before them [17, 18, 21] led to the authors’ deduction

that a more atomistic phenomenon must be taking place at the stressed interface

between the two sliding bodies. The insight that the kinetics of the molecules in the

contact govern the interfacial behavior had also previously been applied to the study of

friction at the nanoscale [25, 26]..

Gotsmann and Lantz proposed the following model, inspired by Reaction Rate

Theory, for the rate of height loss with respect to time:

(

) 3.2

12

where, is the height of the tip, is the lattice parameter, is the activation energy

(the same as in the Eyring derivation), and is the attempt frequency. Their

contributing insight was that, similar to a catalytic reaction, the activation energy would

be reduced by the stress at the contact because of the bond stretching (that occurs at

adhesive load) coupled with shear stress. They assumed that the dominant stress

component would be the interfacial shear stress (arising from friction). Thus the

energy barrier that an atom must cross in order to bond to an atom on the counter

surface is , where is the “activation volume”, a material property that can

sometimes be considered as the volume an atom sweeps out when it transitions from

the reactant state to the product state [2].

Because the authors were interested in wear as a function of distance, equation 3.2

had to be modified. Additionally, because of the assumption that shear stress

dominates at the contact, there must be an allowance for the change in contact area

that occurs when the AFM tip is worn down. The first problem is fixed through the

relationship between distance and velocity

, and the second borrows from the

work on the study of friction in Langmuir-Blodgett layers performed by B.J. Briscoe and

D.C.B. Evans [7]. As this work claims that shear stress is dependent on velocity, it will

be discussed in more detail section. For now we forgo the details and simply give the

shear stress relation that these researchers propose:

(

) (

)

where and are constants, and the normal load is the sum of the adhesive and

applied loads: .

Given the relationship between the radius of a conical tip and its height, the

velocity-distance relationship, and the shear stress relation given in equation 3.3,

equation 3.2 becomes:

[

(

)]

3.3

3.4

13

With this model Gotsmann et al. found excellent correlation with the data shown in

Figure 4.

Subsequent experimental work by Jacobs and Carpick bolsters the strength of

applying the Eyring equation to model atomistic wear [2, 3]. In that work, the authors

developed a novel procedure whereby they could perform and visually document the

wear process simultaneously, i.e., in situ. They mounted AFM chips on a conductive

holder which itself mounts to a Hysitron (Minneapolis, MN) PI-95 Picoindenter with 3

degrees of freedom, which allows them to slide a flat diamond (111) indenter tip over

an AFM tip, all while being observed in a transmission electron microscope. Figure 6

illustrates their setup.

All wear tests in that study were performed at adhesive load, which means that after

the typical jump-to-contact, the probe was advanced to the neutral position of the

cantilever, i.e., when (i.e., ). After a

certain sliding distance, images were taken of the silicon tip. Additionally, videos of pull-

off tests were taken after each wear increment to later calculate the adhesive force at

the contact, allowing for knowledge of stress at the interface. The authors then directly

measured wear as a function of stress and sliding distance, using the continuum

average Hertzian contact area:

(

)

where is the radius of the tip, is the adhesive force calculated using videos

of the pull off distance and knowledge of the AFM cantilever spring constant, and is

the composite modulus ( [

]

where denote the individual

Poisson’s ratio and individual elastic modulus respectively). Data for four wear tests are

shown in Figure 7 a. Figure 7 b. applies Archard’s wear equation to the data in Figure 7

a. No correlation is evident between the two, so Archard’s law does not hold for these

values of normal force and sliding distance.

3.1

14

Fig 6. A modified in situ indentation

apparatus is used for the sliding tests. a. An

AFM probe is mounted on the sample

surface of a TEM nanoindenter. b. The flat

punch indenter tip is brought into sliding

contact with the nanoscale asperity,

allowing in situ characterization of the

evolving interface. Observing the deflection

of the tip/cantilever enables the applied

normal force to be determined [3].

Fig 7. a. Total wear volume lost versus total sliding distance from wear experiments conducted by

Jacobs et al. on four Silicon AFM probes on a Diamond (111) counter surface. b. Application of Archard’s

law to the same data given in figure a. There is no correlation which means that Archard’s law cannot be

applied in this case [3]

Jacobs and Carpick also proposed a more general atomistic wear equation which can

be applied to any asperity geometry:

{ (

)} (

) 3.5

15

Where has units of number/s, is the internal energy of activation, is

the stress, and is the attempt frequency. They assumed the normal contact stress

to be the main contributing stress to lowering the activation barrier. Because normal

stress is not a function of tangential sliding, there is no explicit dependence of wear on

frictional shear stress in their analysis, but the shear stress component can be

introduced easily. The general equation with a shear stress dependence given by

Briscoe and Evans [7] is here introduced:

{ (

)} (

(

(

) (

))

)

or, using

{

(

)} (

(

)

)

where

is the rate of atom loss versus sliding distance [m-1 units].

A few similar studies were performed previously to probe the dependence of atom-

by-atom wear on stress [4-6]. Each author utilized a different stresses at the interface.

Park et al. used the continuum radial shear stress [4], while Jacobs and Carpick used

continuum average normal stress [3], and both Gotsmann and Lantz [1] and Bhaskaran

et al. [6] proposed the interfacial shear stress.

Despite the differences in the form of interfacial stress, the activation energies and

activation volumes al fall within a range which suggests that there is indeed an

atomistic wear process occurring. Table 1 lists the activation energies found in each

experiment. An approximate area for a silicon atom is on the order of 4 Ǻ2. [27] so this

data gives an approximate activation length (the length the atom sweeps out during

reaction) of 14 Ǻ, which is a value that one can expect intuitively because it is on the

order of several atomic diameters. This value is determined by assuming the activation

volume is equivalent to the area of the atom multiplied by the activation length. Given

3.6

3.7

16

the experimental value of the activation volume for a Silicon wear event of 55 Ǻ3, the

activation length is 55 Ǻ3/4 Ǻ2 14 Ǻ.

Table 1. Experimentally determined values for activation energy ( ) and

activation volume ( ) for several studies [2].

Even with the appearance of a complete model for atom-by-atom wear, some of its

assumptions must be addressed. These assumptions were discussed by Jacobs and

Carpick [2]. Perhaps the most conspicuous of the assumptions is that the state of stress

in all references is claimed to be uniform. This, as Jacobs and Carpick pointed out and

was debated at length by Luan and Robbins [28], cannot be the case. Indeed, different

contact geometries give wildly different states of stress. For example, a step edge will

induce a stress concentration, while a tip terminating in a grain will have a stress field

governed by the crystallographic orientation of the grain.

4. Atomic simulations show that Boltzmann statistics can apply to

the atoms in an elastic contact.

An inherent assumption in the application of RRT to nanoscale wear is that the

atoms in contact achieve a Boltzmann distribution of energies. To explore the validity of

this assumption, molecular dynamics studies (MD) must be performed because of the

difficulty of probing the theory experimentally. The molecular dynamics work of Cheng

and Robbins on defining contact at the atomic scale [29] sheds some light on the

problem of the assumption that the area of contact attains a Boltzmann-like

distribution. Cheng et al.’s study was motivated by the disparities between simulations

and experiments of friction studies, which gave large differences between contact

[4]

[5]

[2]

[6]

17

a. b.

d.

c.

e.

f.

geometry tested

atomically flat substrate

areas, degrees of contact stiffness, and fitted material parameters [30-34]. By focusing

on these inconsistencies, Cheng and Robbins’ results give insight into the reaction rate

theory perspective of wear, specifically in what load and temperature regimes that the

assumption of a Boltzmann distribution of energies in the contact is an accurate one.

The authors performed indentations of several interfacial geometries onto an

atomically flat substrate using the Large-scale Atomic/Molecular Massively Parallel

Simulator (LAMMPS) developed at Sandia National Laboratories. First, three flat-on-flat

contact geometries were used: commensurate, incommensurate, and amorphous. They

also applied all of these contact profiles and a stepped profile to spherically-shaped tips.

Figure 8 shows a cartoon of the flat on flat geometries as well as a depiction of the tip

geometries.

Fig 8. Depictions of geometries in Cheng et al.’s study [29] of a. Commensurate flat on flat, b.

amorphous flat on flat, c. incommensurate flat on flat, d. commensurate and incommensurate tip

geometry, e. amorphous tip geometry, f. stepped tip geometry [28].

18

Additionally, they demarcated both cases of adhesive and non-adhesive surface

interactions. For every geometry a Lennard-Jones (LJ) 12-6 potential was used to

define the interaction between atoms in the substrate. This potential allows for

attraction and repulsion between atoms. The parameters that characterize this potential

are the potential cutoff length , the atomic diameter, and the binding energy. The

cutoff length and binding energies are different for adhesive and non-adhesive cases.

The authors defined contact as the point at which the force on any substrate atom

from its counter surface is repulsive i.e., .

They propose a simple mean-field model for the probability at any instant that an

atom in the contact has a height :

, where ∫

.

This is a statistical mechanical model, where is the partition function of this system

and is the energy that an atom has from being at height .

The authors assume that the interaction between atoms in each surface satisfies

Hooke’s law i.e., :

where is an effective spring

constant for the potential that contacting substrate atoms experience due to substrate

atoms just below the surface, is the equilibrium separation of atoms, and is the

potential due to the rigid, atomically flat counter surface, a.k.a. the “wall potential”. The

authors make the simplifications that the wall potential is negligible and that

deformations are small so that . Under these simplifications the total potential

becomes

where .

The authors found that the MD data for the probability that an atom has a force fit

the model probability ( ) accurately at low loads and high temperatures using the

4.1

4.2

19

potential under the assumptions outlined in the previous paragraph. Figure 9

shows that the MD data follow the trend of exponential decay as a function of

⟨ ⟩ -where

⟨ ⟩ is the average force on all atoms in the contact at a specific moment in time- for

many decades and for a large range of atomic forces , temperatures , and applied

load on the system. The load and temperature were made dimensionless; the total

load is represented by

where is the continuum contact area, is the normal

load, and is the composite modulus of the rigid upper surface and the elastic

substrate. The temperature is represented by , where is Boltzmann’s

constant. Figure 9 shows the state of force for a single moment in time.

Fig 9. Probability density (

⟨ ⟩) as a function of the

force on an atom normalized by the average force

⟨ ⟩. Data are for three flat upper surfaces: a.

commensurate, b. incommensurate, c. amorphous. For

nonadhesive contacts, there are three data sets for a

high dimensionless temperature of = 0.175,

corresponding to dimensionless applied loads

of

(o open circles), (+ pluses), and

0.007 (Δ open triangles), and one data set for a low

dimensionless temperature of = at

( open squares). For adhesive contacts,

there is one data set for = 0.175 and

( X crosses). This corresponds to the highest effective

load (

) , where is the adhesive load. The

dashed line is the proposed model ( (

⟨ ⟩)) using

the potential (

⟨ ⟩). Adhesion causes some deviation

from the mean-field model, with a reduction in the

occurrence finding large forces. The biggest deviation

comes at low temperatures where the atomic forces

follow a narrower distribution, with far fewer

occurrences of high forces [29].

20

The dashed line in Figure 9 is the fit of the mean field statistical model ( (

⟨ ⟩)) .

At high temperatures, the simulation data and model are qualitatively equivalent, with

the strongest agreement for the lower atomic forces. Increasing the load has little

effect on the agreement between simulation and the mean field model. The authors

attribute the deviations of the nonadhesive data at high loads to variations in atomic

separation, whereby changes significantly. The deviations are also due to more

restriction of the surface atoms at higher applied loads, leading to less variability in

contact distance . This can be understood by considering that the Boltzmann

distribution form is, in essence, a Gaussian distribution, which requires randomness of

the physical parameters to be realized.

The data for the low temperature amorphous distribution bolsters this hypothesis;

the randomness of the equilibrium position of atoms in this configuration is enough to

make the force distribution Boltzmann-like for most of the atomic forces, even at low

temperatures (the authors claim that the low temperature here is one-seven

thousandth of the melting temperature of the material). The contact force of each atom

for commensurate and incommensurate flat surfaces is quite evenly distributed about a

mean value when thermal vibration amplitudes are not sufficient to randomize the

atomic forces in the contact at any given instant. Additionally, the adhesive contact

tends to lower the entropy of the contact system, thus making the forces deviate from

a random nature to a more ordered one, which cannot be represented by a Boltzmann

distribution. Finally, the assumption of zero wall potential for an adhesive contact

becomes less applicable at higher loads due to the increasing proximity of

countersurface atoms, which has a significant effect on the potential of a surface atom.

To summarize, Cheng et al. found that one of the main assumptions inherent to the

Eyring equation, that the reacting species in a contact between atomic solids have a

Botlzmann distribution of energies, is accurate under the conditions of low load and

high temperature. Thus, the Eyring model may be appropriate to describe atom-by-

atom wear under these limited conditions.

21

However, for the study of wear in the context of chemical reactions, one would like

to see agreement between the mean field model (which is the form Eyring derived) and

the data at high loads and low temperatures for all contacts (adhesive and

nonadhesive). A potential explanation which works in favor of the model is that

removing the small displacement assumption ( ) will only serve to bring the model

and data into greater agreement.

With the proportionality between and (recall ), the data from the

abscissa and ordinate of Figure 9 c. can be analyzed without the small assumption

and the results, , can be plotted with respect to the along with the mean

field model. This data, calculated for this paper, is shown in Figure 10. It can be seen

that the discrepancy between the two is small over a large range of atomic forces. The

data in this figure is taken from the amorphous-adhesive data of Figure 9 c. (green

dots). The unmodified potential ( , blue dots) which is compared to this data is:

( ) .

Fig 10. , the

probability that an atom at

height has a potential ,

versus . Simulation result

from [29] of the low

temperature adhesive contact

is compared to the mean field

model without the small z

assumption.

4.3

𝒛

𝒍𝒐𝒈 𝑷

𝑼 𝒛

22

Of course, it is always possible that the theory is overestimating the probability to be in

the activated state, meaning that the activation energies and volumes obtained in

studies may be too high.

However, another point that Cheng and Robbins results lends in favor of the mean

field model is that the forces at which the data start to deviate significantly from the

mean field model are many times larger than the average force on all of the atoms in

contact. This signifies that very large deviations from the mean force (and thus energy)

can still be within even the small displacement approximation. Since chemical reactions

take place due to the fact that some atoms’ energies fluctuate to higher values than the

mean, the fact that these energies remain within a simple theory enhances its

predictive power. Whatever the case, more MD studies need to be conducted which

track the complete potential landscape and compare actual probabilities with the

probabilities predicted with the Eyring equation.

The authors extend their study to the contact area of spherical tips of radius

R=100 , where is the atomic diameter, and find similar correlation between theory

and data. Figure 11 shows Cheng’s data for the contact area versus load compared

to the model (bold lines) for commensurate, incommensurate, amorphous, and stepped

tip-surface contacts (the authors leave out the model prediction for the stepped

contact) at different time intervals, applied loads, and temperatures. The Hertzian

prediction is shown as a dashed line. The contact area is determined by counting the

number of top layer tip atoms which touch the opposite surface, where “touching” is

the point at which any interaction between opposing substrate and tip atoms occurs.

This is normalized by the square of the atomic diameter ( ).

The data for different time intervals suggest that the time of contact plays a role in

the number of surface atoms that make contact. However, for all geometries (except

stepped) and all temperatures, the logarithm of the normalized contact area

is linear

with the normalized load (

)

, with the slope of each decreasing only slightly due to

the increasing contact area. The mean field model at the low temperature is also shown

23

as a solid line in the graph for each geometry. Its linearity and the linearity of the MD

data suggests that the Boltzmann distribution remains intact over these time intervals.

Fig 11. Normalized contact area

versus normalized

load (

)

for a spherical tip with different

geometries: a commensurate, b incommensurate, c

amorphous, d stepped. Open and filled symbols are for

and

, respectively. The contact area is

measured by counting the number of atoms in the top

layer of the substrate that interact with the opposite

surface at any instant (o circle) or during time

intervals ( square) or (Δ triangle). The

dashed lines represent the Hertz prediction and are the

same in all panels. Solid lines represent fits for each tip

to the simple harmonic mean-field theory with

and set equal to the number of atoms that

contact at

[29].

Indeed, the low temperature MD data deviates only slightly from the mean field model

during all time intervals. The increase in contact area over time is due to the fact that

longer dwell times equate to more substrate atoms exploring the counter-surface.

Although for amorphous tips this may have the effect of allowing time for lower

coordinated atoms to seek out a lower energy state by bonding with a counter-surface

atom, in general, as Mo et al. have shown, the average number of atoms in contact

over a period of time is nearly constant after a holding or sliding time of a few

picoseconds [35]. The fact that all atoms that can come into contact will within a

timespan of a single atomic vibration is consistent with reaction rate theory’s

assumption that the prefactor is the natural vibrational frequency of the material.

24

To summarize the work by Cheng et al., the distribution of energy among the atoms

in contact for many geometries, including tip geometries relevant to the study of atomic

scale wear, is shown to follow Boltzmann statistics. Thus, there is solid impetus to rely

on the Eyring model to describe atom-by-atom wear. It should be noted that Mo et al.

performed simulations of single asperities and also found that the pressure distribution

at the contact followed a Boltzmann-like distribution [35].

Because of the difficulty in obtaining experimental data for the real contact area of

single asperities, molecular dynamics data must be relied upon, such as those discussed

above, to justify the use of reaction rate theory to explain atomistic wear. However,

some assumptions of RRT can be experimentally tested, such as the dependences of

wear on temperature and velocity. Because there is a dearth of data on atomistic wear

as a function of temperature, we will focus first on the current literature view of the

dependence of wear on velocity, and will present some data taken during this author’s

research.

5. The current theory of atomistic wear as a function of sliding

distance shows no dependence on velocity

In section 3 the derivations of the rate of atom loss per unit time as a function of

shear stress and the rate of atom loss per unit distance as a function of shear stress

were given. The results, equations 3.6 and 3.7, are reproduced below for easy

reference during the following discussion.

{ (

)}

(

) (

)

{

(

)} (

(

)

)

In Gotsmann et al.’s analysis, there is no dependence of the wear rate per unit sliding

distance on sliding velocity (equation 3.7). This is because there is a tradeoff between

the higher rate of wear due to velocity and time spent in contact; higher velocities

spend less time in contact and therefore have less chance to wear, but the shear stress

3.6

3.7

25

is higher because of the higher velocity. Because of the functional form of the shear

stress in Gotsmann and Lantz’s analysis, these effects exactly cancel each other. This

trend has also been observed in MD studies performed by Vargonen et al. [36] for

abrasive wear. Regardless, Gotsmann and Lantz’s picture assumes that, in isochronous

wear experiments under the same conditions, a higher velocity will result in a greater

wear rate. It still remains a question whether or not this is a robust model. To explore

the model in more depth, the origins of Gotsmann and Lantz’s proposed frictional shear

stress dependence on velocity will be shown in the next section.

6. The interfacial shear stress has a velocity dependence in

Briscoe and Evans’ theory.

To make a judgment on the validity of the functional form of the shear stress it is

important to review the derivation that Briscoe and Evans provided [7] to explain the

interfacial shear strength (friction force per unit contact area) of Langmuir Blodgett

films as a function of sliding velocity. Their work was specifically focused on the

frictional sliding of Langmuir Blodgett films, but the approach is general.

Briscoe and Evans deposited mono- and multilayered organic films of carboxylic

acids and their calcium soaps onto muscovite mica substrates using the Langmuir

Blodgett technique [37]. The two mica surfaces are mounted onto crossed cylinders in a

friction apparatus which measures the lateral force (friction) and normal force. It also

has a multiple beam interferometer that measures contact area (Figure 12; note the

interferometer is not shown in the figure).

Fig 12. Schematic of Briscoe and Evan’s friction force apparatus. A: mica surfaces; B: lower support with

vertical adjustment; C: lever area; D: Flexure pivot; E: horizontal friction drive; F: cantilever springs; G:

resistance strain gauge elements; N: dead load [7].

26

Briscoe and Evans measured friction as a function of pressure, temperature, and

sliding speed for a range of circumstances, most notably for carboxylic acids created

under varying pH. The results for the velocity experiments are shown in Figures 13 a.

and b. These results show that the velocity-friction relationship is indeed log-linear, and

that the slope of the plot changes sign as a function of the pH; one can see

that at a pH of 9 the slope reverses and stick-slip behavior dominates, whereas

smoother sliding occurred at lower pH. The friction-pH relationship is actually due to the

manifestation of this stick-slip behavior, as will be discussed later. The authors provide

a series of phenomenological equations for their observations at varying pressures,

temperatures, and velocities:

at constant

at constant

at constant

The constants in equations 6.1, 6.2, and 6.3 are listed in Table 2 and come from fitting

the data.

Fig 13 a. The variation of shear strength with

velocity for stearic acid monolayers deposited from

M calcium chloride at pH 4.5; = 70 MPa, =

21 'C.

Fig 13 b. The variation of shear strength with velocity

for calcium stearate monolayers deposited from

M calcium chloride at pH 9.0. The extent of the

relaxation oscillations is indicated by the error bars.

The behavior of stearic acid monolayers is indicated by

the dashed line [7].

6.1

6.2

6.3

27

Table 2. Constants in the functional relations between shear strength and pressure, temperature, and

velocity, measured by using myristic, stearic,, and behenic acids. [7].

Motivated by an obvious temperature dependence of friction, the authors proposed

an Eyring model for the average time that a molecule overcomes a potential barrier to

slip into a lower potential state as the Boltzmann factor (which encapsulates the

activation energy) multiplied by the vibration frequency of the molecule:

where and are activation volumes. They multiply both sides by the lattice spacing

to get the average velocity of a single molecule. They then reason that the overall

sliding velocity will be proportional to the average sliding velocity and so

, which means that

(

)

and utilizing relations 6.1, 6.2, and 6.3 they arrive at the functional form for the friction

force:

(

) (

).

It is interesting to note that Gnecco [38] derived a similar result (in a different

fashion) for frictional sliding while incorporating the assumption that the energy barrier

decreases linearly with the frictional force (linear creep). They note that this

6.4

6.5

6.6

28

approximation fails in high speed experiments and one must resort to the assumption

that

so that equation 6.6 becomes

(

)

(

)

Where √

is the critical velocity which demarcates the two velocity regimes

(k is the effective spring constant of the system).

Briscoe and Evans note that some of the results obtained experimentally are not

consistent with the model. For example, the experimentally obtained slope in

equation 6.3 is too small as it would require to obtain a physically unreasonable

value of ~1024 m/s. The model also fails to account for the dependence of on

pressure. The authors do offer an explanation for this: stick-slip motion, in which the

surface or asperity sticks in a lower potential and then slips as the applied lateral force

overcomes this potential gradient of the barrier, may account for the decrease in

surface area, and thus pressure, as velocity increases. Because stick-slip occurs more

rapidly at higher velocities, there is less time in contact, and thus a lower overall

effective pressure, explaining why the data shown in Figure 13 b. has a negative slope.

Equation 6.6 was derived by Briscoe and Evans under the assumption that the shear

stress is a thermally activated process. This formula, utilized by Gotsmann and Lantz,

has a velocity dependence, yet these authors did not explore the relationship between

velocity and wear in their paper. Bhaskaran and Gotsmann use the Briscoe and Evans

model to describe the wear versus distance data in their experiments, which were

similar to Gotsmann and Lantz’s, but with Si-DLC on SiO2 [6]. Indeed, Bhaskaran and

Gotsmann assume that there is no dependence of wear on velocity and their data

appears to support this. Thus it seems that the shear-stress-velocity relationship,

obtained for the highly specific system in Briscoe and Evans’ experiment, can be

generalized to other systems. However, in the following section it will be shown that

data from Bhushan et al., [39] suggests that the analyses given above fails to capture

the trend of volume loss versus sliding velocity seen therein.

6.7

29

7. Experimental studies disagree on the velocity dependence of

wear.

We now turn our attention to a study that, to this author’s knowledge, is the only

one specifically focused on the velocity dependence of atom-by-atom wear for

nanoscale single asperities. Bhushan et al. performed wear tests using Pt/Cr coated

silicon AFM probes on separate diamond like carbon (DLC) and Z-Tetraol(P) coated flat

samples [39]. They used two loads: 50 nN and 100 nN and 5 sliding velocities 0.1, 0.25,

1, 10, and 100 mm/s. All tests were performed up to 1 m of sliding. They characterized

the shape of their AFM tips before and after wear by sliding their tip over a grating

which has an array of ultra-sharp silicon tips. The image produced in the AFM in this

way gives a convolution of the grating’s tips and the AFM tip. Since the grating’s tips

are so sharp, the fidelity of the given AFM tip shape is highly accurate, although not as

accurate as the shape obtained through direct observation of the AFM tip in the TEM.

Results for their wear tests as well as tip profiles before and after wear are shown in

Figure 14.

Figure 14 a. and b. show data for wear volume versus sliding speed and normal

load. The trend in Figure 14 b. shows that for any chosen tip radius (the radii shown in

Figure 14 c. are all within a narrow range of each other), the wear volume increases

with velocity for a fixed sliding distance. This evidence is contrary to the Gotsmann RRT

model as that model predicts that the wear volume will be the same for a given tip

radius at a fixed distance, at any velocity. Indeed, because

and also { (

)} (

(

(

) (

))

) then,

{ (

)} (

(

)

)

{ (

)} (

(

)

). 7

30

Fig 14. a. Wear volumes for three tips after 1 m sliding at 50 nN and after additional 1 m sliding at 100

nN as a function of sliding velocity on unlubricated and Z-Tetraol(P)-lubricated samples. b. Data

presented in Figure 13 a. c. Tip profiles before and after 1 m sliding at 50 nN after additional 1 m sliding

at 100nN on unlubricated and Z-Tetraol(P)-lubricated samples.

We conclude that the volume of material lost for a given distance is due to only material

parameters, the area of contact, the temperature, and the normal force. It should be

noted that the authors suggest a thermally activated (RRT) model at the slower sliding

c.

a.

b.

31

speeds but differentiate with the model represented by equation 7 by assuming that the

wear rate is linear with friction force [40-42].

Keeping within the context of the RRT model, we can analyze the discrepancy

between the data in Bhushan’s work and model represented by equation 7, which, in

the RRT framework, can be due only to the dependence on temperature since all of the

other parameters are either held constant (normal force, material parameters) or are

within a very narrow range of each other so as to not affect the trend (stress, contact

area). As sliding speeds increase it is possible that not all heat energy from friction is

dissipated and the temperature of the tip increases. These phenomena would yield a

monotonically increasing wear volume and monotonically decreasing rate of volume loss

just as we see in the data curves of Figure 14 b. because of the negative inverse

temperature term in the exponential of the Eyring equation. Additionally, a mechanism

other than atom-by-atom wear may be occurring. It is difficult to provide a more

detailed analysis with the data presented in [39] because not all parameters were

given, such as the environmental conditions of the test for example. More experiments

on the dependence of wear on temperature and velocity are needed to elucidate the

origins of the velocity dependence of wear in the data obtained by Bhushan [39].

Jacobs et al. also presented data for wear as a function of velocity as discussed

above in section 3 [3]. Figure 15 below shows data for two different velocities.

Fig 15. Reaction rate versus total stress for two

velocities: 4nm/s (orange diamonds) and 21 nm/s

(remaining data) from wear experiments conducted

by Jacobs et al. on four Silicon AFM probes on a

Diamond (111) counter surface. [3].

32

The orange diamonds in Figure 15 are data points for volume lost versus sliding

distance at 4 nm/s and the remaining sets are at 21 nm/s [43]. Equation 3.6 says that

there should be a noticeable change in the slope of the curve with varying contact

stress. Although there are only two data points for the lower sliding velocity, they lie

along the same curve as the other data taken at 21 nm/s. Bhaskaran et al. also have

data at two different sliding velocities which show a deviation from the RRT model [6].

Besides requiring more data which focuses on the velocity dependence of wear, this

trend suggests that the model may need adjustment in the area of velocity

dependence.

The fact the only data in the literature available either ignores the velocity

dependence of wear or seems to deviate from the Eyring model with the activation

energy reduced by interfacial stress merits an analysis of some of the assumptions that

are made when deriving the model. For one, the model suggests that the temperature

of the contacts is not increasing as velocity increases. As mentioned above, this may

not be the case, and frictional heating may be occurring. Secondly, there will be a

limited range of velocity in which atom-by-atom wear occurs, and research must be

done to explore this range. Last but not least, it cannot be known with confidence

whether the literature results can provide a reliable description of the velocity

dependence of wear as none of the studies were done both in situ and exclusively on

the wear dependence of velocity.

8. Preliminary data at various sliding velocities suggests a velocity

dependence of wear.

Using the technique developed by Jacobs et al. , [3] whereby wear is observed in

situ in a TEM, this author performed wear versus velocity tests for two velocities. After

testing at these two velocities, a software glitch in the Picoindenter caused the AFM tip

to crash

33

Fig 16. Reaction rate [atoms/second] as a function of continuum average stress for

amorphous silica on diamond (111).

into the diamond surface and testing had to be halted. The software glitch has been

fixed very recently. The data obtained so far is presented above in Figure 16.

Although there are few data points in this study, it was clear from the videos that no

net wear could be resolved outside of the detection limits at velocities of 2.6 and 2.5

nm/s. In fact debris accumulated on the tip, producing a negative wear volume.

Furthermore, it was seen wear significantly increased at the two velocities greater than

10 nm/s. For these two velocities, 11 and 14 nm/s, the mean normal stresses were

rather different at .8 and 1.3 GPa respectively. The 14 nm/s and the 2.6 nm/s data

were obtained at nearly the same stress (1.4 and 1.4 GPa respectively) and the higher

velocity data has a far higher rate of atom loss. This data supports the idea that there is

a correlation between sliding speed and atom loss rate.

It must be noted that this author found there to be no wear of the actual crystalline

silicon portion of the tip. The wear shown in Figure 16 is confined to the outer native

oxide of tip. Future studies will focus on wear of the crystalline silicon portion of the tip.

Images of each wear increment with the previous wear increment’s tip shape

highlighted in red are shown in Figure 17.

Figure 18 presents the data from Figure 16 as reaction rate versus velocity.

Although the normal stress values are not accounted for in this figure, the size of the

34

markers conveys the magnitude of the stress, and it appears that there is indeed a

positive correlation between wear rate and velocity. If this is the case, it indicates that

the current theory of the dependence of wear on velocity is incomplete, and a focused

study on this relationship would provide a useful model to industry. Future analysis will

attempt to normalize each variable of interest (stress, velocity) in order to isolate the

effect of each on the wear rate.

This is only a small piece of what will become a more systematic and rigorous study

on wear versus velocity. Several challenges arise when using the in situ procedure. For

one, the silicon tips always have an outer native oxide, making it difficult to determine

the wear of the crystal silicon structure. Another issue is that the raster scanning

software of the PI-95 Picoindenter is not tailored for this type of test.

Fig 17. a. post slide two at 2.5 nm/s overlaid on post slide one (traced in red) at 2.0 nm/s, b. post slide

three at 2.5 nm/s overlaid on post slide two (traced in red) at 2.6 nm/s, c. post slide four at 14 nm/s

overlaid on post slide three (traced in red) at 2.5 nm/s, d. post slide five at 11 nm/s overlaid on post

slide four (traced in red) at 14 nm/s. The same silicon tip was used for all sliding increments. Colors

correspond with data points in Figure 16.

a. 2.5 nm/s b. 2.6 nm/s

c. 14 nm/s d. 11 nm/s

35

Fig 18. Reaction rate [atoms/second] as a function of sliding velocity for amorphous silica on

diamond (111).

When a velocity scan is executed the indenter first seeks out a zero position and then

starts scanning. Unfortunately, when seeking the zero position the indenter is driven at

a velocity of around 10 nm/s, ruining any velocity analysis at speeds other than that. To

solve these problems, the author will become trained on the use of a focused ion beam

(FIB) and will contact Hysitron to seek a solution to the raster scanning issue.

9. Future work is proposed to probe the velocity and temperature

dependence of wear.

The possible inconsistency between the RRT model with velocity and the recorded

data, and the overall paucity of data call for more research. Additionally, no

experiments to date have tested the theorized dependence of atomistic wear on

temperature. It is also not clear what the relationship between friction and contact

heating (frictional heating) spells out for the RRT picture of wear. In this section, I will

propose experimental methods which I intend to use to explore these pressing issues.

As mentioned previously, the bulk of the procedure for observing and measuring

wear in situ, i.e., inside a transmission electron microscope, has been developed by Dr.

Tevis D.B. Jacobs at the University of Pennsylvania for the experiments in reference [3].

An AFM tip is mounted on a Hysitron Pi-95 Picoindenter which allows the user to slide a

diamond (111) indenter tip over the AFM tip, all while viewing and recording the

36

process in the TEM. The Picoindenter has angstrom distance resolution and

micronewton force resolution. Adhesive force measurements are taken after each wear

increment by measuring the distance that the adhesive force between the diamond

indenter tip and the AFM probe pulls the AFM probe before it pulls off. This adhesive

force is used as the normal force value, i.e., no additional normal force is applied.

Images of the tip after each wear increment are used to calculate the volume loss and

radius, thereby allowing approximate calculations of the contact area, and thus stress.

This author will augment this procedure by utilizing a raster scanning feature provided

in the Pi-95 control software. This will allow me to probe a wide range of velocities

accurately.

The schedule of planned velocity experiments is detailed in Table 3 below. This

schedule is optimistic and may need to be modified to have a shorter total distance for

each velocity. If this modification is made the 2000 nm/s velocity may need to be

removed from the experiment as it will be very difficult to perform a single segment of

the test for a time of 4.5 seconds. This author proposes to have several probes ready to

be used during any given TEM session, as probes often crash into the diamond indenter

tip or pick up contamination. If CO2 snow cleaning does not reduce contamination

significantly it may be that a new diamond indenter tip will need to be ordered as the

current one has been used extensively for the last few years and has thus accumulated

debris or become amorphized in areas.

Table 3. Schedule of wear versus velocity experiments. All tests are to be performed at the same total

sliding distance.

total distance (nm)

speed (nm/s)

number of pictures

scan length between pics (nm)

number of scans between pics

scan length (nm)

total time (s)

total time (h)

time per scan (s)

180000 2000 20 18000 90 200 90 0.025 4.5

180000 1000 20 18000 120 150 180 0.05 9

180000 500 20 18000 180 100 360 0.1 18

180000 100 20 18000 180 100 1800 0.5 90

180000 10 20 18000 180 100 18000 5 900

180000 1 20 18000 180 100 180000 50 9000

37

For the temperature control experiments this author proposes to begin with AFM

temperature modified wear tests. Doing so before performing such experiments in situ

will illuminate potential obstacles to performing quality experiments. The procedure will

become more fine-tuned at a quicker pace than TEM experiments because of the

relative ease of conducting AFM sliding tests. When an appropriate level of repeatability

is established, this author will begin wear versus temperature experiments in the TEM.

To allow for current to be applied to the AFM chip this author will develop an adapter

for the electrical contact resistance (ECR) feature of the Hysitron PI-95. This feature

allows the user to control either the voltage or current applied to the sample. The group

of W. P. King at the University of Illinois at Urbana-Champaign manufactures heated

probes, which have a tip on a bridge between two cantilevers. The cantilevers are

silicon, heavily doped with phosphorous, while the bridge carrying the tip is only lightly

doped, thereby making it more resistive and prone to Joule heating than the

cantilevers. Reference [44] details the resistance calibration for these probes. The

temperature calibration requires the use of a Raman spectrometer, which is available at

the Nano Characterization Facility at the University of Pennsylvania. The shape of the

tip will be determined by scanning it over a TGT1 sample which consists of an array of

ultra-sharp tips. The image produced by scanning over a TGT1 sample is a convolution

of the tip image and the TGT1 tip and, since the TGT1 tip is much sharper than the

AFM tip, the image is approximately that of the AFM tip.

Using the temperature-current calibration, the temperature will be modulated in situ

and wear versus temperature data will be obtained. I will rely heavily on my experience

from the wear versus velocity experiments to determine a proper schedule of

experiments.

10. Conclusion

The predictive power and relative ease of use of the Eyring equation has recently

extended into the study of wear of nanoscale contacts. A few studies have probed the

38

relationship between normal stress and wear, and have found that the significant

amounts of data are consistent with the model and provide physically reasonable

activation parameter. Additionally, molecular dynamics studies by Cheng et al., and Mo

et al. suggest that the assumption that nanoscale contacts achieve a Boltzmann

distribution of energies is a reasonably valid one away from low temperatures and high

applied loads. The current hypothesis of the velocity dependence for wear as proposed

by Briscoe, Evan, and Gotsmann was reviewed. Finally a brief analysis of two data sets

which explore this relationship was given.

Unfortunately, due the scarcity of data on wear versus velocity, the modeled velocity

predictions for the velocity dependence of wear cannot be validated as of yet. More

effort is needed to illuminate the dependence of wear on velocity and temperature and

it is this author’s proposal to explore these effects. Such fundamental relationships,

once known, will potentially have a significant impact on the current understanding of

wear on the atomic scale and will give engineers a better idea of the trends they can

expect when manufacturing and deploying M/ NEMS.

39

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43. This data was given to me in a correspondence with the author Dr. Tevis D.B. Jacobs.

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