Multibond Model of Single-Asperity Tribochemical Wear at the

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A multi-bond model of single-asperity tribochemical wear at the nano-scale Yuchong Shao, Tevis D B Jacobs, Yijie Jiang, Kevin T. Turner, Robert W Carpick, and Michael L. Falk ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b08023 • Publication Date (Web): 07 Sep 2017 Downloaded from http://pubs.acs.org on September 11, 2017

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ACS Applied Materials & Interfaces

A multi-bond model of single-asperity tribochemical wear at the nano-scale Yuchong Shao,† Tevis D. B. Jacobs,‡ Yijie Jiang,¶ Kevin T. Turner,¶ Robert W. Carpick,¶ and Michael L. Falk∗,§ †Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 ‡Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261 ¶ Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104 §Department of Materials Science and Engineering, Department of Physics and Astronomy and Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218 E-mail: [email protected] Abstract Single-asperity wear experiments and simulations have identified different regimes of wear including Eyring- and Archard- like behaviors. A multi-bond dynamics model based on the friction model of Filippov et al. [Phys. Rev. Lett. 92, 135503 (2004)] captures both qualitatively distinct regimes of single-asperity wear under a unified theoretical framework. In this model, the interfacial bond formation, wearless rupture and transfer of atoms are governed by three competing thermally activated processes. The Eyring regime holds under the conditions of low load and low adhesive forces; few bonds form between the asperity and the surface and wear is a rare and ratedependent event. As the normal stress increases, the Eyring behavior of wear rate

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breaks down. A nearly rate-independent regime arises under high load or high adhesive forces, in which wear becomes very nearly, but not precisely, proportional to sliding distance. In this restricted regime, the dependence of wear rate per unit contact area is nearly independent of the normal stress at the point of contact. In true contact between rough elastic surfaces, where contact area is expected to grow linearly with normal load, this would lead to behavior very similar to that described by the Archard equation. Detailed comparisons to experimental and molecular dynamics simulation investigations illustrate both Eyring and Archard regimes, and an intermediate crossover regime between the two.

Keywords adhesive wear, nanoscale tribology, tribochemistry, mathematical modeling, atomic force microscopy

1

Introduction

The complex phenomenon of “wear” typically involves removal of material between opposing surfaces. Macroscopically, wear can take a myriad of forms such as erosion, abrasion and fretting, which are often characterized by fracture and plastic flow under extreme conditions and during initial run-in. 1–3 In contrast, wear occurring after the initial running-in stage can often be quite subtle involving the tribochemical transfer of material in small clusters or even on an atom-by-atom basis. 4–8 Studying the wear behavior of single-asperity contacts helps focus investigations on those that can be performed under the most controlled conditions, in the hopes of contributing to a body of fundamental theory that serves as a foundation for a “first principles” understanding of wear. Apart from the theoretical concerns, understanding nano-scale wear is also particularly important in the design of micro- and nanoelectromechanical systems (MEMS/NEMS) as they move from laboratory to application. 9–12 2


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Among the numerous phenomenological models of wear, perhaps the most widely accepted on the macroscopic scale is the Archard equation, which states that the wear volume Q is linearly proportional to the sliding distance d and the load FN , yet independent of sliding speed v and contact area: 13 Q ∝ FN d.

(1)

The Archard equation is empirically useful to predict the wear behavior on the macroscopic level where multi-asperity contact is predominant, and sometimes even on the microscale. 14,15 Recently a molecular dynamics (MD) simulation investigation by Sha et al. 16 reported single-asperity wear behavior between an unpassivated asperity and a flat surface composed of diamond-like carbon (DLC) in the absence of irreversible deformation of the tip. In this investigation, the observation of wear rates proportional to load and sliding distance led the authors to suggest that under certain conditions a form of the Archard relation even extends to the smallest scales, although one could not expect the wear rate on these scales to be independent of contact area due to the absence of roughness. One aim of this theoretical investigation is to assess under what conditions such an Archard-like regime could emerge at the nanoscale. In the discussions to follow we will use the phrase "Archard-like" to denote a wear regime that shares characteristics of the Archard equations, particularly a proportionality of wear with sliding distance and normal load, but which is not true Archard behavior insofar as this response does not arise from plastic mechanisms as theorized by Archard. This is significant because other aspects of the Archard theory, for example that the wear rate should be inversely related to the material hardness, cannot be relevant in an adhesive wear regime in which loads are sufficiently low that plasticity is absent. Thanks to the development of the scanning force microscope (SFM), tremendous progress has been made in understanding friction at the atomic level. 17,18 SFM has also been employed to investigate the mechanisms of single asperity wear between many surfaces of interest. Extremely low wear rates (on the order of one atom per millimeter of sliding) have been 3



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observed under low loads of 1nN to 1µN 5,6 or even zero load with small adhesion. 7 These wear rates are orders of magnitude lower than those measured in the aforementioned MD simulations. The Archard equation is often found not to hold in such experiments: the wear volume per unit sliding distance is typically nonlinearly dependent on applied load and varies with sliding velocity. 5,6,19 Moreover, the Archard equation itself remains purely empirical, making it hard to extend this theory to elucidate some of the striking findings in single-asperity wear experiments. 4–6,20,21 It has been proposed in many experiments 5–7,20,21 that thermally activated atom-by-atom attrition takes place at the nano-scale, rather than the plastic deformation that dominates on the large scale at high loads. A recent study by Jacobs and Carpick 7 provides direct evidence that wear at the atomic level is a thermally activated process assisted by normal contact stress. Using in situ transmission electron microscopy (TEM), silicon tip samples have been gradually worn by a diamond surface without plastic deformation or fracture. The wear rate (total wear volume per unit time, normalized by estimated number of atoms in contact) is observed to obey an Arrhenius equation where the activation barrier is reduced by the normal stress σ, given by

Γatom loss

∆U − σ∆Vact = Γ0 exp − kB T

.

(2)

Here Γ0 is an effective attempt frequency; ∆U is the stress-free activation barrier; ∆Vact is the activation volume for the normal stress component; T is the absolute temperature, and kB is the Boltzmann constant. More recent MD simulation work by Yang, Huang and Shi 22 ascribes this discrepancy in behavior to a transition induced by variation in adhesion and normal load, and notes a transition from Archard like behavior at high adhesion and normal load to Eyring like behavior at low adhesion and normal load. The authors of this prior work propose empirical relations describing the transition, which they attribute to a cross-over from atomic to plastic

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wear. However, the nature of this transition and the physical reason for the emergence of the Archard-like regime remain unaddressed. In this paper, we present a theoretical wear model based on the multi-bond friction model originally proposed by Filippov et al. 23 A compelling aspect of this model is that both regimes of wear behavior (Eyring- and Archard-like) have been identified, mainly depending on the rate of forming interfacial bonds which depends sensitively on the normal and/or adhesive forces that bring the surfaces into contact. In each regime, the sliding speed of the tip also influences the wear rate, as it affects the time scale of interfacial bond detachment, although this effect is more subtle in the high normal force/adhesion limit. We compare the predictions of the model to existing experimental data from Ref. ( 7 ) in the low normal stress regime as well as new data from Ref.( 24 ) in the transition regime and simulation data from Ref. ( 16 ), which we argue is in the high normal stress regime due to strong adhesion.

2 2.1

Model Development The multi-bond model

It is often observed in SFM experiments that over a wide range of materials, the friction shows a non-monotonic dependence upon temperature and sliding velocity, which can not be well explained by the traditional Prandtl-Tomlinson model. 25,26 In this regime, the “multibond" model does a remarkable job of reproducing the experimental findings by taking into account the role of temperature on the formation and rupturing of microscopic contacts. 23,25 In their model, the complex nature of single-asperity contact is coarse-grained into a fixed number of “contact sites” attached with independent bonds modeled by elastic springs with stiffness κ. The rates of the formation and rupture of the bonds are governed by thermally

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activated processes: ron = ω0 exp(−∆Eon /kB T ), and

(3)

roff = ω0 exp(−∆Eoff (1 − fi /fc )α /kB T ),

(4)

where ω0 is the atomic attempt frequency for both processes. ∆Eon represents the energy barrier for bond forming. The energy barrier for bond breaking ∆Eoff is lowered by the tension in each bond, fi , and vanishes when fi reaches the critical value, fc . Here α is chosen as 3/2 in accordance with the “ramped creep” condition where the energy barrier for slipping depends on lateral force as ∆E ∼ (fc − f )3/2 . As a result the average friction depends on velocity as f¯ ∼ | ln v|2/3 . 27,28 The total frictional force is obtained by summing up the tension in all the formed bonds, Fcontact =

X

fi =

i

X

κ∆xi ,

(5)

i

while the movement of the SFM tip is simplified to one-dimensional damped dynamics driven by a spring of stiffness k with a constant velocity v. This is described by the dynamic equation ¨ + η X˙ − Fcontact + k(X − vt) = 0, MX

(6)

where M is the mass of the tip, and η is a damping coefficient responsible for other degrees of freedom that are not explicitly depicted such as the dissipation of kinetic energy to phonons. There are four distinct time scales associated with the model: ron and roff determine the p rate of bond forming and breaking; kv/fc is the rate of forced unbinding and max(k/η, k/M ) is the rate of pulling force relaxation under the influence of the external driving spring. The

first two time scales depend on the temperature, which essentially leads to the transition from stick-slip to continuous sliding as the temperature increases. The latter two time scales depend solely on the mechanical properties of the system.

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where roff and rwear are also dependent on ∆x. These two rates are updated as each bond stretches. Once the rates are large enough to cause either a bond breaking or wear event, the ratio rwear / (rwear + roff ) is compared with a standard uniform random number to determine whether a wear event or wear-less bond uncoupling will take place. The expected wear rate per potential bonding site, Γwear , under the probability distribution of ∆x is

Γwear = −

Z

∞ 0

rwear roff + rwear

non v ∆x

dP (∆x) d∆x

d∆x,

(9)

where non denotes the fraction of potential bonding sites that are currently bonded. In steady-state, non = ron /(ron + roff + rwear ), since n˙ on = (1 − non )ron − non (roff + rwear ). In the numerical calculations, the tip slides over a distance d at temperature T for various choices of ∆Eon and sliding speed v. The other two stress-free energy barriers ∆Eoff and ∆Ewear are fixed. Unless stated otherwise, the relationships ∆Eoff ≤ ∆Ewear and fc ≤ fw hold. For ease of analytical derivation, it is also assumed that wear-less bond rupture and wear have the same critical length lbreak , i.e., lbreak ≡ ∆Eoff /fc = ∆Ewear /fw . The sliding distance d is assumed to be long enough to ensure that the corresponding wear rate Γwear reaches steady state.

2.3

Variation of effective normal stress σ

One of the most important physical quantities in studying wear is the effective normal stress σ upon the contact, which includes the effects of both external load and adhesion. This parameter has been shown to be responsible for the change of the real contact area Areal in experimental, 29 analytical 30,31 and numerical 32–35 work. Since the multi-bond model presented here does not provide direct information about the real contact area, all wear rates are calculated per contact site. In the present model, the effective normal stress σ is not explicitly employed as a parameter; instead, we assume that the barrier for bond forming is linearly dependent on the

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effective normal stress: ∆Eon = ∆U − σ∆Vact , where ∆U is the stress free energy barrier and ∆Vact is the activation volume for bond formation. The effect of changing load is simulated by varying ∆Eon , which controls the coverage of connected bonds. The dimensionless normal stress σ ˜ is then inferred from ∆Eon and ∆U by:

σ ˜≡

σ∆Vact ∆Eon =1− , ∆U ∆U

(10)

and varies from 0 to 1. Of course, in the experiments to which the model is compared the normal stress is the experimentally controlled parameter. Typically the normal stress can be determined from the applied load, the experimentally measured contact area, and the adhesive interaction between the tip and the surface, or, in the absence of independent measurements of contact area, this may be inferred through a contact theory approximation. To connect with the model, the dimensionless normal stress can be obtained by dividing the normal stress by a stress scale obtained by dividing the energy barrier for bond formation by the activation volume for bond formation between the tip and substrate once these have been estimated for the system in question.

3

Results and discussion

The multi-bond model of wear described above is too complex to solve analytically, but two approaches can be used to reveal its predictions. We have used asymptotic analysis to infer the behavior of the model in the low and high normal stress limits. We have also numerically simulated the model. This provides calculated estimates of the model’s predictions limited in accuracy by the computational time available to sample the steady-state behavior. We find that the theory proposed above predicts that the system will exhibit distinct behaviors depending upon the level of effective normal stress; however, normal stress is not the only factor, and sliding velocity can also influence the wear behavior. Both Eyring- and Archard9



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like behaviors arise in the model.

3.1

Low effective normal stress regime

In the low stress regime where only a few bonds can be formed simultaneously, the wear rates appear to follow the Eyring equation as observed by Jacobs and Carpick. 7 The trend is demonstrated in Fig. 2 under a wide range of sliding speeds, from the sliding rate mainly used in Ref. 7 (20 nm/s) to the one in Ref. 5 (1.5 mm/s). The wear rate data from Ref. 7 is also plotted for direct comparison. Note that the measured wear rates of the present work have been normalized by the number of potential bonding sites, consistent with definition of wear rate in Ref., 7 where the number of atoms in nominal contact is the normalization factor amongst results from various tip samples. We use the energy barriers ∆U = 0.9 eV, ∆Vact = 3

6.7 Å , ∆Eoff = 0.3 eV and ∆Ewear = 0.33 eV to fit the simulation data with the experiment. The bond-breaking length lbreak = 3.0Å is chosen to be identical with the original frictional model. 25,26 The selection of other parameters are as discussed in the Methods section at the end of the paper. Fig. 2 shows that the measured wear rates are confined within two limits at high and low sliding speeds. In the low-speed limit, thermal activation is mainly responsible for breaking or wearing a bond, whereas the contribution from the mechanically forced unbinding rate (∼ kv/fc ) becomes negligible. Therefore the rates depicted by (4) and (7) can be 0 0 simplified to roff ≈ roff ≡ ω0 exp(−∆Eoff /kB T ) and rwear ≈ rwear ≡ ω0 exp(−∆Ewear /kB T ).

The normalized wear rate in the low velocity limit is then

v Γlow wear

0 ron rwear ≈ , 0 0 ron + roff + rwear

(11)

which corresponds to the lower dashed line in Figure 2. As the sliding speed v increases, the mechanical unbinding plays a more significant role in tearing interfacial bonds apart, and the crossover behavior of wear rate (see, e.g., the upward

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10

3

20 nm/s 100 nm/s 10 µ m/s 1.5 mm/s low velocity limit high velocity limit EXP data

Γwear (s−1 )

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10

2

10

1

10

0

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

σ ˜ Figure 2: Normalized wear rates (logarithmic scale) versus dimensionless effective normal stress σ ˜ in the low effective normal stress regime at various sliding speeds, in comparison with the experimental wear rate data in 7 (open symbols and error bars). The contact stress σ surveyed (2.7 ∼ 5.4 GPa) is made dimensionless by Eq.(10) using the fitted stress-free energy 3 barrier for bond formation ∆U (0.91 eV) and activation volume ∆Vact (6.7 Å ) from. 7 Dashed lines represent high and low velocity limits and solid symbols are numerical results of the present model. 0 triangles in Fig. 2) begins to arise when v > ω0 λfc ron / (αβc κroff ), where βc ≡ ∆Eoff /kB T , βw ≡ ∆Ewear /kB T and λ ≡ e−βc + e−βw . In the high velocity limit, all the formed bonds

will be quickly stretched to the critical length of unbinding before thermal fluctuations can assist breaking. In this regime roff ≫ ron and rwear ≫ ron , making the formation of a

bond the rate-limiting process. The normalized wear rate in the high velocity limit can be approximated as v Γhigh wear ≈ ron ·

11

fc , fc + fw


(12)


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represented by the upper dashed line in Figure 2. In the low effective normal stress regime, the wear rate generally increases with sliding velocity, which qualitatively agrees with the experiments performed by Park et al , 20 where an SFM tip (Si3 N4 ) has been utilized to study the wear behavior of calcite in an aqueous solution under the dimensionless stress σ ˜ of 0.25 or lower. In addition, the increasing trend saturates at high velocities up to mm/s in our model, suggesting the possibility of identifying the Eyring wear behavior in the low effective normal stress regime even under the condition of fast sliding speed. This prediction is supported by Gotsmann and Lantz’s experiment, 5 where Eyring atom-by-atom wear behavior has been detected using Si tips on polymeric surface, with σ ˜ under 0.28 and the scanning speed of 1.5 mm/s, although in their experiment the nominal contact area of the tip gradually increased due to long sliding distance.

3.2

Intermediate effective normal stress regime

The formation of bonds becomes easier as normal stress increases; on the other hand, the coverage of bonds is also affected by the external sliding speed and the low coverage of interfacial bonds could arise even under high normal stress. Here we define two velocity scales that are determined by the physical properties of the contact:

v1 ≡

0 (roff

v2 ≡ ron ·

+

0 rwear )

fw ακ

·

fw ακ

,

(13) (14)

where fw /(ακ) represents the maximum length scale of stretching; v1 is the velocity at which shear-induced bond breaking overtakes thermally activated bond breaking between the surfaces, whereas v2 represents the velocity at which shear-induced bond-breaking dominates over thermally-activated bonding. The system will shift away from the low effective stress regime as the growing rate of bond 12


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formation ron becomes comparable with or larger than the combined rate of bond breaking and wearing, i.e., (15)

ron & (¯ roff + r¯wear ) .

Since ron varies with the normal stress, we can define σ ˜c , the threshold of intermediate stress regime. Essentially, σ ˜c indicates the normal load or adhesive contact force at which the system begins to form a full coverage of bonds between the tip and the substrate: kB T kB T σ˜c ≡ 1 + ln λ + ∆U ∆U

βw v v1

ln

βw v v1

.

(16)

In the low-speed limit, the influence of mechanical unbinding is negligible while thermally activated processes dominate, and Eq.(16) can be further simplified to σ ˜ & 1 +kB T ln λ/∆U . Hence the threshold decreases with ∆U and ∆Eoff,wear . As the sliding speed becomes more significant, the velocity term in Eq.(16) bumps up the threshold since higher sliding speeds make bond breaking and wearing even more frequent. Using ex situ TEM technique, Liu et al. performed systematic nano-scale wear tests with DLC-coated tips on DLC-coated Si substrates under a series of external loads at a constant sliding speed of 29.5 µm/s. 24 Each wear test consisted of intermittent ex situ TEM examination of tip shape evolution and frequent in situ AFM adhesion monitoring to keep track of the volume loss. The average normal stress within contact was then estimated using DMT contact theory 36 and the measured wear rates are normalized by average number of atoms in contact to account for the changing shape of the tip. As shown in Figure 3, the Eyring behavior of wear rate only holds in the low stress regime. In the range of error bars, the experimental results (open symbols with error bars) reasonably match with the numerical simulation (filled circles) and theory (dashed line). The filled circles in Fig. 3 clearly show the transition from the low to the intermediate effective normal stress regimes in numerical simulations of the model. In these simulations the sliding velocity (30 µm/s) closely matches the one in experiment in Ref. ( 24 ). The fitting parameters are: ∆U = 13



3

0.7 eV, ∆Eoff = 0.72 eV, ∆Ewear = 0.9 eV, Vact = 120.0 Å , lbreak = 4.4 Å, and the other parameters are the same as in Fig. 2. It is important to note that the experimental data in the high dimensionless normal stress regime corresponds the very initial sliding of sharp tips. Collecting accurate experimental data in this portion of the sliding regime is difficult. Although the data is sparse in this high normal stress limit, the data clearly indicates that the wear rate falls far below the prediction one would obtain by extrapolating the Eyring behavior that predominates at low normal stress. 10 3

normalized Γwear (s−1 )

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Simulations tip1 tip2 tip3 tip4 tip5 tip6 Theory

10 2

10 1

10

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ ˜ Figure 3: Change of normalized wear rate from the low to intermediate effective normal stress (˜ σ ) regime. Experimental 24 data, numerical simulation results from an implementation of Eqs.(3-7), and theoretical approximation of the numerical model derived from Eq.(17), are plotted together for direct comparison. For experimental data, σ ˜ = σ · (∆U/Vact )−1 ; for numerical simulation data, σ ˜ is calculated using Eq.(10), where the stress-free bond 3 formation energy barrier ∆U = 0.7 eV and activation volume Vact = 120.0 Å .

3.3

High effective normal stress regime

To further investigate the high stress regime where formation of bonds is not anymore the rate-limiting process, we use Laplace′ s local method 37 and the definitions of v1 and v2 to

14


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approximately calculate the wear rate as:

Γwear ≃

√

2πω0

e1+βw

·

v v1

exp

v1 βw v

ln βvw1v +

βw v v2

(17)

.

When the sliding velocity is in the regime where v1 ≪ βw v ≪ v2 , v is high enough so that the bond breaking is mainly due to the mechanical unbinding, but not too high to overshadow the normal stress and cause a low coverage of bonds, the wear behavior is approximately Archard-like in the sense that the dependence of ∂Nwear /∂d = Γwear /v upon v is very weak, i.e., ∂ ∂v

∂Nwear ∂d

−1 βw v . ∝ ln v1

(18)

Another defining aspect of the Archard law is that ∂Nwear /∂d grows linearly with normal load FN , which can be related to the dependence of wear rate upon normal stress per Eq.(17). Note that the dependence of v2 upon σ is: ∂v2 /∂σ = v2 Vact /kB T , thus ∂ ∂σ

∂Nwear ∂d

∝

βw v βw v ln + v1 v2

−2

.

(19)

In the regime where v1 ≪ βw v ≪ v2 , ∂Γwear /∂σ → 0, which is in line with the rightmost plateau of wear rate observed in Fig. 3. By definition, Γwear is normalized by the real contact area A, therefore the relationship between the un-normalized wear rate AΓwear and the normal load FN is reduced to the one between A and FN . It has been shown in both theoretical 31 and computational 34,35 work that for a spherical tip under high normal stress, Areal can either increases linearly or sub-linearly. Therefore under high normal stress, in the particular regime confined by the two velocity scales v1 and v2 , the system approximately displays Archard-like wear behavior. Using MD simulations, Sha et al. reported the Archard-like wear behavior in a DLC system under external loads from 4 to 104 nN and a constant sliding speed of 20 m/s. 16 Although the external load covers a wide range, it is also crucial to take into account the

15



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strong adhesion between the un-passivated DLC interface when using this particular MD potential. By replicating the configuration in, 16 we found that the work of adhesion between the un-passivated surfaces when utilizing their potential is about 3.9 J/m2 and the pull-off force about 67.2 nN, both of which are much higher than typical experimental measurements for in H-passivated DLC (e.g., 0.07 ∼ 0.15 J/m2 and 1.08 ∼ 2.02 nN reported by 38 ). According to Maugis-Dugdale′ s model of continuum contact mechanics, 39 the strong adhesion leads to a significant increase of the effective normal load FH as given by 40

FH = (

(20)

p p N + Fp + 6πRw − 3Fp )2 ,

where w is the work of adhesion; R is the radius of the tip; N is the external load, and Fp 1

is the pull-off force. The contact radius a is related to FH by a = (3FH R/(4E ∗ )) 3 , where E ∗ = [(1 − ν12 )/E1 + (1 − ν22 )/E2 ]

−1

is the effective modulus with E1,2 and ν1,2 as the elastic

moduli and Poisson’s ratios of the contacting materials. The radial pressure distribution within the contact radius is then given by: E ∗a σ(r) = 2πR

q 1 − (r/a)2 −

s

4E ∗ (2w − Fp /(πR)) . πa (1 − (r/a)2 )

(21)

The average normal stress σ ¯ can be defined according to the average rate of forming a bond within the area of contact:

r¯on

a

∆U − σ(r)Vact rω0 exp − k T 0 B ∆U − σ ¯ Vact ≡ ω0 exp − kB T 2 = 2 a

Z

dr

(22) (23)

It turns out that in 16 σ ¯ is only varied by about 30% between the lowest and highest external loads. Fig. 4 shows that within a reasonable range of parameters, the dependence of wear volume per unit sliding distance upon dimensionless effective normal stress σ ˜ in 16 can be well reproduced, where the same temperature (300 K) and sliding speed (20 m/s) are used 16


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in our simulations. The fitting parameters are: ∆U = 0.48 eV, ∆Eoff = 0.48 eV, ∆Ewear = 3

0.56 eV, Vact = 0.6 Å , lbreak = 4.8 Å and the other parameters are the same as in Fig. 2. 5.5 Numerical data Asymptotic analysis MD data by Sha et al.

5

∂Nwear /∂d(˚ A−1 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.5 4 3.5 3 2.5 2 1.5 1

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

σ ˜ Figure 4: Wear events per sliding distance versus dimensionless effective normal stress σ ˜ in the high stress regime at a sliding speed of 20 m/s. Dashed line is derived in light of Laplace’s local method, given in Eq.(17). Solid symbols represent numerical calculations of the multi-bond model of wear, while the open symbols and error bars are extracted from Ref. 16

3.4

Conclusion

Based on the original multi-bond friction model, we have proposed a single-asperity multibond wear model that depicts the process of wear as interfacial bond breaking governed by shear stress assisted thermally activated processes. The model predicts that under the conditions of low external load and weak adhesion, the wear events manifest a pronounced Eyring behavior. The dependence of wear rate upon the sliding speed of the asperity reveals the high- and low-rate limits in the Eyring regime. The model also predicts that when the external normal load or adhesion becomes sufficiently large, the exponential growth of wear rate with stress saturates and the Eyring behavior breaks down.

17



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We have also shown that in the high effective normal stress scenario where interfacial bonds form easily due to high external load or strong adhesion, the wear volume per unit distance appears to display Archard-like behavior under a confined regime of sliding velocity. In particular, the wear volume per unit sliding distance becomes relatively sliding-rate independent and appears to vary linearly or sub-linearly with load due to the elastically induced changes in contact area. However, this is a combined artifact of thermal activation and mechanical forced breaking of interfacial bonds, whereas the original Archard theory, having been devised for understanding deformation-mediated wear, neglects the role of thermally activated rate processes associated with bond formation and breaking during nanoscale wear phenomena. The picture that emerges from this theory is empirically consistent with the prior work of Yang, Huang and Shi, 22 but ascribes the transition to a kinetic crossover in the underlying tribochemistry rather than to an atomic-to-plastic transition. Consequently, the wear crossover and the emergence of Archard-like behavior can be quantitatively derived from the underlying atomic-scale kinetic model we propose here. Few, if any, physically-based quantitative models of wear exist at any scale. The theories that exist are typically empirically derived in ways that do not relate parameters to the atomistic processes that govern the wear process. As a consequence, while these relations are useful for aggregating measurements in the field, they do not open up the possibility that wear processes could be predicted from more basic calculations or principles. While the model presented here is phenomenological in nature, the elements of the model can be directly related to processes on the atomic scale. As such this model could serve as a foundation for a more first principles approach to the quantification of wear processes in nanotechnology applications or even at larger scales by providing a means for building multi-asperity models starting from a single-asperity perspective.

18


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4

Methods

The authors implemented the dynamics model in MATLAB 41 following the fourth-order Runge-Kutta algorithm for stochastic differential equations developed by Kasdin. 42 Parameter fitting was performed systematically as follows. The attempt frequency ω0 (1014 Hz) is chosen to be the same for all thermally activated processes in accordance to the value selected by Jacobs, et al. 7 Temperature T (300K), and stiffness of the driving spring k(12N/m) are selected based solely on physical considerations or directly derived from known experimental values (see, e.g., Li, et al. 43 ). The parameters M (5 × 10−11 kg), η(5 × 10−6 kg/s), κ(1N/m) are chosen in accord with prior theoretical studies in which the multi-bond model was used to study friction. 25,26 We, furthermore, assume that the critical bond-breaking length lbreak = ∆Eof f /fc = ∆Ewear /fw is the same in wear or wear-less bond breaking, which sets both critical forces once the energy barriers are determined. This length is assumed to fall in the 2 − 5Å range. The energy barrier for wear, ∆Ewear , is always chosen to be 10-50% in excess of the energy barrier for wearless bond breaking, ∆Eof f . For each experimental regime, then, these two energy barriers as well as ∆U , Vact and lbreak are varied to produce a best fit with the expectation that the energy barriers must fall within the range 3

0.1 − 1.0eV and the activation volume is a volume of radius 1 − 4Å, i.e. 4 − 268Å . Notably, in performing the fitting for the passivated DLC experiments, the results of which are shown in Fig. 3, and the unpassivated DLC simulations, shown in Fig. 4, we find we must 3

3

use activation volumes that vary significantly from 120Å in the experiments to 0.6Å in the simulations. This is, however, consistent with our assertion that the lack of passivation leads to a much greater ease of bond formation between tip and substrate, accounting for the shift in the wear regime between experiment and simulation data sets. In principle, one can obtain an estimate to these parameters directly from empirical measurements. The bond-breaking length should be a reasonable fraction (20-50%) of the bond length in the system, and ∆Eof f , which is the energy to dislodge an atom from the surface, can be estimated from thermophysical measurements. The remaining three parameters can 19



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be obtained from the logarithm of the wear rate per unit contact area as a function of the dimensionless normal load, σ ˜ . The activation volume is directly related to the slope of this curve in the low-load Eyring regime. If one has data from a single system in the high-speed limit, the intercept of this curve should be a good approximation of ∆U/kB T . If wear data from both the very low- and very high-speed limits are accessible, the shift between the curves should be a rough approximation of (∆Ewear − ∆Eof f )/kB T . In practice, since data over a very broad range of sliding rates are usually not accessible, an optimization procedure must be used to fit at least one of these parameters to the available data.

Acknowledgement The authors gratefully thank M. O. Robbins for useful discussions. Y. S. and M. L. F. acknowledge support from the National Science Foundation under Grant No. CMMI 0926111. R. W. C. acknowledges support from AFOSR under contract no. FA2386-15-1-4109 AOARD. R.W.C. and K.T.T. acknowledge support from the National Science Foundation under grant CMMI 1200019.

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Multibond Model of Single-Asperity Tribochemical Wear at the

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