# Adhesion Dynamics in Probing Micro- and Nanoscale Thin Solid Films

Authors: Xiaoling He
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

This study focuses on modeling the probe dynamics in scratching and indenting thin solid films at micro- and nanoscales. The model identifies bifurcation conditions that define the stick-slip oscillation patterns of the tip. It is found that the local energy fluctuations as a function of the inelastic deformation, defect formation, material properties, and contact parameters determine the oscillation behavior. The transient variation of the localized function makes the response nonlinear at the adhesion junction. By quantifying the relation between the bifurcation parameters and the oscillation behavior, this model gives a realistic representation of the complex adhesion dynamics. Specifically, the model establishes the link between the stick-slip behavior and the inelastic deformation and the local potentials. This model justifies the experimental observations and the molecular dynamics simulation of the adhesion and friction dynamics in both the micro- and nanoscale contact.


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## Body

## 1. Introduction

The  adhesion  and delamination at the interface of micro- and nanoscale thin films are of
interest in semiconductor thin films, microstructures, and many other
applications. Under stress, defects such as dislocations, twins, and grain boundaries
initiate and nucleate. The accumulation of the defects intensifies the local
stress field, which eventually induces cracks to release energy. Such a stress and relaxation process continues before
the material eventually  breaks down at the limit of the adhesive strength of
the interfacial material to cause delamination, or at the limit of the coherence
strength of the singular material itself that leads to rupture.An experimental study of the thin film’s adhesion property commonly uses
a scratching test to determine the adhesion or coherence strength in terms of
the critical load with respect to the tip’s penetration depth. A diamond stylus
scratches the film to generate stresses up to the critical stress
intensity till fracture [1, 2]. Experimental observation
indicates that many factors, including material properties and the test
conditions, such as the scratching velocity, probing load, and stylus geometry,
play a role in the penetration process of the probing tip. Moreover, the tip
exhibits oscillatory behavior, in contrast to a direct penetration, subject to
the constant driving force or constant driving velocity. This oscillation is
generally referred to as the stick-slip behavior in adhesion and friction dynamics,
a common phenomenon observed in atomic force microscopy or nanoindentation [3]. The stick-slip behavior is
also prevalent in microscale friction dynamics [4]. Savkoor and Briggs also
found that during the scratching, a moderate peeling occurs as a result of an
increasing shear force. This indicates an irreversible inelastic deformation of
the thin film up to fracture. The stick-slip behavior prior to, during, and
after the crack initiations suggests highly nonlinear relations between the inelastic
deformation, material properties, the forcing, and the contact parameters [5, 6].Experimental studies have confirmed
that interfacial adhesion depends on deformations and accumulating defects such
as dislocation, stacking faults, steps, vacancies, pining junctions, and
cracks. Studies of adhesion phenomena have taken different perspectives in
examining the contact mechanisms at the adhesion junctions. Many of the
previous studies have focused on the threshold energy at the initiation of the
cracks subjected to different stress fields. The most highlighted theories on
adhesion strength are the JKR [7] and DMT models [8, 9]. Both the JKR and DMT models examine
the contacting spheres based on Griffith’s concept in normal mode fracture. However, a difference exists between the two
theories in representing the adhesion forces and the contacting areas. This discrepancy
leads to different predicted pull off forces by each theory [10]. Other than these two well-known
models, Bradley [11] also evaluated the adhesion
threshold strength of two contact bodies by incorporating both the surface
energy and the cohesive force based on the Leonard-Jones potential.For shear-mode fracture, Rice deduced that the stress intensity at the
periphery of the contact region is expected to nucleate glide dislocations by
an effective Peierls stress [12]. We should mention that Peierls
stress refers to a periodic relation between the shear stress and the dislocation-induced
sliding at an atomic scale. The Peierls stress arises from the periodicity of
the atomic lattice structure. Rice found that the shear stress increases
proportionally to the contact size, determined by the unstable stacking fault energy
[12]. Hurtado and Kim also
developed a model based on a Peierls stress concept [13]. Their model prescribes a
slip zone circulating the dislocation region driven by an effective Peierls
force.Until recently, studies on the mixed mode crack, due to a shear force
interacting with the normal force in sliding friction, have remained a
challenge, both theoretically and experimentally [14]. In addition to the dislocation-induced
shear mode crack, which is regarded as the major source of the sliding friction
and the initiation of plasticity [12, 13], other defects play a role in
the mixed mode fracture, including steps, twins, vacancies, and interstitials [15]. Johnson [10] studied the interaction
between the adhesion and friction following Hutchinson’s model for joint normal
and shear mode fractures [16]. Johnson concluded that the
relation between the normal and the shear forces cannot be found by continuum mechanics
in a linear model; rather, the response to a combination of normal and
tangential forces involves specific characteristics of the nanoscale adhesion
junction. The reason is that the nature of this interaction comes from the
interface interaction at the periphery of the contact area [10]. Such a mixed mode adhesion
or friction problem is typical of the scratching and the nanoindentation of
thin solid films using a multifaceted Berkovich tip, shown in Figure 1(a). The
driving force pushes the tip into the film as the probe moves forward, which
generates both shear and normal forces around the tip. Other configurations of
tip profile, such as the parabolic or spherical tips,
also give rise to both shear and normal deformation around the tip. The atomic
scale molecular dynamics (MD) simulation suggests that the plasticity due to nucleation and the translation of the dislocations is the source of
the dynamic stress field around the tip of the indenter, which causes penetration of the tip in a stick-slip behavior [6, 17].  The MD simulation verified
the adhesion and friction dynamics at the nanoscale, as observed in experiments.
Moreover, both the experiments and the MD computations confirmed that the strain
rate-dependent plasticity governs the behavior of nanostructure metals, such as
Al and Cu [5, 17].Probe geometry.
(a)
Berkovich tip(b)
Cross-section and overlook of the probe tip(c)
Penetration parametersDifferent formulations have been used to characterize adhesion and
friction dynamics for the stick-slip phenomena of a mechanical system. Persson
and Volokitin used a linear vibration model to account for the friction
oscillation [4]. The nonlinear model for the
friction dynamics by Dankowicz and Nordmark [18] focuses on the chaos and
bifurcations in the stick-slip behavior induced by a discontinuous force. The
model by Heslot et al. [19] predicts that under a
constant driving force, the motion of the contact solids is often unsteady and
oscillatory. Both models support the experimental observation of the contact
dynamics. These studies analyzed the complexity of the nonlinear behavior in
the stick-slip oscillation based on a generalized vibration model. However,
these models are not adaptable to explain the stick-slip behavior associated
with different deformation mechanisms at the contact interface; in particular,
when inelastic deformation is required to account for the strain rate-dependent
behavior. Fundamentally, inelastic deformation occurs prior to the crack
initiation [20–22]. The underpinning of the inelasticity
or plasticity is the irreversible process at the atomic scale due to dislocations
and formation of other types of defect, such as twins and grain boundaries [23, 24], which in part, is the cause
of the stick-slip oscillation.It is  apparent that  the probe dynamics phenomenon of two contacting
solids at the adhesion junction should be justified by the deformation
mechanisms involved. Although an MD computation can simulate the dynamic behavior
of the probe tip, the computation does not prescribe a direct constitutive
relation to explain the adhesion dynamics in relation to their deformation
mechanisms. On the other hand, the existing formulations for the stick-slip
behavior do not incorporate inelastic deformation mechanisms. Therefore, the
objective of this study is to develop a model to explain adhesion dynamics associated
with both the elastic and inelastic deformations of thin solid films at the micro- and nanoscales. This model builds on Griffith’s
concept of fracture and inelastic deformation theory for the mixed mode
deformation around the probe tip when penetrating thin films subjected to constant
driving velocity. By relating the energy release rate to the inelastic
deformation of the solid film, we arrive at a relaxation equation to explain
the stick-slip behavior in the tip’s dynamics at the adhesion junction.In the following, we will first obtain the equation of motion to describe
the dynamics of the probe tip. Further, the stability analysis identifies the
bifurcation conditions that define the transitions between a direct sliding
penetration mode and a stick-slip relaxation oscillation mode. These
bifurcation conditions quantify the dynamics of the tip influenced by a collection of parameters,
determined by the local energy function, the contact parameters, and the
inelastic material properties for both micro- and nanoscale thin films.
Subsequently, the computation illustrates the tip’s dynamics in either a direct
penetration or a stick-slip behavior. Finally, the paper concludes with a
discussion of the adhesion dynamics with respect to the underlying physics of
the deformable solids, and the scale effect.


## 2. Equation of Motion

We assume
that a probe approaches the thin film at a constant driving velocityV,
as shown in Figure 1(c). The probe penetrates the material with both normal and
tangential forces at the contacting surfaces, while the multifaceted probe
keeps the angle θ close to 10 degrees. Denoting v as the actual velocity of the tip along the
penetration direction, ε˙ the strain rate of the thin film, V the driving velocity of the probe, and h the displacement of the tip, the relation
between these parameters is (2.1)V=v+hε˙, where ε=δh/h is the strain of the thin film along the probe
direction, and v=h˙ is the actual velocity of the penetration. The
relation (2.1) comes from the derivative of the total travel distance L of the tip, which is given by (2.2)L=h+hε. This relation holds when the probe traverses through the
atomic lattice and displaces the atoms for dislocations at the nanoscale, or
when the penetration induces inelastic deformation in the creeping of the
microscale thin films. Along with the penetration, defects migrate and
accumulate to produce a high stress field to slow down the tip’s motion, whereas
the probe overcomes the energy barrier to proceed. This process repeats until the
incipient of cracks, which gives the energy release rate per unit contact area as (2.3)Gc=ϕ(v)+φ(v)+I(v˙). Here ϕ(v) is the inelastic energy loss near the crack
tip, which is proportional to the surface energy, φ(v) is the inelastic energy loss away from the
crack tip, and I(v˙) is the inertia energy. Since the far-field
energy release φ(v) is negligible due to the localized deformation
surrounding the tip, the energy release rate simplifies to (2.4)Gc=ϕ(v)+I(v˙). The equivalent mass associated with the inertia in the unit contact
length is (2.5)m=I(v˙)dv˙. This leads to (2.6)dGcdt=mv¨+v˙d(ϕ(v))dv. On the other hand, the strain energy release balances against
the increase in the surface energy [14], which means (2.7)Gc=σh+τΩ=σh(1+η)=σ˜h. Here σ and τ are the normal stress and shear stress, respectively; σ˜ represents  the combined stress of a joint normal
and shear stress field, Ω stands for the total surface area of the
activation volume of the contact plane, and η=τΩ/σh is the shear factor. This factor is an
indicator for the major deformation in either shear or normal mode, which makes
the oscillation represent either the friction or the adhesion dynamics,
respectively. The derivative of the energy release rate with respect to time gives (2.8)G˙c=dGcdt=∂(σ˜h)∂t=(σ˜˙+σ˜ε˙)h. This suggests that the energy release rate of the inelastic
deformation depends on the time-dependent strain rate. In reference to the
theory of time-dependent plasticity in solids by Persson [15], the mean stress filed σ with respect to the mean strain field ε and the strain rate ε˙ can be described by the relation of (2.9)Eε˙=σ˙+λ˜(t)σ. Here E is Young’s modulus; λ˜(t) and η(t) are parameters associated with the deformation
mechanisms and material properties, and they are related by λ˜(t)=E/η(t). The solution to (2.9) as a first-order ordinary differential
equation is in the general form of (2.10a)σ(t)=Eμ(t)[∫0tμ(t)ε(t)dt+σ0],μ(t)=e∫0tλ˜t=eλt. This can be further approximated in a series expression, as: (2.10b)σ(t)=Eλε˙(t)−(Eλ2)ε¨(t)+⋯(−1)n+1(Eλn)ε(n)+σ0,ε(n)=d(n)εdt(n). After dropping the higher-order strain-rate, we obtain (2.11)σ˜≅A0ε+B0ε˙−C0ε¨,A0=E,B0=Eλ,C0=Eλ2, where σ0=Eε represents the yield stress at the limit of elastic
deformation; B0ε˙ and C0ε¨ result from the inelastic deformation that is strain
rate-dependent. To justify the negative strain effect, that is, the strain
softening along with a plastic flow when the tip passes through a trail of
voiding, or the strain hardening due to a high energy barrier while
experiencing a reduced deformation, the coefficient λ, or effectively B0, can be negative. A positive coefficient B0 stands for an increasing deformation along
with an intensified stress field. Both coefficients B0 and C0 can be regarded as time-dependent localized
functions determined by defect formations, localized energy functions, material
properties, and contact parameters, such as the activation volume or contact
length.Ignoring the second-order strain terms, the time
derivative of the energy release rate becomes(2.12a)G˙c=σ˜˙h=(Aε˙+Bε¨−Cε⃛)h, where all coefficients can be time-dependent functions and given
by (2.12b)A=E+B˙0,B=B0−C˙0,C=C0. From the velocity analysis, the time-dependent strain rate is
related to the velocity by (2.13)ε˙=V−vh, which yields (2.14)ε¨=−(V−v)vh2−v˙h. Since (V−v)≪v,
we have (2.15)ε¨≈−v˙h,ε⃛=−v¨h. Using (2.12a) and (2.12b), we obtain  (2.16)dσ˜dt=Aε˙+Bε¨−Cε⃛=A(V−vh)+B(−v˙h)+Cv¨h. Equations (2.16) and (2.6) lead to (2.17)v˙dϕdv+mv¨=[A(V−v)−Bv˙+Cv¨], which is equivalent to (2.18a)v¨+μω(dϕdv+B)v˙+ω2(v−V)=0, where (2.18b)ω2=Am˜,μ=1m˜ω=1m˜A,m˜=m−C. This is Leonard’s equation in a typical relaxation
oscillation that describes the jerky motion of the probing dynamics. The
parameters in this equation quantify the influence of the elastic and inelastic
deformation moduli, contacting parameters, and the driving velocity on the
probing dynamics. The equation indicates that the oscillation frequency is
determined by both the elastic and inelastic deformation moduli and the mass
contained in the activation volume, that is, (2.19)ω2=E+B˙ρΩ−C. The unit mass m=ρΩ is related to the inertia energy release by
(2.5). The forcing of this jerky motion is (2.20)dFdt=ω2V=(E+B˙)VρΩ. Equation (2.20) indicates that an intensified forcing comes
from a reduced contact area or a smaller volume of mass, and from the high
elasticity, a time-dependent variation of inelastic modulus and a high driving velocity. Another term
in (2.18a)
is μω(dϕ/dv+B),
which represents the friction damping in the probing process. The antidamping
coefficient μ is defined as (2.21)μ=1m˜ω=1m˜(E+B˙)=1ρ(E+B˙)Ω>0.Under constant driving velocity, it is
natural to assume that the inelastic energy dissipation at the interface is a
function of the driving velocity and the local energy release. Introducing the functionχ(ϕ)=dϕ/dv,
(2.18a) becomes (2.22)v¨+μω(χ(ϕ)+B)v˙+ω2(v−V)=0.χ(ϕ) measures the inelastic energy release rate at
the tip with respect to the penetration rate. This function can be a
time-dependent function during the penetration, even though the driving
velocity is constant. In conjunction with the inelastic deformation modulus B,(χ(ϕ)+B)  reflects the localized energy dissipation or
energy accumulation rate, influenced by the local energy barriers, and the
inelastic deformation mechanisms at different length scales. For example,
dislocation at the atomic level requires energy that is several orders less
than that for creeping at the microscale. On the other hand, the contact
parameters could also differ by the same order between the two length scales.
Such combinations could produce a product  in μ(χ(ϕ)+B) that is the same for the two different length
scales to prescribe similar oscillation behavior of the tip, as analyzed
below.Note that the energy release rate for cracks
or defect formation implies that(2.23)dϕdv=χ(ϕ)>0. This is because the energy release function ϕ(v)>0 symbolizes an energy loss at the tip, and ϕ(v)<0 an energy gain due to an intensified stress
field. Therefore, the motion at ϕ(v)<0 has a reduced penetration rate due to an
increasing energy barrier, that is, v<0 and ϕ(v)=χ(ϕ)v<0. On the other hand, ϕ(v)>0 corresponds to an increasing penetration rate at v>0 with energy dissipation, and ϕ(v)=χ(ϕ)v>0.


## 3. Stability Analysis

The bifurcation conditions determine the
oscillation patterns and their stabilities. In matrix form, (2.22) becomes (3.1){u˙v˙}=[−μω(χ(ϕ)+B)−ω210]{uv}+{ω2V0}, where u=v˙ is the acceleration. The eigenvalues of this
linear system are (3.2)λ1,2=ω2(−μ(χ(ϕ)+B)±μ2(χ(ϕ)+B)2−4). The eigenvalues indicate that there are several oscillation
patterns in the phase diagram, which are determined by the collective parameter μω(χ(ϕ)+B) [25]. It is apparent that both the
elastic and the inelastic deformation rate influence the oscillation stability
or the bifurcation behavior, as analyzed below.(a)(χ(ϕ)+B)=0, λ1,2=±iω, Periodic Oscillation
This is the condition for the Hopf
bifurcation. The velocity is periodic, given as(3.3)v¨+ω2v=ω2V. The condition χ(ϕ)+B=0 indicates a balance between the inelastic
energy release rate and an inelastic deformation energy variation in a negative
strain effect, B<0.
This occurs when the tip goes through vacancies to cause the reduced stress
intensity due to strain softening, while discharging energy along with the
penetration. Alternatively, a higher energy barrier develops at a reduced
deformation at B<0 for strain hardening when accumulated defects
prevent the tip from further penetrating the film. When χ(ϕ)+B=0, there is insufficient energy to stop the
motion completely by the high energy barrier, or to sustain a continuous
penetration. Instead, a cyclic velocity occurs, alternating between v>0 and v<0,
respectively. This corresponds to two phases of negative strain fields, that is,
the strain softening and hardening, which give rise to the stick-slip oscillation.(b)(χ(ϕ)+B)>0, Re(λ1,2)<0, Stable Oscillations
From (3.2), if μ(χ(ϕ)+B)≥2, there are two real negative eigenvalues. The
oscillation converges to a stable node in the phase diagram, reaching the driving
velocity after a transient oscillation. This allows for a continuous
penetration with energy release through deformation or crack. If 0<μ(χ(ϕ)+B)<2, the complex conjugate eigenvalues, λ1,2=Re(λ1,2)±iIm(λ1,2) with a negative real part,  define the subcritical Hopf bifurcation, in
which the penetration velocity oscillates in a spiral pattern to the nodal velocity
at V.
The spiral motion is the consequence of two alternating stress field: that is, the
intensified stress field due to the strain hardening, and a reduced stress
field associated with a moderate creeping or voiding. In this case, the
penetration is partially constrained by energy barriers from strain hardening
effect.(c)(χ(ϕ)+B)<0, Re(λ1,2)>0, Unstable Oscillations
Ifμ(χ(ϕ)+B)≤−2,
two positive real eigenvalues make an unstable nodal oscillation. On the other
hand, if 0>μ(χ(ϕ)+B)>−2, then λ1,2=Re(λ1,2)±iIm(λ1,2) defines a supercritical Hopf bifurcation in an
unstable spiral motion. In both cases, there is a dominant negative strain
effect to make (χ(ϕ)+B)<0 due to the strain softening. The velocity
keeps increasing deviating from the driving velocity due to the predominant
creeping or voiding, which results in a straight forward penetration. A spiral
oscillatory velocity occurs when the creeping is intertwined with the
intermittent energy barriers or energy releases through cracks.The bifurcation conditions indicate that the
oscillation pattern is determined mainly by the  parameter(χ(ϕ)+B) associated with the local energy barriers and
inelastic deformation. The fluctuation of the collective parameter in the form μ(χ(ϕ)+B) gives rise to transitions among different
bifurcations. The antidamping coefficient μ influences such transitions by variable contact
area and material properties.


## 4. Numerical Computation

The computation is based on the normalized
equation (2.22) using the time scale τ=ωt,
which yields (4.1)d2vdτ2+μ(χ(ϕ)+B)dvdτ+(v−V)=0. The time step is Δτ=0.1s using the 4th-order Runge-Kutta method. The
bifurcation condition is specified in the form of μ(χ(ϕ)+B)=2k,
where k is a constant chosen for each case. Using
different values of μ and (χ(ϕ)+B) for the contact, deformation, and the energy
release rate at different length scales of the Cu thin film, the numerical results
below illustrate the oscillatory patterns determined by the above bifurcation
conditions for contact at both micro- and nanoscales.


### 4.1. Microscale Thin Film Behavior

We chooseE=105[GPa], G=42[GPa]  for the microscale Cu thin film, the same as the bulk material
properties. We also set B a constant,
or B˙=0. The density of the Cu film is ρ=8.9[g/cm3]. The driving velocity is V=20[μm/s]. The contact area is Ac=22.67h02[μm2] at a fixed contact length of h0=0.4[μm]. Figure 2(a) shows the periodic oscillation of
the velocity with respect to the travel distance for the bifurcation case 1: (χ(ϕ)+B)=0. The velocity-distance phase diagram traces
back to the ellipsoid phase portrait for the acceleration and velocity
represented by (3.3). Figure 2(b) is the periodic penetration depth with respect
to time in a stick-slip oscillation, which shows a repeated forward and an
almost standstill motion due to a cyclic velocity variation. The penetration in
a forward motion traverses a distance about 100μm in each stroke. As already explained, such a
periodic behavior indicates a penetration cycle where the creeping deformation
alternates with the strain hardening motion due to the defect-induced energy
barriers. The higher energy barrier pushes the tip to nearly a standstill
before the external energy overcomes the barrier to move the tip forward again.Periodic oscillation.
(a)
Velocity-acceleration phase diagram(b)
Stick-slip penetrationFor the bifurcation case 2:(χ(ϕ)+B)>0.
Figures 3(a) and 3(b) illustrate the oscillation in a stable spiral pattern,  whereas
Figures 3(c) and 3(d) are in a stable nodal pattern, respectively. For the
stable spiral oscillation, the parameters are set as μ(χ(ϕ)+B)=6 at k=3, while the stable node oscillation corresponds
to μ(χ(ϕ)+B)=1 at k=0.5. The velocity approaches the node in both phase
diagrams, reaching v=V=20[μm/s].
Notably, the stable oscillation in either a spiral pattern or a nodal pattern
makes the tip traverse in a direct penetration at the driving velocity at the steady
state. A slight difference exists in the penetration rate between these two
cases, as shown in Figures 3(c) and 3(d), respectively, owing to different
transient oscillations.Stable oscillation forμ(χ(V)+B)>0.
(a)
Velocity-acceleration phase diagram in the subcritical Hopf bifurcation
atk=0.5(b)
Penetration distance in a subcritical Hopf bifurcation atk=0.5(c)
Velocity-acceleration phase diagram in a stable node atk=3(d)
Penetration distance in a stable node atk=3For case 3:μ(χ(ϕ)+B)<0.
Figures 4(a) and 4(b) illustrate the unstable spiral motion while Figures 4(c)
and 4(d) are for the unstable nodal pattern. The unstable spiral is subjected
to μ(χ(ϕ)+B)=−1 for k=−0.5, while the unstable node corresponds to μ(χ(ϕ)+B)=−4 at k=−2.
Both unstable oscillations make the velocity increases continuously. The spiral
velocity introduces a backward and forward stick-slip behavior, as shown in Figure 4(b). On the other hand, the unstable nodal pattern enables a direct
penetration shown in Figure 4(d). Note that nodal-type unstable penetration may
not occur in reality, the unlimited creeping is unlikely to occur to produce an unstable nodal type
penetration, due to defects that induce energy barriers to effectively constrain  tip’s motion.Unstable oscillation forμ(χ(V)+B)<0.
(a)
Penetration distance-velocity phase diagram in the supercritical Hopf
bifurcation atk=−0.5(b)
Penetration distance in a supercritical Hopf bifurcation atk=−0.5(c)
Penetration distance-velocity phase diagram in an unstable node atk=−2(d)
Penetration distance in an unstable node atk=−2


### 4.2. The Influence of Periodic Potentials

The above results confirm that the tip’s
penetration behavior is determined by the deformation mechanisms that are reflected by the
bifurcation conditions. As illustrated in both Figures2(b) and 4(b), the
stick-slip behavior occurs when the velocity is in either a periodic oscillation
or an unstable spiral motion, corresponding to the Hopf bifurcation or the
supercritical Hopf bifurcation, respectively. The stick-slip penetration could
also be a manifestation of the continuous transitions among different
bifurcations due to the variable bifurcation parameters. This could occur because
the localized defects and deformation introduce instantaneous changes of the
local potentials, contact areas, and energy release rate.To simulate the influence of a periodic
variation of the local potentials associated with the energy release rate, the
collective bifurcation parameter is set in a time-dependent trigonometric
function, that is,μ(χ(ϕ)+B)cos(kτ/j)=2kcos(kτ/j). Figures 5(a) and 5(b) show the stick-slip behavior corresponding
to the potential  function at k=2 and j=3, which leads to μ(χ(ϕ)+B)cos(kτ/j) oscillating periodically within [−4,4]. The phase diagram in Figure 5(a) shows a
nonperiodic velocity that is almost all positive, which makes the stick-slip
penetration in a forward-standstill pattern, as shown in Figure 5(b). The
penetration behavior is due to the instantaneous transitions among all five
bifurcations described above in each cycle. By varying the parameters, for
example, k=1 and j=3, identical bifurcation transitions take place,
since the range of μ(χ(ϕ)+B)cos(kτ/j) is [−2,2].
Here three types of the Hopf bifurcations dominate, that is, the oscillation
transits mainly among the stable spiral, unstable spiral, and periodic
oscillations. However, both the backward and forward stick-slip motion are
present in each cycle due to the velocity alternating between positive and
negative values, as shown in Figures 5(c) and 5(d). Each stroke of the slip
motion traverses a different distance. These results demonstrate that a
periodic energy barrier leads to a quasiperiodic stick-slip oscillation owing
to transitional bifurcations. Effectively, the periodic energy release and
inelastic deformation transform the system into a nonlinear relaxation model.
As a result, the oscillation is no longer in a fixed frequency in each cycle;
instead, a variable frequency oscillation occurs.Nonperiodic oscillations
subject to a periodic potentialμ(χ(V)+B)cos(kτ/j).
(a)
Penetration distance-velocity diagram at a periodic potential atk=2 and j=3(b)
Penetration distance at a periodic potentialk=2 and j=3(c)
Penetration distance-velocity diagram at a periodic potentialk=1 and j=3(d)
Penetration distance at a periodic potentialk=1 and j=3The influence of the periodic bifurcation
parameters in another form, that is,μ(χ(ϕ)+Bcos(kτ/j)),
a periodic inelastic deformation that goes along with a constant energy release
rate χ(ϕ),
indicates that an unstable oscillation prevails when the transitional
bifurcation parameter μ(χ(ϕ)+Bcos(kτ/j))<0. For example, an unstable spiral for μ(χ(ϕ)+Bcos(kτ/j))=−4+2cos(kτ/j)<0 generates a stick-slip oscillation, similar to
that shown in Figure 4(b). On the contrary, a stable oscillation emerges to
give rise to a forward penetration when μ(χ(ϕ)+Bcos(kτ/j))=2+2cos(kτ/j)=[4,0].The difference in the forms of the periodic
potential affects both the transient and the steady-state oscillation behavior.
Physically, these periodic variations can be attributed to different material behavior
and factors at the adhesion junction, including the contact area, the elastic
and inelastic deformation moduli, the energy dissipation and the external
forcing, among others. Note that periodic potential variation directly reflects
the atomic potentials of crystals in a periodic structure that gives rise to
Peierls stress in shear mode deformation, which constitutes the energy barrier
for sliding dislocations.We should also point out that a variable
contact area contributes to the dynamic variation of the bifurcation conditions.
This is because the depth of the penetration influences the contact area, which
indirectly affects the parameterμ and the bifurcation condition μ(χ(ϕ)+B).
During the probing, the tip remains in contact with the solid film until the
crack debris peels off from the thin film, upon which the contact length may reduce
to zero, or to a critical length. This is because cracks counterbalance the
increasing activation volume or the contact length around the tip. The critical
contact length defines the transition upon which the penetration process
repeats itself. Therefore, the contact area variation can be regarded as a
source of the periodic or an irregular variation of local energy release rate.
The wider range variations of the local energy potentials can lead to complex
dynamics in the tip’s oscillation, compared to that under the periodic
potentials. This also means that the real tip’s nonlinear stick-slip behavior becomes
more complex, due to the instantaneous fluctuation of the contact parameters and
energy functions.


### 4.3. Contact Dynamics at the Nanoscale

As has been shown, the computation results
phenomenologically agree with the experimental observation of the stick-slip or
the direct penetration behavior in the microscale thin films. By examining
(4.1) it is apparent that an identical relaxation behavior exists with the
nanoscale  thin films at a different energy level. This is because the collective
parameter μ(χ(ϕ)+B) defines the oscillation behavior, even though different energy
function (χ(ϕ)+B) and contact parameter μ are specified. This makes the model adaptable to
describe both the micro- and the nanoscale phenomena. In fact, the model in (2.22)
agrees with the dislocation dynamics equation that accounts for the dislocation
displacement fluctuation at the atomic lattice when overcoming a finite mass at
the pining junction, as described by Osip’yan and Vardanian in [26].To illustrate the tip dynamics when
penetrating a nanoscale thin film, we use a periodic potential in the formμ(χ(ϕ)cos(kτ/j)+B), for the periodic energy releases accompanied
by a constant inelastic deformation module B>0.
Figures 6(a)–6(d) compare the transition of  the penetration behaviors for a
fixed contact area and a continuously variable contact area. The bifurcation
parameters vary in the range μχ(ϕ)(cos(kτ/j)−2)=[18,6] with μχ(ϕ)=2k,k=−3,j=3. For the fixed contact area with a contact
length of h0=0.4nm, the condition defines a stable oscillation
approaching a constant nodal velocity at the steady state, as shown in Figure 6(a), giving rise to a direct penetration as displayed in Figure 6(b). However,
an increasing contact area of the form Ac=αh2 transforms the nodal-type velocity and
acceleration phase diagram into a quasiperiodic oscillation pattern, as shown
in Figure 6(c). This indicates the effective transitions of the bifurcation
parameters due to a decreasing antidamping coefficient μ as a result of an increasing contact area. The
velocity oscillates in a positive but declining range and the penetration is in
a forward-standstill type stick-slip behavior, as shown in Figure 6(d). At a
driving velocity of V=0.06nm/s,
the stick-slip oscillation reveals that each stroke is of the order of the
atomic lattice parameter a=0.364nm for Cu. This suggests that dislocations occur
at the Berger’s vector length, since the Berger’s vector |b→|=a for a perfect dislocation [27].Nanoscale oscillations withμχ(V)(cos(kτ/j)−2),μχ(V)=2k,k=−3, and j=3.
(a)
Velocity-acceleration phase diagram at a constant contact area(b)
Slip penetration at a constant contact area(c)
Velocity-acceleration phase diagram at a variable contact area(d)
Stick-slip penetration at a variable contact areaFor theCu nanoscale thin film interacting with the tip,
the above computation used the ideal shear strength τmax=2.5GPa [28, 29], which renders the
dislocation line energy Eb=2eV/nm,
based on Eb=Gb→2 for a perfect dislocation [27]. This energy level agrees
with established results for a dislocation and void initiation, which is
generally in the order of Eb=5eV per Berger’s vector [30]. The MD simulation of the Cu void growth and dislocation process indicates
that the activation energy is Eb=2.3~3.5eV/nm [31]. Note that the dislocation
energy through dissociation is found to be Ed=1.3~2.7eV from a higher to lower temperature range [32]. The dislocation nucleation
and parallel formation require an energy level that is less than that for a
perfect dislocation, as confirmed by the MD simulation [5, 6]. Similarly, twins and the
stacking faults formed
along with the dislocation nucleation, all of which are activated at a reduced
energy level compared to that for a perfect dislocation. Therefore, the
stick-slip behavior indicates the onset of plasticity due to dislocations and
other defects, including vacancies, twins, stacking faults, and so forth. Alternatively,
it can be said that the stick-slip oscillation is a reflection of the process
whereby defects initiate and nucleate to induce a periodic or an arbitrary fluctuation
of the energy release rate at the atomic scale due the tip’s probe of the thin
film.


## 4.1. Microscale Thin Film Behavior

We chooseE=105[GPa], G=42[GPa]  for the microscale Cu thin film, the same as the bulk material
properties. We also set B a constant,
or B˙=0. The density of the Cu film is ρ=8.9[g/cm3]. The driving velocity is V=20[μm/s]. The contact area is Ac=22.67h02[μm2] at a fixed contact length of h0=0.4[μm]. Figure 2(a) shows the periodic oscillation of
the velocity with respect to the travel distance for the bifurcation case 1: (χ(ϕ)+B)=0. The velocity-distance phase diagram traces
back to the ellipsoid phase portrait for the acceleration and velocity
represented by (3.3). Figure 2(b) is the periodic penetration depth with respect
to time in a stick-slip oscillation, which shows a repeated forward and an
almost standstill motion due to a cyclic velocity variation. The penetration in
a forward motion traverses a distance about 100μm in each stroke. As already explained, such a
periodic behavior indicates a penetration cycle where the creeping deformation
alternates with the strain hardening motion due to the defect-induced energy
barriers. The higher energy barrier pushes the tip to nearly a standstill
before the external energy overcomes the barrier to move the tip forward again.Periodic oscillation.
(a)
Velocity-acceleration phase diagram(b)
Stick-slip penetrationFor the bifurcation case 2:(χ(ϕ)+B)>0.
Figures 3(a) and 3(b) illustrate the oscillation in a stable spiral pattern,  whereas
Figures 3(c) and 3(d) are in a stable nodal pattern, respectively. For the
stable spiral oscillation, the parameters are set as μ(χ(ϕ)+B)=6 at k=3, while the stable node oscillation corresponds
to μ(χ(ϕ)+B)=1 at k=0.5. The velocity approaches the node in both phase
diagrams, reaching v=V=20[μm/s].
Notably, the stable oscillation in either a spiral pattern or a nodal pattern
makes the tip traverse in a direct penetration at the driving velocity at the steady
state. A slight difference exists in the penetration rate between these two
cases, as shown in Figures 3(c) and 3(d), respectively, owing to different
transient oscillations.Stable oscillation forμ(χ(V)+B)>0.
(a)
Velocity-acceleration phase diagram in the subcritical Hopf bifurcation
atk=0.5(b)
Penetration distance in a subcritical Hopf bifurcation atk=0.5(c)
Velocity-acceleration phase diagram in a stable node atk=3(d)
Penetration distance in a stable node atk=3For case 3:μ(χ(ϕ)+B)<0.
Figures 4(a) and 4(b) illustrate the unstable spiral motion while Figures 4(c)
and 4(d) are for the unstable nodal pattern. The unstable spiral is subjected
to μ(χ(ϕ)+B)=−1 for k=−0.5, while the unstable node corresponds to μ(χ(ϕ)+B)=−4 at k=−2.
Both unstable oscillations make the velocity increases continuously. The spiral
velocity introduces a backward and forward stick-slip behavior, as shown in Figure 4(b). On the other hand, the unstable nodal pattern enables a direct
penetration shown in Figure 4(d). Note that nodal-type unstable penetration may
not occur in reality, the unlimited creeping is unlikely to occur to produce an unstable nodal type
penetration, due to defects that induce energy barriers to effectively constrain  tip’s motion.Unstable oscillation forμ(χ(V)+B)<0.
(a)
Penetration distance-velocity phase diagram in the supercritical Hopf
bifurcation atk=−0.5(b)
Penetration distance in a supercritical Hopf bifurcation atk=−0.5(c)
Penetration distance-velocity phase diagram in an unstable node atk=−2(d)
Penetration distance in an unstable node atk=−2


## 4.2. The Influence of Periodic Potentials

The above results confirm that the tip’s
penetration behavior is determined by the deformation mechanisms that are reflected by the
bifurcation conditions. As illustrated in both Figures2(b) and 4(b), the
stick-slip behavior occurs when the velocity is in either a periodic oscillation
or an unstable spiral motion, corresponding to the Hopf bifurcation or the
supercritical Hopf bifurcation, respectively. The stick-slip penetration could
also be a manifestation of the continuous transitions among different
bifurcations due to the variable bifurcation parameters. This could occur because
the localized defects and deformation introduce instantaneous changes of the
local potentials, contact areas, and energy release rate.To simulate the influence of a periodic
variation of the local potentials associated with the energy release rate, the
collective bifurcation parameter is set in a time-dependent trigonometric
function, that is,μ(χ(ϕ)+B)cos(kτ/j)=2kcos(kτ/j). Figures 5(a) and 5(b) show the stick-slip behavior corresponding
to the potential  function at k=2 and j=3, which leads to μ(χ(ϕ)+B)cos(kτ/j) oscillating periodically within [−4,4]. The phase diagram in Figure 5(a) shows a
nonperiodic velocity that is almost all positive, which makes the stick-slip
penetration in a forward-standstill pattern, as shown in Figure 5(b). The
penetration behavior is due to the instantaneous transitions among all five
bifurcations described above in each cycle. By varying the parameters, for
example, k=1 and j=3, identical bifurcation transitions take place,
since the range of μ(χ(ϕ)+B)cos(kτ/j) is [−2,2].
Here three types of the Hopf bifurcations dominate, that is, the oscillation
transits mainly among the stable spiral, unstable spiral, and periodic
oscillations. However, both the backward and forward stick-slip motion are
present in each cycle due to the velocity alternating between positive and
negative values, as shown in Figures 5(c) and 5(d). Each stroke of the slip
motion traverses a different distance. These results demonstrate that a
periodic energy barrier leads to a quasiperiodic stick-slip oscillation owing
to transitional bifurcations. Effectively, the periodic energy release and
inelastic deformation transform the system into a nonlinear relaxation model.
As a result, the oscillation is no longer in a fixed frequency in each cycle;
instead, a variable frequency oscillation occurs.Nonperiodic oscillations
subject to a periodic potentialμ(χ(V)+B)cos(kτ/j).
(a)
Penetration distance-velocity diagram at a periodic potential atk=2 and j=3(b)
Penetration distance at a periodic potentialk=2 and j=3(c)
Penetration distance-velocity diagram at a periodic potentialk=1 and j=3(d)
Penetration distance at a periodic potentialk=1 and j=3The influence of the periodic bifurcation
parameters in another form, that is,μ(χ(ϕ)+Bcos(kτ/j)),
a periodic inelastic deformation that goes along with a constant energy release
rate χ(ϕ),
indicates that an unstable oscillation prevails when the transitional
bifurcation parameter μ(χ(ϕ)+Bcos(kτ/j))<0. For example, an unstable spiral for μ(χ(ϕ)+Bcos(kτ/j))=−4+2cos(kτ/j)<0 generates a stick-slip oscillation, similar to
that shown in Figure 4(b). On the contrary, a stable oscillation emerges to
give rise to a forward penetration when μ(χ(ϕ)+Bcos(kτ/j))=2+2cos(kτ/j)=[4,0].The difference in the forms of the periodic
potential affects both the transient and the steady-state oscillation behavior.
Physically, these periodic variations can be attributed to different material behavior
and factors at the adhesion junction, including the contact area, the elastic
and inelastic deformation moduli, the energy dissipation and the external
forcing, among others. Note that periodic potential variation directly reflects
the atomic potentials of crystals in a periodic structure that gives rise to
Peierls stress in shear mode deformation, which constitutes the energy barrier
for sliding dislocations.We should also point out that a variable
contact area contributes to the dynamic variation of the bifurcation conditions.
This is because the depth of the penetration influences the contact area, which
indirectly affects the parameterμ and the bifurcation condition μ(χ(ϕ)+B).
During the probing, the tip remains in contact with the solid film until the
crack debris peels off from the thin film, upon which the contact length may reduce
to zero, or to a critical length. This is because cracks counterbalance the
increasing activation volume or the contact length around the tip. The critical
contact length defines the transition upon which the penetration process
repeats itself. Therefore, the contact area variation can be regarded as a
source of the periodic or an irregular variation of local energy release rate.
The wider range variations of the local energy potentials can lead to complex
dynamics in the tip’s oscillation, compared to that under the periodic
potentials. This also means that the real tip’s nonlinear stick-slip behavior becomes
more complex, due to the instantaneous fluctuation of the contact parameters and
energy functions.


## 4.3. Contact Dynamics at the Nanoscale

As has been shown, the computation results
phenomenologically agree with the experimental observation of the stick-slip or
the direct penetration behavior in the microscale thin films. By examining
(4.1) it is apparent that an identical relaxation behavior exists with the
nanoscale  thin films at a different energy level. This is because the collective
parameter μ(χ(ϕ)+B) defines the oscillation behavior, even though different energy
function (χ(ϕ)+B) and contact parameter μ are specified. This makes the model adaptable to
describe both the micro- and the nanoscale phenomena. In fact, the model in (2.22)
agrees with the dislocation dynamics equation that accounts for the dislocation
displacement fluctuation at the atomic lattice when overcoming a finite mass at
the pining junction, as described by Osip’yan and Vardanian in [26].To illustrate the tip dynamics when
penetrating a nanoscale thin film, we use a periodic potential in the formμ(χ(ϕ)cos(kτ/j)+B), for the periodic energy releases accompanied
by a constant inelastic deformation module B>0.
Figures 6(a)–6(d) compare the transition of  the penetration behaviors for a
fixed contact area and a continuously variable contact area. The bifurcation
parameters vary in the range μχ(ϕ)(cos(kτ/j)−2)=[18,6] with μχ(ϕ)=2k,k=−3,j=3. For the fixed contact area with a contact
length of h0=0.4nm, the condition defines a stable oscillation
approaching a constant nodal velocity at the steady state, as shown in Figure 6(a), giving rise to a direct penetration as displayed in Figure 6(b). However,
an increasing contact area of the form Ac=αh2 transforms the nodal-type velocity and
acceleration phase diagram into a quasiperiodic oscillation pattern, as shown
in Figure 6(c). This indicates the effective transitions of the bifurcation
parameters due to a decreasing antidamping coefficient μ as a result of an increasing contact area. The
velocity oscillates in a positive but declining range and the penetration is in
a forward-standstill type stick-slip behavior, as shown in Figure 6(d). At a
driving velocity of V=0.06nm/s,
the stick-slip oscillation reveals that each stroke is of the order of the
atomic lattice parameter a=0.364nm for Cu. This suggests that dislocations occur
at the Berger’s vector length, since the Berger’s vector |b→|=a for a perfect dislocation [27].Nanoscale oscillations withμχ(V)(cos(kτ/j)−2),μχ(V)=2k,k=−3, and j=3.
(a)
Velocity-acceleration phase diagram at a constant contact area(b)
Slip penetration at a constant contact area(c)
Velocity-acceleration phase diagram at a variable contact area(d)
Stick-slip penetration at a variable contact areaFor theCu nanoscale thin film interacting with the tip,
the above computation used the ideal shear strength τmax=2.5GPa [28, 29], which renders the
dislocation line energy Eb=2eV/nm,
based on Eb=Gb→2 for a perfect dislocation [27]. This energy level agrees
with established results for a dislocation and void initiation, which is
generally in the order of Eb=5eV per Berger’s vector [30]. The MD simulation of the Cu void growth and dislocation process indicates
that the activation energy is Eb=2.3~3.5eV/nm [31]. Note that the dislocation
energy through dissociation is found to be Ed=1.3~2.7eV from a higher to lower temperature range [32]. The dislocation nucleation
and parallel formation require an energy level that is less than that for a
perfect dislocation, as confirmed by the MD simulation [5, 6]. Similarly, twins and the
stacking faults formed
along with the dislocation nucleation, all of which are activated at a reduced
energy level compared to that for a perfect dislocation. Therefore, the
stick-slip behavior indicates the onset of plasticity due to dislocations and
other defects, including vacancies, twins, stacking faults, and so forth. Alternatively,
it can be said that the stick-slip oscillation is a reflection of the process
whereby defects initiate and nucleate to induce a periodic or an arbitrary fluctuation
of the energy release rate at the atomic scale due the tip’s probe of the thin
film.


## 5. Discussion

(1) The present study discusses a model that distinguishes the scale effect by local energy
functions, namely, the inelastic deformation and energy dissipation, as well as contact
parameters at the adhesion junction. The localized energy function at the microscale is associated with the
creep due to strain rate-dependent inelastic deformation and cracks, while the
energy function at the nanoscale is due to the dislocation-induced plasticity
and nucleation of other defects. The local contact parameter, energy
fluctuations, and inelastic deformation at the adhesion interface define a
collective bifurcation parameter to describe the stick-slip behavior at both micro-
and nanoscales. Although each parameter can be explicitly specified, it is the
collective parameterμ(χ(ϕ)+B) that defines the stick-slip behavior.(2) This model bridges the inelastic deformation mechanism
and local energy release with the tip’s oscillation patterns. By incorporating a
variable energy release along with the inelastic deformation, this model
reveals that the variations of the potential, the inelastic deformation
modulus, or the contact parameters, in a periodic or an arbitrary manner, are
the underlying mechanisms of the nonlinear stick-slip behavior. The
transitional bifurcation behavior justified the underpinning physics of adhesion
dynamics. By relating the stick-slip behavior of the tip to thin film’s deformation
mechanisms, interfacial energy release, defects, contact parameters, material
properties, and forcing conditions at the peripheral of the contact, this model
brings the analytical prediction closer to the experimental observation of the
stick-slip phenomena.(3) The stick-slip oscillation at both the nanoscale and
microscale suggests that the Griffith
concept for the energy release at the crack tip is adaptable to describing the
onset of plasticity by dislocation migration and nucleation at the nanoscale. Furthermore,
the periodic bifurcation condition adopted in the computation represents the
typical energy fluctuation at the atomic scale, since it is consistent with the
Peierls stress concept associated with the sliding dislocations of the atomic
lattice. Such variations also represent the energy barrier at the microscale
deformation influenced by the cyclic contact parameters and repeated energy
release through cracks. In addition to the adaptability of the model to the tip’s dynamics at
both micro- and nanoscale, the model allows for an arbitrary energy fluctuation for
complex interfacial contact behaviors.(4) Although this model describes the adhesion dynamics in a
scratching process, the stress field and the tip oscillation behavior are identical
to the friction dynamics observed in the nanoindentation, when a higher shear
factor characterizes the motion [6]. This is because the friction
force in the nanoindentation involves deformations in both normal and shear
modes. Essentially, this model is capable of describing the nonlinear dynamics
at the tip-solid interface of both the friction and adhesion dynamics.


## 6. Conclusion

This work develops a model to describe the
adhesion and friction dynamics at the interface of two contacting solids for
the tip and thin solid film interaction, such as those during an AFM adhesion
microscopy or nanoindentation process. The bifurcation conditions generated
from this model give a proper account of the interplay between the localized
energy functions, the elastic and inelastic deformations, the material
properties, and the contact parameters that collectively influence the tip’s
dynamics at the adhesion junction. The bifurcation conditions attribute the
stick-slip behavior of the tip to the Hopf bifurcations and the transitional
bifurcations associated with a periodic or an arbitrary potential variation,
defect formations, and inelastic deformations at the contact interface. The
model reveals that these interactive parameters give rise to the nonlinear
oscillation in penetration patterns. The collective parameters determine the
adhesion and friction dynamics at the periphery of the contact of both micro- and
nanoscales.



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*Source: 742569-2009-01-27.xml*