Facolta di Scienze e Tecnologie

Laurea Triennale in Fisica

Nonlinear friction mechanisms

in a 1D model

Relatore: Prof. Nicola Manini

Gabriele Tasca

Matricola n◦ 867302

A.A. 2017/2018

Codice PACS: 68.35.Af

Nonlinear friction mechanisms

in a 1D model

Gabriele Tasca

Dipartimento di Fisica, Universita degli Studi di Milano,

Via Celoria 16, 20133 Milano, Italia

October 11, 2018

Abstract

We briefly review the recent computational and analytical advances in

the understanding of the weak interaction limit of the dissipation mecha-

nisms in a wear-free one-dimensional model for nanoscopic friction.

Then, by molecular-dynamics simulations, we explore the differences

that arise when the interaction is not sufficiently weak and therefore the

analytic weak-coupling formulas no longer hold.

Advisor: Prof. Nicola Manini

Contents

1 Introduction 3

2 The model 4

3 Linear Response Theory 4

3.1 The Weak Coupling Limit . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Friction and Energy Loss . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Linearized Time Evolution . . . . . . . . . . . . . . . . . . . . . . 6

3.4 Fourier Representation and Fluctuation-Dissipation . . . . . . . . 8

3.5 Structure Factor for the Harmonic Chain . . . . . . . . . . . . . . 9

3.6 Friction Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Simulations 14

4.1 The deceleration method . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 The fixed-velocity method . . . . . . . . . . . . . . . . . . . . . . 16

5 Strong-Coupling Effects 17

5.1 Phonon Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Discussion and Conclusion 23

References 30

2

1 Introduction

Friction phenomena affect the dynamics of moving bodies on many very different

scales, and for this reason, reliable models for the prediction and the control of the

features of frictional interactions have great practical importance in many scien-

tific and technological fields. The phenomenological features of sliding friction on

the macroscopic scale, in particular, have been studied from as early as the 15th

century by Leonardo da Vinci, and greater insight on its microscopic causes was

achieved in the early 20th century with the development of the Frenkel-Kontorova

and Prandtl-Tomlinson models, and the pioneering experimental and conceptual

work by F. P. Bowden and D. Tabor [1, 2].

These models still provide a reasonable representation of friction that cap-

tures many of its essential characteristics, but make no attempt to give a quan-

titative description of the thermalization and dissipation of the kinetic energy of

ordered motion. In more recent years the study of the microscopic and nanoscopic

features of friction attracted more attention due to technological advances in the

miniaturization of electronic devices and scientific instruments, such as the AFM

microscope. At the same time, the increase in computing power made possi-

ble numerical simulation of molecular-dynamics models in which friction effects

arise naturally from the energy transfer from ordered motion to internal phononic

modes. However, these models usually require an artificial damping term for the

internal dynamics, to prevent the internal degrees of freedom from overheating.

The simple model studied in this work represents the sliding friction of a

point particle on a linear harmonic chain. Although numerical simulation of this

model still needs viscous damping for the chain atoms, the damping does not

affect the energy of the sliding particle directly. This allows the quantitative

study of the kinetic-energy dissipation rate, and thus of the friction force.

Recent work [3] discovered unexpected non-monotonic features of the depen-

dence of the friction force on the slider velocity in the weak-coupling regime. An

analytical study [4] of the model found an approximate formula that agrees quan-

titatively with the simulation results, provided that the interaction is sufficiently

weak. However, when the coupling becomes stronger, significant deviations from

the weak-coupling formula appear. In this work, after briefly reviewing the deriva-

tion of the formula, we use numerical simulations to explore the main features of

these deviations.

3

2 The model

Our model for sliding friction is the simple 1D model introduced by Apostoli et

al. [3]. It consists of N pointlike particles forming a linear harmonic chain, plus

another pointlike particle that acts as a “slider”. The atoms in the chain are

constrained to move in its horizontal direction, and the slider moves parallel to

the chain along a line at a fixed distance d. The atoms in the chain have equal

mass m and positions xj, and interact harmonically with their first neighbours

in such a way that the equilibrium positions of the atoms are equispaced with

spacing constant a. The slider has mass mSL and position xSL, and interacts with

all atoms in the chain with a Lennard-Jones potential, transferring its kinetic

energy to the phononic excitations of the chain, thus experiencing a net friction

force. Periodic boundary conditions are enforced, identifying the (N+1)-th atom

with the first, so that the slider potential shifts smoothly from the tail of the chain

to the head. In particular, the total slider-chain interaction is given by

Vtot =N∑

n=1

VLJ(rj), (1)

where

rj =√d2 + (xSL − xj)2 (2)

is the distance between the slider and the j-th atom, or its periodic replica nearest

to xSL, and VLJ is the Lennard-Jones potential

VLJ = ε

[(σ

Rj

)12

− 2

(σ

Rj

)6]

(3)

with equilibrium distance σ. The harmonic chain potential is given by

Vharm(x1, . . . , xN) =1

2K

N−1∑j=1

(xj − xj+1 − a)2, (4)

where K is the spring constant characterizing the interaction.

The physical quantities that characterize the harmonic chain define natural

units for many physical quantities of interest, listed in Table 1. In many of

the graphs in the following sections, the values of the quantities represented are

referred to these units.

3 Linear Response Theory

Previous work by E. Panizon et al. [4] found an analytical expression for the net

friction force dependence on the slider speed in the weak-coupling limit. That

4

Physical quantity Natural units

length a

mass m

spring constant K

energy Ka2

time (m/K)12

speed a(K/m)12 ≡ vs

force Ka

Table 1: Natural units for some physical quantities in a harmonic chain.

vs is the speed of acoustic (sound) vibrations of the chain.

analysis relies on the application of Linear Response Theory (LRT) to the har-

monic chain when perturbed by a weak external potential. We briefly review

the fundamentals of LRT and their application, highlighting the approximations

made.

3.1 The Weak Coupling Limit

If the slider-chain interaction is weak, we can imagine that each individual colli-

sion with an atom in the chain has a small, almost negligible effect on the slider’s

total kinetic energy, so that the measurement of the friction force can be made

in a time interval during which the slider speed is almost constant. Of course

this approximation, analogous to the Born approcimation of scattering theory,

cannot be made for all velocities, no matter how weak the coupling is: it is only

applicable to the region in which the typical interaction energy V0 is much smaller

that the slider kinetic energy 12mSLv

2SL. Indeed, as our macroscopic experience of

friction suggests, at low velocities friction is dominated by nonlinear phenomena

such as stick-slip and a nontrivial transition from static to dynamic friction [6].

With this constant-speed approximation, we can forget about the slider as a

point particle altogether, and model its coupling with the chain with the action

of a time-dependent potential that captures the motion of the slider through a

x − vSLt dependence. This way, the j-th particle in the chain is subject to an

external potential Vext(xj, t) = VLJ(√

(xj − vSLt)2 + d2), where VLJ(rj) is the

usual Lennard-Jones interaction. The total Hamiltonian of the chain can thus be

written as

Htot(t) = Hchain +

∫ ∞

−∞Vext(x, t)n(x)dx, (5)

where Hchain is the Hamiltonian of the unperturbed harmonic chain, and n(x) =

5

∑Nj=1 δ(x−xj) is the chain atom density. This formulation is ideal to apply LRT.

To be consistent with the usual LRT formalism, we use a quantum description

of the system: this is a technical choice which should not affect the results too

strongly in a system of harmonic oscillators.

3.2 Friction and Energy Loss

Though the slider itself disappeared from our picture, we know that it exerts a

force on the chain equal and opposite to the friction force that the chain exerts on

it, and that the work done by that force per unit time is related to the variations

of the internal energy of the chain by

FvSL =∂

∂tEchain. (6)

Applying the Hellmann-Feynman formula with t as a parameter to the full Hamil-

tonian (5), which depends explicitly on time through the Vext term, we find

∂

∂tEtot ≡

∂

∂t⟨Htot⟩ =

⟨∫ ∞

−∞

∂Vext(x, t)

∂tn(x)dx

⟩=

∫ ∞

−∞

∂Vext(x, t)

∂t⟨n(x)⟩ dx,

(7)

since Vext is external and remains constant through quantum and thermal

averages.

Expanding the internal and external terms of the full Hamiltonian (5) in the

time derivative,

∂

∂t⟨Htot⟩ =

∂

∂tEchain+

∫ ∞

−∞

∂Vext(x, t)

∂t⟨n(x)⟩ dx+

∫ ∞

−∞Vext(x, t)

∂ ⟨n(x)⟩∂t

dx. (8)

and substituting the previous result for the total energy, leads to the following

expression for the time derivative of the internal energy of the chain:

dEchain

dt=

∫ ∞

−∞Vext(x, t)

∂ ⟨n(x)⟩∂t

dx. (9)

3.3 Linearized Time Evolution

We need to calculate the term ⟨n(x)⟩ and its time dependence, i. e. ⟨n(x)⟩t. Thisis done through the LRT formalism. In LRT, the time-evolution U(t, t0) derived

from the Hamiltonian written in the form

Htot = H0 + ϕ(t)B, (10)

where ϕ(t) is a generalized force and B is the operator to which it couples, like

Vext(x, t) and n(x) in Eq. (5), is separated in two parts, one due to the internal

6

energy of the system and one due to the perturbation. The Schrodinger-like

equation for the perturbative part is then solved to first order, leading to the

expression

U(t, t0) ≃ e−iℏH0t

(I− i

ℏ

∫ t

t0

ϕ(t)B(t′ − t0)dt′). (11)

In a state characterized by a density matrix∑

n Pn

⏐⏐ψn

⟩⟨ψn

⏐⏐, this result and its

hermitian conjugate can be used to compute the expectation value⟨A⟩of another

operator A from the definition. One eventually finds

⟨A⟩t=

∑n

Pn

⟨ψn(t)

⏐⏐A⏐⏐ψn(t)⟩=

⟨A⟩0− i

ℏ

∫ t

t0

ϕ(t′)⟨[A(t), B(t′)]

⟩0dt′, (12)

where A(t) = eiℏH0tAe−

iℏH0t and B(t) = e

iℏH0tBe−

iℏH0t are Heisenberg represen-

tations of A and B at time t and t′, and⟨. . .

⟩0represents the average over

the equilibrium state. Accordingly, a linear response function (or susceptibility)

χAB(t− t′) is defined, as

χAB(t− t′) = − i

ℏθ(t− t′)

⟨[A(t), B(t′)]

⟩0, (13)

so that we can write

⟨A⟩t − ⟨A⟩0 =∫ ∞

−∞χAB(t− t′)ϕ(t′)dt′, (14)

using the additional assumption that the strength of the perturbation tends to

zero for t → −∞, so that the system can be considered to lie in unperturbed

equilibrium in that limit. The time derivative is then computed as

∂

∂t⟨A⟩t =

∫ ∞

−∞

∂χAB

∂t(t− t′)ϕ(t′)dt′. (15)

Some care is needed when substituting the x-dependent Vext(x, t) and n(x) of our

problem, because n(x, t) and n(x′, t′) represent operators distinct from each other

for values of x different from x′. Taking this into account, the calculation step

leading to Eq. (12) now yields

∂

∂t⟨n(x)⟩t =

∫ ∞

−∞dx′

∫ ∞

−∞dt′

∂χnn

∂t(x, x′, t− t′)Vext(x

′, t′), (16)

with

χnn(x, x′, t− t′) = − i

ℏθ(t− t′)

⟨[n(x, t), n(x′, t′)]

⟩0. (17)

7

This can be substituted directly into Eq. (9) to find

∂

∂tEchain(t) =

∫ ∞

−∞dx

∫ ∞

−∞dx′

∫ ∞

−∞dt′ Vext(x, t− t′)

∂χnn(x, x′, t− t′)

∂tVext(x

′, t′).

(18)

From this result and from Eq. (6), the average friction force F can be ob-

tained by averaging the dissipated power over one time period τ = a/vSL, namely

the time interval in which the slider advances by exactly one lattice spacing a:

F = − 1

vSL

∫ τ

0

∂

∂tEchain(t)dt. (19)

The above steps introduce two approximations. The first is the linearization

of the perturbative part of the time evolution operator: this is the heart of the

LRT method and its accuracy is directly linked to the weakness of the perturbing

potential. The less obvious one is related to the state of the system previous

to the perturbation: if we take the LRT formalism literally, the chain started in

an unperturbed equilibrium state at t → −∞, and shifted its state to the one

described by LRT in an infitely slow transient lasting from t′ = −∞ to t′ = t.

Although explicit references to this equilibrium state were dropped out when

taking the time derivative, it remains in the definition of χnn (17) as the state

over which the ensemble average is performed.

3.4 Fourier Representation and Fluctuation-Dissipation

To take advantage of the form of Eq. (18), we write the integrated factors in

Fourier representation, so that only the infinite-wavelength components survive

the integrations from −∞ to ∞.

Because of the discrete translational invariance of the crystal, the Fourier

representation of the density-density linear response function χnn(x, x′, t − t′)

takes the form [4]

χnn(x, x′, t− t′) =

∑G

∫ ∞

−∞

dQ

2π

∫ ∞

−∞

dω

2πe−iω(t−t′)eiQxχnn(Q,Q+G,ω)ei(Q+G)x′

,

(20)

where χnn is the Fourier transform of χnn over space and time, and G = n2πaare

the reciprocal-lattice “vectors”.

Since the potential Vext depends on space and time only through x − vSLt,

its Fourier representation is

Vext(x, t) =

∫ ∞

−∞

dq

2πeiq(x−vSLt)Vext(q) (21)

8

where Vext(q) is the Fourier transform of the potential that acts on the chain

particles at t = 0:

Vext(r, t = 0) = VLJ

(√x2 + d2

). (22)

When substituting the Fourier representation in Eq. (18) and combining the

exponential factors, as anticipated, the integrations over x, x′ and t′ yield Dirac

delta distributions for q + Q, q′ + Q + G and then ω − (Q + G)vSL. Similarly,

when calculating F from Eq, (19), the time integration over the period a/vSLleads to a Dirac delta for G from the resulting eiGvSLt factor, since for all other

values of G the period of the exponential function is exactly commensurate to

the integration domain, so that it integrates to zero. The result is

F (vSL) = − 1

vSL

∫ ∞

−∞

dQ

2π(−ivSLQ)χnn(Q,Q,QvSL)

⏐⏐⏐Vext(Q)⏐⏐⏐2 , (23)

where vSLQ is the frequency that survives the t′ integration.

By using the standard symmetry properties of Fourier-transformed response

functions, one can show that only the imaginary part of χnn contributes. The

imaginary part can then be related to the equilibrium properties of the system

by use of the Fluctuation-Dissipation Theorem [7]:

Imχnn(Q,Q, ω) = −1

2(1− e−βℏω)Snn(Q,Q, ω). (24)

Here which Im denotes the imaginary part, and Snn(Q,Q, ω) is the dynamical

structure factor of the system [7]

Snn(Q,Q, ω) =1

ℏlimV→∞

1

V

∫V

eiωt ⟨nQ(t)n−Q(0)⟩ dt, (25)

the time Fourier transform of the correlation function between the spatial Fourier

transform of the density operator and its Hermitian conjugate, nQ(t) and n−Q(t).

In our 1D model, the “volume” is intended to be the length Na of the chain.

Using these results, Eq. (23) takes the form

F (vSL) = −1

2

∫ ∞

−∞

dQ

2πQ(1− e−βℏvSLQ)Snn(Q,Q, vSLQ)

⏐⏐⏐Vext(Q)⏐⏐⏐2 . (26)

3.5 Structure Factor for the Harmonic Chain

To find an expression for the structure factor (25), we start from the density

operator n(x, t) =N∑j=1

δ(x− xj(t)). Writing xj(t) = ja+ uj(t), where uj(t) is the

9

operator for the displacement of the j-th atom from its equilibrium position, the

Fourier transform of the density takes the form

nQ(t) =

∫ ∞

−∞e−iQx

N∑j=1

δ(x− xj(t))dx =N∑j=1

e−iQxj(t) =N∑j=1

e−iQjae−iQuj(t). (27)

Substituting in the structure factor and expanding, we find

Snn(Q,Q, ω) =1

ℏlim

N→∞

1

Na

∫V

eiωt

⟨N∑j=1

e−iQjae−iQuj(t)

N∑k=1

e+iQkae+iQuk(0)

⟩dt

=1

ℏlim

N→∞

1

Na

∫V

eiωtN∑

j,k=1

⟨e−iQjae−iQuj(t) e+iQkae+iQuk(0)

⟩dt.

(28)

Because of the lattice translational invariance of the equilibrium state of

system, we have ⟨f(j)f(k)⟩ = ⟨f(j − k)f(0)⟩ for any function depending on the

lattice index j. The N2 j, k sums in the above equation are divided into N equal

sums for each j′ ≡ j − k, itself spanning N values. We obtain

Snn(Q,Q, ω) =1

ℏlim

N→∞

1

Na

∫V

eiωtNN∑

j′=1

e−iQj′a⟨e−iQu′

j(t) e+iQu0(0)⟩dt. (29)

To expand the averages present in this expression, we resort to some results

from the standard theory of the harmonic crystal. First, we use the identity [8]⟨eAeB

⟩= e

12⟨A2⟩e

12⟨B2⟩e⟨AB⟩, (30)

valid for any operators A,B linear in the displacement and momentum operators

of the atoms in a harmonic crystal, to write⟨e−iQuj(t) e+iQu0(0)

⟩= e

12

⟨−Q2uj(t) uj(t)

⟩e

⟨Q2uj(t) u0(0)

⟩e

12

⟨−Q2u0(0) u0(0)

⟩. (31)

The first and the last terms are equal because of space and time translational

invariance. Eq. (31) thus simplifies to

e−Q2⟨uj(t) uj(t)

⟩eQ

2⟨uj(t) u0(0)

⟩. (32)

Then, we use the representation of uj(t) in terms of the standard phononic cre-

ation and destruction operators bk and b†k, using Heisenberg time evolution:

uj(t) =1√N

∑k =0,∈BZ

eikja

√ℏ

2mω(k)(e−iωktbk + eiωktb†−k), (33)

10

where ωk is the angular frequency that corresponds to k through the dispersion

relation

ωk = 2

√K

m

⏐⏐⏐⏐ sin Ka2⏐⏐⏐⏐ . (34)

As discussed at the end of Sect. 3.3, the choice of the equilibrium ensem-

ble where the averages are calculated is a significant approximation step when

applying LRT predictions to a real system. However it is worth noting that this

is the first time we are forced to select it explicitly. We choose the canonical

ensemble: this is coherent with the thermodynamic limit of an infinitely long

chain, where perturbations induced by the slider (which usually excites only a

few specific phononic modes) affect the overall thermodynamics of the chain in a

negligible manner. However this may result in deviations when comparing with

the results of finite-size simulations.

Since the canonical ensemble is composed of eigenstates of the Hamiltonian,

on which the action of bk and b†k is known, the expansion of the thermal average

is now straightforward. The calculation yields [5]

⟨uj(t)uj(t)⟩ =1

N

∑k =0,∈BZ

ℏ2mωk

(2nBk + 1)

⟨uj(t)u0(0)⟩ =1

N

∑k =0,∈BZ

ℏ2mωk

[nBkeiωkt + (nBk + 1)eiωkt]eikja,

(35)

where

nBk =1

eβℏωk − 1(36)

is the Bose distribution factor.

Substituting these expressions in Eq. (29), and taking the thermodynamic

limit N → ∞, which leads to dense k points in the Brillouin zone, we find [4]

Snn(Q,Q, ω) =1

ℏa

∫ +∞

−∞eiωt

+∞∑j=−∞

e−iQaje−Q2Φj(t,β)dt

Φj(t, β) = a

∫ +πa

−πa

dk

2π

ℏ2mωk

×[(2nBk + 1

)−(nBk e

iωkt + (nBk + 1) e−iωkt)eikaj

].

(37)

To write this expression in a practically usable form, we approximate the real

exponential with its first-order expansion:

e−Q2Φj(t,β) ≃ 1−Q2Φj(t, β). (38)

11

This approximation corresponds to considering only one-phonon processes [8].

For detailed study of its applicability to our model, see [5, 4]. After substituting

Eq. (38) into Φ(t, β) in (37), the terms in Φ that do not contain an explicit time

dependence do not survive the time integration, for any ω = 0. For ω = 0, one

finds elastic contributions, which vanish in 1D [4].

The approximate result for the inelastic (ω = 0) structure factor is thus

Snn(Q,Q, ω) ≃ Q2

∫ +∞

−∞dt

∫ +πa

−πa

dk

2π

nBk + 1

2mωk

ei(ω−ωk)t

+∞∑j=−∞

ei(k−Q)ja. (39)

The sum over j is the Fourier series representation of the periodic delta distribu-

tion for the reciprocal lattice points in k space:

+∞∑j=−∞

ei(k−Q)ja =2π

a

∑G

δ(k −Q−G), (40)

with G = 2πna

and n ∈ N. This sets k = Q + G, but since k now appears only

through the dispersion relation ωk, and since G is a reciprocal lattice vector,

we can refer back to the first Brillouin Zone and set k = Q. Finally, the time

integration gives a delta for ω = ωk: the result is

Snn(Q,Q, ω) ≃ πQ2

maωQ

(nBQ + 1) δ(ω − ωQ) =πQ2

maωQ

δ(ω − ωQ)

1− e−βℏωQ. (41)

3.6 Friction Force

We can now use Eq. (26) and Eq. (41) to obtain a final approximate expression

for the friction force. We obtain

F (vSL) ≃ −1

2

∫ ∞

−∞

dQ

2πQ (1−e−βℏvSLQ)

πQ2

maωQ

1

1− e−βℏωQ

⏐⏐⏐Vext(Q)⏐⏐⏐2 δ(vSLQ−ωQ).

(42)

The two Bose distribution factors coming from the Fluctuation-Dissipation The-

orem and from the thermal averages in the structure factor itself cancel out,

leading to a temperature-independent result. It should be noted that this would

not have happened if we kept terms beside first order in the expansion of the

exponential in Eq. (37).

We can then eliminate the final integral using the delta function property

δ(f(Q)) =∑Qj

δ(Q−Qj)

f ′(Qj), (43)

where the Qj are the solutions of f(Qj) = 0. In our case, f(Q) = vSLQ−ωQ, and

f ′(Q) = vSL− dwdQ

≡ vSL−v(Q), where dwdQ

≡ v(Q) is computed from the dispersion

12

0.2 0.4 0.6 0.8 1.0vSL [vs]

100

101

102

F[Ka]

Figure 1: LRT friction F (vSL), calculated from Eq. (46) with the the

slider-chain potential of Eq. (3) with ε = Ka2, σ = 0.5 a, slider-chain

distance d = 0.475 a, and damping rate γ = 0.1√K/m.

relation (34), and gives the group velocity of the Q-wavelength phonons in the

chain. Thus, the final result is

F (vSL) =1

2mavSL

∑j

Q2j⏐⏐vSL − v(Qj)

⏐⏐ ⏐⏐⏐Vext(Q)⏐⏐⏐2 , (44)

where the sum extends on the solutions Qj of the equation

vSLQ = ωQ. (45)

Eq. (44) is the final result for the friction force, considering an infinite

harmonic chain. However, when comparing the theory with finite-size numerical

simulation results, it is practically necessary to include a damping force Fvisc,j =

−mγxj acting on each atom j. In principle, all results of Section 3.5 are calculated

from the conservative linear chain Hamiltonian, and are not applicable to the

dissipative case. However, if the damping rate γ is sufficiently small compared to

the typical frequency√K/m, it is a reasonable approximation to assume that the

damping only contributes to the correlation functions through a decaying time

exponential term. This has the effect of replacing the delta distributions in Eq.

(42) with Lorentzian curves [4]. The result is

F (vSL) =1

2mavSL

∫ ∞

−∞

dQ

2πQ2

⏐⏐⏐Vext(Q)⏐⏐⏐2 γ/2

(vSLQ− ωQ)2 + (γ/2)2. (46)

13

Figure 1 reports a numeric evaluation of this final formula.

4 Simulations

As was done in Refs. [6, 3], we use classical molecular-dynamics simulations to

explore the friction phenomenology of our system. The simulation are performed

by solving numerically the Newton-Hamilton equations of motion for all particles

in the system, using a fourth-order adaptive Runge-Kutta-Fehlberg algorithm,

and keeping track of the positions and velocities of all particles and of the forces

acting on them as needed. To make numerical simulation possible, some precau-

tions have to be taken:

1. The usual Lennard-Jones potential is replaced by a truncated version:

VLJ(Rj) =

{V ThLJ (Rj)− V Th

LJ (Rcutoff ), if Rj < Rcutoff

0 otherwise(47)

where V ThLJ is the standard form (Eq. (3))

V ThLJ = ε

[(σ

Rj

)12

− 2

(σ

Rj

)6]. (48)

The truncation is necessary when using periodic boundary conditions, so

that the potential has a finite range and the number of interactions to be

computed is not infinite.

2. As anticipated, the atoms in the chain are also subject to a weak dissipative

force

Fvisc,j = −mγxj, (49)

where γ is the damping rate, chosen to be small compared to the typical

phonon frequencies. Once again the introduction of the unphysical viscous

force, somewhat problematic for an investigation of friction, is made neces-

sary by the use of periodic boundary conditions, to avoid the possibility that

the phononic waves generated by the slider-chain interaction may reach the

end of the supercell and travel all the way back to the position of the slider,

influencing its dynamics. The angular frequency ω and the wave vector k

of the phononic waves in a linear harmonic chain with no damping follow

the dispersion relation (34). In the long-wavelength limit

ω(k) ≃ a

√K

m|k| , (50)

14

one obtains the speed of sound

vs =dw

dk= a

√K

m. (51)

Thus, without the damping force, the excited waves will eventually return

to their starting point in a time of the order of Na/vs = N√m/K. This

represents a severe limitation on the time intervals that the simulations

could reliably explore, because it can only be increased by increasing the

number of atoms N , which comes at a great computational cost.

In the simulations of this work, unless otherwise specified, the parameters

have the following values:

– Number of atoms in the chain: N = 500,

– Lennard-Jones potential equilibrium distance: σ = 0.5a,

– Distance between slider and chain: d = 0.475a,

– Damping rate: γ = 0.1√K/m .

To extract the F (vSL) dependence from the simulations, Giusti [9] and Apos-

toli [6] adopted two different methods, described in the following subsections.

4.1 The deceleration method

In the deceleration method, the slider is initially kept moving at a fixed starting

speed, in order to bring the chain to a steady state. After this initial time the

slider is left free to decelerate until it stops. If the deceleration is slow enough, it

is then possible to consider time intervals in which the variation of vSL is small

and approximately linear in time, but that still cover several a/vSL time intervals,

i.e. many slider-chain particle collisions. In each interval one then can evaluate

the average acceleration ⟨aSL⟩ by calculating the slope of vSL(t), and then the

average friction force as −⟨aSL⟩ /mSL. Over the same interval one evaluates the

average velocity ⟨vSL⟩, thus obtaining one data point for the F (vSL) dependence.

This method is a quite realistic representation of friction, as the direct effect

of a friction force is the deceleration of moving bodies. In practice this method

introduces a dependence on the slider mass mSL, which did not enter the picture

in the LRT analysis. In principle, since different masses lead to entirely different

trajectories for the slider, the results obtained for F (vSL) can be different, and

thus this is an extra parameter that needs to be controlled. In practice, however,

15

there is often very little freedom when adjusting its value, if one is to obtain a

simulation which covers a wide range of velocities and is at the same time slow

enough in the sense of the last paragraph, but fast enough to be computationally

tractable in reasonable times.

It should also be noted that in all simulations the center of mass of the chain

is accelerated forward due to the interaction with the slider, since it experiences

a force equal and opposite to the friction force on the slider. The damping force

(49) prevents the chain from accelerating infinitely, so that eventually it drifts

with a velocity vCM . Since LRT is evaluated in the reference frame where the

chain is at rest, to better compare theory and simulation we will report the speed

of the slider relative to the chain:

vrelSL ≡ vabsSL − ⟨vCM⟩ , (52)

where vabsSL is the absolute speed of the slider in the “laboratory” rest frame of the

simulation, and the average ⟨vCM⟩ of the chain is taken on the same time interval

relevant to the calculation at hand. From now on, we take vSL to stand for vrelSL.

Although it is in principle possible to use the istantaneous values of vCM

recorded from the simulation, the average can be calculated in a faster and more

accurate way from the values of xCM . If [t, t + τ ] is the interval over which the

average is computed,

⟨vCM⟩ = 1

τ

∫ t+τ

t

vCM(t′)dt′ =1

τ

(xCM(t+ τ)− xCM(t)

). (53)

4.2 The fixed-velocity method

In the fixed-velocity method, the slider simply moves at fixed speed vabsSL , and

the total force acting on it is recorded at every simulation step. After an initial

transient, the force starts following a periodic pattern, usually with the known

a/vSL period. The average value of F over an integer number of periods, together

with vSL, is added as a data point to F (vSL).

Rather than using the recorded values of F , the average friction force is con-

veniently computed from the values of vCM , similarly to Eq. (53). By definition,

the average force on the chain center of mass, calculated over a time interval

[t, t+ τ ], is given by

⟨FCM⟩ = 1

τ

∫ t+τ

t

FCM(t′)dt′ =1

τ

∫ t+τ

t

mtotvCM(t′)dt′ =mtot

τ

(vCM(t+τ)−vCM(t)

),

(54)

with mtot =∑j

mj = Nm. In the steady state, integrating over an integer

number of periods τ of the pseudo-periodic motion, this average force vanishes,

16

as vCM(t + τ) = vCM(t). This on-average vanishing force can be seen as the

resultant of two contributions:

FCM = Fopp + Fvisc. (55)

Here Fopp is a force equal and opposite to the friction force F on the slider,

resulting from Newton’s third law, and Fvisc is the sum of the unphysical viscous

forces Fvisc,j = −mγvj acting on all the chain atoms:

Fvisc = −∑j

mγvj. (56)

Its average value is

⟨Fvisc⟩ =

⟨∑j

−mγvj

⟩= −mγ

⟨∑j

vj

⟩= −Nmγ ⟨vCM⟩ . (57)

This quantity coincides with the average friction force experienced by the slider,

which, following Eq. (53), can be expressed as:

F = −⟨Fvisc⟩ =Nmγ

τ(xCM(t+ τ)− xCM(t)). (58)

The center of mass of the chain is also dragged forward in the fixed-velocity

method. Therefore the same correction as described in Eq. (52) is applied here

as well.

Unlike the deceleration method, in which the slider’s deceleration naturally

explores the continuous range of velocities from its starting speed to zero, in

the fixed-velocity method a different simulation is needed for each vSL value.

This can make the fixed-velocity method computationally more intensive than

the deceleration method, if high resolution over wide velocity ranges is needed.

However, it is more versatile for exploring specific narrow velocity ranges, and

avoids the problem of the arbitrariness of the mSL parameter discussed in the

last paragraph.

In particular, the fixed-speed method can be considered as the mSL → ∞limit of the deceleration method. However one should consider that the infinite-

mass picture is intrinsically inadequate for studying phenomena such as stick-slip

and static friction, in which vSL needs to be allowed to change in time.

5 Strong-Coupling Effects

In this section we use the simulation protocols described above to explore how the

F (vSL) dependence deviates from the LRT formula when the coupling becomes

17

0.0 0.2 0.4 0.6 0.8 1.0vSL [vs]

100

101

F ε2[1 Ka

3]

fixed-velocitydeceleration

Figure 2: F (vSL) calculated with the two methods, for a fairly high

coupling ε = 0.10 Ka2. For the deceleration method curve, a mass

mSL = 10000 m was used. The other parameters have the standard

values σ = 0.5 a, d = 0.475 a.

stronger and the approximations no longer hold. The most direct way of changing

the coupling strength is through the ε parameter of the Lennard-Jones potential.

LRT predicts that the friction force F depends on the slider velocity with a

fixed shape independent of ε, and it simply rescales as ε2. In the following, we

will therefore report the quantity F/ε2 rather than F , thus making the changes

in shape more evident.

Figure 2 shows the results of two simulations performed respectively with

the constant-speed and the deceleration methods, but with the same parameters.

The two methods give very similar results for all the used values of ε, provided

that the slider mass used for the deceleration method is chosen high enough to

obtain a sufficiently slow deceleration. To avoid the additional complication and

arbitrariness brought in by a finite slider mass mSL, as discussed in Sect. 4.1, we

adopt the fixed-velocity method for all the following analysis.

We come now to the main results of this thesis: the numerically simulated

variation of the F (vSL) dependence for different values of ε. Figures 3 , 4, and 5

represent the F (vSL)/ε2 curves with ε ranging from 0.005 to 0.14 Ka2, compared

with the LRT curve of Eq. (46).

Remarkably, the overall F ∝ ε2 dependence predicted by LRT remains ap-

proximately and qualitatively correct across the examined speed range, as the

18

0.2 0.4 0.6 0.8 1.0vSL [vs]

100

101

102

F ε2[1 Ka

3]

ε = 0.05 Ka2

ε = 0.07 Ka2ε = 0.08 Ka2

LRT

Figure 3: Comparison of the simulated (ε = 0.05 Ka2, ε = 0.07 Ka2 and

ε = 0.08Ka2) F (vSL)/ε2 friction curves with the LRT formula Eq. (46).

The other parameters have the standard values σ = 0.5 a, d = 0.475 a.

simulation curves for F/ε2 remain “close” to the analytical profile. However, the

shape of the curve varies significantly with increasing ε. As shown in Ref. [4],

for ε < 0.005 Ka2 simulations and LRT are essentially indistiguishable. Up to

ε ≃ 0.05 Ka2 the difference is quite small, and the qualitative features of the

curve remain similar. The quantitative deviations, however, are not trivial: near

the first two resonant peaks and in the ultrasonic region F/ε2 is lower than in

the LRT curve, while it becomes larger around the third peak and for most of

the non-resonant velocities in the 0.3− 0.9vs range. In the 0.75− 0.95vs region,

in which the slider travels nearly at the speed of sound vs and stays close to

the wavefront of the excitation that it generates, the friction force increases sig-

nificantly. At about vSL ≃ vs, the friction force drops sharply, as the excited

wavefront lags behind the slider.

The resonant peaks of the LRT curve remain distinctly visible up to ε ≃0.09 Ka2. The peaks evolve into discontinuities of the curve for increasing ε.

In the vSL ≲ 0.15 vs region, to the left of the last LRT peak, more complex

qualitative features appear, as shown is Figure 6.

Another interesting feature emerging from this analysis is a new peak not

predicted by LRT in the vSL = 0.3−0.4 vs region. The presence of this peak is also

confirmed by the deceleration method, as visible in Figure 2. For ε ≃ 0.08Ka2,

this peak is pictured in detail in Figure 7: interestingly, its shape is not Lorentzian

19

0.2 0.4 0.6 0.8 1.0vSL [vs]

100

101

102

F ε2[1 Ka

3]

ε = 0.09 Ka2

ε = 0.10 Ka2

ε = 0.11 Ka2

LRT

Figure 4: Same as Figure 3, but for ε = 0.09 Ka2, ε = 0.10Ka2, and

ε = 0.11 Ka2.

as it is for the LRT resonance peaks.

Figure 8 depicts the variation in F (vSL)/ε2 as a function of ε, computed for

a few fixed values of vSL. Interestingly, for many values of vSL this variation is

not monotonic.

5.1 Phonon Analysis

As shown in Ref. [3], the phonon modes of the chain excited by the interac-

tion with the slider in the weak-coupling regime are the ones that satisfy the

momentum-conservation condition

vSLQ = ω(Q), (59)

where ω(Q) is the given by the chain dispersion relation, Eq. (34). In the LRT

picture, this is clear from Eq. (44) and from the form of the one-phonon structure

factor (41), where the delta distribution ensures that only the phonon modes that

solve Eq. (59) contribute to the friction.

In the following, we use the methods of Ref. [3] to explore the nonlinear

deviations in the phonon excitation patterns that occur in the non-weak-coupling

regime. To analyze the excitation of the phononic modes, we compute the spatial

20

0.0 0.2 0.4 0.6 0.8 1.0vSL [vs]

100

101

102

F ε2[1 Ka

3]

ε = 0.12 Ka2

ε = 0.13 Ka2

ε = 0.14 Ka2

LRT

Figure 5: Same as Figure 3, but for ε = 0.12 Ka2, ε = 0.13Ka2, and

ε = 0.14 Ka2.

0.04 0.06 0.08 0.10 0.12 0.14 0.16vSL [vs]

3.5

4.0

4.5

5.0

5.5

6.0

F ε2[1 Ka

3]

Figure 6: Detail view of the low-speed portion of the simulated

F (vSL)/ε2 curve, for ε = 0.12 Ka2. In this region several small non-

Lorentzian peaks absent in the LRT curve appear. The F (vSL)/ε2 axis

is in linear scale.

21

0.325 0.330 0.335 0.340 0.345vSL [vs]

2.5

3.0

3.5

4.0

4.5

5.0

F ε2[1 Ka

3]

Figure 7: Close-up of the peak not predicted by LRT, for ε = 0.08Ka2.

As is Figure 6, the F (vSL)/ε2 axis is in linear scale.

Fourier transform vj of the chain atom velocities at a given moment,

vj(k, t) =N∑j=1

eiktvj(t), (60)

and we take its squared modulus |vj(k, t)|2. Since this quantity varies periodically

with time, usually with period a/vSL, we then take its average over a period.

Figure 9 illustrates how the the solutions of Eq. (59), brought back into

the first Brillouin zone, correctly identify the positions of the phonon excitation

peaks for a very small ε = 0.005 Ka2.

Remarkably, applying this method to larger values of ε, significant multi-

phonon deviations from the one-phonon rule of Eq. (59) only appear in the

0.3 vs ≲ vSL ≲ 0.4 vs region, where the nonlinear peak mentioned in the previous

subsection is located.

Figure 10 shows how, for a velocity outside the peak region, the excited

phonon modes remain the same as those predicted by LRT, even for moderately

high values of ε. However, the relative amplitudes of the excitation peaks change

significantly when the coupling strength grows.

Figures 11 and summarizes the results of the analysis for vSL = 0.333 vs, at

the center of the nonlinear peak for various values of ε.

Figures 12 and 13 show that for values of vSL near the center of the peak,

nonlinear deviations in the phonon excitation spectrum arise for greater values of

ε, because as ε grows the region covered by the peak becomes wider, as visible in

22

Figure 4. However, the nonlinear features that arise are different for each different

velocity.

6 Discussion and Conclusion

In this thesis, we provide an initial description of the nonlinear friction phe-

nomenology of our simple 1D model. Although many nonlinear features have

plausible qualitative explanations, such as the close interaction with the excited

wavefront of the excitations of many phonons at once, the quantitative laws that

regulate these effects remain unclear. In particular, we could not find a way to

explain why the most significant multi-phonon effects are confined in the narrow

(but growing with ε) velocity range around the peak at vSL ≃ 0.333 vs, or to

predict which phononic modes are excited beside those predicted by the linear

one-phonon theory.

In further investigations, a more systematic study of the k values of the

excited phonon modes and the corrisponding oscillation frequencies could help in

identifying a generalization of the rule based on Eq. (59) that predicts them.

On the analytic front, further insight into these mechanisms could be achieved

by using a less drastic approximation for the chain structure factor (37). Interest-

ingly, a higher-order expansion is likely to lead to a dependence on temperature

and on ℏ in the result for the friction force, because the Bose distribution factor

in Eq. (37). This could make comparisons with classical molecular-dynamics

simulations problematic.

As a further step, it would be interesting to compare both the linear and

nonlinear features with the results of the same analyses in more realistic models.

For example, the slider-chain distance d could be allowed to vary, and the model

could be extended to 2D or 3D.

23

6

8

10

F ε2[1 Ka

3]

(a) vSL=0.21vs

2

4

6

8(b) vSL=0.333vs

5

10

15

20(c) vSL=0.8vs

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14ε [Ka2]

0.25

0.30

0.35

(d)

vSL=1.1vs

Figure 8: Variation in F (vSL)/ε2 as a function of ε, computed for the

indicated fixed values of vSL. The sharp jumps of the curves in (b) and (c)

correspond to the sudden appeareance of the nonlinear features visible

in Figure 3.

24

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

⟨ v2⟩ [1

0−⟩Ka

2m

]

⟨a

0 5 10 15 20 25 30k [a−1]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ω[√

K m]

⟨bω⟨kω= vSL|k| = 0.1 vs |k|

Figure 9: (a): Squared modulus of the spatial Fourier transform of the

chain atom velocities, averaged over a period a/vSL. The vertical dotted

lines correspond to the solutions of Eq. (59), brought back to the first

Brillouin Zone. The velocities are recorded from a fixed-speed simulation

with a very small ε = 0.005 Ka2, vSL = 0.1 vs, and the standard values

of the other parameters σ = 0.5 a, and d = 0.475 a. (b): Graphical

representation of the solutions of Eq. (59).

25

0.0002

0.0004

0.0006

0.0008

0.0010 ε=0.005 Ka2

0.02

0.04

0.06

0.08ε=0.05 Ka2

0.2

0.4

0.6

⟨ v2⟩ [K

a2 m]

ε=0.10 Ka2

0.0 0.5 1.0 1.5 2.0 2.5 ⟨.0k ⟩a−1]

0.25

0.50

0.5

1.00

1.25ε=0.15 Ka2

Figure 10: Same as Figure 9 (a), for a velocity vSL = 0.2 vs, for a velocity

well below the nonlinear peak, and for the indicated values of ε.

26

0.000005

0.000010

0.000015

0.000020 ε=0.005 Ka2

0.05

0.10

0.15

0.20 ε=0.05 Ka2

0.25

0.50

0.75

1.00

⟨ v2⟩ [K

a2 m]

ε=0.10 Ka2

0.0 0.5 1.0 1.5 2.0 2.5 ⟨.0k [a−1]

0.2

0.

0.⟩

0.8

1.0ε=0.15 Ka2

Figure 11: Same as Figure 9 (a), but at the resonant velocity vSL =

0.333 vs, at the center of the nonlinear peak, and for the indicated values

of ε.

27

0.000005

0.000010

0.000015

0.000020ε=0.005 Ka2

0.05

0.10

0.15 ε=0.05 Ka2

0.5

1.0

1.5

⟨ v2⟩ [K

a2 m]

ε=0.10 Ka2

0.0 0.5 1.0 1.5 2.0 2.5 ⟨.0k [a−1]

0.2

0.

0.⟩

0.8 ε=0.15 Ka2

Figure 12: Same as Figure 9 (a), for vSL = 0.30 vs, at the lower end of

the nonlinear peak.

28

0.000005

0.000010

0.000015

0.000020ε=0.005 Ka2

0.05

0.10

0.15

0.20 ε=0.05 Ka2

0.25

0.50

0.75

1.00

1.25

⟨ v2⟩ [K

a2 m]

ε=0.10 Ka2

0.0 0.5 1.0 1.5 2.0 2.5 ⟨.0k [a−1]

0.2

0.

0.⟩

0.8 ε=0.15 Ka2

Figure 13: Same as Figure 9 for (a), vSL = 0.34 vs, at the upper end the

nonlinear peak.

29

References

[1] O. M. Braun and Yu. S. Kivshar, The Frenkel-Kontorova Model: Concepts,

Methods, and Applications (Springer, Berlin, 2004).

[2] A. Vanossi, N. Manini, M. Urbakh, S. Zapperi, and E. Tosatti, Rev. Mod.

Phys. 85, 529 (2013). N. Manini, G. Mistura, G. Paolicelli, E. Tosatti, and

A. Vanossi, Adv. Phys. X 2, 569 (2017).

[3] C. Apostoli, G. Giusti, J. Ciccoianni, G. Riva, R. Capozza, R. L. Woulach, A.

Vanossi, E. Panizon, and N. Manini, Beilstein J. Nanotechnol. 8, 2186 (2017).

[4] E. Panizon, G. E. Santoro, E. Tosatti, G. Riva, and N. Manini, Phys. Rev. B

97, 104104 (2018).

[5] G. Riva, Teoria della risposta lineare per un modello minimale di attrito di-

namico, diploma thesis (University Milan, 2017), http://materia.fisica.

unimi.it/manini/theses/riva.pdf.

[6] C. Apostoli, Phonon analysis of the dynamic to static friction crossover in a

1D model, diploma thesis (University Milan, 2015/2016), http://materia.

fisica.unimi.it/manini/theses/apostoli.pdf.

[7] G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cam-

bridge Univ. Press, 2005).

[8] N. Ashcroft and M. Mermin, Solid State Physics (Holt-Saunders, Philadel-

phia, 1976).

[9] G. Giusti, Meccanismi microscopici di dissipazione in un modello unidimen-

sionale, diploma thesis (University Milan, 2015), http://materia.fisica.

unimi.it/manini/theses/giusti.pdf.

30