A. Benassi !in collaboration with M. Ma, A. Vanossi, M. Urbakh

How long can the superlubricity go?

New atomically precise elongated nano-structuresProgresses in nano-synthesis techniques allow to create defect free quasi 1D structures of considerable length that can be picked up and manipulated:

Kawai et al. PNAS 111 (2014) 3968

Polymer chains ~100nm

double-walled carbon nanotubes ~1cm

R. Zhang et al. Nature Nanotech. 8 (2013) 912

Graphene nanoribbons ~50nm

work in progress with Meyer & Kawai

Understanding their tribological properties is important for: !

• Nano-manipulation and production of nano structures for mechanics or electronics purposes

• Bringing peculiar nano-scale friction concepts at larger scales (i.e. superlubricity)

The breaking of superlubricity

Assuming we can model these nanostructures as 1D incommensurate systems our Frenkel-Kontorova model must take into account: !• Finite size • Edge driving (non-uniform driving) !!that destroy the ideal superlubric state (in the standard Aubry sense). However, at small N, we are left with a state of very low (dynamic) friction that we still call superlubric. !We know that, also for defect free systems, superlubricity is expected to vanish as we move to the macro scale: !

Muser Europhys. Lett. 66 97 (2004) Consoli et al. PRL 85 302 (2000) van de Ende J. Phys. Condens. Matter 24 445009 (2012)

!Heuristically one might think that in a finite size incommensurate FK model edges plays a major role and, when N increases, the effective stiffness of the chain decreases leaving more freedom to the edges of sitting in a minimum of the potential increasing friction. !!!!!however for non-uniformly driven systems the way in which superlubricity is broken is non-trivial…

1/Keff = N/K

The breaking of superlubricity

Assuming we can model these nanostructures as 1D incommensurate systems our Frenkel-Kontorova model must take into account: !• Finite size • Edge driving (non-uniform driving) !!that destroy the ideal superlubric state (in the standard Aubry sense). However, at small N, we are left with a state of very low (dynamic) friction that we still call superlubric. !We know that, also for defect free systems, superlubricity is expected to vanish as we move to the macro scale: !

Muser Europhys. Lett. 66 97 (2004) Consoli et al. PRL 85 302 (2000) van de Ende J. Phys. Condens. Matter 24 445009 (2012)

!Heuristically one might think that in a finite size incommensurate FK model edges plays a major role and, when N increases, the effective stiffness of the chain decreases leaving more freedom to the edges of sitting in a minimum of the potential increasing friction. !!!!!however for non-uniformly driven systems the way in which superlubricity is broken is non-trivial…

1/Keff = N/K

- infinite size!- incommensurate condition!- stiff chain!- uniform driving

zero static friction force

A critical length

A critical length

A critical length exist above which friction increases

A critical length

A critical length exist above which friction increases

Sudden jumps occur for chains longer than the critical length

A critical length

A critical length exist above which friction increases

Sudden jumps occur for chains longer than the critical length

A strong dissipation takes place above the critical length but is extremely localized in a specific region

A kink of commensuration We can define a local index of commensurability as:

and look how does it behave along the chain in time…

d(i) =Xi �Xi�1

as

A narrow region in which the chain is commensurate nucleates at the driving edge ad propagates up to a specific position. !!The atoms pertaining to this region undergo a regular stick-slip with strong dissipation.

We can define a local index of commensurability as:

and look how does it behave along the chain in time…

d(i) =Xi �Xi�1

as

A narrow region in which the chain is commensurate nucleates at the driving edge ad propagates up to a specific position. !!The atoms pertaining to this region undergo a regular stick-slip with strong dissipation.

A kink of commensuration

d(i) =Xi �Xi�1

as

P (i) = m⌘ limT!1

Z T

0

x

2i

m⌘v

20

dt

F (t) = Kdr(V0t�XN (t))

A kink of commensuration

d(i) =Xi �Xi�1

as

P (i) = m⌘ limT!1

Z T

0

x

2i

m⌘v

20

dt

F (t) = Kdr(V0t�XN (t))

Nucleation of the commensuration kink at the driving edge

incommensurate

incommensurate

commensurate

A kink of commensuration

A simple analytical theory

The nucleation of the kink of commensurability starts with the commensuration of the rightmost couple of atoms, the external force needed for such a commensuration can be calculated exactly:

F

ext

= K(as

� a

c

) +2⇡u0

a

s

sin

✓2⇡x

N

a

s

◆

Fmin

ext

= K(as

� ac

)� 2⇡u0

as

Fmin

ext

= hF i

applied to the rightmost atom, is minimum value is

the nucleation occurs when the average friction force exceed this minimum threshold

how to calculate the average friction force?

A simple analytical theoryThe average rate of energy pumped in by the external driving must equal the total energy dissipated by each particle due the viscous force:

hF iv0 = m⌘ limT!1

1

T

Z T

0

NX

i=0

X2i dt

hF i = mN⌘v0

1 + lim

T!1

1

NT

Z T

0

NX

i=0

✓Xi

v0� 1

◆2

dt

�

Xi = hXii+ �Xi = v0 + �Xi1

1

A simple analytical theory

center of mass!viscous damping

⌘

The average rate of energy pumped in by the external driving must equal the total energy dissipated by each particle due the viscous force:

hF iv0 = m⌘ limT!1

1

T

Z T

0

NX

i=0

X2i dt

hF i = mN⌘v0

1 + lim

T!1

1

NT

Z T

0

NX

i=0

✓Xi

v0� 1

◆2

dt

�

Xi = hXii+ �Xi = v0 + �Xi1

1

A simple analytical theory

center of mass!viscous damping

⌘

dissipation of the internal !degrees of freedom

The average rate of energy pumped in by the external driving must equal the total energy dissipated by each particle due the viscous force:

hF iv0 = m⌘ limT!1

1

T

Z T

0

NX

i=0

X2i dt

hF i = mN⌘v0

1 + lim

T!1

1

NT

Z T

0

NX

i=0

✓Xi

v0� 1

◆2

dt

�

Xi = hXii+ �Xi = v0 + �Xi1

1

A simple analytical theory

center of mass!viscous damping

⌘

dissipation of the internal !degrees of freedom

The average rate of energy pumped in by the external driving must equal the total energy dissipated by each particle due the viscous force:

hF iv0 = m⌘ limT!1

1

T

Z T

0

NX

i=0

X2i dt

hF i = mN⌘v0

1 + lim

T!1

1

NT

Z T

0

NX

i=0

✓Xi

v0� 1

◆2

dt

�

Xi = hXii+ �Xi = v0 + �Xi

✏ =Kint

K

Xi = X0i + ✏X1

i +O(✏2) X0i = v0

hF i = mN⌘v0

1 + ↵

✓Kint

K

◆+O(✏3)

�

Kint = u0

✓2⇡

as

◆2

1

1

A simple analytical theory

center of mass!viscous damping

⌘

dissipation of the internal !degrees of freedom

The average rate of energy pumped in by the external driving must equal the total energy dissipated by each particle due the viscous force:

hF iv0 = m⌘ limT!1

1

T

Z T

0

NX

i=0

X2i dt

hF i = mN⌘v0

1 + lim

T!1

1

NT

Z T

0

NX

i=0

✓Xi

v0� 1

◆2

dt

�

Xi = hXii+ �Xi = v0 + �Xi

✏ =Kint

K

Xi = X0i + ✏X1

i +O(✏2) X0i = v0

hF i = mN⌘v0

1 + ↵

✓Kint

K

◆+O(✏3)

�↵ = 2.64

Kint = u0

✓2⇡

as

◆2

1

1

A simple analytical theorySubstituting the average force expression in the nucleation condition we can calculate the critical chain length above which superlubricity is broken:

the agreement with the numerical simulation is good…

Nc =K(as � ac)

m⌘v0

✓1� �

Kint

K

◆1 + ↵

✓Kint

K

◆2�� =

as2⇡(as � ac)

with:

ac/as =1 +

p5

2but our model shows very good agreement with simulations done for different in commensuration ratio . And also reversing the ratio…

ac/as

Up to now we used the golden ratio:

the fitting parameter is the same as before:

!!!

within the confidence error

↵ = 2.69± 0.1

Testing the model with existing experiments

DWCNTs GNRs

R. Zhang et al. Nature Nanotech. 8 (2013) 912

`c = Ncac = 0.5m

50 times > than the maximum NT size

Under investigation

Moire’ pattern and ribbon rippling

Orientation dependent friction

Still according to static friction force measurements the system is superlubric up to 30 nm:

Can the edge driving allow the nucleation

of a kink of commensurability ?

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35

per a

tom

forc

e [p

N]

ribbon length [nm]

Static friction force per atom VS GNR length (0 degs)

MD simulation

More information at: https://sites.google.com/site/benassia/

Thank you!

Modeling material properties !

at different length scales