Friction as the energy lost in each slip

Work, friction and heat are one number

The scanner advances at speed $v$ and the spring pulls on it with force $F$ , so the scanner does work $Fv$ each second. In steady state the mean energy of the tip no longer changes, and all that work has to leave as heat. So the mean friction force, the work per period and the heat per period differ only by a factor $a$ .

The substrate force averages to zero over one period, so friction does not come from the force the tip feels as it rubs past the atoms. If $\eta<1$ , a slowly dragged tip returns nearly all the work it receives to the scanner and only viscous drag remains; Vanossi and co-workers write that friction is then viscous and vanishes as the speed goes to zero.

If $\eta>1$ , every slip comes with an energy drop. Vanossi and co-workers cite Helman and co-workers: kinetic friction at low speed equals the energy drop from the instability point to the next valley, divided by $a$ . For a jump of $n$ sites, the divisor becomes $na$ .

\langle F\rangle\,v=\frac{\Delta W_{\mathrm{drive}}}{\Delta t},\qquad F_k(v\to0)=\frac{U(x_c,X_c)-U(x_{n},X_c)}{n\,a}

Left: the steady-state energy balance. Right: friction in the low-speed limit, where

x_n

is the valley reached after the slip.

Source: A. Vanossi · Y. Sang

One site or two

When the valley disappears, the tip rushes forward. Which valley it stops in depends on how fast the energy of the rush is carried off. Vanossi and co-workers cite Medyanik and co-workers: jumps over several sites are possible only for $\eta>4.604$ , and that is only a necessary condition.

For $\eta>4.604$ , Vanossi and co-workers split the behaviour with a damping scale $\gamma_*$ : above it the tip only jumps to the neighbouring valley; below it the tip may jump several sites and even overshoot the lowest valley.

The parameters of Sang and co-workers give $\gamma_*=1.11\times10^{6}\,\mathrm{s^{-1}}$ , while the damping they used is $8.9\times10^{5}\,\mathrm{s^{-1}}$ , a little below that scale. By this criterion the tip “may” jump several sites, yet in the course’s zero-temperature simulation it jumps one site every time. Only when the damping drops to $2.67\times10^{5}$ or $8.9\times10^{4}\,\mathrm{s^{-1}}$ does the tip jump two sites each time. Once again the criterion is necessary, not sufficient.

\gamma_{*}=\frac{2\pi}{a}\sqrt{\frac{U_0}{m}}=1.11\times10^{6}\,\mathrm{s^{-1}}

The number uses the parameters of Sang and co-workers. Above

\gamma_*

the tip only jumps to the neighbouring valley; below it, several sites are possible. The boundary only matters when

\eta>4.604

; for smaller

\eta

the landscape holds too few valleys for a jump of several sites.

Source: A. Vanossi · Y. Sang

The energy books of the simulation

The course’s Labs use the splitting integrator written down by Sivak and co-workers: the velocity first exchanges energy with the heat bath, then position and velocity advance under the force, with the scanner moved in the middle. Sivak and co-workers show that the energy change of each step then splits cleanly into three parts: heat exchanged with the bath $Q$ , protocol work done by moving the scanner $W_{\mathrm{prot}}$ , and shadow work from the finite time step $W_{\mathrm{shad}}$ .

Shadow work is not physical; it is the book entry of the integration error. Dragging for 10 periods at zero temperature, the scanner does 13.77 eV of work and shadow work is only $-5.2\times10^{-4}$ of it; each halving of the time step shrinks that fraction by 4.0007 and 4.0003, exactly the $\Delta t^{2}$ a second-order method should give. So the heat per period that the Lab reads out can be taken as real dissipation.

Vanossi and co-workers also warn that $\gamma$ in the model stands in for the real process in which phonons carry heat away; it changes the dynamics, yet there is no clear way to choose its value. The two-site slips of the last section show exactly that.

\Delta E=Q+W_{\mathrm{prot}}+W_{\mathrm{shad}}

Eq. (5) of Sivak and co-workers.

Q

is the energy the bath gives the system, negative when the tip releases heat.

Related on this site: Work and heat along one pull

Source: D. A. Sivak · A. Vanossi

Worked example: how much heat one slip releases

Problem. With the parameters of Sang and co-workers ( $U_0=0.27\,\mathrm{eV}$ , $a=0.4\,\mathrm{nm}$ , $k=0.93\,\mathrm{N\,m^{-1}}$ ), at zero temperature and slow drive, find the low-speed friction when the tip jumps one site and when it jumps two.

Step 1, the fold. Chapter 2 found $X_c=2.084\,a$ and $x_c=0.264\,a$ . The total energy there is $U(x_c)=0.024+1.538=1.561\,\mathrm{eV}$ (substrate term plus spring term).

Step 2, the valleys after the slip. Holding $X_c$ fixed, the next valley sits at $x_1=1.091\,a$ with energy $-0.227+0.457=0.230\,\mathrm{eV}$ ; the one after that sits at $x_2=2.007\,a$ with energy $-0.267\,\mathrm{eV}$ .

Step 3, friction. One site: $(1.561-0.230)\,\mathrm{eV}/0.4\,\mathrm{nm}=1.331\,\mathrm{eV}/0.4\,\mathrm{nm}=0.533\,\mathrm{nN}$ . Two sites: $1.828\,\mathrm{eV}/0.8\,\mathrm{nm}=0.366\,\mathrm{nN}$ .

Checks. The zero-temperature simulation dragged at 10 nm/s gives a mean friction of 0.5338 nN, 0.15% above the energy-drop prediction; the difference comes from the delay before the slip (chapter 2). With the damping lowered to $2.67\times10^{5}\,\mathrm{s^{-1}}$ , the simulation gives 0.3666 nN, 0.13% above the prediction.

Explanation. In a two-site jump the tip releases more energy at once, but the scanner must travel one more site before the next slip, so the energy per period is smaller. At room temperature the Lab’s default state reads 0.809 eV per period, which is exactly 0.324 nN times 0.4 nm; thermal noise lets the tip slip before the fold, so the energy drop is smaller. That is the subject of chapter 4.

Source: A. Vanossi · Y. Sang

Try it in the Lab

Two lines: work and heat

Predict first: after a few periods, how will the work by scanner and heat to substrate lines in the Where the work goes chart move relative to each other?
Watch the default preset for a few periods, note the slopes of the two lines, then read Heat per lattice period and Mean friction force.
Turn the mean friction into work per period with 0.4 nm, compare it with the heat reading, and explain why both lines step up together at every slip.

Light damping makes the tip jump two sites

Predict first: with the damping cut to a tenth, how many sites will the tip jump each time? Will the mean friction rise or fall?
Choose Long slips, read the mean slip length and the mean friction force, and look at the period of the sawtooth in the force chart.
Explain the readings with the two-site energy drop from the worked example, then turn the damping slider back slowly and find where the tip goes back to one-site jumps.

Exercises with explained answers

A tip with $\eta<1$ is dragged extremely slowly. Where does the scanner’s work go, and what is the mean friction?
Show the answer
The total energy has only one valley, and the tip follows the scanner continuously. The energy stored in the spring while it stretches is returned to the scanner in the other half of the period, so the net work per period is zero. Only the viscous drag $m\gamma v$ remains, and it vanishes as the speed goes to zero.
The Lab’s default state drags at 100 nm/s with a mean friction of 0.324 nN. How much heat does the tip release per second? How many slips of 0.809 eV is that per second?
Show the answer
The power is $P=Fv=0.324\times10^{-9}\times10^{-7}=3.24\times10^{-17}\,\mathrm{W}$ . The scanner advances 250 periods per second at 0.809 eV each, and $250\times0.809\times1.6\times10^{-19}\,\mathrm{J}=3.24\times10^{-17}\,\mathrm{J}$ , which agrees.
If you halve the time step, how should the ratio of shadow work to scanner work change? If it does not change, what does that tell you?
Show the answer
Shadow work of the splitting integrator falls as $\Delta t^{2}$ , so halving the step should shrink the ratio about fourfold; the course’s check gives 4.0007 and 4.0003. A ratio that stays fixed points to another error in the books, for example a wrong force or a wrong scanner update.

Sources and what we recomputed

A. Vanossi, N. Manini, M. Urbakh, S. Zapperi and E. Tosatti, Colloquium: Modeling friction: from nanoscale to mesoscale, arXiv:1112.3234v4; Reviews of Modern Physics 85, 529 (2013)
Version 4, 3 April 2013. Sec. I: the Coulomb–Amontons laws. Sec. II.A: the Prandtl–Tomlinson model (Eq. 1), the stick-slip parameter, the fold where a slip starts, the barrier that shrinks as the three-halves power of the distance to it, multiple slips when that parameter exceeds 4.604 and the energy-drop rule for low-speed friction. Sec. II.C: the Langevin equation with its noise (Eqs. 2 and 3), the ratio of pulling rate to hopping rate, thermolubricity (Eq. 4), the logarithmic law (Eq. 5) and the disagreement with low-temperature experiments. Sec. II.D: the Frenkel–Kontorova chain (Eq. 6) and the Aubry transition. Sec. II.E: superlubricity, the criticism of the term, the graphite-flake experiments it reports (below 50 pN in most orientations, about 250 pN near the matching ones, and a flake turning back into the matching orientation), Figs. 4 and 5 and the rule that the angular width of a friction peak is about one over the flake diameter. Sec. III.A: thermostats and the damping rate that no one knows how to choose. Sec. III.B: the times and sizes a simulation can reach and the gap of up to six orders of magnitude between experimental and simulated stick-slip speeds; its printed figure of one microsecond per simulation day is about nine times smaller than its own arithmetic gives. Sec. VI: conclusions. We recomputed the stick-slip parameter, the fold force and the angular widths with our own models; we did not digitise any figure.
Y. Sang, M. Dubé and M. Grant, Thermal effects on atomic friction, arXiv:cond-mat/0109168v1; Physical Review Letters 87, 174301 (2001)
Version 1, 10 September 2001. Eq. (1): ramped creep, in which friction falls with the two-thirds power of temperature times the two-thirds power of the logarithm of speed over temperature. Eqs. (2) and (3): the Langevin equation and the tip potential, a cosine with its minima on the lattice sites. Eq. (4): the fold position. The parameters fitted to the NaCl data of Gnecco et al.: lattice period 0.4 nm, effective stiffness 0.86 N/m, corrugation 0.27 eV, damping rate 8.9 times 10 to the 5 per second, tip mass 8.7 times 10 to the minus 12 kg; footnote 21 gives a spring constant of 0.93 N/m. The text prints the spring resonance as 52 MHz; with this mass and spring it is 52.0 kHz, so we read MHz as a typo. Eq. (5): the Kramers rate. Eqs. (6) and (7): curvature and barrier near the fold. Eq. (8): the slip distribution. Eq. (9): the scaled speed. Eqs. (10) to (12): the closed form of the mean friction and its constants, quoted as 0.55 nN in theory against 0.54 nN, with an uncertainty of 0.01 nN, from simulation and experiment. We recomputed the scaled speed, the constants and the mean friction with our model; we did not digitise Figs. 1 to 3.
D. A. Sivak, J. D. Chodera and G. E. Crooks, Using nonequilibrium fluctuation theorems to understand and correct errors in equilibrium and nonequilibrium simulations of discrete Langevin dynamics, arXiv:1107.2967v5; Physical Review X 3, 011007 (2013)
Version 5, 30 January 2013. Eq. (3): a Langevin step split into deterministic and heat-bath substeps. Eq. (5): the change in energy over a step equals heat plus protocol work plus shadow work, the error that a finite time step adds. We used this split to keep the energy ledger of the stick-slip simulation and checked that the shadow work shrinks fourfold when the step is halved.