NOTES TO ACCOMPANY FIRST ATOMIC FRICTION DEMO

 

Friction starts from the variation in energy of atoms on one surface as they move over atoms on the opposing surface.  The first simulation illustrates this for a single atom (blue) moving over a regular crystalline surface (red atoms). The atoms interact with a simple Lennard-Jones potential.  The blue atom is attached to a spring and the other end of the spring is pulled back and forth at a fixed speed.  The main panel shows the motion of the blue atom and compression of the spring. The two floating panels show plots of the atomic coordinate x as a function of time t and of the spring force F vs. the atomic coordinate x.  This is a simple model of an atomic force microscope (AFM) with one atom on the AFM tip interacting with the substrate and the spring representing the cantilever that pulls the tip across the substrate.  Atomic force microscopes can actually resolve changes in force as the tip moves over an atom.

 

The main panel and plot of coordinate vs. time show that the blue atom tends to get stuck in local energy minima between two red atoms.  The pulling end of the spring keeps moving, leading to a growing force on the atom.  When the force is large enough to lift the atom out of the energy minimum it jumps rapidly to the next minimum and the process repeats. The plot of spring force vs. coordinate shows this rise in pulling force while the atom is stuck in a minimum and the rapid drop as it moves to the next minimum.  The friction force opposes motion and has the opposite sign from the pulling force from the spring.  The pulling force is positive when the atom is pulled to the right and the friction force is negative.  When the atom is pushed to the left, both forces change sign.

 

The plot of spring force vs. coordinate illustrates both the static and kinetic friction force. The static friction Fs is the magnitude of the force needed to initiate motion.  In this case it is the magnitude of the force peak where the atom pops to the next energy minimum.  If a constant force of smaller magnitude was applied, the atom would remain trapped in an energy minimum.   Once the applied force is bigger than the largest opposing force from the substrate, the atom will keep on moving.  The static friction is this maximum opposing force.

 

The kinetic friction Fk is the average force needed to maintain sliding at the driving velocity. For the initial parameters the average pulling force is clearly nonzero.  Taking a time average gives a mean force that is about half of the peak force, or a kinetic friction of about half the static friction.  The work done by the pulling spring is the kinetic friction times the distance traveled back and forth.  This work is converted into heat, which is why you can make fire by rubbing sticks together.  Activity II below shows that rapid pops between minima generate lots of heat and kinetic friction.

 

Two parameters can be varied in the simulations to show how friction depends on the spring constant k of the pulling spring and the force pushing the atom into the substrate which is called the load.  It will take a little while for the system to settle down after you change these parameters.

 

The following activities help in understanding the molecular origins of friction and illustrate the main points of the demo.  There are also some references for further reading. Real contacts have more than one atom.  The next three demos illustrate how that changes friction.

 

Activity 1: Static Friction and Load

 

The friction laws we teach in introductory physics classes date back to da Vinci and Amontons (1699).  They say that the kinetic and static friction forces are both proportional to the load pushing the surfaces together.  The ratio of the force to load is called the friction coefficient.  Let’s test how this works at the atomic level.

 

Vary the load using the slide bar on the right.  For each load, write down the magnitude of the peak force or static friction.  The peak value Fs is output at the bottom left of the force vs, coordinate panel.  The peak heights are a little different as the direction is changing and then settle to a more constant value.

 

Plot Fs as a function of load.  You should find that Fs rises linearly from about 2.6 at load=0 to 11.2 at load=20.  This type of behavior is observed for nearly all types of interaction, not just the simple Lennard-Jones potential used here.  Theorists and experimentalists often quote the slope of this line (~0.4) as the friction coefficient, but the presence of friction at zero load seems to contradict the laws taught in introductory physics.

 

The force at zero load comes from the adhesive forces pulling the surfaces together.  It would vanish if the interactions were purely repulsive.  Many macroscopic surfaces are so rough and stiff that the fraction of the surface in contact is very small.  In this case the adhesive forces are small and the friction is nearly zero at zero load.  Tape, putty, and caulk are common household materials that deform to opposing surfaces, giving high contact and adhesion.  The fact that tape sticks to a vertical wall implies that there is friction at zero load, just as in our atomic example.  These common exceptions to the “laws” taught in introductory physics have been well known for centuries and are included in work by the famous physicist Coulomb.  It is unfortunate that they are left out of the classroom.  One reason might be that they make it harder to solve the friction problems that are presented near the start of introductory courses.

 

Activity 2: Friction and Spring Stiffness

 

Put the load back to 10 and try varying the spring stiffness.  At the lowest stiffness, the plot of coordinate vs. time has pronounced steps where the atom is stuck in a potential energy minimum.  The atom spends most of its time stuck, and the hops between minima are rapid.  As the stiffness increases, the coordinate changes more smoothly.  At the largest stiffness, the energy cost for changing the length of the spring is much larger than that of moving in the substrate potential.  The atom follows the spring very closely, showing only weak wiggles in the coordinate.

 

Now look at the changes in the panel showing force vs. coordinate.  The peak value corresponding to the static friction is just the maximum force from the energy wells produced by the substrate.  This is independent of the spring stiffness.  What does change with stiffness is the shape of the force curve and its average value, the kinetic friction.  At low stiffness there is a lot of hysteresis – the force produced on moving to the right is almost always more positive than any force produced when moving to the left.  The area between the curves represents energy that is converted into heat by sliding.  It corresponds to the distance traveled times the kinetic friction.  As the stiffness decreases, the hysteresis and the kinetic friction both decrease toward zero.  (They don’t go exactly to zero at any finite pulling velocity, but do vanish as velocity goes to zero.)

 

The kinetic friction Fk is output at the bottom right of the force vs. coordinate plot.  Try making a plot of Fk and Fs as a function of stiffness.  You should see that Fs is nearly constant and Fk drops rapidly as the stiffness exceeds a threshold value.  See how the plot of coordinate vs. time is different above and below the threshold stiffness.

 

Activity 3: Kinetic Friction and Load

Put the spring stiffness back to 15 and the load to 20.  For this case the atom moves in sharp pops between energy minima and the force changes a lot as the direction changes.  Now try stepping the load down to zero in steps of 5.  You’ll see that the steps in the coordinate vs. time become less pronounced.  The spring force also changes less with the direction of motion.  The kinetic friction is nearly zero.  What’s going on?

 

The variations in the energy as the atom moves get weaker as the load decreases, while the spring stiffness remains the same.  Thus decreasing the load leads to the same transition as increasing the stiffness did in Activity 2 – it’s as if the spring gets stiffer when you decrease the load.  Try plotting the kinetic friction Fk vs. load for different spring constants and see how this transition depends on k.

 

Activity 4: A Simple Analytic Model for the Algebra Savvy

 

The behavior illustrated in this simulation is captured by an even simpler model called the Tomlinson model.  In this model the atom can’t move up and down and the potential from the substrate is a simple sinusoidal function: v(x) = v0cos(2πx/a), where a is the spacing between atoms.  When k > (2π)2v0/a2 the atom moves smoothly and the kinetic friction goes to zero at low velocities.  For smaller k, the atom moves in a sequence of rapid pops that dissipate lots of energy.  The low velocity kinetic friction is comparable to the static friction.  

 

To understand this behavior, consider the sum of the force from the substrate potential and the spring.  Show that for k > (2π)2v0/a2 there can only be one value of x that gives zero force, while for smaller k the atom can sit at rest at more than one minimum.  You may find it useful to solve the problem graphically.  Plot the force from the substrate potential minus the force from the spring.  The intersections are where the total force vanishes.  Consider the maximum slope of the force vs. x that you can get from the substrate and compare to the spring.  After solving this simple case, you might consider other potentials with more complicated functional forms than a pure sine wave.

 

Main Points to Remember

 

 

 

 

 

 

 

Technical details: The tip atom and substrate atoms interact with something called a 6-12 Lennard-Jones potential.  The energy for a pair of atoms separated by distance r is V(r)=4ε [(σ/r)12-(σ/r)6], where ε is the binding energy and σ the diameter of the atoms.  All of our plots are in the natural units for this potential.  In addition to the above energy and length scale, there is a time τ = (m σ 2/ ε)1/2, where m is the mass of an atom.  In terms of real units the typical energy scale is about 10-20 Joules and the length scale is 3·10-10 meters.  This gives a force unit of about 3·10-11 N or 30 pN and a unit spring constant of about 0.1 N/m – not untypical for an AFM.  The time constant for a copper atom (~10-25 kg) is about 10-12 s or 1ps and the pulling velocity of 0.2 corresponds to about 300 m/s – pretty fast compared to an AFM, but this allows the simulations to run while you watch.

 

A small damping force is added to the motion of the atom.  This represents the coupling to phonons and electrons in the substrate.  The jiggling of the atom as it pops between energy minima transfers energy to the phonons and electrons.  This random energy represents heat and flows off into the substrate. The simulation is effectively at zero temperature.  At finite temperature, static friction corresponds to changes in free energy instead of energy.

 

Potentially Useful References:

 

Pedagogical:

J. Ringlein and M. O. Robbins, “Understanding and illustrating the atomic origins of friction,” Am. J. Phys. 72, 884-891 (2004).  doi:10.1119/1.1715107. 

 

Ugo Besson, Lidia Borghi, Anna De Ambrosis, and Paolo Mascheretti, “How to teach friction: Experiments and models,” Am. J. Phys. 75, 1106-1113 (2007).  doi:10.1119/1.2779881.

 

A More Technical Review:

M. H. Müser, M. Urbakh, and M. O. Robbins, “Statistical Mechanics of Static and Low-Velocity Kinetic Friction,” Advances in Chemical Physics, 126, 187-272 (2003).

 

 

Use Policy:

This text is copyrighted by Mark Robbins and Oleg Tchernyshyov.  Non-profit educational use is permitted with appropriate acknowledgement.