First principles–based theory of collective creep

To scientists, creep refers to the very slow motion of a solid, or that of an extremely high viscous fluid, in response to an external stimulus. For creep to occur, atoms must move in a concerted or collective way.

Creep motion matters in a variety of fields: In materials science, creep of grain boundaries makes it challenging to predict the longevity of material properties with potential implications for nuclear power plants or railway wheels. In Earth science, creep plays a critical role in the motion of tectonic plates. In physics, whether glasses creep or not when they have been slowly cooled below their glass transition temperature has been philosophized about for many decades. Such questions are difficult to tackle scientifically because experimentalists are too impatient to wait 2,000 years to determine whether church windows flow slowly because of gravity. Theorists, however, suffer from another handicap: They cannot apply their beloved tools—such as linear-response theory—to creep phenomena in a straightforward fashion. This is why it is difficult to develop a theory for creep motion, which is precisely what Reguzzoni et al. (1) succeeded in doing for a specific system in this issue of PNAS. Their work is impressive because they not only derive the functional relationship between creep motion and external force, but they manage to ascertain the correct material-specific parameters with surprisingly simple and thus elegant analytical calculations.

The work by Reguzzoni et al. was motivated in part by a recent experiment on the microscopic origin of friction: Coffey and Krim (2) had found that a monolayer of xenon atoms adsorbed onto a crystalline copper surface could be transformed into relative sliding motion merely by oscillating the copper substrate back and forth. If the registry between xenon and copper had not been perfect, this result would not have come as a surprise, because registration defects can move with relatively little energy through a system, which results in mass transport (3–5). However, as the two systems did interlock perfectly and as the maximum possible relative kinetic energy per Xe atom in such an experiment is much less than the corrugation energy barrier that one atom needs to overcome when moving from one energy minimum to the next, the question arises why sliding was initiated.

For creep to occur, atoms must move in a concerted or collective way.

Ever since the ingenious work by Ludwig Prandtl on solid friction and plastic deformation published almost 100 years ago (6), one knows how thermal fluctuations change the frictional response of a system from Coulomb (barely velocity dependent) friction to viscous (linear in velocity) friction: One only needs to give the relevant (single or collective) degree of freedom enough time to overcome the appropriate energy barrier U and should not ramp up the external force too quickly. The average jumping or waiting time for a particle to initiate motion—and also the linear-response viscosity— then both are proportional to a factor exp(U/k_BT), where T is the temperature and k_B the Boltzmann constant. In naive applications of this theory to our monolayers, one would assume that U grows linearly with the number of atoms (4), because the Xe layer is in perfect registry with the Cu substrate.

Thus velocity should be exponentially small in the (incredibly large) number of adsorbed atoms, i.e., zero for all practical and measurable purposes. This thought knocks us back to square zero in our attempt to explain why commensurate adsorbed monolayers can depin under the influence of a small inertial force.

Nucleation of a Critical Droplet as the Trigger of Creep

Reguzzoni et al. (1) recognized that atoms in the adsorbed layer do not move in parallel, but that motion is initiated collectively by fractions of the layer, which one may call droplets. Specifically, they suggested that frictional slip occurs by the nucleation of a small commensurate domain that then expands by displacing a domain wall. To form a small droplet, energy will be required. However, once a critical droplet radius r has been reached, the droplet will keep growing spontaneously without much further ado, as will become evident from the following nucleation scenario suggested by Reguzzoni et al.: The energy of a droplet ΔE has two main contributions: In its inside, each atom has gained an energy equal to the external force per atom F times lattice constant a. As the number of atoms inside the droplet is the product of the area of the droplet, πr², times the surface density of atoms, σ, the system has roughly gained an energy − πr²σFa. At the “grain boundary” between the inside and outside of the droplet, the system must pay an energy penalty, because atoms at that boundary do not sit in potential energy minima and furthermore are (elastically) compressed at the front of the droplet and stretched at its trailing end. On average there will be an energy penalty Γ per unit length resulting in an overall grain boundary energy of 2πΓr. Thus ΔE is approximately

When r exceeds the value where ΔE takes its maximum value, the droplet will gain energy by growing. This happens at a critical radius of r_c = Γ/(σFa), for which the net energy would be

This is the appropriate value for the energy barrier U and not the naive expectation of U growing linearly with particle number, unless F is so small that the value of U exceeds the energy required to form a vacancy or vacancy/interstitial pair (in which case another flow mechanism would become dominant). In Eq. 2, a constant energy term was added, because so far only energies that grow with the first or second power of r were retained. It was parameterized such that the energy barrier disappears when an (athermal) layer is exposed to the static friction force F_s. The authors then demonstrated the validity of Eq. 2 by showing that the activation time t₀ ∝ exp(U/k_BT) predicted in the theory matches the one obtained within molecular dynamics simulations of a model, which was parameterized with a highly realistic force field (7).

Relating Atom- and Continuum- Mechanics-Based Descriptions of Frictional Slip

An impressive aspect of ref. 1 is that the value for Γ was estimated accurately for a simplified geometry, i.e., a pair of linear domain walls, with analytical calculations. This was done by mapping the interaction potential (which itself was parameterized from quantum chemical calculations) onto the Frenkel Kontorova (FK) model (5), which consists of a harmonic chain with nearest neighbor interactions that moves in a simple sinusoidal external potential. Exploiting the solutions of the FK model in the continuum limit is what turns the theory into a Hamiltonian or first-principles-based theory. Molecular dynamics simulations showed that the mapping was indeed quite accurate.

It did not escape the attention of the authors of ref. 1 that defects accelerate the depinning dynamics and that their effect may have to be incorporated for quantitative purposes. The defects may be, as mentioned above, vacancies or interstitials in the adsorbed layer but also surface steps or chemical impurities in the substrate. The latter could change the nucleation process from homogeneous to heterogeneous. All these deviations from the idealized model could matter qualitatively—in particular at small external forces. The theory is still sufficiently realistic to predict the relevant transport coefficient or friction in a broad range of conditions—or it would not have matched the simulations as nicely as it did.

An intriguing consequence of Eq. 2 is that the activation energy and thus the slip time, which ultimately determines the (effective) viscosity and thus dissipation, cannot be expanded into positive powers of F. This implies that the model shows a highly nonlinear relationship between dissipation and driving force, very similar to the situation for flux lines in type-II superconductors (8). As a matter of fact, the superconductor community strictly requires creep to be a response that cannot be treated within a perturbative expansion (such as linear-response theory). Conversely, the term creep is used in a more lax fashion in materials science, where it simply refers to astronomically slow motion. Appreciating the difference between nonequilibrium creep and extremely slow but viscous flow might benefit tribologists and glass physicists alike in analyzing their results. For example, the latter should not ask if glasses flow (because they do—just like crystalline solids/adsorbed layers), but how they flow in response to a (small) external force. Homework is also given to theorists: Mathematical physicists with a fetish for bookkeeping of special functions can now extend the calculations from linear to circular grain boundaries, but let’s not call them creeps.