Noise and Dislocation Formation in Atomistic Friction Simulations

C.J. Olson Reichhardt

T-12, Los Alamos National Laboratory, Los Alamos, NM, USA

ABSTRACT
We use large-scale atomistic simulations to study the work-hardening process that occurs when two metals slide against one another. Dislocations form at the interface between the work pieces and then migrate into the bulk. We examine the relationship between the velocity noise signature at the atomistic level and the number of dislocations present. We compare these signatures to those observed in a system of a single particle dragged through a lattice, where local melting can occur.

Keywords: friction, colloids, voltage noise, melting

1. INTRODUCTION
There has been considerable recent interest in the microscopic origins of friction,¹ due to a combination of the availability of new experimental probes capable of resolving nanometer length scales and new simulation models running on high speed computers. Despite its long history and its technological importance,² nanotribology remains poorly understood. Even the seemingly simple case of dry friction between two metal surfaces, when no lubricant material is present, produces a rich array of behavior such as work hardening, which occurs when the originally crystalline sliding surface develops a complex structure of nanocrystalline grains.³

A number of simple models have been developed to treat dry friction in a simple way.^4-8 We focus here on a two-dimensional system in which the two materials sliding against each other are crystals composed of identical atoms but with differing orientations. In the lower surface, the close-packed crystal direction is parallel to the interface, while in the upper surface, the close-packed direction is perpendicular to the interface. (See Fig. 1.) Consider the motion of the atoms in the top layer of the lower surface. When the surfaces move against each other at relatively low velocities, the motion of the atoms in the lower surface can be approximated by a Frenkel-Kontorova model, in which the upper surface is replaced by a sinusoidal potential and the top layer of atoms in the lower surface are connected to each other by springs. A major limitation of this model is the fact that, in the actual system, the effective potential produced by the atoms in the upper layer is not static, but responds to the motion of the atoms in the lower layer. It may be possible to represent this effect by adding noise to the model; however, although the effect of noise on friction has been studied,⁹ the noise resulting from friction at the microscopic level is poorly characterized.

Here we use large-scale computer simulations to measure the noise experienced by a single atom moving at the interface at relatively low velocities. We also measure the noise in the dislocation density as a function of time as the sliding velocity increases. We compare the behavior of this friction system to the noise measured in a simulation of a single colloid driven through a colloidal crystal, where local melting occurs at the interface between the crystal and the moving particle.¹⁰

2. MODEL
To represent two metals sliding against each other at high speeds in perfect vacuum, we use the SPaSM code developed by the Los Alamos group¹¹ to perform large-scale two dimensional non-equilibrium molecular dynamics friction calculations. Details of the geometry of the computational cell are given in Ref.^4,12; a schematic is shown in Fig. 1. The cell is periodic in the direction of sliding (x direction), and has open boundaries in the y direction that are connected to thermostatted reservoir regions. In non-reservoir regions, the system evolves according to Newton's equations of motion. The velocity of the upper crystal is +u_p and the velocity of the lower crystal is -u_p, so that the relative velocity difference at the interface is 2u_p. The interactions between atoms take the form of a Lennard-Jones spline potential, described in detail in Ref.^13,14 We consider two sample sizes, a larger sample with 29064 atoms, and a smaller sample with 3448 atoms. The initial temperature is taken to be very low, well below the melting temperature of the crystal lattice.

To measure the noise experienced by a single particle, we identify a single atom located on the interface between the two workpieces. We measure the acceleration in the x or sliding direction, a_x, experienced by this atom as a function of time. We are also interested in the fluctuations of the number of dislocations present in the system as a function of time, since this quantity shows interesting properties near a 2D melting transition.¹⁵ It is nontrivial to identify dislocations in a system with tens of thousands of atoms, and previous methods used approximate schemes based either on the local potential energy or the local crystal orientation surrounding an atom. We have adapted an algorithm given in Ref.¹⁶ to perform a Voronoi tessellation on the entire system and directly identify atoms which do not have six neighbors. Our method can be used on 2D systems of arbitrary size, and we have tested it with up to 975,000 atoms. The defect identification is also highly valuable for data reduction, since we can record the positions of only the defected atoms, and construct high-resolution animations of the defects that contain only a fraction of the total number of atoms in the system.

3. LOW SLIDING VELOCITY
In Fig. 2 we show the locations of defects in a system with 3448 atoms as the sliding velocity is increased from u_p=0.2 to u_p=1.0. Sixfold-coordinated atoms are not plotted in the figure, but fill the remainder of the boxed area. At low velocities, the interface between the workpiece on top (with close-packed crystal direction perpendicular to the interface) and the workpiece on the bottom (with close-packed direction parallel to the interface) remains well defined and the behavior of the system is elastic. For higher velocities, local heating and plastic distortion begins to occur at the interface. Dislocations proliferate at the interface and begin to move outward into the surrounding crystal.^5,17 This has the effect of broadening the interface region, and results in local mixing of the atoms from the original crystals.

We measure the acceleration in the sliding direction, a_x, of a single particle located at the interface for velocities u_p < 0.5 in the elastic limit for a system containing 29064 atoms. The power spectrum of the single atom motion has a signature, with ~ 3/4, as shown in the left panel of Fig. 3. We integrate S() over the first octave of frequencies to obtain the noise power S₀,^18,19 which we plot in the right panel of Fig. 3 as a function of sliding velocity u_p. We observe that the noise power increases roughly as the square of the sliding velocity in the elastic regime at lower u_p. This noise dependence could be incorporated into simple models for sliding friction in the elastic limit.

4. DISLOCATION NOISE

As the driving speed is further increased, dislocations proliferate at the interface, as shown in Fig. 2(b-e). We measure the percentage of sixfold-coordinated atoms, P₆, as a function of time for increasing sliding velocity in a sample containing 3448 atoms. The power spectrum S( of P₆(t), plotted in the left panel of Fig. 4, is broad, with a signature where 1.5. We find a dramatic increase in noise power when the velocity is increased above the elastic limit, as illustrated by the gap between the spectra in the left panel of Fig. 4. This is more clearly shown by plotting the corresponding noise power S₀ (right panel of Fig. 4). The noise in the dislocation density increases monotonically with increasing sliding velocity once the dislocations begin to proliferate at the interface, which occurs for u_p > 0.2 in a sample of this size.

5. HETEROGENEITY AND NOISE

We compare the strongly heterogeneous structures observed in the high-speed dry friction simulations with heterogeneity introduced in a 2D colloidal crystal by driving a single colloid. Colloids are micron-sized particles suspended in a liquid solution. Each colloidal particle carries an electrostatic charge, leading to a repulsive interaction between the colloids with a range that can be adjusted by varying the salt concentration of the solution. The colloids can be confined to two dimensions by means of laser trapping, and can be manipulated individually using optical tweezers^20-23 or magnetic inclusions.²⁴ An important advantage of studying the colloidal system is that the colloids can be imaged directly using video microscopy techniques due to their large size. By studying colloids moving through a periodic substrate created by other colloids, it may be possible to gain insight into related phenomena such as the role played by dislocations in friction and depinning.

To model the colloids, we perform numerical Langevin simulations of the type employed in Refs.^15,25,26 We consider overdamped particles in a two-dimensional (2D) sample with periodic boundary conditions in the x and y directions. The initial configuration of the colloids is a triangular lattice. An additional colloid is placed in the system, and a constant driving force f_d is applied only to that particle. The equation of motion for colloid i is

Here is the interaction force from the other colloids. The colloids interact with a screened Coulomb interaction,

Here q_i(j) is the charge on colloid i(j) and 1/ is the screening length, which can be adjusted experimentally by varying the salt concentration of the solution. We take =2/a, where a is the lattice constant. All the particles have the same charge q=Q except the driven particle, which we take here to have q=3Q. The thermal noise f_i^T arises from random Langevin kicks with < f^T(t) >=0 and < f^T_i(t)f^T_j(t') > = 2k_BT(t-t'). Further details of the simulation appear in Ref.¹⁰

When a single particle is driven along the zero degree angle with respect to the background lattice, we find three distinct phases for increasing temperature. For low temperatures, as in Fig. 5(a), the flow is elastic and the particle moves along a 1D path while causing small distortions in the surrounding lattice. Although a group of defects surrounds the driven particle, which is an extra or interstitial particle in the triangular lattice, there are no other dislocations induced elsewhere in the lattice. At intermediate temperatures, as in Fig. 5(b), we find a transition to plastic flow where the particle no longer moves strictly in a straight line, and defects form in the surrounding lattice. A localized molten region surrounds the moving particle, and ejects 5-7 defects into the lattice over time. Above the bulk melting temperature of the lattice, as in Fig. 5(c), defects are generated in the sample even in the absence of the driven particle. The colloid velocity drops once dislocations begin to proliferate in the lattice, as shown in Ref.¹⁰

We measure the fraction of sixfold-coordinated particles, P₆, as a function of time at different temperatures. Representative noise spectra from each of the three phases are illustrated in Fig. 6. The noise is broad with a signature, where ~ 1.2. In previous work on a system with no driven particle, we observed a sharp increase in noise power of P₆(t) upon crossing the melting temperature T_m.¹⁵ When local melting is induced by a single driven particle, we find that this transition is rounded, as illustrated in the left panel of Fig. 7. The increase in noise power S₀ as a function of temperature occurs at the same temperature where local melting begins, as indicated by the right panel of Fig. 7 where P₆ is plotted as a function of T.

This increase in the noise of P₆ when a dislocated region forms around the driven particle is similar to the increase in noise observed in the friction system when dislocations begin to proliferate at the interface. It will be interesting to further explore possible connections between these two systems in the future.

6. CONCLUSION
We have studied the noise produced in two heterogeneous processes: the proliferation of dislocations at a friction interface for an atomic friction system at different sliding velocities, and the similar proliferation of dislocations in a colloidal crystal when a single colloid is dragged through the crystal at different temperatures. Both systems undergo a combination of effective local heating and local plastic flow. In both cases, we find broad noise signatures and an increase in the noise power as either the sliding velocity or the temperature increase.

ACKNOWLEDGMENTS
The author thanks T.C. Germann, J.E. Hammerberg, B.L. Holian, C. Reichhardt, and J. Roder for useful discussions. This work was supported by the US Department of Energy under Contract No. W-7405-ENG-36.

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