Onset of chaotic dynamics in a ball mill: Attractors merging and crisis induced intermittency

Abstract

In mechanical treatment carried out by ball milling, powder particles are subjected to repeated high-energy mechanical loads which induce heavy plastic deformations together with fracturing and cold-welding events. Owing to the continuous defect accumulation and interface renewal, both structural and chemical transformations occur. The nature and the rate of such transformations have been shown to depend on variables, such as impact velocity and collision frequency that depend, in turn, on the whole dynamics of the system. The characterization of the ball dynamics under different impact conditions is then to be considered a necessary step in order to gain a satisfactory control of the experimental set up. In this paper we investigate the motion of a ball in a milling device. Since the ball motion is governed by impulsive forces acting during each collision, no analytical expression for the complete ball trajectory can be obtained. In addition, mechanical systems exhibiting impacts are strongly nonlinear due to sudden changes of velocities at the instant of impact. Many different types of periodic and chaotic impact motions exist indeed even for simple systems with external periodic excitation forces. We present results of the analysis on the ball trajectory, obtained from a suitable numerical model, under growing degree of impact elasticity. A route to high dimensional chaos is obtained. Crisis and attractors merging are also found. Among the others, the scarce knowledge of the dynamics of ball milling devices, where milling bodies undergo a huge number of collisions during the processing. Under such conditions, indeed, any experimental measurement of the exact number of impacts and of the energy transferred to powders at collisions is greatly hindered. It becomes therefore impossible to relate the degree of structural evolution to the mechanical energy dissipated, maybe the most important macroscopic parameter used to characterize the yield of a mechanochemical reaction. Any information on the dynamics of milling bodies is then extremely valuable in order to quantitatively describe and rationalize the kinetic features of mechanically induced transformations. Preliminary investigations have already shown the possible occurrence of chaotic regimes during milling treatments. 6,7 On the other hand, modeling results and experimental evidences demonstrate the occurrence of regular dynamical regimes allowing for the direct measurement of both the average collision frequency and impact energy. 8 It becomes therefore important to study the transition from periodic to chaotic regimes in order to understand when and why transition takes place, so as to avoid it. Experimentalists are indeed mainly interested in periodic and regular regimes, which permit the full control of experimental parameters.

Multifractal properties of ball milling dynamics

“…A classical period-doubling cascade route to chaos is initially observed, then replaced by a crisis scenario where two symmetric chaotic attraction basins merge, resulting in an intermittent dynamics. 19,20,29 Based on the above mentioned observations, it can be reasonably expected that the ball dynamics exhibits a similar dependence on other milling parameters. To suitably verify this expectation, we studied the response of the milling dynamics to the variation of the parameters h 0 , a 0 , and in the ranges [p/8, p/16], [p/8, p/16], and [8.0, 22.5] Hz, respectively.…”

“…Previous work 19,20 has clarified the role of the restitution coefficient as a parameter able to critically affect the system dynamics. At low values of the restitution coefficient, i.e., in the regime of inelastic collisions, the ball trajectories exhibit a periodic behavior.…”

“…The FFT spectra refer to the oscillatory behavior described by the milling ball along the x direction, which represents a suitable observable to characterize the transition to chaos. 2,19,20 In both cases, the FFT spectra (see black pixels) point out a periodic behavior in the inelastic region (0, 0.2] of the restitution coefficient f, dominated by the oscillation frequency of the device, here equal to 18. 3 Hz.…”

“…2 and Ref. 20). In the plot, the 3D data obtained with the set of parameters a 0 ¼ h 0 ¼ p/12 and ¼ 18.…”

“…2,6,7 For instance, the elasticity of collisions governs the transition from chaotic to periodic behaviors, thus allowing for the promotion of ordered patterns in the ball trajectories. 19,20 In this respect, it is worth noting that spatio-temporal organization in chaotic systems can also be sustained with a noise-driven stabilization of unstable orbits within the related attractors in the phase-space. 21 More in general, strange attractors that typically underlie chaotic dynamics embed themselves ordered and highly non-trivial patterns within their compact topology.…”