The Kovacs memory effect in a thin granular layer: experimental evidence and its physical origin

Abstract

We report the experimental observation of memory effects in a vertically vibrated thin granular layer. Following a quench in the input acceleration, the granular temperature exhibits an anomalous Kovacs memory effect confined to the initial fast relaxation stage. This memory vanishes shortly thereafter, yielding a time-dependent memoryless regime governed solely by the instantaneous temperature before the system reaches its final steady state. We develop a kinetic theory framework that quantitatively captures these features by identifying the initial memory and subsequent memoryless regimes with the kinetic and hydrodynamic states, respectively (that are well established in kinetic theory). Our analysis reveals that memory emerges during fast transients through coupling between horizontal and vertical temperatures, a mechanism that fundamentally constrains the accessible memory phenomenology and precludes observation of the standard Kovacs effect in this system. Molecular dynamics simulations provide independent confirmation of all experimental and theoretical findings.

Figures (5)

• Figure 1: Schematic representation of the Kovacs cooling protocol and the two kinds of memory effects considered. The granular fluid initially at stationary temperature $T_0$ (the energy input being $\Gamma_0$) is quenched by lowering the energy input to $\Gamma_s$ at time $t_0$ and relaxes monotonically toward a new stationary state $T_s < T_0$, that would be reached at a long time $t_s$ (green solid line). However, at an intermediate waiting time $t_w$ (where $t_0 < t_w< t_s$), the energy injection is adjusted to $\Gamma_w$ so that the new stationary temperature $T_s^*$ equals the instantaneous temperature: $T(t_w) = T_s^* > T_s$. The normal Kovacs effect is produced in the form of an upwards hump of the temperature, and hence $\mathrm{sg}(\mathrm{d}T/\mathrm{d}t|_{t_w^-}) \neq \mathrm{sg}(\mathrm{d}T/\mathrm{d}t|_{t_w^+})$. In this case, the system transitorily tends to reach the initial state that the system had at $t=t_0$ (red dot line). On the contrary, the anomalous Kovacs effect is characterized by a downwards hump and now $\mathrm{sg}(\mathrm{d}T/\mathrm{d}t|_{t_w^-}) = \mathrm{sg}(\mathrm{d}T/\mathrm{d}t|_{t_w^+})$ where the system tries to continue cooling down (blue dot line), as if it remembered the previous relaxation process instead. In both cases, the system remembers features of the past. The top left inset shows the corresponding changes of the input signal $\Gamma$.
• Figure 2: Schematic representation of the experimental setup. The thin granular layer of steel spheres is vertically vibrated. The input acceleration $\Gamma = A\omega^2/g$ is monitored with an accelerometer, where $A$ is the vibration amplitude, $\omega$ the angular frequency, and $g=9.8~\mathrm{m/s^2}$.
• Figure 3: Horizontal temperature (blue) relaxation curves for a Kovacs protocol in experiments and molecular dynamics simulations. (a) Experiments: $\nu = 180~\mathrm{Hz}$; initial and final acceleration inputs are $\Gamma_0 = 7.0$ and $\Gamma_f = 3.15$ respectively, while intermediate values of the input acceleration at different waiting times $t_w$ are $\Gamma_w = \{6.5; 6.0; 5.5; 5.0; 4.5; 4.0\}$. (b) Molecular dynamics: $\nu = 643.22~\mathrm{Hz}$. $\Gamma_0 = 12.5$ amd $\Gamma_f = 5.0$ respectively, and $\Gamma_w = \{11.5; 10.5; 9.5; 8.5; 7.5; 6.5; 5.5\}$. For both experiment and simulation, darker blue in the curves corresponds to higher $\Gamma_w$. Arrows mark the $t_m$ values extracted from the time evolution of the kurtosis.
• Figure 4: (a) Relaxation curves $T$ vs. $t$ from experiments, for a step-down protocol with $\Gamma_s = 3.15$, and $\Gamma_0 = \{6.5; 6.0; 5.5; 5.0; 4.5; 4.0 \}$. Here $t_{\Gamma}$ denotes the translation in the time performed to each curve to make the curves collapse. (b) Relaxation curves of $T_z$ vs. $T$ from simulation data, for a series of values of initial input accelerations $\Gamma_0 = \{2.5; 3.5; 4.5; 5.5; 6.5; 7.5; 8.5; 9.5; 10.5; 11.5; 12.5 \}$ to a final state with $\Gamma_s = 5.0$.
• Figure A1: Experimental setup in our laboratory (Instituto de Computación Científica Avanzada, ICCAEx, Badajoz, Spain). Inset: top-view close-up of the granular layer.