Getting hotter by heating less: How driven granular materials dissipate energy in excess

We investigate how the kinetic energy acquired by a dense granular system driven by an external vibration depends on the input energy. Our focus is on the dependence of the granular behavior on two main parameters: frequency and vibration amplitude. We find that there exists an optimal forcing frequency at which the system reaches the maximal kinetic energy: if the input energy is increased beyond this threshold, the system dissipates more and more energy and recovers a colder and more viscous state. Quite surprisingly, the nonmonotonic behavior is found for vibration amplitudes which are sufficiently low to keep the system always in contact with the driving oscillating plate. Studying dissipative properties of the system, we unveil a striking difference between this nonmonotonic behavior and a standard resonance mechanism. This feature is also observed at the microscopic scale of the single-grain dynamics and can be interpreted as an instance of negative specific heat. An analytically solvable model based on a generalized forced-damped oscillator well reproduces the observed phenomenology, illustrating the role of the competing effects of forcing and dissipation.

Figure 1

Vane kinetic energy $K_v$ in experiments as a function of $f$, for different values of $A$. Error bars are $\pm 1$ standard deviation. Inset: $K_v$ measured in numerical simulations. Sketch of the system setup: green circles show granular particles, red circles represent the vane, and yellow circles the lid. Numerical parameters are $k_n=12\times {10}^7$ Pa and ${\gamma }_n=2.9\times {10}^7$ (ms)${}^{-1}$.

Figure 2

Total kinetic energy $K$ of the granular medium versus $S$, varying $f$ at fixed $A$, for several values of ${\gamma }_n$, with $k_n=6.1\times {10}^7$ Pa and $A=0.026$ mm.

Figure 3

Mean translational kinetic energy of the whole system $K$ versus the driving frequency $f$ at fixed $A=0.026$ mm (simulations without the vane). (a) Comparison between the translational kinetic energy, the rotational energy, and the total energy (translational plus rotational). We show two curves reported in Fig. 2 as a function of $f$, instead of $S$. The rotational degrees of freedom follow the same nonmonotonic behavior but with smaller absolute values, so that their contribution does not affect the qualitative behavior of the total kinetic energy. (b) Study of the mean kinetic energy for different values of nonlinear stiffness: $k_n$ is varied at fixed ${\gamma }_n=2.9\times {10}^7$ (ms)${}^{-1}$. The three curves have the same shape (the peak position does not change) but are vertically ordered by $k_n$. This implies a higher kinetic energy and therefore less dissipation in the system with a higher stiffness.

Figure 4

Total kinetic energy $K$ of the grains (a) and mean angular velocity $\langle \mathrm{\Omega }\rangle$ of the blade (b) versus $S$, varying $A$ for three fixed frequencies (200, 400, and 800 Hz) and varying $f$ for one fixed amplitude ($A=0.026$ mm). Parameters are $k_n=6.1\times {10}^7$ Pa and ${\gamma }_n=2.9\times {10}^7$ (ms)${}^{-1}$.

Figure 5

Mean angular velocity of the blade $\langle \mathrm{\Omega }\rangle$ versus the driving frequency $f$. (a) ${\gamma }_n$ is varied with a fixed $k_n=1.5\times {10}^8$ Pa. (b) ${\gamma }_n$ with a fixed $k_n=6.1\times {10}^7$ Pa. (c) $k_n$ is varied with a fixed ${\gamma }_n=2.9\times {10}^7$ (ms)${}^{-1}$.

Figure 6

Rescaled single-particle MSD on the $XY$ plane versus the rescaled time, for different values of the driving frequency. Inset: Single-particle MSD on the $XY$ plane versus the time. Simulation parameters are $A=0.026$ mm, $k_n=6.1\times {10}^7$ Pa, and ${\gamma }_n=2.9\times {10}^7$ (ms)${}^{-1}$.

Figure 7

Comparison between the average particle speed on the $XY$ plane obtained from the mean kinetic energy of the whole system and from the ratio between the cage size $S_c$ and the trapping time ${\tau }_c$. Bottom inset: $S_c$ as a function of the input energy $S$, varying $f$ at fixed $A$. Top inset: ${\tau }_c$ as a function of $f$. Same parameters as in Fig. 6.

Figure 8

(a) Single-particle MSD on the $XY$ plane rescaled by $S_c^2$ as a function of the rescaled time $t/{\tau }_c$. Different curves correspond to different driving amplitudes, while the frequency is fixed at $f=200$ Hz. (b) Size of the cage $S_c$ versus the driving amplitude $A$. Inset: MSD not rescaled. Simulations with $k_n=6.1\times {10}^7$ Pa and ${\gamma }_n=2.9\times {10}^7$ (ms)${}^{-1}$.

Figure 9

Coarse-grained description of the single-particle dynamics. The springs $k$ and the viscosity $\gamma$ represent the cage that mediates the vibration of the shaker.

Figure 10

Mean kinetic energy from Eq. (5) for different values of $a$, with $2\pi f_k=11905$ ${\mathrm{s}}^{-1}$, $A=0.026$ mm, and $m_{\text{eff}}=912.6$ g, compared with $K$ measured in numerical simulations with $A=0.026$ mm and $k_n=6.1\times {10}^7$ Pa.

Figure 11

Different regions of the parameter space investigated in our work with respect to Refs. [30, 31].

Figure 12

Collision rate between grains and bottom plate as a function of $f$ with $A=0.026$ mm.

Figure 13

Experimental data for the spectra of the vertical $z$ coordinate of the top plate for different shaking frequencies $f$ and $A=0.053$ mm.