Nonvibrating granular model for a glass-forming liquid: Equilibration and aging

We studied experimentally a model of a glass-forming liquid on the basis of a nonvibrating magnetic granular system under an unsteady magnetic field. A sudden quenching was produced that drove the system from a liquid state to a different final state with lower temperature; the latter could be a liquid state or a solid state. We determined the mean-squared displacement in temporal windows to obtain the dynamic evolution of the system, and we determined the radial distribution function to obtain its structural characteristics. The results were analyzed using the intermediate scattering function and the effective potential between two particles. We observed that when quenching drives the system to a final state in the liquid phase far from the glass-transition temperature, equilibration occurs very quickly. When the final state has a temperature far below the glass-transition temperature, the system reaches its equilibrium state very quickly. In contrast, when the final state has an intermediate temperature but is below that corresponding to the glass transition, the system falls into a state that evolves slowly, presenting aging. The system evolves by an aging process toward more ordered states. However, after a waiting time, the dynamic behavior changes. It was observed that some particles get close enough to overpass the repulsive interactions and form small stable aggregates. In the effective potential curves, it was observed that the emergence of a second effective well due to the attraction quickly evolves and results in a deeper well than the initial effective well due to the repulsion. With the increase in time, more particles fall in the attractive well forming inhomogeneities, which produce a frustration in the aging process.

Figure 1

Scheme of the experimental setup.

Figure 2

Scheme of a sphere with a magnetic moment in a region with a vertical magnetic field. A trajectory and the parameters needed to describe the dynamics of the ball are shown.

Figure 3

Relation between the effective temperature $T_E$ of the system as a function of the magnetic field, which is the control parameter in the system.

Figure 4

Maxwell-Boltzmann velocity distribution for different temperatures. The labels indicate the effective temperature obtained from the fitting of the experimental data.

Figure 5

$W\left(t\right)$ curves for several effective temperatures. Note the linear behavior of the curves corresponding to high temperature. We can observe that the emergence of a plateau indicates arrested behavior at low temperatures.

Figure 6

Average diffusivity as a function of the effective temperature. The black continuous line is a linear fitting to $D$ values corresponding to high temperatures. The dotted line is a linear fitting to $D$ values corresponding to low temperatures.

Figure 7

Typical trajectories observed in the system. (a) Diffusive trajectories corresponding to a system with an effective temperature above the glass transition. (b) Arrested trajectories corresponding to a system with an effective temperature below the glass-transition temperature.

Figure 8

Slowing down of the dynamics as the effective temperature approaches the glass-transition temperature. $T_E/D$ is proportional to the effective viscosity. The solid line is a fit to the Arrhenius law.

Figure 9

$W\left(t\right)$ curves for several temporal windows after the quick quenching drives the system into a region where particles diffuse with lower energy. The curve labeled $t_a$ corresponds to an initial high temperature before the quenching.

Figure 10

Radial distribution function curves for several temporal windows. It can be seen that the curves reached stationary states very quickly. The curve labeled $t_a$ corresponds to a state at high temperature, before the quenching.

Figure 11

$W\left(t\right)$ curves for several temporal windows for the case in which the system has an effective temperature below but near the glass transition. We can observe a relatively slow evolution of the curves. As time progresses, the $W\left(t\right)$ curves show lower displacement values. The curve labeled $t_a$ corresponds to a state at high temperature before the quenching.

Figure 12

Radial distribution function curves for several temporal windows. First, the $g\left(r\right)$ curves evolve to more ordered configurations; however, around 66 s they undergo a noticeable change, and a second peak starts to grow. The curve labeled $t_a$ corresponds to the state at high temperature before quenching.

Figure 13

Comparison of the mean-square displacement taken at 1 s as a function of the waiting time. For case B (see the text), aging is observed.

Figure 14

Evolution of the peaks of the radial distribution function as a function of the waiting time for cases A and B. In case B there are two peaks, where one of them increases as the other decreases.

Figure 15

$W\left(t\right)$ curves in the case in which the final temperature is zero. The curves immediately evolve to stationary states. The curve labeled $t_a$ corresponds to a state at high temperature before the quenching.

Figure 16

The radial distribution function curves also evolve very quickly toward stationary states. The curve labeled $t_a$ corresponds to a state at high temperature before the quenching.

Figure 17

$F(q\sigma ,t)$ curves for several time windows after quenching starts. The curves exhibit a quick loss of temporal correlations. The curve labeled $t_a$ corresponds to a state at high temperature before the quenching. The curve labeled $t_b$ corresponds to a state at the initial position before quenching occurs, which undergoes quenching at the point indicated with the arrow. $q\sigma =0.72$ for the $t_a$ curve and $q\sigma =0.65$ for the other curves.

Figure 18

$U_E\left(r\right)/T_E$ curves in the case in which the system falls into a state of lower effective temperature in a diffusive state. It can be seen that both approximations yield similar behavior. A weak well is observed of around 1.5 particle diameter. Of course, the repulsive part of the potential governs the dynamics of the system.

Figure 19

$F(q\sigma ,t)$ for the case in which the system is in the glassy state near the glass-transition temperature. The temporal correlation is lost; however, as the waiting time grows, the curves uncorrelate faster. The curve labeled $t_a$ corresponds to a state at high temperature before the quenching. The curve labeled $t_b$ corresponds to a state at the initial position before quenching occurs, which undergoes quenching at the point indicated by the arrow. $q\sigma =0.75$ for the $t_a$ curve and $q\sigma =0.57$ for the other curves.

Figure 20

$U_E\left(r\right)/T_E$ curves for different waiting times obtained using the HNC approximation in the case in which the system is near but below the glass transition. The growth and evolution of a second well of around one-particle diameter are observed, although values of the second well are all positive.

Figure 21

$U_E\left(r\right)/T_E$ curves for different waiting times obtained using the PY approximation. The growth and evolution of a second well of around one-particle diameter is observed. Using this approximation, the second well takes values below the zero value.

Figure 22

$F(q\sigma ,t)$ values taken at 1 s as a function of the waiting time. While in case A the values are low, in case B the values are almost the unit that shows high temporal correlation.

Figure 23

$F(q\sigma ,t)$ curves in the case in which the final effective temperature is zero. The curves are totally correlated because particles get arrested immediately. The curve labeled $t_a$ corresponds to a state at high temperature before quenching. The curve labeled $t_b$ corresponds to a state at the initial position before quenching occurs, which undergoes quenching at the point indicated by the arrow. $q\sigma =0.72$ for the $t_a$ curve and $q\sigma =0.56$ for the other curves.

Figure 24

$U_E\left(r\right)/T_E$ curve for the case in which the final temperature is zero. The effective potential shows a kind of periodic well.